^ ionaj lity THE LIBRARY OF THE UNIVERSITY OF CALIFORNIA LOS ANGELES ' ' ( ' .\ ^ I fr':r\i '( )^^ jf,K,T\ '' ^i , > I ',:-. 1'^ .7 //:'/>/ :'/',/'/' 'i .!/,.. liW/r ,...,'//<, ,,,,, /A .,./., ,//,/ . , ^ ,,...;./.; TRACTS ON MATHEMATICAL f| AND PHILOSOPHICAL SUBJECTS; COMPRISING, AMONG NUUEBOCS IMPORTANT ARTICLES, THE THEORY OF BRIDGES, WITH SEVERAL PLANS OF RECENT IMPROVEMENT. ALSO THE RESULTS OF NUMEROUS EXPERIMENTS ON THE FORCE OF GUNPOWDER, WITH APPLICATIONS TO THE MODERN PRACTICE OF ARTILLERY. IN THREE VOLUMES. BY CHARLES HUTTON, LL.D. and F.R.S. &c. Late Professor of Mathematics in the Royal Miiitarj' Academy, Woolwich. VOL. I. LONDON: I'RINTED FOR F. C. AND J. RIVINGTON ; G. WILKIE AND J. ROBINSON ; J. WALKER ; LACKIKGTON, ALLEN, AND CO. ; CADELL AND DAVIES; J. CUTHELLJ B. AND R. CROSBY AND CO. J J.RICHARDSON; J. M. RICHARDSON J R.BALDWIN; AND G. ROBINSON. 1812. T. DAVISON, Lombard-Street, Wbitefriara, London. hi7t, v./ PREFACE. Having been, for a long series of years, in the constant habit of preserving original Tracts and dissertations on sci- entific subjects ; and now enjoying, at a very advanced pe- riod of life, some degree of leisure, in consequence of my retirement from the laborious duties of the Royal Military Academy; I have anxiously embraced the opportunit}^ of selecting, and revising, such of those papers as were likely to be most useful, and of presenting them to the public. Some few parts of these Tracts have been already printed in the Philosophical Transactions, and in other works; but most of them are quite new; and such as are not so, having been recast and greatly improved, may be also considered in some measure as original compositions. These papers, being necessarily of a miscellaneous nature, are here arranged nearly according to the order of time in which they were composed ; and the description of them, is briefly as fol- lows. VOLUME I. Tract I, is on the Principles of Bridges. The original of this paper was a small pamphlet on the same subject, first pub- blished by me on a particular occasion at Newcastle, in the year 1772. It was also republished at London in 1801, nearly in the same state. But it has been now recomposed, and greatly enlarged with many additional propositions, as also numerous observations, both practical and scientific. An Appendix is also added, containing my report to the Committee of Parliament on the project for a new iron 1250725 IV PREFACE. bridge, of only one arch, proposed to be thrown over the river Thames at London ; with several other appropriate ar- ticles, as below. Tract ii, exhibits some curious queries concerning Lon- don Bridge, proposed in the year 1746 by the magistrates of the city; with the ingenious answers given to the same, by Mr. George Dance, surveyor-general of the city works, being the result of that gentleman's examination concerning the state of the bridge at that time. Tract hi contains texperiments and observations to be made on the state of London bridge; being the report of a committee of the members of the Royal Society, addressed to the common council of the city of London. Tract iv treats of the effects which might be produced on the tides in the river Thames, in consequence of erecting a bridge at Blackfriars. This was an ingenious report, drawn up by the late Mr. John Robertson, at the request of the city of London. Tract v consists of answers, given by me, to questions proposed by the Select Committee of Parliament, relative to a proposal, made by Messrs. Telford and Douglas, for erect- ing a new iron bridge, of a single arch only, over the river Thames, instead of the present London bridge. Tract vi exhibits a brief history of the original invention, and subsequent improvements of iron bridges, as practised of late years in this country. Tract vii is a dissertation on the nature and value of iur finite series; explaining the properties of several forms of such series, as converging, diverging, and neutral. Tract viii is a new method for the valuation of numeral infinite series, that have their terms ahernately plus and niirms ; which is performed by taking continual arithmetical means betwetjii the successive sums, and between the means ; a method bv which tiie value or sum of any such series is very easily and quickly obtained. PREFACE. Y Tract ix is a method of summing the series a-\-ba;-\-cJt:^ + dx^-{-e.r'^-{-Sic, in the case when it converges very slowly^ namely, when a; is nearly equal to 1, and the coefficients a, i, c, d, &c, decrease very slowly; the signs of all the lenns being plusor positive : a method which has been considered a great desideratum in intinite series. Tract x contains the investigation of certain easy and general rules, for extracting any root out of a given immber ; exhibiting a general and very easy formula, to serve for all roots whatever. Tract xi is a new method of finding, in general and finite terms, near values of the roots of equations of this form, x^ px^~ -\-qx^~ &c = ; namely, having the terms alternately plus and minus : being one method more to be added to the many we are already possessed of, for deter- mining the roots of the higher orders of equations. Tract xii treats of the binomial theorem ; exhibiting a demonstration of the truth of it in the general case of frac- tional exponents. The demonstration is of this nature, that it proves the law of the whole series in a formula of one single term only : thus, p, a, r, denoting any three succes- sive terms of the series, expanded from the given binomial (1 + x)" , and if -f P = a, then is r Q = R, Avhich denotes ^ ' ' h ' k+n ' the general law of the series, being a new mode of proving the law of the coefficients of this celebrated theorem. But, be- sides this law of the coefficients, the very form of the series is, for the first time, here demonstrated, viz, that the form of the series for the developement of the binomial (1 + x)" , with respect to the exponents, will be 1 -\- ax -\- bx^ -j- cx^ -{-dx'^ + &c, a form which has heretofore been assumed without proof. Tract xiii treats on the common sections of the sphere and cone: with the demonstration of some other new pro- perties of the sphere, which are similar to certain known properties of the cijrcle. The few propositions which form VI PREFACE. part of this tract, is a small specimen of the analogy, and even identity, of some of the more remarkable properties of the circle, with those of the sphere. To which are added some properties of the lines of section, and of contact, be- tween the sphere and cone: both of which can be further extended as occasions may offer. Tract xiv, on the geometrical division of circles and el- lipses into any number of parts having equal perimeters, and areas either all equal or in any proposed ratios to each other: constructions which were never before given by any author, but which, on the contrary, had been accounted impos- sible to be effected. Tract xv contains an approximate geometrical division of the circumference of the circle. Tract xvi treats on plane trigonometry, without the use of the common tables of sines, tangents, and secants : resolv- ing all the cases in numbers, by means of certain algebraical formulae only. . Tract xvii is on Machin's quadrature of the circle; be- ing an investigation of that learned gentleman's ver}'^ sim- ple and easy series for that purpose, by help of the tangent of the arc of 45 de":rees : which series the author had given without any proof or investigation. Tract xviii, a new and general method of finding sim- ple and quickly-converging series ; by which the proportion of the diameter of a circle to its circumference may easily be computed to a great many places of figures. By this method are found, not only Machin's series, noticed in the last Tract, but also several others that are much more simple and easy than his. Tract xix, the history of trigonometrical tables, &c : being a critical description of all the writings on trigono- metry made before the invention of logarithms. Tract xx, the history of logarithms; giving an account of the inventions and descriptions by several authors on the different kinds of logarithms. PREFACE. Vii Tkact XXI, on the construction of logarithms ; exhibiting the various and peculiar methods employed by ail the dif- ferent authors, in their several computations of these very- useful numbers. Tract xxii, treats on the powers of numbers ; chiefly re- lating to curious properties of the squares, and the cubes, and other powers of numbers. Tract xxiii, is a new and easy method of extracting the square roots of numbers ; very useful in practice. Tract xxiv, shows how to construct tables of the square- roots, and cube-roots, and the reciprocals of the series of the natural numbers ; being a general method, by means of the law of the differences of such roots and reciprocals of num- bers. Tract xxv, is an extensive table of roots and recipro- cals, constructed in the above manner, accompanied also with the series of the squares and cubes of the same numbers. VOLUME II. Tract xxvi, an account of the calculations made from the survey and measures taken at mount Shichallin, in order to ascertain the mean density of the earth : being the result of a laborious calculation, the first ever made to ascertain that density ; by which it is shown to be nearly equal to 5 times the density of water, or almost double the density of the rocks at the surface of the earth, and that consequently the interior of the earth must consist of immense quantities of metals or metallic ores. Tract xxvii, consists of calculations to determine at what point, on the side of a hill, its attraction will be the greatest. This is inserted as an appendix to the preceding tract, and intended to direct operations of any future at- tempt to ascertain such density, or to corroborate the fore- going statement; and, by this determination, it is shown that the best situation is generally at about ^ of the altitude of the hill. Vlll PREFACE. Tract xxviii, is an extensive treatise on cubic equations and infinite series: showins^ tlieir nature, properties, and so- lutions, both in finite formulas and by expressions in infinite series. Tract xxix contains a curious project for a new division ,of the qiiaclrantal arc of the circle, with a view to trigono- metrical and other purposes : being intended for tlie novel desiii;n of constructing tables of the sines, tangents, and se- cants of arc^, to equal purts of the radius of the circle ; or exjircssiiig all these lines, as well as the arcs themselves, in sucii parrs. Tract xxx, on the sections of spheroids and conoids: showing that all such plane sections are the same as conic sections; and that all the parallel sections, in each of these solids, are like and similar figures. Tract xxxi, on the comparison of curves of the same species ; showing their mutual relations. Tract xxxii contains a theorem for the cube-root of an algebraic binomial, one of the terms being a quadratic radi- cal ; useful in the solution of certain cubic equations by Cardan's rule. Tract xxxiii, is a complete history of algebra; tracing its origin and practice among the ancient Greeks, the Indi- ans, Persians, and Arabians; with particular details of the various peculiarities and improvements, made among dif- ferent people, and b)^ several eminent individuals, especially among the European authors, namely, the Italians, Spaniards, French, Germans, and the English ; in which all the dis- coveries and improvements are ascribed to the proj^er au- thors. Tract xxxiv, exhibits the results of new experiments in ArtUery, for determining the force of fired gunpowder, the initial velocity of cannon balls, the ranges of projectiles at different elevations, the resistance of the air to their motions ; the effect of different lengths of guns, and of different quan- PREFACE. titities of powder, &,c, &c : giving a complete detail of all the circumstances attending these very numerous and ac- curate experiments, with many useful philosophical and prac- tical inferences deduced from them; the whole formino- as it were a new era in the progress of this curious and important branch of knowledge. VOLUME III. Tract xxxv, on a new Gunpowder Eprouvette ; show- ing its construction and use, by means of which the strength and quality of gunpowder may be proved and evinced, in a way far more exact and easy than by any other machine. Tract xxxvi, on the Resistance of the Air to bodies in motion, as determined by the Whirling Machine : showing the exact quantity of the air's resistance to all forms of bodies, moved through it with slow and moderate motions ; the effects of which, combined with those of the very high motions of cannon and musket shot, furnish us with a com- plete and uniform series of resistances to all degrees of ve- locit}' , from the very slowest perceptible motions, to those of the highest and most violent. Tract xxxvii, on the Theory and Practice of Gunnery, as dependent on the Resistance of the Air. This tract is employed in stating the deductions abstracted from all the preceding- experiments, and applying them in many pro- blems, to the important purposes of i\rtillery and projectiles. Here are given complete tables of the quantity of resistance to balls moving with every degree of velocity; with correct rules for ascertaining those that are proper to all other sizes of balls. Here are also given general rules and alge- braic formnlse, for expressing the resistance to any size of ball in terms of the velocity ; with a great variety of pro- blems for determining the motions of balls in all directions, upwards, downwards, or obliquely, touching their velocities and times in motion, with the ranges of projectiles in the air, tc preface. and practical applications to the cases of gunnery, in a great variety of useful instances. Tract xxxviii, being the last, contains a miscellaneous collection of practical questions, illustrating several of the principles in the preceding Tracts, with the solutions at large. Such are the outlines of a work, which is the result of many years assiduous study and persevering research ; and which it is presumed will be found to contain several new articles, on civil and military science, that may be deemed of national importance. It is, in all probability, the last original work that I may ever be able to offer to the notice of the Public, and I am therefore the more anxious that it should be found worthy of their acceptance and regard. To their kind indulgence, in- deed, is due whatever success I may have experienced, both as an Author and Teacher for more than half a century : and it is no small satisfaction to reflect, that my humble endea- vours, during that period, have not been wholly unsuccessful in the diffusion of useful knowledge. To the same liberal encouragement of the Public must like- wise be ascribed, in a great measure, the means of the com- fortable retirement which I now enjoy, towards the close of a long and laborious life : and for which I have every reason to be truly thankful. CHA. HUTTON. London, July, 1812. CONTENTS TO VOL. 1. Tract i. The Principles of Bridges. - - _ i II. Queries concerning London-bridge, proposed in the year 1746, by the Magistrates of the City. -..---- 115 III. Experiments and Observations proposed to be made on London-bridge, by the Committee of the Royal Society. - - - - 120 IV. Consequences in the Tides in the River Thames, by erecting Blackfriars-bridge. 122 v. Answers to Questions proposed by the Com- mittee of Parliament, on a Bridge of one Arch over the Thames at London. _ - _ 127 VI. History and Practice of Iron Bridges. - 144 VII. On the Nature and Value of Infinite Series. 167 viii. New Method for the Value of Infinite Se- ries. --___-- 176 IX. On summing the very slow series a -\- bx { ex' -I- &c. 203 X. General Rules for the Extraction of all Roots. 210 XI. On the Roots of the Equation .r" J5.r"~' -f- qx"-^ - &c. = 0. - - - - - 218 XII. Demonstration of the Binomial Theorem. 228 XIII. On the Common Sections of the Sphere and Cone. - - - _ - - - 245 XIV. Division of Circles into- any Parts of equal Areas and Perimeters. _ _ _ - 254 XV. Geometrical Division of the Circumference of a Circle. ------ 25S XVI. Trigonometry without the Tables of Sines, &c. _.--_-, 260 XVII. Investigation of Machin's Quadrature of the Circle. ------ 266 XVIII. On quickly-converging Series for the Circle. ----- - - 268 XIX. Hi tory of Trigonometrical Tables. - 278 XX. The History of Looarithms. - - 306 CONTENTS. Tract xxi. On the Construction of Logarithms. - 340 XXII. Certain Properties of the Powers of Num- bers. _______ 454 XXIII. New Method for the Square-roots of Numbers. ______ 457 XXIV. Construction of Tables of the Square and Cube Roots and Reciprocals of all Numbers. 459 XXV. Tables of Squares, Cubes, Reciprocals, Square-roots, and Cube-roots of Numbers. 466 CORRECTIONS. VOLUME I. Note, b denotes counted from the bottom, Page: Lme 8 9h E = Ds. 12 \b horizontal line CH. 19 in the cut, at the bottom of the lines appended from the points h, c, d, e, /, set the letters i, k, I, m, n. 19 DH =dh, 57 lb supposing jr. 89 Mb channel. 264 5h 2.4.or^. 267 13i \\%. 430 14i ifS. 26i3 1 4a^6^ VOLUME II. 153 4 f jr Hill Street, read Portland Place. 311 \2b and elsewhere, for Bloomjield, read Bloviejldd. 379 lib foi pendulum, rfdd gun. VOLUME 111. 46 llh ddeo/. lOi for such, read some. 101 2 for 15, read 18. 303 9 for 128, cead 1:28(). TRACT I. THE PRINCIPLES OF BRIDGES: CONTAINING THE MATHEMATICAL DEMONSTRATION OF THE PROPERTIES OF THE ARCHES, THE THICKNESS OF THE PIERS, THE FORCE OF THE WATER AGAINST THEM, &C. WITH PRACTICAL OB- SERVATIONS AND DIRECTIONS DRAWN FROM THE WHOLE, 1 HIS Tract, on bridges, originated from the circumstance; of the fall of Newcastle bridge, in the year 1771; which, with other particulars relative to the Tract, are noticed in the preface to that Edition of it; which was as follows : THE ORIGINAL PREFACE. A large and elegant bridge, forming a way over a broad and rapid river, is justly esteemed one of the noblest pieces of mechanism that man is capable of perforaiing. And the usefulness of an art which, at the same time that it connects distant shores by a way over the deep and rapid waters, also allows those waters and their navigation to pass smooth and uninterrupted, renders all probable attempts to advance the theory or practice of it, highly deserving the encour; gement of the public. This little book is offered as an attempt towards the im- provement of the theory oC this art, in which the more es- sential properties, dimensions, proportions, and other rela- VOL. I. B 2 THE ORIGINAL PRILFACE. TRACT 1, tions of the various parts of a bridge, are strictly demon- strated, and clearly illustrated by various examples. It is divided into five sections: the 1st treats on the projects of bridges, containing a regular detail of the various circum- stances and considerations that are cognizable in such pro- jects. The 2d treats on arches, demonstrating their various properties, Avith the relations between t!ieir intrudes and ex- trados, and clearly distinguishing the most preferable curves to be used in a bridge; the first two or three propositions being instituted after the manner of two or three done by Mr. Emerson in his Fluxions and Mechanics. The 3d sec- tion treats on the piers, demonstrating their thickness ne- cessary for supporting any kind of an arch, springing at any height, both vvhcn part of the pier is supposed to be im- mersed in water, and when otherwise. The 4th demonstrates the force of the water against the end or face of the pier, considered as of ditlerent forms; with the best form for di- viding the stream, &c. and to it is added a table, showing the several heights of the fall of the water under the arches, arising from its velocity and the obstruction of the piers; as it was composed by Tho. Wright, Esq. of Auckland, in the county of Durham, who informs me it is part of a work on which he has spent much time, and with which he intends to favour the public. And the 5th and last section contains a Dictionary of the most material terms relating to the sub- ject: in which many practical observations and directions are given, which could not be so regularly nor properly in- troduced into the former sections. The whole, it is pre- sumed, containing full directions for constituting and aciapt- inrr to one another, the several essential parts of a bridge, so as to make it the strongest, and the most convenient, both for the passage over and under it, which tlie situation and other circumstances will admit: not indeed for the actual methods of disposing the stones, making of mortar, or the external ornaments, &c. those things are not here attempted, but are left to the discretion of the practical architect, as being no part of the plan of this undertaking ; and for the THE ORIGINAL PREFACE. % same reason also here are not given any views of bridges, but only prints of such parts or figures as are necessary in explaining the elementary parts of the subject. As my profession is not that of an architect, very probably I should never have turned my thoughts to this subject, so as to address the public upon it, had it not been for the occasion of an accident in that part of the country in which 1 reside, viz. the fall of Newcastle and other bridges on the river Tyne, on the 17th of November, 1771, occasioned by a high flood, which rose about 9 feet higher at Newcastle than the usual spring tides do. This occasion having furnished me with many opportunities of hearing and seeing very absurd notions advanced on the subject in general, I thought the demonstrations of the relations of the essential parts of a bridge, would not be unacceptable to those architects and others, who may be capable of perceiving their force and effects. Newcastle, 1772. The original edition, of 1772, being out of print, and th book being much asked for, a new edition was printed in ISOl, at a time when the project of a cast-iron bridge of one arch, proposed to be built over the Thames at London, by- Messrs. Telford and Douglass, was the subject of much con- versation : on which occasion the following addition was made to the Preface ; viz. This little work, which was hastily composed on a parti- cular occasion, having been long out of print, is now as sud- denly reprinted'in the same form, on the present occasion, of the report of a new bridge proposed to be thrown across the Thames, at London: reserving the long intended edition, on a much larger and more improved plan, till a more conve- nient opportunity. Royal Military Academy, Jan. 12, 1801. It may here be added, that the whole tract has been now [uite re-cast and composed, and greatly enlarged with more B 2 < THE PRINCIPLES OF BRIDGES. TRACT I. propositions, and numerous observations, both practical and scientific. To the end is also added an Appendix, being the author's report to the Committee of Parhament, on the pro- ject for a new cast-iron bridge, of one arch, over the river at London ; and several other appropriate appendages. SECTION I. ON THE PROJECTS OF BRIDGES; WITH THE DESIGN, THE ESTIMATE, &C. When a bridge is deemed necessary to be built over a river, the first consideration is the place of it; or what par- ticular situation will contain a maximum of t'le advunta;:;cs over the disadvantages. In agitating this important ques- tion, every circumstance, certain and probable, attending or iikely to attend the bridge, shoidd be separately, minutely, and impartially stated and examined; and the advantage or disadvantage of it rated at a value proportioned to it ; then the difference between the whole advantages and disadvan- tage!^, will be the net value of that particular sitnation for M'hich the calculation is made. And by doing the same for other situations, all their net values will be found, and of consequence the most preferable situation among them. Or, in a competition between two places, if each one's ad- vantage over the other be estimated or valued in evory cir- cumstance attending them, the sums of their advantages will show which of them is the better. And the same being dour for this and a third, and so on, the best situation of all will be obtained. In tliis estimation, a great number of particulars must be included; nothintr beinir omitted that can be found to make a part of the consideration. Among these, the situation of the town or place, for the convenience of which the bridge SECT. I. THE PROJECTS OF BRIDGES, &C. S is chiefly to be made, will naturally produce an article of the first consequence ; and a great many others, if necessary, ought to be sacrificed to it. If possible, the bridge should be placed where there can conveniently be opened and made passages or streets from the end of it in every direction, and especially one as nearly in the direction of the bridge itself as possible, tending towards the body of the town, v/ithout narrows or crooked windings, and easily communicating with the chief streets, thoroughfares, &c. And here every person, in judging of this, should divest himself of all partial regards or attachments whatever ; think and determine for the good of the whole only, and for posterity as well as for the present. The banks or declivities towards the river are also of par- ticular concern, as they affect the conveniency of the passage to and from the bridge, or determine the height of it, ou which in a great measure depends the expense, as well as the convenience of passage. The breadth of the river, the na- vigation upon it, and the quantity of water to be passed, or the velocity and depth of the stream, form also considera- tions of great moment ; as they determine the bridge to be higher or lower, longer or shorter. However, in most cases, a wide part of the river ought rather to be chosen than a nar- row one, especially if it is subject to grc.it tides or floods: for, the increased velocitv of tlie stream in the narrow part, being again augmented by the further contraction of the breadth by the piers of the bridge, will botli incommode the navigation through the arches, and undermine the piers and endanger the whole bridge. The nature of the bed of the river is also of great concern, it having a great influ(;nce on the expense; as upon it, and the derjth and velocity of the stream, depend the manner of laying the foundations, and building the piers. These are the chief and capital articles of consideration, v.diich will branch themselves out into other dependent ones, and so lead to the required estimate of tlie whole. Having resolved on the place, the next considerations are, the form, the estimate of the expense, and the manner of 6 THE PRINCIPLES OF BRIDGES. TRACT I. execution. With respect to the form; strength, utility, and beauty ought to be regarded and united ; the chief part of which lies in the arches. The form of the arches will depend on their height and span ; and the height on that of the water, the navigation, and the adjacent banks. They ought to be made so high, as that they may easily transmit the water at its greatest height, either from tides or floods; and their height and figure ought also to be such as will easily allow of a convenient passage of the craft through them. This, and the disposition of the bridge above, so as to render the passage over italso convenient, make up its utility. H.iving fixed ihe heights of the arches, their spans are still necessary for deter- mining their figure. Their spans will be known by dividing the whole breadth of the river into a convenient number of arches and piers, allowing at least the necessary thickness of the piers out of the whole. In fixing on the number of arches, let an odd number always be taken; and few and large ones, rather than many and smaller, if convenient: For thus we shall have not only fewer foundations and piers to make, but fewer arclies and centres, which will produce great savings in the expense; and besides, the arches them- selves will also require much less materials and workmanship, and allow of more and better passage for the water and craft through them; and will appear at the same time more noble and graceful, especially if constructed in elliptical, or in cy- cloidal forms ; for the truth of which, it may be sufficient to refer to that noble and elegant bridge lately built at Black- friars, London, by Mr. Mylne ; which might perhaps be ac- counted incomparable, at least in England, if the piers were of equal excellence: but these are too thick, and clumsy, and their appearance is made still less graceful by the double co- lumns placed before them. So that Blackfriar's arches and the Westminster's piers united, would be prercrable to cither briiige separately. If the top of the bridge be a straight horizontal line, the arches may l)c made all of a size; if it be a little lower at the ends than the middle, the arches must proportionally de- SECT. I. THE PROJECTS OF BRIDGES, &C. 7 crease from the middle towards the ends ; but if higlier at the ends than the middle, which can seldom happen, they may then increase towards the ends. A choice of tlie most convenient arches is to be made from some of the foilowincp propositions, where their several properties and effects are demonstrated and pointed out: Among these, the elliptic, cycloidal, and equihbrate arch, will generally claim the pre- ference, as well on account of the strength, and beauty, as cheapness or saving in materials and labour: Other particu- lars also concerning them may be seen under the word Arch in the Dictionary in the last section. Next find what thickness at the keystone or top will be necessary for the arches. For which see the word Keystone in the Dictionary in the 5th section. Having thus obtained all the parts of the arches, with the height of the piers, the necessary thickness of the piers themselve:^ are next to be computed. This done, the chief and material requisites are found; the elevation and plans of the design can then be drawn, and the calculations of the expense thence made, in- cluding the foundations, with such ornamental or accidental appendages as shall be thought fit ; which, being no part of the plan of this undertaking, is left to the fancy of the Architect and Builder, together with the practical methods of carrying the design into execution, I shall however, in the Dictionary, in the last section, not only describe the terms, parts, machines, &c, but also speak of their dimensions, pro- perties, and any thing else material belonging to them ; and to which therefore I from hence refer for more explicit in- formation in each particular article, as well as to these im- mediately following propositions, in which the theory of the arches, piers, &,Cj are fully and strictly demonstrated. THE PRINCIPLES OF BRIDGES. TRACT I. SECTION II. OF THK ARCHES. rROPOSlTION I. Let there be any number of lines ab, bc, cd, be, ATc. all in the same vertical phme, connected together and moveable about the joints or angles A, b, c, d, e, f; the two extreme points A and g being fixed: It is required to determine the pro- portion of the weights to he laid upon the angles b, c, d, isc. so that the whole mat/ remain in equilibrio. Solution. From the several singles, having dvawn the lines lib, cc, Da', &.C. per{)cndicular to the horizon ; about them, as diagonals, ''V constitute parallcio- A grams sacli, tliat those sides of every two tliat are at the op- posite ^nds of the given lines, niay be equal to each other ; viz, l;aving made one parallelogram 7nn, take cp b?2, and form tlic parallelogram pq ; then take nr = cq, and make the paralleicgram rs; and take zr = t>s, and form t!je parallelo- gram tv; and so on : Then the ^..id vertical diagonals r/^, cc, r>d, Ee, &.C, of those parallelograms, will be ])roporti()nal to the weigiits, as required. ])i'tnonstratioyi. l^y the rcsf;'iL!t;on of forces, each of tli? weights or i'orces Y.b, cc, nd, S^c, in the di.igonals oiiiie pa- ra!lelog'-a;ii ,, is equal to, and ;u,;v he I'e.^olvcd into, two force.;, e\pressed by two adj.icent ^ide.^ of tlii'. p..i'aiulograMi ; - viz. the lorce lib may be resoi-.i".! into the tuo iovccr, h;n, i>n. SECT. 2. OF THE ARCHES. 9 and in those directions; the force cc, into the two forces cp, cq, and in those directions ; the force nd, into the forces or, DS, and in those directions; and so on. Then, since two forces that are equal, and in opposite directions, do mutually balance each other; therefore the several pairs of forces mi and cp, cq and Br, gs and e/, &c, being equal and o])posite, by the construction, mutually destroy or balance each other; and the extreme forces Bm, ev, are balanced by the opposite resistances of the fixed points A, G. There is no force there- fore to change the position of any one of the lines, and con- sequently they will all remain in cquilibrio. CoroHarij. Hence, if one of the weights and the positions of all tlic lines be given, all the other weights may thence be found, as well as all the oblique forces in the direction of the bars or lines. And the weight which is given, may either be that at the lower extremity, as Bb, or it may be that at the vertex Tid, or it may be any of the intermediate ones, as cc; for, whichever of these is given, it will serve, as a diago- nal, to form the parallelogram about it ; then the sides of this parallelogram will give the sides of the. two next parallelo- grams, on each side of the former ; and so on through the whole collection of the bars. Thus, if the uppermost ver- tical weight, or diagonal d^, be the given one : Then draw dr parallel to de, and ds to dc, so forining the parallelogram rosd : then make cq =. Dr, and E^ = us: and, having drawn the several indefinite vertical lines b^, cf, Ee, at the angles, form the parallelograms pq and iv, by drawing qc parallel to EC, and cp to cd, and te to ef, and ev to de, Lastly, take B = pc, and make the parallelogram nm, by drawing nb pa- rallel to AB, and b}ii parallel to Bc. And so on through the whole. 10 THE PRINCIPLES OF BRIDGES. TRACT I. PROP. H. Jf any number of lines^ that are connected together and vioveable about the points of cn)inectio?i, be kept in equilibrio by weights laid on the u?igles, as in the last proposition: Then will the weight 07i any angle c be universally proportional to sine of the Z. bcd , . ,. , , . . , 1 ; that IS, directly as trie sine ot tnat s. z. BC6- X s. z. ctn -' -^ angle, and reciprocally as the siyics of the t'xo parts or angles into which that angle is divided by a line drawn through it perpendicular to the horizon. Sec the forruer figure. Demonstration. By the last proposition the weights are as B^, cc, Dd, &c, where e ~ pc, cy ?"d, Di" = tE, &.c. But, since the angle ab6 is the angle Bbn, and the angle Bcc = the angle ccg, kc, these being always t!ie alternate angles aiade by a line catting two other parallel lines ; also the sine of the Z. aec s. Z. Bnb, and s. A. bcd = s. Z. cqc, these being supplements to each other ; by plane trigonoaieiry we shall have, B^ X s. Z. AB^ cf X s. Z. con ( B = ) = ( CD = ) , ^ ' S. Z. ABC ^ ^ ' ^. jL BCD CC X S. Z. BCC D(/ X S. Z. c/dE [cq ) =r ( Dr = ) , ^ ^ ' S. Z. BCD ^ ^ S. Z. CDE Dr/ X S Z. Q.T)d . Ef X S. Z. (^KF ^ ' S. Z. CDE ^ ^ S. Z. DEF iind so on. Ilenct>, Y,b : cc cc : D^ S. Z. AEC s. z. AB^ S. Z. BCD s. z. BCf S. A CDE S. Z- BCD s. Z. ccd' S. Z. CDE s, Z. ^/be' . _ ^ S. Z. DEF Df/: EC : : , : , ac. S. Z. CDtt s. Z. CtF Or, by dividing the latter terms of the first of these pro- portions each by s. Z. 6bc, and then compounding togetlier two of the proportions, then three of them, &c, striking out vhe common fiictors, and observing that the s. Z. ^ec is = SECT. 2. OF THE ARCHES. 11 s. z. Bcc, the s. Z. ccD = s. cdc/, &c, we shall have the fol- lowing proportions ; viz, S. A ABC S. Z. BCD BU : cc : : -, ; : = , S. Z. AB6> X S. Z. OBC S. Z. BCC X S. Z. CCD , , S. Z. ABC S. Z. CDE Bb -.Bel : : -. -. : r ; , S. Z. ABt X S. Z. OBC S. Z. cud X S. Z. (^DE , S. Z. ABC S. Z. DEF Bb : Ee : : r -. : , S. Z. ABO X S. Z. ^BC S. Z. DEC X S. Z, ^-Ef and SO on. Since cp or bw : bw or nb : : s. z. b6, or s. Z- ABO : s. Z. obc or s. Z. bcc : ; : r : S. Z. BCf S. Z. ABO and c/? or ^rc : eg or or : : s. Z. cct/ or s, Z. corf : s. Z. ccy or 1 1 s. z. Bcc : : s. z. BCC ' s. z. cdJ' it is clear that cp is as ; that is, the forces 77iB, pc. ^ S. Z. BCC rOy &c. are always reciprocally as the sines of the angles which they make with the vertical line. . , . cp X S. Z. CpC cp X S. Z. BCD And since cc = ^ r: : fe. Z. CCp S. Z CCD 1 r r , S. Z BCD tncreiorc any rorce or weip-lit cc is as , ' ^ s. z ccB X s. z cca And this is the same as the property in corol. 4 to the 3d proposition following. Corol. If DC be produced to h; then, the sine of the angle hcB being equal to the sine of its supplement BCD, the same weight or force cc will be always proportional to : which three angles toeether make up S. Z BCC X S, Z DCC O J3 1 two right angles. Properties similar to the foregoing are otherwise deter- mined in the following propositions. 12 THE PEINCllM.KS OF BRIDGES. TRACT I. PROP. III. Let there be any number ofliih's, or bars, or beams, ab. bc, CD, DE, h\c. all in the same vertical plane, connected together and freely moveable about the joints or a^igles a, k, c, d, e, &'c, and kejyi in equilihrio by their oivn xceights, or by iceights only laid on the angles : It is required to assign the proportion of those weights: as also the force or push in the direction of the said lines; and the horizontal thrust at every angle. Solution. Through an}- point, as D, draw a verti- cal line aDHg, &:c : to \vi)ich, f'roiii any point, as c, draw lines in the direction -^ of, or parallel to, the given lines or beams, viz, ca parallel to AB, and cb parallel to bc, and ce to de, and cf to ef, and eg to FG, &c; also CH parallel to the horizon, or perpendicular to til.: vertical line uDg, in wliicii also all these parallels ter- minate. 1 hc'.i v> ;!! all thr.^e lines be exactly proportional to the forces actii-g or cxL-rtcd in the directions to which they arc parallel, and of all the llirt-e kind>, viz. vertical, horizontal, and obiiqne. That is, tiie obh;v.ie lorces or thrusts in direc- tion of the bars A7i, bc, co, de, ef, fg, are proportional to th.eir parahel;, . . cr/, cb, CD, c, cf, eg; and the vertical wciglits on t.he angles B, Cj d, E; f, &C, are as the parts of the vertical .... ab, bv., dc, cf, fg, and the weight of the uhole frame ai;ci3EFG, . . . is proportional to t'jc snpi of ail the verticals, or to ag ; also the horizontal thrust, ;it cverv angle, is every where the same constant qtianti: v, and is eKj)rc>sed l)v the constant ho- SKCT. 2. OF THE ARCHES. 13 Demonstration. All these proportions of the forces derive and follow immediately from the general well known pro- perty, in Statics, that when any forces balance and keep each other in equilibrio, they are respectively in proportion as the lines drawn parallel to their directions, and terminatino- each other. Thus, the point or angle b is kept in equilibrio by three forces, viz, the weiglit laid and acting vertically downward on that point, and by the two oblique forces or thrusts of the two beams ab, cb, and in these directions. But ca is parallel to ab, and cb to bc, and ab to the vertical weight ; those three forces are therefore proportional to the three lines ab^ ca, cb. In like manner, the angle o is kept in its position by the weight laid and acting vertical!}^ on it, and by the two ob- lique forces or thrubts in the direction of the bars bc, cd: conseqiiently these three forces are proportional to the three lines ^D, cbj cd, which are parallel to them. Also, the tlircc forces keeping the point d in its position, are proportional to their three parallel lines oe, cd, ce. And the three forces balancing the angle e, are proportional to their three parallel lines ef, ce, cf. And the three forces balancing the angle f, are proportional to their three parallel lines ^"", cf, eg. And so on continually, the oblique forces or tLirnsts in the directions of the bars or beams, beino- al- ways proportional to the parts of the lines parallel to them, intercepted by the common vertical line ; while the vertical forces or weights, acting or laid on the angles, are propor- tional to the parts of this vertical line intercepted by the two lines parallel to the lines of the corresponding angles. Agaiii, with regard to the horizontal force or thrust: since the line DC represents, or is proportional to the force in the direction dc, arising from the weight or pressure on the angle d ; and since the oblique force dc is equivalent to, and resolves into, the two dh, hc, and in those directions, by the resolution of forces, viz, the vertical force dh, and the horizontal force HC ; it follows, that the horizontal force or li THE PRINCIPLES OF BRIDGES. TRACT I. thrust at the angle d, is proportional to tlie line ch ; and tlie part of the vertical force or weight on the angle d, which produces the oblique force DC, is proportional to the part of the vertical line dh. In like manner, the oblique force c^, acting at c, in the di- rection CB, resolves into the two Ah, hc; therefore the hori- zontal force or thrust at the angle c, is expressed by the line CH, the very same as it was before for the angle d ; and the vertical pressure at c, arising from the weights on both d and c, is denoted by t!ie vertical line bii. Also, the oblique force ac, acting at the angle b, in the direction ba, resolves into the two an, hc ; therefore again the horizontal thrust at the angle b, is represented by the line CH, the very same as it was at the points c and d ; and the vertical pressure at b, arising from the weights on b, c, and D, is expressed by the part of the vertical line an. Thus also, the oblique force ce, in direction de, resolves into the two ch, ue, being the same horizontal force with the vertical ue; and the oblique force c^ in direction ef, re- solves into the two CH, a/; and the oblique force c^, in di- rection FG, resolves into the two ch, h^,- and the oblique force c^, in direction fg, resolves into the two ch, H.y ; and so on continually, the horizontal force at every point being expressed by the same constant line ch ; and the vertical pressures on the angles by the parts of the vertical, viz, an the whole vertical pressure at b, from the weights on t!iti angles b, c, d : and bn the whole pressure on c from the weights on c and d; and dh the part of the weight on d causing the oblique force dc ; and He the other part of the weight on d causing the oblique pressure de ; and uf the whole vertical pressure at e from the weights on d and e; and Kg the whole vertical pressure on f arising from the weights laid on d, e and f. And so on. So that, on the whole, AH denotes the whole weight on the points from d to a; and h^ the whole weight on the jjoints from d to c ; and a^ the whole weight on all the points on both sides; SECT. 2. OV THE ARCHES. 15 while ab, bv>, De, ef,fg express the several particular weights laid on the angles b, c, d, e, f. Also, the horizontal thrust is every where the same con- stant quantity, and is denoted by the line CH. Lastly, the several oblique forces or thrusts, in the direc- tions AB, BC, CD, DE, EF, FG, are expressed by, or are propor- tional to, their corresponding parallel lines, ca, c3, CD, ce, cfyCg. Corollary 1. It is obvious, and remarkable, that the lengths of the bars ab, bc, &c, do not affect or alter the pro- portions of any of these loads or thrusts; since all the lines ca, c3, ab, &c, remain the same, whatever be the lengths of AB, EC, &c. The positions of the bars, and the weights on the angles depending mutually on each other, as well as the horizontal and oblique thrusts. Thus, if there be given the position of dc, and the weights or loads laid on the angles D, c, B ; set these on the vertical, dh, d3, ba, then cb, ca give the directions or positions of CB, BA, as well as the quantity or proportion ch of the constant horizontal thrust. Coral. 2. If CH be made radius; then it is visible that na is the tangent, and ca the secant of the elevation of ca or ab above the horizon ; also nb is the tangent and cb the secant of the elevation of cb or CB ; also hd and CD the tangent and secant of the elevation of CD ; also ue and ce the tangent and secant of the elevation of ce or de ; also h/ and cf the tan- gent and secant of the elevation of EF ; and so on ; also the parts of the vertical ab, bu, ef,/g, denoting the weights laid on the several ano-les, are the differences of the said tangents of elevations. Hence then in general, 1st. The oblique thrusts, in the directions of the bars, are to one another, directly in proportion as the secants of their angles of elevation above the horizontal directions; or, which is the same thing, reciprocally proportional to the co- sines of the same elevations, or reciprocally proportional to 16 THE PRINCIPLKS OF BRIDGES. TRACT 1. the sines of the vertical angles, cr, b, y>, e,fy g, &c, made by the vertical line with tlic several directions of the bars ; be- cause the secants of any angles are always reciprocally in proportion as their cosines. 2, The weight or load laid on each angle, is directly pro- portional to the diirerence between the tangents of the ele- vations above the horizon, of the two lines which form the angle. 3. The horizontal tlirust at every angle, is the same con-' stant quantity, and has the same proportion to the weight on the top of the uppermost bar, as radius has to the tangent of the elevation of that bar. Or, as the whole vertical ag, is to the line CH, so is the weight of the whole assemblage of bars, to the horizontal thrust. Other properties also, concerning the weights and the thrusts, might be pointed out, but they are less simple and elegant, than the above, and are therefore omitted ; the following only excepted, which are inserted here on account of their usefulness. Corollary 3. It may hence be deduced also, that the Aveight or pressure laid on any angle, is directly proportional to the continual product of the sine of that angle and of the secants of the elevations of the bars or lines which form it. Thus, in the triangle ^cd, in which the side bn is propor- tional to the weight laid on the angle c, because the sides of any triangle are to one another as the sines of their opposite angles, therefore as sin. d ; c6 : : sin. ^cd : ^d ; that is, bn is as sin. bcT> J . , . r ^ 1 f 7 X CO ; but the sme or angle D is the cosine oi the sm. D ^ elevation DCH, and the cosine of any angle is reciprocaJIy proportional to the secant, therefore ^d is as sin, bcu x sec. DCH X c^ ; and ch being as the secant of the ang'e /;cn of the elevation of ^c or T5C al)ove the horizon, therefore ^d is as sin. bci:) x sec. hen x sec. dch; and the sine of bcu being the siune as the sine of its supplcncnt bcd ; therefore the weight on the angle c, which is as bn, is as the sin. bcd SECT. 2. OF THE ARCHES. 17 X sec- DCH X sec. ben, that is, as the continual product of the sine of that angle and the secants of the elevations of its two sides above the horizon. Corol. 4. Further, it easily appears also, that the same weight on any angle c, is directly proportional to the sine of that angle ecd, and inversely proportional to the sines of the two parts bcp, dcp, into which the same angle is divided by the vertical line cp. For the secants of angles are reci- procally proportional to their cosines or sines of their com- plements : but BCP = cbn, is the complement of the eleva- tion ^CH, and DCP is the complement of the elevation dch ; therefore the secant of ^ch x secant of dch is reciprocally as the sin. bcp x sin. dcp ; also the sine of ^cd is rz the sine of its supplement bcd ; consequently the weight on the angle c, which is proportional to sin. ^cd x sec. bcu x , . , sin. BCD sec. dch, is also proportional to -. , when * sin. bcp x sin. dcp the whole frame or series of angles is balanced, or kept in equilibrio, by the weights on the angles j the same as in the preceding proposition. Scholium. The foregoing proposition is very fruitful in its practical consequences, and contains the whole theory of arches, which may be deduced from the premises by sup- posing the constituting bars to become very short, like arch stones, so as to form the. curve of an arch. It appears too, that the horizontal thrust, which is constant or uniformly the same throughout, is a proper measuring unit, by means of which to estimate the other thrusts and pressures by, as they are all determinable from it and the given positions; and the value of it, as appears above, may be easily computed from the uppermost or vertical part alone, or from the whole as- semblage together, or from any part of the whole, counted from the top downwards. The solution of the foregoing proposition depends on this consideration, viz, tliat an assemblage of bars or beams, VOL. I. c 18 THE PRINCIPLES OF BRIDGES. TRACT 1, being connected torrether by joints at tlieir extremities, and freely moveable about thein, maybe placed in such a vertical position, as to be exactly balanced, or kept in equilibrio, by their mutual thrusts and pressures at the joints ; and that the eflfect will be the same if the bars tiiemsclvesbe considered as without wcioht, and the angles be pressed down by laying on them weights which shall be equal to the vertical pressures at the same angles, proiluccd by the bars in the case when they are considered as endued with their own natural weights. And as we have found that the bars may be of any length , without aftccting the general properties and proportions of the thrusts and i)rcssnres, therefore by suppo^sing them to be- come short, like arch stones, it is plain that we shall then have the same principles and properties accommodated to a real arch of equilibration, or one that supports itself in a p<;r- fect balance. It may be further observed, that the conclu- sions here derived, in this proposition and its corollaries, exactly agree with those derived in a very different way, in the former editions of the princi})lesof bridges, viz, in props. 1 'and 2, and their corollaries ; and which have been here re- peated, in the foregoing prop. 2. PROP. IV. If the u'holc figure in the tJiird proposition i>e inverted, or turned round the Iborizonial line ag as an axis, till it be eo)n' pletehj reversed, or in ike same vertical plane below the first position, each angle D, d, bsc, being in the same plmnb line ; and ifxi'eights i, k, /, m, n, which are respectively equal to the weights laid on the angles b, c, d, b, f, of the first figure, be now suspended by threads fronp tlie corresponding angles b, c, d, f'i./\ of the lower figure ; then will those weights keep this figure in exact equilibrio, the same as the former, and all the tensions or forces in the latter case, whether vertical or horizontal or oblique, nvitl be exactly equal to the corresponding forces of weight or pi cssure or thrust in the like directions of the firstfigurc. SECT. 2. OF THE ARCHES. 19 This necessarily happens, from the equality of the weights, and thesi- milarity of the po- sitions and actions of the whole in both cases. Thus, from the equality of the correspond- ing weights, at the Jike angles, the ratios of the weights, ab, bd^ dh^ he, &c, in the lower figure, are the very same as those, ab^ bo, dh, ue, &c, in the upper figure ; and from the equality of the constant horizontal forces ch, cA, and the similarity of the positions, the corresponding vertical lines, denoting the weights, are equal, namely, ab =: ab, bn = bd, dh = -DH, &c. The same may be said of the oblique lines also, ca, cb, &c, which being parallel to the beams a^, he, &c, will denote the tensions of these, in the direction of their length, the same as the oblique thrusts or pushes in the upper figures. Thus, all the corresponding weights and actions, and positions, in the two situations, being exactly equal and similar, changing only drawing and tension for pushing and thrusting, the balance and equilibrium of the upper figure is still preserved the same in the hanging fes- toon or lower one. Scholiimi. The same figure, it is evident, will also arise, if the same weights, i, k, ly ?n, w, be suspended at like dis- tances, A^, be, &c, on a thread, or cord, or chain, &c, having in itself little or no weight. For the equality of the weights, and their directions and distances, will put the whole fine, when they come to equilibrium, into the same festoon shape of figure. So that, whatever properties are inferred in the c 2 20 THE PBINCIPLES OF BRII?GES. TRACT I. corollaries to the 3d prop, will equally apply to the festoon or lower figure hanging in equilibrio. This is a most useful principle in all cases of equilibriums, especially to the mere practical mechanist, and enables him in an experimental way to resolve problems, which the best mathematicians have found it no easy matter to effect by mere computation. For thus, in a simple and easy way he ob- tains the shape of an equilibrated arch or bridge ; and thus also he readily obtains the positions of the rafters in the frame of an equilibrated curb or mansard roof^ a single instance of which may serve to show the extent and uses to which it may be applied. Thus, if it should be required to make a curb frame roof having a given width AE, and consisting of four rafters ab, BC, CD, DE, which shall either be equal or in any given proportion to each other. There can be no doubt but that the best form of the roof will be that which puts all its parts in equilibrio, so that there may be no un- balanced parts, which may require the aid of ties or stays, to keep the frame in its position. Here the mechanic has nothing to do, but to take four like but small pieces, that are either equal qr in the same given proportions as those proposed, and connect them loosely together at the joints A, B, c, D, E, by pins or strings, so as to be freely move- able about them J then suspend this from two pins, a, Cj fixed in a horizontal line, and the chain of the pieces will arrange itself in such a festoon or form, ahcde, that all its parts will come to rest in equilibrio. Then, by inverting tlie figure, it will exhibit the form and frame of a curb roof flCy Je, which will also be in equilibrio, the thrusts of the pieces now balancing each other, in the same manner as was done bv the mutual pulls or tenbiou::< SECT. 2.. OP THE ARCHES. 21 of the hanging festoon abcde. By varying the distance ae^ of the points of suspension, moving them nearer to, or farther off, the chain will take different forms ; then the frame abode may be made similar to that form which has the most pleas- ing or convenient sliape, found above as a model. Indeed this principle is very fruitful in its practical conse- quences. It is easy to perceive that it contains the whole theory of the construction of arches : for each stone of an arch may be considered as one of the rafters or beams in the foregoing frames, since the whole is sustained by the mere principle of equilibration, and the method, in its application, vi'iW afford some elegant and simple solutions of the most dif- ficult cases of this important problem ; some examples of which will be shown hereafter. PROP. V, To form mechatikalli) a balanced Festoon arch, on the prin- ciples of (he last proposition ; having a gi-veii pitch or height und span, and also a given height and form of wall or roadway over it. Let AM be the given r proposed span ot the arch, DQ its pitch or greatest height, dk the thickness at the crown, and ALKXM the given an- terior form of the wall : in order to determine the form of the curve Adm which shall put that wall in equilibrio. Invert the whole figure alknm, as in the opposite posi- tion Al/aiM, or construct this latter figure, on the lower side of AM, exactly equal and similar to the proposed upper one; the point d answering to the point D, and the point k to the 22 THE PRINCIPLES OF BRIDGES. TRACT I- point K, &c. Let a very fine and thin, but strong line, such as a fine silkencord, or abricklayer's working hne, or perhaps a very fine and slender chain of small links, be suspended from the extreme points a and m, and of such a length, that its middle point may hang at the point d, or a little below it. Divide the given span or width am into a number of equal parts, the more the better, as at the points 1, 2, 3, 4, 5, &c; from which draw vertical lines, cutting the festoon chain or cord in the corresponding points 1, 2, 3, 4, 5, &c. Then take short pieces of another chain, and suspend them by these points of the festoon I, 2, 3, &c, as represented by the dotted verticals in the lower part of the figure. This will somewhat alter the form of the curve. If now the neu' curve should correspond with the point d, and all the bottoms of the vertical pieces of appended chain also coincide with the given line of roadway /An, the business is done. But if both those coincidences do not take place, then alterations must be made, by trials and by judgment, in lengthening or short- ening either the festoon A(^m, or the appended vertical pieces of chain, or in both, till such time as those coincidences are accomplished, namely, the bottom of the arch with the point d^ and the bottom of the appended pieces v/ith the boundary I k n. Then re-invert the whole figure, or otherwise trace out the upper curve adm exactly like or the same as the lower one Af/M, and there will be obtained an arch sustixining the wall above in perfect equilibrium. Scholium. Thus then, as explained by professor Robison, we have an easy and practical way, bv which any common in^ telligent workman may readily construct for himself the form of a real balanced arch, to an}' proposed design for a bridge. In this method, the thinner and lighter the festoon line is, so us to bear but a small proportion to the weight of the ap- jjcnded j)ieces of chain, so much the more exact will the conclusion be obtained, when the superincumbent wall is of uniform weight of njasonry. But as the festoon line rejjre- sents ttjc hne of voussoirs or arch stones, in the conslructuU ECT. 5. Of THE AnCllES. ' 25 arcli, if these only are solid, and the rest of the wall or matter above them be looser and lighter, then there ought to be an equality of proportion between the weights ot the festoon chain and the string or rib of arch stones, and between the superior wall and the appended pieces of chain; a circum- stance of equality to be obtained by mutual accommodations and calculations adapted to the real circumstances of the case. The chief objection to the curv^e found in this way is a Avant of elegance, and perhaps too of convenience and of economy, because it does not spring or rise at right angles to the horizontal line, but at a much smaller angle ; and which indeed is the case with all curves of equilibration. However, this is a circumstance which can be very safely and prolitably remedied ; for in the part of the flanks near the piers, it may be cut away to hollow the arch out to any form we please, so as, for instance, to resemble the elliptical arch, which is one of the njost graceful of all; because the masonry is so solid and strong in that part. And this will be not only more agreeable to the eye, but will also leave more room for water and boats to pass, and will be a saving in the expcnce of masonry. To accomplish this end with more re- sularitv and method, instead of dividing the horizontal line into equal parts at the points 1, 2, 3, &,c, if the festoon chaia itself be so divided, viz, into equal parts, and the pieces of chain be appended at these, in the manner before mentioned, then the greater number of these pieces being thus near the extremities, they will draw the arch more down in that part, and thus hollow it out there in a more regular and uniform manner, making the shape more pleasing and commodious, and yet leaving it sufficiently near a true balance. The following proposition is here added, to determme the figure of a balanced arch, on the supposition that the voussoirs are at liljcrty to slide on each other. A principle indeed having no real foundation in fact, though it has been much insisted on by some persons. 24 THE PllINCIPLES OF BRIDGES. TRACT I. PROP. VI. // is proposed to determine the nature and properties of a balanced archy as derived from the property/ of the wedge, or hy considering the voussoirs or arch-stones as frustrums of wedges. Let ACEGi &c, be the inner or lower curve of an arch, formed of the voussoirs, or wedge pieces, the vertical sections of which are the quadrila- terals AD, OF, EH, GK, &C, considered as so many ele- mentary parts of the arch, the upper sides of them forming the exterior or outer curve BDFHK, and their butting sides making the joints ab, cd, ef, GH, IK, &c, which joints produced, meet in the point o, of the vertical hne oab. Through any point 6, in that line, draw the horizontal line bdfhk, or perpendicular to the ver- tical line oab, and cutting the directions of the joints in the respective or corresponding points b, d,f, A, k, &c. Now every wedge in the balanced arch, supposing its sides polished, must be kept in equilibrio, in its place, by the mu- tual action of three forces, viz, by its own weight acting in a direction perpendicular to the horizon, and by the thrust or pressure of the two adjacent wedges, one on each side, in directions perpendicular to their sides, or to the joints: So, for instance, the wedge ad is balanced, or kept in equilibrio, by its own weight acting in the vertical direction bo, and by two forces acting perpendicularly to ab and cd ; and tlie stone CF, by its weight in the vertical direction, and by two forces perpendicular to cd and ef; also the stone eh, bj'its weight acting vertically, and by two forces perpendicidar to EF and GH ; also the stone gk, by its weight verticuily, and SECT. 2. OF THE ARCHES. 25 by two forces perpendicular to gh and ik ; and so on, the weights ail acting in tlie vertical direction parallel to bao. But, whenever three forces balance one another, they have then to each other the same ratios as the sides of a triano-je drawn perpendicular to the directions of the forces. There- fore the three forces balancing the wedge ad, are propor- tional to the three sides of the triangle obd, these sides being respectively perpendicular to those forces, viz, the side bd perpendicular to the vertical direction of gravity, also oh perpendicular to the force against the joint ab, and oxl per- pendicular to the force against the joint cd. For the same reason the wedge cf is balanced by three forces proportional to the three sides df, od, of, of the triangle odj"; and the wedge EH by forces proportional to the three sides //ij of, oh, of the triangle of/i ; and the wedge gk by forces pro- portional to the three sides hk, oh, ok, of tlie triangle ohk ; and so on. So that, in all these cases, the weights of the wedges, and their oblique push perpendicular to the joints, will have these following ratios, viz, the weights of the wedges - - - ad, cf, eh, gk. Sac, as the parts of the horizontal - - bd, df, fh, hk, &c, and the push at the joints as - - oh, od, of, oh, &,c, also the sums of the wedges, or the parts, ad, af, ah, ak, are proportional to the perpendiculars bd, bf, bh, bk, which are the tangents of the angles bod,bof,boh, bok, &c, of which the oblique thrusts od, of, oh, ok, are the secants, to the radius ob, which denotes tlie constant push in the hori- zontal direction at every wedge, or every point of the arch. Which, on the whole, amounts to this, viz, that the weights of any part of the balanced arch, or set of wedges, com- mencing from the vertex, are directly proportional to the tangents of the angles which the joints make with the vertical line or direction, while the oblique thrusts, in the directions of the arch at the extremity, or perpendicular to the joints, are proportional to the secants of the same angles ; the con- stant horizontal push, at every point, being proportional to the radius. ff THE PRlNXTPLns OF BRIDGES- TRACT I. Awd this property comes to the very sairie tiling as the properties in tlie foreii;oiiif; propositions, b(;cause tiie angles I'f elevation of the curve at every point, or of the direction of the tangents there, or of the curve itself, are equal to tlie an:;!es in this proposition, which the joints form with the ver- tical direction. So that, all the three theories in these fonr propositions are all one and the same in effect, amounting to the very same thing, and yielding the same conclusions. And therefore, whatever consequences may further be drawn from any one of them, may be understood as deduced from tiie whole. Scholium. ^In the practice of bridge-building, the key piece, or wedge at the crown, is a solid, having its magni- tude and weight half on each side of the middle vertical line ; whereas, in this proposition, it has been su[)poscd that this wedge is divided and actually separated in two by that line ab: this however will cause no difference in the theory, nor yet in the practice; for, in any calculations that niay be re- quired, it is only necessary to suppose the key [)iece divided cxactlv in the middle, then taking half its weight for the weight of the piece ad, and computing all the other weights and angles from the nnddle line ab. It has also been su])posed, in all the three theories that have been contemplated, that the constituent parts are formed of mateiials perlectlv smooth and polished, and [)ut together without cement, and without all kinds of ties or bars, so as to leave them quite at liberty to slide over each other, the parts being kc[it in a perfect balance bv means of their shape, weight, and dis[)osiiion only. This, it must Ix; acknowledged, is not the case in real practice; as here ail the materials are quite rough, which very much prevents them from sliding bv each other, even when their abutting surfaces are laid at a considerable slope or angle. But this circumsrance however, so far from being a disadvantage, by thus deviating from the theory, is on that very account of great use and benefit. For, the equilibrium among the con- SECT. 2. OF THE ABCIIES. wj stituent paits of the arch, established by the fore'Toinn- the- ories, is of that nice and critical nature, that the whole han, therefore, extending 2 degrees and a half on each side of the vertical line passing through the centre; and have n other wedges, of equal angle (5) on each side of the key, making in all S5 wedges, which, at 5 degrees each, will form an entire arch of 175 degrees. In this case, the angle which the sides of the middle wedge forms with the middle vertical line, will be that of half the breadth of the wedge, or 2 ^ degrees ; and the angles which the sides of the other wedges, on each hand of the crown or key wedge, form with the vertical direction, will be found by adding continually so THE PRINXIPLES OF BRIDGES. TRACT I, the breadth of each wedge (5 degrees), to the said 2^ de- grees; b}' wliich it will he found that the angles at the cen- tre, formed with the vertical, by the said lower edges of the arch pieces, in order after the key, will be as follows, viz, that of the 2d wedge 7f degrees; that of the 3d, 12^ de- grees; that of the 4th, I ly degrees; and so on to the 17th or last on each side the key, which will have its lower edge making an angle of 81f degrees with the vertical direction: all which angles, of inclination to the vertical, are ranged in the 2d column of the following tablet, the first, or half the middle wedge, making an angle of 2^ degrees. We shall also suppose the weight of t!ie middle wedge at the crown to be a certain given quantity, represented by unity or I , and express the several other weights and pressures, as in the other columns of the said tablet, in terms of that unit: so that all these proportional numbers for the other weights and pressures, Avill require to be multiplied by any other weight of middle wedge which may happen to occur in any Qther case. Now, in regard to the rule for computing all the other "weights and pressures, according to the conclusions from the preceding theory, it is very easy and simple indeed, viz, that the weight of any part of the arch, counted from the vertex or crown downward, is always proportional to the tangent of the angle of inclination of the lower wedge to the vertical, while the oblique push or pressure, in direction of the curve, is proportional to the secant of the same angle, and the constant horizontal thrust is proportional to the ra- dius. For which reason it is, as former! v observed, that the constant horizontal thrust is a proper radical measuring unit, by means of which to compute the two other circumstances, namely, the weight of the arch, and the oblique push or pressure in the direction of the curve: for, the horizontal thrust being taken for radius, then the weight of the semi- arch will be the tanocut of the anirle witii the vertex, and the oblique pressure the secant of the same angle, to that mdius Coiisetpientlv, if the constant horizontal push be called hy then the weight of t)ic semiarch will he h x tj or h seCT. 2. OF THE ARCHES. 31 multiplied by the tangent of the side's inclination to the ver- tical, and the oblique pressure of the arch will be h x s, or h multiplied by the secant of the same angle. So that, in calculating the said several weights and oblique pushes of the arches, we have nothing to do but to take out, from a trigonometrical table, the tano-ents and secants of the seve- ral angles of inclination to the vertical, as contained in the 2d column of the tablet, and multiply all the tangents and secants by the number expressing the constant horizontal thrust, for all the values of the several weights and pres- sures, as arranged in the 3d and 4th columns of the tablet j tlie products of the tangents being the several weights of the half arches, in the 4th column, and the products of the se- cants being the oblique pressures of tlie same in the arch's direction, as in the 3d column. This calculation will be rendered still easier by using the log. tangents and secants; for there will then be nothing to do, but to take out all the log. tangents and secants; then to each of them add the cou- stant log. of the horizontal thrust ; lastly, take out the na- tural numbers answering to these sums, and they will be tlte required weights and pressures. As to the uniform horizontal thrust, which is the constant multiplier, its value is easily found thus: It has been shown that this horizontal thrust is every where in the same pro- portion to the weight of half the middle or key wedge, a* radius is to the tangent of half the angle of that weds:e; that is, as / : 1 : : 4-^' : 4^' -i- t = h the horizontal thrust, put- ting zo for the weight of the key piece, and t for the tangent of half its angle; or, if we put its weinht :c' 1, thou tliis will beconie ^ ~ t = h the horizontal thrust. Now, in the example, the angle subtended by the key is 5 degrees, the half of which is 2|- degrees, and tlie tangent of this is 043660y; then i or -5 -0436609 = 11-4.5 1 SS3 = h the constant horizontal thrust, that is, 1 1 times the weight of the key piece and nearly one half more; or, the same mav be easier found from the cotangent of the same angle 2 r degrees, which is 22*903766, the cotangent of any angle being equal to the reciprocal of its tangent, to the radius 1 ; 92 THE PRINCIPLES OF BRIDGES. TRACT I* therefore, in general, -^ -4- tang. = ^ the cotang. is = h the horizontal thrust, and in the present instance the half of the cotangent 22*903766 is 11-451883 the same value of tl>e horizontal thrust as before. Hence then the constant number 1 1*45 1883 is to be mul- tiplied by the tangents of all the vertical angles, to give the weights of the semiarch, in the 4th column, and by the se- cants of the same angles, to give their oblique pressures, as in the 3d column; or else, to work by the logarithms, the log. of the constant number 11*451883, which is r0588769, is to be added to all the log. secants and tangents of the said angles, then the corresponding natural numbers taken, and ranged in the 3d and 4th columns of the table. The differences of the numbers in the 4th column are taken, and ranged in the 5th or last column, for the weights of the single wedge pieces taken separately, making the whole of the first or key wedge equal to 1. The table is as follows. Vertical No. of aneles of Oblique pressures, Wts. of half arches, Wts. of the sec- sections. the joints, or Z s 0. = A X sec.z.O. = /(Xtan.z.O. tions or wedges. de2;roc8. 1 21 ir4627i> 0*5 r 2 7^ 11-55070 1*50767 1-00767 3 124 11*72993 2-53882 1-03115 4 17i 12*00763 3-61076 1-07194 3 22t 12-39543 4-74352 1-13276 6 27t 12-91065 5-96147 1-21795 7 32t 13-57S37 7-29565 1-33418 8 37t 14-43478 8-78734 1-49169 V 42t 15-53267 1049372 1-70638 10 47t 16-95094 12-49753 2*00381 11 52i 18-81177 14-92439 2-42686 12 57t 21-31377 17-97585 3-05146 1'5 G2i- 24-80112 21-99886 4-02301 It 67^ 29-92521 27-64727 5-64841 15 72- 38-08334 36-32073 8-67346 16 -^Tl 52-91028 51-65611 15-33538 17 821 87-73628 86-98568 35-32957 18 874- 262-54113 262-29125 175-30557 BCl*. 2. OP THE ARCHfiS. 33 From this calculation, as well as from the theorems by which it is made, it is manifest how greatly the weight and the pressure of the semiarch increase towards the bottom or the extremity, where the position of the joint approaches towards the horizontal direction, or the angle it makes with the vertical approaches towards a right-angle; and when that angle actually becomes a right-angle, or the joint quite horizontal, then the weight and pressure become equal and infinite, which must naturally be expected^ both because the tangent and secant of the angle (being a right one) are then infinite, and also because it must require an infinite weight or pressure to balance there the constant giv^en horizontal thrust, which is perpendicular to the former. We may here, by the way, stop to examine a little in what manner the preceding calculation of the weights of the voussoirs may be employed to give a familiar and easy me- chanical construction, that may approach very near to a true balanced arch. In order to this, we are to consider, that since the bases, or extents of the under sides, of all the voussoirs, are equal, it will thence happen that their weights will have to each other nearl}' the same ratios as their lengths, from the under to the upper side of them, or taken in the direction of the radius, that is perpendicular to the under curve or intrados, at least when tlie breadth or angle of these^ wedges is very small, which is the case iti real practice, the approach to equality being the nearer indeed as their breadth is the smaller. And though the angle of 5 degrees, em- ployed in the preceding calculation, be not such a small breadth as to render the equality and the construction per- fect, it will yet strve to show the manner of proceeding in such a way of forming the arch,ajid will besides approach tolerably near to the truth. As it is most proper that the joints between the wedge^ mM" - ''^'--''i"!S SKCT. 2. OF THE ARCHES. ' 35 passing through the middle of every second wedo-e the first OAB passing through tl)e middle of the keT-piece. Then on these radii produced, set off, from the arc of the semi- circle, AB, GH, &c, every second number in the last column of the table, when multiplied b}' 6, the assumed length of ab; then, drawing with the hand a curved line tlirough the extremities of all the exterior lines, it will be the extrados required, exhibiting the form and limit of the wall built of uniform materials, above the circular soffit, so as to constitute an arch of equilibration nearly as in the annexed fiof. L M Where it is seen that the extrados follows nearly a course parallel to the intrados for about 30 degrees on each side of the vertexj after which, it begins to bend the contrary way, having there a contrary flexure during the rest of its course, going off to an infinite distance on each side parallel to the base, making the voussoirs at last of an infinite length, and composing all together a form of arch very unfit for adop- tion in practice. We shall now show, in the next proposition, that, by another very strict and genuine construction, an exterior curve is derived exactly similar to the curve here obtained: in the determination of which, some part of the mode of reasoning in the demonstration of the last prop, is here again necessarily repeated. PROP. VII, If ACEGi ^c, be ail arch, supporting a wall abki, formed of the voussoirs or arch stones ad, cf, Kc, lying aslope, on smooth surfaces, and having the joints ab, cd, bsc, every where perpendicular to the curve of the arch ace ^c. It is required to find the lengths of these arch stones, so that the whole fab r if mdj/ be balanced, or kept in equilibrio. D 2 3a he principles of bridges. tRACT 1- Let A be the vertex of the inner curve of the proposed arch; ab the given thickness j^ of the wall at the crown, or lenjjthofthearchstonethere; also BAO, Dco, &c, the joints produced, making ao the ra- dius of curvature at a, and co at c, and eo at e, &c; the bases of the stones ac, ce, eg, gi, &c, being so many elements or small parts of the arch; and the vertical sections of the stones, or the areas of the qua- drilaterals AD, CF, EH, gk, being proportional to the weights of them. Now every stone in the balanced arch will be kept in equilibrio by three forces, viz, by its own weight acting perpendicular to the horizon, and by the pressures of the two adjacent stones, in directions perpendicular to their sides, or to the two adjacent joints : So, for instance, the stone ad is balanced, or kept in equilibrio, by its own weight, and by two forces acting perpendicularly to ab and cd; and the stone cf, by its weight, and by the two forces perpen- dicular to CD and ef; also the stone eh, by its weight, and b}- the two forces perpendicular to ef and gh ; also the stone GK, by its weight, and by the two forces perpendicular to gh and ik; and so on; all these v.'eightn acting in the ver- tical direction bad. But whenever three forces balance one another, they have then the same ratios as the sides of a triangle drawn per- pendicular to tlieir directions. Therefore, if there be con- structed another figure obd/hk, having b/c horizontal, or j)er- pendicular to a given vertical line ob; and having od parallel to CD, and rf to or, and oh to oh, and ok to ok, &c: then the three forces bakuuing tlie stone ad are proportional to the three sides of tlie triangle obd, these sides being respect- ively perpendicular to those forces; for the same reason, 'the stone cf is liulaiii-ec! by the three forces (If, od, of; also the stone eh by the three ^/A, of, oh; and the stone gk by SECT. 2. OF THE ARCHES. fjl the three hk, oh, ok; and so on; in all these cases the Aveights of the stones being proportional to the bases bd, df^ fh, hky of the triangles obd, odf, qfh, ohk. But as these tri- angles have all the same common altitude oh, they have the same ratios as their bases bd, df^ &c, which bases, it has been shown, are proportional to the weights of the stones, which have also been found proportional to the quadrilateral areas ad, of, &c; therefore the quadrilaterals ad, cf, eh, gk, are respectively proportional to the triangles obd,odf,ofh^ohk. But, as these small triangles have their angles respectively equal to the angles of the corresponding sectors, because their sides are parallel by the construction; that is, the an- gle bod = the angle bod, &c ; their areas arc therefore pro- portional to the squares of their corresponding sides; viz. tiif sectors obd, oac, obd, proportional to ob% oa^, ob^i and the sectors odf, oce, odf, proportional to od^, oc'', od^-, and so on. Therefore, by taking the differences, AD : obd : : ob^ oa^ : ob-, and CF ; odf : : od* oc^ : o^% and EH '. ofh '.'. of^ oe* : o/^, and GK : ohk : : oh* oo* : oh-, &.c. Hence, if oh^ be taken = OB" -- oa*, then od* is = od" oc", and of^ is = of' oe*, and oh~ is = oh" og'-, &c. Or, by transposing, ob* = oa'' + o3% and od^ = cc? + o^% and OF^ rr oe' -j- of-, and OH^ =r OG^ + oh-, &c. Which "-ives us the following geometrical constractiwn, viz, Produce the joints till oa, oc, oe, og, &c, be equal to the several radii of curvatuie at the corresponding points, A, c, E, &.c; to which also ^ViLVt the par.iliels ob, od, of, &c. Then take oh = ^^ o^ ~ oa^, and draw ^c(/i^/c perpendicular 38 THE PRINCIPLES OF BRIDGES. TRACT I. too^. Lastly, make OD = ^/ oc' + 0^/% and of = -v/oe^ + ^ = -y/oA- -{ L/)*, which gives a point H in the curve. And the curve thus constructed gives the very same as the fig. p. 35, formed on the principles of prop. 6, as might be expected. Examples of other curves, besides the circle, might be here taken, but the above case may suffice, as none of them are of a nature to be suitable for, or to hold good, in the construction of arches, at least for the ordinary purpose of bridges. Because, that in such arches, the parts do not en- deavour to sHde down in the oblique direction of the joints, both on account of the roughness or friction there, and because, when the parts are cemented together by the mor- ter, or keyed together by pieces within side, the weights then all act perpendicular to the horizon, being each fixed to the other parts of t!ie arch, after tlie manner supposed in the 9th and 10th propositions; and according to the examples to the latter of these, it will therefore be expedient to make such calculations as may occur in cases of real practice. PROP. VIII. IFhen a curve is kept in equilibria^ in a vertical position^ by loads or weights bearing on every point of it : then the load or vertical pressure on every pointy is directly proportional to tlie product of the curvature at that point, and the square of the secant of the elevation above the horizon of the tangent to the curve at the same point, the radius being I. That is, the load or vertical pressure on any point c, is directly as the cur- vature at c, and as the square of the secant of the angle bcit^ made by the tangent be and the horizontal line en. 40 qfHK PWNCIPtEIS OF BRIDGES. TRACT I- Thisj property will be de- duced as a corollary from the properties in the 2d and 3d propositions, according to the idea mentioned in the conclu- sion of the scholium there, by conceiving the bars or lines kept in equilibrio to become iixlc- finitely small; for, by this means, those bars will form a con- tinued curve line, after the manner of the arch stones in a bridge, constituting an arch of equilibration, by weights pressing vertically on every small or elementary part of the arch. Now the consequence of the above idea, namely, of the bars becoming very small, and forming a continued curve, is, that the angle ^cd becomes the angle of contact of the curve and tangent, and the angles ^ch, dch become equal to each other ; consequently, the vertical load on the point c, which, in the 3d corol. prop. 3, was proportional to the sin. ^CD X sec. ^CH X sec. dch, will be here proportional to the sin. bcD X sec^. ^ch, or as the angle bcD x sec'. ^CH, since a small angle [ben) has the same proportion as its sine. But the angle of contact ^CD, in any curve, is the measure of the curvature there ; therefore, lastly, the vertical load or pres- sure, at any point c, in the curve of equihbration, is propor- tional to the curvature multiplied by the sec*, of ^ch ; that is, proportional to the curvature at that point, and also to the square of the secant of the elevation of the curve or tangent above the horizon. Co7'0l. Because the curvature at any point in a curve, k reciprocally proportional to the radius of curvature at that point; it follows, therefore, that the vertical load or Aveiglit sec'', bcfi on any ]ioint c, is as ' , where r denotes the radius of curvature at the point c ; that is, directly proportional to the square of the secant of elevation, and inversely proper* lional to the radius of curvature to the same poipt. SECT. 2. OF THE ARCHES. 41 PROP. IX. fVhen an upright wally bounded by a curve beneath, is kept in equilibrio by the mutual id eight and pressure of its parts and materials ; then the height of the wall above every point of the curve, is directly proportional to the cube of the secant of eleva- tion of the tangent to the curve there, and also directly propor^ iional to the curvature at the same point, or else, which is the same things inversely proportional to the radius of cunaturc there. By the last proposition, the load or pressure on every ele- mentary or small portion, cc, of the curve, is pro- sec^, bcii portional to ' . Now this load on every such small equal part of the arch, as cc, is a mass of solid matter ciic, incumbent on that part of the curve, and pressing it verti- cally ; and which may be considered as made up of a number of equal heavy lines standing vertically on it ; the number of which lines may be expressed by the breadth ca of the said pillar ci of heavy materials : but the breadth ca is = C6' cc 1 , , , = ; , or as 7 , because the element cc is sec. fca sec.ocH sec.d?CH supposed given, or always of the same length, that is, ca is reciprocally as the secant of the angle of elevation. Hence CI sec . bcu then the vertical load, or ci, or is as con- sec. bciV r sequently the altitude ci of the wall aklm, at the point c, is SeC^. ^CH , , 1 rr^l as , or as sec\ och x curvature there. That is, r the height of the Avail above every part of the arch of equili- bration, is directly proportional to the cube of the secant of the curve's elevation at that part, also directly proportional 42 THE PRINCIPLES OF BRIDGES. TRACT I, to the degree of curvature there, or else inversely as the ra- dius of curvature at the same part. Corollary 1 . Hence, if the form of the arch, or the nature of the inner curve abcdm, be given; then the form or nature of the outer line kil, bounding the top of the wall, or form- ing vviiat is therefore called the extrados, may be found, so as that the intrados abcdm shall be an arch of equilibration, or be in equilibrio in all its parts, by the weight or pressure of the .superincumbent wall. Fur, since the arch or nature of the curve is given, by the supposition, the radius of cur- vature and position of the tangent, at every point of it, will be given, and thence also the proportions of the verticals ci, &c. So that, by assuming one of them, as the miilJlc one VD for instance, or making it equal to an assigned length, the rest of the verticals will be found from it, and will be in pro- portion as it is greater or less ; and then the extrados line KIVL may be drawn through all their extremities. Or, on the other hand, if the extrados kivl. or line bound- ing the top of the wall, be given ; then tiic nature of the correspondent curve of equilibration abcdm may be found out. And the manner of the practical derivation ot both these curves, mutually the one from the other, will be shown in the following propositions. Corollary 2. If the intrados curve abcd should be a circle; then the radius of curvature will be a constant quantity, and equal to the semidiameter of that circle; also the angle ben will be always measured by the arc dc, from the vertex d of the curve; and then the height ci of the wall, will be ever\ where proportional to the cube of the secant of the arch dc. Corollary 3. Hence also it fol- lows, that if between tlie intrados and extrados curves, an interme- diate curve kivl, be drawn through the middle of tiie wall, bisecting all the verticals dv, ci, &c, or indeed SECT. 2. OF THE ARCHES. 4^3 dividing them in any ratio whatever, so as that it may be everywhere dv : Dt; ; : ci : ci; then if acdm bean arch of equiHbration to the wall akvlm, it will be an arch of equili- bration to the inner wall a/cvIm also. PROP. X. Having given the Intrados or Soffit, of a Balanced Arch; to find the Extrados. That is, having given the nature or form (fan arch ; from thence to find the nature of the line forming the top of the seperincumbent wallj hy the pressure of which the arch is kept in equilibrio. The solution of this problem is to be made out generally from the last proposition and its corollaries, by adopting ge- neral values of the lines there employed, which belong to all curves whatever : or otherwise by making use of the peculiar values proper to any individual curve, for the solution of particular cases. For the general solution, in fig. pa. 41, kvl represents the extrados, the form of which is required, and abcdm the given intrados or soffit of the arch, tlie vertex of which is d, and DV the height or thickness of the wall there, which is com- monly a dimension that is known from the particular circum- stances of the case. Now if we make the arch dc = z, its element cc = x, the absciss dh = x, its element ca = x, the ordinate CH y^ its element ca = j, the height or thickness of wall at the vertex dv = a, and the radius of curvature at any point c ~ r, that at the vertex d being = r. Then, because the height ci, at any point c, is as sec^ ^CH or of cca , . i ^ . . , , ,, , by the last proposition, and because the cc X secant of fca is = = , the radius being 1, therefore CI is as rr, or as r, , because x= cc= x/ca- + Crt' = rj^ O 44 THE PRINCIPLES OF BBIDGBS. TRACT I. z' a {x-+f)^ a Or, the ceneral viilue of ci is -rr x = :t X - > to yi y yi y where q denotes a certain given or constant quantity, the value of which may be determined by making the general expression equal to a or dv, the height at the crown of the arch. Corollary 1. Because, at the vertex of the curve d, the angle of elevation is nothing- or its secant zz = 1 the radius, and the radius of the curvature there being r; there- fore the general expression for the height, becomes there DV = a = ; consequently a = ctr, which is the general value of a for all curves whatever, expressed in terms of the height a at the crown, and n the radius of curvature at the same point. Hence then, substituting this value of q in- stead of it, the general expression or value of ci becomes -7T X = r: X . ji r y r Carol. 2. Because, in all curves that are referred to an axis, the general value of the radius of curvature r, is = TT. rr. ; therefore, by substituting this value for r in the last expression, the general value of the height ci then be- comes r; X <;(R = ~ x Q, or = rr- X a w hen x is yi yi yi constant. For, as either x or 3^ may be supposed to flow uniformlj-, and wlien, consequently, either of their second fluxions may- be taken equal to nothing, which will cause one of tlie terms in the numerator of the above value of ci to vanish ; there- fore, by striking out either of those terms, and then extermi- nating either of the unknown quantities by means of the equation to the curve, the particular value of the height i SECT. 2. OF THE ARCHES. will be obtained: as is done in the following examples, ex- cept in some certain cases, where more peculiar methods may suit them better, as when the radius of curvature is a known quantity, &c. EXAMPLE 1. To find the Extrados of a Circular Arch, That is, ACDM being a cir- cular arc, of which aq or QD is the semidiameter, a the centre, and d the vertex of the given circular arch ; also k the vertex of the extrados kig, and the other lines as in the figure. Making a = dk, r xq. -^ ao the radius of the circle, which is also equal to the radius of curvature throughout, or r = r ; also a: =. dp, and y zz PC :=. Ri, and s = the arch dc. Then, because = =1, r r ^ and T7 := the cube of the secant of elevation at c, which is ca^ or Da^ ; therefore the general value of ci, incorol. 1, r pa^ becomes ci DK -' Da^ r or, as pa^ Da* or ca' : : DK : CI. X a = V r^ y Carol. 1. This expression for the value of ci, affords a very simple mode ot calculation for the case of a circular arch ; viz, to the constant logarithm of the height a, add triple the logarithm secant of the elevation, or of the arc dc ; then the natural number answering to the sum will be th<^ value of the height ci, at every point c. 46 THE PRINCIPLES OF BRIDGL^. TRACT T. Carol. 2. It gives also a very simple construction by scale and compasses, which is as follows : Join ac ; draw pf per- pendicular to QC, and fg perpendicular to qp ; then shall Q^ : QC : ; gp^ : qc^ ; because, by similar triungles, ag : qf : : o./ : qp and : : qp : qc, or q^, qf, qp, qc arc four terms in continued proportion, in which case the first q^ is to the fourth qc, as qp^ to qc% the cube of tlie third to the cube of the fourth. Hence, if ci be taken a fourth projjor- tional to Q^, QC, dk, it will be the length of the vertical line sought. And this fourth proportional will be easily deter- mined in the following manner: viz, Join eg, and in the vertical line ic downward take ch = dk, and draw hi pa- rallel to eg, so shall CI be equal to ci the fourth proportional to Qg, QC, DK, or to QP% Qc% DK, as required. Carol. 3. The extrados line in this figure is accurately drawn according to the above construction and calculation, when the thickness dk at the cro^vn is the exact 15th part of the span am. It falls more and more below the horizontal line, from the crown all the way till the arch be between 30 and 40 degrees, where it takes a contrary fiexm-e, tending upwards, passing the point i very obliquely, and thence rising very rapidly to an unhmited height, in an inhnite cuive, to which the vertical line AG is an asymptote ; a circumstance wiiich must always be the case with every curve, which, like AC, springs perpendicularly from the horizontal line aqm. This curve cuts the horizontal line nearly over the point of 50 degrees. If dk were taken greater than the 15th part of AM, all the other vertical lines ci would be greater in the same proportion, and the curve kig would cut the horizontal line drawn through k in some point still nearer to k ; but the reverse, or farther off, if dk were taken less than the 15th part. Hence it appears, that a circular arch cannot be put in equilibrio by budding on it up to a horizontal line, what- ever its span may be, or whatever be the thickness at tlie crown. And consequently it may generally be inferred, that SECT. 2. OF THE ARCHES. 47 the circle is not a curve well suited to the purposes of a bridge which requires an outline quite horizontal, but may answer tolerably well when that line bends a little down- wards, from the crown toward the extremities ; and then a great variety of proportions between the thickness at the crown and the span of the arch might be assigned, which would put the circular arch in equilibrio, nearly. Now these cases will happen in general when kr vanishes, or is of no length, and then ci must be equal to pk, or nearly so ; with which genei-al condition many particular cases may be found to agree nearly. But it may be proper here first to make out a general rule for such cases, which may be done in the followinsr manner : By the premises, the general value of CI being dk x sec^. DC, or as 1 : sec^ do : : dk : CI ; then, by taking ci = pk, in order to cause the outer curve Ki to cross the horizon- tal line KI at the point i, that proportion becomes 1 : sec^ DC : ; DK : PK or dk -j- dp, 1 i 1 L^ K im ! 1 j'^ P^ i, ' IP ^ P C /vi 1' a u or sec' DC 1 : 1 : : dp : dk = dp sec^ DC 1 , the radius being 1, Now, by taking the arch dc of various magnitudes, from DA or 90, to o or nothing at d, the several thicknesses dk, at the crown, will be found by this theorem, corresponding to the several heights dp, or span cc, as here following, so as to make cdc a balanced arch very nearly. Thus, 1st. If DC be taken = da or 90 : then its height is dq = r, its span am = 2/', and its secant is infinite; consequently = o. That is, the thickness at the crown comes DK = infin. out equal to nothing in this extreme case. 2d. If DC be taken = 75 : then its height dp = '741 18r, the span cc = r93l85?', and the sec. dc =: 3'8637 ; there- 48 THE PRINCIPLES OF BRIDGES. TRACT 1, DP CC fore DK = r -OlSOSr = . That is, the thick- sec^ 1 148 tiess at the crown would be the 148th part of the span, being also much too small for common practice. 3d. If DC be taken = 60" : then its hciglit dp = ^r, the DP span CC = T^Z, the sec. do = 2 ; therefore dk = ~ CO CC CO , ^, = iDP = -r^*' = = = r nearly. That ^ ^ 14v^3 24-2487 24^ ^ is, the thickness at the crown would be rather less than the 24th part of the span: which is still too small in ordinary bridges. 4. If DC be taken = 54 : then its height dp = 4122r, the span cc = reiSr, and the sec. dc = 1'7013; therefore DP CC DK = ; = 10504r = ' . That is, the thickness sec3.-l *^ 15*41 at the crown would be between the 15th and 16th part of the span ; which is nearly the proportion allowed in common bridges. 5. If DC be taken = 45 : then its height dp = r ^Vs/I^ the span cc = rv'2, the sec. do = v'2 ; therefore dk = DP 1 4-\/2 r cc cc r sec^-1 2x/2-l 2 + 3-^,72 G-{2^2 8-8284 ice nearly. That is, the thickness at the crown would be more than the 9th part of the span : which in common cases is too much. 6. If DC be taken = 30 : then its height dp = r \i'\/'l. n the span cc = r, the sec. do = ^7 ; therefore dk = DP r-^r^/S 6v/3 - 9 6^/3-9 cc -r = cc - s&c^. \ 8 16-6v/3 16-6v/;j 4-03 3^3 ~ ^ |cc nearly. That is, the thickness at the crown would then be almost the 4th part of the span. 1. If DC be taken = 13": then it? hcitrht DP :r- -OSiOTrj SECT. 2. OF THE A&CHES. 49 the span cc = -sner, and the sec. dc = 1*0353 ; therefore DP _ DP .3 DK sec^:^ = rpf = 9DP = '^r =; fee nearly, or * of the span. From all which it appears, that a whole arch cdc of about 108 or 110 degrees, is the part of the circle which may be used for most bridges with the least impropriety, the thick- ness at the crown being nearly the 16th part of the span, with a horizontal straight line at top. EXAMPLE 2. To determine the Extrados of an Elliptical Arch of Equi- libraiion. Suppose the curve in this figure to be a semiellipse, with either the longer or shorter axe horizontal : put- ting h to denote the horizon- tal semiaxe aq, and r the vertical one dq, also x = dp, y =. vCf and a = dk, as usual. Then, by the nature of the ellipse, r : h h , , . hx : y ; therefore J/ = v2rx xx, and j = x ^ p Q, \/ 'Irx XX r x also>' =: -h y/ (2rar - xx) r ^{"Zr xxY by making x constant. Hence the general value of ci, viz, r^ x a, becomes {2rxxx)T: irx-Q. {2rx xx}-^ But at the vertex of the TO. h'i' ^ {r-xY h'{r-xf curve D, Avhere x is :=: o, this expression becomes only , , which must be = dk or a ; therefore the value of q is =: , which being substituted for it in the above general value VOL. 1. 50 THE PRINCIPLES OF. BRIDGES. TRACT r. DK X Da* whicli is the pa"* of CI, this becomes ci = r- z=. very same expression as the value of ci in the case of the circle in the former example, and which belongs equally to the ellipse in both positions, that is, both with the longer axe vertical, and with the shorter one vertical, as it is in the figure to this example. Hence it appears, that the flat ellipse is more nearly ba- lanced by a straight horizontal back or wall at top, than the circle is; but the circle more nearly than the sharp ellipse: the want of balance being least in the flat ellipse, but most in the sharp one, and in the circle a medium between the two. EXAMPLE 3. To determine the Extrados of a Cycloidal Arch of Equili- bration. Let Dza be the circle from which the cycloid ACD , is generated ; and the other lines as before. I Put a = DK, X = DP, and J/ = cp = IR, as usual; al^o put r = dq the diameter of the circle, and z =. the circular arc dz. Then, by the nature of the cycloid, cz is always equal to dz =r z ; and, by the nature of the circle, pz is = ^^ rx xx ; tlicrefore pc or j/ ( = cz-f pz) is = 2; + ^/7-x-xx. Hence j- ir-x . . .y'i by the 11a- = z 4- ^/{rx-xx^ X X ; but z is A^/ (rx xx) rx ture of the circle; therefore j is = -- ~ - x x x V (;'.r ^-.r) r - X , .. - ri"^ . ^' ; then J .-, making a- constant. Hence X CI is ^.fv'' (r.r xx) xy and x = ^UTy ^Y making j constant, _,, M 2ra , Then ci = 77 X ais = -7y- = a constant quantity = a; that is, CI is every where equal to kd. Consequently kr is = dp ; and since ri is = pc, it is evident that ki is the same parabohc curve with dc, and may be placed any height above it, always producing an arch of equilibration. B 3 52 THE PRINCIPLES OF BRIDGES. TRACT I. EXAMPLE 5, To find the figure of the Extrados for an Hj/perbolic Arch of Equilibration. Here putting, as before, a =: KD, r = the semi-transverse, and h = the horizontal or semi-con- jugate axe, also jr =: dp, and j/ = PC = Ri. Then, by the na- ture of the hyperbola, J/ = ~^'2rx + xx ; hence > = -^ r -\- X . . . . hrx^ \/{2rx-\- xx) Therefore ci or - - , and, by making x constant, jp = 7''Q X Q IS =- y h"- X (r -\-xy vertex d, where x = o, this expression becomes r>.r,r'^ 1.3.5.7^^ o ^ , 1- 4. &C) : hence, uivulii!;r )y y d X \.",x^ 1.0.,i.r' r, we ijuve = ^/ - X ( I -7.-7 + - /, - TTa'T^^Ii '^~ SECT. 2, OF THE ARCHES. 55 1.3.5.7^* \ , ,- V d 2.4.6.8. 9/^ ^^ ' ^' ^' writing v for J, and w for V-, it 1 , 1.3 1.3.5 , 1.3.5.7 '2:iw ^ 2A.5w^ 2A.6.1w= 2.4.6.8. 9:^;' ^' Then, by reverting this series, we have w znv + ^ ^ ^ ' 6y 360x;* 547 337 ^ 5040^" 5'60(hr7 ^''- K^"^^' ^y squaring, &c, and re- storing the original letters, it is {\d =. ^xw"- z=.) c = \x x ,y\ 1 8^* 69 Lr* 2385 iT^ o \ ^^ + J - 45^ + 3780^- T^^Em/^ ^')' ''^''''' " ^^^" ^ the first terms are sufficient to determine the value of c pretty nearly. Now, for an example in numbers, suppose the height of the arch to be 40 feet, and its span 100, which are nearly the -dimensions of the middle arch of Blackfriars Brid 47 39-126 12 7 330! 21 lO-.'iSj 30 15-98() 39 25-505 48 41-293 13 7-571; 22 I()-85S 31 16-811 40 26-894 49 43-581 14 7-834| 23 11-368 32 17-693 41 28-364 50 46-000 The above numbers may either be feet, or any other lengths, of whieli dq is 40 and qa is 50. But when dq is to QA in any otiier proportion than that of 4 to 5, or when dk is not to DQ as 6 to 40 or 3 to 20; then the above numbers will not answer; but others must be found by the same rule, to construct the curve by. In the beginning of the table, as far as 12, the value of Kf is made to difier by 2, because the value of CI in that part increases so very slowly. After- wards they diiTer by units or 1. Other examples of given cxtrados might be taken; but as there can scarcely ever be any real occasion for them, and as the trouble of calculation would be, in most cases, very great, they are omitted. As the theory for arch vaults, before laid down, will so easily apply to the arches for domes or cupolas also, a prcr- position or two may be here added for that purpose, as follows. PROP. XII. IVhen a regular Concave Surface Do)ne, or Vaults formed by the rotation of a curve turned about its axis^ is kept in equi- libria by the pressure of a solid limit built on every part of it ; theyi the Height of the ~u:all over any part ^is directly proportional to the cube of the secant of elevation there, and inversely pro- portional to the radius of curvature^ and to the diameter or width of the dome at the same part. SECT. 2. OF THE ARCHES. 63 That is, VI being the form of the exterior surface of a balanced shell, the interior surface of which is formed by the rotation of the curve DCA about its axis dh ; the eleva- tion of any part c being the angle ^H, and CH tlie ordinate or semi- diameter of the dome at the point c, also r the radius of curvature to the same point: then the height or vertical thickness of the shell over the point c, or ci, is proportional sec^ ^CH to r X CH Let ACDCB be a small part of the inner surfaccj like a curved sector or gore, dca and DeB being two near positions of the generating curve. Now the vertical load on any part c of a balanced arch, in a shell or dome, in the present case, is a solid pillar, ci, whose height is ci, its breadth ca, and thickness ce, and consequentiy is = ci x ca x oe. But CH 1 ca is as j or as ; and ce is alv.ays iji the same CO sec. ben proportion as ch ; therefore the pillar c/, or ci x c x c^ CI X CH is as ; M'hich load, by the 8th prop, is also propor- sec. bcu. tional to bcii therefore CI X CH sec. ben sec^ ben is as- sec', ben -;coniie- That is, the vertical quently the heig-ht ci is as ^ -^ ^ r X cH height of the vv^all over every part of a balanced shell, or dome, or vault, is directly as the cube of the secant of the curve's elevation at that part, and inversely as the radius of curvature, and also inversely as the width of the dome at the same place. And here may be also understood several corollaries and observations exactly similar to those to the Sd, and the 9th propositions, and which therefore need not be repeated in this place. 64 THE PRINCIPLES OP BRIDGES. TRACT I. PROP. XIII. ' Having given the form of the Inner Surface of a balanced Shell or Dome; to determine that of the Exterior or Outer Surface. That is, having given the nature or form of an inner shell; thence to find the nature of the outer or bounding turf ace of the superincumbent wall, by the pressure of which, the shell is kept in equilibria . By reasoning here exactly as in the lOtli proposition, it will be found that the ge- neral value of the height CI of the wall, will be pro- portional to the following forms or quantities, viz, sec\ bcH CI is either as or as as yx xjf ryr -, or as r X CH ry f or as ryy^ or yy^ when X is considered as invariable, or as yy when> is invariable: in which the letters have the usual values, namely, x = dh the absciss, j/ rz ch the ordinate, and 2 = DC the curve, also r the radius of curva- ture at the point c. Or the general value of ci will be equal to any of these forms midtiplied by a certain constant quan- tity and 15 + which were above calculated to be 40-6 and 15-3. Secondly, to apply our example to the 5ld theorem, ^ef. fg* IQ_, GP REH IK the same methods of de- termining the posi- tion of the centre of gravity i may be em- ployed. If the me- chanical method of balancing and measurement on a scale be used, we may then measure, not only the lines ik, ok,kc, but all the other Unes also depending on it, as oi, or, ti, tr, te, KQ, la, &c, excepting only such lines as depend on the un- known breadth fg of the pier. But, instead of that, we shall calculate the accurate value of all the lines wanted by strict mathematical principles, as follows. In the example are given OB = OC = DE = 45, OD = CE =r. 52, CF = 20, EF = 72 ; and just above we have found by calculation ok = 29*7, kc = 15*3, IK = 40*6, and the area bdec or a = 750 ; and we bave to compute iq and gp. Now oi = v'(ok^ + ik') = ^(29-7^ 4- 40-6') = 50-3;. then by similar triangles oi : OK : ; IK : IQ = 23-97. Again, to get an expression for gp, put the required thickness of the pier kh or fg = ~ ; then, because by similar triangles, ik : ok : : od : dr = 3804, and IK : lo : : on : or 2= 64-42, hence or 01 = 14*12 = ir, and OK : 01 : : IR : TR = 23-91 , also de OR = C-96 = re, hence TR 4- RE =r 30'S7 = TE, and th = te + eh =TE-f ;r, then IK : OK : : th : hv ~ 1^2-58 -)- 0-7:; 15z, and GH iiv = Gv = 49-42 -73152, lastly 01 : IK : : GV : GP = 39-89 - -5904^. These valuer licinsi now substituted in the 2d theorem Irr . SECT. 3. OF THE PIERS. 81 TG^ la. GP IK . a, give 36z* = 17664-9 -26 1-50, or 2* +7*263 = 490*69 ; the root of which quadratic equation gives = 18'82 = EH or PG, the thickness of the pier sought. It may be presumed that this theorem brings out the thick- ness of the piers very near the truth, and very near what would be allowed in practice by the best practical engineers, as may be gathered from a comparison of the two cases of AVestminster and Blackfriars bi-idges, in the former of which the centre arch is a semicircle of 76 feet span, and 17 feet thickness of piers, and in the latter it is a semiellipse, of 100 feet span, 40 feet in height, and 19 feet thickness of piers. EXAMPLE 2. Suppose the span to be 100 feet, the height 40 feet, the thickness at top 6 feet, and the height of the pier to the springer 20 feet, as before. Here the figure either is, or may be considered as, a scheme arch, or the segment of a circle, in which the versed sine ob is = 40, and the right sine oa or oc = 50 ; also db = 6, cf z: 20, and ef = 66. Now, by the nature of the circle, whose centre is w, the ra- dius WB or wc = OB^ + oc^ 40^ + 50* 20B 80 = 51^; hence ow = 51^-40 = 11^; and the area of the semiseo^ment OBC is found to be 1491 ; which being taken from the rectangle odec = od x oc = SO X 46 = 2300, there remains 809 = a, the area of the space BDECB. Hence, by the method of balancing this space, and measuring the lines, there will be found, kc = 18, ikl = 34-6, IX 42, KX = 24, OX = 8, IQ = 19-4, TE = 35*6, and TH = 35*6 + Z^ putting z = eh, the breadth of the pier, VOL. I. G S2 THE PRINCIPLES OF BRIDGES. TRACT I. as before. Then iit : kx : : th : hv = 24*7 + 0*7z; hence gh hv = 41-3 0-T~ r: GV, and ix : IK : : GV : GP = 34'02 O'oSs. Tliese values beiiii'- now substituted , , , iQ . GP. a . in the theorem ^ef . fg = , uue 33;''- = 15431 '47 IK } b' - 263*092, or z" + Sz = 467*62, the root of which quadra- tic equation gives 2 r: 18 eh or FG the breadth of the pier, and which it may be presumed is sufficiently near the truth. These two cases it may be expected are sufficient to ex- emplify this ntethod of determining the proper dimension of the piers ; a method, the propriety of which is thus confirmed by conclusions that arc so confonnable to the practice of the best engineers. In all cases it apjjears to be the easiest course, and sufficiently correct, to construct accurately the semiarch and superstructure above it ; then find its centre of gravity by the method of balancing it in two positions per- pendicular to each other, viz. in lines parallel and perpendi- cuhir to the base ac ; next through that centre i draw a line iw perpendicular to the curve of the arch, or in the direction of the arch joints there, and meeting the base line in the point x; next, through i draw Tvp perpendicular to ix, and ik perpendicular to ac, and kq })erpendicu]ar to tp. Then measure by the scale as many of these lines as are necessary in the inlended calculation, and as are used in workins: the 2d example above, viz, the lines ik, kx, te, iq, and compute the area bdec a, v.hich may be sufficiently done in a me- chanical manner, and to an apj)roximate degree, whatever niay be the figure (j1 the curve, and shape of that area. After this, continue to complete the rest of the calculation as in tli? example above. { 83 ) .^ SECTION IV. THE FORCE AND FALL OF THE WATER, &C. PROP. XV. To determine the Form of the Ends of a Pier, so as to make the Least Resistance^ or be the Least subject, to the Force of the Streani of Water. Let the following figure represent a horizonta.1 section of the pier, ab its breadth, cd the given length or projection of the end, and adb the line required, wiiether right or curved; also let ef represent the force of a particle of water acting on ad at the point f, in the direction parallel to the axis CD : produce ef to meet ab in G, and draw the tangent FH ; also draw eh perpendicular to fh, hi perpendicular to EF, and FK perpendicular to DC. Now the absolute force ef of the particle of water may be resolved into the two forces eh, hf, and in those directions; of these, the latter one, acting parallel to the face at f, is of no effect; and the former eh is resolved into the two ei, ih; so that EI is the only efficacious force of the particle to move the pier in the direction of its axis or length : That is, the absolute force is to the efficacious force, as ef is to ei. Then, since ef is the diameter of a semicircle passing through h, by the nature of the circle it will be, as ef : ei : : ef" : eh"' : : (by similar triangles) hf' : nf and : ; the square of the fluxion G 2 84 THE PRINCIPLES OP BRIDGES. TRACT I; of the curve or line : tlie square of the fluxion of the ordinate FK, because hf, hi are parallel to the line and ordinate. Therefore, putting the abscissa dk = x, the ordinate kf = J/, and the line df = z, it uill be, as i* :_y' : : 1 (the force ef) : 77 = the force of the particle at p to move the pier in the direction efg. But the nunnber of particles striking against the indefuiitely small part of the line, is as j ; this drawn into tlie above lound force of each, we have = t; r, ' ^ ** + J?* for the fluxion of the force, or the force acting against the small part z of the line. But, by the proposition, the uhole force on dfa must be a minimum, or the fluent of-, r, must be a minimum, when tliat of X becomes equal to the constant quantity dc; in which case it is known that ~~ rrr; must be always equal to some constant quantity q ; and hence xj^ = q x {x'' +i^)*- Now, in this equation, it is evident that * is to j in a con- stant ratio ; but when two fluxions are always in a constant ratio, their fluents .r, j/, it is known, are also in a constant ratio, which is the property of a right line. Therefore df^ is a right line, and the end adb of the pier must be a right- lined triangle, that the force of the water upon it may be the least po'=;siblc. PROP. XVI. To determine the 2uantitij of the liesistanee of the End of a. Pier against the Stream of Water. Using here the same figure and notation as in the last proposition, by the same it is found, that tha fluxion of th force of the stream against the face dt, s .. -; and since the fluxion of the force against the base is>, it follows, that SECT. 4. THE FORCE AND FALL OF THE WAITER, &C. [^5 the force of the stream against the base ab, is to the force against the face adb, as ( y) the fluent of 7, is to the fluent of -; . .; . That is, the absolute force of the stream, is to the efficacious force against the face of tlie pier, as its breadth is to double the fluent of rr, when y is equal to half the breadth. Corollary 1 . If the face adb be rectilineal. Putting DC = a, AC = b, and ad = ^/{aa + bb) = c ; then, as tf : ^ : : ^ : 3/ by similar triangles ; hence x ~ ay oy . j-y and ;e = -7- ; this being written for it in the general , . , bh'y bby ^ , o expression above, it becomes 7-^r = , for the fluxion ^ ' aa-\- bb cc of the force on ad ; the fluent of which, or ^, is the force ce itself. Consequently the force on the flat base ab, is to that on the triangular end adb, as j/to ^, or as cc to bb, that is, as ad' to Ac'. And if AC be equal to cd, or adb a right angle, which is generally the case, then ad* = 2ac% and the force on the base will be to that on the face, as 2 to 1. Moreover, as the force on adb, when adb is a right angle, is only half of the absolute force, so it is evident that the force will be more than one-half when adb is greater than a right angle, and less when it is less ; and also, that the longer ad is, the less the force is, it being always inversely as the square of ad. Corollary 2. If adb be a semicircle. The radius ac = cd = a ; then 2ax xx = yy^ or x = yy y^ f. -- v/ (aa -~ yi/)} acd #= 7-" -^ ; hence irr-7-r, becomes S6 THE PRINCIPLES OF BRIDGES. TRACT X. ..dz. X j, the fluent of which is ^^ x y ; and there- a ' ' aa fore the force on the base is to the force on the circular end, aaUjy as J/ IS to -' X 1/, or as aa to aa }j/j/, or as :iaa to 3aa yy. And when ?/ = a = ac, the piojiortion becomes that of 3 to 2. So that, only one-third of the absolute force is taken off by making the end a seniicirclc. Corollary 3. When the face adb is a parabola. Then, the notation being as before, viz, DC = a, and ac ... ,, , V// , . 'laijy = 0. It IS a '. X : : bb : iixi ; hence x ,-> and x = ,, - ' "' ' bb ^ bb ^ which being written in the general expresMOii, the iiuent of it becomes the circular arc whose radius is and tanorcnt ?/, bb . , , , -^'''/ or = - - X arc whose raduis is 1 and taniient - , , ; sotnat '2a '^ bo the absolute force is to the force on tlie paraboUc end, as jj is to the arc whose tangent is j/ and radius ; that is, as the tangent of an arc is to the arc itself, the r^.dius being to the 2ay ay , , , , , tangent, as 1 to -7'^, or as 2 to . And whenj/ /;, the ra- tio of the tan<>eut to radius, is that of 2 to ; or that of 2 a to 1 when dc ca. In which ca'-e, the whole force is to the force on the parabolic en(i, a- the tangent, which is double the radius, is to the corresi oneiing arc ; tnat is, as the tangent of 6S 2b' 4" to the arc of the sunie, or as 2 to r 107 14 ; whicii is a less force than on the circle, but grcwirr ihan ;.n the triangle. And so on for otii; r cur\-e.>; lu w iiich it will be found, that the near.-r they appf , tiie less t -e f .rce wili be, and th.. it > .'.'a^t ot'.ii.i ni tiie tn.mg e, in which it is oue-half oi tile wLo'c aosoiute iorci* when r,g;it- anirlud. SECT. 4. THE FORCE AND FALL OFTHEWATEH, kc. 87 It must be noted, however, that in determining the best form of the end of the pier to be a right-lined triangle, the water is supposed to str'ke every part of it witii thv- s. me velocity: had the variably increased velocity been used, the. form of the ends would come out a little curved ; but as the increase of the velocity in the best bridges is very smai), the difference in them is quite imperceptible. PROP. XVII. To determine the Fall of the Water in the Arches, Having, in the foregoing propositions, treated of the re- sistance made by the piers to the current of water, it will now be proper to contemplate the effects of that resistance, ar.d of the contraction of the passage they produce in the water- ^vay. These effects are, a fall, or sudden steep descent, and an increase of velocitv in the stream of water, just under th.e arches, more or less in proportion to the quantitv of tiie ob- struction ; being somewhat observable at the p]ace of all bridges, even where the arches are very large and the piers small, but in a high and extraordinary degree at London bridge, and some others, Avhere the piers, and the st^i'lings, are so very large, in proportion to the arches. Now, in an open canal or river, an equal quantity of water passing in every part, in the same time, if in any part the passage be narrower, there, the bottom continuing the same, the velo- city of the stream must be so much the greater, and a corre- spondent rise in the surface must also take place, to produce that increased celerity. Similar effects also occur in a river when any obstacles, as the piers of a bridge, are placed in the way of a stream. This is resisted and obstructed by the piers ; of courtxi the water rises against them, and conse- quently the stream from thence descends the more rapidly. And this is the case, not only in such canals or rivers Avhere the stream runs always the same way, but in tide rivers also, both upward and downward. During the time of dood, when 88 THE PRINCIPLES OF BRIDGES. TRACT I. the tide is flowing upward, the rise of the water is against the under side of the piers ; but the difference between the two sides gradually diminishes as the tide tiows less rapidly towards the conclusion of the flood. When this has attained its full height, and there is no longer any current, but a still- ness prevails in the water for a short time, the surface assumes an equal level, both above and below bridge. But, as soon as the tide begins to ebb again, the resistance of the piers against the stream, and the contraction of the water-way, cause a rise of the surface above and under the arches, with a fall and a more rapid descent in the contracted stream just below. The quantity of this rise, and of the consequent ve- locity below, keep both gradually increasing, as the tide con- tinues ebbing, till at quite low water, when the stream or natural current being the quickest, the fall below the arches is the greatest. And it is the quantity of this fall whicli it is the object of this problem to determine. Now, the motion of free running water is the consequence of, and produced by tiie force of gravity, as well as that of any other falling body. Mence the height due to the velo- city, that is, the height to be freely tallen by any body to acquire the observed velocity of the natural stream, in the river a little above the bridge, becomes known. From the same velocity also will be found that of the increased stream in the narrowed way of the arches, by taking it in the reci- procal proportion of the breadth of the river above, to the contracted way in the arches; viz. bysa3-ing, as the latter is to the former, so is the first velocity, or slower motion, to the quicker. Next, from this last velocity, will be found the height due to it as before, that is, tiie height to be freely fallen tlnough by gravity, to produce it. Then the differ- ence of tiiese two heights, thus freely fallen by gravit}', to produce tlie two velocities, is the recjuired quantity ol the water-fall in the arches; allowing however, in the calcula- tion, lor the contraction of the stream, in the narrowed pas- sage, at the rate as observed bv Sir I. Newton. Such then are tue elcoiCDts and principles on which the solution of the SECT. 4. THE FORCE AND FALL OF THE WATER, kc. 89 problem is to be made out ; and which it is now easy for any one to perform. But, as it may be desirable to exhibit tlic manner of the solution of this curious problem, by some former noted au- thors, in this instance I shall give the solution from some ma- nuscripts that have now been many years in my possession: viz, one solution by the celebrated Wm. Jpnes, Esq. the friend of Sir I. Newton, and father of the late Sir Wm. Jones ; which is in Mr. Jones's own hand writing, and which I had from the late Mr. John Robertson, many years clerk and li- brarian to the Royal Society, who had the paper from Mr. Jones himself. Another solution is by the same Mr. Robert- son himself, from a paper found among a great number of other manuscripts which I purchased at the sale of his books, after his death in the year 1776 ; and among which papers there are also other solutions that have never been published. The solutions here inserted, are given in the same words and pe- culiar manner as in those authors, in order to show their dif- ferent forms and modes of stating and working. And first the solution by Mr. Jones, done in his usual manner, which was always remarkably concise, neat, and accurate. The Solution of Wm. Jones, Esg. " Lemma. In a chanel, whose stream runs with such an uniform velocity, in any given time, as is acquired by falling from a certain bight {h) ; if an obstacle should contract the passage of the water, in any place, the water above the ob- stacle will rise to such a hight (h) as to acquire a velocity that will discharge the stream as it comes ; but will occasion a fall at the obstacle: and the difference (h //} between these bights, is the measure of that fall. " In a chanel of running water, whose breadth [b feet), and the velocity of its stream (,: feet in l), being given: To determine the quantity of the fall, occasioned by an obstacle that takes up p feet of the breadth of the chanel. '^ Let the hight fallen (near the surface of the earth) in l" 90 THE PRINCIPLES OF BRIDGES. TtlACT I. of time, be {a i'cet) ; and the contractioa of streams, in the water-way, be as r to 1. Put c = 7 : d D'cc: Then ^ o~-p the quantity of the fall is ci 1 x vv x -- feet. h p ** For, the watcr-"svay takes up w (7) part of the breadth of the chanel. But streams are found to be contracted in tlie water-way, in the proportion of r to 1. Therefore the water-way contracted will be ( - = ) (= vi). But the current above the obstacle moves v feet in \" of time; and the velocities of water through difierent passages, of the same Light, arc as the reciprocals of the breadth of those passages. Therefore the current, in the true water-way, must move V 1 ( = t,' =r ) nv feet in l" of time. ^ m Vi " Now, since {a) feet is the bight fallen in 1" of time to acquire a velocity to move uniformlv the length 2a in that time : Let x and z feet be the bights fallen to ac([uire a ve- jocity to move uniformly the lengrhs v and ?iv fe(,'t in l" of time: and because bights fallen are ns the squares of their velocities: therefore = , and -^^ = - -. cousctiuently 'iv X nin-j 'z ^ -^ t'v - nniv rv . , X , anct z =3; . That is, leet is tne linnt of 'hi 'la ' 'ki ^ water necessary to produce, in the elianel, a current that n)Qves r feet in l" of time. And '- feet, is the bight of water necessary to produce, in th.e wuter-wav, a current that mr.vcs ?2:.' in tliat time, '^i'hen the diiference z^// -- 1 x - 'id of tiicse bights, is the fall in feet. But ;? ( r^ ) rr, tliereforc nn ~ rrcc d per supposition. Thert;fore dl .-' - X y ; feet, is tlie quantity of tlie fall. a. i". d. SECT. 4. THE FORCE AND FALL OF THEAVATER, kc. 91 ^' Hence, putting a =: l. , b = lt, c = l.c, d =: 2 x E -t- c = L.(^: Then ud I -\- 2l . v + a = Log. of the quantity of the fall, in feet*. ** Now, if the length of a pendum vibrating seconds, is 39*126 inches, then will a = 160899 feet; and, according to Ne'.vton, r = I4-" consequently A = 2.1913861 ; and b = 0.0757207." Such is the solution of this problem as given by Mr. Jones. And as there is contained in the same paper with this, a short solution of another kindred problem, it is here inserted, as follows. " The length, p inches, of a pendulum that performs one vibration in l" of time, at a given place, being known ; the altitude (a) fallen from, in 1" of time, will be ^/'Tftf inches, or -^^p^Ti feet, at that place. T- ^'"^^ ^^f ^" r c TiTi . . " For (-: : = ) = - = ; therefore Hime in \p ' t d 1 ' rr a cc liTi ^tt " ' ij) "^ ^ dd -~ '\' " Consequently a = -pTt-r: inches = -j-'^pTTTr feet. *' And putting n = (l.^Vtttt =: 2L.7r - l.21 =T.6 140885 ; then L.a = l./; + n." Proceed we now to Mr. Robertson's solution of the pro- blem, which is on the principles, but more in detail, than Mr. Jones's. This solution was published by Mr. R. in the Philos. Trans, vol. 50, or in my new Abridgement, vol. 13, from which it is chiefly here extracted. Mr. John EobertsorCs Solution of the Problem. *' Sometime before the year 1740, the problem about the fall of water, occasioned by bridges built across a river, was * This is the theorem, adapted to working by logarithms, given by Mr, Jones to Mr. Gardiner, and printed in p. 12 of his Logarithms in 4toj the latter L denoting logarithm, in the theorem. 92 THE PRINCIPLES OF BRIDGES. TRACT X. much spoken of at London, on account of tlie full that was supposed would be at the new bridge to be built at West- minster. In Mr. Hawksmoor's and Mr. Label3-e's pamphlets, the former published in 1736, and the latter in 1739, the result of Mr. Labelye's computations was given: but neither the investigation of the problem, nor any rules, were at that lime published. *' In the year 1742 was published, Gardiner's edition of Vlacq's Tables; in which, among the examples there prefixed, to show some of the uses of those tables, drawn up by the late Wm. Jones, esq, there are two examples, one showing how to compute the ftill of water at London-bridge, and the other applied to Westminster-bridge : but that excellent mathe- matician's investigation, by which those examples were wrought, was not printed, though he comnujnicatcd copies of it to several of his friends. Since tliat time, it seems as if the problem had in general been forgotten, as it has not made its appearance, to my knowledge, in any of tlie subsequent publications. As it is a problem somewhat curious, though not difficult, and its solution not generally known, (having seen four different solutions, one of them very imperfect, extracted from the private books of an office in one of the departments of engineering in a neighbouring nation), I thought it might give some entertainment to tlie curious in tiicsc matters, if the whole process were published. ' PKINXIPLES. *' 1. A heavy bodv, that in the first second of time has fallen the height of a feet, has acquired such a velocity, that, moving uniforndy with it, will in the next second of time move the Icnfrth of 2fl feet, " 2. The spaces run through by falling bodies are propor- tional to one another as the squares of their last or acquired velocities. Tiiese two principles are demonstrated by the writers on mechanics. '* ?. W'xUw forced out of a larger chanci, through one or ECT. 4. THE FOUCE AND FALL OF THE WATER, &.c. 95 more smaller passages, will have the streams through those passages contracted in the ratio of 25 to 21. This is shown in the 36th proh. of the 2d book of Newton's Principia. " 4. In any stream of water, the Telocity is such, as would be acquired by the fall of a body from a height above the surface of that stream. This is evident from the nature of motion. " 5. The velocities of water through different passages of the same height, are reciprocally proportional to their breadths. For, at some time, the water must be delivered as fast as it comes ; otherwise the bounds would be over- flowed. At that time, the same quantity, which in any timo flows through a section in the open chanel, is delivered in equal time through the narrower passages; or the momentum in the narrow passages must be equal to the momentum iu the open chanel; or the rectangle under the section of the narrow passages, by their mean velocity, must be equal to the rectangle under the section of the open chanel by its mean velocity. Therefore the velocity in the open chanel is to the velocity in the narrower passages, as the section of those pas- saores is to the section of the open chanel. But, the heiohts in both sections being equal, the sections are directly as the breadths. Consequently the velocities are reciprocally as the breadths. ''6. In a running stream, the water above any obstacles; put therein, will rise to such a height, that by its fall the stream may be discharged as fast as it comes. For the same body of water, which tiowed in the open chanel, must pass through the passages made by the obstacles ; and the nar- rower the passages, tb.e swifter will be the velocity of the water : but the swifter tlie velocity of the water, the greater is the height, from which it has descended: consequently the obstacles, which contract the clianel, cause the water to rise against them. But the rise will cease, when the Avater can run off as fast as it comes : and this must happen when, by the fall between the obstacles, the water will acquire a velo- city in a reciprccal proportion to tlr.it in the open chanel, as 94 THE PRINCIPLES OP BRIDGES. TRACT I. the breadth of tl^e open chanel is to the breadth of the nar- row passages. " 7, The quantity of the fall, caused by an obstacle in a running stream, is measured by the dilference bct.vcen the heights fallen from, to acquire tiie velocities in the narrow passages and open chanel. For, just above the fall the velo- city of the stream is sucl), as would be acquired by a body falling from a height higher than the surface of the water: and at the fall, tim velocity of tijc strt:ani is such, as would be acquired by the fall of a heavy body from a lieight more elevated than the top of the falling stream ; and conse(]uently the real fall is less than this height. Now as the stream comes to the fall with a velocity belonging to a full above its sur- face ; consequently the height belonging to the velocity at the fall, must be diminished by the height belonging to the velocity with which the stream arrives at the fall. *' PROBLEM. *' In a chanel of running water, whose breadth is con- tracted by one or more obstacles; the breadth of the chanel, the mean velocity of the whole stream, and the breadth of the water-way between the obstacles, being given ; to find the quantity of the tall occasioned by those obstacles. " Let b breadth of the chanel in feet ; V = mean velocity of the water in feet per second ; c- = breadth of the water-way between the obstacles. Now 25 : 21 :: c : |^c, tlie water-way contracted, by prin. 3. 25b And -IjC : b : : v : ~."^'> the veloc. in the contr. way, prin. 5. w Also {2ay -.WW a: , height fallen to gain the vcl. t', 1 and 2, 2. lb , 25b vv And {Oaf : (--.v/ : : a : (- )^ x , ditto for the vclo- citv -I', by priuc. 1 and 2. ECT.'4. THE FORCE AND FALL OF THE WATER, &C. 9} _, 25b w TV . . ., Then - x ' 7- is the measure 01 the tali required, prin. T. 21c 4a 4a i ' i Or f ( Y~ 1] X is a rule for computinq; the fall. * ^21f -'4a f o Here a = 16,0899 feetj and 4a = 64,3596. ** EXAMPLE 1. For London-Bridge. ** By the observations made by Mr. Labelye in 1 746, Tiie breadth of the Thames at London-bridge is 926 feet; Sum of water-ways at the time of low water is 236 feet; Mean veloc. of stream just above bridge is d~ f. per sec. Under almost all the arches there are great numbers of drip- shot piles, or piles driven into the bed of the water-way, to prevent it from being washed away by the fall. These dri]>- shot piles considerably contract the water-ways, at least |- of their measured breadth, or about 39y feet in the whole. So that the water-way will be reduced to 196 j- feet. " Now b = 926; c = l9Gf; v = 3^; 4a = 64,3396. 2ob 23150 Then ; r: = 5,60532; its square = 31,4196: 2 If 4130 ' ' ' 251^ And 31,4196 - 1 = 30,4196 = (- '2\C' 19, 361 , , rv 361 Then 30,4196 x 0,15581 = 4,739 f. = 4 f . 8,868 inc. the fall required. " By the mo^t exact observations made about the year 1736, the measure of the full was 4 feet 9 inches." " EXAMPLE 2. For JVestminsier-Bridge. *' Though the breadth of trie river at Westminster-bridge is 1220 feet; yet, at the time of the greatest fall, therein water through only the 13 large arche?, which amount to 820 feet : to Avhich adding the breadth of the 12 intermediate piers, equal to 17 'r feet, give? 9i-i- for the breadth of the river 96 THE PRINCIPLES OF BRIDGES. TRACT I. at that time ; and the velocity of the water just above the bridge, from many experiments, is not greater than 2^ feet per second. " Here b = 99^ ; c = 820 ; v = 2\ ; 4a = 64,359G. 25b 24850 Now -- ; = "p;^ lj4;43; and Its square = 2,082; 25b Hence 2,082 - 1 =z 1,082 = { )' 1. Also z. = (f)= = ^i ; .nd ": =. ^^--IL-^ = 0,0786. Then 1,082 x 0,0786 = 0,084 f. =: 1 inch, the fall required; and is about half an incli more than the greatest fall observed by Mr. Labelye." Among the old papers of Mr. Robertson 1 find several other solutions of the same problem, by different persons, and on somewhat different principles. Several of the papers also, which are of a miscellaneous nature, relate to other branches of the subject of bridges ; some of which, being curious, I shall avail mvself of, by insertion in the appendix to tiiii Tract. Thefollowmg table shows, at one view, the quantity of fall in the water under the arclies, in conseqiience of its obstruction and contraction by the piers, according to seve- ral rates of velocity and quantity of obstacles ; as computed on the foregoing principles. - w _ i,' . o . Oi CO -o 1 c 1-5 >-s ^--H i 3 = = X 'Z .- s -?'= jT^S-a Stages c ccumul tion in Floods. 5e^ CO b. s n -S-^ = - i D c g fe ^ _-0 5_=. 5 c - _ = < ^ ^-^ V 1 '^ ^ ,0 3 c > uc CO ?0 ^ c^ c ^ "*> : pi = *3 W /-^ ^ .^ o,- -^ V y - ^ ? - o; _ -^ jj . tf: T O -T 2 ,-5 "o ^o CO c-1 O -H CO CO Oj - - " 2 5 - ? .~ H GO c * ^ "^ -- " "X '-' > t/3 I .= 'O CO CO o : o liliii?! d o o -^ (S CO .' Lo i < o 00 CO ;^. s c o ^ .9 V 1 i j^ 00 CI C-1 'O co' CO as X C ^ " X C - 3.orj i o CO O 00 C CO 2 -c i c^. o ,^ Oj ~. p ec o y; H coj in - c - T)< t^ o UO i 1 IlilSil V3 1 " o o o o' ' i 0^ " ii''sil = y-: r-3 CO CTi WJ CO C-, .:; o 5 .2 rt ^ CO 9 7 -* -t s||S.|? ;-. - O I' - O - r? -0 t^ 2 -c 5 ii M = ^~ii= P^ d O O c 3 o < ./; XI lO x> ;^, j^ m CD " "O CO M CO 'O C-? -+ Jt'ts.gi^ ^ " "^ o ^ p c:) 1;^ *" ^^ ^ - -^ - ii. " X) ^ (D H ^0 = - = 5 I = o o 1 r^ -,-> "-O C -P -= t",_'I^ |] d o o ^ _P_ O Q ii; < ' --5^'E 2 5' c Velocity of the - o ^ _^ _^_J ^ ^ -l~) _!_, X _5 v! ' :, 5 T " - jj - S = _ c = l i i J 0) UO 'O ~ 9 ^ > -^ -a s ; cji c-= C - 7> J / '"" CO :o -.- c> ci 1 c i^ . ; ' .2 ^ y * S S*' CI rj CO Oi ^^ c^ ~ .i -J 1 w ^ ^^ c _ LO T-^ I '*" i' " ^- = J SCO "^ > ^ ^ r: - ^ ^"^ S S Ti ^ -* -'---" JZ ^ ' c^ 5 ^^ CJ "-^ O = c o -* M 'CO -T 1^ #lll=l '3 u- "C -^ 3S .- .i: - ^ J- T C -i Tf. _,7^ -. d o o c C O O 1 2i 5J= 5 r ;-; J VOL. I. ( 98 ) SECTION V. OF THE TERMS OR NAMES OF THE VARIOUS PARTS PECU- LIAR TO A BRIDGE, AND THE MACHINES, &C, USED ABOUT IT; DISPOSED IN ALPHABETICAL ORDER. Abutment, or Butment, which see in its place below. Arch, an opening of a bridge, through or under which the water and vessels pass, and which is usually supported by piers or by hutments. Arches are denominated circular, elliptical, cycloidal, catenarian, &c, according to the figure of the curve of them. There are also other denominations of circular arches, according to the different parts of a cir- cle: So, a semicircular arch, is half the circle ; a scheme or skeen arch, is a segment less than the semicircle ; and arclies of the third and fourth point, or gothic arches, consi of two circular arcs, excentric and meeting in an angle at top, each being l-3d or l-4th, &c, of the whole circle. The chief properties of the most considerable arches, with regard to the extrados they require, &c, may be learned from the second section. It there appears, that none, but the arch of equilibration in the 2d example to prop. 5, can admit of a horizontal line at top: that this arch is not only of a graceful, but of a convenient form, as it may be made higher or lower at pleasure with the same opening: that, with a horizontal top, it can be equally strong in all its parts, and therefore ought to be used in all works of much conse- quence. All the other arches require tops that are curved, either upward or downward, some more and some less. Of these, the elliptical, or the cycloidal arch, seems to be the fittest to be substituted instead of the balanced one, with the least degree of impropriety : it is in general also the best form for most bridges, as it can be made of any neight to the same span, or of any span to the same height, while at the same time its flanks are sufficiently elevated above the SECT. 5. A DICTIONARY OF THE TERMS. 99 water, even wlien it is pretty flat at top; a property of which the other curves are not possessed in an equal degree : and this property is the more valuable, because it is remarked that, after any arch is built, and the centering struck, it set- tles more about the hanches than the other parts, by which other curves are reduced near to a straight line at the flanks. Elliptical arches also look bolder, are really stronger, and require less materials and labour than the others. Of the other curves, the cycloidal arch is next in quality to the elliptical one, for all the above properties. And, lastly, the circle. As to the others, the parabola, hyperbola, and ca- tenary, they ma}'^ not at all be admitted in bridges of several arches ; but may in some cases be used for a bridge of one single arch, which is to rise very high, because then rtot much loaded at the flanks. We ma}' hence also perceive the fallacy of those arguments which assert, that because the catenarian curve supports itself equally in all its parts, it will therefore best support an)^ additional weight laid upon it: for the additional building made to raise the bridge to a horizontal hne, or nearly such, b}' pressing more in one part than another, must force those parts down, and the whole must fall. Whereas, other curves will not support themselves at all, without some additional parts built above them, to balance them, or to reduce their parts to an equi- librium. Archivolt, the curve or line formed by the upper sides of the voussoirs or arch stones. It is parallel to the intrados or underside of the arch when the voussoirs are all of the same length; otherwise not. By the archivolt is also some- times understood the whole set of voussoirs. Banquet, the raised foot path at the sides of the bridge next the parapet. This ought to be allowed in all bridges of any considerable size: it should be raised about a foot above the middle or horse passage, being made 3, 4, 5, 6, 7, &c, feet broad, according to the size of the bridge, and paved with large stones, of a length equal to the breadth of the walk. 100 THE PFxIXCIPLES OF BRIDGF.S. TRAfcT I. BATTARDrAU, or Coffer-dam, a case of piling, &.c, Avith- oiit a bottom, fixed in tlie bed of the river, water-tight or nearly so, by which to lay the bottom dry for a space large enough to build the pier on. Wlien it is fixed, its sides reaching above the level of the water, the wat('r is pumped out of it, or drawn ofi"by engines, till the included space be laid drv; and it is kept so, by the same means, if there are leaks whicii cannot be stopped, till the pier is built up in^t; and then the materials of it arc drawn up again. Battardeaux are made in various manners,, either by a sin- gle inclosure, or by a double one, with clay or chalk rammed in between the two, to prevent the water from coming tlirough the sides. And these inelosures are also made, either with piies only, (hiven close by one another, and sometimes notclied or dove-tailed into each other; or with piles, grooved in the sides, and driven in at a distance from one another, with boards let down between tliem in the grooves. The method of building in buttardeaux cannot well be \ised where the river is either deep or rapid. It also re- quires a very good natural bottom of solid earth or clav: for, though the sides be made water-tight, if the bottom or bed of the river be of a loose consistence, the water will ooze up through it, in too great abundance to be evacuated by tlie engines. It is almost needless to remark, that the sides must i)e made verv strong, and well propt or braced on the inside, to prevent the ambient water from pressing the sides in, and forcing its way into the battanleaux. Bridgi-:, a woik of carpentry, masonry, or iron, built over a river, canal, S:c, for the conveniency of crossing the same. A bridge is an editiee forming a way over a river, &.c, sup- ported by one arch, or by several arches, and these again suj)ported by ])roper piers or hutments. A stately hriJoCj, over a large river, is one of the most noble and striking pieces of human ait. To behold huge and bold arches, (-(niiposed of an immense (|uiintity of small materials, as stones, bricks,, \c, so disposed and united together, that they seem to foriu SECT, 5. A DICTIONARY OF THE TERMS. - 101 but one solid compact body, affording a safe passage for men and carriages over large waters, Avhich witli their navi- gation pass free and eii^y under them at the same time, is a sight tmly surprizing and affecting. To the absolutely necessarv parts of a bridge, already mentioned, viz, the archc'. piers, and abutments, may be added the paving at toji, the })arapet wall, either with or without a balustrade, &c; also the banquet, or raised foot way, on each side, leaving a sufficient breadth in the middle for horses and carriages. The breadth of a bridge for a great -city should be such as to allow an easy passage for three carriages and two horsemen a-breast in the middle way, and for three foot passengers in the same manner on each banquet. And for other less bridges, a less breadth. As a bridge is made far a wav or jiassage over a river, &.c, so it ought to be made of such a height, as will be quite convenient for that passage ; but vet so as to be consistent with the interest and concerns of the river itself, easily ad- mitting through its arches the craft that navigate on it, and all the water, even at liigh tides and floods. The neglect of this prece]it has been the ruin of many bridges, and parti- cularly that at Newcastle, over the river Tyne, on the l7th of November 1771. So that, in determining its height, the conveniences both of the passage over it, and under it, should be considered, and the height made to answer the best for them both, observing to make the convenient give place to the necessiny, when their interests are opposite. Bridges are generallv placed in a direction perpendicular to the stream in a direct Ime, to give free passage to the water, &,c. But some think they should be made, not in a straight line, but convex towards the stream, the better to resist floods, &.c. And some such bridges have been really made. A"-ain, a bridije should not be made in too narrow a part of a navigable river, or one subject to tides or floods: because the breadth being still uiore contracted by the piers, will increase the depth, velocity, and fall of the water under the ai"ches, and endanger the whole bridge and navigation. 102 THE PRINCIPLES OF BRIDGES. TRACT I. Bridges are usually made with a odd number of arches, as one, or three, or five, or seven, &c; either that the mid- dle of the stream or chief current may flow freely without the interruption of a pier; or that the two halves of the bridge, by gradually rising from the ends to the middle, may were meet in the highest and largest iircii ; or else, for the sake of grace, that by being open in the middle, the eye, in viewing it, may look directly through there, as one always expects to do in looking at it, and without which opening we generally feel a disappointment in viewing it. If the bridge be equall}' high throughout, the arches, being all of a height, are made all of a size; which causes a great saving of centring. If the bridge be higher in the middle than at the ends, the arches are made to decrease from the middle towards each end, but so, as that each half may have the arches exactly alike, and tliat they decrease in span, pro- portionally to their height, so as to be always the same kind of figure, and similar parts of that figure; thus, if one be a semieircle, the rest should be semicircles also, but propor- tionally less; if one be a segment of a circle, the rest should be similar segments of other circles ; and so for other figures. The arches being equal at equal distances, on both sides of the middle, is not only for the strength and beauty of the bridge, but that the centring of the one half may serve for the other also. But if the bridge be higher at the ends than the middle, which is a very uncommon case, the arches ought to increase in span and pitch from the middle towards the ends. When the middle and ends are of different heights, their difference however ought not to be great in proportion to the length, that the ascent may be easy; ami then also it is more beautiful to make the top one continued curve, like Blackfriars, than two inclined straight lines, from the ends towards the middle, like that of Westminster bridge. Bridges should rather be of few and large arciies, than of many and small ones, if the height and situation will allow of it; for this will leave more free passage for the water and navigation, and be a great .saving in materials and labour, as SECT 5. A DICTIONARY OP THE TERMS. 103 there will be fewer piers and centres, and the arches them- i^lves will require less materials. And, one large single arch only is best, when it can be executed. For the fabric of a bridge, and the proper estimate of the expence, &c, there are generally necessary three plans, three sections, and an elevation. The three plans, are so many horizontal sec- tions, viz, the first a plan of the foundation under the piers, with the particular circumstances attending it, whether of gratings, planks, piles, &c : the second, is the plan of the piers and arches, &c : the third, is the plan of the super- structure, with the paved road and banquet. The three sections, are vertical ones: the first of them a longitudinal section, from end to end, and through the middle of the breadth; the second, a transverse one, or across it, and through the summit of an arch : and the third also across, but taken on a pier. The elevation, is an orthographic pro- jection of one side or face of the bridge, or its appearance as viewed at a great distance, showing the exterior aspect of the materials, and the manner in which they are worked and decorated. Other observations are to be seen in the first section. BuTMENTs, or Abutments, are the extremities of a bridge, by which it joins to, or abuts on, the land or sides of the river, &c. These must be made very secure, quite immovable, and more than barely sufficient to resist the drift of its adjacent arch. So that, if there are not rocks or very solid banks to raise them against, they must be well reinforced with proper walls or returns, &c. The thickness of them, which will be barely sufficient to resist the shoot of the arch, may be calculated as that of a pier by prop. xi. When the foundation of a butment is raised against a sloping bank of rock, gravel, or good soHd earth, it will produce a saving of materials and labour, to carry the Mork on by returns at different heights against it, like steps of stairs. And if the foundation, and all the courses, parallel to it, be laid, not horizontal, but rising backwards, so as to be perpendicular to the springing and pressure of the arch, H 2 104 THE PRINCIPLES OF BRIDGES. TRACT I. it will be less liable to slide or be forced back by the push of the arch. j^ Caisson, a kind of Chest, or flat-bottomed boat, in which a pier is built, then sunk to the bed of the river, and the sjdps loosened and taken off from the bottom, by a contri- vance for that purpose; the bottom of it being left under the pier as a foundation. It is evident therefore, that the bottoms of caissons must be made very strong, and fit for foundations of the piers. The caisson is kept afloat till the pier be built to about the height of low-water mark; and, for that purpose, its sides must cither be made of niore than that height at fjrst, or else gradually raised to it as it sinks by the weight of the work, so as alwaj-s to keep its top above water. And therefore the sides must be made very strong, and be kept asunder by cross timbers within, le.>t the great pressure of the ambient water should crush the sides in, and so not only endanger the work, but also drown the men who work within it. The caisson is made of the shape of the pier, but some feet wider on every side, to make room for the men to work: the whole of the sides arc of two pieces, both joined to the bottom quite around, and to each other at the salient angles, so as to be disengaged from the bottom, and from each other, when the pier is raised to the desired height, and sunk. It ia also convenient to have a small sluice made in the bottom, occasionally to open and shut, to sink the caisson and pier sometimes by, before it be finished, to try if it bottom level and rightly; for, by open- ing the sluice, the water v.ill rush in and fill it to the height of the exterior water, and the weight of the work already built will sink it; then, by shutting the sluice, and pumping out the water, it will be made to float again, and tije rest of the work mav be completed : but it must not be sunk e.xcej)t wliL'ii t!)e sides are high enougli to reach al)ove the surface ot tiie water, otherwise it cannot be raised and laid drv again. ^h. I.abelye savs, that the caissons in which he !)uilt ^o;ln.; of the. piers of \\'estuunster bridge, contained .iliu\e I.IO load of lir timber, ol' 40 eubie feel each. SECT. 5. A DICTIONARY OF THE TERMS. 103 and was of more tonnage, or capacity, than a 40 gun ship of war. Centres, and Centri?-.'G, or Centering, are the timber frames erected in the spaces of the arches, to turn them on, by building on them the voussoirs of the arclies. As the centre serves as a foundation for the arch to b^ bi; a on, wb.en the arch is completed, that foundation is : uck ironi under it, to make v.-ay for the water and navigatioii, und then the arch will stand of itself from its cur^^id {ig;-^-e. A centre must therefore be constructed of the ex.xt ngt?rc of the intended arch, convex as the arch is concave, to receive it on as a mould. If the form be circular, the curve is stru -k from a central point by a radius: if it be elliptical, it ought to be struck with a doubled cord, passing over two pins or nails fixed in the foci, as the mathematicians and gardenevs describe their elliT)ses. Very often, in practice, an oval is employed, as made of three circular arcs. Tliis verv nearly resembles the true geometrical ellipsis, being formed of two equal arcs of small circles at the extremities, having betv.een them a longer arch of a much larger circle, the ends of these arches being made to butt and join to each other, that they seem like the same curve only continued. As this mecha- nical oval will have nearly the same properties and etTect as the true ellipsis, and can be more conveniently worked by the builders, as it requires the voussoirs to be cut only to two moulds, or for two centres, Avhile those for the true elli[)sis have them all different, we shall add in this j)lace some of the most approved methods of describing these ovals. These methods indeed are, and must be, various, according as the length or span is required to be more or less, in pro- portion to the breadth or height. But in all of them, the centres of the large and small arcs must be so taken, that the right line passing through them, may also, when continued, pass through exactly the point where the ends of those arches butt and join together; for by this means they will )uve the same common tangent at that point, and conse- 106 THE PRINCIPLES OP BRIDGES. TRACT I. quently they will unite together, or run into each other, like parts of the same curve produced. fiRST METHOD. When the Length and Breadth differ not very much. Divide the given length or span AB into three equal parts, at the points c and d. With one of those parts, CD, as a radius, and from the two centres c, d, describe two circles, intersecting each other in IC^^'^^^ -e . :" :.:-^^^! the two points e and f. Through these two points e, f, and the two centres c, d, draw four lines zcG, edh, fdi, FCK, cutting the two circles in the four points g, h, i, k. Lastly, with one of these lines, as a radius, and from the two centres e, f, describe the two arches gh, KI, and they will complete the oval, forming a figure so much resembling a true ellipse, that the eye. cannot perceive the difference be- tween them. In this oval it is evident that the radius of the larger circular arch is just double of that of the smaller arches. SECOND METHOD. /or a Nuvrower Otal. Divide the length or span AB into four equal parts; then, with one of those parts as a radius, and from the three points of division, c, D, E, as centres, describe three circles. Find the uppermost and lowest points, f, g, of the middle circle; or through the middle point d draw a perpendicular to ab, which will give the points f, g, or construct the square cgef, which will give the centres of the larger arch. Through these two points f, g, and the two c, E, draw four lines fh, fi, gk, CL; with any one of which as a radius, and the two cen- SECT. 5. A DICTIONARY OF THE TERMS. 107 tres F, G, describe the other two arcs hi, kl, to complete the oval j which does not rise so high as the former. THIRD METHOD. Cr O Other ovals may be made to the same length, or any other length, but rising still less in the crown, in any degree whatever, if, after having described the two smaller or end circles from the centres c and e, as in the second method, instead of forming the right angled triangles cge, cfe, these be described with acute an- gles at F and g, by making the equal lines cf, cg, ef, eg, longer than before in any ratio at plea- sure; these being then produced to the little circles at the four F points H, I, K, L, from the centres f, g, describe the other two arches hi and kl, to complete the ovals, narrower and narrower at pleasure. The little circles also at the ends, may have their radius taken smaller to any degree, or a less portion of the whole span ; and indeed it is evident that its radius ought always to be less than the pitch or height of the arch. There are other methods of making such ovals, but those above given are some of the best. The last method is gene- ral too, and will serve to accommodate an oval to any length and breadth whatever, at pleasure. Having thus described the half of such an oval to any span and pitch proposed, for any arch of a bridge, &c, the whole of the voussoirs may be cut by two mold boards only, viz, one for the voussoirs for the arch ah and ib, and the other for those in the arch hi. But if the arch be of any other form, the several abscisses and ordinates ought to be calculated ; then their correspond- lOS THE rniNTClPLES OF BKIDGF.S. TRACT I. ing- lengths, tninsferrod to the centring, vill give so many points of the curve, and exactly by these points bending a bow of pliable matter, the curve may be drawn close by it. The centres arc constructed of beams, &c, of timber, firmly pinned and bound together, into one entire compact frame, covered smooth at top Avith planks or boards to place the voussoirs on, the Avholc supported by offsets in the sides of the piers, and by piles driven into the bed of the river, and capable of being raised and depressed by wedges, con- trived i'or that pm'pose, and for taking them down when the arch is completed. They ought also to be constructed of a stren2;th more than sufficient to bear the weight of the arch. In taking down the centring; it is first let down a little, all in a piece, by easing some of the wetlges ; it js there let to rest a few hours, or days, to try if the arch make any v.i'- forts to fall, or any joints open, or stones crush or crack, &c, tliat the damage may be repaired before the centring is entirely removed, which is not to be done till the arch ceases to make any visible eflbrts. In some bridges the centring makes a considerable part of theexpence, and therefore all means of saving in this article ought to be closely attended to; such as making lew arches, and as nearly alike or similar as possible, that the centring of one arch may serve for others, and at least that the same centre may be used for each pair of equal arches, on both sides of the middle. Chks'J", the same as Caisson. CoFiKKDAM, the same as Battkrdf.au. Drift, Shoot, or Thrust, of an arc^h, is the push or force which it exerts in the direction of tlie length of the bridge. This force arises from the perpendicular gravita- tion or weight of the stones of tlie arch, which, being kept from di'scending in' the form of the arch and the resistance of the ])i('rs, exert their force in a latenil direction. This force ii cinupuied in j)r(>p. xj, where tiro tisickiu'-^s of the SECT. 5. A DICTIONARY OF THE TERMS. 109' pier is determinecl which is necessar}'^ to resist it ; and is the greater as the pitch is lower, cateris paribus. Elevation, the orthographic projection of the front of a bridge, on the vertical plane, parallel to its length. This i* necessary to show the form and dimensions of the arches, and other parts, as to height and breadth, and therefore it has a plain scale annexed to it^ to measure the parts by. It also shows the manner of working up and decorating the fronts of the bridoc. ExTRADOs, the exterior curvature or line of an arch. Ir> the propositions of the second section, it is the outer or upper line of the wall above the arch ; but it often means only the upper or exterior curve of the voussoirs. Foundations, the bottoms of the piers, &c, or the bases. on which they are built. These bottoms are always to be made with projections, greater or less according to the spaces on which they are built. And according to the nature of the ground, the depth and A'elocity of water, &c, the foundation are laid, and the piers built, after ditferent manners, either iti caissons, in battcrdeaux, or on stilts with sterlings, &c ; for the particular methods of doing which, see each under its re- spective term. The most obvious and simple method of laying the founda- tions, and raising the piers up to water*mark, is to turn the river out of its course above the place of the bridge, into a new channel, cut for it near the place where it makes an el- bow or turn ; then the piers are built on dry ground, and the water turned into its old course again, the new one l)eing se- curely banked up. This is certainly the best method, when the new channel can be easily and conveniently made.; but which however is very seldom the case. Another method is, to lay only the space of each pier dry, till it be built, by surrounding it with piles and planks driven down into the bed of the river, so close together as to exclude the water from coming in ; then the water is pumped out of the inclosed space, the pier buih in it, and lastly the piles and planks drawn up. This is cofferdam work; but it evidently 1 10 THE PRINCIPLES OP BRIDGES. TRACT 1. cannot be practised when the bottom is of a loose consist- ence, admitting the water to ooze and spring up through it. When neither the whole nor part of the river can be easily laid dry, as above, other methods are to be used ; such as, to build either in caissons or on stilts, both which methods are described under their proper words ; or yet by another me- thod, which hath, though seldom, been sometimes used, with- out laying the bottom dry, and which is thus : the pier is built upon strong rafts or gratings of timber, well bound to- gether, and buoyed up on the surface of the water by strong caWes, fixed to other floats or machines, till the pier is built; the whole is then gently let down to the bottom, which must be made level for the purpose. But of these methods, that of building in caissons is the best. But before the pier can be built in any manner, the ground at 'the bottom must be well secured, and made quite good and safe, if it be not so naturally. The space must be bored into, to try the consistence of the ground ; and if a good bottom of stone, or firm gravel, clay, &c, be met with, within a mo- derate depth below the bed of the river, the loose sand, &c, must be removed and digged out to it, and the foundation laid on the firm bottom, on a strong grating, or base of tim- ber, made much broader every way than the pier, that there may be the greater base to press on, to prevent its being sunk. But if a solid bottom cannot be found at a convenient depth to dig to, the space must then be driven full of strong piles, the tops of which must be sawed off level, some feet below the bed of the water, the sand having been previously digged out for that purpose ; and then the foundation, on a grating of timber, laid on their tops as before. Or, Avhen the bottom is not good, if it be made level, and a strong grating of tim- ber, two, three, or four times as large as tlie base of the pier, be made, it will form a good base to build on, its great size in a great measure, preventing it from sinking. In driving the piles, the method is, to begin at the middle, and proceed outwards, all the way to the borders or margin : the reason of which is, that if the outer piles were driven first, the earth SJiCT. 5, A DICTIONARY OF THE TERMS. Ill of the inner space would be thereby so jammed together, as not to allow the inner piles to be driven at all. And besides the piles immediately under the piers, it is also very prudent to drive in a single, double, or triple row of them, around and close to the frame of the foundation, cuttmg them off a j^ttle above it, to secure it from slipping aside out of its place, and to bind the ground under the pier the firmer. For, as the safety of the whole bridge depends much on the founda- tions, too much care cannot be used to have the bottom made quite secure. Jettee, the border made around the stilts under a pier; being the same with Sterling. Impost, is the pra-t of the pier on which the feet of the arches stand, or from which they spring. Keystone, the middle voussoir, or the arch stone in the crown, or immediately over the centre of the arch. The length of the keystone, or thickness of the archivolt at top, is allowed to be about l-15th or l-16th of the span, by the best architects. Orthography, the elevation of a bridge, or front view, as seen at a great distance. Parapet, the breast wall made on the top of a bridge, to prevent passengers from falling over. In good bridges, to build the parapet only a little part of its height close or solid, and on that a balustrade to above a man's height, has an ele- gant and useful effect. Piers, are the walls built for the support of the arches, and from which they spring as their bases. These ought to be built of large blocks of stone, solid throughout, and cra:mped together with iron, or otherwise, which will uiake the whole like one solid stone. Their faces or ends, from the base up to high-water mark, ought to project sharp out with a salient angle, to divide the stream. Or perhaps the bottom of the pier should be built flat or square up to about half t'le height of low- water mark, to allow a lodgment against it for the sand or mud, to cover the foundation; lest, by being left bare, the water should in time undermine, and so ruin or 112 THE PRINCIPLES OF BRIDGES. TRACT 1. injure it. The best form of the ])rotection for ilividing tlie stream, is the triangle ; and the longer it is, or the more acute the salient angle, the better it will divide it, and the less will the force of the wiiter be against the [)ier ; but it may be suf- ficient to make that angle a right one, as it will make the Avork stronger, and in that case the perpendicular projection will be equal to half the breadth or thickness of the pier. In rivers on which large heavy craft navigate, and pass the arches, it may ])erhaps be better to make the ends semicircu- lar; for though It docs not divide the water so well as the triangle, it will both better turn oBand bear the shock of the craft. The thickness of the piers ought to be sucli, as will make them of weight or strength sufficient to support their interja- cent arch, independent of anv otlier urchc-s. The thickness, in most cases of practice, may be made about l- of the span of the arch. And then, if the middle of the pier be run up to its full height, the centring may be struck, in order to be used in another arch, before the haiiches are filled up. The whole theory of the piers may be seen in the third section. They ought to be made with a broad bottom on the foundation, and gradually diminished in thickness bv offsets, up to low-water mark. The methods of laying their foundations, and build- ing them up to the surface of the water, are given under the "word Foundation. PiLKS, are timbers driven into tlie bed of the river for va- rious purposes, and are either round, square, or flat like planks. They may be of any wood which will not rot under "\vater, but elm, oak, and fir are mostly used, especially the latter, on account of its length, straightness, and ciicapness. They are shod with a pointed iron at the bottom, the better to penetrate into the ground ; and are bound with a strong iron band or ring at to]), to prevent them from being split by the violent strokes of the ram by which they are driven down. It is said, that the stilts, or piles, under London- bridge, are of ehu, which lasts a long time in the w ater. Piles are either used to build the foundations on, or are SECT. 5. A DICTIONARY OF THE TERMS. ' 113 driven about the pier as a border of defence, or to support the centres on ; and in this case, when the centrmg is re- moved, they must either be drawn up, or sawed off very low- under water ; but it is perhaps better to saw them off, and leave them sticking in the bottom, lest the drawing of them out should loosen the ground about the foundation of the pier. Those to build on, are either such as are cut off by the bottom of the water, or rather a few feet within the bed of the river ; or else such as are cut off at low-water mark, and then they are called stilts. Those to form borders of defence, are rows driven in close by the frame of a foundation, to keep it firm ; or else they are to form a case or jettee about the stilts, to keep within it the stones that are thrown in to fill it up ; in this case, the.piles are grooved, driven at a small dis- tance from each other, and plank piles let into the grooveii between them, and driven down also, till the whole space is surrounded. Besides using this for stilts, it is also sometimes necessary to surround a stone pier with a sterling or jettee, and fill it up with stones to secure an injured pier from being still more damaged, and the whole bridge ruined. The piles to support the centres may also serve as a border of piling to secure the foundation, cutting them off low enough after the centre is removed. Pile Driver, is an engine for driving down the piles. It consists of a large ram or square block of iron, shding per- pendicularly down between two guide posts ; which beino- drawn up to the top of them, and there let fall from a great height, it comes down on the top of the pile with a violent blow. It is worked either by men or horses, and either with or without wheel work. That which was used at the build- ing of Westminster-bridge, is perhaps one of the best kind. Pitch, of an arch, is the perpendicular height from the spring, or impost, to the keystone. Plan, of any part, as of the foundations, or piers, or su* perstructure, is the orthographic projection of it on a plane parallel to the horizon. Push, of an arch, the same as drift, shoot,, or thrust, VOL. I. I ll-* THE PBIN.CIPLES OF BRIDGES. TRACT J. Salient Angle, of a ])ier, is the projection of the end against the stream, to divide it. The right-hncd angle best divides the stream, and the more acute the better for that purpose ; but tlie right angle is generally used, as making the best masonry. A semicircular end, though it does not divide the stream so well, is sometimes better in large navi- gable rivers, as it carries the craft the better off, or bears their shocks the better. Shoot, of an arch, is the same as drift, thrust, &c. Span, of an arch, is the extent or width at the bottom, or on the level at its springing. Spandrels, or Spandrils, are the spaces about the flanks or haunches of the arch, above the curve or intrados. Springers, are the first or lowest stones of an arch, being those at its feet, bearing immediately on the impost. Sterlings, or Jettees, akind of case, made of stilts, &c, about a pier, to secure it. It is particularly described under the next word Stilts. Stilts, a set of piles driven into the space intended for the pier, whose tops being sawed level off about low-water mark, the pier is then raised on them. This method was formerly used, when the bottom of the river could not be laid dry ; and these stilts were surrounded, at afew^ feet dis- tance, by a row of piles and planks, &c, close to them like a coffer-dam, and called a sterling or jettee ; after which, loose stones, &c, are thrown or poured down into the space, till it be filled up to the top, by that means forming a kind of pier of rubble or loose work, w^hich is kept together by the sides of the sterlings : this is then paved level at the top, and the arches turned upon it, This method was formerly much used, most of the large old bridges in England being constructed in that way ; such as London-bridge, Newcastle-bridge, Ro- chester-bridge, &c. But the inconvenicncies attending it arc so great, that it is now quite exploded and disused : for, be- cause of the loose composition of the piers, they must be made very large or broad, otherwise the arch would push them over, and rush down as soon as the centre should be drawn: which SECT 5. A. PICTIONARY OF THE TERMS. 1 15 V great breadth of piers and sterlings so much contracts the passage of the water, as not only very much incommodes the navigation through the arch, from the fall and quick motion of the water, but from the same cause also the bridge itself is in much danger, especially in time of floods, when the quantity of water is too much for the passage. Add to this, that besides the danger there is of the pier bursting out the sterlings, they are also subject to much decay and damage by the rapidity of the water, and the craft passing through the arches. Thrust, the same as drift, shoot, &c. VoussoiRS, the stones which immediately form the arch, their under sides constituting the intrados or soffit. The middle one, or keystone, ought to be, in length, about -^-j or 1^ of the span, as has been observed ; and the rest should iiv crease in size all the way down to the impost ; the more they increase the better, as they will the better bear the great weight which rests upon them, Avithout being crushed, and also will bind the firmer together. Their joints should also be cut perpendicular to the curve of the intrados. TRACT II. QUERIES CONCERNING LONDON BRIDGE: WITH THE ANSWERS, BY GEORGE DANCE, ESQ. AS an Appendix to the foregoing Tract, on the Principles of Bridges, a few smaller papers, on kindred subjects, are inserted in this and some of the Tracts immediately follow- ing. The present paper is one, among several of a curious nature, which I purchased at the sale of Mr. Robertson's books, in the year 1776, and appears to contain circumstances of too much importance to be kept private. It seems to have ori- I 2 116 THE PRINCIPLES OF BRIDGES. TRACT 2? ginated from enquiries formerly made, for improving the bridge and the port of London, in the year 1746. It consists of queries proposed by the magistrates of the city ; and answers to those queries, by Mr. George Dance, the Surveyor Gene- ral of all the works of the city of London, who was the father of that excellent architect the present City Surveyor. It seems also that the queries had been proposed to the public in general, to solicit answers from any ingenious engineers or architects ; for the paper remarks that, " The persons who are to answer these queries, may add to their answers what further remarks and observations they shall think proper, to the same purpose as these queries. In the middle of every arch there are driven down piles, called dripshot piles, in order to prevent the waters from gullying away the ground. I am of opinion, from the nature of the work, that the bridge was not so wide originally as it now is; and that the points of the piers have been much extended, in order to erect houses thereon. I observe likewise, that in some of the piers, there are fresh casings of stone, before the original ashler. " July the 9th, 1746. George Dance." ** Query 1. What are the shapes and dimensions of the stone piers, the sterlings, and the openings at high and low water ? N. B. This will be best answered by figured sketches, or plans, correctly laid down from an exact men- suration by a scale, provided that scale be not smaller than 8 or 10 feet to an inch." " Answer. I have described the shapes and dimensions of the stone piers, sterlings, and openings at high and low water, in a figured plan, which I delivered to Mr. Comptroller." " Query 2. What are the depths of water, just above, under, and just below the arches, or Jocks, at a common low water? N. B. These depths may be marked on the plans or sketches." " Answer. The depth of water, beginning at the south ud of the bridge, is as follows ; viz. TJlACT 2. ON LONDON BRIDGE. 117 On the west side. Under the arch, Oo the east side. ft. inc. ft. inc. ft, inc. 1st lock . 16 .... 5 9 ... . 8 10 2d , . . 14 6 . .90..., 10 4 3d . . . 22 3 . ..30.:. 14 4th . . . 14 . .70.... 15 7 5th . . . 18 9 . . . 10 3 . . . 18 7 ^th . . . 17 7 . .87.... 15 11 7th . . 18 1 . . 8 10 . . . 15 U 8th . . . 25 1 . .92.... 18 3 9th . . 17 8 . ..59... 18 6 10th . . . 21 2 . , .56.... 17 8 nth . . 18 11 . .35... . 12 8 12th . , .no.. .24..* 22 13th . . 24 6 . ..89... 20 14th . . 22 3 . ..90... . 17 4 15th . . . 23 9 . .69... 20 7 16th . , . 19 9 . . 6 11 . . . 21 10 nth . . . 20 3 . .46... 21 10 18th . . 19 4 . ..79... 14 1 19th . . . 10 10 . ..40.... 13 10 20th . ..67. ..61.... 10 10 I have likewise described the dimensions in the plan afore- said." " Query 3. At what height, above low-water mark, and at what depth below the surface of the sterlings, is the under- bed, or lower side of the first course of stones ?" " Answer. The height of the underbed of the first course of stones, is various: some being 2 feet 4 inches, some 1 ft. 1 1 inc., some 1 ft. 10 inc., some 1 ft. 3 inc., some ift. 1 inc. above low-water mark j and some are 6 feet, some 5 ft. 8 inc., some 4 ft. 6 inc., some 4 ft. 1 inc., and some 4 feet below the surface of the sterlings. These are the dimensions, as far as I am able to get them : there being no opportunity to make observations but when a breach happens to any of the piers." *' Query 4. What is there between the stones and the heads of the piles .'' Is it one row of planks only ; or two rows, 113 THE PRIWCIPLES OF BRTDGES. TRACT 2." crosslaid; or timber: what wood are they made of, and what are their dimensions or scantlings?" ** Answer. In cceneral I find nothinfj between the stones ind piles, but sometimes pieces of plank, mostly of oak, and ft little of elm, some of which is 6 inches and 4 inc. in thick- ness ; which I apprehend Avere not oricjinally placed there, but only when reparations have been made, on which account they were fixed, in order to wedge up tiglit to the stone- work ; it being impossible to make sound work in that case by any other method." *' Query 5. Arc the piles wliich surrounded the founda- tions of the piers, before the sterlings were added, square or round, rough or hewn, driven as close as possible, or at a distance? If they touch one another, are they fastened to- gether with a dovetail, or by any other contrivance of the same nature; and if they do not touch, at what distance arc they at a mean?" " y'insxi)cr. These piles are round, rough, and unhewn : they are driven close, and toucli one another: tliey do not seem to be fastened together by any contrivance, except that some have planks upon them, and some have none. But these observations I have made where breaches have hap- pened, so that one might get 1, 2, or 3 feet witliin the sur- face of the piers: but how they are in the middle of tlie piers, is impossible to determine." " Query 6. Are the heads of those surrounding piles fas- tened together by any kirb or capcile ? If there be any, let it be described, and its dimensions, by a figured sketch." *' ylnswe)'. They arc fastened by no kirb or capcile. There are only planks upon some of them, as I mentioned iu the former answer." " Query 7. Are the inside piles, on which the founda- tions of the piers are laid, round or square, hewn or rough, very close, or at what distance at a mean ; of what timber, and size ; are they shod or not?" *' Answer. This query is very difficult to answer. I can only siiy, that I have had an opportunity to examine one TRACT 2. 'LONDON BRIDGE. H^ pier, about 7 feet within. It is the south pier of the datn lock ; a great part of which Mas undermined, by some of the sflWings being carried away, and leaving it defenceless there. I observe that the piles are round, rough, unhewn, and driven close together ; and they arc chiefly elm, of about one foot diameter. Some of these piles, being taken up, were shod with iron ; and I think it is reasonable to suppose they are all so." *' Query 8. Whether the foundations of the piers, before the sterlings were added, extended beyond the naked line of the stone-work: and if so, as it is most likely, describe how much, at a mean, and the manner, by a figured sketch ?" " Aiiszcer. There is, to every pier, a setoif, or foundation, which extends about 7 inches beyond the naked line of the pier ; and that setoff or foundation is of stone. But I am of opinion that sterUngs were fixed at the first erecting of the bridge ; because I think it impossible for the piers to stand long without some such defence. But whether they were so much extended, or in the same shape they are now, is not easy to determine." " Query 9. Are the piles, that are under the foundations of the piers, much decayed and galled by the action of the currents of waters, before the sterlings were added ?" " Answer. All those piles under the foundations of the piers, which I ever saw, are very sound at heart. But about one inch of their surface hath been decayed : but these were piles which had been for some time exposed to the violence of the flood, by the breaches made in the sterlings. But I apprehend that cannot be the case with the piles which go farther under, or in the middle of the piers; because water cannot act upon them." " Query 10. What is the inside of the stone piers made of ? whether of the same sort of stone as the outside ; cut and laid regular, or only common rubble stones, laid in very bad mortar, as it is in Rochester-bridge r" " Answer. I have seen, in several breaches, the texture 120 THE PRINCIPLES OF BRIDGES. TRACT 3. of the piers : and by them it appears to me, that the insides of the said piers are filled with rubble ; and the external faces are formed with ashler laid in courses : but the rubble ap- pears to be laid with good mortar. " George Dance.'* TRACT III. EXPERIMENTS AND OBSERVATIONS TO BE MADE ABOUT LONDON BRIDGE. THIS is another of the papers, relating to the state of London bridge, bought at the sale of the late Mr. John Robertson's books. It appears to be an answer given to certain queries, addressed to the Royal Society from the Committee of Common Council of the City of London. This answer is signed by the President, the Vice-Presidents, and several other respectable members of the Royal Society ; viz. by Martin Folkes, esq. the president, and by Wm. Jones (father of the late Sir Wm. Jones), James Jurin, M. D., Geo. Lewis Scott, esq., Benj. Robins, esq., and John Ellicott, esq., all names highly respectable for their eminent scientific la- bours. Their report is in the following words: *' In order to answer the queries proposed by the Com- mittee, with regard to the alterations of London bridge, we apprehend it will be necessary, " 1st. To have an exact level taken, between some fixed point on the west side of London bridge, and another point on the east side of Westminster bridge ; as also, to take the like level between some fixed point on the east side of London brido^e, and another point at some convenient place about 2 ^lilcs below the bridge. " 2. To take the perpendicular height of each of those 4 TRACT 3. ON LONDON BRIDGE. 121 points jibove the surface of the river at low-water, and like- wise at every quarter of an hour before and after low-watfer ; and to observe the time when the low- water happens at those places; and the same for high- water. " 3. To take the height of the fixed point on the west side of London bridge, above the surface of the river, at the low still water, and high still water under the drawbridge, with the time of each. \ " 4. To take the height of the same point, above the sur- face of the river, just above the sterling, at the time of low- water below bridge. " 5. To take the depth of the water in all the gullets, or t least in that under the drawbridge, at the time of low still water. "6. To ascertain in how many of the arches the dripshot piles are driven ; how close together ; and how far the tops of them are below low still water mark. "7, To know particularly at what time the sterlings are first intirely covered, and when first intirely uncovered. ** 8. To know exactly the time of low and high water mark, and the height the water rises to, at the Nore, Graves- end, and Woolwich. " 9. That all the foregoing observations of the tides, be made at some one spring tide, and likewise at some one neap lime. Was signed, M. Folkes; Wm. Jones ; Jas. Jurinj Geo. L.Scott j Benj. Robins ; John ElUcott." ( 122 ) TRACT I^. ON THE CONSEaUENCES TO THE TIDES IN THE RIVER THAMES, BY ERECTING A NEW BRIDGE AT LONDON. BY MR. JOHN ROBERTSON. WHILE it was in cniitemplation to erect the new bridge over the river at Blackfriurs, there was much pubhc conversa- tion and speculation on the probable effects of such erection, relative to the tides in the river, and other matters connected with it. On this occasion, the magistrates of the city of London consulted many scientific men and practical engi- neers, touching those points. Among others, they requested the advice and opinion of Mr. John Robertson, then master of the Ro3'al Mathematical School in Clirist's Hospital, by a special letter from the Town Clerk, as follows. " To Mr. Rcbcrison at Chrisfs Hosjntal. *' Sir, The Committee of Common Council appointed to consider, whether the Navigation of the river Thames will in an}' and w})at n)anner be affected by a new Bridge, intend to meet at Guildhall, on Thursday the 12th instant, at 10 o'clock in the forenoon, and desire you will be so kind as to favour them with jour conipany at that time, in order to give them your opinion and assistance therein. I am. Sir, *' Your most obedient, humble Servant, " James Dobson." Town Clerk's Office, Guildhall, 5 Dec. 1754. //;. lioberlson^s Ansicer. " Before I deliver my opinion concerning the question proposed, 1 think it necessary to preuiise some few principles TRACT 4. NEW BRIDGES. l2$ relating to the Tides, and particularly those which affect the river Thames ; because a just solution to this question depends chiefly on the phenomena of the tides. *' 1 . It is now well known that the tides are regulated by the motion of the moon ; and that this planet takes something less than 25 hours,between the times of its departing from any me- ridian, to its return to the same; in which time she causes two floods and two ebbs ; so that in most parts of the earth there is a new time in every revolution of about 12 hours and a half. " 2. There is a flood tide which flows round the northern parts of Europe, and thence proceeds southward through the western ocean : a branch of this tide runs southward along the German sea, and makes high water to all the eastern coasts of Great Britain, in a successive order, in regard'to the time the moon has passed the meridians of those places: this branch of the tide runs but a little to the southward of the mouth of the river Thames. " 3. While the said branch is running down the German sea, the grand body of the tide is marching southward along the western coasts of Ii-eland,and thence flowing partly south- ward, partly south-eastward ; one branch runs up St. George's Channel, and another branch flows eastward, up the English Channel, and makes, in a successive order of time, the high waters upon all the southern coasts of England : this branch extends something to the northward of the mouth of the river Thames. " 4. Tile said tides, meeting near the mouth of the river Thames, contribute to send a powerful tide up that river; and so long as the said southern and northern branches con- tinue to flow, so long will the Avaters continue to accumulate at the mouth of this river, and make their way up it, in order 10 restore the waters to a level. " 5. The flowing of the tide up the river Thames is greater or less, in proportion only to the accumuhition of the waters at its mouth ; and therefore, in the common course of things, there is, relative to the moon's age, a fixed quantity of tide which the river Thames is to receive ; and therein to be 124 '' THE PRINCIPLES OF BRIDGES. TRACT 4- disposed of in the best manner that its situation will admit. ** 6. On account of the water being confined between the banks of the river, the tide must flow up higher, in propor- tion as the river becomes narrower, till the fixed quantity is received. But then it must be observed, that when the tide acts against the stream of a river, the tide up that river be- comes progressively stronger and stronger, for a time, ac- cording as the velocity of the natural stream is checked ; and in this manner the river waters themselves by degrees obtain a contrary direction, and run up with the tide, and so may be considered as waters coming in with the tide of flood, and part of the fixed quantity which that river is to receive. ** 1. The return of the tide, or the time of ebbing, is not every where performed in the same time as it took to flow in. For, in the ebb tide there is to be discharged, not only the waters which were brought in by the tide, but also all the river water which has been retarded by it. ** 8. Whatever obstacles are laid in the way of the tide, across any channel, the utmost rise, or the high-water mark, at different times, will be respectively the same : because tiie water will continue to rise till the fixed quantity of tide is disposed of, and no longer. And, in like manner, the low- water mark will not be affected by such obstacles. Indeed, between the limits of high and low-water marks, the water will be raised higher against those obstacles, both in the flood and ebb tides, than they would be in those places, were the obstacles removed. For, as the velocity of the current must, on both sides of the obstacle, be equal, in order for one part of the water to run away, as fast as the successive ones fol- low ; therefore the waters must rise on that side of the ob- stacle which they run against, till they be so high, that by their fall they acquire a velocity sufficient to carry them off, as fast as they arise at the obstacle. ** These principles being premised, the solution to the question proposed naturall3^ follows. And in order to this, let us for the present suppose, that betvreen London and TRACT 4. NEW BRIDGES. 12fJ Westminster bridges, another bridge were built ; and to show what might be the consequences in the worst case, let us sup- pose it occasioned as great a fall as at London bridge. " Consequences during the time of the Flood Tide. ** The flood tide, meeting with the obstruction of the new bridge, would accumulate on the eastern side thereof, much in the same manner as it does now at London bridge : this would cause the flood tide at first to run through London bridge with less velocity than it does at present. For, the new bridge, by penning up the water, would throw some of it back again, towards London bridge; and consequently the Waters on the eastern side of London bridge, would rise higher than they now do, that they might run off with the same ve- locity, with which they came to the bridge. " The tide would not run up the river so far as it now does ; and consequently the tide of flood would be sooner spent, than at present : nevertheless the rise of the waters would not, at any place, be lessened beneath the present standard. For, the more obstacles any moving body has to encounter with, the sooner will its motion be destroyed. But the fixed quantity of the tide being in no wise diminished, the waters must necessarily rise as many feet high, either above or below the bridges, as they would, were there no bridges over the river. " Consequences duri?ig the Tide of Ebb. '^ The ebb tide would be obstructed, on the western side of the new bridge, in the same manner as it is now at London bridge; but the rise of the water at the new bridge would be highest*. For, as London bridge, by penning up the water, * It is manifest that all this reasoning, by Mr. Robertson, we must remember, has been on die supposition, that the new bridge would be built with piers and sterlings, like London bridge, and so eause a similar obstruction to the currents. 126 THE PRINCIPLES OF BRIDGES. TRACT 4. would cause it, at tlic beginning of the ebb, to revert or fall back again towards the new bridge : consequently the waters on the western side of the new bridge must rise higher, on account of the |)en below, that they might run away as fast as they were succeeded by the following wuter. *' The length of the tide of ebb would be greater than it as at present, by as much time as the tide of flood would be shortened. For though the same quantity of flood tide, being- poured through London bridge, would spend its force sooner |JaaQ,t present, yet the time of the return of the aggregate of the flood tide, and the retarded land waters, Avould be greater ; in proportion as the obstacles, they would have to pass by, were increased. ** From what has been said, I apprehend it is evident, that a new bridge, built between London and Westminster bridges, cannot alter the present high and low- water marks ; even though this new bridge should be so constructed, as to oc- casion a fall of the waters, equal to what they have at London bridge. *' But experience has shown, how a bridge may be built, so as to cause no sensible fall : and were such a bridge sub- stituted in the place of that we have before supposed, the consequences already remarked would become so inconsider- able, in respect to the tides, that I believe, and it is my opi- nion, that there would ensue no apparent alteration in the present state r.f the navigation of the river Thames, either above or below London bridge." John Robertson. ( 127 > TRACT V. ANSWERS TO QUESTIONS, PROPOSED BY THE SELECT COM- MITTEE OF PARLIAMENT, RELATIVE TO A PROPOSAL FOR ERECTING A NEW IRON' BRIDGE, OF A SINGLE ARCH ONLY, OVER THE RIVER THAMES, AT LONDON, INSTEAD OF THE OLD LONDON BRIDGE. AMONG the various means of improving the port of Lon- don, which have lately been devised, was one by removing the old inconvenient London bridge, and erecting another in its stead, which might be more commodious, and better accord- ing with the improved state of the port. Several projects were given in to the Committee of Parliament, appointed to consider those improvements, among which was one pro- posed by Messrs. Telford and Douglass, to be of a single aixh, made of cast iron, which the Committee so far noticed, as to order engravings to be made of the design, and, for more safety, to issue a set of questions, concerning this ex- traordinary project, to be sent to several ingenious profes- sional and literary men, requesting their answers to all or any of them, within a limited time. The present tract contains my answers, which were de- livered in, to those questions, and for which I was honoured with the thanks of the Committee ; wliich answers are here given as a proper appendix, among other articles, to the essay on bridges in the first Tract. The situation proposed for this new bridge, is about 200 yards above the old bridge, v.hich brings it to run nearly in a line with the Royal Exchange, and with the wide part of the main street of the Borougli of Southwark. This is the narrowest part of the river, being here but 900 feet over. It was also proposed to narrow the river stili more in this 12 THE PRINCIPLBS OP BRIDGES. TRACT 5. part, by building strong abutments of masonry, running 150 feet into the river on each side, against which to abut the proposed arch of cast iron, vhich consequently was to be of 600 feet span, extending across the river at one stretch. The height of the arch at the crown or key piece, was to be 65 feet above high- water, to allow ships of considerable burden, with their top masts only struck, to sail through beneath it, up to Blackfriars bridge; to load or unload by the side of new wharfs, to be built into the river, on both sides of it, all the way up io Blackfriars. The width of the bridge, to be 45 feet in the middle, and from thence widening all the way, in a curved form, till it should become enlarged to 90 feet at the extremities. The letter of the Committee is here given first, with the set of questions, followed by the answers as delivered in con- sequence of that requisition. THE ORDER OF THE COMMITTEE. " Luna? 23 die Martii 1801, ** At the Committee for the further improvement of the Port of London ; " Charles Abbot, Esq. in the Chair : *' Ordered, That the Print, Drawings, and Estimates of an Iron Bridge, of a single arch, 600 feet in the Span, together with the annexed Queries, be sent to Dr. Hutton, requesting that he will, on or before the 25 th of April next, transmit to Mr. Samuel Gunnell, the Clerk to this Committee, his opinion upon all of these queries, or such of them as he may be disposed to consider. " Charles Abbot, Chairman. " To Dr. Hutton, ** Military Academy, Woolwich'^ tRACT 5. ARCH FOR LONDON BRIDGE, 129 " Estimate. . . 20,000 *' Gettirm out and securing: the foundation of the two abutments - - - - 432,000 cubic feet of granite or other hard stone 86,400 20,029 cubic yards of brickwork, at 20s. - - 20,029 19,200 cubic feet of timber in tyes, at 3s. 6d. 3,360 6,500 tons of cast iron, including scaffolding? -.^r. r.r.n. and putting up, at 20l. - - 3 Making roadways and footpaths - _ - - 2,500 ^26-2,289" *' 2uestio7is respecting the Construction of the annexed Plate and Drawings of a Cast Iron Bridge of a Single Arch, 600 feet in the Span, and 65 feet Rise. " 1. What parts of the bridge should be considered as wedges, which act on each other by gravity and pressure, and what parts as weight, acting by gravity only, similar to the walls and other loading, usually erected upon the arches of stone bridges. Or, does the whole act as one frame of iron, which can only be destroyed by crushing its parts ? " 2. Whether the strength of the arch is affected, and in what manner, by the proposed increase of its width towards the two extremities, or abutments ; Avhen considered verti- cally and horizontall3^ And if so, what form should the bridge gradually acquire? " 3. In what proportions should the width be distributed from the centre to the abutments, to make the arch uniformly strong ? "4. What pressure will each part of the bridge receive, supposing it divided into any given number of equal sections, the weight of the middle section being given. And on what parts, and with what force will the whole act upon the abut- ments ? VOL. I. K J30 A SINGLE ARCH TRACT 5, " 5. What additional weight will the bridge sustain ; and Avhat will be the effect of a given weight placed upon any of the before mentioned sections ? " 6. Supposing tlie bridge executed in the best manner, Avhat horizontal force will it require, when applied to any particular part, to overturn it, or press it out of the vertical plane ? " 7. Supposing the span of the arch to remain the same, and to spring ten feet lower, what additional strength would it give to the bridge. Or, making the strength the same, what saving may be made in the materials. Or, if instead of a circular arch, as in the plate and drawings, the bridge should be made in the form of an elliptical arch, what would be the difference in effect, as to strength, duration, convenience, and expences ? *' 8. Is it necessary or adviseable, to have a model made of the proposed bridge, or any part of it, in cast iron. If so, Avhat are the objects to which the experiments should be directed ; to the equilibration only, or to the cohesion of the several parts, or to both united, as they will occur in the in- tended bridge? ' 9. Of what size ought the model to be made, and what relative proportions will experiments, made on the model, bear to the bridge, when executed? " 10. By what means may ships be best directed in the middle stream, or prevented from driving to the side, and striking the arch, and what would be the consequence of such a stroke ? " 11. The weight and lateral pressure of the bridge being given, can abutments be made in the proposed situation for London bridge, to resist that pressure? *' 12. The weight and lateral pressure of the bridge being given, can a centre or scaffolding be erected over the riv(?r, sufficient to carry the arch, without obstructing the vessels wiiich at present navigate that part? " 13. WJiether it would be most adviseable to make the bridge of cast and wrought iron combined, or of cast iron TRACT 5. FOR LONDON BRIDGE. 131 only. And if of the latter, whether of the hard white metal, or of the soft grey metal, or of gun metal? " 14. Of what dimensions oug-ht the several members of the iron work to be, to cfive the bridge sufficient strensth? " 1 5. Can frames of cast iron be made sufficiently correct, to compose an arch of the form and dimensions as shown in the drawings No. 1 and 2, so as to take an equal bearing as one frame ; the several parts being connected by diagonal braces, and joined by an iron cement, or other substance ? N. B. The plate is considered as No. 1. " 16. Instead of castino- the ribs in frames, of considerable length and breadth, as shown in the drawing, No. 1 and 2, would it be more adviseable to cast each member of the ribs in separate pieces of considerable lengths, connecting theui together by diagonal braces, both horizontally and vertically, as in No, 3 ? " 17. Can an iron cement be made, which shall become hard and durable. Or can liquid iron be poured into the joints? " 18. Would lead be better to use in the whole or any part of the joints? " 19. Can any improvement be made in the plan, so as to render it more substantial and durable, and less expensive. And, if so, what are those improvements? " 20. Upon considering the whole circumstances of the case, and agreeable to the resolutions of the Committee, as stated at the conclusion of their third report : Is it your opinion, that an arch of 600 feet in the span, as expressed in the drawings produced by Messrs. Telford and Douglass, or the same plan, with any improvements you may be so good as to point out, is practicable and adviseable, and capable of being made a durable edifice ? " 21. Does the estimate communicated herewith, accord- ing to your judgment, greatly exceed or fall short of the probable expence of executing the plan proposed, specify- ing the general grounds of your opinion ? " The Resolutions referred in No. 20, are as follow, k2 132 A SINGLE ARCH TRACT 5. " Ist. That it is the opinion of this Committee, that it i:* essential to the improvement and accommoJation of the port of Loudon, that London Bridge should be rebuilt, upon such H construction, as to ]iermit a free passage at all times of the tide, for shij)s of such a tonnage, at least, as the depth of the river would admit of, at present, between London Bridge and Blackfriars Bridge. "2d. That it is the opinion of this Couiniittec, that an Iron Bridge, having its centre arch not less than 65 feet high in the clear, above high-water mark, will answer the intended purpose, and at the least expence. " 3d. That it is the opinion of this Committee, that the most convenient situation for the new bridge, will be imme- diately above St. Saviour's Church, and upon a hue from thence to the Royal Exchange. " Charles Abbot. " To Dr. llution, IVoolnnchy The Answers to the foregoing Queries, were as follow; where each question is repeated immediately before its an- swer, to preserve the connection more close and imme- diate. jinswers to the Questions coucery^ing the proposed New Iron Bridge y of one arch, 600 feet in the span, and 65 feet high. Quest. 1. What parts of the bridge should be considered as wedges, which act on each other by gravity and pressure, and w hat parts as weight, acting by gravity only, similar to the walls and other loading usually erected upon the arches of stone bridges. Or, does the whole act as one frame of iriiii, which can only be destroyed by crushing its parts ? jiiisii-er. It "is my opinio!!, that all tlie small frames or parts (Hight to l)e so connected together, at least verticallv, as that the whole may act as one iVame 'of iro!i, which can o'l-: I);- (Us;ro\c(i by crushing its parts, For, by t'jis niean.', the pressure and strain will be taken otH'from every particular TRACT 5; FOR LONDON BRIDGE. 133 arch or course of voussoirs, and from every single voussoir or frame, and distributed uniformly throughout the whole mass. Hence it will happen, that any particular part which may by chance be damaged, or be weaker than the rest, will be relieved, and prevented from a fracture, or, if broken, prevented from dropping out and drawing other parts after it, which may be next to it, either above or on the sides of it. By this means also, the effect of any partial or local pressure, or stroke, or shock, whether vertical or horizontal, will be distributed over or among a great number of the ad- jacent parts, and so the effect be broken and diverted from the immediate y^lace of action. By this means also Avill be obviated, any dangerous effects arising from the continual expansion or contraction of the metal, by the varying tem- perature of the atmosphere, in consequence of which the bridge will, all together, in one mass, in a small and insensible degree, keep perpetually and silently rising or sinking, as the archlengthensbythe expansion, or shortens by the contraction of the metal. This unity of mass will be accomplished, by connecting the several courses of arch pieces together verti- cally, or the lower courses to the next above them, and also by placing the pieces together in such a way as to break ioint, after the manner of common or wall masonry, and that perhaps in the longitudinal and transverse joints, as well as the vertical ones. Quest. 2. Whether the strength of the arch is affected, and in what maimer, by the proposed increase of its width towards the two extremities, or abutments; when considered vertically, and horizontally ; and if so, what form should the bridge gradually acquire ? Answer. There can be no doubt but the bridge will be greatly strengthened by an increase of its width towards the two extremities, or abutments, especially if the courses or parts be connected together in the manner above mentioned, in the answer to the first question. For thus, the extent of the base of the arch at the impost being enlarged, the strength or resistance of the abutment will be increased in a much higher degree than the weight and thrust of the arch, and I54f A SIKGLE ARCH TRACT 5y^ consequently will resist and support it more firmly. The arch iiaelf will thus also acquire a great increase of strength and stability, both from the qnaniity and disposition of the materials, as well vertically as horizontally, by which, in the latter direction in particular, the arch will be better enabled to preserve its true vertical position, and to resist tne force or shock of any thing striking against it in the horizontal di- rection. And, for the better security in these particulars, considering the immense stretch of the arch, it will perhaps be adviseable to enlarcre the width in the middle to 50 feet^ instead of 45, and at the extremities to 100 feet, instead of 90, as proposed in the design. As to the form of this width or enlargement, the side of the arch might be bounded either by a circular arch, or by any curve that will look most grace- ful : perhaps a verj' exccntrie ellipse will answer as well as any other curve, or better. Quest. 3. In what proportions should the weight be dis- tributed from the centre to the abutments, to make the arch uniforaily strong ? Anszi'er. To make the arch uniformly strong throughout, it ought to be made an arch of equilibration, or so as to be equally balanced in every part of its extent. When the ma- terials of the arch are uniform and solid, then, to find the M'eight over every part of the curve, so as to put the arch in equilibrio, is the same thing as to find the vertical thickness of the arch in every part, or the height of the extrados, or back of the arch, over every point of the intrados or soffit of the under curve of the arch : the rule for determining and pro- portioning of which, is described at large in my Treatise on Bridges, particularly in prop. 4"*^, and the examples there given to the same. But in the case of the present proposed design for a bridge, a strict mathematical precision is not to be ex- pected or attained by . "r? calculation, on account of tlie open frame Wi\k of iron, in parts of various shapes and sizes. We must therefore be c":-,tent with a near approach to that point of perfection ; which can be acconqdished in a deurc? * The same as prop, lo, tract l, of this volume. 'rRACf 5. FOR LONDON BRIDGE. 135 sufficient to answer all the purposes of safety and conveniencCi Now this can be conveniently done, by a comparison of the present design of a bridge, with the example of a similar in- trados curve in the book above mentioned, and which is the case of the first example to the said 4th prop., being that with a circular soffit. By that example it appears, that the weight above every jpoint in the soffit curve> should increas. exactly in proportion as the cube of the secant of the number of degrees in the arch, from the centre or middle, to the se- veral points in going toward the abutments* This propor-* tion, though it require an infinite weight or thickness at the extremities of a whole semicircle, where the arch rises per- pendicular to the horizon ; yet for a small part of the circle near the vertex, the necessary increase of weight or thick- ness, toward the extremities, is in a degree very consistent with the apnvenient use and structure of such a bridge ; us will be evident by a glance of the figure and cr.rve to that example. For, as the whole extent of the soffit arch, in the present design for an iron bridge, is but a'jout 43" 54', or 24 27' on each side, from the middle point to the abutments, that is, little more than the fourth part of the arch in that example; tiierefore, by cuctuig out the fourth part of that arch, it wiil giVe us a tolerable idea of the requisite shape of the whole structure, and increase in the thickness where the materials are solid, or at least the increase in weight over every yjoint in the soffit ; that is, the figure exhibits a curve for the scale of such increase. Or, if we compute the numeral va- lues of the weights or thickness, by the rule in that example, in the pro- poriio.) of t!ie cube of the secants, they will be as in the annexed tablet; which is compuicd for every degree in the arch, Dtf Wt;or height Dee. Wt.orlifit,i 10-000 IS 10-81U 1 10-000 14 10-947 2 10-018 15 11-096 3 10-041 16 11-25S 4 10-073 17 11-434 5 10-115 18 11-625 6 10-166 19 11-831 7 10-227 20 12-052 8 10-298 21 12 J90 9 10-:'79 22 12-546 10 10-470 23 12-821 11 10-5-72 24 13-116 12 10-685 244- 13-272 136 -A SINGLE ARCH TRACTS. from the middle, supj)osinn; the middle thickness or weight to be 10. And the true representation of the figure, as con- structed from these nuntbers, or the extrados curve deter- mining the true scale of weight or thickness, over everv such point in the soffit curve, is as is here exhibited below. Where the thickness or hciglit in the middle being supposed 10, the vertical thickness or h.eight of the outer curve, above the inner, at the extremities, is 13-272, or nearly 13^, and the other intermediate thicknesses, at every degree from the ver- tex, are as denoted by the numbers in the latter column of the table. If the thickness at top be supposed 7, or 8, or 12, or any other number, instead of 10, all the other numbers must be changed in the same proportion. Now the upper curve in tliis figure is constructed from these computed tabular numbers, and exhibits an exact scale of the increase of weight or thickness, so as to make the whole an arch of equilibration, or of uniform strengtli throughout, when the materials are of uniforni shape and weight. And in this case the upper curve does not sensibly differ from a circular arc in any part of it. But, ar, the convenient passage over the bridge requires that the height or thickness at the extremities, or imposts, should he a gr{ at deal more than in proportion to these numbers denoting the equilibrium of weight, it therefore follows, that the frame work of the pieces above the arch, in the filling up of the flanks, ouglit to be lighter and lighter, or cast of a form more and more light and open, as in the engraved design, so as to bring the loading in those parts as near to the equilibrium weight, as the strength and stability of the iron frames will permit. Quest, l. \Vli;it pressure will eacli ])art of the bridge re- ceive, supposing it (iividcd into any given number of equal sections, the wcigh.t of the midLlle section Oeing given ; and TRACT 5. FOR LONDON BRIDGE. 13T on what part, and with what force, will the whole act upon the abutments ? Ans'dver. By the equal sections, mentioned in this question, may be understood, cither vertical sections of equal weight, or those perpendicular to the curve of equal weight, or of equal length ; and whichever of these is intended, their thrust or pressure in direction of the curve may be easily computed, if wanted for the purpose of making experiments on the strength of the frames, to know whether they will bear those pressures, or what degree of pressure they will bear, without being crushed in pieces. But as it is evident that the frames next the abutments will suffer the greatest pressure of any, I shall here give a computation of the actual pressure there, which may be sufficient, since if the frames at the abutments are capable of sustaining that greatest pressure, we may safely conclude, that all the others, from thence to the vertex, w'll be more than capable of sustaining the lesser loads or pres- sures to which they are subject ; and this computation will answer the latter and most essential part of the question, viz. " on what part, and with what force, will the whole act on tlie abutments." Now, from the nature of an arch, it appears that the whole pressure on the abutments, will be chiefly on the lower part of the impost, where the lower frame rests on it, and where we shall therefore, in our computation, suppose it to act. And in the calculation, the whole weight of the half arch ao must be supposed united in its centre of gravity N. Then, if a vertical line mn be drawn tiirough the centre of gravity n, by computation it is found that dm is nearly equal to 160 feet, and consequently me equal to 140 feet: also, if NO be perpendicular to the impost, or in the direction of the arch at oe ; v.e shall have this proportion, viz, as mn (60), is to the weight of the half arch (3250 tons), so is NO (152), to the pressure on the impost in the direction of the arch at o, and so is me (140), to the horizontal thrust or pressure in, the direction me ; this gives 8233 tons for the pressure on the impost at o '\^ direction of the arch, and 7583 tons for the horizontal thrust in direction me j benig ISS A SINGLE ARCH TRACT 0^ the pressures at each end of the bridge. M'c may therefore estimate the greatest pressure on the last or abutment frame, at about 8 or 9 thousand tons. Quest. 5. What additional weight will the bridge sustain, and what will be the effect of a given weight placed upon any of the before mentioned sections ? Answer. It is perhaps not possible to pronounce exactly what additional weight the bridire will sustain, withovit break- ing, as it depends on so many circumstances, some of which are not known. But, considering the great dimensions and strength of the arch frames, and of the whole fabric, we arc authorized to conclude, that there is no possible weight which can pass over any part of the bridge, even heavy loaded wag- gons, whose pressure can be great enough to cause any dan- ger to such strong and massy materials, and especially when it is considered that, by connecting all the frames together, by proper bond and otherwise, as mentioned in the answer to the first question, the local additional pressure will soon be distributed through the whole series of the iron framing. Quest. 6. Supposing the bridge executed in the best manner, what horizontal force will it require, when applied to any particular part, to overturn it, or press it out of tlie vertical plane ? Answer. This question will be much better answered by means of experiments, made on a proper model, than b}' theoretical calculations a priori. But when the bridge is executed in the best manner, with the frames properly bonded and connected together, it seems more likely that any violent horizontal shock, such as a ship driving against it, would break any particular frame, rather than overturn such a mass of bonded materials, or even move it sensibly out of the ver- tical position. QuKST. 7. Supposing the span of the arch to remain the same, and to spring ten feet lower, what additional strenL',t]i would it give to the bridge. Or, making the strength the same, what saving may be made in the materials. Or, if instead of a circular arch, as in the plate and drawing^;, the TRACT 5. FOR LONDON BRIOGE. 189 bridge should be made in tbe form of an elliptical arch, what would be the difference in effect, as to strength, duration, convenience and expence? jiiiswer. Should the arch spring ten feet lower than in the design, the bridge would be more stable, because the thrust or pressure on the abutments would be directed lower down, and more into the solid earth: and in general, the lower the springing of the arch, the more firm the abutments and stable the bridge, if the height of the crown above the springing of the bridge be the same. But the greatest ad- vantage would be, by making the bridge in the form of an elliptical arch, instead of the circular one, in all the articles of strength, duration, convenience, and expence. For, as the elliptical flanks require less filling up than the circular, this will produce a great saving in the iron frame work : and this suuie reduction of materials in the flanks, toward the abutments, is the very cause of greater strength, by reducing the weight there nearer to the case of equilibration ; since that very extraordinary mass employed in the flanks of the circular arch destroys the equilibrium of the whole, by an overload in that part. The elliptical arch will be also much more convenient, as it will allow^ of a greater height of navi- gation Avay betv.cjn the water and the soflEit of the arch. The elliptical arch is also a much more graceful and beautiful form than the circular arch. Quest. 8. Is it necessary or adviseable, to have a model made of the proposed bridge, or any part of it, in cast iron. If so, what are the objects to which the experiments should be directed ; to the equiHbration only, or to the cohesion of the several parts, or to both united, as they will occur in the in- tended bridge ? Ansuier. It appears to be very adviseable, to have a model made of the whole of the proposed bridge, in cast iron, as well for the greater safety and satisfaction, as for the benefits and improvements to be derived from the experiments to be made with it, and from the experience and knowledge de- 140 A SINGLE AllCH TRACT 5. rived from the casting and making it. Tiie objects to which the experiments should be directed, might be, the equilibrium of the whole, the cohesion and fitting of the several parts, the effects of a vertical load on every part separately, aiid the effects of a horizontal blow or shock against every part in the side of the arch. Also wliat weight would be requisite to break or to crush the model frames. Quest. 9. Of what size ought the model to be made, and what relative proportions will experiments, made upon the model, bear to the bridge, when executed ? Ansxver. The greater the size of the model, the more sa- tisfactory the experiments and conclusions will be. For this purpose, it seems adviseable, that the model be not less than the 20th part or dimensions of the bridge, that is, of 30 feet in length. Now, as the solid contents of similar bodies arc in the same proportion as the cubes of tlieir linear dimen- sions, such a model would require only the 8 thousandth part of the weight or metal in the bridge, because the cube of 20 is 8000. So that, as it is estimated the bridge -will require 6500 tons of metal, it follows, that about 3 quarters of a ton weight of metal will suffice for the model of 30 feet in length. As to the relative proportions of experiments made with the model: those relating to the equilibrium, will be in the same direct proportion with the masses of the model and bridge , as well as those relating to loads or shocks. But the strength of any particular bar or frame will be only as the square of the scantling, while the stress upon it will be barely in the same proportion as the length. Quest. 10. By what moans may ships be best directed in the middle stream, or prevented fronj driving to the side, and striking the arch; and what would be the consequence of such a stroke? Ansu'er. Some kind of fences migiit be placed in tiie river, to direct the navigati(Mi to the projier opening in the uiiddle. The ciR-ct of the stroke or shock of a vessel, strikin-;- the side of the bridge, if very heavy, might endanger the breaking TRACT 5. FOR LONDON BRIDGE. 141 of the particular frame or bar so struck. But, the whole being well bonded and connected together, none of the others would probably be displaced. Quest. 11. The weight and lateral pressure of the bridge being given, can abutments be made in the proposed situa- tion for London bridge, to resist that pressure ? Answer. No doubt of it ; and especially if the courses of masonry have the joints directed towards the centre of the arch. Quest. 12. Tiie weight and lateral pressure of the bridge beinof oiven, can a centre or scafFoldins be erected over the river, sufficient to carry the arch, without obstructing the vessels which at present navigate that part ? Ansxi^er. I doubt not that the requisite centring or scaffold- ing can be erected, without obstructing the present naviga- tion. Quest. 13. Whether it would be most adviseable to make the bridge of cast iron and wrought iron combined, or of cast iron only ; and if of the latter, whether of the hard white metal, or of the soft grey metal, or of gun metal? Answer. It appears most adviseable to make the bridge of cast iron onlv, and that of the soft gre}' metal, the bars and frames of which will be less liable to fracture by a blow or shock, than the hard metal. The mixture of wrought iron with the cast metal, would be very improper, as the sorts are of unequal expansion and contraction hv heat and cold, and as the several arch frames should not be tied or bolted together, but suffered to have a little play lengthways, in their butting grooves, so as that no one part be more confined than another. Quest. 14. Of what dimensions ought the several mem- bers of the iron work to bo, lo give the bridge sufficient strength ? Answer. This question will be best answered hy experi- ments made on the metal. Quest. ! 5. Can frames of cast iron be made sufficiently correct, to ccmpose an arch of the form and dimensions a* 142 A SINGLE ARCH TRACT 5. shown in the drawings No. 1 and 2, so as to take an equal bearing as one frame, the several parts being connected by diagonal braces, and joined by an iron cement, or other sub- stance ? N. B. The plate is considered as No. 1. Answer. There can be no doubt that cast iron frames may be made sufficiently correct to compose an arch of any form whatever, and give them an equal bearing ; because the wooden moulds, from which the metal is cast, can be made or cut to any shape desired. Quest. 16. Instead of casting the ribs in frames, of con- siderable length and breadth, as shown in the drawing No. 1 and 2, would it be more adviseable to cast each member of the ribs in separate pieces of considerable lengths, connect- ing them together by diagonal braces, both horizontally and vertically, as in No. 3 ? Answer. It is, in my opinion, better to cast the ribs in frames, of considerable length and breadtli. Quest. 17. Can an iron cement be made, which will become hard and durable, or can liquid iron be poured into the joints ? Quest. 18. Would lead be better to use in the whole, or any part of the joints ? Answers to Questions 11 and 18. The joints might either be filled with an iron cement ; or liquid iron might be poured into the joints, having a furnace near at hand for that pur- pose ; or, melted lead may be run in, which will be best of all ; because, being a soft metal, it will yield to, and accom- modate itself to the inequalities of jjressure or of shape, forming a sound and soft bond or bearing between frame and frame ; and preventing their fracturing each other by a too hard and unequal bearing ; in some respect perforniiiig the same office as the cartilages between the joints of the bones n\ tlic animal frame. (^UEST. 19. Can any improvement be made in the plan, so as to render it more substantial and durable, and less ex- pensive. And if so, what are those improvements ? TRACT 5. FOR LONDON BRIDGE. 143 Answer. Although the plan appears to possess a very ex- traordinary degree of excellence, I am of opinion, that it is not incapable of some further improvements, so as to render it more substantial and durable, as Avell as less expensive. The circumstances which, it appears to me, would be im- provements, are as follow : 1st. To make the vertical arch or curve of the bridge elliptical, instead of circular ; which will be an improvement in stability, in convenience, in beauty, and in saving ex- pence. 2d. To make the width of the bridge 50 feet in the middle, and 100 feet at the extremities; which will add greatly to its stabihty and security. 3d. To make the thickness of the arch at the crown, or the height of the middle or key frame there, to be not less than 10 or 12 feet, inste;id of 6 or 7 as proposed ; because, in so extended and massy a fabric, that seems to be the least thick- ness that can afford a rational ground for security and sta- bility. 4th. I would tie or connect every course of frames to those next above them, so as that the whole bridge may rise or settle together as one mass, by expansion or contraction. Yet I would not tie or bolt the frames together lengthways, but W'Ould simply make the edge, or the tenons, of the side of each frame, fit into the groove or the mortice holes of the next, going into each other two or three inches ; by which means the arch frames will always sit or fit close together, in every degree of temperature, without straining or tearing asunder at the ties. 5thly. I would place the frames of the whole fabric so to- gether, as to make a proper bond, in the manner of good masonry, by making them all to break joint both longitu- dinally and transversly : by which means, everv shock or pressure on any part, would be broken and divided, or shared, among a great many, and any openings be prevented, which might arise from the manner of placing the frames tvjth straight joints continued quite through. 144 HISTORY OF TnACT 6. Quest. 20. Upon considering- the whole circumstances of the case, and agreeable to the resolutions of the Select Committee, as stated at the conclusion of their Tliird Re- port, Is it your opinion that an Arch of 600 feet in the span, as expressed in the drawings produced by Messrs. Telford and Douglass, or the same j)]an, with any improvements you may be so good as to point out, is practicable and adviseable, and capable of being rendered a durable edifice ? Annucer. On considcrins the Avhole circumstances of the case, It is my opinion, that an Arch of 600 feet in the span, as expressed in the drawings produced by Messrs. Telford and Douglass, especially when combined with the improve- ments above mentioned, is practicable and adviseable, and capable of being rendered a durable edifice. Charles Hutton. Woolwich, April 21, 1801. TRACT VI. HISTORY OF IRON BRIDGES. A General History of all Arches and Bridges, both an- cient and modern, and constituted of either wood, or stone, or iron, would be a very curious and important Avork. It should contain a particular account of every circumstance re- lating to them : such as their history, date, place, artificer, form, dimensions, nature, properties, &c. Such a work, in a chronological order, would make a considerable volume, and much too large to form a part of the present work. I con- fine my views, therefore, in the present Tract, to a short account of the novel invention of Iro;; Bridges, in several instances that have recently been xecuted or proposed ; some iew of which have been lately noticed in the new edition of Dr. Recs's Encyclopedia. TRACT 6. IRON BRIDGES. 145 Bridges of cast iron appear to be the exclusive invention of British artists. The first that was executed on a large scale, is that on the river Severn, at Colebrook Dale, which was erected in the year 1779, by Mr. Abr. Darby, iron- master at that place. This bridge is composed of five ribs; and each rib of three concentric rings or circles, which are connected together by radiated pieces. The inner I'ing, of each rib, forms a complete semicircle : the others only seg- ments, being terminated and cut off at the road-way. These rings pass through an upright frame of iron, which stands on the same plate as the ribs spring from ; which not only acts as a guide to the ribs, but also supports a part of the road- Avay. Between the inner upright of this frame and the outer ring of the ribs, in the haunches, is a circular ring of iron, of about 7 feet diameter ; and between the outer upright of the frame, and the ribs, are two horizontal pieces, which act as abutments between the stonework and the ribs. There are also two diagonal stays, to keep the ribs upright. The roadway is covered with cast iron plates ; and it has an iron railing on each side. The inner or under ring, of each rib, is cast in two pieces, each of which is about 78 feet in length, VOL. I. L 146 HISTORY OF TRACT 6, the arch being 100 feet 6 inches span : and the whole of the ivon in it weighs 178 ^ tons. Whoever judiciously examines the construction of this bridge, will see, that its fame has arisen chiefly from the cir- cumstance of its having been the first of the kind : for the construction is very bad. The cast iron indeed is in the best state of preservation; but the stone-work has cracked in several places. It is probable, therefore, that its duration will not be long ; though not from any deficiency in the iron-work. The second iron bridge which has come to my knowledge^ is that which was designed by the noted Mr. Thomas Payne. This arch was set up in a bowling-green, at the public- house called the Yorkshire Stingo, at Lisson-Green, in the year 1790. This bridge was intended to be sent to America ; but, owing to Mr. Payne's being unable to defray the ex- pense, the arch was taken down by Messrs. Walker of Ro- therham, the persons who made it, and some of the materials were afterwards employed in the bridge at Wearmouth and Sunderland, next following. The third iron bridge that has come to our knowledge, was that executed on the river Wear, at Sunderland, by Rowland Burdon, Esq. M. P. for the county of Durham, by the assistance of Messrs. Walker tlie founders, Mr. Wilson, and several other persons : and for erecting bridges on simi- lar principles, the first gentleman took out a patent in the year 1794. This bridge was begun in the year 1793, and completed in August 1796. The stone abutments are 70 feet high, above the ordinary surface of the low-water in Sunder- land harbour, to the spring of the arch. The iron arch is 2?jG feet span; and the springing stones project about 2 feet beyond the face of the masonry: so that the whole span, from abutment to abutment, is 240 feet. The versed sine of the arch is 30 feet : its soffit is therefore 100 feet above the surface of low-water in Sunderland harbour. The arch is composed of 6 ribs ; and each rib of 3 con- centric tings, or seguicnts of circles. Each ring is 5^ inches Tract 6. IRON BRIDGES. 147 deep, by 4f inches thick ; and these rings are connected by radii, 4f inches by 2i ; the rings being at such a distance from each other, as to make the whole depth of a rib 5 feet. The ribs are composed of pieces of about 2^ feet long \ and worked iron bars are let into grooves in the sides of the rings, and fastened by rivets. These ribs are connected transversely by hollow iron tubes, or pipes, with flanches on their ends, and fastened to the ribs by screw-bolts : there are also diagonal iron bars, to prevent the ribs from twisting. The haunches are filled with circular rings ; and the top is covered with a frame of wood, and planked, to sustain the roadway. It has also an iron railincr on each side. The construction of this bridge is thought to be superior to that at Colebrook Dale ; and its weight is much less, in proportion to the length, the whole being only 250 tons, of which 210 tons are cast iron, and 40 tons of worked iron. Yet it is considered in no small danger of falling, the arch having settled several inches, as well as twisted from a straight direction, and the whole vibrating and shaking in a remark- able manner in passing over it. The fourth iron bridge that has been executed, is that over the river Severn at Buildwas, about 2 miles above Colebrook L2 H* HISTORY OP TRACT 6. Dale. It was begun in the year 1795, and finished in 1796, the iron work by the Colebrook Dale Company, under the direction of Mr. Thomas Telford. The arch is 130 feet span, with a versed sine or height of only 17 feet ; and it is but 18 feet wide to the outside. This bridge seems to have been contructed on the principle of the famous Avooden bridge at Schaufhausen. The ribs under the roadway are segments of a large circle, each cast in two pieces : but, on each side of the railing, there is a rib, cast in 3 pieces, which springs from a base, 10 feet lower, then crosses the others, and rises as high as the top of the railing : and from the upper part of these outer side ribs, the other ribs, which bear the covering plates, are suspended by king-posts : the covering plates, which are 46 in number, each extending quite across the bridge, have flanges 4 inches deep, and act as an arch. The outside ribs are 18 inches deep, and 2| inches thick ; the middle ribs 15 inches deep, and 2^ thick; and the whole weight of iron is about 174 tons. Perhaps this may not be the most favourable construction that might be contrived : the tendency of the rib aa, when it expands, being to raise the ribs BB a little higher than they TRACT 6. IRON BRIDGES. H9 would by their own expansion, and to depress them lower when it contracts: which is not the case in a wooden bridge, this material not being so affected by heat and cold. About the same time as the bridge at Buildwas was erected, an iron bridge was thrown over the river Tame in Hereford- shire ; but its parts were so slender, and so ill disposed, that no sooner was the wooden centring taken from under it, than the whole gave way, and tumbled into the river. In the same year also as the Buildwas bridge was begun, another was erected by the Colebrook Dale Company, over the river Parret, at Bridgewater. The arch of this bridge is an ellipsis of 75 feet span, with 23 feet rise. The haunches are filled with circular rings of iron, and other fanciful figures : it is composed of ribs connected together by cross ties of iron ; and the roadway is supported by plates. This bridge is very neat, and thought to be exceedingly firm and durable. From the completion of the above bridge, few of any note were executed in this country, till about the year 1800, when the stone bridge erected over the Thames, at Staines, gave way. On this occasion the magistrates of the counties of Middlesex and Surrey came to a resolution to erect an iron bridge there, on the abutments of the stone bridge, the piers of which had failed ; and Mr. Wilson, the agent of I\Ir. Bur- don, was employed for this purpose. He accordingly under- took the construction of an iron arch of 181 feet span, with 16f feet rise or versed sine ; the arch being the segment of a circle. In this bridge the ribs were similar to those of Wear- mouth : but instead of having the blocks, of which the ribs are composed, kept together by worked iron bars, let into grooves in their sides, the rings of the ribs were cast hollow, and a dowel was let into the hollow ring at each joint; so that the two adjacent blocks were fixed together by this dowel, and by keys passing through the rings. The ribs were also connected transversely by frames, instead of pipes as in the Sunderland bridge. The haunches were filled with iron rings, and the whole was covered with iron plates. 150 HISTORY OP TRACT 6, It is to be noted, that an iron arch, in small blocks, is not set up after the manner of a stone one, by beginning at the abutments, and building upwards; but is begun at the top, and continued downwards; it being easier to join the stone totthe iron, than to cut the iron at the top, if it should not fit. It is somewhat remarkable, therefore, that when these ribs were put together, and before they joined the masonry, it was so nicely balanced, and its parts were so firmly locked together, that after all the supports were taken out, except those next the abutment, the whole was moved by a man, with a crowbar under the top, and it seemed to have little tendency to push the abutments asunder. This, however, turned out unfortunately not to be the case. The centring was taken away, and the bridge was opened for the use of the public, about the end of the year 1801, or beginning of 1802. At first it seemed to stand firm, and tiie public were much pleased with its light and elegant appearance. But in a short time it was found that tlie arch was sinkino: : and soon after it had gone so much, that it was obliged to be shut up, and the old bridge opened again. The sinking of the arch broke several of the transverse frames, and many of the radii at the haunches ; which left no doubt that the abutments had given way. But on examination there appeared no visible sign of such failure : there was not a crack in the masonry, nor had they gone out of the upright, After much investi- gation however, it appeared that the whole masonry of the abutments, to the very foundation, had slidden horizontally backwards, still preserving the perpendicular, or upright position. The failure took place in the south abutment, ^vhich was supposed to be owing to a cellar, that had been made in it. The inhabitants of Staines therefore, by the ad- vice of an engineer whom they consulted, had this abutment strenfithened : but no sooner was this done, than the north one failed : and they had intended to strengthen this also ; but their funds being nearly exhausted, they came to the re- solution to take the whole doAvn, and erect a wooden bridge jn its stead. TRACT 6. IRON BRIDGES. 151 Before the completion of the iron bridge at Staines, another was begun of the same dimensions, and on the same principle, over the river Tees at Yarm. This bridge was completed also : but, instead of gradually yielding, as that at Staines had done, the whole suddenly tumbled into the river at once. From the accidents above described, and from several others of less note, iron bridges have lost a good deal of their cele- brity, but probably on no just grounds. Those failures that have happened, have not been through any intrinsic defi- ciency in the iron material, but from the injudicious manner in which they have been constructed. An opinion has gone forth, not only among the practical builders of iron bridges, but among some men of science, that the lateral pressure of iron bridges, in consequence of their parts being so firmly bound together, is comparatively small, to that of stone arches. But, on a due consideration of their principle, I believe it will be found quite different, and that an iron arch, of the same weight as one of stone, requires much stronger abutments, to resist its lateral pressiire or push, than the stone arch does. And this we shall here endeavour to account for. Stone may, in a great measure, be considered as an un- elastic substance, being very little subject to expansion or contraction. When, therefore, an arch is composed of this material, and the abutments are sufficiently strong, to support it, when left to itself, there is little probability of its failure. No ordinary load upon it will excite a tremulous motion ; nor will it change b}^ heat or cold. The lateral pressure on the piers or abutments is therefore uniform. But iron is an elastic substance, and is greatly affected by heat or cold, expanding with the one, and contracting by the other. When, therefore, a heavy load acts upon an iron bridge, such as a loaded waggon, the whole is put in motion, and the arch vibrates like the string of a violin, contracting and expanding while its parts are in the act of vibration. Thus at one part of the vibration it pulls the abutments to- 152 HISTORY OF TRACT 6, gether, and at the other it pushes thern asunder, with a force compounded of>the quantity of niattcr in motion, and the Te- locity with which it moves. When it exj)ands, the uliole Aveightof the arch is raised, abd the pressure on the abut- ments is compounded of the matter and velocity of the weight raised. No such pressure, or rather impulsive momentum, takes place in a stone bridge : therefore the strength of the abutments of an iron bridge should be such, as not only to sustain the weight of the arch, but also the additional push arising from the causes above stated. The abutments of Staines bridge were only 1-1 icet thick ; whereas they ought to have been at least 25 feet. Tfierc were also other causes which contributed to the failure of this bridge, such as the improper manner in which the foundations were made. The abutments of Yarm bridge were made still weaker than those of Staines : no wonder, therefore, that its failure was more sudden. I am therefore most decidedly of opinion, from what has happened in the bridges above described, and in several others, that no part of the failure is attributable to the iron material, at least respecting its strength. I do not however mean to say, that iron is generally to be preferred to stone: on the contrary, I tiiink a stone bridge is preferable to an iron one, when it can be executed with propriety and conveniency. But there are many cases where stone would not answer the purpose; in which cases therefore iron is most valuable. The cases here chiefly alluded to, are when the foundations cannot be made within the width that a 5-tone arch can with convenience be erected ; or when the requisite rise would be very inconvenient for a stone bridge, or in places where stone cannot easily be procured. The bridge atWearmouth is an example of the former, as stone juers would have very much obstructed the navigation of the river ; and of the latter, as the arch is a segment of a circle of about 500 feet diameter. The bridge at Boston, in Lincolnshire, is another exam})ie, though of less extent: the banks of the Witham are very low, and the houses are built close to the river ; the rise of tide is TRACT 6. IRON BRIDGES. IS$ great, and barges navigate under it : therefore, to render the access easy over the bridge, it became necessary to make it flat ; and to admit of headroom under the arch, flatness again was necessary. This bridge was therefore made of cast iron. Its span is 86 feet, and its versed sine only 5^ feet. The abutments have been well secured ; and though many of the radii of the ribs broke, when the pavement was put on it, yet the rings are quite entire, and the bridge is as firm as can be wished. In the course of the late improvements in Bristol harbour, two handsome cast iron bridges were erected over the New River there, in the years 1805 and 1806, under the direction of Messrs. Jessop. These two bridges are equal and alike in all respects. The arcli in each is a circular segment, of 100 feet span, with a versed sine or rise of only 15 feet: the width of the bridge about 31 feet : the whole is of cast iron, of the strongest grey metal ; amounting to 150 tons, viz. 100 tons in the ribs, pillars, bearers, balustrade, &c, and 50 tons in the plates for the roadway. The arch consists of two concentric circular rings or segments, firmly connected and bound toge- tlier. Each of these is formed of 6 ribs, at 6 feet distance from each other, tied together by cross bars, at intervals of about 9x feet ; as appears in the plan of the fabric here an- nexed on the following page. On the upper ring, of each rib, stand a number of pillars, in an upright position, or per- pendicular to the horizon, their tops formed hke a T, as bearers to support the plates for the roadway. All which, with the railing, or balustrade, as well as the disposition and coursing of the abutments, with piling underneath, appear in the represented elevation following; the courses of masonry very judiciously being laid inclining, as we have elsewhere recommended; and the whole seems otherwise very properly contrived. It would lead us too far here to enumerate all the ingenious particulars in the construction of this arch, with the dimensions of all the parts, and the practical methods of put- ting them together, and securing the whole in the firmest manner, as prescribed to the iron masters for their direction. 154 HISTORY OF TRACT 6. TRACT 6. IRON BRIDGES. 155 Suffice it therefore to observe, that, from the mode of putting the bridge together, it is so contrived, that if any part be in- jured, it can be taken out, and replaced, without disturbing the main body of the bridge. The cost of one bridge, independent of the digging and earth work, and making the roads to it, was nearly as below. . Piles - - 250 Masonry, 3200 yards, at 18s. including stone - - 1600 Iron work, 100 tons, at 9l. 18s. and 50 tons, at 9l. - 1440 Covering with gravel, and paving, &c. - - - 292 Expences of erection and painting - - - - 418 .4000 Thus has been given a short history of such iron bridges as have come to my knowledge : aware however that many others have been built, both for roads and for aqueducts in ca- nals, &c : but none of these, that I have heard of, are remark- able either for their span or construction : so that it appears unnecessary to enter into any particular description of them. The projects also that have been made foi* bridges of this kind, but not executed, are numerous, and a short account may here be added of some of the more remarkable designs that have come to our notice ; though our researches have not enabled us to trace any of them to a period prior to the execution of the bridge at Colebrook Dale. A design was made in the year 1783, by whom, does not appear, for an arch, chiefly of iron, of 400 French feet in span, and 45 feet in the versed sine ; answering to a circle of about 934 feet diameter. This design, with a memorial on the advantages of using iron, in the construction of bridges, was presented by the author to the unfortunate Louis of France, on the 5th of May 1783. It had two large ribs, partly of iron and partly of wood. These ribs were 30 feet deep at the springs, and 15 feet at the midde of the arch. Each rib was composed pf 4 ringS;, drawn from different centres, the inner ring 156 HISTORY" or TRACT 6. "being the sti'ongest ; and they were connected together by pieces of iron in various fanciful forms, httle adapted to give strength to the arch. Between the ribs were cills, or logs of timber, laid transversel}-, resting on the interior ring; and a floor of wood was proposed to cover them. So that the road was suspended by the ribs; and the upper part of the ribs was to answer the purpose of a parapet, similar to the wooden bridges in Switzerland. It appears that this project possessed Jittle merit beyond the boldness of its design ; and we have never heard that any bridge has been constructed on this principle. In the year 1*791 a project was made by Mr. John Rennie, Civil Engineer, for. an iron bridge, intended for the isle of Nevis. The span of the arch was to be 110 feet, and its versed sine IS-} ; answering to a circle of 234 feet diameter. It was proposed that this arch was to have 6 ribs ; each rib to consist of 3 rings, which were to be connected together by radii. The depth of the rib at tlie middle was 3^ feet, and at the springs 6 feet. 1 he ribs were to be connected together by transverse frames of iron, placed in the joints of the blocks of wliich the ribs were compose:'; ; the haunches to be filled with circular rings of iron ; and the whole was to have been covered with plates of iron, to support the road. In April 1794, he made another design for the same island of Nevis, in which the span was 80 feet, and the rise or versed sine 94 feet. This design was formed on the same principles as the former, except that the rib was 1 1 ^ deep at the springs, though still only 3^ in the middle. The radii were continued to the roadway ; and tlic whole was to be covered with iron plates, as the former. Neither of these designs however was executed, as the French got possession of the island. From the above period, no projects for iron bridges, except those above described, have come to my knowledge, till a|)- plications were made to parliament, for the purpose of im- proving the port of London, by means of wet docks. The House of Commons, after having heard a great deal of evi- dence, on the inadequacy of the Thames to accommodate the TRACT 6. IBON BRIDGBS. 157 shipping, appointed a select committee, to take the whole into their consideration, and to report to the house the best means for giving relief to the extensive commerce of the me- tropolis. This committee, after having recommended the construction of the West India and London Docks, took up the consideration of the state of the Thames, and of London Bridge, which forms the great obstruction to the influx of tide, and greatly injures the navigation of this very important commercial river ; and in the year 1799 they directed plans of London bridge to be made out, with correct descriptions of its construction and state of repair ; from which it appeared to them, that a new bridge, of more Avaterway, was impe- riously required : and in consequence encouragement w^s held out to artists, to bring forward designs, for the construction of a new bridge, instead of the old one. On this occasion many designs were made out, and presented to the committee. Some were for stone bridges, and some for iron. But as the object of this account relates to projects for iron bridges only, we shall here confine our attention to these last alone. The encouragement held out, by the Select Committee, brought forward four designs of this kind : namely, one by Mr. Wilson, formerly mentioned, of 3 arches ; the middle one of which was 240 feet span, having a versed sine of 37 feet ; the two side arches of 220 feet span each, and their versed sine 30 feet. The height of the soffit of the middle arch 80 feet above the high-water of an ordinary neap tide. The principles of this design were so nearly the same as those of Sunderland bridge, that it is unnecessary to enter into any minute description of it. Two other designs were brought forward by Messrs. Tel- ford and Douglass : one to consist of 5 arches across the river, and the other of 3. The middle arch of the former was 180 feet span, with a versed sine of 38 feet; also two arches, each of 140 feet span, and two of 120 feet span each. The other had a middle arch of 240 feet span, with a versed sine of 48 feet ; and two side arches, of 220 feet span each : the height of the soffit of the middle arcl^ being SO feet above 158 iiisTOKY OF TnAcr C. the high water of fteap tides, the same as that of Mr. Wilson's design. The arches of both the designs of Messrs. Telford and Douglass were constructed in the same manner ; therefore a description of one will serve for both. They were composed of ribs ; each rib having an outer and inner ring : the inner ring much stronger than the outer, and they were connected together by radiated bars, which extended quite to the pieces that supported the roadway. In the large arches there were two portions of rings, to stay the radiated bars in the haunches; but in the small arches only one. Of how many pieces the ribs were composed, or in what manner to be joined, was not shown in the designs, nor mentioned in the descriptions. The great height given to these bridges, to admit of vessels pass- ing under them, renders it necessary, particularly on the south side of the river, where the land is under the level of spring tides, that long approaches, or inclined planes, as the designers called them, should be made ; and these they pro- posed to support on iron arches, constructed in a manner similar to those of the bridge. By the section it appears that there will be a rise of about 1 foot in 19, on the main approach from the Borough ; so that, taking the height of the road- way on the bridge at 60 feet above the wharf of the Thames, this approach will extend 1140 feet into the Borough, High- street. Now a rise of 1 in 19 is almost double the rise in Ludgate-hill : so that, if it were to be made the same rise as Ludgate-hill, it would extend to a distance not much short of half a mile. The side approach upward, it appears albo, would come within about 260 yards of Blackfriars bridge, and that downwards would extend to nearly opposite the Tower. So that a considerable part of the Borough would probably be subjected to great inconveniences and expences by these far extended approaches, which appear unavoidable. The additional labour too that would by this means be occasioned, would probably cost more, to the inhabitants of London and the Borough of Southwark, than all the advantage that might arise by bringing vessels up to Blackfriars bridge. These ob- TRACT 6. IRON BRIDGES. K9 jections are not applicable to these designs alone, but in an equal degree to Mr. Wilson's also. There can be no doubt but that both designs could be exe- cuted ; whatever may have been the opinion of artists on the skill exercised in their mechanical construction. We have before shown, that the true principle on which an arch ought to be constructed, is to increase the depth of the voussoir, as it is called in masonry, towards the spring of the arch, so that the arch, with its load upon it, shall be in equilibrio in all its parts. This being accomplished, it does not appear that any good can result from extending the radii further ; for as the roadway presses perpendicularly on the arch, it appears not the strongest mode to support this perpendicular load by in- clined pieces ; but rather the contrary. It seems proper, therefore, that the roadway should be sustained by upright pillars of iron, instead of inclined radii, though less elegant in appearance to the eye : nay we might even prefer the circular rings or eyes of Mr. Wilson, to this mode : though we are aware that a circle, pressed on four points, is by no means calculated to bear a very great pressure. The Select Committee of the House of Commons, not be- ing satisfied with any of the three designs, that have been described, directed Messrs. Dance and Jessop to report, whether any, and what advantages, would accrue to the na- vigation of the Thames, if it were to be considerably con- tracted. Accordingly these gentlemen reported, that if, in- stead of the channel of the Thames at London bridge being 740 feet wide, as it was proposed to be when the above de- signs were made, it were reduced to 600 feet, that great ad- vantages would result to the navigation ; since, by diminish- ing the width, the depth would be much increased. It might be foreign to the purpose of the present work, to enter into any discussion on the propriety of this measure ; for which reason we may leave that discussion to a future opportunity. In consequence of this opinion, Messrs. Telford and Douglass presented to the Committee a very elegant and magnificent design, for an arch of 600 feet span, having its versed sine 166 HISTORY OP TRACT 6. about 65 feet; so that the circle of which this arch is a seg- ment, must be about 1450 feet diameter. The arch was composed of seven ribs ; and each rib may- be said to have 6 rings, the 3 lower concentric, and about 8 feet deep. The dimensions of the iron cannot be correctly taken by measurement from the plan, this being on a small scale. These rings were connected by radii about 18 inches asunder ; the outer and inner are the strongest, and that in the mjddle appears light, and seems intended, it is presumed, chiefly to stift'en the radii, though doubtless it will also add to the strength of the bridge. The ribs are composed of frames of iron, each about 10 feet long, which extend quite to the entablature of the cornice. The other 3 rings are not concentric with those 3 lower, but each drawn from a larger radius than the other. The lowest of these three terminates in the upper ring of the three lower, at about 120 feet from the key, or the middle of the arch. The two above this unite at about the same distance from the middle of the arch, and are thence continued in one ring, till they reach within about 35 feet of the middle or key of the arch, where they join the said upper rib of the lower three. These three upper ribs are united to the third or upper ring, of those first de- scribed, by means of radii; but the spaces between these radii include the space of two of the lower radii ; and, instead of being stiffened by a light ring, as the lower radii are, that object is effected by Gothic tracery. These seven ribs, above described, are set parallel to each other ; and, to brace them horizontally, there are six others, or diagonal ribs, four of which cross the former diagonally, two terminating in the middle rib, and two in the adjoining ribs ; and there are two outside ribs, that terminate each on the face of the exterior ones. So that, in fact, two of the seven have no diagonal rib terminating at their top. The whole of these last described ribs are therefore side or diagonal braces, to keep the seven principal ribs in their vertical position, and prevent the arch from racking sideways, as happened at Sunderland or Wear- mouth bridge, before mentioned. All these vertical and TRACT 6. IROX ERIDGT.S. 161 diagonal ribs are connected together by transverse frames, at tihe joints of each of the radiated frames or voussoirs. The top or platform, under the roadway, is covered, in the usual manner, with iron plates ; and there is a light iron railing on each side, with-Gothic ornaments. The breadth of the road- way at the top, or middle of the arch, is 45 feet, and at the haunch or extremit}^ of the arch 82 feet wide. The arch springs from large frames of iron, set in abutments of ma- sonry ; and its approaches are similar to those before de- scribed for the designs of Messrs. Telford and Douglass. The principles on which this arch is designed, maybe found in a work published at Leyden, in the year 1721, entitled ** Recueil de plusieurs machines de nouvelle invention, ouv- rage posthume de M. Claude Perrault, &c. &,c." and is de- scribed in pages 712, 13, 14 of that work, and represented in plates 10 and 11. It is described, *' Pont de bois d'une seule arche de trente toises de diametre, pour traverser la Saine visavis le village de Sevre, ou I'on proposoit de la contruire.'* It may also be seen in the 1st vol. of the Machines approved by the Academ}^ of Sciences, pa. 59, pi. 14, It may appear perhaps doubtful to some persons, whether this design is so proportioned as to be in perfect equilibrio, being remarkably heavy at the haunches ; and that, were such an arch as there described to be erected over the Thames, whether it would permanently support itself. The extension of the radii to the roadway has been before noticed as not well adapted to sustain the perpendicular pressure, with which it would be charged, and that unless its parts were in perfect equiUbrio, the joints of the frames might open in such a manner, as to deranse the whole fabric, and accelerate its destruction,'! That an iron arch of 600 feet span might be constructed in such a manner, as to become a firm and stable fabric, it is not meant to be denied ; but, according to the principles we have laid down, it should be rather differently constructed from that we have described. Indeed, if the weight of iron, men- tioned in the estimate, be correct, the parts must be very slender indeed ; and were the whole to be in equilibrio, this TOL. I. M 162 HISTORY OF TRACT 6. Aveight of the structure it>elf luiglit beud the parts in such a nianuer, as in sou)e moas\ir(.' to cuduncer its downtalL We imagine that three distinct objects were proposed to be obtained by the improvements which the public have in view. These are, 1st. The maintaining of deeper water, from the lower part of the Thames to Blackfriars bridge, and upward. 2d. More clear bpace for the navigation of vessels under the bridge. 3d. Effecting this object with the least rise of road over it. In respect to the first question, I have already declined en- tering into it ; being of opinion it is a discussion rather fo- reign to the purpose of a book on bridges. The second ap- pears to come fully under the scope of the prmciples we have treated on. The arch here proposed, as we have before seen, is of 600 feet span, with a versed sme or rise of 65 feet. Now, at the distance of 100 feet from the middle, the height is 58 feet ; at 150 feet from the middle the heigiit is 49 feet ; and at 200 feet it is 37 feet in height. So that, only about 200 feet, or 4- of the width of the river, can be accounted fit for the navigation of coasters : about another third may befit for the ordinary barges; and the remaining third Avill be for little other purpose than the lug boats and w^herries that ply on the river. Vessels, therefore, in departing from the wharfs, must be drawn out nearly to the middle of the river, before they can take the advantage of the tide downwards: and those coming to a wharf, must letch up in the river till they are hauled into it. This might do for vessdb that frequent wharfs situated a considerable distance above the bridge : but those for wharfs that might be near it, must experience much trouble and in- convenience ; and it is to be feared that they w^ould fre- quently sustain damage in their masts and rigging, by strik- ing against it, and might probably injure the bridge itself, Mr. llennie has very properly noticed this, in his answer to one of the queries proposed by the Select Committc'e of the House of Commons : but he follows up his observations by saying, that, as the strength of the current will be chiefly in TRACT 6. IRON BRIDGES. 163 the middle of the river, the vessels will generally pass in that track, Now we may admit that, for a vessel sailino- up or down the river, and going to some wharf near Bii.cktriars bridge, or departing from thence downward, that this will be the case : but when going to, or sailing from wharfs near the new bridge, it will be very much otherwise ; as may be observed by any one who will attend to the vessels sailing to or from the wharfs below London bridge : and \vc should fear that, in order to prevent the accidents above noticed, dolphins, or some such contrivance, will be found absolutely necessary, to keep the vessels in the proper track, in passing- through this arch. Now, if we be right in our cpnjecture, it would probably be better to have two piers, and a bridge of three arches, than a bridge of one only ; by which the height or space under the bridge, for vessels to pass, might be very much increased ; and those wharfs which lie near the bridge not be subject to the inconveniences, nor the vessels to the risk before mentioned. Thirdly, A bridge of three arches will not require the ribs to be so deep at the top, as a bridge of one arch, by at least 5 feet ; and therefore so much will be gained in tlic height of the roadway over it. On the whole thei*efore it seems, that the design in question is not completely calculated to attain the objects the Select Committee of the House of Commons had in view : but, on the contrary, that it will appear to most thinking men, rather an injudicious idea, to effect by a great work, that which can at least as well, if not better, be ac- complished by a work of less expence, and of more probable stability. Our observations have been hitherto confined to the possi- bility and propriety of executing an iron arch, of 600 feet span, according to the design given with the report of the House of Commons. We may now add some observations on the practicability of building abutments, in this situation, suf- ficiently strong to resist the lateral pressure of this arch ; which, according to our calculation, made on the supposi- tion that the arch would be similar to one of stone, acting m2 I(i4 MISTOIIY OV TRACT G. v,it!! ;i regular and unii"onr> jjicssiirc upon it, \vould be of alxnit yOOU tons. J>iit whrii the effects of tlic vibration, which must necessarily take place in an arch of this magnitude, arcs taken into consideration, the lateral pressure, or rather vibrating push, will far exceed that quantity ; and for this effort, as has been before noticed, provision must be made in the strength of the abutments : and though the thickness of these in the design, name!}' 85 feet, seems to be great, yet I am inchned to think it would be found too small, especially at the south end of the brido-e, where I am informed the ground is very bad, being moorlog and soft mud to a con- siderable depth. Indeed I should fear that something of the kind of what happened at Staines would be likely to take place here, namely, the whole mass of masonry be forced back ho- rizontally, by the great lateral push of the arch, in spite of every precaution that could be taken to prevent it. But we must observe, as we have before done in answer to theQueries in the Report of the Committee of the House of Commons, that thefoundations of the abutments should be laid inclining towards the centre of the circle to which the arch is drawn, as a more likely mode of preventing them from sliding out- wards, than if laid horizontally : but even with this precau- tion, if the substratum be moorlog or soft mud, it Avill be likely to give way ; and if this ever take place, the abutment and arch must follow it. The following is a rough sketch, on a very small scale, on the design, at least very elegant, which was given along with the above project. TRACT 6. IROJJ BRIDGES. 165 o ^Sfl ^ \f\ if ll* i if J CiIttT uTl mij ffn 1 u ifli 1 LjL jjf 1 jMf 1 1 / T fJJ tT ID 1 JJ 11 J Jul UL J t tnf Tj T T 1 jn j| it 1 J It 1 Tf 1 It pfT T LL " 1 ^^ : 1 ]6o HISTORY OF IRON BRIDGES. TRACT 6. As in some degree and nature related to the foregoing account of iron arches, properly so called, we may here add a few words, just to notice two ingenious works lately exe- cuted, b^ng a kind of straight or flat arch, for an iron aque- duct, supported on pillars, carried over rivers. These were, both of them, designed by Mr. Thomas Telord, engineer, and executed under his direction. The former Avas a small aqueduct of cast iron, the first for a navigable canal, which was constructed in the year 1795, on the Slirewsbury canal, near Wellington in Shropshire. It is 1 80 feet in length ; and the surface of the water in the aqueduct is about 20 feet above that of low water in the river. The supporting pillars, in this case, are also of cast iron. There are no ribs under the bottom plates, these being connected with the side plates, shaped like the stones in a Hat arcii, Avhich is also the case in the second instance, at Pontcysylte. The iron work of this aqueduct was cast at Ketlcy foundery, by Messrs. Keynolds. The second instance was erected in the year 1805, at the Pontcysylte aqueduct. It having been found necessary to carry the Ellesmere canal across the river Dee, at the eastern termination of the vale of Llangollen, at the height of 126 feet 8 inches above the surface of low water in the river, Mr. Telford conceived the bold design of effecting this by means of an aqueduct constructed of cast iron, supported by stone pillars. These are 20 in number, including the abutments : the length of the aqueduct is 1020 feet, and the breadth across it 12 feet. It has been in constant use for the purposes of navigation ever since it was first opened, on the 26th of November 1805, and it answers every pur- pose perfectly well. The iron work was cast, and set up, by Mr. William Hazledine, of Si)rewsbury. A small view of the elevation of this elegant structure is as here below. i^fcw^fet^ ( 167 ) TRACT VII. A DISSERTATION ON THE NATURE AND VALUE OF INFINITE SERIES. 1. About the year 1780 I discovered a very general and easy method of valuing series, wiiose terms are alternately positive and negative, which equally applies to such series, whether they be converging, or diverging, or their terms all equal ; together with several other properties relating to certain scries : and as there may be occasion to deliver some of those matters in the course of these tracts, this opportu- nity is taken of premising a few ideas and remarks, on the nature and valuation of some of the classes of series, which form the object of those communications. This is done with a view to obviate any misconceptions that might perhaps be made, concerning the idea annexed to the tcnu value of such series in those tracts, and the sense in w?)ich it is there always to be understood ; which is the more necessary, as many con- troversies have been warmly agitated concerning these mat- ters, not only of late, by some of our own countrymen, but also by others among the ablest matiiematicians in Europe, at different periods in the course of the last century ; and all this, it seems, through the want of specifying in what sense the term value or sum was to be understood in their disser- tations. And in this discourse, I shall follow, in a great measure, the sentiments and manner of the late celebrated L. Euler, contained in a similar memoir of his in the fifth volume of the New Petersburgh Commentaries, adding and intermixing here and there other remarks and observations of my own. 2. By a converging series, is meant such a one whose terms continually decrease ; and by a diverging series, that 168* THE NATURE AND VALUE TRACT 7. whose terms continually increase. So that a series whose terms neither increase nor decrease, but are all equal, as they neither converge nor diverge, may be called a neutral series, as a a -\- a a -^ &c. Now converging series, being sup- posed infinitely continued, may have their terms decreasing to as a limit, as the scries 1 4 + ^ ^+ &c, or only decreasing to some finite magnitude as a limit, as the serie* r + r i + &c, which tends continually to 1 as a limit. So, in like manner, diverging series may have their terms tending to a limit, that is either finite or infinitely great : thus the terms 1 2 + 3 4 + &c, diverge to in- finity ; but the diverging terms f |. + .|. |-f- &c, only to the finite magnitude 1. Hence then, as the ultimate terms of series which do not converge to 0, by supposing them continued in infinitum, may be cither finite or infinite, there will be two kinds of such series, each of which will be further divided into two species, according as tlie terms shall either be all affected with the same sign, or have alternately the signs + and . We shall, therefore, have altogether four species of series which do not converge to 0, an example of each of which may be as here follows : 1. - 3. 4. - Cl -f 1 + 1 + I U+1 + l- 4- i C 1 + 2 4- 3 + 4 + 5 + 6 -j- &c. d 1 + 2 4- 4 + 8 + 16 + 32 + .Scc. ri_24.3_4-{- 5 6 + &C. CI 5 + 4 - 8 + 16 - 32 + &c. \- 1 + 1 + &c. 4- 1+ -^ + &c. 1 + 1 - 1 + 1 - 1 + &c. + I - I + I - T + -^C. 3. Now concerning the sums of these species of series, ihere have been great dissensions among juathematicians ; sou^e affirming that they can be expressed by a certain sum, while others deny it. In the first place, however, it is evi- dent that the sums of such series as come under the ('rst < f these species, will be really infinitely great, since l)y --.-ually I-RACT 7. OF INFINITE SERIES. 169 collecting the terms, we can arrive at a sum greater than any proposed mimber whatever: and hence there can be no doubt but that the sums of this species of series may be exhibited by expressions of this kind . It is concerning the other species, therefore, that mathematicians have chiefly differed ; and the arguments which both sides allege in defence of their opinions, have been endued with such force, that neither party could be hitherto brought to yield to the other, 4. As to the second species, the celebrated Leibnitz was one of the first who tre-ated of this series 1 l-j-i i-f. 1 1 + &c, and he concluded the sum of it to be = f , relying on the following cogent reasons. And first, that this series arises by resolvingr the fraction- into the series ^ =* 1 + a 1 a + a* a^ + * tf^ + &c, by continual division in- the usual way, and taking the value of a equal to unity. Secondly, for more confirmation, and for persuading such as are not accustomed to calculations, he reasons in the follow- ing manner : If the series terminate any where, and if the number of the terms be even, then its value will he = ; but if the number of terms be odd, the value of the series will be = 1 : but because the series proceeds in infinitunit and that the number of the terms cannot be reckoned either odd or even, we may conclude that the sum is neither = 0, nor 1, but that it must obtain a certain middle value, equidiHerent from both, and Avhicli is therefore = f. And thua, he adds, nature adheres to the universal law of justice, giving no partial preference to either side. 5.. Against these arguments the adverse party make use of such objections as the following. First, that the fraction. ; ; is not equal to the infinite series 1 a + a* a^ 4- 1 -}- &c, unless a be a fraction less than unity. For if the division be any where broken off, and the quotient of the remainder be added, the cause of the paralogism wiJl be manifest; no THE NATURE AND VALUE TRACT 7. 1 for we shall then have =1 a + a* a'-f- i fl" T - ; and that, although the number n should be made infinite, yet the supplemental fraction ^ 1 + a ought not to be omitted, unless it should become evanescent, which happens only in those cases in which a is less than 1, and the terms of the series converge to 0. But that in other cases there ought always to be included this kind of supple- a" + ' ment ^ - ; ; and thoucrh it be affected with the dubious 1 + sign ^ , namely or -f according as 7i shall be an even or an odd number, yet if n be infinite, it may not therefore be omitted, under the pretence that an infinite number is neither odd nor even, and that there is no reason why the one sign should be used rather than the other ; for it is absurd to sup- pose that there can be any integer number, even though it be infinite, which is neither odd nor even. 6. But this objection is rejected by those who attribute de- terminate sums to diverging series, because it considers an infinite number as a determinate number, and therefore either odd or even, when it is really indeterminate. For that it is contrary to the very idea of a series, said to proceed in infi- nitum, to conceive any term of it as the last, though infinite : and that therefore the objection ab vo-meniioned, of the supplement to be added or subtracted, naturally falls of itself. Therefore, since an infinite series never terminates, we never can arrive at the place where that supplement must be joined; and therefore that the supplement not only may, but indeed ought to be neglected, because there is no place found for it. And these arguments, adduced either for or against the sums of such series as above, hold also in the fourth species, which is not otherwise embarrassed with any further doubts peculiar to itself. 7. But those who dispute against the sums of such scries. TRACT 1. OF INFINITE SERIES. 171 think they have the firmest hold in the third species. For though the terms of these series continually increase, and that, by actually collecting the terms, we can arrive at a sum greater than any assignable number, which is the very definition of infinity ; yet the patrons of the sums are forced to admit, in this species, series whose sums are not only finite, but even negative, or less than nothing. For since the fraction , by evolving it by division, becomes 1 + a + a* + a' + * + &c, we should have = 1 = 1+2 + 4+ 8 + 16 + &C, 1 2 1 = i-=l+S + 9+ 27 + 81 + &c, 1 - 3 which their adversaries, not undeservedly, hold to be absurd, since by the addition of affirmative numbers, we can never obtain a negative sum ; and hence they urge that there is the greater necessity for including the before-mentioned supple- ment additive, since by taking it in, it is evident that 2"+ ^ 1 is =1+2+4+8 2" + y ^, though 71 should be an infinite number. 8. The defenders therefore of the sums of such series, in order to reconcile this striking paradox, more subtle perhaps than true, make a distinction between negative quantities ; for they argue, that while some are less than nothing, there are others greater than infinite, or above infinity. Namely, that the one value of 1 ought to be understood, when it is conceived to arise from the subtraction of a greater number a + 1 from a less a ; but the other value, Avhen it is found equal to the series 1 +2 + 4 + 8+ &c, and arising from the division of the number 1 by 1 ; for that in the former case it is less than nothing, but in the latter greater than infi- nite. For the more confirmation, they bring this example of fractions 2_J_J_JI_ 1 1 1 1 4' 3> T' "o"' -1' -S' -3' ' 112 THE NATURE AND VALUE TRACl" 1. which, evidently increasing in the leading terms, it is inferred will continually increase ; and hence they conclude that -- is greater tban^, and - greater than -, and so on: and therefore as is expressed by 1 , and ^ by - , or infinity, 1 -will be greater than <-> , and much more will = _i be greater than -> . And thus they ingeniously enough repel- led that apparent absurdity by itself. 9. But though this distinction seemed to be ingeniously devised, it gave but little satisfaction to the adversaries; and besides, it seemed to affect the truth of the rules of algebra. For if the two values of 1, namely 12 and -, be really different from each other, as we may not confound then), the certainty and the use of the rules, which we follow in making calculations, would be quite done away ; which would be a greater absurdity than that for whose sake the distinction was devised : but if 1 2 = -, as the rules of algebra require, for by multiplication 1 x (12)= 1 + 2=1, the matter in debate is not settled; since the quantity -- I , to which the series 1+2 + 4 + 8 + &c, is made t(|'iu.]j is less than nothing, and therefore the same difficnltv still re- mains. In the mean time however, it seems but agieeiibie to truth, to say, that the same quantities which are below nothing, may be taken as above infinite. For we know, not only from algebra, but from geometry also, that ther^ are two ways, by which quantities pass from positive to negative, the one through the cypher or nothing, and the other throutvi infinity: and besides, that quantities, either by increa.>iiii or decreasing from the cypher, return again, and revert to t,.c same term 0; so that quantities more than infinite arc r. c same with quantities less than nothing, like as quantities less than infinite agree with quantities greater than nothing. 10. But, further, those who deny the truth of the sums TRACT 1. OP INFINITE SERIES. iTS that have been assigned to diverging series, not only omit to assign other values for the sums, but even set themselves ut- terly to oppose all sums whatever belonging to such series, as things merely imaginary. For a converging series, as suppose this 1 +f + :^+4 + &c, will admit of a sum = 2, because the more terms of this series Ave actually add, the nearer we come to the number 2: but in diverging series the case is quite different; for the more terms we add, the more do the sums which are produced differ from one an- other, neither do they ever tend to any certain determinate value. Hence they conclude, that no idea of a sum can be applied to diverging series, and that the labour of those per- sons who employ themselves in investigating the sums of such series, is manifestly useless, and indeed contrary to the very principles of analysis. 1 1 . But notwithstanding this seemingly real difference, yet neither party could ever convict the other of any error, when- ever the use of series of this kind has occurred in analysis ; and for this good reason, that neither party is in an error, the whole difference consisting in words only. For if in any calculation we arrive at this series 1 l-fl l-f iScc, and that we substitute f instead of it, we shall surely not thereby commit any error ; which however we should cer- tainly incur if w-e substitute any other number instead of that series; and hence there remains no doubt but that the series 1 1 + 1 ~ i + ^c, and the fraction 4-, are equivalent quantities, and that the one may always be substituted instead of the other without error. So that the whole matter in dis- pute seems to be reduced to this only, namely, whether the fraction f can be properly called the^MWi of the series 11 + 1 1 -f &c. Now if any persons should obstinately deny this, since they Avill not however venture to deny the fraction to be equivalent to the series, it is greatly to be feared they will fall into mere quarrelling about words. 12. But perhaps the whole dispute will easily be compro- mised, by carefully attending to what follows. Whenever, in analysis, we arrive at a complex function or expression, 174 THE NATURE AND VALUK TRACT 7. either fractional or transcendental ; it is usual to convert it into a convenient series, to which the remaining calculus may be more easily applied. 7\nd hence the occasion and rise of infinite series. So far only then do infinite series take place in analytics, as they arise from the evolution of some finite expression ; and therefore, instead of an infinite series, in any calculus, we may sn'uUitute that formula, from whose evo- lution it arose. And i:i:nce, for performing calculations with more ease or more benefit, like as rules are usually given for converting into infinite series such finite expressions as are endued with less proper fonij:- , so, on the other hand, those rules are to be esteemed not less useful, by the help of which we may investigate the finite expression from which a pro- posed infinite series would result, if that finite expression should be evolved by the proper rules: and since this ex- pression may always, without error, be substituted instead of the infinite series, they must necessarily be of the same- value : and hence no infinite series can be proposed, but a finite expression may, at the same time, be conceived as equivalent to it. 1 3. If, therefore, we only so far change the received notion of a sum as to say, that the sum of any series, is the finite expression by the evolution of which that series maybe pro- duced, all the difficulties, which have been agitated on both sides, vanish of themselves. For, first, that expression by Avhose evolution a converging series is produced, exhibits at the same time its sum, in the common acceptation of the term : neither, if the series should be divergent, could tlie investigation be deemed at all more absurd, or less proper, namely, the searching out a finite expression which, being evolved .according to the rules of algebra, shall ]iroduce that scries. And since that expression may be substituted in the calculation instead of this series, there can be no doubt but that it is equal to it. WMiich being the case, we need not necessarily deviate from the usual mode of sjjcaking, hut might be permitted to call that expression al^o t!ic sinn, which is equal to any series whatever, provided however, TRACT 7. OF INFINITE SERIES. 175 that, in series whose terms do not converge to 0, we do not connect that notion with this idea of a sum, namely, that the moi'e terms of the series are actually collected, the nearer we must approach to the value of the sum. 14. But if any person shall still think it improper to apply the term sum, to the finite expressions by whose evolution all series in general are produced ; it will make no difference in the nature of the thing ; and mstead of the word sum, for such finite expression, he may use the term value, or func- tion, or perhaps the term radix would be as proper as any other that could be employed for this purpose, as the series may justly be considered as issuing or growing out of it, like as a plant springs from its root, or from its seed. The choice of terms bemg in a great measure arbitrary, every person is at liberty to employ them in whatever sense he may think fit, or proper for the purpose in hand ; provided always that he fix and determine the sense in which he understands or employs them. And as I consider any series, and the finite expression by whose evolution that series may be pi'oduced, as no more than two different ways of expressing one and the same thing, whether that finite expression be called the sum, or value, or function, or radix of the series ; so in the follow- ing paper, and in some others which may perhaps hereafter be produced, it is in this sense I desire to be understood, when searching out the value of series, namely, that the ob- ject of the enquiry, is the radix by whose evolution the series may be produced, or else an approximation to the value of it in decimal numbers, &.c. { ne ) TRACT Vlir. A NEW METHOD FOR THE TALUATION OF NUMERAL INFI- NITE SERIES, WHOSE TERMS ARE ALTERNATELY (+) PLUS AND ( ) minus; BY TAKING CONTINUAL ARITH- METICAL MEANS BETWEEN THE SUCCESSIVE SUMS, AND THEIR MEANS. ARTICLE 1. The remarkable difference between the facility -which mathematicians have found, in their endeavours to determine the values of infinite series, whose terms are alternately affirm- ative and negative, and the difficulty of doing the same thing with respect to those series whose terms are all affirmative, is one of those striking circumstances in science which we can hardly persuade ourselves is true, even after we have seen many proofs of it ; and Avhich serve to put us ever after on our guard not to trust to our fust notions, or con- jectures, on these subjects, till we have brought them to tlie I test of demonstration. For, at first sight it is very natural to imagine, that those infinite series whose terms are all affirm- ative, or added to the first term, must be much simpler in their nature, and much easier to be summed, than those whose terms are alternately affirmative and negative ; which, how- ever, we find, on examination, to be directly the reverse; the methods of finding the sums of the latter series being nu- merous and easy, and also very general, whereas those that have been hitherto discovered for the summation of the former sc-ries, are few and difficult, and confined to series whose terms are generated from each other according to some par- ticular laws, instead of extending, as the other methods do, TRACT 8. TilE VALUATION OP INFINITE SERIES. Ill to all sorts of series, whose terms are connected together by addition, by whatever law their terms are formed. Of this remarkable difference between these two sorts of series, the new method of finding the sums of those whose terms are al- ternately positive and negative, which is the subject of the present tract, will afford us a striking instance, as it possesses the happy qualities of simplicity, ease, perspicuity, and uni- versality ; and yet, as the essence of it consists in the alter- nation of the signs + and , by which the terms are con- nected with the first term, it is of no use in the summation of those other series whose terms are all connected with each other by the sign +. 2. This method, so easy and general, is, in short, simply this : beginning at the first term a of the series a b -\- c - d -\- e / + &c, which is to be summed, compute several successive values of it, by taking in successively more and more terms, one term being taken in at a time ; so that the first value of the series shall be its first term a, or even or nothing may begin the series of sums ; the next value shall be its first two terms a b, reduced to one number ; its next value shall be the first three terms a 6 + c, reduced to one number ; its next value shall be the first four terms a b -}- c d, reduced also to one number; and so on. This, it is evident, may be done by means of the easy arithrnetical operations of addition and subtraction. And then, having found a sufficient number of successive values of the series, more or less as the case may require, interpose between these values a set of arithmetical mean quantities or proportionals; and between these arithmetical means interpose a second sec of arithmetical mean quantities ; and between these arith- metical means of the second set, interpose a third set of arithmetical mean quantities ; and so on as far as you please. By this process we soon find either the true vaUie of the series proposed, when it has a determinate rational value, or otherwise we obtain several sets of vahies approximating^ nearer and nearer to tl)e sum of the scries, both in the co- lumns and in the lines, either horizontal or obliquely dc- VOL, I. N 178 THE VALUATION OP TRACT ^. scending or ascending ; namely, both of the several sets of means themselves, and the sets or scries formed of any of their corresponding terms, as of all their first terms, of their second terms, of their third terms, &c, or of their last terms, of their penultimate terms, of their antepenultimate terms, &c : and if between any of these latter sets, consisting of the like or corresponding terms of the former sets of arithmetical means, we again interpose new sets of arithmetical means, as "we did at first with the successive sums, we shall obtain other sets of approximating terms, having the same properties as the former. And thus we may repeat the process as often as we please, which will be found very useful in the more difficult diverging series, as we shall see hereafter. For this method, being derived only from the circumstance of the al- ternation of the signs of the terms, -\- and , it is therefore not confined to converging series alone, but is equally appli- cable both to diverging series, and to neutral series, by which last name I shall take the liberty to distinguish those series, whose terms are all of the same constant magnitude ; namely, the application is equally the same for all the three following sorts of series, viz. Converging, i_i-j-| ^ + |_|^ + &c. Diverging, 1 2+3 4 + 5 6 + &c. Neutral, 1 1 + 1 1 + 1 1 + &c. As is demonstrated in what follows, and exemplified in a variet}' of instances. It must be noted, however, that by the value of the series, I always mean such radio:, or finite expression, as, by evolu- tion, would produce the series in question ; according to the sense we have stated in the former ])aper, on this subject; or ;in approximate value of such radix ; and which radix, as it may be substituted instead of the series in any operation, I call the value of the series, 3. It is an obvious and well-known property of infinite series, with alternate signs, that when we seek their value bv collecting their terms one after another, wo obtain a series of successive suras, which approach continually nearer and TRACT S. INFINITE SERIES. 179 nearer to the true value of the proposed series, when it is a converging one, or one whose terms always decrease by some regular law ; but in a diverging series, or one whose terms as continually increase, those successive sums diverge always more and more from the true value of the series. And from the circumstance of the alternate change of. the signs, it is also a property of those successive sums, that when the last term which is included in the collection, is a positive one, then the sum obtained is too great, or exceeds the truth ; but when the last collected term is negative, then the sum is too little, or below the truth. So that, in both the converging and diverging series, the first term alone, being positive, ex- ceeds the truth ; the second sum, or the sum of the first two terms, is below the truth ; the third sum, or the sum of the three terms, is above the truth ; the fourth sum, or the sum of four terms, is below the truth ; and so on ; the sum of any even number of terms being below the true value of the series, and the sum of any odd number, above it. All which is ge- nerally known, and evident from the nature and form of the series. So, of the series a b + c d + e f -\- &c, the first sum a is too great ; the second sum a b too little ; the third sum a b -\- c too great ; and so on as in the fol- lowing table, where s is the true value of the series, and is placed before the collected sums, to complete the series, being the value when no terms are included : Successive suras. 5 is greater than ^- is less than a s is greater than a b s is less than a b -{ c s is greater than a b -f c d s is less than a b -f c d \- e &c. &c. 4. Hence the value of every alternate series s, is positive, and less than the first term a, the series being always sup- posed to begin with a positive term a ; and consequently, if the signs of all the terms be changed, or if the series begin N 2 ISO THE VALUATION OF TRACT 8. with a negiitive term, the value s will still be the same, but negative, or the sign of the sum will be changed, and the value become -~ s zz: a -\- b c -\- d &c. Also, be- cause the successive sums, in a converging series, always ap- proach nearer and nearer to the true value, while they recede always farther and farther from it in a diverging one ; it follows that, in a neutral series, a a -\- a a -\- &c, which holds a middle place between the two former, the successive sums 0, a, 0, a, 0, a, &c, will neither converge nor diverge, but will be always at the same distance from the value of the proposed series a a -\- a a -\- &c, and consequently that value will always be = fa, which holds every where the mid- dle place between and a. I am not unaware that, though a a \- a aH &c, may be produced by evolving by actual division, it will also arise by evolving several other functions in like manner ; as fl^ fl' o * + a' + a^ + &c : , or ; ; ; , &C, Or ; r ) a-fia + fl a-\-a-^a-\-a a -i- a \- a -\- a -\- olc or any other similar function, in which the numerator has fewer terms than the denominator. Yet the preference among them all seems justly due to the fust = -=:=: 4^, lor tins reason, besides what a -\- a 2a 2 is said above, viz, put s for the value of the series a a + a a + &c : since then s = a a -{ a a -\ &c, and a = a, take tiie upper equ. from the under, therj a s a a -\- a ' a -\- &c = s by sup. thercf, a s :zz s, and 2^ = a, or s = la, as above. 5. Now, with respect to a converging series, a d-i-c d+ &c ; because is below, and rt above ^, the value of the series, but a nearer than to the vahie s, it follows that 6- lies be- tween a and ^a, and that -^a is less than s, and so nearer to s than is. In like manner, because a is above, and a b below the value s, but a b nearer to that value than a is. TRACT 8. INFINITE SERIES. 181 it follows that s lies between a and a b, aiul that the arith- metical mean a f 6 is something above the value of s, but nearer to that value than a is. And thus, the same reason- ing holding in every following pair of successive sums, the arithmetical means between them will form another series of terms, which are, like those sums, alternately less and greater than the value of the proposed series, but approximating nearer to that value than the several successive sums do, as every term of those means is nearer to the value 5, than the corresponding preceding term in the sums is. And, like as the successive sums form a progression approaching always nearer and nearer to the value of the series ; so, in like man- ner, their arithmetical means form another ])rogression, com- ing nearer and nearer to the same value, and each term of the progression of means nearer than each term of the successive sums. Hence then we have the two following scries, namely, of successive sums and their arithmetical means, in which each step approaches nearer to the vahie of s than the former, tlie latter progression being however nearer than the former, and the terms or steps of each alternately below and above tho value s of the series a b -\- c d -\- &c. Successive sums. Arithmetic; xl means. -3 n ia c- a c- a ^b -2 a b -D a b \- Ic c- a b ^ c r- a b ^ c -id -T3 a -^ b + c d -D a b ^ c d^\c c- a b + c d -\-e cr a b -{ c - d+ c &c. 3tc, u where the mark ^, placed before any step, signifies that it is too little, or belov/ the value s of the converging series a b -\- c d -^ &c; and the mark c- signifies the con- trary, or too great. And hence -^a, or half the first term of such a converging series, is less than 5 the value of th^ series. 182 THE VALUATION OP TRACT 8. 6, And since these two progressions possess the same pro- perties, but only the terms of the latter nearer to the truth than the former ; for the very same reasons as before, the means between the terms of these first arithmetical means, will form a third progression, whose terms will approach still nearer to the value of 5 than the second progression, or the first means ; and the means of these second means will ap- proach nearer than the said second means do ; and so on con- tinually, every succeeding order of arithmetical means, ap- proaching nearer to the value of s than the former. So that the following columns of sums and means will be each nearer to the value of s than the former, viz. Sue, sums. a a b a b-\-c a b-\-c d 1st means. 6 -2 a-6 + 2 a 6-f C-- a b-\-kc. 2d means. 3a 6 4 5b-c , 3c d a b-\-c- a-b + kc. )de 3d means. a Tb-\c + i lc ^d+ e a b + a b-i-ckc. a b-{-kc. Where every column consists of a set of quantities, ap- proaching still nearer and nearer to the value of 5, the terms of each column being alternately below and above tliat value, and each succeeding column approaching nearerthan the pre- cedin? the proposed series. 10. Or, by taking the first terms of all the orders of means, we find the same set of theorems, namely a Za-b la-U + c 15a- lib + 5c- d "' ~T"' 8 ' 1 6 ' ' *' ^" general, ^x [(2"-l)a-{A-^n)b+ {ji ji.'-^) c - &c], which will be alternately above and below s, the value of the series, till the divergency is overcome. Then this series, which consists of the first terms of the several orders of means, maybe treated as the succcessive sqms, taking several orders of means of these again. After which, the first terms of these last orders may be treated again in the same manner ; and so on as far as we please. Or the series of second terms, or third terms, &c, or sometimes, the terms ascending obliquely, may be treated in tiie same manner to advantage. And with a little practice and inspection of the several series, whether vertical, or horizontal, or oblique, for they all tend to the detection of the same value s, we shall soon learn to distinguish where- abouts the required quantity > cr- nr ^ 1 to -"^^g ^ 1 10 _^ , to *- Or ' to to ^1 S 1 t^ c ^ ^ to O) ^ rr - <=C ^ ^ ?) g SI tc ,.,>- lO to >- ^ ^ ^ " 00 cr. -J Oi < 10 ^ K, I to Go 4^ Ot > .T^OOOtCXOO " '^' O OJ to (T> to r to c o to 03 to o - ^ '^ 4^ C^ CTi > CT> 10 o to '^ 03 .^ ,1 tooooo*-* ^ ^^ CD O to -J J OS o CO Or to ^ *- Oi -I -J CO 00 lOOOOOOCOO 1 '^COOCTitOCOkf'OO^ to to Oi ^ ^ K- CO O CD CTi I?) <3^ to o; oo -J Ol cc CO CO 05 , - to to 00 OS JS CI to -1 o to ^ - c; *; T g c to to CO t. o *> J to OC W w to to to Oi -3 -J f" O^ Oi ~3 lO -J .- I toooorOto^CTiOcriO^ i ^C^5^X^**^O^^dCC034^0TOTOi i-',oocoS?I;QO'^t~G<^o^CoooioOt ~*-j--i'~^^coooOi*'Ot'-oocO'-o:> ^^'OCOOOOOO COCO'OCOOn to o ^- - ^ S s g Tt i^ i' ^ S ^ ^ to ^ C5^2?OOCi Oi GCi^00tO>f- 03> jV-JCjO Cl-t-OCOC-jtO-IOCCCOO oio_'J*:^o>>-ooostooooiOt ^ Cl X ,?< CO Oi I Ci to CO I -t 00 to Cl Ot --1 ^ ^ C C C N OP TRACT S. The last mean '822467 is true in the last figure, the more accurate value of the series 1 |- + |- tV + ^*^5 being 8224670 &c. , 17. Let the diverging series ^ t + i y + ^*^' ^ proposed ; where the terms are the reciprocals of those iu Art. 13. Arithmetical means. Terms. Sue. sums. + 5 + 5 666666 _ ^66606 + 75 + 583333 8 _ 216666 + 833333 a. 616666 857143 240476 + 875 + 634524 888889 _ 254365 + 9 + 645635 909091 263456 + 9166G7 + 65321 1 188095 197024 190080 195635 191090 194878 192560 193552 192853 193362 192934 193056 193205 193110 193173 131 157 142 144 150 147 Here the successive suras arc alternately + and , as wel! as the terms themselves of the proposed series, but all the iirithmetical means are positive. The numbers in each co- lumn of means are alternately too great and too little, but so as visibly to approach towards each other. The same mu- tual approximation is visible in all the oblique lines from left to right, so that there is a general and mutual tendency, in all the columns, and in all the lines, to the limit of the value of the series. But with this dilTerencc, that all the numbers in any line descending obUquely from left to right, are on one side of the limit, and those in the next line in the same direction, all on the other side, the one line having its num- bers all too great, while those in tlic next line are all too little; but, on the coiitriirv, the lines Vvhich ascend from be- low obliquely towards the right, have their numbers alter- nately too great and too little, after the manner of those in the columns, but approximating quicker than those in the columns. So that, ;ilter having continued the columns of aritlimctical means to any convenient extent, we may tlien ^elect the terms in the last, or any other line obliquely as- cending from left to right, or rather beginning with the last found mean on the right, and descending towards the left; then arrange those terms below one another in a column, and TRACT 8. tNPINlTE SERIES. 193 take their continual arithmetical means, like as was done with the first succesive sums, to such extent as the case may re- quire. And if neither these new columns, nor the oblique lines approach near enough to each other, a new set may- be formed from one of their oblique lines which has its terms alternately too great and too little. And thus we may pro- ceed as far as we please. These repetitions will be more necessary in treating series which diverge more ; and having here once for all described the properties attending the series, with the method of repetition, we shall only have to refer to them as occasion shall offer. In the present instance, the last two or three means vary or differ so little, that the limit may be concluded to lie nearly in the middle between them, and therefore the mean between the two last 144 and 150, namely 147, may be concluded to be very near the truth, in the last three figures ; for as to the first three figures 193, repetition of them is omitted after the first three columns of means, both to save space, and the trouble of writing them so often over again. So that the value of the series in question may be concluded to be '193147 very nearly, which is = 4- + the hyp. log. of 2 ; or 1 less than its reciprocal series in Art. 13. 5 5.7 5.7.9 S. 7.9. 11 18. Take the diverging series 7-;^ + ; 4 4.6 ' 4.6.8 4.6.8.10 ' &.C. Here, first using some of the formulae, we have by the a 1st, - = -625. ^ 3a-b 2d, ;: = -57292. 4 , la Ab + c 3d, = '56966. o 4th, -^ = -56917. 16 . ?,\a-26h-\-\6c-6d+e 5th, = 'o690i. &c. ' 32 Or, thus, taking the several orders of means, &c. o 2 196 THE VALUATION OF f RAOr 8. Terms. + 1-25 - 1-438333 + 1-640625 - 1-804688 + 1-955079 - 2-094'727 + 2-225647 - 2-S4y294 Sue. sums. + 1-25 - 0-208333 + 1-432292 - 0-372396 + 1-582683 - 0-512044 + 1-713603 - 0-635691 Arithmetical means. 520833 611980 529948 605144 535320 600780 538956 566406 570964 567546 570232 568050 5G9868 8685 9255 8889 9141 8959 8970 9072 9015 9050 021 043 033 032 033 033 Here the successive sums are alternately + ^n^ ^^^ the arithmetical means are all -{-. After the second column of means, the first two figures 56 are omitted, being common ; and in the last three columns the first three figures 569, which are common, are omitted. Towards the end, all the num- bers, both oblique and vertical, approach so near together, that we may conclude that the last three figures 035 are all true; and these being joined to the first three 369, we have 369033 for the value of the series, which is otherwise found = - = -56903359 kc. 19. Let us take the diverging series V i'-hf V + &c, or 4 I + V or 4 4^ Terms. + 4 - 4-5 5-333333 6'25 7-2 8' 166666 9142857 10- 125 11-111111 12 1 13090909 14-085333 Arithmetical means. + 5^ 6^ + 7 j 8i + c^c. Sums. + 4- - 0-5 + 4'833333 - 1-416666 + 5-783333 2383333 + 6-759524 - 3365476 + 7-745635 4-354365 + 8-736544 5-34678S.' 21S8096 1-697024 2-190080 1-695635 2-191089 1-694877 1-942560 1-943557 1 -942^57 1 -943302 1 -942983 059 207 no 173 128 158 1 42 + c^C, 143 150 1.47 After the second column of means, the first four fi(J k .05 I , + ,,'f- + ILG'25 '^..J+ -^^.61328 ,,..^| - 4-9531-'|-l.?:,^ 1^.3.322461 + l-^-^'^2^^-l + 22-29296'.+ S^^-'^-^- + 5 _ + 875 _ 1-125 + 4-53125 14-4375 + 59-0234381 Here the same approximation in tlie lines and columns, to- wards the value of tlie series, is ooservable again, (viiv in a higher degree ; also the terms in the colunms and oblique descending linos, are again ahf-rnati^ly too great and too little, but now within narrower l.mits, and the signs of the terms arc more of them posit. ve ; also tht; terms in each ob- fujue asce"iidiijg line, are still eitiu'r all above or all below the value of tiie series, and that alternately one line after another, as Ijcfore. But the descending lines will again be tiie fittest to use, because the terms in eaca are alternatelv above and below the vahie souglit. Taking therefore again the hrst of these oblique descending lines, and treating it as before, we TRACT 8. INFINITE SERIES. 20 1 obtain sets of terms approaching still nearer to the viilue, thus : 25 343^5 25 361328 U)4336 492066 296875 296875 30j6:>4 277832 3-i5201 296875 301271 291"4S 310516 299073 296509 301132 297791 293821 29830S Here the approach to an equality, among all the lines and columns, is still more visible, and the deviations restricted within narrower limits, the terms in the oblique ascending lines still on one side of the value, and gradually increasing, while the columns and the oblique descending hnes, for the most part, have their terms alternately too great and too little, as is evident from their alternately becoming greater and less than each other : and from an inspection of the whole, it is easy to pronounce that the first three figures of the number sought, will be 298. Taking therefore the last four terms of tiie first descending line, and proceeding as before, we have 296875 299073 297791 293306 297974 298432 298048 298203 298240 298222 And, finally, taking the lowest ascending line, because it has most the appearance of being alternately too great and too little, proceed with it as before, thus : 298306 298048 29S24u 298222 29S177 293144 298231 298187 ' where the numbers in the lines and columns gradually ap- proach nearer together, till the last mean is true to the nearest unit in the last figure, giving us -298174 for the value of the proposed hypergeonietrical series 1 3 + 12 60 -{- 360 2520 + 20160 - ccc. And in like manner are we to proceed with any other series whose terms have alternate signs. Roval Military Academy, Woolwich, May, 1780. 202 THE VALUATION OF INFINITE SERIES. TRACT 8. POSTSCRIPT. Since the forciroin^ method was discovered, and made known to several friends, two passa<]^es liave been olTercd to my considiM-ation, Avhich I shall here mention, in justice to their authors, Sir I. Newton, and the late learned Mr. Euler. The first of these is in Sir Isaac's letter to Mr. Oldcnbm-g, dated October 24-, 1 616, and may be seen in Collins's Com- merciiwi Epistoliciuny p. 177, the last paragraph near the bottom of the piip;e, namely. Per sericm I.eihnitii etiam, si ultimo loco dimidium termini adjiciatur, et alia qucedam simi- ha artificia adhibeanfur, potest computum produci ad multas figuras. The series here alluded to, is 1 f + ' "" 7 "i'i~7T+ &c, denoting the area of the circle whose diameter is I ; and Sir Isaac here directs to add in half the last term, aft .r hav- ing collected all the foregoing, as the means of obtaining the sum a little exactcr. And this, indeed, is equivalent to taking one arithmetical mean between two successive sums, but it does not reach tlie idea contained in my method. It appears also, both by the other words, et alia qucedam similia artificia adhibeantur, contained in the above extract, and by these, f;/zV/i- artes adhibuissem, a little higher up in the same ];a. 177, t'lat Sir Isaac Newton had several other contrivances lor obtaining the sums of slowly converging series; but what tliey were, it may perhaps be now impossible to detenuine. The other is a passage in the A'ov/ Co;j:vn'nt. Pctropol. tom. V. p. 226, where Ah\ Euler gives an in t:iiice of taking one set of arithmetical means h::t\vce;i a slt'.cs of (juanlities which are gradually too little ami too great, to obtain a nearer value of the sum or a series in (|ues'[;on. Init neither doe-, this reicli the idea containc;! in onr uu.-thod. However, 1 have tliouuht it biit ;u^i:ice to ti:;: ciiaracters of these f.vo eminent meii, to n^a'.^ this mention (.!' tiieir ideiis, v.liich li;;vc some rehition to my own, t'neiigb i::i!^i!owr: to w.c ;.' tb.e time A my discovery. ( 203 ) TRACT IX. A METHOD OF SUMMING THE SERIES a -^ bx + CJf^ { dx^ -p ex"^ + S(c, WHEN IT CONVERGES VERY SLOWLY, NAMELY, WHEN X IS NEARLY EQUAL TO 1, AND THE COEFFICIENTST a, b, C, d, S(c, DECREASE VERY SLOWLY : THE SIGNS OF ALL THE TERMS BEING POSITIVE. ARTICLE 1. When there is occasion to find the sum of such series as that above-mentioned, having the coefficients a, b, c, d, &c, of the terms, decreasing very slowly, and the converging quantit)^ x pretty large ; the sum cannot be found by col- lecting the terms together, on account of the immense num- ber of them which it would be necessary to collect ; neither can it be summed by means of the differential series, because the powers of the quantity will then diverge faster than the differential coefficients converge. In such case then we must have recourse to some other method of transforming it into another series which shall converge faster. The follow- ing is a method by which the proposed scries is changed into another, which converges so much the quicker as the original series is slower. 2. The method is thus. Assume = the cfiven series D ^ a -\- bx -{- ex- -{- dx' + Sec. Then shall Dbe = --, ; TT-^y whicn, by actual division, is=:a /r a-\-ox-{-cx--{-&iC - ' b^^ , , 'Ibc b\ , , obd+c' Sbr b' Sec. Consequently a- divided by this series will be eqn.al to the series proposed , and this new scries will be very easily 204' OF SLOW SERIES WITH TRACT 9. sunrno:], in comparison with tlie original one, because all the coc'ilicients after the Sv'coiicl term are evidently very small ; and indeed thev are so uiiich the smaller, and fitter for sum- jnation, by how mucli the coefficients of the original series are nearer to ecjiuility ; so that, when these a, b, c, d, 8cc, are quite e(|ua!, then the third, fourth, &e, coefficients of the new ggries become equal to nothing, and the sum accurately equal to 7 = = ; which IS also known to be true* a-s~. bx a ax \~ x from other principles. 3, Though the first two terms, a bx, of tlie new series, be very great in comparison with each of the following terms, yet these latter may not always be small enough to be entirely rejected when much accuracy is recpiircd in the su.nmation. And in such case it will be necessary to collect a great num- ber of them, to obtain their sum jjretty near the truth ; be- cause their rate of converging is but small, it being indeed pretty nmch like to the rale of the original series, but only the terms, each to each, are much sm.dier, and that commonlv in a degree to the hundredlh or thousandth part. 4, The resemblance of this new series however, bcginnin with the third term, to the original one, in tlie law of pro- gression, intimates to us that it will be best to sum it in th(^ very satne manner as tlie former. Hence then putting b-" a c , a 2 be b' b' = d + -z. 2bd + f^ 3b'c a ' ir b- &c, and consequently the proposed series a -\- bx -\- ex'- -f- &^c. a bx a f^ 1) x^ ( .c '&c a bx xx [u -\-b x-^c x^&.C) by taking the .-{1111 of the series a' -{- b'x + f-i' + &e, by the TRACT 9. ALL TliEIR TERMS PdSlTl\'-E. 205 very same theorem as before, the sum s of the original setieiS will then be expressed thus, s == ^ '^-^^- ^1 --^> p a' - b'x-'((f r)^^ - id' + -ir^r^-Scc; a a a* where the series in the last denominator, having attain the same properties with the former one, will have its first two terms very large in respect of the following terms. But these tirst two terms, a'b'x, are themselves very small in compa* rison with the first two, a bx, of the former series ; and therefore much more are the third, fourth, &c, terms of this last denominator, very small in comparison with the same a bx: for which reason tliey may now perhaps be small enough to be Tieglected. 5. But if these last terms be still thought too large to be omitted, then find the suui of them by the very same theorem : and thus proceed, by re})eating the operation in the same manner, till the required degree of accuracy is obtained. Which it is evident, will happen after a small number of re- petitions, because that, in each new denominator, the third, fourth, &c, terms, iire couimonly depressed, in the scale of numbers, two or three places lower than the first and second terms are. And the general theorem, denoting the sum s when the process is continually repeated, \vill be this, aa a'a'xx a bx -77-,-/ b':. a X- a'"~b"'x a" a}'' XX b""x &c. a 6. But the general denominator d in the fraction , which is assumed for the value of s or a -f- bx -f <'.i'^ + &c, mav be otherwise found as below ; from which the general law of 206 OP SLOW SERIES WITH THACT 9. the values of the coefficients will be rendered visible. Assume s or a -{- bx -\- ex"- -\- &c, a"- a* or -p-; _; _ ; then shall D a bx ~ a X- b x^ ~ c x^ tec a'm= a -\- bx + cx^ + &c x a bx ax'' ^'.r' &c = a*-\-abx + acx^ + adx^ + aex* + qfx^ + &c ab bb be bd be a a db uc a'd b'a b'b b^c ca c'b d'a. Hence, by equating the coefficients of the like terms to no- thing, we obtain the following general values : bb a = c , a bd-{-cb a ' bb'-\-ca'4-db c = e d'=/- a be' -{-eb'-\- da -\-eb a ' bd'-\-ec'+db'-\-ea'-\-/b a ice. Wlierc the values of the coefficients arc the same in effect as before found, but here the law of their continuation is manifest. 7. To exemplify now the use of this method, let it be proposed to sum the very slow scries x -|- -^.r^ + f-^^ + ^'^ Avhen ^ = T^ = '^) denoting the hyp. log. of ,or,in this case, of 10. Now it will be ])roper, in the first place, to collect a few of the fir^t terms together, and then apply the theorem to the remaining term?, which, bv being nearer to an equality than the terms are near the beginning of the scries, will be TRACT 9. ALL THEIR TERMS POSITIVE. 20T fitter to receive the application of the theorem. Thus to collect the first 12 terms : No. Powers of *. The first 12 terms, found by dividins: x, x', x^, I 9 - - - - 9 &c, by the numbers 1, 2, 3, &.c. 2 81 - - - - 405 3 729 - - - - 243 4 6561 - - . 164025 5 59049 - - - 118098 G 531441 - - 0885735 1 4782969 - - 06832812857 8 43046721 - - 05380840125 9 387420489 043046721 10 3486784401 - 034867S4401 11 31381059609 - 02852823601 12 282429536481 02353579471 13 2541865828329 2"1708U62555 the sum of 12 terms. Then we have to find the sum of the rest of the terms after these first 12, namely of jt'^ x {t^+'^^x' + ^t^v^ +to-^^ + ^^')f in which x = -9, and x'^ ~ -2541865828329 ; also a =r i^j, I) =. tV) c rz -^-^, &c, and the first appHcation of our rule, gives, for the value of -^3- + -j^-^x -{- y-^x- + &c, or s, (-rL.)-' = -0059 171 59763 &c . _. 012637363-x''' X :'OO03-iO136+ 000279397x+ 000233o92a-'+ &c * the second gives -00591715976 . -0003401 36".r- * 01263/O6J .00U088678 - x2 X : -UUOOO-i.OSd + -000003000;? + &,o ' the third gives 00591715976 012G37363 'O003l0136\r'' -000088678 00000408 7\i'" 000001333 -^'-x : 00000008 9 + ike the fourth gives 00591715976 012637363-- -000340136'^- 00008867S-- O0a004087lr- 000001333- UU()O00089V' ~O000OOU344 208. OP SLOW SERIES WITH TRACT 9. Or, when the terms in the numerators are squared, it is 00591715976 '012637363 - 000000093710985 000088678 - 000000000013526212 000001333 0000000000000064 1 6 0000000344. Or, by omitting a j^roper number of ciphers, it is 0591715976 12637363- 0093710985 88678-- 013526212 1333 0064 1 6 344- z An unknown quantity :: is here placed after the last denomi- nator, to represent the small quantity to be subtracted from the said denominator 344. Now, rejecting the small quan- tity z, and beginning at the last fraction to calculate, their values will be as here ranq;ed in the first annexed column. Fractions. 518200000 1218931 11799 187 l.Ra. 2. Ra. 425 4-01 106 1-68 63 137 - 167 44 1^43 3. Ratio. 2-S9 1-68 X 187 63z 1-18 4. Ratio. 2-39 X 63.2 l-68xl87 2-03 placing z below them for the next unknown fraction. Divide then every fraction by the next below it, placing the quotients or ratios in the next column. Then take the quotients or 2^39 X 63z ratios of these : and so on till the last ratio --^ ^; which, l"68x 187 from the nature of the series of the first terms of every co- lumn, nmst be less tiian the next preceding one 2*39: con- , , , resx 187 ,, . ,, ^aquentiy z must be less tiian , or less than o. Jjut, from the natra-e of the series in the vertical row, or column of first ratios, - ' must be less than 63 ; and consennently z z must be greater tl.an \,y, or greater than 3, Since then TRACT 9. ALL THEIR TERMS POSITIVE. 209 z is less than 5 and greater than 3, it is probable that the mean value 4 is near the truth : and accordingly taking 4 for 2, or rather 4*3, as z appears to be nearer 5 than 3, and taking the continual ratios, as placed along the last line of the table, their values are found to accord very well with the next preceding numbers, both in the columns and oblique rows. Hence, using -043 for in the denominator 344 of the last fraction of the general expression, and computing from the bottom, upwards through the whole, the quotients, or values of the fractions, in the inverted order, will be 213 12079 1223397 518414000 of which the last must be nearly the value of the series -h + -r?*^ + -rr-^^ + &c, when x = -9. Then this value 518414 of the series, being multiplied by x'^ or -2541865828329, gives 1317738 for the sum of all the terms of the original series after the first 12 terms; to which therefore the sum of the first 12 terms, or 2'17081162, being added, we have 2*30258542 for the sum of the original series X + ix* + i.r^ + ^x^ + &c. Which value is true within about 3 in the 8th place of figures, the more accurate value being 2'30258509 &c, or the hyp. log. of 10. N. B. By prop. 8 Stirling's Summat. ; and by cor. 4, p. 65 Simpson's Dissert, the series a -^^ bx -^ cx^ -{- dx^ + &c, transforms to -^X[a-D(-^)+D'(-^r-D"(-^)' + D"'(-^r-]. lx ^ix' ^lx' M T ^lx' And thus the series x + ^.r^ + \x^ + &c, becomes -^ X [1 - i(-^ ) + ^{-^y-^{-^Y + &c], which \x ^ ^ix ^^lx ^^\x may be summed by our method. VOL. I. ( io ) TRACT X. THE INVESTIGATION OF CEUTAIN EASY AND GENERAL RULES, FOR EXTRACTING ANY ROOT OF A GIVEN NUMBER. 1. The roots of given numbers are commonly to be found, with much ease and expedition, by means of logarithms, when the indices of such roots arc simple numbers, and the roots are not required to a great number of figures. And the square or cubic roots of numbers, to a good practical degree of ac- curacy, may be obtained, by inspection, by means of my tables of squares and cubes, published by order of the Com- missioners of Longitude, in the year 1781. But when the in- dices of such roots are complex or irrational numbers ; or when the roofs are required to be found to a great many places of figures; it is necessary to make use of certain ap- proximating rules, by means of the ordinary arithmetical computations. Such rules as are here alluded to, have only been discovered since the great improvements in the modern algebra: and the persons who have best succeeded in their enquiries after such rules, have been successively Sir Isaac Newton, Mr. Raphson, M. de Lagney, and Dr. Halley; who have shown, that the investigation of such theorems is also useful in discovering rules for approximating to the roots of all sorts of compound algebraical equations, to which the former rules, for the roots of all simple equations, bear a con- siderable affinity. It is presumed that the following short tract contains some advantay;es over any other method that has hitherto been given, both as to the ease and universality of the conclusions, and the general way in which the investi- gations are made: for here, a theorem is discovered, Vihicli, though It be general for all roots whatever, is at the same time TRACT 10. RULE FOR EXTRACTING ROOTS. 2H very accurate, and so simple and easy to use and to keep in mind, that nothing more so can be desired or hoped for; and further, that instead of searching out rules severally for each root, one after another, our investigation is at once for any indefinite possible root, by whatever quantity the index is expressed, whether fractional, or irrational, or simple, or compound. 2. In every theorem, or rule, here investigated, N denotes the given number, whose root is sought, n the index of that root, a its nearest rational root, or a" the nearest rational power to N, whether greater or less, X the remaining part of the root sought, which may be either positive or negative, namely, positive when n is greater than a", otherwise negative. Hence then, the given number , N is = (a + x)", and the required root n" = a + .r. 3. Now, for the first rule, expand the quantity {a-\-x)''hy the binomial theorem, so shall we have 721 N == (a 4- t)" = a" + na^-^x + n . cC'^x' + &c. Subtract a" from both sides, so shall n \ N - a" = n a^-Kv + n . a'-^r" + &c. Divide by ncC~^, so shall N " N a" n~\ x- 71 I 71 2 x^ T~? or r- X a 2= .r -1 - 4- . - . - + &c. n a"-' 71 a" 2 a^ 2 3 a"- Here, on account of the smallness of the quantity x in respect of a, all the terms of this series, after the first term, will be very small, and may therefore be neglected without much error, which gives ~a for a near value of x, being only a small matter too great. And consequently N-f(?J-lV - , f , 1*1 a + X z=. ^ a IS nearly = n the root sought. And na" ^ * this may be accounted the first theorem. p2 312 A OtNERAL RULE TRACT 10. 4. Again, let the equation n = a" + 7i a""' jr f &c, be multiplied by n - 1, and a" added to each side, so shall we have (Ml) N + rt" = no" 4- ( - 1) .na'-'r + &c, for a divisor: Also multiply the sides of the same equation by a and subtract tf*+ ' from each, so shall we have n\ (n o") a = 71 a" X -f n . a"~^x"' |- &c, for a dividend : Divide now this dividend by the divisor, so shall T^ a" n\ x'^' ?il7i 2x^ -a = X 4- - + occ. (w-ON-f-rt" 2 a 2 3 a Which will be nearly equal to x, for the same reason as be- fore ; and this expression is about as much too little as the former expression was too great. Consecjuently, by adding a, we have a 4- .r or n ncarlv = ;; :, lor a second {n Ijs + a theorem, and which is nearly as much in defect as the former was in excess. 5. Now because t'ncr two foregoing theorems differ from the truth by nearly equal small quantities, if we add toge- ther the two numerators and the two denominators of the foregoing two fractional esprc^ssions, namel}- N+(W-I)a'= 72N -mi r a and -r cf,tlicsums will be the numera- n a [>i Ij.N f fl tor and denominator of a new fractioUy whicli Avill be much nearer than either of the former. The fraction so found is ?rf l.y \-n-l .a' - i -ii i i , f a ; Avhicii will be verv nearly equal to N y ?i 1 . N V f 1 ^' ' ^ or a ^ X, the root souglit ; for, by division, it is found to be ?i-l 7i\-\ x^ . , ... equal to ^7 + .z' *- --r + hich contiiins the square of .r, and the following terms are very small. And tins is tiic third theorem. G. A fourtli tlieorcm might be found by taking the arith- metical un.-an bclwc^-n tlie first and second, which Avould hc' TRACT 10. FOR EXTRACTING ROOTS. 213 , s + n-l .a" WN . rt ,. , , 1 ^ , \ -f r) X ; which will be nearly of the na" w l.N + a" 2 -^ same value, though not so shnple, as the third theorem ; for this arithmetical mean is found equal to 711 n 2 x^ ^ 1- >r* + ^.^ + &c. 7. But the third theorem may be investigated in a ftiore general way, thus: Assume a quantity of this form i^t " I J g-N + pa with coefficients p and q to be determined from the process ; the other letters n, a, n, representing the same things as be- fore ; then divide the numerator by the denominator, and make the quotient equal to a + x, so shall the comparison of the coefficients determine the relation between p and g re- quired. Thus, 71 1 pt; + ga''= {p + q)a:'+ pna^'^v^-pn . ^ a''-'.r^+ &c. 71 1 ^N -f pa'= {]) + g)a''-\-g7ia''-^x + qn .;^a'~'- x'' + &c. then dividing the former of these by the latter, we have pN + qa pq p q 711 qil .x'^ - . -, or a-\-x a \ --nx\n ( - - -; + &.c. yN-f/> V^H PVl 2 p-\-q a Then, bv ecjuating the corresponding terms, we obtain these three equations a = Oy p-q = 1, p\-q 71 1 qn ~2~ ~ JVq 0. From which we find - and p : q :: n ^ 1 \ yi 1. p-Vq n So that, by substituting n \- 1 and 7i 1, or any quanti- tities proportional to them, for p and q, we shall have 71-1-1 .N + 72-1 .a" r. , , c .1. J \ ; ;; zo for the value or the assumed quantity SI* A GENERAL RULE TRACT lO. pN-\-ga* , , , . ^i"} which is supposed nearly equal to a; + .r, the re- quired root of the quantity n. , , . , , w 4- 1 . N 4- w 1 . fl" f S. Now this third theorem : a = n , 711 . N + 71-1- 1 . ^" which is general for roots, whatever be the value of n, and whether a" be greater or less than n, includes all the rational formulas of De Lagney and Halley, which were separately investigated by them ; and yet this general formula is per- fectly simple and easy to apply, and easier kept in mind than any one of the said particular formulas. For, in words at length, it is simply this: to n+ 1 times n add w 1 times o", and to M 1 times n add ?i 4- 1 times a", then the former sum multiplied by a and divided by the latter sum, will give the root n" nearly ; or, as the latter sum is to the former sum, so is Oy the assumed root, to the required root, nearly. Where it is to be observed that a" may be taken either greater or less than n, but that the nearer it is to it, the better. 9. By substituting for n, in the general theorem, severally the numbers 2, 3, 4, 5, &c, we shall obtain the following particular theorems, as adapted to the 2d, 3d, 4tli, 5th, &c, roots, namely, for the 3n 4- c' J 2d or square root, ; ttto - - - - = ^ ' ^ N 4- 3a- 4n '\- 2a' 2n f a' ^ 3d or cube root, a, or tTTI.^ = ^' 2N 4- 4a^ N -f '2a^ 6n ^- 3fl* i 4th root - - - ^ a - - - - = n jN 4- 5a^ 6n 4- 4^'^ 3n 4- 2fl* 5th root - - - - 7-.^, or - -ri,a = n 'iN 4- Sa" -h 6th root - - - - . ,a - - - - N 5n 4- la'' 8n 1- 6a7 4n- f 3fl' 7th root - - - -a, or - r-;^ = n' Gn 4- 8' ' 3n 4- 4^7 ^c. then as 1448 : 1456 : : 19 : - = 19^1^ = 19-10497 &c. TRACT 10. FOR EXTRACTING ROOTS. 215 10. To exemplify now our formula, let it be first required to extract the square root of 365. Here N = 365, n = 2, the nearest square is 361, whose root is 19. Hence 3n + a^ = 3 X 365 + 361 = 1456, and N f 3fl^ = 365 + 3 x 361 = 1448; 19X 182 TsF Again, to approach still nearer, substitute this last found 19 X 182 root :; for tf, the values of the other letters, remain- 18 1 , , 19* X 182^ 3458* , mg as before, we have a^ = = , ,, ; then 3458* 47831059 3N + = 3 X 363 + _=^^ , 3x3458* 47831057 , N + 3.* = 365 + --:=-^, hence 19x182 3458 3458x47831059 47831057:47831059:: -or:-^^^^ zz the root of 365 very exact, which being brought into de- cimals, would be true to about 20 places of figures, 11. For a second example, let it be'proposed to double the cube, or to find the cube root of the number 2. Here n = 2, n = 3, the nearest root a:=.l, also a^ = 1. Hence 2n + ^^ = 4 + 1 = 5, and N + 2a^ = 2 + 2 = 4 ; 5 , ^ . . then as4:5::l: = 1*25=: the first approximation. 5 , , , 125 Again, take = t"j ^"" consequently a^ =z ; 125 381 Hence 2n + ^ = 4 + = -- , 64 64 250 378 andN+2.3 = 2 + = ; ^ 5 5 127 635 then 3-78:381, or as 126: 127 :: 4 = 4X Y^ = 5^=l'25992l, for the cube root of 2, which is true in the last figure, 216 A GENERAL RULE TRACT 10. 635 And by taking for the value of a, and repeating the process, a great many more figures may be found. 12. For a third e^cample let it be required to find the 5th root of 2. Here n = 2, w = 5, the nearest root a ^ I. Hence 3n + 2^^ = 6 f 2 = 8, and 2n + Sa^ = 4 + 3 = 7 ; g then as 7 : 8 : : 1 : = 1 j^ for the first approximation. Again, taking a = , ^ve have 65536 166378 98304 165532 then 165532 : 166378 : 4 i/ii!2=^ l.'^=?i21L 7 -7^82766 7 41383 289681 = 1-148698 &o, for the 5th root of 2, true in the last figure. 13. To find the 7th root of 126^. Here n = 1 26|, n = 7, the nearest root a = 2, also a' = 1 28. Hence 4n + Sa^ = 5044 -f 384 = 888i = , 5 4453 and 3n -f- 4a' = 378|. + 512 = 8904 = ; 8888 then 4453 : 4444 : : 2 : -^ = 1-995957, root very exact by 4453 " "^ one operation, being true to the nearest unit in tlie last figure. 14. To find the 365th root of i-05, or the amount of 1 pound for 1 day, at 5 per cent, per annun), comjiound in- terest. Here n = 1-05, n = 365, a zz 1 the nearest root. Hence' 366N -{- ^ify\u = 748 3, and 364N + 366a = 748-2; TRACT 10^ FOR EXTRACTING ROOTS. 217 ^483 then as 748*2 : 748-3 : : 1 : = 1^^ = 1*00013366, the root sought, very exact at one operation. 15. Required to find the value of the quantity (5^)"^ or (^^* )7. Now this may be done two ways ; either by finding the |. power or | root of y at once ; or else by finding the 3d or cubic root of ^ , and then squaring the result. By the first way : Here it is easy to sec that a is nearly = 3, because 3^ =v/ 27 = 5 + some small fraction. Hence, to find nearly the square root of 27, or V27, the nearest power to which is 25 = a^ in this case : Hence 3n + a^ = 3 x 27 + 25 = 106, and N -}- 3a' = 27 4- 3 X 25 = 102 ; 5 X 53 265 then 102 : 106, or 51 : 53 : : 5 : =-^=v/27 nearly. 21 3 f 265 Then having n n: ^, n ~ , a = 3, and a' =-rr nearly; . ... , ^ -I 5 21 1 265 6415 It will be|N + ia = X -\ 'X and iN 4- 4 a^ = - X -r + -^ x 51 408' 265 -6371 2 4^2 51 408' 19245 ^ , hence 6371 : 6415 : : 3 : --- = 8-5^7^ = 3-020719, for the value of the quantity sougtit nearly, by this wa}'. Again, by the other method, in finding first the value of I (-_?)7^ or the cube root of y . It is evident that 2 is the nearest integer root, being the cube root of 8 = a^. Hence 2n 4- a^ = 2_i _j_ g 7^4^ and N 4- 2a^ = V 4- 16 = V J 1^-8 7 , ^, , . 7 then 85 : /4 : : 2:--, or = -nearly. Then taknig for a, 21 343 1015 we have 2n 4- a' = + TTp^ ~;n~* 2 o4- 64 21 2.343 1022 and N 1 2a^ = 4- -rr = "TT" ' 4 64 64 21S NEW METHOD FOR TRACT 11. 7 7 145 ' hence 1022 : 1015, or 146 : 145 ::- : -x =(V)^ nearly. 4 4 146 2 Consequently the square of this, or (V)^ ^^'i'l ^^ = 7' 145^ 1030225 4^ ^ Tii^= 341056 = ^7l^^ = 3-020690, the quantity sought more nearly, being true in the last figure. TRACT XI. A NEW METHOD OF FINDING, IN FINITE AND GENERAL TERMS, NEAR VALUES OF THE ROOTS OF EQUATIONS OF THIS FORM, JV" p.v"~^ + qx"~'^ &C = ; NAMELY, HAVING THE TERMS ALTERNATELY PLUS AND MINUS. 1. The following is one method more, to be added to tlu- many we are already possessed of, for determining the roots of the higher equations. ]3y means of it we readily find a root, which is sometimes accurate ; and when not so, it is ;ir least near the truth, and that by an easy finite forn:iula, which is general for all equations of the above form, and of the same dimension, provided that root be a real one. This is of use for depressing the equatiofi down to lov/cr dimensions, and thence for finding all the roots, one after anotiier, when the formula gives the root sufficiently exact ; and when not, it serves as a ready means of obtaining a near value of a root, by which to commence an approximation still nearer, b}' the previously known methods of Newton, or Halley, or others. This method is further useful in elucidating the nature of equations, and certain properties of numbers ; as will appear in sonwi of the following articles. Vic have already easy me tliods for finding the roots of simple and quadratic equations TRACT 11. THE ROOTS OP EaUATIONS. 219 I hall therefore begin with the cubic equation, and treat of each order of equations separately, in ascending gradually to the higher dimensions. 2. Let then the cubic equation x^ px^ + ^:r r = be proposed. Assume the root t =: a, either accurately or ap- proximately, as it may happen, so that x a =.0, accu- rately or nearly. Raise this ;r a = to the third power, the same dimension with the proposed equation, so shall x^ 3ax' + 3a^x a^ = 0; but the proposed equation is x' /;.r* + 9^ ^' =^ 0; therefore the one of these is equal to the other. But the first term (.r^) of each is the same ; and hence, if we assume the second terms equal between themselves, it will follow that the sum of the two remaining terms will also be equal, and give a simple equation by which the value of x is deter- mined. Thus, 3ax'^ being =. px^^ or a = ^p, and ^a-x a^ = qx r, we hence have a' r ('n)3_r p'o'jr \ , . . X z=. - ; = - ~ = ' X by substitutmrj ^ar-q 3x{-^py--q p^-3q 9 ^ " -\-py the value of a, instead of it. 3. Now this value of x here found, will be the middle root of the proposed cubic equation. For because a is assumed nearly or accurately equal to x, and also equal to ^p, there- fore X i^ = j-p nearly or accurately, that is, j- of the sum of the three roots, to which tiie coefficient p, of the second term ot the equation, is always equal ; and thus, being a medium among the three roots, it will be either nearly or accurately equal to the middle root of the proposed equation, Avhenthat root is a real one. 4. Now this value of x will always be the middle root ac- curately, whenever the three roots are .in arithmetical pro- gression; otherwise, only approximately/. For when the three roots are in arithmetical progression, 4-p or x of their sum, it is well known, is equal to the middle term or root. In the other cases, therefore, the above-found value of x is only near the middle root, 220 NEW METHOD FOR TRACT 11. 5. Wl>en the roots are in arithmetical progression, because the middle termor root is then =4p, and also =-x- - j ^'^ 9 p^ 3q 1 /)' 27?- therefore |/j = - x n^ r~ "'" -P^ ~ ^P1 ~ 27r = 9 x (pg 3r) , ah equation expressing the genera! relation of p, g, and r ; where p is tljc sum of any three terms in arithmetical pro- gression, g the sum of their three rectangles, and r the pro*, duct of all the three. For, in any equation, the coefficient p of the second term, is the sum of the roots ; tiie coefficient ^ of the third term, is the sum of the recta\)g!es of the roi)ts ; and the coefficient r of the fourth term, is the sum of the solids of the roots, which ii' the case of the cubic equation is only one: Thus, if the roots, or arithmetical terms, be 1,2,3. Here/> =:l+2 + 3 = 6,g = l x2 + lx3-r2x3 = 2 + 3+6=11, r=l X 2x 3=6; then '2p^ 2 X 6' rz 432, and 9 x {pg ~ 3r) = 9 x 48 = 432 also. 1 p^2lr 6. To illustrate now the rule x = x -, bv some 9 p"- 'ig " examples ; let us in the first place take the equation x^ 6.r^ + llx 6=0. Here /)=C, j subDtituliug 11;- for x in the proposed eqiiatiou, the TRACT II. THE ROOTS OF EGUATIONS. 221 negative terms are found to exceed the positive terms by 5, thus showing that llf is very near, but something above, the middle root, and that therefore the roots are not in arith- metical progression. It is therefore probable that 1 1 may be tlie true value of the root, and on trial it is found to succeed. Then dividing x^ - 39^' + 479.r - 1881 by .r - 11, the quotient is x- 2Sx + 171 =0, the roots of which quadra- tic equation are 9 and 19, the two other roots of the pro- posed equation. 8. If the equation be x^ 6x^ + 9r 2 = ; we shall have p =i 6, q =: 9y and r rr 2 ; then x = 1 p^ 2lr_ 1 6^-27x2_ 2'-2_ 6 _ T ^ p^-~Sq ~Y ^ 6'- 3x9 "^ 12 -9 ~ T ~ ^* This value of x being substituted for it in the proposed equa- tion, causes all the terms to vanish, as it ought, thus showing that 2 is the middle root, and that the roots are in arithmeti- cal progression. Accordingly, dividing the given quantity x^ 6\r' -f 9.r 2 by ^ 2, the quotient is x- 4^+1 =0, a quadratic equation, whose roots are 2 + ^/2 and 2 V2, the two otiier roots of the equation proposed. 9. If the equation be x^ 5x^ + 5x 1 z= ; we shall have p = 5, q = 5, and r I ; then .r r: 1 5^-i?''Xl_ 1 125-27 _ I 98 _ 49 _ '9^5-- Vy'x5~~j'^ 25\5~~9^To~T3~ ^^' From which one might guess the root ought to be l,and which being tried, is found to succeed. But without such trial, we inigiit maue use of this value 1^^-, or l^-x nearly, and approximate with it in the common way. Havin"- found the middle root to be 1, divide the given quantity x^ 5.r^ + 5.r 1 by .r 1, and the quotient is X- 4x + 1=0, the roots of which are 2 + y/2, and 2 V 2y the two other roots, as in the last article. 10. If the equation be .r^ 7.r- + IS^ 18 = 0; we shall have /; =: 7, q = IS, and r = IS ; then x 1 7^--27xJ8 1 343-486 143 X = X ^-1 ', -" = 3 A or 3 nearly. 9^7- 3xia 9^ 49-54 4o *' ^ 222 NEW METHOD FOR TRACT II. Then trjdng 3 for x, it is found to succeed. And dividing x^lx^-\-l^x-\S by .r-3, the quotient is ^^ 4^ + 6 =0, a quadratic equation whose roots are 2 f ^ 2 and 2 V 2, the two other roots of the proposed equation, which are both impossible or imaginary. 11. If the equation be .r^ 6^* + 14:r 12 = ; we shall havep = 6, * 2.5fo)] ^ = 8 X {lr4q) ' or if we put a = |/;- 4^, B = p^ 16r, c = /?* ~ 256^, , .,, , B \/(b- AC) the two roots will be x =- . 8 A 15. It is evident that the same property is to be understood here, as for the cubic c(juation in Art. 0, namel}', that the two roots above found, arc the middle roots of the four which belong to the biquadratic equation, when those roots are real ones ; for otherwise the formulae are of no use. Uut how- ever those roots will not be accurate, when the sum of the two middle roots, of the proposed equation, is equal to the sum of the greatest and least roots, or when the four roots arc in arithmetical progression ; because that, in this case, ^py the assumed value of a, is neither of the middle roots exactly, but only a mean between them. .r rr re .X- p 1 B^(B^ + Ac) 16. 1 o exempluy this formula x=- :; , let the ^ -^ 8a proposed equation be x'* I2x'+ i-Ox'-lS.r 1 10=0. Then TRACT 11. THE ROOTS OF EaUATIONS. 225 A.=|./?* 4^=l2*x|-~ 4x49= 216 196= 20, B= p3 l6r=I2' 16X78= 1728 1248= 480, C= />* 256^=12* 256x40 = 20736 10240=10406. _ Bi V(B'-Ac)_48O^/(480^-20xlO496) __ 8a 8 X 20 15 + V/40 , 1,1 = 3 + li nearly, or 4^ and 1|. nearly, or nearly 4 and 2, whose sum is 6. And trying 4 and 2, they are both found to answer, and therefore they are the two middle roots. Then {x 4) x {.r 2) x"- 6r + 8, by which divid- ing- tlie given equation x* 12;r^ 4- 49.r^ I8x + 40 = 0, the quotient is x' 6x -\- 5 := 0, the roots of which quadra- tic equation are 5 and 1, and which therefore are the greatest and least roots of the equation proposed. 17. If the equation be x*12x^ + 'i7x'-~72x-\-36 0; then A-ip^ 4y=]2^X|--- 4x47=x 216 188= 28, B= p3 I6r = 125 16x72= 1728 1152= 576, C= p* 2565=12'* 256x36=20~36 9216 = 11520. Bv/(b'-Ac) 576-tx/(576' 28 X 11520) Hence x =. - =: =: 8a 8x28 . 18 4-3 z=. 3 and 2-f, or 3 and 2 nearly ; both of which an- swer on trial ; and therefore 3 and 2 are the two middle roots. Then (r-3)x (.r-2) rr x^'-jx + 6 = 0, by which divid- ing the given quantity :r* I'Jx^ -\- 47^^ 72r + 36 = 0, the quotient is x^ 7^' -{- 6 = 0, the roots of which quadratic equation are 6 and 1, which therefore are the greatest and least roots of the equation proposed. 18. If the equation be ^'^ 7.r3 + 15^^ 1 lx + 3 = ; then A = -i.// 4^ = 7^x1 4x15= 73I- 60= 13^, B= ]P I6r = 7^ 16x11= 34'i 176 rz 167, Q p* 2565=7^ 256 X 3 = 2401 768=1633. B v/(B' - AC) _ 167 4-^(167-15-:^ 1633) _ 8a ~" ~8^']Jl " = 2i and -^-^ nearly, or nearly 2 and 1 j both which Q Hence x = 167 + 76 108 VOL. I. 22 S NEW METHOD TOU tRACT ttv are found, on trial, to answer ; and therefore 2 and 1 are the two middle roots sought. Then {x-2) x {x l) = x^ Sx + 2, by which divid- ing the giveit equation x* Ix^ 4- I5jr* ^ 1 lor + 3 = 0, the Quotient \s x^ - 4je + 1 =0, the roots of which quadratic equation are 2+^/2 and 2 y/2, and which therefore ar6 the greatest and least roots of thd proposed equation. 19. But if the eqaa.be ^*9jr^ + 30jr^ 46^ + 24=0; then k = ip'- 4q=9^xi-^ 4x30= 121i 120= H, B= p^ 16- = 9^ 16x46= 729 736 = -7, e= p* 256s = 9* 256x24 = 6561 6144=417. ^- B^/(B*-AC) -7 ^/(49-625i) Hence .r = ^ = 8a 8xli - -y an imaginary quantity, showing that the two middle roots are imaginary, and therefore the formula is of no use in this case^ the four roots being 1,2 + y' 2, 2- V 2, and 4. 20. And thus in other examples the two middle roots will be found when they are rational, or a near value when irra- tional, which in this case will serve for the foundation of a nearer approximation, to be made in the usual way. We miglit also find another formula for the biquadratic equation, by assuming the last terms as equal to each other; for then the sum of the 2(1, 3d, and 4th terms of each would be equal, and would form another quadratic equation, whose roots would be nearly the two middle roots of the biquadratic proposed. 21. Or a root of the biquadratic equation may easily be found, by assuming it cquy]x^ - 2ah [a + b) x + ^"^' = 0. For, comparing the terms of this with the terms of tlie eqi'.ation proposed, in this manner^ Jiatnely, making the second terms equal, then the third terms equal, and lastly the sums of the fourth and fifth terms equal, these enrations will determine a near value of jr by a simpl aquation. For those equations are ITRACT 11. THE ROOTS OP EftUATIONS. ffif? p=2(a + b), or ipz=a+ b, q = 2ab -{-{a -{ b)- = Q,ab + J-p*, or 2ab = q ^p*, rx$ = 2ab{a + b) x O-b'- - ^y {q-~^p^)r~-}{q ^p')\- Then the values of ab and fl + ^, found from the first and second of these equations, and substituted in the third, give X = . rr '- -r ^-rr, a general formula for one of the roots of the biquadratic equation x'^ px^ + ox^ rx -{ s = 0. 22. To exempUfy now this formula, let us take the same equation as in Art. 17, namely, x^ 12.r^ + 41x' 72.r + 36 = 0, the roots of which were there found to be 1, 2, :i, and 6. Then, by the last formula we shall have x = e4s-{4q-p y' 64 X 36 -(4x47-12 ")- _ 64x 36 -44x44 64r -8/>(4^-p")~64X 72-96(4 X 47- lF)"~64x 72 -96 x 44 = !, or nearly 1 , which is the least root. 23. Again, in the equation r^ 7^1^+1 5.r 11x^+3 = 0, whose roots are 1 , 2, 2 4- -y/S, and 2 V2, Ave have x = 64 x3- (60-49)- 64x3 llx 11 ^ 92 --J12J_ _ ^ ^ _ 64x 11 56(6d-49)~64x ll-56x 11~704-616 ~^^~^ nearly, which is nearly a mean between the two least roots 1 and 2 ^2 or | nearly. 24. But if the equation be x* g.r^ + SO^'* 46.r + 24=0, which has impossible roots, the four roots being 1, 2 + y^ 2, 2 x/ 2, and 4 ; we shall have x = 64x24 (120-81)* _64x24-39x39 _ 1536-15 21 ^ 64x46-72(120 81) "64x46-72x39 "~ 2944- 2808 ~ ^5^ = -^ nearly, which is of no use in this case of imaginary roots. 25. This formula will also sometimes fail when the roots are all real. As if the equation be ^'^ 12.r' + 49jr- 78:r + 40=0, the roots of which are 1, 2, 4, and 5. For here x =. 64x40 (196 144)' 64x40-52x52 16x10 -13x13 64x78 -'96(1 96 -144) ~ 64x78-96 x 52 ~ 16X19^-24x13 160-169 -9 , . , p 1 ^ = = , which IS or no use, bemgmnnite. 312-312 ' > o a2 K8 OF THE BINOMIAL THEOREM. TRACT f2. 26. For equations of higlier dimensions, as the 5th, the 6tb, the 7th, &c, we might, in imitation of this last method, com- bine other forms of quantities together. Thus, for the 5th power, we might compare it either with (r g)* x {r b), or with (jT ay x (.r bf, or with [x a)' x (x b) x {x c), or with (,r a^ x (.r b)- X {x c). And so for the other powers. TKACT XII. OF THE BINOMIAL THEOREM. WITH A DEMONSTRATION OF THE TRUTH OF IT IN THE GENERAL CASE OF FRAC- TIONAL EXPONENTS. 1. It is well known that this celebrated theorem is called binomial, because it contains a proposition of a quantity con- sisting of two terms, as a radix, to be expanded in a series of equal value. It is also called emphatically the Newtonian theorem, or Newton's binomial theorem, because he has com- monly been reputed the author of it, as he was indeed for the case of fractional exponents, which is the most general of all, and includes alt the other particular cases, of powers, or divisions, &c. 2. The binomial, as proposed in its general form, was, by m Newton, thus expressed p -f pq ; where p is the first term of the binomial, q the quotient of the second term divided bv the first, and consequently pq is the second term itself; or pa may represent all the terms of a multinomial, after the fust term, and consequently q the quotient of all those terms, except the iirst term, divided by that first term, and may be either positive or n(>gative ; also represents the exj)onent of the binomial, and may denote anv (juantitv, intcral or TRACT 12. OF THE BINOMIAL THEOREM. 229 fractional, positive or negative, rational or surd. When the exponent is integral, the denominator n is equal to 1, and the quantity then in this form (p -\- pq)'", denotes a binomial to be raised to some power ; the series for which Mas fully determined before Newton's time, as will be shown in the course of the 19th Tract of this volume. When the ex- ponent is fractional, m and n may be any quantities what- ever, m denoting the index of some power to which the binomial is to be raised, and w the index of the root to be extracted of that power: and to this case it \vas first extended and applied by Newton. When the exponent is negative, the reciprocal of the same quantity is meant ; as _^ 1 (p-fpa) " is equal to / (p -1- pq)~ 3. Now when the radical binomial is expanded in an equi- valent series, it is asserted that it will be in this general form, namely (p -|- pa)" or p* x (1 + a) " = ^ mm m n m m n m 2 p" X 1 + -Q 4- - . -^Q^' + - ^7" -57-^' + &c), ^ m m n m In m 3 or p X 1 + AG + ~^Ba -I- "sT'ca + -j;^ dq + &c. where the law of the progression is visible, and the quanti- ties p, ??i, n, Q, include their signs -f or , the terms of the series being all positive when a is positive, and alternately positive and negative when a is negative, independent how- ever of the effect of the coefficients made up of m and n : also A, B, c, D, &c, in the latter form, denote each preceding term. This latter form is the easier in practice, when we want to collect the sum of the terms of a scries; but the former is the fitter for showing the law of the progression of the terms. 4. The truth of this series was not demonstrated by New- ton, but only inferred by w^ay of induction. Since his time however, several attempts have been made to demonstrate it, with various success, and in various ways; of which however 230 OF THE BINOMIAL THEOREM. TRACT 12/ those are justly preferred, wbich proceed by pure algebra, and without the help of fluxions. And such has been es- teemed the difficulty of proving the general case, independ- ent of the doctrine of fluxions, that many eminent mathe- maticians to this day account the demonstration not fully accomplished, and still a thing greatly to be desired. Such a demonsti-ation I think is here effected. But before deliver- ing it, it may not be improper to premise somewhat of the history of this theorem, its rise, progress, extension, and de- monstrations. 5. Till very lately the prev^ailing opinion has been, that the theorem was not only invented by Newton, but first of all by him ; that is, in that state of perfection in which the terms of the series, for any assigned j)Ower whatever, can be found independently of the terms of the preceding |)owers ; namely, the second term from the first, the third term from the second, the fourth term from the third, and so on, by a general rule. Upon this point I have already given an opi- nion in the history to my logarithms, above cited, and I shall here enlarge somewhat further on the same head. That Newton invented it himself, I make no doubt. But tliat he was not the first inventor, is at least as certain. It was described by Briggs, in his Trigonometria Britannica, long before Newton was born; not indeed for fractional ex- ponents, for that was the application of Newton, but for any integral power whatever, and that by the general law of the terms as laid down by Newton, independent of the terms of the powers preceding that which is required. For as to the generation of the coefficients of the terms of one power from those of the preceding powers, successively one after another, it was remarked by Vieta, Oughtred, and many others, and v.as not unknown to much more earlv writers on arithmetic ami algebra, as will be manifest by a slight inspection of their v.'orks, as well as the gradual advance the property made, both in extent and perspicuity, under the hands of the suc- cessive masters in arithmetic, every one adding somcwhai more towards the perfection of it. TRACT 12. OP THE BINOMIAL THEOREM. 231 .6. Now the knowledge of this property of the coefficients of the terms in the powers of a bingmial, is at least as old as the practice of the extraction of roots; for this property was both the foundation and the principle, as well as the means of those extractions. And as the writers on arithmetic be- came acquainted with the nature of the coefficients in powers still higher, just so much higher did they extend the extrac- tion of roots, still making use of this property. At first it seems they were only acquainted with the nature of the square, which consists of these three terms, 1, 2, 1 ; and accord- ingly they extracted the square roots of numbers by means of them ; but went no further. The nature of the cube next presented itself, which consists of thetje four terms, 1, 3, 3, 1 ; and by means of these they extiMcted the cubic roots of num- bers, in the same manner as we do at present. And this was the extent of their extractions in the time of Lucas de Burgo, an Italian, who, from 1470 to 1500, wrote several tracts on arithmetic, containing the sum of what was then known of this science, which chiefly consisted in the doctrine of the proportions of numbers, the nature of figurate numbers, and the extraction of roots, as far as the pubip root inclvjr s;vely. 7. It was not long however before the nature of the co-f /efficients of all the higher powers became known, and tables formed for constructing them indefinitely. For in the year 1544- came out, at Norimberg, an excellent treatise of arith- metic and algebra, by Michael Stifelius, a German divine, and an hojiest, but a weak, disciple of Luther. In this work, Arlthmetka Integra, of Stifelius, are contained several curious things, some of which have been ascribed to a much later date. He here treats, pretty fully and ablv, of progressional and figurate numbers, and in particular of the following table for constructing both them and tiie coefficients of the terms of all powers of a binomial, which has been so often used since his time for these and other purposes, and which more than a century after was, by Pascal, otherwise called the. 232 OF THE BINOMIAL THEOREM. TRACT 12. arithmetical triangle, but who only mentioned some addi- tional properties of the table. 1 2 3 3 4 6 t 5 10 10 6 15 20 7 21 35 35 8 28 56 70 9 36 84 126 126 10 45 120 210 252 11 55 165 330 462 462 12 66 220 495 792 924 13 78 286 715 1287 1716 1716 14 91 364 1001 2002 3003 3432 15 105 455 1365 3003 5005 6435 16 120 560 1820 4368 8008 11440 17 136 680 2380 6188 12376 19448 6435 12870 24310 Stifelius here observes that the horizontal lines of this table furnish the coefficients of the terms of the correspondent powers of a binomial ; and teaches ho\y to use them in ex- tracting the roots of all powers whatever. And after him the same table was used for the same purpose, by Cardan, and Stevin, and the other writers on arithmetic. I suspect how- ever, that the nature of this table was known much earlier than the time of Stifelius, at least so far as regards the pro- gressions of figurate numbers, a doctrine amply treated of by Nichomachus, who lived, according to some, before Eu- clid, but not till long after him according to others. His work on arithmetic was published at Paris in 1538 ; and it is supposed was chiefly copied into the treatise on tijc same subject by Boethius : but I have never seen either of these two works. Though indeed Cardan seems to ascribe the in- vention of the table to Stifelius; but I suppose that is only to be understood of its application to the extraction of root-:. Sec Cardan's Opus Novum de Proportionibiis, where he quotes it, and extracts the table and its use from Stifeiius's l)ook. Cardan also, at p. 185, ct seq. of the same work, makes uj^e TRACT 12. OP THE BINOMIAL THEOREM. 23S of a like table to find the number of variations of thinsrs, or conjugations as he calls them. 8. The contemplation of this table has probably been at- tended with the invention and extension of some of our most curious discoveries in mathematics, both in regard to the powers of a binomial, with the consequent extraction of roots, the doctrine of angular sections by Vieta, and the dif- ferential method by Brisrgs and others. For, one or two of the powers or sections being once known, the table would be of excellent use in discovering and constructing the rest. And accordingly we find this table used on many occasions by Stifelius, Cardan, Stevin, Vieta, Briggs, Oughtred, Mer- cator, Pascal, &c, &.c. 9. On this occasion I cannot help mentioning the ample manner in which I see Stifelius, at fol. 35, et seq. of the same book, treats of the nature and use of logarithms, though not under the same name, but under the idea of a series of arith- meticals, adapted to a series of geometrical s. He there ex- plains all their uses ; such as, that the addition of them, an- swers to the multiplication of their geometiicals; snl)traction to division ; multiplication of exponents, to involution; and dividing of exponents, to evolution. And he exemplifies the use of them in cases of the RuIe-of-Three, a!id in finding mean proportionals between given terms, and such like, ex- actly as is done in logarithms. So that he seems to have been in the full possession of the idea of logarithms, and wanted only the necessity of troublesome calculations to in- duce him to make a table of such numbers. 10. But thougii the nature and con^truction of this table, namely of figurate numbers, was thus early known, and em- ployed in the raising of powers, and extracting of roots; yet it was only by raising the numbers one from another by con- tinual additions, and then taking them from the table for use when wanted; till Briggs first pointed out the way of raising any horizontal line in the foregoing table by itself, without any of the preceding hnes ; and thus teaching to raise the terms of any power of a binomial, independent of any other JJ34 OF THE BINOMIAL THEOREM. TRACT 12. powers ; and so gave the substance of the binomial series in words, wanting only the notation in symbols ; as it is shown at large in the 19th Tract, in this Toiume. 11. Whatever was known however of this matter, related only to pure or integral powers, no one before Newton hav- ing thought of extracting- roots by infinite series. He hap- pily discovered, that, by considering powers and roots in a continued series, roots being as powers having fi'actional ex- ponents, the same binomial series v.'ould equally serve for them all, whether the index should be fractional or integral, or the series be finite or infinite. 12. The truth of this method however Mas long known only by trial in particular cases, and by induction from ana- logy. Nor does it appear that even Newton himself ever attempted any direct proof of it. But various demonstrations of this theorem have been since given by the more modern mathematicians, of which some are by means of the doctrine effluxions, and others, more legally, from the pure principles of algebra only. Some of which I shall here give a short ac count of. 13. One of the first demonstraters of this theorem, was Mr. James Bernoulli. His demonstration is, among several other curious things, contained in this little work called Ars Co)7Jectandi, which has been improperly omitted in the col- lection of his works published by his nephew Nicholas Ber- noulli. This is a strict demonstration of tlic binomial theorem in the case of integral and affirmative powers, and is to this effect. Supposing the theorem to be true in any one power, as for instance, in the cube, it must be ti lie in the next higher power ; which he demonstrates. 13ut it is true in the cube, in the fourth, fifth, sixth, and seventh powers, as will easily ii[)pcar by trial, that is by actually raising those jiowers by continual multiplications. Therefore it is true in all highcu' powers. All this he shows in a regular and logitin)ate n)an- )ii;r, from the principles of multiplicatiun, and wilhcut; the 1['RAC3T 12. OF THE BINOMIAL THEOJIEM. 235 help^of fluxions. But he could not extend his proof to the other cases of the binomial theorem, in which the powers ar^ fractional. And this demonstration has been copied by Mr. John Stewart, in his commentary on Newton's quadrature of curves. To which he has added, from the principles of fluxions, a demonstration of the other case, for roots or frac- tional exponents. 14. In No. 230 of the Philosophical Transactions for the year 1697, is given a theorem, by Mr. De Moivre, in imita- tion of the binomial theorem, which is extended to any num- ber of terms, and thence called the multinomial theorem ; which is a general expression in a series, for raising any multinomial quantity to any power. His demonstration of the truth of this theorem, is independent of the truth of the binomial theorem, and contains in it a demonstration of the binomial theorem as a subordinate proposition, or particular case of the other more general theorem. And this demon- stration may be considered as a legitimate one, for pure powers, founded on the principles of nmitiplication, that is, on the doctrine of combinations and permutation^. And it proves that the law of the continuation of the terms, must be the same in the terms not computed, or not set down, as in those that are written down. 15. The ingenious Mr. Landen has given an investigation of the binomial theorem, in his Discourse concerning the Re- sidual Analysis^ printed in 1758, and in t\\it Residual Analysis itself, printed in 1764-. The investigation is deduced from this lemma, namely, if ? and n be any integers, and ^ 21 then is f m m x'^ V" --I \ -\- a {- q"- -\- q^ - - (w) x" X X V '" ^"' '-J'" ^+q" +q"-\-q'^ - - () which theorem is made the principal hasis of his Residual Analvsis. 236 OF THE BINOMIAL THEOREM, TRACT 12. The investigation is thus : the binomial proposed li^eing m (1 + x)n, assume it equal to the following series I -^ ax -\- bx^ -\- cx^ &c, Avith indeterminate coefficients. Then for the same reason as (1 + x)~ is =r 1 + ax + bx^ -\- cx^ &c, m will ( 1 + y)~ be = 1 + cry + 3j/^ + cy^ &c. Then, by subtraction, m m (1 +.r)"^- (1 +.y)~= a {x-T/)-\- b (x^ y) 4- 0(^3 -y) &c. And, dividing both sides by xy, and by the lemma, we I have '+->7_<; + '^'" =(l+..r X 14"-^ 1 + '^ 1 -j- a "1 2m 3m =a-\-b{x +j/) + c iv' -\- xy +y) ^d{x^-\- x'-y + xy"- -\-y'^) &c. Then, as this equation nuist hold true whatever be the value of y, take y rr x, and it will become m -i^ X (I + x)~~^ a ^ 2bx ^ 'icx- + \cx'^ &c. n Consequently, multiplying by 1 + x , v.e have -- X (1 + x)" , or its equal by the assumption, VIZ. 4- ax A bx^ + ~cx^ is^c. n ^ n n n , 2b) , 3c i ^ , 4d) , . Then, b}' comparing the homologous terms, the value of the coefficients a, b, c, &:c, are deduced for as many terms as are compared. A large account is also given of this investigation by tlic learned Dr. Hales, in his Analysis Equationum, lately ))ub- lished at Dublin. Mr. Landen then contrasts this investigation with that bv TRACT 12. OP THE BINOMIAL THEOREM. 237 the method of fluxions, which is as follows. Assume as before ; m (1 + xf - \ -{- ax { bx^ + cx^ + dx* &c. Take the fluxion of each side, and we have X {I + x) X X = ai -{- 2bxi + 'icx^i &c. Divide b)' x, or take it =z 1 , so shall m ~ X {\ -\- xf ' = a 4- 2bx + 3c^- + Ad.v^ &c. Then multiply by 1 + x, and so on as above in the other way. 16. Besides the above, and an investigation by the cele- brated M. Euler, which are the principal demonstrations and investigations that have been givenofthis important theorem,! have been shown an ingenious attempt of Mr. Baron Maseres, to demonstrate this theorem in the case of roots or fractional exponents, by the help of De Moivre's multinomial theorem. But, not being quite satisfied with his own demonstration, as not expressing the law of continuation of the terms which are not actually set down, he was pleased to urge me to at- tempt a more complete and satisfactory demonstration of the general case of roots, or fractional exponents. And he fur- ther proposed it in this form, namely, that if q be the coeffi- cient of one of the terms of the series which is equal to (1 -f- jr)", and p the coefficient of the next preceding term, and R the coefficient of the next following term ; then, if a be =: 7- X P, it is required to prove that r will be = t - X Q. This he observed would be quite perfect and satis- factorv, as it would include all the terms ot* the series, as well those that are omitted, as those that are actually set down. And I was, in mv demonstration, to suppose, if I pleased, the truth of the binomial and multinomial theorems for in- tegral powers, as truths that had been previously and per- fectly proved. 3$ OP THE EIN-OMIAL THEOREM. fRACt l^; In consequence I sent him soon after the substance of the following demonstration ; with which he was quite satisfiedj and which I now proceed to explain at large. 17. Now the binomial integral is (I -}- x)" = a b c d , ,n nni nnln2 nnln2n3 , *+i' + r ^'+r '^'+r ^^''' w nl , w 2, , w 3 Qrl+-^-l ~-fl.r-4- ~-bx^ + 7-cx^ + &c, 1 J 6 4 where a, ^, c, &c, denote the whole coefficients of the 2d, 3d, 4th, &c, terms, over which they are placed; and in which the la w^ is this, namely, if p, a, r, be the coefficients of any three terms in succession, and if Xp be = Q, then is ^^I-a = r ; as is evident : and which, it is granted, has been proved. 18. And the binomial fractional is (1 + x)" = a b c d n 2n n 2fi 1 1-w , l-2;z , , l-3n &c, or 1 + -x-] - ax"- -\ bx'-] ex* + &c ; n 2n '6n 4n in which the law is this, namely, if p, a, r be the coefficients of three terms in succession ; and if ^p be zz Q, then is ^-(i. = R. Which is the property to be proved. 1 . n nl + , , AC ^1 2 / n TRACT 12. THE BINOMIAL THEOREM* 239 19. Again, the multinomial integral ( I +Ajr+B4r*+cT'&c)% is = - - - 1 , . , n n nl n 2 w 3 (a) +-A.r +_.-_.__._aV , n n-l , , n nl n-2 . {*) + yB - 71 nl n2 , , + T-T- *'-' M n-l 71 n l n 2 n3 n 4> ^12 3 4 5 n nl 71 2 n 3 n n 1 n2 ^12 1 . . n n 1 n 2 n n 1 + y. AD W 711 -f-.-^BC &c. Or, if we put a, b, c, c?, &c, for the coefficients of the 2d, 3d, 4th, 5th, &c, terms, or powers of x, the last series, by sub- stitution, will be changed into this form, 240 OF THE BINOMIAL THEOREM. TRACT 12. (1+- tiX + BX^ 4- c.r^ + &c)' ' n: - - _ - - 1 {a) WA + X ib) + 2nB + (n- 2 -l)Aa (0 + 2>nc-\-{2n- 3 -2)a6 , {d) 4nD + f3?7- l)ctf + (2- 2)R^+(n -3)AC^, ^ 4 [e) + 5n-E-{-(\7l l)Drt \-[?>7l 2)ci + (2w- 3)Bc + (n -4)Arf.i 5 &c. 20. Now, to find tlie series in Art. 18, assume the proposed binomial equal to a series with indeterminate coefficients, as ( 1 + .r)" rr I 4- AX -f B.r* + c.r^ + dx* + &c. Then raise each side to the n power, so shall 1 + ;t' = (1 + A;r + ]ix^ + cx^ + &c)''. But it is granted that the multinomial raised to any integral power is proved, and known to be, as in the last Art. viz, l+^=:(l + AX + Bx'- -f cx' + &c) = a If c r^^ ( ^ ( ^ ' N 7iA 2Bf(?z- l)Ar/ :i)!C \ (2?i-'l)-Ba-{-hi-2)Ab , 1 + Y^ + .vV-H ^' &c. It follows then, that if this last series be ecjua! to 1 -{- x, by equating the homologous coefficients, all the terms after the second must vanish, or all the coefficients b, c, d, &c, after the second term, must be each = 0. Writing ttiereforc, in this series, for each of the letters Z*, c, d, &c, it will become of this more simple form, viz, 1 + .r a b = c ~ r-^ ( ^ , , -^ > riA 2/;b f (;? l)Art 37?c + (2- 1 )Btf , , . \+x + ^ 'x^ + '- -x' + &c. TRACT 12. OF THE BINOMIAL THIORIM. 241 Put now each of the coefficients, after the second term, = 0, and we shall have these equations 2wB + {In 1) Aa = 3nc + (2n l) Ba = 4wD+ {3n i) ca 5nE 4- (4w . 1 ) Da = &c. The resolution of which equations gives the following values of the assumed indeterminate coefficients, namely, 1 n 1 2 1 3 1 4n <, E = -Aa, c = - Bfl, D = ca, E = - Da, &c : 2 3 4ra 5 ' which coefficients are according to the law proposed, namely, when 4 P is = Q, then is f-r-Q = R. a. e. d. 21. Also, by equating the second coefficients, namely, 1 a = MA, we find An. This beinor written for a in the above values of b, c, d, &c, will give the proper series for the binomial in question, namely, (1 + ^)" = 1 + A^ + BJr* 4- cx^ + &c, t 1 re - , 1 2n , . = 1 + -X + ax^ + -:r-bx^ + &c, ' n n 2n ' n 2n 3n Of the Form of the Assumed Series. 22. In the demonstrations or investigations of the truth of the binomial theorem, the butt or object has always been the law of the coefficients of the terms : the form of the series, as to the powers of x, having never been disputed, but taken for granted, either as incapable of receiving a demonstration, or as too evident to need one. But since the demonstration of the law of the coefficients has been accomplished, in which the main, if not the only, difficulty was supposed to consist, we have extended our researches still further, and have even doubted or queried the very form of the terms themselves, VOL. I. H 242 OF THB BINOMIAL THROREM. TRACT \2, namely, l 4. ^jr 4. Br* -f cx^ -\- nr* -J- &c, increasing by the regular integral series of the powers of .r, as assumed to denote the quantity (1 + ;)", or the n root of 1 + r. And in consequence of these scruples, I have been required, by a learned friend, to vindicate the propriety of that assumption. Which I think is effectually done as follows. 23. To prove then, that any root of the binomial I + jt can be represented by a series of this form 1 -i- x -j' .r^ { x^ + or* &c, where the coefficients are omitted, our attention being now employed only on the powers of x ; let the series representing the value of (1 + ^r)" be 1 + a + B + C I- d + &c ; where a, b, c, &c, now represent the whole of the 2d, 3d, ^th, &c, terms, both their coefficients and the powers of Xf whatever they may be, only increasing from the less to the greater, because they increase in the terms 1 -|- .r of the given binomial itself ; and in which the first term must evidently be 1, the same as in the given binomial. Raise now (1 + -^^Y, anJ its equivalent series 1 + A + B + c + &c, both to the 71 power, by the multinomial theorem, and we shall have, as before, 71 71 71 i 7inl71 2 1 + r = 1 + -A +y.-j-A' + _ .-_. ^3 + &c, 71 71 77} 71 'J'hen equate the corresponding terms, and we have the first term 1 = 1. Again, the second ttrm of the series - a, must be equal to the second term x of the binomial. For none of the other terms of the scries are equipollent, or contain the same power of X, witli the term a. Not an}' of the terms a*, a', a% &c ; for they are double, triple, quadruple, &c, in power to A^ Nor yet any of the terms containing b, c, d, &c ; be- TRACT 12. OP THE BINOMIAL THEOREM. 243 cause, hy the supposition, they contain all different and in- creasing powers. It follows therefore, that a makes up the whole value of the second term x of the given binomial. Consequently the second term A of the assumed series, con- tains only the first power of x ; and tlie whole value of that term a is = x. n But all the other equipollent terms of the expanded series must be equal to nothing, which is the general value of the terms, after the second, of the given quantity 1 -f jr or 1 +x + + 0-f"0 + &c. Our business is therefore to find the several orders of equipollent terras of the expanded series. And these it is asserted will be as they are arranged above, in which b is equipollent with a^, c with a% d with a"*, and so on. Now that B is equipollent with a*, is thus proved. The value of the third term is 0, But . ^ a* is a part of the third term. And it is only a part of that term : otherwise would be = 0, which it is evident cannot happen in every value of 7Z, as it ought ; for indeed it happens only when w is = 1 . Some other quantity then must be equipol- lent with A", and must be joined with it, to make up the whole third term equal to 0. Now that supplemental quantity can be no other than b : for all the other follow- ing terms are evidently plupollent than e. It follows there- fore, that B is equipollent with a^, and contains the second n 721 n power of ;r ; or that r~A*-f- b=0, and consequently n~\ \n \n 1 \ii A^+B 0, orB= A*= - A^ = ^7 -r'. Again, the fourth term must be r: 0. But the quantities A^ + AB are equipollent, and make 12 3 12 ^ ' up part of that fourth term. They are equipollent, or a^ equipollent with ab, because A* and B are equipollent. And R 2 34;!^ OF THB BINOMIAL THEOREM. TRACT 12, they do not constitute the whole of that term ; for if they fl ji 1 71 2 71 11 1 did, then would a' ^ ab be =0 in all ' 12 3^12 values of , or -^ a^ + b = : but it has been iust shown above, that p-A- + b = ; it would therefore follow that would be = , a circumstance which can only hap-^ pen when n = l, instead of taking place for every value of n. Some other quantity must therefore be joined with these to make up the whole of the fourth term. And this supplemental quantity can be no other than c, because all the other following quantities are evidently plupollent than a^ or ab. It follows therefore, that c is equipollent with A^, and therefore contains the 3d power of .r. And the whole value of c is lnn 2 1 I 2n l-2 1 1-7? 1 2^3 And the process is the same for all the other following terms. Thus then we have proved the law of the whole series, both Avith respect to the coefficients of its terms, and to the powers of the letter x. Since the above account Mas first written, almost 30 years ago, other demonstrations have been given by several inge- nious and learned writers ; which may be seen in some of the later volumes of the Philos. Trans, and elsewhere. ( 245 ) TRACT XIII. ON THE COMMON SECTIONS OF THE SPHERE AND CONK. WITH THE DEMONSTRATION OP SOME OTHER NEW PRO- PERTIES OF THE SPHERE, WHICH ARE SIMILAR TO CERTAIN KNOWN PROPERTIES OF THE CIRCLE. The study of the mathematical sciences is useful and pro- fitable, not only on account of the benefit derivable from them to the affairs of mankind in general ; but are most eminently SO, for the pleasure and delight which the human mind feels in the discovery and contemplation of the endless number of truths, that are continually presenting themselves to our view. These meditations are of a sublimity far above all others, whether they be purely intellectual, or whether they respect the nature and properties of material objects; they methodize, strengthen, and extend the reasoning faculties in the most eminent degree, and so fit the mind the better for under- standing and improving every other science ; but, above all, they furnish us with the purest and most permanent delight, from the contemplation of truths peculiarly certain and im- mutable, and from the beautiful analogy Avhich reigns through all the objects of similar inquiry. In the mathematical sciences, the discovery, often accidental, of a plain and simple pro- perty, is but the harbinger of a thousand others of the most sublime and beautiful nature, to which we are gradually led, delighted, from the more simple, to the more compound and general, till the mind becomes quite enraptured at the full blaze of light bursting upon it from all directions. Of these very pleasing subjects, the striking analogy that prevails among the properties of geometrical figures, or figured extension, is not one of the least. Here we often f)iid that a plain and obvious property of one oi: the simplest m 246 PROPERTIES OF THE TRACT 13. figures, leads us to, and forms only a particular case of, a property in some other figure, less simple ; afterwards this again turns out to be no more than a particular case of an- other still more general ; and so on, till at last we often trace the tendency to end in a general property of all figures whatever. The few properties which make a part of this paper, con- stitute a small specimen of the analogy, and even identity, of some oi the more remarkable properties of the circle, with those of the sphere. To which are added some properties of the lines of section, and of contact, between the sphere and cone. Both which may be further extended as occasions may offer ; like as all of these properties have occurred from the circumstance, mentioned near the end of the paper, of considering the iimcr surface of a hollow spherical vessel, as viewed by an eye, or as illuminated by rays, from a given point. PROPOSITION 1. All the tangents are equal, which are drawn, from a given point without a sphere, to the surface of the sphere, quite around. Demons, For, let ft be anv tangent from the given point p ; and (h'aw PC to the centre c, and join tc. AUo let cta be a great circle of the sphere in the plane of the triangle tpc. Tlieii, cp and CT, as well as the angle r, wh'.i-h is right (Eucl. iii. 18), being constant, in every position of the tangent, or of the point of contact T ; the square of pt will be everywhere equal to the diilcrence of the squares of the constan: lines CP, CT, and tiiercfore constant, and consequently the .iMcor tangent pt iticlf of a constant length, in every j)Obition, quite round the surface of the sphere. TRACT 13. SPHERE AMD COm. I4T PROP. 2, If a tangent be drawn to a sphere, and a radius be drawn from the centre to the point of contact, it will be perpendi- cular to the tangent ; and a perpendicular to the tangent will pass through the centre. Demons. 'Y ox, let ft be the tangent, tc the radius, and CTA a great circle of the sphere, in the plane of the triangle TPc, as in the foregoing proposition. Then, pt touching the circle in the point T, the radius to is perpendicular to the tangent pt, by Eucl. iii. 18, 19. PROP. 3. If any line or chord be drawn in a sphere, its extremes terminating in the circumference; then a perpendicular drawn to it, from the centre, will bisect it: and if the line drawn from the centre, bisect it, it is perpendicular to it. Demons, Yov, a plane may pass through the given line and the centre of the sphere ; and the section of that plane with the sphere, will be a great circle (Theodos. i. 1), of which the given line will be a chord. Therefore (Eucl. iii. 3) the perpendicular bisects the chord, and tlje bisecting line is perpendicular. Corol. A line drawn from the centre of the sphere, to the centre of any lesser circle, or circular section, is perpendi- cular to the plane of that circle. For, by the proposition, it is perpendicular to all the diameters of that circle. PROP. 4. If from a given point, a right line be drawn in any position through a sphere, cutting its surface always in two points ; the rectangle contained under the whole line and the external part, that is the rectangle contained by the two distances be- tween the given point, and the two points where the line meets the surface of the sphere, will always be of the same constant magnitude, namely, equal to the square of the tan- gent drawn from the same given point. 248 PROPERTIES OF THE TRACT 13. Demons.- L.et r be the given point, and AB the two points in which the line pab meets the surface of the sphere ; through PAB and the centre let a plane cut the sphere in the great circle tab, to which draw the tangent ft. Then the rectangle PA . PB is equal to the square of pt (Eucl. iii. 36) ; but pt, and consequently its square, is constant by Prop. 1 ; therefore the rectangle pa PB, which is always equal to this square, is every where of the same constant magnitude. PROP. 5. If any two lines intersect each other within asphere, and be terminated at the surface on both sides; the rectangle of the parts of the one line, will be equal to the rectangle of the parts of the other. And, universally, the rectangles of the two parts of all lines passing through the point of intersec- tion, are all of the same magnitude. Demons. Through any one of the lines, as ab, conceive a plane to be drawn through the centre c of the sj)here, cutting the sphere in the great circle abd ; and draw its diameter DCPF through the points of intersection p of all the lines. Then the rectangle ap . pb is equal to tht^ rectangle dp . pl (Eucl. iii. 35). Again, through any other of tlie intersecting lines gii, and the centre, conceive anotlier plane to pass, cutting the sphere in anotber great circle dgfh. Then, because the points c and p are in this latter plane, the line cp, and consequently the whole diameter dcpf, is in thesanv r ^')e; and therefore it is a diameter of the circle dgfh, of w nich gph is a chord. Therefore, again, the rectangle gp . ph is equal to the rect- angle DP . pf (Eucl. iii. 35). Consequently all the rectangles ap . pb, gp . ph, &c, are equal, being each equal to the constant rectangle dp . pf. TRACT 13. SPHERE AND CONE. 249 Corol. The great circles passing through all the lines or chords which intersect in the point p, will all intersect in the common diameter dpf. PROP. 6. If a sphere be placed within a cone, so as to touch it in two points; then shall the outside of the sphere, and the inside of the cone, mutually touch quite around, and the line of contact will be a circle. Demons. Let v be the vertex of the cone, c the centre of the spliere, t one of the two points of contact, and TV a side of the cone. Draw ct, cV. Then tvc is a triangle right-angled at t (Prop. 2). In like manner, t being another point of contact, and ct being drawn, the triangle two will be right-angled at ^. These two triangles then, tvc, i\c, having the two sides ct, tv, equal to the two ct, tY (Prop. 1), and the included angle t equal to the included angle t, will be equal in all respects (Kucl. i. 4), and consequently have the angle tvc equal to the angle ^VC. Again, let fall the perpendiculars tp, tp. Then the two triangles tvp, tvv, haviiig the two angles tvp and tpv equal to the two /vp and ^pv, and the side tv equal to the side tv (Prop. 1), will be equal in all respects (Eucl. i. 26) ; conse- quently tp is equal to tv, and vp equal to vp. Henc(; pt, p^ are radii of a little circle of the sphere, whose plane is per- pendicular to the line cv, and its circumference every where equidistant from the point c or v. This circle is therefore a circular section both of the sphere and of the cone, and is therefore the line of their mutual contact. Also cv is the axis of the cone. Corol. 1. The axis of a cone, when produced, passes through the centre of the inscribed sphere. Corol. 2. Hence also, every cone circumscribing a sphere, so that their surfaces touch quite around, is a right cone; 250 PROPERTIES OF THE TRACT 13. nor can any scalene or oblique cone touch a sphere in that manner. PROP. 7. The two common sections of the surfaces of a sphere and a right cone, are the circumferences of circles, if the axis of the cone pa;)S througli the centre of the sphere. Demons. Let v be the vertex of the cone, c the centre of the sphere, and s one point of t!ie less or nearer section ; draw the lines cs, cv. Then, in the tri- angle CSV, the two sides cs, cv, and the included angles cv, are constant, for all positions of the side vs ; and therefore the side vs is of a constant length for all positions, and is consequently the side of a right cone hav- ing a circular base ; therefore the locus of all the points s, is the circumference of a circle perpendicular to the axiscv, that is, the common section of the surfaces of the sphere and cone, is that circumference. In the same manner it is proved tb.at, if A be any point in the farther or greater section, and ca be drawn ; then va is constant for all positions, and therefore, as before, is the side of a cone cut off by a circular section whose plane is perpen- dicular to the axis. And these circles, being both perpendicular to the axis, are parallel to each other. Or, thev are parallel because they are both circular sections of the cone. Carol. 1. Hence sa = sa, because va = va, and vs zzys. Carol. 1. All the intercepted equal parts sa, sa, &c, are equally distant from the centre. For, all the sides of the tri- angle sca are constant, and therefore the perpendiculiir cp is constant also. And thus all the eijual right jinf.s or chords in a sph.'.'rc, are equally distant from the centre. Covol. ?i. The s(xtions are not circles, and thcrerore not in plunes, if the axis pass not through the cenre. Fur then TRACT 13. SPHERE AND CONE. 251 some of the points of section are farther from the vertex than others. PRQP. 8. Of the two conmmon sections of a sphere and an oblique cone, if the one be a circle, the other will be a circle also. Demons. Let s\as and Asva be sec- tions of the sphere and cone, made by a common plane passing through the axes of the cone and the sphere ; also s^, /^a the diameters of the two sections. Now, by the supposition, one of these, as Afl, is the diameter of a circle. But the angle vs-s = tiie angle wax (Eucl. i. 13, and iii. 22), therefore S5 cuts the cone in sub-contrary position to AC ; and consequently, if a plane pass through S5, and per- pendicular to the plane AVrt, its section with the oblique cone will be a circle, whose diameter is the line S5 (Apol. i. 5). But the section of the same plane and the sphere, is also a circle whose diameter is the same line S5 (Theod, i. l). Con- sequently the circumference of the same circle, whose diame- ter is S5, is in the surface both of the cone and sphere ; and therefore that circle is the common section of the cone and sphere. In like manner, if the one section be a circle v/hose dia- meter is sa, the other section will be a circle whose diameter is ^A. Carol. 1. Hence, if the one section be not a circle, neither of them is a circle; and consequently they are not in planes; for the section of a sphere b}- a plane, is a circle. Corol. 2. When the sections of a sphere and oblique cone are circles, the axis of the cone does not pass through the centre of the sphere, (except when one of the sections is a great circle, or passes through the centre). For, the asispasi^cs through the centre of the base, but not perpendicularly ^ whereas a line drawn from the centre of the sphere to the 252 PROPERTIES OF THE TRACT l'^. centre of the base, is perpendicular to the base, by cor. to prop. 3. Carol. 3. Hence, if the inside of a bowl, Avhich is a hemi- sphere, or any segment of the sphere, be viewed by an eye not situated in the axis produced, which is perpendicular to the section or brim ; the lower, or extreme part of the inter- nal surface which is visible, will be bounded by a circle of the sphere ; and the part of the surface seen by the eye, will be included between the said circle, and border or brim, which it intersects in two points. For the eye is in the place of the vertex of the cone ; and the rays from the eye to the brim of the bowl, and thence continued from the nearer part of the brim, to the opposite internal surface, form the sides of the cone ; which, by the proposition, will form a circular arc on the said internal surface; because the brim, Avhicli is the one section, is a circle. And hence, the place of the eye being given, the quantity of internal surface that can be seen, may be easily determined. For the distance and height of the eye, with respect to the brim, will give the greatest distance of the section below the brim, together with its magnitude and inclination to the plane of the brim ; which being known, common mensuration fur- jiishes us with the measure of the surface included bet\\ ecn them. Thus, if ab be the dia- meter in t'ic vertical plane pass- ing tiiVouj;!! the eye ot e, also AFB the cT'ciion of th^ howl by the same ',:'; ncj and a:t? the supplement of that arc. Drav; r.w, EiB, cuttine: this vertical circle in F and i ; and join if. Then shall if be the diame- ter of the section or extremity of ttii; visible surface, and Mv its greatest distance below the hriirij an arc wiiich measures an a-jgk: double the angle at a. Coi'ol. 4 Hence aiso, and fiom Proposition 4, it lollows, thiit if througl) every [ioiiit in t!ie circumference </(r* - f) = dh ; also r r . ' 1' x =z y X , y X . Hence then x -. v : : : arx V{r^y') w 'u) r^ + arw r+aw .... ., , . , , , r: : : 1 : ar X -, r;; which is evidently a variable a'(a/--f ti;)- {ar-{wf ratio. Therefore there is no such fixed point e, as that men- tioned in the problem. Corollary 1. Hence then it appears, that the common method of finding the side of a polygon inscribed in a circle, by drawing a line from a certain fixed point e, through f and G, making a f to a c as 2 is to the number of sides of the poly- gon, is not generally true. Corol. 2. But such a point e maybe found, as shall render that construction at least nearly true, in the following man- ner. Suppose the line efg to revolve about e, from b to a: at B, the arc bg and the line df arise in the ratio of be to DE; and at A they are in the ratio of BA to ad or db ; there- fore make these two ratios equal to each other, and it will determine the point e, so as that the ratios in all the inter- mediate points, or situations, will be nearly equal: thus then, BE : de : : BA : AD : : J5 : 2, making p = 3-1416 ; or bd : DS : : o 2 : 2 : hence DE :r x ED = 1-152 bd r= ^-bd ^ ' p2 -^ very nearly. If, therefore, de be taken to da as 7 to 4 ; then any line drawn from e, to cut the diameter ac, and the semi- circumference ABC, it will very nearly cut them proportion- ally. Therefore, if a polygon is to be inscribed, or if the whole circumference is to be divided into any number of equal parts ; first divide the diameter into the same number s 2 260 GEOMETRICAL DIVISION OF THE CIRCLE. TRACT 15* of parts, and through the 2d point of division draw kfg, so will AG be one of the equal parts very nearly. Corol. 3. The number 1*752 being equal to v^S nearly, for V3 = 1*732; therefore, if de be taken to da as ^3 to 1, the point e will be found answering the same purpose as before, but not quite so near as the former. And here, be-^ cause DA : DE : : 1 : ^^3, therefore de is the perpendicular of an equilateral triangle described on ac. Hence then, if with the centres a, c, and radius ac, two arcs be described, they will intersect iu the point E, nearly the same as before. And this is the method in common practice ; but it is not so rear the truth as the construction in the 2d Corollary. Corol. i. Hence also a right line is found equal to the arc of a circle nearly : for bg is = y dp nearly. And this is the same as the ratio of 1 1 to 7, which Archimedes gave for the ratio of the semicircumference to the diameter, or 22 to 7 the ratio of the whole circumference to the diameter. But the proportion is here rendered general for any arc of the circle, as well as for the whole circumference. TRACT XVI. ON PLANE TRIGONOMETRY WITHOUT TABLES. The cases of trigonometry are usually calculated by means of tables of sines, tangents or secants, either of their natural numbers, or their logarithms. But the calculations may also be made without any such tables, to a tolerable degree of ac- curacy, by means of the theorems and rules contained in the following propositions and corollaries. tROPOSlTION. If 2a denote a side of any triangle, A the number of degrees contained in its opposite angle, and r the radius of the circla TRACT 16. TRIGONOMETRY WITHOUT TABLES. 261 circumscribing the triangle : Then the value of a is equal to 47-2957795 x :-+ + + + &c. r 2.3H 2.4.5r5^2.4.6.1r7^2.4.6.8,9r' For, since 2a is the chord of the arc on which the angle, whose measure is a, insists ; a will be the sine of half that arc, or the sine of the angle to the radius r, since an angle in the circumference of a circle is measured by half the arc on which it stands; now it is well known that the said half arc 2 is equal to a' 3a* 3.5a' . , . , a+ r-?H -A r-^ &c ; and, 3'14159r denoting ^ 2.3r* 2.4.5r* 2.4.6.7r^ ' ^ half the circumference of the same circle, or the arc of 180 degrees, it will be 1802 57-2957795Z as 3-14159r i 180' : : z : r= 3-14159r r ^ , a a^ Sa^ 3.5a' . , = 57-2957795 X + ,+ ~ ^4- 7r=-, &c,) the ^ r ^ 2.3r^ 2.4.5?-^ 2.4.6.7/-' ' degrees in the angle or half arc. Corollary 1. By reverting the above series, we obtain a _ a a^ a' a' T~'n~' 2.3w^ 2.3.4.5s " 2,3.4.5. 6.7n' ^' 180 putting^ ^^ = 57*2957795 = --, . ^ "" 3-14159 &c. Corollary 2. If 2a be the hypothenuse of a right-angled triangle, a will be = r, and then the general series will be,. come n X (a + 13 3.5 ^ 90 90x3 14159 &.C + -T^ + ;:; &c) = 90, or = -^ r- = 2.3 2.4.5 2.4.6.7 ' ' n 1^0 3.14159 &c . . 1 . 3 . 3.5 2,5.7 [8.9 ^ 2.3 ^ 2.4.5 ^ li.4.6.7 ^ 2.4.6.^ '^ Coral. 3. Since the chord of 60 degrees is = the radius, or the sine of 30 degrees = half the radius, putting a for \r in the ,..,,. 11 3 3.5 general series, will give n x (--}- -x- x. ^ ' ^ ^2^2.3.-^^*^2.4.5.25^2.4.6.7.2^ ^c =; 30 ; and hence the sum of the infinite series 262 JLANE TRIGONOMETRY TRACT 16 2_ 1 _3 3.5 2 ^ 2.3.2^ ^ 2aJ^' ^ 2.4.6.7.2' ^^' ' _ 30__ 30x 3-14159 &c_ 3-14159 &c _ "~ w ~ Tso ~ 6 ~ ^th of the circumference of the ciixle whose diameter is 1. Corol. 4. It might easily be shown, from the principles of common geometry, that the sine of 60 degrees is to tle radius, as ^v/3 is to 1 ; substituting then j^7W2 for a in the general 1 3 3.3* 3.5.3-' series, we shall have ?2a/3x ( 4-- rr^ 1 '- ;4- '- =-;r; ' ^ ^ 2 ^ 2.3.2^ 2.4.5.25^ 2.4.6.7.2^ &c) = 60; and hence the sum of the infinite series 1 3 3 3* 3 5 3' 'T + 2:5:^3 + ix^, + iX6?r^' ^"' '''^^ ^" CO 60 X 3-14159 &c 3-14159 &c , . , ^ =.I7"3= r8o73 = 37?-' ""^ '' '^"''"^'"" *^ the infinite series in the 3d corollary, as 2 is to ^/3. Corol. 5. If by c be the halves of the other two sides of the triangle, and b, c the degrees contained in their opposite angles j smce b = n x ( + -^j^^ + ^jj^ &c), and c = c c^ 71 X ( H ^j &c, and the 3 angles of any triangle are equal to 180 degrees; we shall have iS0rrA-f-B+ c -=. n x .avb\-c a'Arb^-\-c^ ^ ^ ^ c x n \ -\ &c), or the sum or the innnite series r 2.3r' ' a^b c J fl'+Z>5+c3 3 a^^-b^^-c^ 3.5 fl'4-^^+f' r ""^2.3' r^ '^2.4.5" r^ "*" 2.4.6.7 r? 180 180x3-14159 &c kc, ^x\\\ be = = = 3.14159 &c = the n 180 circumference of a circle whose diameter is 1 ; a, by c, being the halves of the three sides of any triangle, and 7- the radius of its circumscribing circle. aa cc Corol. 6. Since, by theor. 3, b : a -\- c :: a-c : -. = half the difference of the segments of the base {b) made by a TRACT 16. WITHOUT TABLES. 263 perpendicular demitted from its opposite angle, and b -{- aa cc aa-\-bb cc . .... , ., 7 = r = the segment adjoining to theside 2a, we ' {aa-\-bb- cc)\ ^(4.a'b^-{aa + bb-ccy) shall have v'^ (4a -rr ) = t -- for the value of the said perpendicular to the base ; and hence x/ {^a^b'' - { aa + bb- ccf) _ 2abc I : 2fl : : c : ^^ {^a'b^ {aa-{-bb - ccY)~'' the radius of the circumscribing circle. Having now found the value of r, we can calculate all the cases of trigonometry without any tables, and without re- ducing oblique triangles to right-angled ones; for, having any three parts (except the three angles) given, we can find the rest from these five equations following : 2abc ^ * ^ ~ y/ (4a'6^ - {aa ^bb ccf)' _ a a^ 3flS 3.5a^ 3.5.7 fl^ 2. A-wx(-+^^3+,7y;^^ + 2^g^^,7 + ^^^y^-^&c.) _ b M 3/)5 3.5^7 3.5.7/^' 3. B = wx( + ^-^3 + ^^-T^ + 2^ ^ + ^-^-^ &c.) c c^ 3c' 3.5f7 3 5 7c' 5. A -}- B + c r= 180. And, for the more convenience, we may add the three fol- lowing, which are derived from the 2d, 3d, and 4th, by rever- sion of series. 6.,= ,X(^-^3 + ^-^^-^-^;;^,&C.) 1 B B^ B^ B^ , 7 b =1 r Y /^ A &c \ ' "^ Wi 2 in^ ^ 2.3.4.5W' 2..-;.4.o.b.7^ ' ' c c^ c' c' 8. c =r X ( - -g-3 + 2_3.4,_5^js - 2.3.4.5.6.7/Z7 ^"^"^ Where ?2. = 57-2957795 &c. EXAMPLE. Suppose we take here the following example, in which are given the two sides 2b 345, 2c zz 232, and the angle op- 26i PLANE TRIGONOMETRY TRACT 16. posite to '2c = 37 20' rr 37^ degrees = c. Then since C 374-X 3-14159 &c , 232 zr ^ ; = -631589587, we have c =: 71 180 ' 2 = 116 = r X (-651589587 -04610744 -f -00097879 000009894 + -000000058 &c) r X (-652568435 046117334) ;=-6064511r. Hence r = --^.^ ^ . , = 191-27677; ' -6064311 J h 345 X -6064511 and . -9018346. r ~ 2x116 Again, b = 57 2957795 x 1-12402 (the sum of the series in the 3d equation) = 64-4016 degrees = 64 24'. And A = 180 - 37j-- 64-4016 = 180 - 10r735 78-265 = 78 16' nearly. A . . 78-265 Lasth^, being = -r-r-TzzT^ = 1-365982, and r zz -" n ^ 57-2957795. 191-27677, from the 5th equation we have a = 191-27677 x (1-365982 - -4247992 + -0396379 - -0017607 + '0000288 - -0000005) = 191-27677 x -9790883 = 187-27684. And hence 2a = 374-55368 =the third side of the triangle. Corol. 7. As the series by which an angle is found, often converges very slowly, I have inserted the following approxi- mation of it ; viz, A = w X (^^^(2 - 2s/{\ - ^ ) 77) nearly ; where the letters denote the same quantities as in the above series. For aa a a^ la^ 5,nceP = v'{2-2v/(l--)).s = - + 3+ &c, , A . a a? 3a^ n and IS = -i 1 1 &c, n r 2.3r3 2.4.5^ we shall have, by taking the former of these from the latter, A a^ 13a5 p = : + -- r &c. But, from the first series, n 24;-^ 640r^ a a? 1a^ , , , 11 ip - = r + : xc : hence, bv subtracting the lat- ^ 3r 24;-3 ^ 384r5 > > . o ter from the former, it gives TRACT 16. WITHOUT TABLES. 265 A ^ a a a a} ^ p 4-P f ;:-= -tP H = r &c ; and A = X (ip- ~=nx{Wi2-2^{l-^) ) - ^) nearly. 4 1 Corol. 7. And again, since x (p - ? i?^ = ^y' &c ; where o is = : by subtractins; this from 7: = "T^fi &c, and reducing, there will be obtained a = 777- x (l44p-39^-i^3)=-^x(l44v/2-2>/(l-y^))-395'-i-^S which will commonly give the angle exact to within a minute of the truth. Where note, that the constant quantity is = '54567409. And from the whole may be drawn the fol- lowing general problem. PROBLEM. To perform all the Cases of Trigonometry/ without any Tables. Having any three parts of a triangle given, except the three angles, the other three parts ma}' be found, by some of the following six general theorems. 1. A = i X (*v/(^ - 2v'( -tJ ) r) nearly. Or A= x(144v/(2- 2^(l--,))-39 - ) more nearly. _ a a^ 3a^ 3.5a'' 3 5.7a' ^ 2.A = nx (7-+ ^7^3 + ^X575 "^2.4.6.7r7'^2.4.6.S.9r' ' 3. - ^ X (^^ - 2.1^3+ 2.3.4. 5n5 2.3.4.0. 6.7 /i^ '^ a 4- r = ri n '2.3.n}'^ 2.3A.5n^ 2, 3. 4.5. 6. 7^^ ^'^* 266 PLANE TRIGONOMETRY. TRACT 16. 2abc r = 2abc ^ \{a-^b-\-c)x{a\b^c) x {a-b + c) x {-a-{-b + c)Y "- Wz 2.3;i^ 2.3. 4. an^ ' Where a, b, c, are the halves of the three sides of the tri- angle, and A tlie number of degrees in the angle opposite the side 2a, and c the degrees in the angle opposite the side 2c; also r is the radius of the circumscribed circle ; and n = r-~- = 57-2957795, or = -54567409. 3-14159 ' 105 EXAMPLE. Thus, if the three sides be given, as for example a = 13, 6=14, c = 15. Then is r = 16^, and the angles by these theorems come out as follow ; viz. Angles by the Theor. The true Angles. 53 7' - - angle a - - 53 7'^ 59 28 - - angle b - - 59 29| 67 19 - - angle c - - 67 224 179 54 sum of all 180 00 TRACT XVII. ON MACHIn's QUADRATURE OF THE CIRCLE. Since the chief advantage of this method consists in taking- small arcs, whose tangents shall be numbers easy to manag*.', Mr. Machin very properly considered, that as the tangent of 45 is 1 ; and that the tangent of any arc being given, the tangent of double that arc can easily be found ; if there be assumed some small simple number for the tangent of an arc, tRACT 11. MACHINES QUADHATURE OF THE CIRCLE. 267 and then the tangent of the double arc be continually taken, till a tangent be found nearly equal to 1, the tangent of iS"*, by taking the tangent answering to the small difference be- tween 45 and this multiple, there would be obtained two very small tangents, viz. the tangent first assumed, and th.e tangent of the difference between 45 and the multiple arc;, and that therefore the lengths of the arcs corresponding to these two tangents being calculated, and the arc belonging to the tan- gent first assumed being as often doubled as the multiple de- notes, the result increased or diminished by the other arc, would be the arc of 45, according as the multiple arc should be below or above it. Having thus thought of his method , by a few trials he was lucky enough to find a number, and perhaps the only one, proper for this purpose, viz, knowing that the tangent of ^ of 45 is nearly = 4, he assumed ^ as the tangent of an arc : then since, if i be the tangent of an arc, the tangent of the double arc will be the radius being 1 ; the tangent of an arc double to that of which ^ is the tangent, will be - 1- = ^, and the tangent of the double of this last 1 ~ ^ T t4 120 is -; ^TT" 7^ y "^^'^ich, being very near equal to 1, shows that the arc wliich is equal to 4 times the first, is very near 45. Then, since the tangent of the difference between 45 . T 1 and an arc whose tangent is t, is , we shall have the tan- T+ 1 gent of the difference between 45 and the arc whose tangent 120 , 44^-1 120-119 1 is equal to - = = . 119 ^ 4ff+-l 120+119 239 Now by calculating, from the general series, the arcs wliose tangents are | and ^^, which may be quickly done, by rea- son of the smallness and the simplicity of the numbers, and taking the latter arc from 4 times the former, the remainder will be the arc of 45. And this is Mr. Macliin's ingenious quadrature of the ciixle. But it was by means of Dr. Hailey's method that Mr. 268 MACHINES aUADRATURE OF THE CIRCLE. TRACT 17. Machin found the circumference of a circle, whose diameter is I , to be 3- 141.59265335, 8;)7.Q323846,2643383279,. 502884197 1, 693919375 10, 582Q974944,3923078 1 64,0628620899,8628034825,3421 170679 +. true to above 100 places of figures. Or, by substituting the above numbers in Machin's series, 16 4 , 1 ,16 4 ^ 1 ,16 4 . .ve get the scnes(-- )--(-- 3) + -(--&c, equal to the semicircumfcrcnce whose radius is 1, or the whole circumference whose diameter is 1. Being the series published by Mr. Jones, and which he acknowledges he re- ceived from Mr. Machin. But because the arc whose tangent is -^, is r= 2 times the ftrc whose tangent is Tf%> iriinus the arc to tangent -^-].-^ ; (for ^-^ = = taniient of twice the arc to tangent t^. and ^O ' - ^ = T-i-r =E tanfr. of diff. between the arcs whose tan- 1 -4-,^ * ^ gents are || and f ) ; therefore 8 times arc to tangent ^ig. 4 times arc to tang. -^-^ arc to tang, ^j-^ rr arc of 45, or whose tang, is 1. Which is much easier than Machin's way. And various other methods may easily be discovered from the ^ame nrincii)los. TRACT XVIII. A KEW AND GKNERAL METHOD OF FINDING SIMPLE AND QUICKLY-CONVKRGING SERIES; BY WHICH THE PROPOR- TION OF THE DIAMETER OF A CIRCLE TO ITS CIRCUMFER- f.rxXE MAY EASILY EE COMPUTED TO A GREAT MANY PLACES OF FIGURES. Ik examining the methods of Mr. Machin and others, for cuinputliig tiic proportion of the diameter of a circle to its circuuuerciice, 1 discovered the method explained in this impi r. '1 lub method is very general, and discovers many l-RACT 18. CONVERGING SERIES FOR THE CIRCLE! 26^ series that are fit for the abov^ementioned purpose. The ad vantage of this method is chiefly owing to the simphcity of the series by which an arc is found from its tangent. For, if t denote the tangent of an arc a, the radius being 1, then it is well known, that the arc a is denoted by the infinite series, i ii^ + 1^* f^^ + -1^' &c ; where the form is as simple as can be desired. And it is evident that nothing further is required, than to contrive matters so, as that the value of the quantity t^ in this series, may be both a small and a very- simple number. Small, that the series may be made to con- verge sufficiently fast ; and simple, that the several powers of t may be raised by eas)?^ multiplications, or easy divisions. Since the first discovery of the above series, many authors have used it, and that after different methods, for determin- ing the length of the circumference to a great number of figures. Among these were, Dr. Halley, Mr. Abra. Sharp, Mr. Machin, and others, of our own country ; and M. de Lagney, M. Euler, &c, abroad. Dr. Halley used the arc of 30, or^\th of the circumference^ the tangent of which being = y/-3-, by substituting ^^ for t in the above series, and mul- tiplying by 6, the semicrcumference is = \/t X (1 -r-r+ z IT- ;r:r,-\ r^ &.0 ; which series i^, ^ ^ ^ 3.3 5.3^ 7.3^ 9.3-* ' ' ' to be sure, very simple ; but its rate of converging is not very great, on which account a great many terms must be used to compute the circumference to many places of figures. By this very series however, the industrious Mr. Sharp com- Jjuted the circumference to 72 places of figures; Mr. Machin extended it to 100; and M. de Lagney, still by the same series, continued it to 128 places of figures. But though this series, from the 12th part of the circumference, does not converge very quickly, it is perhaps the best aliquot part of the cir- cumference which can be employed for this purpose ; for when smaller arcs, which are exact aliquot parts, are used, their tangents, though smaller, are so much more complex, as to render them, on the whole, more operose in the anpli. Ration; this will easily appear, by inspecting some instances 270 QUiCK CONVERGING SERIES TRACT 18. that have been given in the introductions to logarithmic' tables. One of these methods is from the arc of 1 S*', the tangent of which is V{l'2V\); another is from the arc of 22i, the tangent of which is v'2 1 ; and a third is from the arc of 15, the tangent of which is 2 VS. All of which are evidently too complex to alford an easy apphcation to the general series. In order to a still further improvement of the method by the above general series, Mr. Machin, by a very singular and excellent contrivance, has greatly reduced the labour natur- ally attending it. I have given an analysis of his method, ot a conjecture concerning the manner in which it is probable Mr. Machin discovered it, in my Treatise on Mensuration ; which, I believe, is the only book in which that method has been investigated, as it is repeated in the foregoing Tract; For though the series discovered by that method were pub- lished by Mr. Jones, in his " Synopsis Palmariorum Ma- theseos," which was printed in the year 1706, he has given them merely by themselves, without the least hint of the manner in which they were obtained. The result shows, that the proportion of the diameter to the circumference, is equal, to that of 1 to quadruple the sum of the two series, T^('-r?+^^-7:i^ + ^^^)""^ 1 1 1 i_ I 239 ^ V^ ~ 3^39"' ^ T23f^ ~ -^239*' ^ 9.2S9* ^'' The slower of which series converges almost thrice as fast as Dr. Halley's, raised from the tangent of 30. The latter of these two series converges still a great deal quicker; but then the large prime number 239, by the reciprocals of the powers of which the series converges, occasions such long and tedious divisions, as to counter-balance its quickness of converTency ; so that the former series is summed with ra- ther more ease than the latter, to the same number of places of figures. Mr. Jones, in his '' Synopsis," mentions other series besides this, which he had received from Mr. Machin fur the same purpose, and drawn from the same principle- TRACT 18. rOR THE CIRCLE. 27 1 But we may conclude this to be the best of them all, as he did not publish any other besides it. M. Euler too, in his " Introductio in Analysinlnfinitorum,'* by a contrivance something like Mr. Machin's, discovers, that ^ and i are the tangents of two arcs, the sum of which is just 45; and that therefore the diameter is to the circumference, as 1 to quadruple the sum of the two following series, ^(^-^ + ^-7:^ + ^^^'^^""^ 3 ^ ^^ ' 3.9 "^ 5^ ~" 7.9^ + 9.9* '' Both Mhich series convercre much faster than Dr. Hallev's, and are yet at the same time made to converge by the powers of numbers producing only short divisions; that is, divisions performed in one line, or without writing down any thing besides the quotients. I come now to explain my own method, which indeed bears some little resemblance to the methods of Machin and Euler; but then it is more general, and discovers, as particular cases of it, both the series of those gentlemen, and many others, some of which are fitter for this purpose than theirs are. This method then consists in finding out such small arcs, as have for tangents some small and simple vulgar fractions, the radius being denoted by 1, and such also that some mul- tiple of those arcs shall differ from an arc of 45, the tangent of which is equal to the radius, by other small arcs, which also shall have tangents denoted by other such small and simple vulgar fractions. P'or it is evident, that if such a small arc can be found, some multiple of which has such a proposed difference, from an arc of 45% then tlic lengths of these two small arcs v, ill be easily computed from the general series, because of the smallness and simplicity of their tajigents; after which, if the proper multiple of the first arc be in- creased or diminished by the other arc, the result will be the length of an arc of 45, or ith of the circumference. And the manner in which I discover such arcs is thus : Let T, t, denote any two tangents, of which x is the 272 QUICK CONVERGING SERIES TRACT 1^^ greater, and i the less : tlien it is known, that the tangent of T f tlie difference of the corresponding arcs is equal to ^ Hence, if t, the tangent of tlie smaller arc, be successively denoted by each of the simple fractions i, 4* vj tj ^^j ^^^ general expression for the tangent of the difference betweeft the arcs will become respectively 2T-1 3T-1 4T-1 5T-1 , .1, . -f u , , . , &c : so that it t be ex- 2 t-T ' 3-f-T ' 4 + T ' 5-{-t' ' pounded by any given number, then these expressions will give the tangent of the difference of the arcs in known num- bers, according to the values of/, severally assumed respect- ively. And if, in the first place, t be equal to 1, the tangent of 45, the foregoing expressions will give the tangent of an arc, Avhich is equal to the difference between that of 45 and the first arc ; or that of which the tangent is one of the num- bers i, -f, .J-, I, &c. Then, if the tangent of this difference, just now found, be taken for x, the same expressions will give tiie tangent of an arc, which is equal to the difference between the arc of 45 and the double of the first arc. Again, if for T we take the tangent of this last found difference, then the foregoing expressions will give the tangent of an arc, equal to the difference between that of 45 and the triple of the first arc. Ami again taking this last found tangent for t, the same theorem will produce the tangent of an arc equal to the dif- ference l)etween that of 45*^ and the quadruple of the first arc ; and so on, always taking for t the tangent last found, the same expressions will give the tangent of the difference between the arc of 45 and the next greater multiple of the first arc ; or tliat of which the tangent was at first assumed equal to one of the small numbers --, -f, i, -, &c. This ope- ration, being continued till some of the expressions give such a fit, small, and simple fraction as is required, is then at an end, for we have then found two such small tangents as were required, viz, the tangent last found, and the tangent lirst assumed. Here follow the several operations adapted to the several TRACT 18. FOR THP CIRCJ-P. 27S values of t. The letters a, 6, c, 7 ^ ^^ -..49^5.49^ 7.49^ "^9.49* -^' The former of these vahies of a is the same with that before mentioned, as given by M. Euler; but the latter 'is much better, as the power.s of -^-^j- converge much faster than those ofx. Corol. From double the former of these values of A, sub- tracting tlie latter, the remainder is, l^^^-TJ + -^^-7ir + ^^-> ^-^ I 1 1 1 , which is a much better theorem than either of the fonner. St 1 2. If t be taken = ^, then the expression ^ ^ / - gives a = ij 3 = i. Here the value of a \ gives the same ex- pression for the value of a as the first in the foregoing case, and the value of ^ = j^ gives the value of a the very same as, in the corollary to the case above. 4t 1 8. Taking ? = i, the expression - gives a = -f , 6 = 4+ T 5V> <^ = ^Vt* Where it is VOL. I 5?y, c = -5^, (^ = Jj^.. Where it is evident that the value 274 aUICK CONVERGING SERIES TRACT 18. offrr^5^ is tlje fittest number afforded by this case; and hence it appears, that the arc of 45 is equal to the sum of the arc of which the tangent is -^^j and the triple of the arc of which the tangent is ~. 3 . 1 1 Ion Or that A = < ^ ^z ^4 ;^6 '' 99 ''^ ^^ ^ 3^"^ 5^99^" 7799^"*" ^^^' Which is the best theorem that we have 3'et found, because the number 98 resolves into the two easy factors 9 and 11. 4. Let now t be taken = \; then the expression - -gives 2 7 1 a =. -Tj-j ^ = T=5 c = , d ^7:77:. ^V''llere it is evident that the last number, or the value oi d, is the fittest of those produced in this case; and from which it appears, that the arc of 45 is equal to the difference between the arc of which the tangent is -jj-^-, and quadruple the arc of which the tan- gent is ^. Or that ^=^^j_ i_ __i i__ Which is the very theorem that was invented by Mr. rvlachin, as we have before mentioned. 5. Take now / rr i ; then the expression -,~- gives a - ,b==YvC--^,d = -^,e= pp^. Of which numbers it is evident that none are fit for our purpose. 6. Again, take ^ = |, and the expression - will give ^ ~ "T ^ ~ 31' ^ ~" ^' '^ ~" 205' ^ ~ "742' -^ ~ 5265- Neither are any of these numbers fit for our purpose. 7. In like manner take t =. x.. so shall give ii ' 8 + T " a 9,6 , c g^^, d -^^^ e 4^^2y' J ~ 58SU79* TIUCT 18. FOR THE CIRCLE. 175 S. And if/ be taken = -^j the expression -^^ will give a ^ , 6 ^g, C 2^., a 2239J ^ io475> ^^* 9. Also, if we take t = -^^ the expression - ' will give _ ^ A _ Z. _ ^''^ J 5441 _ 410-19 '^ iT' " ~ Uy' ^ == 1269' ~ 13361> ^ "^ 1390al '^* 1 0. Further, if we take t = -^, the expression gives _ ^ , _ 49 _ 234 _ 2159 _ _9475 rZ _ ^ , 6 _ , C _ , d = ^^, e ^^, otc. 1 1. Lastly, if we take / =z -jL-, the expression ~f^ gives ^ ~ Fs > ^ ~ iT?' ^ ~ 73"' ^ 9T7' '^ ~ ri423 ^^' - Here it is evident, that none of these latter cases afford any numbers that are fit for this purpose. And to try any other fractions less than -jV for the value of t, does not seeni likely to answer any good purpose, especially as the divisors after 12 become too large to be managed in the easy way of short division in one line. By the foregoing means it appears then, that we have dis- covered five different forms of the value of a, or ^th of the semicircumfercnce, all of which are very proper for readily computing its length; viz, three forms in the first case and its corollary, one in the 3d case, and one in the 4th case. Of these, the first and last are the same as those invented by Euler and Machin respectively, and the other three are quite new, as far as I know. But another remarkable excellence attending the first three of the before mentioned series, is, that they are capable of being changed into others which not only converge still faster, but in which the converging quantity shall be -j'^, or some multiple or sub-multiple of it, and so the powers of it raised with the utmost ease. The series, or theorems, here meant are these three ; T 2 276 QUICK COHVERGINQ sehies traqt hB-* 1st, A = 2dly, A = 3dly, A = 1 1 1 1 . O X I 1 1 1 - 3.4 *" 5.4^ - ^.4= + ^= -4xc- 1 1 3.49 ^"5.49' ' 1 '-rl 7.49 ' ^'^' |m'- I 1 3.9 "*" S.O"- - r9' + ^^' + ~x (1 - 1 1 3.49"^ 5.49' -rL^ + '^^'- Now if eacli of these be transformed, by means of the dif- ferential series, in cor. 3 p. C4 of the Jate Mr. Simpson's Mathematical Dissertations, they will become of these very commodious forms, viz, 4 , , 4 8a 12? w><(' + iiTo+:5To+Tro + ^^<=) 4 , 4 8a 12? ><(' + ino-+T-iF + -7.7o +**' T 4 "^ -7 1 2a? 50^^' ^^^+5:1^ ^^To5+^'^^^ 2dly, A = < i) :).lly, A = } - 2 4 ee + 5li >< (' i^ + 5:i^ ^ ^ + ^^^>- Where a, if, y, &c, denote always the preceding terms iri each series. Now it IS evident that all these latter series are much easier than the former ones, to which they respectively correspond; li>r, because of the powers of 10 here concerned, we have little more to do than to divide by the series of odd numbers I, 3, 5, 7, 9, kc. Of all these three forms, the 2d is the fittest for comput* TRACT 18. FOK Tfi CIRCLE. 277 ing the required proportion ; because, of tlie two series of which it consists, the several terms of the one are found from the Jike terms of the otlier, by dividing these latter by 10, and its several successive powers, lOO, 1000, &c ; that is, the terms of the one consist of the same figures as the terms of the other, only remoted a certain number of places farther towards the right hand, in the decuple scale of numbers; 6nd the number of places by which they must be removed, is the same as the distance of each term from the first term of the series, viz, in the 2d term the figures must be moved one place lower, in the 3d term two, in the 4th term three, &c; so that the latter series will consist of but about half the number of the terms of the former. Thus then this method may be said to effect the business by one series only, in which there is little more to do, than to divide by the :t printed in folio at Nuremberg, 1543. It is remarkable that be docs not call these lines nines, but semisses siibtcnsarinny namely of the double arcs. Copernicus was born at Thorn in 1 ITS, and died in 1543. Ill 1553 was published the " Canon Fcrcundus," or table of tanoents, of Erasmus Reinhold, professor of mathematics in ihfr aeadi^niv of Wurtemburg, He was born at Salheldt in I p!)cr Saxony, in the year 1511, and died in 1552, Tfi Francis Maurolyc, abbot of Alessina in Sicily, we owe the introduction of the " Tabula Benehca," or canon of se- TRACT 19. TRIGONOMETRICAL TABLES, kc. 233 cants, which came out about the same time, or a little before. But Lansberg erroneous!}" ascribes this to Rlicticus. And the tangents and secants are both ascribed to Rcinhoid, by Briggs, in his " Mathematica ab antiquis minus cognita," (p. 30, Appendix to Ward's Lives of the Professors of Gre- sham College.) Francis Vieta was born in 1 j40 at Fontenai, or Fontenai- ]e-Comte, in Lower Poitou, a province of France. He was master of requests at Paris, where he died in 1603, being the 63d 3-ear of his age. Among other branches of learning in which he excelled, he was one of the nvost respectable ma- thematicians of tiie IGth century, or indeed of any aoe. His writings abound with marks of great originality, and the finest genius, as well as intense applic.ition. Among them are se- w.rn\ pieces relating to trigonometry, whicli may be found in the collection of his works published at Leyden in 1646, by Francis Schooten, besides another large and separate vo- lume in folio, published in the author's lifetime at Paris in 1579, containing trigonometrical tables, with their construc- tion and use; very clegantl}- printed, by the king's mathe- matical printer, with beautiful types and rules; the differences of the sines, tangents and secants, and some other parts, being printed with red ink, for the better distinction ; but it is inaccurately executed, as he himself testifies in page 323 of his other works above mentioned. The first part of this curious volume is entitled " Canon Mathematicus, sen ad Triangula, cum Appendicibus," and it contains a great va- riety of tables useful in trigonometry. The first of tiiese is what he more peculiarly calls " Canon iNIatliematicus, sen ad Triangula," which contains all the sines, tangents, and secants for every minute of the quadrant, to the radius 100,000, with all their differences; and towards the end of the quadtant the tangents and secants are extended to 8 or 9 places of figures. They arc arranged like our tables at present, increasing on tl>e left-hand side to 45 degrees, and then returning upwards by the right hand side to 90 degrees ; so that each niiuiber $84 HISTORY or TRACT 19. and its complement stand on ti)e same line. But here the canon of what we now call tangents isdenominate(iy(a?tt;?rfM5, and that of the secants fa'cundissimus. For the general idea prevailing in tiie form of these tables, is, not that the lilies represented by the luiiiibers are those which are drawn in and about a circle, as sines, tangents, and secants, but the tliree sides of right-angled triangles ; this being the way in which those lines had alwavs been considered, and which still con- tinued for some time longer. Hence it is that he considers the canon as a series of plane right-angled triangles, one side being con>tantly 100, OCO ; or rather as three series of such triaufi^les, for he makes a distinct series for each of the three varieties, namely, according as the hypotenuse, or the base, or the ])erpendicular, is represented by the constant number 100,000, which is sinnlar to the radius. Makiiig each side constantly 100,000, the other two sides are computed to every magnitude of the acute angle at the base, from 1 minute up to 90 degrc(^s, or the whole quadrant. Each of the three scries therefore consists of two parts, representing the two variable sides of the triangle. When the hypotenuse is made the constant numbt^r 100,000, tlie two variable sides of the triangle are the pirpcnJicuLir unci base, or our sine and co- sine; when the base is 100,000, the perpendicular and hypo- tenuse are the variable jjarts, forming the canon facciindus et fcvcundissimus, or our tangent and secam ; nnd when the perpendicular is made the constant 100,000, the series con- tains the variable base ajid hypotenuse, or also cavonfaecundus effcecundtssimus, or oiir cotangent and cosecant. Of course, therefore, the table e(msi-,ts of 6 columns, 2 for each of the three series, bc-ides the two columns on the riaht and left for minutes, from to GO in each d.-gree. The second of tiiesc tables is similar to the first, but all in rational numbers, consisting, like //, of three series of two colunms each ; t!ie radius, or constant side of the triangle, m e.ich series, being 100,000, as before; and the otiicr two sides accuratchj expressed in integers and rational viilgur TRACT \9. TRIGONOMETRICAL TABLES, &.C. 285 fractions. So that we have here the canon of accurate sines, tangents, and secants, or a series of about 4300 rational right- angled triangles. But then the several corresponding arcs of the quadrant, or angles of those triangles, are not expressed. Instead of them, are inserted, in the first column next the margin, a series of numbers decreasing from the beginning to the end of the quadrant, which are called numeri primi basecs. It is from these numbers that Vieta constructs the sides of the three series of right-angled triangles, one side in each series being the constant number 100,000, as before. The tiieorems by which these series of rational triangles are computed from the Jiiimeri primi baseoSy or marginal num- bers, are inserted all in one page at the end of this second table, and in the modern notation they may be briefly ex- pressed thus : Let p denote tlie primary or inarginal number pn any line, and r the constant radius or number 100,000 : then if r denote the hypotenuse of the right-angled triangle, the perpendicular and base, or the sine and cosine will be respectively, j-rrr and r ri :, (which last we may reduce to 1\ ,r). When r denotes the base of the right-angled triangle, the perpendicular and hypotenuse, or the taugeiit and secant, are expressed b}- Tii'~: and r \- r~i Tj (wiucli last we mav reduce to ? :r) and when r denotes the perpendicular of the right-angled triangle, the base and hj'potenuse, or the cotangent and co- secant, are then expressed by ipr - J (or r), and ^pr -|- -j- (or ~f-r). So that Vieta's general values will be as we have here co-- lected them together in the foHowing expressions, imme- diately under the words sine, cosine, iic ; and just below Vieta's forms I have here phiced the others, to which thev reduce and are equivaiciit, which are more contractecl, though not so well adapted to t])o expciditious computatinu as Vieta's form<;. 2S6 HISTORY OP TRACT 19. Sine __P__ r Cosine Tangent Secant <2r '+1: ic'-^' Cotangent Cosecant r All these expressions, it is evident, are rational ; and by as- Ruminir p i,f different values, from the first theorems Vieta com[)nted the corresponding sides of the triangles, and so expressed them all in integers and rational fractions. To the foregoing principal tables arc sul)joined several other smaller tables, or short specimens of large ones: as, a table of the sines, tangents and secants, for every single de- gree of the quadrant, with the corresponding lengths of the arcs, t!ie radius b!'ieSj will be as follows: perimeter of the inscrib, polygon 314,159,265,35, periinet(:r of the circum. ]Jol\gon 314,159,265,37, and th.it therefore the circumference of the circle lies be- I ween those tv o numbers. TRACT 19. TRIGONOMETRICAL TABLES, &.C. 287 Though no author's name appears to the volume we have been describing, there can be no doubt of its being the per- formance of Vieta ; for, besides bearing evident marks of his masterly hand, it is mentioned by himself in several parts of his other works collected by Schooten, and in the preface to those works by Elzevir, the printer of them ; as also in Mon- tucla's " Histoire des Mathematiques ;" which are the only notices I have ever seen or heard of concerning this book, the copies of which are so rare, that I never saw one besides that whicli is in my own possession. In the other works of Vieta, published at Leyden in 1646, b}^ Schooten, as mentioned alx)ve, there are several other pieces of trigonometry; some of which, on account of their originality and importance, are very deserving of particular notice in this place. And first, the very excellent theorems, liere first of all given by our author, relating to angular sec- tions, the geometrical demonstrations of which are supplied by that ingenious geometrician, Alexander Anderson, then professor of mathematics at Paris, but a native of Aberdeen, and cousin-german to Mr. David Anderson, of Finzaugh, whose daughter was the mother of the celebrated Mr. James Gregory, inventor of the Gregorian telescope. We find here, theorems for the chords, and consequently sines, of the sums and diiterences of arcs ; and for the chords of arcs that are in arithmetical progression, namely, that the 1st or least chord is to the 2d, as any one after the 1st is to the sum of the two next less and greater : for example, as the 2d to the isum of the 1st and 3d, and as the 3d to the sum of the 2d and 4th, and as the 4th to the sum of the 3d and 5th, &c ; so that the 1st and 2d being given, all the rest are found from them by one subtraction, and one proportion for each, in which tlic 1st and 2d teruis are constantly the same. Next are given theorems for the chords of any multiples of a given arc or angle, as also the chords of their supijlements to a semicircle, which are similar to the sines and cosines of the multiples of given angles ; and the conclusions from them are expressed 28S HISTORY OP TRACT 19. Arcs Chords of the Sup. la C 2a 6'* -2 ?>a c^ - 3c 4a c^-4cH2 5a c- 5c-^-{- oc 6 a c'-Cc^ + 9c--2 la c' - 76-5 + 1 4f' ~ 7f kc. &c. [ in this manner : 1st, that if c be the cliord of the supplemont of a given arc a, to the radius 1 ; then the chords of the sup- jilemcnts of the multiple arcs will be as in the annexed table: M'herc the author observes, that the signs are alternately + and ; that the vertical columns of numeral coefficients to the terms of the chords, arc the several ordcrsof tigurate numbers, which he calls triangular, pvidinidal, triangulo-triangular, triangulo- pyramidal, &c, generated in the ordinary "way bij continual addi- tions -, not indeed from unity, as IN THE GENERATION OF POWERS, but beginning with the number 2; and that tlic powers ob- serve always the same progression: secondly, that it the chord of an arc a be called 1, and d the chord of tlie double arc 2^, then the chords of the series of multiple arcs will be as in this table ; where the au- thor remarks as before on the law of the powers, signs, and coeiHcients, these being the orders of figurate numbers, raised from unity by continual additions, after the manner of the genesis, of poxcers, which generation in that way he speaks of as a thing g(;nerally known, but without giving any Flint how the coefficients of the terms of any ]:)ower may be found from one another only, and indejKmdent of t!ios(! of any other power, as it was afterwards, and first of all, 1 be- lieve, done by Henry Briggs, about the y-ar 1600: and 3dly, that if c be the chord of any arc a, to the radius I, Aros CaortW. \a 1 2a d 3 a d'~l 4-a d'^'ld 5a d^-Zd'-k- i ea d'^-W-V od la d''~5d'^ 6d'- -1 Sa d'' - 6d' + lOd^ - -4d &c. kc. TRACT 19. TRIGONOMETRICAL TABLES, &;c. 289 then the series of the chords and supplemental chords of the multiple arcs will be thus ; where the values are alternately Arcs Chords and Chords of Sup. la Chord = + c 2a Sup. ch. = - c* + 2 3a Chord = - c^ + 3c 4!a Sup. ch. = + c* - 4c^ + 2 5a Chord = + c* - 5c3 + 5c 6a Sup. ch. = - c** + 6c* - 9c' 4- 2 la Chord = - c' + Ic' - 14c' + 7c &c. &c. chords, and chords of the supplennients of the arcs on the same line, and the law of the powers and coefficients as be- fore, but every alternate couplet of lines having their signs changed. Another curious theorem is added to the above, for finding the sum of all these chords drawn in a semicircle, from one end of the diameter to every point in the circumference, those points dividing the circumference into any number of equal parts ; namely, as the least chord is to the diameter, so is the sum of the said least chord and diameter and greatest chord, to double the sum of all the chords, including the diameter as one of them. As the above theorems are chiefly adapted for the chords of multiple angles, a few problems and remarks are then added (whether by Vieta or Anderson does not clearly ap- pear, but I think by the latter) concerning the application of them, to the section of angles into submultiples, and thence to the computation of the chords or sines, or a canon of tri- angles. The general precept for the angular sections is this ; select one of the above equations adapted to the proper num- ber of the section, in which will be concerned the powers of the unknown or required quantity, as high as the index of the section; and from tliis equation find that quantity by the known methods for the resolution of equations. Examples VOL. I. U 290 HISTORY or TRACT 19'. are given of three different sections, namely, for 3, 5, and 7 equal parts, the forms of which are respectively these, 3c c^ . . , . = g 5c 5c^ + & . . rr ^ 7c llc^ + c' c^ rr if where g is the cliord of the given arc or angle, and c the re- quired chord of the 3d, 5tii, or 7th part of it. And it is shown, geometrically, that tiie first of these equations has 2 real positive roots, the second 3, and the last 4; also, from the same principles, the rekitions of these roots are pointed out. The method br.cn anUvrixcd for constructing the canon of sines, from the foregoing theorems is t;,us : By dividing the radius in extremc-and-niean ratio, is obtained the sine of 1& degrees; this quinquisected, gives the sine of 3 36'. Again, by trisecting the arc of 60', there is obtained the sine of 20; this again trisecLcd gives that cjf 6"^ 40'; and this bisected gives that of 3 20' : Then, l)y the theorem for the difference of two arcs, there will be fou.nd the sine of 16', the dilTerence be- tv/cen 3 3G' and 3 20': Lastly, !)y four successive bisections^ will at length be found th.e sines of 8', 4', '/, and l'. This last being found, the sines of its uudtiples, and again of the multiples of thffse multiples, &c, tliroughout the quadrant, are to be taken by tlie proper theorems before laid down. ^ And the same subject is still favtlicr piysned and explained, in the tract contaiiiiiig the iinswer given by Vieta, to the problem pro})nsed to the Avhole world by Adrianus Romanus. In the samec(jil''ctioii of Vieta's works, iVom page 400 to 432, Is given a complete treatise on practical trigonometry, con- taining rules for rc^olvin:;:, all the cases of jjiane and spherical triangles, by tlie Canon MalJicmaticus, or table of sines, tan- gents and secants. The next authors whose labours in this way have been printed, are Rheticus, Otlio, and Vitiscus : to all of whom we, owe very great improvements in trigonometry. George Joachim RI:cticus, professor of mathematics in tlie univer- iity of Witiemberg. and sometime jiupil to Copernicus, died TRACT 15. TRIGONOMETRICAL TABLES, &.C. Ist in 1576, in .the 60th year of his age. He conceived, and executed, the great design of computing the triangular canon for every 10 seconds of the quadrant, to the radius 1000000000000000, consisting of 1, followed by 15 ciphers. The series of sines which Rheticus computed to this radius, for every 10 seconds, and for every single second in the first and last degree ofthe quad rant, was published in folio at Franc- fort, 1613, by Pitiscus, who himself added a few of the first sines computed to the radius 10000000000000000000000, But the large work, or whole trigonometrical canon com- puted by Rheticus, was pubhshed in 1596 by Valentine Otho, mathematician to the Electoral Prince Palatine. This vast work contains all the three series for the whole canon of right-angled triangles (being similar to the sines, tangents and secants, by which names I shall call them), Avith all the differences of the numbers, to the radius 10000000000. Prefixed to these tables, are several books on their con- struction and use, in plane and spherical trigonometry, &c. Of these, the first three are by Rheticus himself; namely, book the 1st, containing the demonstrations of 9 lemmas, concerning the properties of certain lines drawn in and about circles : the 2d book contains 10 propositions, relating to the sines and cosines of arcs, together with those of their sums and differences, their halves and doubles, kc. The 3d book teaches, in 13 propositions, the construction ofthe canon to the radius 1000000000000000. By some ofthe common pro- perties of geometry, having determined the sines of a few principal arcs, as 30, 36, &c, in the first proposition, by continual bisections, he finds the sines of various other arcs, down to 45 minutes. Then, in the 2d proposition, by the theorems for the sums and differences of arcs, he finds all the sines and cosines, up to 90 degrees, in a series of arcs differ- ing by 1 30'. And, in the 3d proposition, by the continual addition of 45', he obtains all the sines and cosines in the series whose common difference is 45'. In the 4th proposition, be- ginning wath 45', and continually bisecting, he finds the sines and cosines of the series of half arcs, tijl he arrives at the arc U 2 992 HISTORY OF TRACT 19>. of 14^'" 19", the sine of which is found to be 1, and its cosine 999999999999999. In the 5th proposition are com- puted the sine and cosine of 30", or half a minute. In the 6th and 7th propositions are computed the sines and cosines for every minute, from l' to 45', as well as of many larger arcs. The 8th proposition extends the conijiutation for single minutes much farther. In propositions 9 aiid 10 are com- j^tcd the tangents and secants for ^11 arcs in tlie scries whose common difference is 45' ; and these are deduced from the sines of the same arcs by one proportion for each. In the remaining three propositions, 11, 12, 13, are computed the tangents and secants for several small angles. And from all these primary sines, tangents, and secants, the wiiole canon is deduced and completed. The remaining books in thii work art; by the editor Otho j namely, a treatise, in one book, on right-angled plane tri- angles, tlie cases of which are resolved by the tables: then right-angled spherical trigonometry, in four books; next ob- lique spherical trigonometr}', in five books; and lastly several other books, containing various spherical })roblems. Next after the above are placed the tables themselves, con- taining the sines, tar.gcnts, and secants, for every 10 seconds in the quadrant, with all the differences annexed to each, in a smaller charactei'. The numbers however are not called sines, tangents, and secants, but, like Vieta's, before de- scribed, they are considered as representing the sides of right-angled triangle.-, and are titled accordingly. They are also, in like manner, divided into three series, namely, ac- cording as the radius, or constant side of the triangle, is made the hypotenuse, or the greater leg, or the less leg of the tri- angle. When the hypotenuse is made the constant radius 10000000000, the two columns of this case, or series, are called the perpendicular and base, which are our sine and cosine; when the greater le^ is the constant radius, the two columns on this series are titled hypotenuse and periiciidicu- lar, which are our secant and tangent; and when the less leg \ constant, the two columns in this case are called hypotcnu'iC TXACT 1^. TRIGONOMETRICAL TABLES, &C. 29S and base; which are our cosecant and cotangent. After this Jarge canon, is printed another smaller table, Avhicli is said to be the two columns of the third series, or cosecants and co- tangents, with their differences, but to 3 places of figures less, or to the radius 10000000. But I cannot discover the reason for adding this less table, even if it were correct, which is very far from being the case, the numbers being uniformly erroneous, and different from the former through the greatest part of the table. Towards the close of the 16th century, many persons wrote on the subject of trigonometry, and the construction of the triangular canon. But, their writings being seldom printed till many years afterwards, it is not easy to assign their order in respect of time. I shall therefore mention but a few of the principal authors, and that without pretending to any great precision on the score of chronological prece- dence. In 1591 Philip Lansberg first published his " Geometria Triangulorum," in four books, Avith the canon of sines, tan- gents, and secants; a brief, but very elegant work; the whole being clearly explained : and it is perhaps the first set of tables titled with those words. The sines, tangents, and secants of the arcs to 43 degrees, with tiiosc of their comple- ments, are each placed in adjacent columns, in a very com- modious manner, continued forwards and downwards to 45 degrees, and then returning backwards and upwards to 90 degrees: the radius is 10000000, and a specimen of the first page of the table is as follows : 1 2 3 4 5 Sinus Tangens Secans 60 59 58 57 56 55 &c. 2909 5818 10000000 9999999 9999998 2909 5318 infinitum. 34377466738 1 1718S731915 10000000 10000000 1000U002 10000004 10000007 100000 11 infinitum. 34377468193 17188734824 8727 11636 14544 9999996 9999993 9999989 1 8727 ! 11636 14544 11459152994 8594365048 687548S693 11459157357 8594368866 637549596e |i y?| 294 HISTORY OF TRACT 19. Of this work, the first book treats of the magnitude and relations of such Hnes as are considered in and about the circle, as the chords, sines, tangents, and secants. In the second book is delivered the construction of the trigonome- trical canon, by means of tlie properties laid down in the first book : After which follows the canon itself. And in the third and fourth books is shown the application of the table, in the resohuion of plane and spherical triangles. Lansberg, who was born in Zealand 1561, was many years a minister of the gospel, and died at Middleburg in 1632. The trigonometry of Bartholomew Pitiscus was first pub- lished at Francfort in the year 1599. This is a very com- plete work; containing, besides the triangular canon, Avith its construction and use in resolving triangles, the applica- tion of trigonometry to problems of surveying, altimetry, architecture, geography, dialling, and astronomy. The construction of the canon is very clearly described : And, in the third edition of the book, in the year 1612, he boasts to have added, in this part, arithmetical rules for finding the chords of the 3d, 5th, and other uneven parts of an arc, from the chord of that arc being given ; saying, that it had been heretofore thought impossible to give such rules : But, after all, those boasted methods are only the application of the double rule of False-Position to the then known rules for finding the chords of muitiple arcs; namely, making the supposition of some number for the required chord of a sub- multiple of any given arc, then from this assumed number computing what will be the chord of its multiple arc, which is to be compared with that of the given arc; then the same operation is performed with another supposition; and so on, as in the double rule of position. The canon contains the sine, tangent, and secant, for every minute of the quadrant, in some parts to 7 places of figures, in otliers to 8; as also the diffc'.rences for every 10 seconds. The sines, tangents, and secants, are also given for every 10 seconds in the first and last degree of the quadrant, for every 2 seconds in the hrst and last 10 minutes, and for every single second in the first and last minute. In this table, the sines, tangents, and se- TACT 19. TRIGONOMETRICAL TABLES, &C. 296 cants, are continued downwards on the left-band pages, as far as to 45 degrees, and then returned upwards on the right- hand pages, so that the complements are always on the same line in the opposite or facing pages. The mathematical works of Christopher Clavius (a Ger- man Jesuit, who was born at Bamberg in 1537) in five large folio volumes, were printed at Moguntia, or Mentz, in 1612, the year in which the author died, at the age of 75. In the first volume we find a very ample and circumstantial treatise on trigonometr}', with Regiomontanus's canon of sines, for every minute, as also canons of tangents and secants, each in a separate table, to the radius 10000000, and in a form con- tinued forwards all the way up to 90 degrees. Tlie expla- nation of the construction of the tables is very complete, and is chiefly extracted from Ptolemy, Purbach, and Regiomon- tanus. The sines have the differences set down for each second, that is, the quotients arising from the differences of the sines divided by 60. About the year 1600, Ludolph van Collen, or a Ceulen, a respectable Dutch mathematician, wrote his book '* de cir- culo et adscriptis," in which he treats fully and ably of the properties of lines drawn in and about the circle, and especi- ally of chords or subtenses, with the construction of the canon of sines. Tlie geometrical properties from which these lines are computed, are the same as those used by former writers; but his mode of computing and expressing them is different from theirs; for they actually extracted all the roots, &.c, at every step, or single operation, in decimal numbers ; but he retained the radical expressions to the last, making them how- ever always as simple as possible : thus, for instance, he de- termines the sides of the po- lygons of 4, 8, 16, 32, &c, sides, inscribed in the circle whose radius is 1, to be as in the table here annexed : where the point before any figure, as V .2 signifies the Nu. of Sides. Length of each side. 4 V^2 8 16 ^.2-v'2 v/.2--/.2 + v/2 32 &c. v/.2-^/.2 + v/-2-v'2 &c. 296 HISTORY OF TRACT 19. root of all that follows it ; so the last line is in our notation the same as ^ 2 - ^^'l + \/ 2 -V2. And as the perfect management of such surds was then not generally known, he added a very neat tract on that subject, to facilitate the computations. These, together with other dissertations on similar geometrical matters, were translated from the Dutch language, into Latin, by Willebrord Snell, and published at (Lugd. Batav.) Leyden in 1G19. It was in this work that Ludolph determined the ratio of the diameter to the circum- ference of the circle, to 36 figures, showing that, if the diameter be 1, the circumference will be greater than 3-14159 2G535 89793 23846 26433 83279 50288, but less than 3-14159 26535 89793 23846 26433 83279 50289, which ratio was, by his order, in imitation of Archimedes, engraven on his tomb-stone, as is witnessed by the said Snell, pa. 54, 55, '' Cyclometricus," published at Leyden two years after, in which he treats the same subject in a similar manner, recomputing and verifying Ludolph's numbers. And, in the same book, he also gives a variety of geometrical approxi- mations, or mechanical solutions, to determine very nearly the lengths of arcs, and the areas of sectors and segments of circles. Besides the ** Cyclometricus," and another geometrical work (Apollonius Battavus) published in 1608, the same Snell wrote also four others " docrriii triangulorum ca- nonicae," in which is contained the canon of secants, and in which the construction of sines, tangents, and secants, toge- ther with the dimension or calculation of triangles, both plane and spherical, are briefly and clearly treated. After the au- thor's death, this work was published in 8vo, at Leyden, 3627, by Martinus Hortensius, who added to it a tract on surveying and spherical problems. Willebrord Snell was born in 1591 at Royen, and died in 1626, being only 35 years of age. He was professor of mathematics in the university of Leyden, as was also his father Rodolj)h Snell. Also in 1627, Francis van Schooten published, at Amster- TRACT 19. TRIGONOMETRICAL TABLES, &C. 297 dam, in a small neat form, tables of sines, tangents,, and se- cants, for every minute of the quadrant, to 7 places of figures, the radius being 10000000; together with their use in plane trigonometry. These tables have a great character for their accuracy, being declared by the author to be without one single error. This hov/ever must not be understood of the last figure of the numbers, which I find are very often errone- ous, sometimes in excess and sometimes in defect, by not being alwavs set down to the nearest unit. Schooten died in 1659. while the second volume of his second edition of Descartes* geometry was in the press. He was also author of several other valuable works in geometry, and other branches of the mathematics. The foregoing are the principal writers on the tables of sines, tangents, and secants, before the invention of loga- rithms, which happened about this time, namel}', soon after the year 1600. Tables of the natural numbers Avere now all completed, and the methods of computing them nearl}' per- fected : And therefore, before entering on the discovery and construction of logarithms, I shall stop here awhile to give a summar}^ of the manner in which the said natural sines, tan- gents, and secants, were actually computed, after having been gradualhamprovedfrom Hipparchus, IMenelaus, and Ptolemy, who used only the chords, down to the beginning of the 11th century, when sines, tangents, secants, and versed sines were in use, and when the method hitherto employed had received its utmost improvement. In this explanation, we may here first enumerate the theorems by which the calculations were made, and then describe the application of them to the com- putation itself. Theorem 1. The square of the diameter of a circle, is equal to the sum of the squares of the chord of an arc, and of the chord of its supplement to a semicircle. 2. The rect- angle under the two diagonals of any quadrilateral inscribed in a circle, is equal to the sum of the two rectangles under the opposite sides. 3. The sum of the squares of the sine and cosine, hitherto called the sine of the complement, is equal -93 HISTORY OF TRACT 19. to the square of the radius. 4. The difference between the sines of two arcs that are equally distant froTii 60 degrees, or I- of the whole circumference, the one as much greater as the other is less, is equal to the sine of half the difference of those arcs, or of tlie difference between either arc and the said arc of 60 degrees. ,5. The sum of the cosine and versed sine, is equal to the radius. 6. The sum of the squares of the sine and versed sine, is equal to the square of the chord, or to the square of double the sine of half the arc. 7. The sine is a mean proportional between half the radius and the versed sine of double the arc. 8. A mean pro})ortional between the versed sine and half the radius, is equal to the sine of half the arc, 9. As radius is to the sine, so is twice the cosine to the sine of twice the arc. 10. As the chord of an arc, is to the sum of the chords of the sinf^le and double arc, so is the difference of those chords, to the chord of thrice the arc. 11. As the chord of an arc, is to the sum of the chords of twice and thrice the ai'c, so is the difference of those chords, to the chord of five times the arc. 12. And in general, as the chord of an arc, is to the sum of the chords of n times and 71 -\- I times the arc, so is the difference of those chords, to t!ic chord of 2?z + 1 times the arc. 13. The sine of the sum of two arcs, is equal to the sum of the products of tiie sine of each multiplied by the cosine of the other, and divided by the radius. 14. The sine of the difference of two arcs, is equal to the diiFercnce of the said two products divided by radius. 15. The cosine of the sum of two arcs, is equal to the difference between the products of tlieir sines and of their cosines, divided by radius. 16. The cosine of the difference of two arcs, is equal to the sum of the said })roducts divided bv radius. 17. A small arc is equal to its chord or sine, nearly. IS. As cosine is to sine, so is radius to tangent. ['.). Jladius is a mean proportional between the tangent and cotangent. 20. Radius is a mean projiortional between the secant and cosine. 21. Jiaduis is a mean proj)nrtionul be- tween the sine and cosecant. 22. Half the diluTcnce be- tiveen the tangenl and cotangent of an arc, ii cquul to tljc TKACT 19. TRIGONOMETRICAL TABLES, &C. 299 tangent of the difference between the arc and its complement. Or, the sum arising from the addition of double the tangent of an arc with the tangent of half its complement, is equal to the tangent of the sum of that arc and the said half comple- ment. 23. The square of the secant of an arc, is equal to the sum of the squares of the radius and tangent. 24. The secant of an arc, is equal to the sum of its tangent and the tangent of half its complement. Or, the secant of the differ- ence between an arc and its complement, is equal to the tan- gent of the said difference added to the tangent of the less arc. 25. The secant of an arc, is equal to the difference be- tween the tangent of that arc and the tangent of the arc added to half its complement. Or, the secant of the diH'er- ence between an arc and its complement, is equal to the dif- ference between the tangent of the said difference and the tangent of the greater arc. From some of these 25 theorems, extracted from the writers before mentioned, and a few propositions of Euclid's ele- ments, they compiled the whole table of sines, tangents, and secants, nearly in the following manner. By the eiement;> were computed the sides of a few of the regular figures in- scribed in a circle, which were the chords of such parts of tliC whole circumference as are expressed by the number of sides, and therefore the halves of those chords the sines of the halves of the arcs. So, if the radius be lOCOOOOO, the sides of the follov,-ing figures will give the annexed chords and sines. The figure. Arcs sub- tended Its chord or side. Halt- arc. Its sine or 4 chord. Triangle 120 17320508 60 8660254 Square 90 14U2I3b 45 7071C6S Pentagon 72 11753705 56 5877853 Hexagon 60 10000000 30 5000C00 Decagon 36 61S0340 18 30S0170 QuindecaETOn 24. 4158234 12 9079117 Of some, or all of these, the sines of the halves were con- tinually taken, by theorem the 6th, 7thj or 8th, and of their 300 HISTORY OP TRACT 19. complements by the 3d ; then the sines of the halves of these, and of their complements, by the same theorems ; and so on, alternately, of the halves and complements, till they arrived at an arc which is nearly equal to its sine. Thus, beginning with the above arc of 1 2 degrees, and its sine, the halves were obtained as follows : The halves. Sines. 6 t 1045285 3 523360 I 30 20' 1769 45 130896 The Comp. of these. 84 9945218 87 9986295 88 30 9996573 89 15 9999143 The halves of these. 42 6691306 21 3583679 10 30 1822355 5 15 915016 43 30 6883545 21 45 3705574 44 15 6977905 The como. Sines. of these. 43 7431448 69 9335804 79 30 98325+9 84 45 9958049 46 30 7253744 68 15 9288095 45 45 7163019 The lialves of these. 24 4067366 34 30 5664062 17 15 2965416 39 45 6394390 23 15 3947439 The comp. 66 9l35i:55 55 30 8241262 72 45 9550199 50 15 7688418 66 45 9187912 The halves. Sines. 33 5446390 16 30 2840153 8 15 1434926 27 45 4656145 Comps. 57 8386706 73 30 9583197 81 45 9896514 62 15 8849876 Halves. 28 30 4771588 14 15 2461533 36 45 5983246 Comps. 61 30 8788171 75 45 9692309 53 15 8012538 Half. 30 45 5112931 Comp. 59 15 8594064 The sines of small arcs are then deduced in this manner. From the sine of 45', above determined, arc found the halves, which will be thus : 45' 0" - - - - 130896 22 30 - - - - G.'5449,4 11 15 - - _ _ 32724,8 Now these hist two sines bciniT evidently in the same ratio as their arcs, ttie sines of all the less single minutes will he found by single proportion. So the 45th part of the sine of 45', TRACT 19.' TRIGONOMETRICAL -yABLES, &C. 301 gives 2909 for the sine of l' ; which may be doabled, tripled, &e, for the sines of 2', 3', &c, up to 45'. Then, from all the foregoing primary sines, by the theorems for halving, doubhng, or tripling, and by those for the sums and differences, the rest of the sines are deduced, to complete the quadrant. But having thus determined the sines and cosines of tlte first 30" of the quadrant, that is, the sines of the first and last 30, those of the intermediate 30 are, by theor. 4, found by one single subtraction for each sine. The sines of the whole quadrant being thus completed, the tangents are found by theor. IS, 19, 22, namely, for one half of the quadrant by the 18th and 19th, and the other half, by one single addition or subtraction for each, by the 22d theorem. And lastly, by theor. 24 and 25, the secants are deduced from the tangents, by addition and subtraction only. Among the various means used for constructing the canon of sines, tangents, and secants, the writers above enumerated seem not to have been possessed of the method of difi'erences, so profitably used since, and first of all I believe b}- 13riggs, in computing his trigonometrical canon and his logarithms, as we shall see hereafter, when we come to describe those works. They took however the successive differences of the numbers, after they were computed, to verily or prove the truth of them; and if found erroneous, by any irregularity in the last differences, from thence they had a metiiod of cor- recting the original numbers themselves. At least, this me- thod is used by Pitiscus, Trig. lib. 2, where the differences are extended to the third order. In page 44 of the same book also is described, for the first time that I know of, tiie common notation of decimal fractions, as now used. And this same notation was afterwards described and used by baron Napier, in positio 4 and 5 of his posthumous works, on the construction of logarithms, published by his son in the year 1619. But the decimal fractions themselves may be consi- dered as having been introduced by Regiomontanus, by his degimal division of the radius, 6cc, of the circle; and from 302 -HISTORY OF TRACT 19. that time gradually brought into use; but continued long to be denoted after the manner of vulgar fractions, by a line drawn between the numerator and denominator, which last however was soon omitted, and only the numerator set down, with the line below it: thus, it was first 3l^VV, then 3li4. ; afterwards, omitting the line, it became 31", and lastly 3I3 5, or 3J.35, or 3r35: as may be traced in the works of Vieta, and others since his timcj gradually into the present century. Having often heard it remarked, that the word sine, or in Latin and French sinaSy is of doubtful origin; and as the va- rious accounts which I have seen of its derivation are very different from one another, it may not be amiss here to em- ploy a few lines on this matter. Some authors say, this is an Arabic word, others tliat it is the single Latin word sinus ; and in Montucla's *' Ilistoirc des Mathcmatiqucs" it is con- jectured to be an abbreviation of two Latin words. The conjecture is thus expressed by the ingenious and learned author of that excellent histor}', at p. xxxiii, among the addi- tions and corrections of the first volume: " A I'occasion des sinus dont on parle dans cette page, comme d'une invention des Arabes, voici une etymologie de ce nom, tout-a-fait heu- reuse et vraisemblahlc. Je ladois a M. Godin, de I'Academic Royale des Sciences, Directeur de I'Ecolc de Marine de Cadix. Lcs sinus sont, comme I'oii scait, des moities de cords; et les cordes en Latin se nomment ijiscripttv. Les sinus sont done semisscs inscriptarum, ce que })robabIement on ecrivit ainsi pour abreger, S. Ins. Dela en.suitc s'cst fait par abus le mot de sinus." Now, ingenious as this conjecture is, there ap- pears to be little or no probabiUty for the truth of it. For, in t!ie tirst place, it is not in tlic least supported by quotations from any of the more early books, to show that it ever was the practice to write or ]irint the Avords tlius, .5'. Iiis. upon which t!ie conjecture is founded. Again, it is said tlie chords are call- ed in Latin itiscriptce ; and it is true that they soinetiu)es are so: but I think they are more frequently called .subtnurc, and the sines scviisscs sub'cnsariwi of tlie double arcs, vliicii will not abbreviate itito the word si7xus. This conjecture the learned TRACT 19. TRIGONOMETRICAL TABLES, &C. " 303 author has relinquished in the new edition of his history. But it may be said, what reason have we to suppose that this word is either a Latin word, or the abbreviation of any Latin words whatever? and that it seems but proper to seek for the ety- mology of wort/^ in the language of the inventors of the things. For which reason it is, that we find the two other words, iangens and secans, are Latin, as they were invented and used by authors who wrote in that language. But the sines are acknowledged to have been invented and introduced by the Arabians, and thence by analogy it would seem probable that this is a word of their language, and from them adopted, to- gether with the use of it, by the Europeans. And indeed Lansberg, in the second page of his trigonometry above- mentioned, expressly sa}^, that it is Arabic : His words are. Vox sinus Arabica est, et proinde barbara; sed cum longo usu approbata sit, et commodior non suppciat, nequaquam repudi- anda est : faciles enim in verbis nos esse oportet, ciim de rebus coiwenit. And Vieta sa3^s something to the same purport, in page 9 of his *' Universalium Inspectionum ad Canonem Mathematicum Liber:" His words are. Breve sinus vocabulum, dim sit art is, Saracenis prcvsertim quam fayniliare, non est ab artifcibus explodaidum, ad laterum semissium inscriptorum deyiotationem, &fr. Guarinus also is of the same opinion : in his *' Euclidcs Adauctus," &c. tract xx. pa. 307, he says, Sinus vero est 7iomen Arabicum usurpatuni in hanc signijicationcm a mathe- viaticis; though he was av.-are that a Latin origin was as- cribed to it by Vitalis, for he immediately adds, Licet Vitalir, iyi suo Lexico Mathematieo ex eo velit sinum appeUatum, quod chiudat curvitatan arcus. Loner before I either saw or heard of anv conjecture, or observation, coticerniog the etymology of the word sinus, I remember that I imagined it to be taken from the same Latin ward, signifying breast or bosom, and that our sine was so called allegoricallv. I had observed, that several of the terms in trigonometry were derived from a bow to shoot with, and its appendages; as arcus the bov,\ chorda the strinc^, and 'J04- HISTORY Of TRACT 19. sagitta the arrow, by which name the versed snie, which re- presents it, was sometimes called; also, that the tan,fens was so called from its office, being a line touching the circle, and secans from its cutting the same : I therefore imagined that the sinus was so called, either from its resemblance to the breast or bosom, or from its being a line drav/n Avithin the bosom (sinus) of the arc, or from its being that part of the string (chorda) of a bow (arcus) which is drawn near the breast (sinus) in the act of shooting. And perhaps Vitalis's defini- tion, above-quoted, has some allusion to the siimc similitude. Also Vieta seems to allude to the same thing, in calling sinus an allegorical word, in page 4 17 of his works, as pub- lished by Schooten, where, Avith his usual judgment and precision, he treats of the propriety of the terms used in trigonometry for certain lines drawn in and about the circle ; of which, as it very well deserves, I shall here extract the principal part, to show the opinion and arguments of so great a man on those names. " Arabcs autem semisscs inscriptas duplo, numeris praesertim a>stimatas, vocaverunt allegorice Sinus, atque ideo ipsam semi-diametrum, qua) maxima est scmissium inscriptarum, Sinum Totum. Et dc iis sua me- thodo canones exivenint qui circumferuntur, supputante praesertim Regiomontano bene juste et accurate, in iis ctiam particulis qualium semidiameter adsumitur 10,000,000. *' Ex canonibus deinde sinuum derivaverunt recentiorci canonem semissium circumscriptarum, quem dixere Foecun- dum; et canonem edactarum e centro, quem dixere Foccun- dissimum et Benejicum, hypotenusis addictum. Atque adecN ?cmisses circumscriptas, numeris prassertiin a;stimatas, voca- verunt Faecundo.^, Sinus nunierdsve videlicet; quanquam nihil vctat Fd'cundi nomen substantive accipi. Ilypotenusas autem flenehcas, vel etiam simpliciter Hypotenusas: quoniam hy- [)0tenusa in ])rima sevie sinus totius nomen retinet. Itaquc iic novitate verborum res adumbretur, et alioqui sua artihci- bus, eo nomine dibita, praeripiatur gloria, jirffposita in Canone Malhematico canonicis numeris inscriplio, candide admonet primam sericm esse Canonem Sinunni. hi secund.i TRACT 19. TRIGONOMETRICAL TABLES, ScC. 305 vero, partem canonis foecundi, partem canotiis foecundissimi, cotineri. In tertia, reliquam. *' Sane prseter inscriptas et circumscriptas, circulum etiam adficiunt aliae lineae rectae, velut Incidentes, Tangentes, et Secantes. Verum illae voces substantivae sunt, non periphe- riarum relativae. Ac secare quidem circulum linea recta tunc intelligitur, cum in duobus punctis secat. Itaque non loquuntur bene geometrice, qui eductas e centro ad metas cir- cumscriptarum vocant secantes improprie, cum secantes et tangentes ad certos angulos vel peripherias referunt. Imm6 vero artem confundunt, cum his vocibus necesse habeat uti geometra abs relatione. " Quare si quibus arrideat Arabum metaphora; quae quidem aut omnino retinenda videtur, aut omnino explodenda ; ut semisses inscriptas, Arabes vocant sinus ; sic semisses circum- scriptae, vocentur Prosinus Amsinusvej et eductge e centro Transinuosae. Sin allegoria displiceat, geometrica sane in- scriptarum et circumscriptarum nomina retineantur. Et cum eductae e centro ad metas circumscriptarum, non habeant hactenus nomen certum neque elegans, voceantur sane pro- semidiametri, quasi protensae semidiametri, se habentes ad suas circumscriptas, sicut semidiametri ad inscriptas." Against the Arabic origin however of tliis word CsinusJ may be urged its being varied according to the fourth de- clension of Latin nouns, like Dianus; and that if it were an Arabic word latinized, it would have been ranked under either the first, second, or third declension, as is usual in such adopted words. So that, upon the whole, it will perhaps rather seem pro- bable, that the term sinus is the Latin Avord answering to the name by which the Saracens called that line, and not their word itself. And this conjecture seems to be rendered still more probable by some expressions in pa. 4 and 5 of Otho'i " Preface to Rheticus's Canon," where it is not only said, that the Saracens called the half-chord of double t\\Qdirc sinus y but also that they called the part of the radius lying between the sine and the arc sinus lersus, vel sagitta, which are evi- VOL. I. X 506 HISTORF OF TRACT 20. dntly Latin words, and seem to be intended for the Latin translations of the names by which the Arabians called these lines, or the numbers expressing the lengths of them. And this conjecture has been confirmed and realised, by a reference to Golius's Lexicon of the Arabic and Latin lan- guages. In consequence I find that the Arabic and Latin writers on trigonometry do both of them use those words in the same allegorical sense, the latter being the Latin trans- lations of the former, and not the Arabic words corrupted. Thus, the true Arabic word to denote the trigonometrical sitie is L-^a:^, pronounced Jeib, (reading the vowels in the French manner), meaning sinus indusii, vcstisque, the bosom part of the garment; the versed sine is *^j, Sehmi, which hsagiita, the arrow; the arc is rvwr^, which is arcus, the arcj and the chord is >J^, Vit)\ that is chorda, the chord. TRACT XX. HISTORY OF LOGARITHMS. The trigonometrical canon, of natural sines, tangents, and secants, being now brought to a considerable degree of per- fection; the great length and accuracy of the numbers, toge- ther with the increasing delicacy and number of astronomical problems, and splicrical triangles, to the solution of which the canon was applied, urged many persons, conversant in those matters, to endeavour to discover some means of dimi- nishing the great labour and time, requisite for so many mul- tiplications and divisions, in such huge numbers as the tables then consisted of. And their chief aim was, to reduce the multi{)lications and divisions to additions and subtractions, as nmcli as possible. For this purpose, Nichohis Ravnicr Urstis Dithmarsus in- vented an ingenious method, which serves for one case in the TRACT 20. LOGARITHMS. 307 sines, namely, when radius is the first term in the proportion, and the sines of two arcs are the second and third terms ; for he showed, that the fourth term, or sine, would be found by only taking half the sum or difference of the sines of two other arcs, which should be the sura and difference of the less of the two former given arcs, and the complement of the greater. This is no more, in effect, than the following well-known theorem in trigonometry : as half radius is to the sine of one arc, so is the sine of another arc, to the cosine of the differ- ence minus the cosine of the sum of the said arcs. The au- thor published this ingenious device, in 1588, in his " Fun- damentum Astronomise." And three or four years afterwards" it was greatly improved by Clavius, who adapted it to all proportions in the solution of spherical triangles, for sines, tangents, secants, versed sines, &c; and that whether radius be in the proportion or not. All which he explains veiy fully in iem. 53, lib. 1, of his treatise on the Astrolabe. See more on this subject in Longomont. Astron. Danica. pa. 7, et seq. This method, though ingenious enough, depends not on any abstract property of numbers, but only on the relations of certain lines, drawn in and about the circle ; for which rea- son it was rather limited, and sometimes attended with trouble in the application. After perhaps various other contrivances, incessant endea- vours at length produced the happy invention of logarithms, which are of direct and universal application to all numbers abstractedly considered, being derived from a property inhe- rent in numbers themselves. This property maybe considered, either as the relation between a geometrical series of terms and a corresponding arithmetical one, or as the relation be- tween ratios and the measures of ratios, which comes to much the same thing, having been conceived in one of these ways by some of the writers on this subject, and in the other by the rest of them, as well as in both ways at different times by the same writer. A succinct idea of this property, and of the probable reflections made on it by the first writers on loga- rithms, may be to the following effect: X 2 308 HISTORY or TRACT 20. The learned calculators, about the close of the IGth, and beginning of the 17th century, finding the operations of mul- tiplication and division by very long numbers, of 7 or 8 places of figures, wliieh they had frequently occasion to perform, in resolving problems relating to geograpliy and astronomy, to be exceedingly troublesome, set themselves to consider, whether it was not possible to find some method of lessening this labour, by substituting other easier operations in their stead. In pursuit of this object, they reflected, that since, in every multiplication by a whole number, the ratio, or pro- portion, of the product to the multiplicand, is the same as the ratio of the multiplier to unity, it will follow that the ratio of the product to unity (which, according to Euclid's defini- tion of compound ratios, is compounded of the ratios of the said product to the multiplicand and of the multiplicand to unity), must be equal to the sum of the two ratios of the mul- tiplier to imity and of the multiplicand to unity. Conse- quently, if they could find a set of artiticial numbers that should be the representatives of, or should he proportional to, the ratios of all sorts of ruimbers to unity, the addition of the two artificial numbers that should represent the ratios of any multiplier and nuiltiplic:and to unity, would answer to the multiplication of the said nuiltiplicand by tljc said multiplier, or the sum arising from the addition of the said representative numbers, would be the representative munber of the ratio of the product to unity; and consecpientlv, the natural number to which it should be found, in the table i)f the said artificial i)r re[)r>u-.sentative nnmbi;rs, that the said sum belonged, would he tlie produft of the ^aid mnlliplicaiid and multiplier. Having settled this piinci[)le, us the foundation of their vvished-for method of abridging the labour of calculations, tliey resolved to couiijOsc a table of such aitificial numbers, ur numbers t!iat should be representatives of, or ])roportional tu, the ratios of all the common or natural numbers to unity. The fir^t observation that naturally occurred to them in the pursuit of this scheme was, that whatever artihcial numbers should be chosiMi to represent the ratios of other whole nuin- TRACT 20. LOGARITHMS. 809 bers to unity, the ratio of equality, or of unity to unity, must be represented by ; because that ratio has properly no mag- nitude, since, when it is added to, or subtracted from, any other ratio, it neither increases nor diminishes it. The second obser\'ation that occurred to them was, that any number whatever might be chosen at pleasure for the representative of the ratio of any given natural number to unity ; but that, when once such choice was made, all the other representative numbers would be thereby determined, because they must be greater or less than that first represen- tative number, in the same proportions in which the ratios represented by them, or the ratios of the corresponding na- tural numbers to unity, were greater or less than the ratio of the said given natural number to unity. Thus, either 1 , or 2, or 3, &c, might be chosen for the representative of the ratio of 10 to 1. But, if 1 be chosen for it, the representa- tives of the ratios of 100 to 1 and 1000 to 1, which are double and triple of the ratio of 10 to 1, must be 2 and 3, and can- not be any other numbers; and, if 2 be chosen for it, the re- presentatives of the ratios of 100 to 1 and 1000 to 1, will be 4 and 6, and cannot he any other numbers; and, if 3 be cho- sen for it, the representatives of the ratios of 100 to 1 and 1000 to 1, will be 6 and 9, and cannot be any other numbers; and so on. The third observation that occurred to tliem Avas, that, as these artificial numbers were representatives of, or propor- tional to, ratios of the natural numbers to unity, they must be expressions of the numbers of some smaller equal ratios that are contained in the said ratios. Thus, if 1 be taken for the representative of the ratio of 10 to 1, then 3, which is the representative of the ratio of 1000 to 1, v.ill express the num- ber of ratios of 10 to 1 that are contained in the ratio of 1000 to 1. And if, instead of 1, we make 10,000,000, or ten mil- lions, the representative of the ratio of 10 to 1, (in which case I will be the representativ'e of a very small ratio, ovratiuncula, which is only the ten-millionth part of the ratio of 10 to 1, or will be the representative of the 10,000,000th root of 10, $10 HISTORY OF TRACT 20". or of the first or smallest of 9,999,999 mean proportionals interposed between 1 and 10), the representative of the ratio of 1000 to 1, which will in this case be 30,000,000, will ex- press the number of those ratiuncul^e, or small ratios of the 10,000,000th root of 10 to I, which are contained in the said ratio of 1000 to 1. And the like may be shown of the repre- sentative of the ratio of any other number to unity. And therefore they thought these artificial imnibers, which thus represent, or are proportional to, the magnitudes of the ratios of the natural numbers to unity, might not improperly be called the Logarithms of those ratios, since they express the numbers of smaller ratios of which they are composed. And then, for the sake of bre\nty, they called them the Logarithms of the said natural numbers themselves, M-hich are the antece- dents of the said ratios to unity, of which they are in truth the representatives. The foregoing method of considering this property leads to much the same conclusions as the other way, in which the relations between a geometrical series of terms, and their ex- ponents, or the terms of an arithmetical scries, are contem- plated. In this latter way, it readily occurred that the addi- tion of the terms of the arithmetical series corresponded to the multiplication of tlie terms of the geometrical series; and that the arithmetical would therefore form a set of artificial numbers, %vl]ich, when arranged in tables, with their geome- tricals, would answer the purposes desired, as has been ex- plained above. From this property, by assuming four quantities, two of them as two terms in a geometrical series, and the others as the two corresponding terms of the arithmcticals, or artificials, or logarithms, it is evident that all the other terms of both the two series may thence be generated. And therefore there may be as many sets or scales of logarithms as we please, since they depend entirely on the arbitrary assumption of the first two arithmcticals. And all possible natural numbers may bo supposed to coincide with some of tlie terms of any geometrical progrc-sion whatever, the logiuithms or arith- TRACT 20. LOGARITHMS. Sll meticals determining which of the terms in that progression they are. It was proper however that the arithmetical series should be so assumed, as that the term in it might answer to the term I in the geometricals ; otherwise the sum of the loga- rithms of any two numbers would be always to be diminished by the logarithm of 1, to give the logarithm of the product of those numbers : for which reason, making the logarithm of 1 , and assuming any quantity whatever for the value of the logarithm of any one number, the logarithms of all other num- bers were thence to be derived. And hence, like as the mul- tiplication of two numbers is effected by barely adding their logarithms, so division is performed by subtracting the loga- rithm of the one from that of the other, raising ol" powers b^ multiplying the logarithm of the given number by the index of the power, and extraction of roots by dividing the loga- rithm by the index of the root. It is also evident that, in all scales or systems of logarithms, the logarithm of will be in- finite ; namely, infinitely negative if the logarithms increase with the natural numbers, but infinitely positive if the contrary ; because that, while the geometrical series must decrease through infinite divisions by the ratio of the progres- sion, before the quotient come to or nothing; the logarithms, or arithmeticals, will in like manner undergo the correspond- ing infinite subtractions or additions of the conmion equal difference; which equal increase or decrease, thus indefinitely continued, must needs tend to an infinite result. This however was no newly-discovered property of num- bers, but what was always well known to all mathematicians, beino; treated of in the writinos of Euclid, as also by Archi- medes, who made great use of it in his Arenarius, or treatise on the number of the sands, namely, in assigning the rank or place of those terms, of a geometrical series, produced from the multiplication together of any of the foregoing terms, by the addition of the corresponding terms of the arithmetical series, which served as the indices or exponents of the former. Stifelius also treats very fully of this property at folio 35 et 313 HISTORY OF TRACT 20. seq. and there explains all its principal uses as relating to the logarithms of numbers, only without the name ; such as, that addition answers to multiplication, subtraction to division, multiplication of exponents to involution, and dividing of exponents to evolution; all which he exemplifies in the rule- of-three, and in finding several mean proportionals, ike, exactly as is done in logarithms. So that he seems to have been in the full possession of the idea of logarithms, but without the necessity of making a table of such numbers. For the reason why tables of these numbers were not sooner composed, was, that the accuracy and trouble of trigonome- trical computations had not sooner rendered them necessary. It is therefore not to be doubted that, about the close of the sixteenth and beginning of the seventeenth century, many persons had thoughts of such a table of numbers, besides the few who are said to have attempted it. It has been said by some, that Longomontanus invented logarithms: but this cannot well be supposed to have been any more than in idea, since he never published any thing of the kind, nor ever laid claim to the invention, though he lived thirty-three years after they were first published by baron Napier, as he died only in 1647, when they had been long known and received all over Europe. Nay more, Longo- montanus himself ascribes the invention to Napier: vid. Astron. Danica, p. 7, &c. Some circumstances of this matter are indeed related b}^ Wood in his " Athenae Oxonienses," under the article Briggs, on the authority of Oughtrcd and Wingate, viz. " That one Dr. Craig, a Scotchman, coming out of Denmark into his ow n country, called upon .Joh. Neper baron of Marcheston near Edenburgh, and told him among other discourses, of a new invention in Denmark (by Longo- montanus as 'tis said) to save the tedious multiplication and division in astronomical calculations. Neper being solicitous to know farther of him concerning this matter, he could give no other account of it, than that it was by proportionable numbers. Which hint Neper taking, he desired him at his return to call upon him again. Craig, after some weeks had TRACT 20. LOGARITHMS. 313 passed, did so, and Neper then showed him a rude draught of that he called Canon mirabilis Logarithmorum. Which draught, with some alterations, he printing in 1614, it came forthwith into the hands of our author Briggs, and into those of Will. Oughtred, from whom the relation of this matter came." Kepler also says, that one Juste Byrge, assistant astronomer to the landgrave of Hesse, invented or projected logarithms long before Neper did ; but that they had never come abroad, on account of the great reservedness of their author \\ith re- gard to his own compositions. It is also said, that Byrge computed a table of natural sines for every two seconds of the quadrant. But whatever may have been said, or conjectured, concern- ing any thing that may have been done by others, it is certain that the world is indebted, for the first publication of loga- rithms, to John Napier, or Nepair*, or in Latin, Neper, baron of Merchiston, or Markinston, in Scotland, who djed the 3d of April 1618, at 67 years of age. Baron Napier added con- siderable improvements to trigonometry, and the frequent numeral computations he performed in this branch, gave occasion to his invention of logarithms, in order to save part of the troul)ie attending those calculations; and for this rea- son he adapted his tables peculiarly to trigonometrical uses. The origin of which name, Crav.furd informs us, was from a (less) \>ec\less action of one of his ancestors, viz. Donald, second son of the eail of Lenox, in the time of David the Second. " Some English writers, mistaking the import of the term baron, having called this celebrated person lord Napier, a Scotch noble- man. He was not indeed a peer of Scotland : but the peerage of Scotland in- forms us, that he was of a very ancient, honourable, and illustrious family ; that his ancestors, for many generations, had been possessed of sundry baronies, and, amongst others, of the barony of Merchistoun, which descended to him by the death of his father in 1608. Mr. Briggs, therefore, very properly styles him Baro Merchesionii. Now, according to Skene, de verborum sigmjicalione, ' In this realm (of Scotland) he is called a Baronne, quha haldis his landes immediatelie in chiefe of the king, and hes power of Pit and GaUow.'*; Fos^a et Furca; quhilk was first institute and granted be king Malcomc, quha gave power to the Barroncs to have ane Pit, quhalrin wemcn condemned for thi( ft suld be drowned, and ane (iallows, whereupon men thicres and trf spas.-owres suid be ha-nged, confoime to 314 HISTORY OF TRACT 20. This discovery he published in 1614, in his book iiititled " Mirifici Logarithmorum Canonis Descriptio," reserving the construction of the numbers till the sense of the learned con- cerning his invention should be known. And, excepting the construction, this is a perfect work on this kind of logarithms, containing in effect the logarithms of all numbers, and the logarithmic sines, tangents, and secants, for every minute of the quadrant, together with the description and uses of the tables, as also his definition and idea of logarithms. Napier explains his notion of logarithms by lines described or generated by the motion of points, in this manner : He first conceives a line to be generated by the equable motion of a point, which passes over equal portions of it in equal small moments or portions of time: he then considers another line as generated by the unequal motion of a point, in such man- ner, that, in the aforesaid equal moments or portions of time, there may be described or cut off, from a given line, parts jvhich shall be continually in the same proportion with the respective remainders, of that line, which had before been left : then are the several lengths of the first line, the loga- rithms of the corresponding parts of tlie latter. Which description of tiiem is similar to this, tliat the logarithms are a series of quantities or numbers in arithmetical progression, adapted to another series in geometrical progression. The the floome given in tVie Earon C )nrt thercanent.' So that a Scotch baron, though no peor, was nevertheless a very roiisiderable persoiiag'C, both in dignity auil power." lieid's /'sioy on Lot^a'ill.ms. The name of the ilhistiioiis inventor of logaiithms, has been variously wiit'.en at (Hfftrent times, and on different occa- sions. In his own Latin works, and in (perhaps) all other books in Latin, it is Ne/tr, or Neptrus Baro Merc/nsUnii: By Brings, in a letter to Archbistiop Usher, he is called N^iper, lord of Mnrkinslmi : In Wright's translation of the logarithms, which was revised by the autiior riimself, and piibIislie/1 in 1616, he is called Nepair, baroi of Meidihiun ; and the same by Crawfurd and some others : But M'Kcnzic and others write it Napier, baron of Mrrcfiiston ; which, being also the orthography now used by the family, I shall adopt in this work. I obstrve also, that the Scotch Compendium of Honour says he was only Sriggs, not less esteemed for his great probity, and other eminent virtues, than for his excellent skill in ma- thematics, was, at the time of the publication of Napier's logarithms, in 1614, professor of geometry in Gresham col- lege in London, having been appointed the first professor after its institution : which appointment he held till January 1(J20, when he was chosen, also the first, Saviiian professor of gcouietry at Oxford, where he died January the 26th, 1G37-, ;;ged about 74 years. On tiie publication of Napier*s logarithms, Briggs imme- diately applied himself to the study and improvement of them. In a letter to Mr. (afterwards Archbishop) Usher, dated the TRACT 20. LOGARITHMS. S2t loth of March 1615, he writes, " that he was wholly taken up and employed about the noble invention of logarithm^, lately discovered." And again, " Napier lord of Markinston hath set my head and hands at work with his new and admir- able logarithms: I hope to see him this summer, if it please God ; for I never saw a book which pleased me better, and made me more wonder." Thus we find that Briggs began very early to compute logarithms: but these were not of the same kind with Napier's, in which the logarithm of the ratio of 10 to 1 was 2.3025851 &c ; for, in Briggs's first attempt he made 1 the logarithm of that ratio; and, from the evidence we have, it appears that he was the first person who formed the idea of this change in the scale, which he presently and liberally communicated, both to the public in his lectures, and to lord Napier himself, who afterwards said that he also had thought of the same thing ; as appears by the following ex- tract, translated from the preface to Briggs's " Arithmetica Logarithmica :" " Wonder not (says he) that these logarithms are different from those which the excellent baron of Marchi- ston published in his Admirable Canon. For when I explained the doctrine of them to my auditors at Gresham college in London, I remarked that it would be much more convenient, the logarithm of the sine total or radius being O (as in the Canon Mirificus), if the logarithm of the 10th part of the said radius, namely, of 5^44*21", were 100000 &c; and con- cerning this I presently wrote to the author; also, as soon as the season of the year and my public teaching would permit, I went to Edinburgh, Avhere being kindly received by him, I staid a whole niontli. But when we began to converse about the alteration of them, he said that he had formerly thought of it, and wished it ; but that he chose to publish those that were already done, till such time as his leisure and health would permit him to make others more convenient. And as to the manner of the change, he thought it more expedient tliat should be made the logarithm of l,and 100000 he the logarithm of radius; which I could not but acknowledge was jr.uch better. Therefore, rejecting those which I bad before 32S Kl&TORT OF T'KACT 20. prepared, I proceeded, at his exhortation, to calculate these: and the next summer I went again to Edinburgh, to shew him the principle of them; and should have been glad to do the same the third summer, if it had pleased God to spare him 80 long." So that it is plain that Briggs was the inventor of the pre- sent scale of logarithms, in which 1 is the logarithm of the ratio of 10 to l , and 2 that of TOO to I , &c ; and that the share which Napier had in them, was only advising Briggs to begin at the lowest number 1, and make the logarithms, or artificial numbers, as Napier had also called them, to increase with the natural numbers, instead oi decreasing ; which made no alter- ation in the figures that expressed Briggs's logarithms, but only in their affection or signs, changing them from negative to positive; so that Briggs's first loga- rithms to the numbers in the second column of the annexed tablet, would have been as in the first column ; but after they were changed, as they are here in the third column; which is a change of no essential difference, as the logarithm of the ratio of 10 to 1, the radix of the natural system of numbers, continues the same ; and a change in the logarithm of that ratio being the only circumstance that can essentially alter the sy; tern of logarithms, the logarithm of 1 being 0. And the reason why Briggs, after that interview, rejected what he had before done, and began anew, was probably because he had adapted his new logarithms to the approximate sines of arcs, instead of to the round or integer numbers; and not from their being logarithms of another system, as were those of Napier. On Briggs's return from Edinburgh to London the second time, namely, in 1617, he printed the first thousand loga- rithms, to eight places of figures, besides the index, under the title of " Lo^aritlnnorum Chilias Prima." Thouoh tliese seem not to have been published till after death of Napier, B Num. N n 01" n 3 001 -3 2 01 -2 1 '.: -1 1 -1 10 1 2 100 2 -3 1000 3 n 10" n TRACT 20. LOGARITHMS. $2$ which happened on the 3d of April 1618, as before said ; for, in the preface to them, Briggs says, *' Why these logarithms differ from those set forth by their most illustrious inven- tor, of ever respectful memory, in his ' Canon Mirificus,' it IS TO BE HOPED his posthumous work will shortly make ap- pear." And as Napier, after communication had with Briggs on the subject of altering- the scale of logarithms, had given notice, both in Wright's translation, and in his own " Rabdo- logia," printed in 1617, of his intention to alter the scale, (though it appears very plainly that he never intended to compute any more), without making an}- mention of the share which Briggs had in the alteration, this gentleman modestly gave the above hint. But not finding any regard paid to it in the said posthumous work, published by lord Napier's son in 1619, where the alteration is again adverted to, but still without any mention of Briggs; this gentleman thought he could not do less than state the grounds of that alteration himself, as they are above extracted from his work published in 1624. Thus, upon the whole matter, it seems evident that Briggs, whether he had thought of this improvement in the construc- tion of logarithms, of making 1 the logarithm of the ratio of 10 to 1, before lord Napier, or not (which is a secret that could be known only to Napier himself), was the first person who communicated the idea of such an improvenicnt to the world; and that he did this in his lectures to his auditors at Gresham college in the year 1615, very soon alter his peru- sal of Napier's " Canon Mirificus Logarithmorum," published in the year 1614. He also mentioned it to Napier, both by letter in the same year, and on his first visit to him in Scotland in the summer of the year 1616, when Napier approved the idea, and said it had ah-eady occurred to himself, and that he had determined to adopt it. It appears therefore, that it would have been more candid in lord Napier to have told the world, in the second edition of this book, that Mr. Briggs had mentioned this improvement to him, and that he had thereby been confirmed in the resolution he had already taken, before 330 HISTORY OF TRACT 20. Mr. Briggs*s communication with him (if indeed that wa^ the fact), to adopt it in that his second edition, as being better fitted to the decimal notation of aritlimetic which was in ge- neral use. Such a declaration would have been but an act of justice to Mr. Briggs; and the not having made it, cannot but incline us to suspect that lord Napier was desirous that the world should ascribe to him alone the merit of this very useful improvement of the logarithms, as well as that of hav- ing originally invented them ; though, if the having first com- municated an invention to the world be sufficient to entitle a man to the honour of liavinp; first invented it, Mr. Briggs had the better title to be called the first inventor of this happy improvement of logarithms. In 1620, two years after the <^ Chi Has Prima" of Briggs came out, Mr. Edmund Gunter publisiicd liis " Canon of Tri- angles," which contains the artificial or logarithmic sines and tangents, for every minute, to seven places of figures, besides the index, the logarithm of radius being lO'O kc. These logarithms arc of the kind last agreed upon by Napier and Briggs, and they were the first tables of logarithmic sines and tangents that were publisiicd of this sort. Gunter also, in 1623, reprinted the same in his book " l)e St^ctoreet Kadia," together with the " Chilias Prima" of his old colleague Mr. Briggs, he being professor of astronomy at Gresham college -when Briggs was prof^-ssor of geometry there, Gunter having been elected to that office the 6th of March 1(1 19, and enjoyed it till his death, whicii ha])pencd on t!ic 10th of December I62G, about the forty-fifth year of his age. In 1623, also, Gunter applied these logarithms of numbers, sines, and tan- gents, to straight lines drawn on a ruler; with which, pro^ portions in common nuuibers and trigonometry were resolved by the mere application of a pair of compasses ; a method founded on this property, that the logarithms of the terms of efjual ratios are equidifierent. This instrument, in the form of a two- foot scale, is now in common use for navigation and other purposes, and is commonly called the Gunter. He also greatly improved the sector for the same uses. Gunter \\'as TRACT 20. LOGARITHMS. 331 the first who used the word cosine for the sine of the comple- ment of an arc. He also introduced the use of arithmetical complements into the logarithmical arithmetic, as is witnessed by Briggs. chap. 15, Arith. Log. And it has been said, that he started the idea of the logarithmic curve, which was so called because the segments of its axis are the logarithms of the corresponding ordinates. The logarithmic lines were afterwards drawn in various other ways. In 1627, they were drawn by Wingate on two separate rulers sHding against each other, to save the use of compasses in resolving proportions. They were also, in 1627, applied to concentric circles, by Oughtred. Then in a spiral form, by a Mr. Milburne of Yorkshire, about the year 1650. And, lastly, in 1657, on the present sliding rule, by Setli Partridfre. The discoveries relatins: to logarithms were carried to France by Mr. Edmund Wmgate, but not first of all, as he erroneously says in the preface to his book. He published at Paris, in 1624, two small tracts in the French language; and afterwards at London, in 1626, an English edition of the same, with improvements. In the first of these, he teaches the use of Gunter's rules; and in the other, that of Briggs's logarithms, and the artificial sines and tangents. Here are contained, also, tables of those logarithms, sines, and tan- gents, copied from Gunter. The edition of these logarithms printed at London in 1635, and the former editions also I suppose, has tlie units figures {lis])osed along the tops of the columns, and the tens down the margins, like our tables at present; with the whole logarithm, which was only to fix places of figures, in the angle of meeting: which is the first instance that I have seen of this mode of arrangement. But proceed we now to the larger structure of logarithms. Briscsis had continued from the beginning to labour with fjreat industrv at the computation of those logarithms of which he before published a short specimen in small numbers. And, in 1624, he produced his " Arithmetica Logarithmica" a stu- pendous work for so short a time ! containing the logarithms 332 HISTORY OF TRACT 2a of 30000 natural numbers, to fourteen places of figures be- sides the index, nauielv, from 1 to 20000, and from 90000 to 100000; together witli the differences of the logarithms. Some writers say that there was an()therfAi7zW,namely, from 100000 to lOIOOO; but none of the copies that I have seen have more than the 30000 above mentioned, and they were all regularly terminated in the usual way with the word finis. The preface to these logaritlinis contains, among other things, an account of the alteration made in the scale by Napier and himself, from which we have given an extract; and an earnest soli- citation to others to undertake the coniputation for the inter- mediate numbers, offering to give instructions, and paper ready ruled for that purpose, to any persons so inclined to contribute to the com])letion of so valuable a Mork. In the introduction, he gives also an ample treatise on the construc- tion and uses of these logarithms, which will be particularly described hereafter. By this invitation, and other ineans, he bad hopes of collecting materials for the logarithms of the intermediate 70000 numbers, while he should employ his own labour more immediately on the canon of logarithmic sines and tangents, and so carry on both v.orks at once; as indeed thev were both equally necessary, and lie himself was now pretty far advanced in years. Soon after this however, Adrian Vlacq, or Flack, of Gouda in Holland, completed the ir.terinediate seventy chiliads, and republished the " Arithmetica Logarithmica" at that place, in 1G27 and lG2Sj witli those intermediate numbers, making in the whole the logarithms of all numbers to 100000, but only to ten places of figures. To these was added a table of artificial sines, tangents, ai:d secants, to every niinule of the quiidrant. 3>riggs himself lived also to complete a table of logarithmic siiu's ;itul tangents for the hundredth part of every degree, to louvtec:; i)Lices of tigures besides the index; together with a tabl'j oi' iiatLu-al sines for tlie same parts to fifteen places, and the taii'jcnts and secarns i'or the same to ten places; with the eo:iSt.rutL-.un uf the wijole. These tables were printed at TRACT 20. LOGARITHMS. SSS Gouda, unJer the care of Adrian Vlacq, and mostly finished oft" before 1631 , though not pubhshed till 1633. But his death, which then happened, prevented him from completing the application and uses of them. However, the performing of this office, when dying, he recommended to his friend Henry Gellibrand, who was then professor of astronomy in Gresham college, having succeeded Mr. Gunter in that appointment. GeUibrand accordingly added a preface, and the application of the logarithms to plain and spherical trigonometry, &c ; and the whole was printed at Gouda by the same printer, and brought out in the same year, 1633, as the '* Trigonometria Artificialis" of Vlacq, who had the care of the press as above said. This work was called " Trigonometria Britannica ;'* and besides the arcs in decrees and centesms of de":rees, it has another column, containing the minutes and seconds an- swering to the several centesms in the first column. In 1633, as mentioned above, Vlacq printed at Gouda, in Holland, his " Trigonometria Artificialis ; sive Magnus Canon Triangulorum Logaritinnicus ad Decadas Secundorum Scru- pulorum constructus." This work contains the logarithmic sines and tangents to ten places of figures, with their differ- ences, for cverv ten seconds in t!ie (juadrant. To them is also added Briggs's table of the first 20000 logarithms, but carried only to ten places of figures besides the index, with their dif- ferences. The whole is preceded by a description of the tables, and the application of them to plane and spherical trigonometry, chiefly extracted from Briggs's " Trigono- metria Britannica," mentioned above. Gellibrand published also, in 1635, '' An Institution Trigo- nomctricall," containing the logarithms of the first 10000 numbers, with the natural sines, tangents, and secants, and the logarithmic sines and tangents, for degrees and minutes, all to seven places of figures, besides the index ; as also other tables proper for navigation ; v> ith the uses of the whole. Gellibrand died the 9th of February 1636, in the 40th year of his age, to the great loss of the mathematical world. Besides the persons hitherto mentioned, who were mostly 334 HISTORY OF TRACT 20. computers of logarithms, many others have also pubiished tables of those artificial numbers, more or less complete, and sometimes improved and varied in the manner and form of them. We may here juit advert to a few of the principal of these. In 1626, D. Henrion published, at Paris, a treatise concern- ing Brigg's logarithms of common numbers, from 1 to 20000, to eleven places of figures; with the sines and tangents to eight places only. In 1631, was printed, at London, by one George Miller, a book containing Briggs's logarithms, Mitli their difFcreiices, to ten places of figures besides the index, for all numbers to lOOOOO ; as also the logarithmic sines, tungcrUs, an;' secants, for every minute of the quadrant; with the explanation and uses in English. The s.ime vcar, 1631, Richard Norwood published his *' Trigonometria ;" in which we find Briggs's logarithms for all numbers to IOOOO,and for the sines, tangents, and secants, to every minute, both to seven places besides the index. la t!ic conclusion of the trigonoujetry, he coniioiains of the un- fair practices of printing Vlacij's book in 1627 or 1628, and the book mentioned in the last article. His words are, " Now, whereas I have here, and in sundry places in this book, cited i\Ir. Briggs his ' Arithmetica Logaritlnuica,' (lest I may seem to abuse the reader) you are to understand not the book put forth about a month since in English, as a translation o! his, and witli the same title ; being nothing like his, nor worthy his name; but the book which hin}self put forth witli this title in Latin, b(^ing printed at London ainio 162 k And here I have just occasion to blame the ill dealing of these men, both in the mutter before; mentioned, and in printing a second edi- tion of his ' Arithmetica Logarithmica' in Latin, whilst he livetl, against his mind and liking; and brought them over to yell, when the first v/ere unsold; so frustrating those additions which Mr. Briggs intended in his second edition, and more- over leaving out some things that wore in the (irst edition, of special moment : a practice of very ill consequence, and TRACT 20. LOGARITHMS. 335 tending to the great disparagement of such as take pains in this kind." Francis Bonaventure Cavalerius published at Bologna, in 1632, his " Directorium Generale Uranometricum," in which are tables of Briggs's logarithms of sines, tangents, secants, and versed sines, each to eight places, for every second of the first five minntes, for every five seconds from five to ten mi- nutes, for every ten seconds from ten to twenty minutes, for every twenty seconds from twenty to thirty minutes, for every thirty seconds from 30' to 1 30', and for every minute in the rest of the quadrant j which is the first table of logarithmic versed sines tluit I know of. In this book are contained also the logarithms of the first ten chiliads of natural numbers, namely, from 1 to 1 0000, disposed in this manner: all the twenties at top, and from 1 to 19 on the side, the logarithm of the sum being in the square of meeting. In this work also, I think Cavalerius frave the method of finding- the area or spherical surface contained by various arcs described on the surface of a sphere; which had before been given by Albert Girard, in his Algebra, printed in the yeav 1629. Also, in the " Trigonometvia" of the same author, Cava- lerius, printed in 1643, besides the logarithms of numbers from 1 to 1000, to eight places, wit'u their differences, we find both natural and logarithmic sines, tangents, and secants, the former to seven, and the latter to eig])t places; namely, to every 10" of the first 30 minutes, to every 30" from 30' to 1; and the same for their complements, or backwards through the last degree of the quadrant; the intermediate SS" being to every minute only. Mr. Nathaniel Roe, " Pastor of Benacre in SufTolke," also reduced the logarithmic tables to a contracted form, in his *' Tabula- Logarithmica>," jjrinted at London in 1633. Here we have Briggs's logarithms of numbers fiom 1 to 100000, to eight places; the fifties placed at top, and from 1 to 50 on the side; also the first four figures of the logarithms at top, and the other four down the columns. They contain also the 336 HISTORY OP f RACT 20. logarithmic sines and tangents to every 100th partof degrees^ to ten places, Ludovicus Fiobcnius published at Hamburgh, in 1634, his *' Clavis Universa Trigonometriae," containing tables of Briggs's logarithms of numbers, from 1 to 2000; and of sines, tangents, and secants, for every minute; both to seven places. But the table of logarithms of common numbers was re- duced to its most convenient form by John Newton, in his *' Trigonometria Brit.innica," printed at London in 1658, having availed himself of both the improvements of Wingate and Roe, namely, uniting Wingate's disposition of the natural numbers with Rou's contracted arrangement of the logarithms, the numbers being all disposed as in our best tables at pre- sent, namely, the units along the top of the page, and the tens down the left-hand side, also the fii'st three figures of each logarithm in the first column, and the remaining five figures in the otlier columns, the logarithms being to eight places. Tills work contains also the logarithmic sines and tangents, to eight figures besides the index, for every 100th part of a decree, with their diflerences, and for lOOOth parts in the first three degrees. In the preface to this w-ork, Newton takes occasion, as Wingate and Norwood liad done before, as weH as Briggs himself, to censure the unfair practices of some other publishers of logarithms. He says, " In the second part of this institution, thou art presented with Mr. Gellibrand's Tri- gonometric, faithfully translated from the Latin copy, that which the author himself published under the title of ' Trigo- nometria Britannica,' and not that which Vlacq the Dtttchman styles ' Trigonometria Artificialis,' from whose corrupt and imperfect copy that seems to be translated which is amongst us generally known by the name of ' Gellibrand's Trigono- metry ;' but those who either knew him, or have perused his writings, can testify that he was no admirer of the old sexa- genary way of working; nay, that he did preferre the decimal way before it, as he hath abundantly testified in all the ex- amples of this his TKigonometrv, which differs from that othec TRACT 20 tOGARITHMS. S37 which VJacq hath pubHshed, and that which hath hitherto borne his name in EngUsh, as in the foroi, so hkewise in the matter of it; for in the two last-mentioned editions, there is something left out in the second chapter of plain triangles, the third chapter wholly omitted, and a part of the third in the spherical; but in this edition nothing: something we have added to both, by way of explanation and demonstration." In 1670, John Caramuel published his " Mathesis Nova," in which are contained 1000 logarithms both of Napier's and Briggs's form, as also 1000 of what he calls the Perfect Loga- rithms, namely, the same as those which Briggs first thought of, which differ from the last only in this, that the one in- creases while the other decreases, the radix or logarithm of the ratio of 10 to 1 being the same in both. The books of logarithms have since become very numer- ous, but the logarithms are mostly of that sort invented by Briggs, and which are now in common use. Of these, the most noted for their accuracy or usefulness, besides the works above mentioned, are Vlacq's small volume of tables, parti- cularly that edition printed at Lyons, in 1670; also tables printed at the same place in 1760 ; but most especially the tables of Sherwin and Gardiner, particularly my own im- proved editions of them. Of these, Sherwin's " Mathematical Tables," in 8vo, formed, till lately, the most complete col- lection of an}', containing, besides the logarithms of all num- bers to 101000, the sines, tangents, secants, and versed sines, both natural and logarithmic, to every minute of the quadrant, though not conveniently arranged. The first edition was in 1706; but the third edition, in 1742, which was revised by (^ardiner, is et^teemed the most correct of any, though con- taining many ttiousands of errors in the final figures, as well as all the former editions: as to the last or fifth edition, in 1771, it is so erroneously printed that no dependance can be placed in it, being the most inaccurate book of tables I ever knew ; 1 have a list of several thou-and errors which I have corrected in it, as well iis in Garuinei's octavo edition, and in Sherwin's edition. VOL, I. z 5JS HISTORY OP TRACT 20 Gardiner also printed at London, in 1742, a quarto volume of " Tables of Logarithms, for all numbers from 1 to 102100, and for the sines and tangents to every ten seconds of each degree in the quadrant; as also, for the sines of the first 72 minutes to every single second: with other useful and neces- sary tables ;" namely a table of Logistical Logarithms, and three smaller tables to be used for finding the logarithms of numbers to twenty places of figures. Of these tables of Gardiner, only a small number was printed, and that by sub- scription; and they have always been held in great estimatioi* for their accuracy and usefulness. An edition of Gardiner's collection was also elegantly printed at Avignon in France, in 1770, with some additions, namely, the sines and tangents for every single second in the first four degrees, and a small table of iiyperbolic logarithms, copied from a treatise on Fluxions by the late ingenious Mr. Thomas Simpson : but this is not quite so correct as Gardi- ner's own edition. The tables in all these books are to seven places of figures. Lastly, my own Mathematical Tables, being the most ac- curate and best arranged set of logarithmic tables ever before given; preceded also by a large and critical history of Tri- gonometry and LogaritluTis, and terminating with a copious list of the errors discovered in the princi-jial other tables of this kind. There have also lately appcired the following accurate and elegant books of logarithms; viz. 1." Logarithmic l^ibles," by the late Mr. ?\Iicli;iel Taylor, a pupil of mine, and author of " The Sexagesimal Table." His work consists of three tables; 1st, The Logarithms of Common Numbers from 1 to 1260, each to 8 places of figures; 2dlv, The Logarithms of all Numbers from 1 to 101000, each to 7 places; 3dlv, The Logarithmic Sines and Tangents to every Second of the Qua- drant, also to 7 places of figures: a work that must ])rove higlily useful to such persons as may be employed in very nice and accurate calculations, such as astroncunical t;Jjlcs, dkc. The author dying when the tables were nearly all j)rintc(l oil'. TRACT 20. LOGARITHMS. 330 the Rev. Dr. Maskelyne, astronomer royal, supplied a pre- face, containing an account of the work, with excellent pre- cepts for the explaimtion and use of the tables: the whole very accurately and elegantly printed on large 4to, 1792. 2. " Tables Portatives de Logarithmes, publiees a Londres par Gardiner," &c. This work is most beautifully printed in a neat portable 8vo volume, and contains all the tables in Gardiner's 4to volume, with some additions and improve- ments, and with a considerable degree of accuracy. Printed at Paris, by Didot, 1793. On this, as well as several oti)er occasions, it is but justice to remark the extraordinary spirit and elegance with which the learned men, and the artisans of the French nation, undertake and execute works of merit. 3. A second edition of the " Tables Portatives de Loga- rithmes," &c. printed at Paris with the stereotypes, of solid pages, in 8vo. 1795, by Didot. This edition is greatly en- larged, by an fextension of the old tables, and many new ones ; among which are the logarithm sines and tangents to every ten thousandth part of the quadrant, viz. in which the qua- drant is first divided into 100 equal parts, and each of these into 100 parts again. 4. Other more extensive tables, by Borda and Delambre, were published at Paris in 1801. Besides the usual table of the logavithms of common numbers, and a large introduction, on the nature and construction of them, this work contains very extensive tables of decimal trigonometry, arranged in a new and carious way, and containing the log. sines, tangents, and secants, of the quadrant, divided first into 100 degrees, each degree into 100 minutes, and each minute into 100 seconds. The logarithmic canon serves to find readily the logarithm of any assigned number; and we are told by Dr. Wallis, in the second volume of his Mathematical Works, that an anti- logarithmic canon, or one to find as readily the number cor- responding to every logarithm, was begun, he thinks, by Harriot the algebraist, who died in 1621, and completed by Walter Warner, the editor of Harriot's works, before 1640; z 2 ^'^^ CONSTRUCTION OF TRACT 21. which ingenious performance, it seems, ^vas lost, for want of encouragement to publish it. A small specimen of such numbers was published in the Philosophical Transactions for the year 1714, by Mr. Long of Oxford ; but it was not till 1742 that a complete antiloga- rithmic canon was published by Mr. James Dodson, wherein he has computed the numbers corresponding to every loga- rithm from 1 to 100000, for 11 places of figures. TRACT XXI. THE COXSTRUCTION OF LOGARITHMS, &C, Haying, in the last Tract, described tlie several kinds of logarithms, their rise and invention, their nature and proper- ties, and given ome account of tlic principal early cultivators of them, with the chief collections that have been published of such tables; proceed we now to deliver a more particular account of the ideas and methods employed by each author, and the peculiar modes of construction made use of by them. And first, of the great inventor himself, Lord Na))icr. Napier s Construclion of Logarithms. Tiie inventor of logarithms did not adapt them to the series of natund numbers 1, 2, 3, 4, 5, &c,as it was not his principal idea to extend them to all arithmetical operations in general ; but he confined his labours to that circumstance which first sufvrrested the necessity of the invtuition, and adajjtcd his lo- iraritimis to the approximate inmibers which express the na- tural sines of every minute in t!ie quadrant, as they had been set down by former writers on trigonometry. . The same restricted itiea was pursued through his method of constructing the logarithms. As the lines of the .sines of all arcs are parts of the radius, or sine ci the quadrant, which TRACT 21 . LOGARITHMS. 341 Sin3S. Log. AjO I 2 vriLs therefore called the simis tottts^ or whole sine, he conceived the line of the radius to be described, or run over, by a point movingalong it in such a manner, that in equal portions of time it generated, or cut off, parts in a decreasing geometrical pro- gression, leaving the several remainders, or sines, in geome- trical progression also ; while another point, in an indefinite line, described equal parts of it in the same equal portions of time; so that the respective suras of these, or the whole line generated, were always the arithmeticals or logarithms of these sines. Thus, flz is the given radius oh which all the sines are to betaken, and a&c the indefinite line containing the logarithms; these lines being each generated by the motion of points, beginning at A, a. Now, at the end of the 1st, 2d, 3d, &.c, moments, ar equal small portions of time, the mov- ing points being found at the places marked 1, 2, 3, &c; then za, zl, z2, zS, &c, will be the series of natural sines, and aO, or 0, a1, a2, a3, &c, will be their logarithms ; supposing the point which gene- rates as to move every where with a velocity de- creasing in proportion to its distance from z, namely > its velocity in the points 0, 1, 2, 3, Sec, to be re- spectively as the distances 2O, zl , z2, z'i, &c, Avhile the velocity of the point generating the logarithmic line a&c remains constantly the same as at first in the point a or 0. Hitherto the author had not fully limited his system or scale of logarithms, having only supposed one condition or limita- tion, namely, that the logarithm of the radius az should beO: whereas two independent conditions, no matter Avhat, are necessary to limit the scale or system of logarithms. It did not occur to him that it was proper to form the other limit, by affixing some particular value to an assigned number, or part of the radius : but, as anotlicr condition was necessary, he assumed this for it, namely, that the two generating points should begin to move at a and a with equal velocities; or that the increments fil and a1, described in the first moments, should be equal ; as he liiought this circumstance would be 7 &c. &c. 542 CONSTRUCTION OF TRACT 21. attended with some little ease in the computation. And this is the reason that, in his table, the natural sines and their 1q- garithms, at the complete quadrant, have equal differences ; and this is also the reason why his scale of logarithms happens accidentally to agree with whathaye since been called the hy- perbolic logarithms, which have numeral differences equal to those of their natural numbers, at the beginning; except only that these latter increase with the natural numbers, and his on the contrary decrease; the logarithm of the ratio of 10 to 1 being the same in both, namely, 2-30258509. And here, by the way, it may be observed, that Napier's manner of conceiving the generation of the lines of the natural numbers, and their logarithms, by the motion of points, is very similar to the manner in which Newton afterwards considered the generation of magnitudes in his doctrine of fluxions; and it is also remarkable, that, in art. 2, of the " Habitudines Logarithmorum et suorum naturalium numerorum inviccm," in the appendix to the " Constructio Logarithmorum," Napier speaks of the velocities of the increments or decrements of the logarithms, in the same way as Newton does of his fluxions, namely, where he shows that those velocities, or fluxions, arc inversely as the sines or natural numbers of the logarithms ; "which is a necessary consequence of the nature of the gene- ration of those lines as described above ; with this alteration, however, that now the radius az must be considered as gene- rated by an equable motion of the point, and the indefinite line A&c by a motion increasing in the same ratio as the other before decreased ; which is a supposition that Napier must have had in view when he stated that relation of the fluxions. Having thus limited his system, Napier proceeds, in the posthumous work of 1619, to explain his construction of the lofjarithmic canon; and this he eflects in various wavs, but chiefly by generating, in a very easy manner, a series oi pro- portional numbers, and their arithmeticals or logarithms; and then finding, i)y proportion, the logarithms to the natural sines, from those of the nearest numbers among the original proportionals. TRACT 21. LOGARITHMS. 343 After describing the necessary cautions he n^ade use of, to preserve a sufficient degree of accuracy, in so long and com- plex a process of calculation ; such as annexing several ciphers, as decimals separated by a point, to his primitive numbers, and rejecting the decimals thence resulting after the operations M'ere completed ; setting the numbers down to the nearest unit in the last figrure ; and teaching the arithme- tical processes of adding, subtracting, multiplying, and divid- ing theliqiits, between which certain unknown numbers must lie, so as to obtain the limits between which the results must also fall; I say, after describing such particulars, in order to clear and smooth the way, he enters on the great field of calculation itself. Beginning at radius 10000000, he first constructs several descending geometrical series, but of such a nature, that they are all quickly formed by an easy conti- nual subtraction, and a division by 2, or by 10, or 100, &c, "which is done by onl}' removing the decimal point so many places towards the left-hand, as thei-e are ciphers in the divi- sor. He constructs three tables of such series : The first of these consists of 100 numbers, in the pro])ortion of radius to radius minus 1, or of 10000000 to 9999999 ; all which are found by only subtracting from each its 10000000th part, which part is also found by only removing each figure seven places lower: the last of these 100 proportionals is found to be 9999900-0004950. The 2d table contains 50 numbers, which are in the continual propor- tion of the first to the last in the first table, namely, of 10000000-0000000 to 9999900 0004950, or nearly the proportion of 100000 to 99999; tliese tlierefore are found bv only removing tlie figures of each number 5 places lower, and subtracting them from the same number : the last of these he finds to be 999500r222927. And a spe- cimen of these two tables is here annexed. No. First Table. Second 'I'abi.e. 1 10000000.0000000 10000000.000000 2 9999999.0000000 9999900.000000 3 9999998.0000001 9999800.00IO0O i 9999997.0000003 9999700.003000 &c. &c till the 100th &c to tlie 50th 60 term, which will be term. 1(10 9999900.0004950 999500 1.2'22927 344 CONSTRUCTION OP TRACT 21. The 3d table consists of 69 columns, and each column of 21 numbers or terms, which terms, in every column, are in the continual proportion of 10000 to 9995, that is, nearly as the first is to the last in the 2d table ; and as 10000 exceeds 9995 by the 2000th part, the terms in every column will be constructed by dividing each upper number by 2, removing the figures of the quotient 3 places lower, and then subtract- ing them ; and in this way it is proper to construct only the first column of 21 numbers, the last of which will be 9900473-5780 : but the 1st, 2d, 3d, &c, numbers, in all the columns, are in the continual proportion of 100 to 99, or nearly the proportion of the first to the last in the first co- lumn; and therefore these will be found by removing the figures of each preceding number two places lower, and sub- tracting them, for the like number in the next column. A specimen of this 3d table is as here below. The Third Table. Teims 1st Column, 2d Column, l 3(1 Column. &c till the 69th Col. 1 10000000.0000 9900000.0000 9801000.0000 &c for 50488,58.8900 2 9995000.0000 9895050.0000 979(3099.5000 the 4th 5046334.4605 3 9990002.5000 9890102.4750 9791201.4503 5th, 6th, 5043811.2932 4 99S5007.4987 9885157.42,37 9786305.8495 7th, &c 5041289.3879 b 9980014.9950 9880214.8451 9781412.6967 col. till 50387G8.7435 &c &c till &c &c tlie last &c 21 9900473.5780 9801468.8423 9703454.1.539 or 499S609.4034 Thus he had, in this 3d table, interposed between the radius and its half, 68 numbers in the continual jiroportion of 100 to 99 ; and interposed between every two of tli'jse, 20 numbers in the proportion of 10000 to 9995: and again, in the 2d table, between lOOOOOOO and 9995000, the two first of the. 3d table, he had 50 numbers in the proportion of 100000 to 09999; and lastly, in the 1st tabic, between 10000000 and 9999900, or the two first in the 2d table, 100 numbers in the proportion of 10000000 to 9999999 ; tluit is in ail, about IGOO proportionals; all found in tlic most simple luannci-, by iittiu TRACT 21. LOGARITHMS. S45 more than easy subtractions ; which proportionals nearly coin- cide with all the natural sines from 90 down to 30''. > To obtain the logarithms of all those proportionals, he de- monstrates several properties and relations of the numbers and logarithms, and illustrates the manner of applying them. The principal of these properties areas follow : 1st, that the logarithm of any sine is greater than the difference between that sine and the radius, but less than the said difference when increased in the proportion of the sine to radius* ; and 2dly, that the difference between the logarithms of two sines, is less than the difference of the sines increased in the proportion of the less sine to radius, but greater than the said difference of the sines increased in the proportion of the greater sine to radius f. Hence, by the 1st theorem, the logarithm of lOOOOOOO, the radius or first term in the first table, being 0, the logarithm of 9999999, the 2d term, will be between 1 and 1-0000001, and will therefore be equal to 1*00000005 ver}' nearly : and this will be also the common difference of all the terms or proportionals in the first table; therefore, by the continual addition of this logarithm, there will be obtained the loga- rithms of all these 100 proportionals; consequently 100 times the said first logarithm, or the last of the above sums, will * By this first tlieoiem, t being radius, tlie logarithm of the sine s is between r~s and r; and therefore, when s differs but little from r, the logarithm of j s (r + j) X (r will be near! V equal to , the arithmetical mean between the limits 2^ r J and r; but still nearer to [r j)V or v'''''5^'''^Seometricalmeaa S .1 s between the said limits. -j- B\' this second theorem, the difterence between the logarithms of the two S J S s sines S and j, lyim? between the limits r and -r, will, when those sines s S differ but httle, be neariv equal to --:'- or : r, their arithmeti- S J . S J cal mean : or nearly r, tlie seometrical mean ; or nearly = 2r, bv sub- y^/is ' ' S + S ' stituting in tht- last dtiioininator, ^ (? + /) for ^Si, to wliich it is near!v equal. ^46 CONSTRUCTION OF TRACT 21. give 100-000005, for the logarithm o^ 9D99900-0(}Oi950, die Jast of the said 100 proportions. Then, by the 2d theorem, it easily appears, that '0004950 is the difference between the logarithms of 9999900-0004950 and 9999900, the last term of the first tabic, and the 2d term of the second table; this then being added to the last loga- rithm, gives 1 00-0005000 for the logarithm of the said 2d term, as also the conmion difference of the logarithms of all the propoitions in the second tabic; and therefore, by conti- nually adding it, there ^vill be generated the logarithms of all these proportionals in the second table ; the last of which is 5000-025, answering to 9995001-222927, the last term of that table. Again, by the 2d theorem, the difference between the loga- rithms of this last proportional of the second table, and the 2d term in the first column of the third table, is found to be 1-2235387; which being added to the last logarithm, gives 5001-2485387 for the logarithm of 9995000, the said 2d term of the third table, as also the common difference of the loga- rithms of all the proportionals in the first column of that table; and that this, therefore, being continually added, gives all the logarithms of that first column, the last of which is 100024-97077, the logarithm of 9900473-5780, the last term of the said column. Finally, by the 2d theorem again, tlie difference between the logarithms of this last number and 9900000, the 1st term in the second column, is 478'3502; which being added to the last logarithm, gives 100503-3210 for the logarithm of the said 1st term in the second column, as well as the common difference of the logarithms of all the numbers on the same line in every line of the table, namely, of all the 1st terms, of all tliC 2d, of all the 3d, of all the 4th, sxe, teruis, infill the columns; and which, therefore, being continually added Lu the logarithms in the first column, will give the corre-^jjoiiclii;;:, logarithms in all the oilier columns. And thus is completed wliat the author calls I'ne liuiieal Kvble, in which lie retains only one decimal place in the Joga- TRACT 21. LOGARITHMS. S47 rithms (or artificials, as be always calls them in Iiis tract on the construction), and four in the naturals. A specimen of the table is as here follows: Radical Table. Terms 1st C'lumn. 2d Column. 69th Column. Natural.--. Artificals Naturals. Artific, Naturals. Artifici.Tl-' 1 10000000.0000 :9900000.0000 9895050.0000 100503.3 5048858.8900 6834225. S 2 9995000.0000 5001.2 105504.6 5046333.4605 6839227.1 3 9990002.5000 10002.5 :9890102.47J0 110505.8 5043311.2932 6844228.3 4 9985007.49S7 15003.7 ,9885157.4237 115507.1 15041289.3879 6849229.6 5 9930014.9950 20005.0 9880214.8451 120508.3 5038768.7435 6854230.S &c &c till &c &c &c &c &c 21 99004-73.5780 10O025.O 9801468.8423 200528.2 4998609.4034 6934250.8 Having thus, in the most easy manner, completed the radi- cal table, by little more than mere addition and subtraction^ both for the natural numbers and logarithms; the logarithmic sines were easily deduced from it by means of the 2d theorem, namely, taking the sum and dilferencc of each tabular sine and the nearest number in the radical table, annexing 7 ci- phers to the difference, dividing the result by the sum, then half the quotient gives the difference between the logarithms of the said numbers, namely, between the tabular sine and radical number ; consequently, adding or subtracting this difference, to or from the given logarithm of the radical num- ber, there is obtained the logarithmic sine required. And tiius the logarithms of all the sines, from radius to the half of it, or from 90" to 30, were perfected. Next, for determining the sines of the remaining 30 de- grees, he delivers two methods. In the first of these he pro- ceeds in this manner : Observing that the logarithm of the ratio of 2 to 1, or of half the radius, is 693 1469 '22, of 4 to 1 is tiie double of this, of 8 to 1 is triple of it, &.c; that of 10 to 1 is 23025842.34, of 20 to 1 is the sum of the logarithms of 2 and 10; and so on, by composition for the logarithms of the ratios between 1 and 40, 80, 100, 200, &c, to lOOGOOOO; he multiplies any given sine, for an arc less than 30 degrees, 34& CONSTRUCTION OF TRACT 21. by some of these numbers, till he finds the product ucarly (]ual to one of the tabular numbers ; then by means of this and the second theorem, the logarithm of this product is found ; to which adding the logaritlim tliat answers to tiie multiple above mentioned, the sum is the logarithm sought. But the other method is still much easier, and is derived from this property, which he demonstrates, namely, as half radius is to the sine of half an arc, so is the cosine of the said half arc, to the sine of the whole arc ; or as y- radius : sine of an arc : : cosine of the arc : sine of double arc ; hence the loga- rithmic sine of an arc is found, by adding together the loga- rithms of half radius and of the sine of the double arc, and then subtracting the loscarithmic cosine from the sum. And thus the remainder of the sines, from 30" down to 0, are easily obtained. But in this latter way, the logarithmic sines for full one half of the quadrant, or from to 45 degrees, he observes, may be derived ; the other half having already been made by the general method of the radical table, by one easy division and addition or subtraction for each. We have dwelt the lono;er on this work of the inventor of logarithms, because I have not seen, in any author, an account of his method of constructing his table, though it is perfectly different from every other method used by the later compu- ters, and indeed almost peculiar to his species of logarithms. The whole of this work manifests great ingenuity in tlie de- signer, as well as much accuracy. But notwithstanding tlic caution he took to obtain his logarithms true to the nearest unit in the last figure set down in the tables, by extending the numbers in the computations to several decimals, and other means; he had been disa|)})ointed of that end, cither by the inaccuracy of his assistant computers or transcribers, or through some other cause ; as the logarithms in the t;i!)lf are commonly very inaccurate. It is remarkable too, that in this tract on the construction of the logarithms, Lord Napu i never calls them logaritlnns, but every where ariiiichi/--, ;;.s opposed in idea to the nati'.ral numbers: and this notinn, (if Tiatural an.d nnificial numbers, I take to ha\e Ikh-u hi^ Hv-t TRACT 21. I.OGARITH'ms. :U9 idea of this matter, and that he altered the word artificials to Joganthms in his first book, on the description of them, when he j)rinted it, in the year 1614, and that he would also have altered the word every where in this posthumous work, if he had lived to print it: for in the two or three pages of appen- dix, annexed to the work by his son, from Napier's papers, he again always calls them logarithms. This appendix relates to the change of the logarithms to that scale in which 1 is the logarithm of the ratio of 10 to 1, the logarithm of I, with or without ciphers, being 0; and it appears to have been written after Briggs communicated to him his idea of that change. Napier here in this appendix also briefly describes some methods, by which this new species of logarithms may be constructed. Having supposed to be the logarithm of 1, and 1, with any number of ciphers, as 10000000000, the logarithm of 10; he directs to divide this logarithm of 10 and the successive quotients, ten times by 5 ; by which divi- sions there will be obtained these other ten logarithms, viz. 2000000000, 400000000, 80000000, 16000000, 3200000, 640000, 128000, 23600, 5120, 1024: then this last logarithm, and Its quotients, being divided ten times by 2, will give the.s<*. otjjer ten logarithns, 512, 256, 128, 64, 32, 16, 8, 4, 2, 1. And the numbers answering to these twenty logarithms, v.e are directed to find in this manner ; namely, extract the 5th root of 10, with ciphers, then the 5th root of that root, and so on, for ten continual extractions of the 5th root; so shali these ten roots be the natural numbers belonginsi: to the fir.->t ten logarithms, above found in continually dividing by 5 next, out of the last 5th root we are to extract the square root, then the square root of this last root, and so on, for ten successive extractions of tlie square root ; so shall these iasr ten roots be the natural numbers corresponding to the loga- rithms or quotients an^irig from the last ten divisions by the number 2. And from tb.ese twenty logarithms, 1 , 2, 4, 8, I -a, &c, and their natural numbers, the author observes that other logarithms and their numbers may be formed, namely, ov adding the logarithms, and niultiplying their correspond!;)-' iSO CONSTRUCTION OF TRACT 21. numbers. It is evident that this process would generate ra- ther an antilogarithniic canon, such as Dodson's, tlian the table of Briggs; and that the method would also be very laborious, since, besides the very troublesome original ex- tractions of the .^th roots, all the numbers would be very large, by the multiplication of which the successive secondary natural numbers are to be found. Our author next mentions another method of deriving a few of the primitive numbers and their logarithms, namely, by taking continually geometrical means, first between 10 and I, then between 10 and this mean, and again between 10 and the last mean, and so on ; and taking the arithmetical means between their corresponding logarithms. He then lays down various relations between numbers and their loga- rithms; such as, that the products and quotients of numbers answer to the sums and differences of their logarithms, and that the powers and roots of numbers answer to the products and quotients of the logarithms by the index of the power or root, &c ; as also that, of any two numbers whose logarithms are given, if each number be raised to the power denoted by t.'ic logarithm of the other, the two results will be equal. He then delivers another method of making the logarithms to a few of the prime integer numbers, which is well adapted for constructing the common table of logarithms. This method easily follows from what has been said above; and it depends on this property, that the logarithm of any number in this scale, is 1 h;ss than the number of places or figures contained in that power of the given number whose exponent is 1 0000000000, or the loirarithm of 10, at least as to inteoer numbers, for they reall}' differ by a fraction, as is shown by Mr. Briggs in his illustrations of these properties, printed at tlie end of this appendix to the construction of logarithms. We shall here just notice one more of these relations, as the maimer in which it is expressed is exactly similar to that of tUixicuis and fluents, and it is this: Of any two numbers, as the greater is to the less, so is the velocity of the increment or decrement of the logarithms at the less, to the velocity of TRACT 21. LOGARITHMS. S5l the increment or decrement of the logarithms at the greater : that is, in our modern notation, as X : V :: ji to x, where * and J! are the fluxions of the logarithms of X and Y. Kepler's Constnidion of Logarithms. The logarithms of Briggs and Kepler were both printed the same year, 1624; but as the latter are of the same sort as Napier's, we may first consider this author's construction of them, before proceeding to that of Briggs's. We have already, in the last Tract, described the natura and form of Kepler's logarithms ; showing that they are of the same kind as Napier's, but only a little varied in the form of the table. It may also be added, that, in general, the ideas which these two masters had on this subject, were of the same nature ; only they were more fully and methodically laid down by Kepler, who expanded, and delivered in a regular science, the hints that were given by the illustrious inventor. The foundation and nature of their methods of construction are also the same, but only a little varied in their modes of applying them. Kepler here, first of any, treats of loga- rithms in the true and genuine way of the measures of ratios, or proportions*, as he calls them, and that in a very full and scientific manner : and this method of his was afterwards fol- lowed and abridged by Mercator, Halley, Cotes, and others, as we shall see in the proper places. Kepler first erects a regular and purely mathematical system of proportions, and the measures of proportions, treated at considerable lengtli in a number of propositions, which are fully and chastely demonstrated by genuine mathematical reasoning, and illu- strated by examples in numbers. This part contains and demonstrates both the nature and the principles of the struc- * Kepler almost always uses tlie term proportion ijjstead of ratio, which ws shall also do in the account of his worR, as well as conform in expressions and notations to his other peculiarities. It may also be here remarked, that I observe the same practice in describing the works of other authors, the better to convey the idea of their several methods and style. And this may serve to account for 6ooae seeming inequalitiei in the language of this history. 352 CONSTRUCTION OF TRACT 21. ture of logarithms. And in the second part the author applies those principles in the actual construction of his table, which contains only 1000 numbers, and their logarithms, in the form as we before described : and in this part he indicates the va- rioug contrivances made use of in deducing the logarithms of proportions one from another, after a few of the leading ones had been first formed, by the general and more remote prin- ciples. He uses the name logarithms, given them by the in- ventor, being the most proper, as expressing the very nature and essence of those artificial numbers, and containing as it were a definition in the very name of tliem ; but without taking any notice of the inventor, or of the origin of those useful numbers. As this tract is very curious and important in itself, and is besides very rare and little known, instead of a particular de- scription only, we shall here giv^e a brief translation of both the parts, omitting only the demonstrations of the proposi- tions, and some rather long illustrations of them. The book is dedicated to Philip, landgrave of Hesse, but is without cither preface or introduction, and commences immediately with the subject of the first part, which is intitled " The De- monstration of tlie Structure of Logarithms ;" and the con- tents of it are as follow. Postulate 1. That all proportions that are equal among themselves, by whatever variety of coiij)lets of tern.s thev may be denoted, are measured or expressed by the same quantity. Axio)n 1 . If there be any number of quantities of the same kind, the proportion of tlie extremes is understood to be com- posed of all the proportions of every adjacent couplet of terms, from the first to the last, 1 Proposition. The mean proportional between two term-:, divides the proportion of those terms into two equal \)vn- porlions. ylxiovi 2. Of any number of (juantities regularly increas- ing, thi> means divide the ])roponion of tiie extremes into one proportion mure than the number of the means. TRACT 21. LOGARITHMS. SS Postulate 2. That the proportion between any two terms is divisible into any number of parts, until those parts become less than any proposed quantity. An example of this section is thenjnserted in a small table, in dividing the proportion which is between 10 and 7 into 1073741 8:24 equal parts, by as many- mean proportionals wanting one, namely, by taking the mean proportional be- tween 10 and 7, then the mean between 10 and this mean, and the mean between 10 and the last, and so on for 30 means, or 30 extractions of the square root, the last or 30th of which roots is 99999999966782056900 ; and the 30 power of 2, which is 1073741824, shows into how many parts the proportion between 10 and 7, or between 1000 &c, and 700 &c, is divided by 1073741824 means, each of which parts is equal to the proportion between 1000 &c, and the 30th mean 999&C, that is, the proportion between lOOO&c, and 999&C, is the 1073741824th part of the proportion between 10 and 7. Then by assuming the small differ- ence 00000000033217943100, for the measure of the very small element of the proportion of 10 to 7, or for the measure of the proportion of lOOO&c, to 999&C, or for the logarithm of this last term, and multiplying it by 1073741824, the number of parts, the product gives 35667.49481,37222.14400, for the logarithrti of the less term 7 or 700 &c. Postulate 3. That the extremely small quantity or element of a proportion, may be measured or denoted by any quan- tity whatever ; as for instance, by the difference of the terms of that element. 2 Proposition. Of three continued proportionals, the dif- ference of the two first has to the difference of the two latter, the same proportion Avhich the first term has to the 2d, or the 2d to the 3d. 3 Prop. Of any continued proportionals, the greatest terms have the greatest difference, and the least terms the least. 4 Prop. In any continued proportionals, if the difference of the greatest terms be made the measure of the proportion between them, the difference of any other couplet will be less than the true measure of their proportion. 5 Prop. In continued proportionals, if the difference of the greatest terms be made the measure of their proportion, then the measure of the proportion of the greatest to any other term will be greater than their difference. 6 Prop. In continued proportionals, if the difference of the greatest term and any one of the less, taken not immediately VOL. I. A A S54 CONSTRUCTION OF TRACT 21. next to it, be made the measure of their propoftion, then the proportion whicli is between the greatest and any other term greater than the one before taken, will be less than the differ- ence of those terms ; but the proportion which is between the greatest term, and any one less than tliat first taken, will be greater than their difference. 7 Prop. Of any quantities placed according to the order of their magnitudes, if any two successive proportions be equal, the three successive terms which constitute them, will be con- tinued proportionals. 8 Prop. Of any quantities placed in the order of their mag- nitudes, if the intermediates lying between any two terms be not among the mean proportionals Avhich can be interposed between the said two terms, then such intermediates do not divide the proportion of those two terms into commensurable proportions. Besides the flemonstrations, as usual, several definitions are here given ; as of commensurable proportiuns, &,c. 9 Prop, AVhen two expressible lengths are not to one an- other as two figurate numbers of the same species, such as two squares, or two cubes, there cannot fall between them other expressible lengths, which shall be mean proportionals, and as many in number as that species requires, namely, one in the squares, two in the cubes, tiiree in the biquadrats, &.c. 10 Prop. Of any expressible quantities, following in the order of their magnitudes, if the two extremes be not in the proportion of two square numbers, or two cubes, or two other powers of the same kind, none of the intermediates divide the proportion into commensurables. 1 1 Prop. All the proportions, taken in order, which are between expressible terms that are in arithmetical propor- tion, are incommensurable to one another. As between 8. 13, 18. 12 Prop. Of any quantities placed in the order of their magnitude, if the diilercuce of the greatest terms be made the measure of t!ieir proportion, tlicn the difference between any two others will be less than the measure of thtir propor- TRACT 21. LOGARITHMS. 355^* tion ; and if the difference of the two least terms be made tHe^ measure of their proportion, then the differences of the rdst' will be greater than the measure of the proportion btt\een- Mr terms. ::9^i.J ^u oicai , " ^ i; i-i Corol. If the measure of the proportion betsfle^W the greatest' exceed their difference, then the propof tiofr'&ij this ttieasure to the said difference, will be less thanT th^t dfa following measure to the difference of its terms. Because proportiorials- have the same ratio. ' "- ": . '^ * 1 3 Prop. If three quantities follow oaie another in thf* drdfer of magnitude, the proportion of the two Idast will be con- tained in the proportion of the extremes, aJ Jess- number 6f times than the difference of the two least is contained in- tliej difference of the extremes: And, on the cdnt4*ary, the pro- portion of the two greatest will be contained in the proportioii of the extremes, oftener than the difference of the former iV contained in that of the latter. '.-... ,:.:> Corol. Hence, if the difference of the two greater be equal to the difference of the two less terms, the proportion between' the two greater will be less than the proportion between th' two less. ' " 14 Prop. Of three equidifferent quantities, taken in order, the proportion between the extremes is more than double the proportion between the two greater terms. Corol. Hence it follows, that half the proportion of the extremes is greater than the proportion of the two greatest terms, but less than the proportion of the two least. 15 Prop. If two quantities constitute a proportion, and each quantity be lessened by half the greater, the remainders will constitute a proportion greater than double the former. 16 Prop. The aliquot parts of incommensurable proportions are incommensurable to each other. n Prop. If one thousand numbers follow one another in the natural order, beginning at 1000, and dilTermg all by unity, viz. 1000, 999, 998, 997, kc ; and the proportion be- tween the two greatest 1000, 999, by continual bisection, be cut into parrs thai are smaller than the excess of the propor- A A 2 9iS9' CONSTRUCTION OF TRACT 21. tiou betweea the next two 999, 998, over the said proportion between the two greatest 1000, 999 ; and then for the mean sure of that small element of the proportion between 1000 and 99.9, there be taken the difi'erence of 1000 and that mean proportional which is the other term of the element. Again, if the proportion between 1000 and 998 be likewise cut into double the number of parts which the former pro- portion, between 1000 and 999, was cut into; and then for the measure of the small element in this division, be taken the difference of its terms, of which the greater is 1000. And, in the same manner, if the proportion of 1000 to the following numbers, as 997, &c, by continual bisection, be cut into particles of such magnitude, as may be between 4 and i of the element arising from the section of the first proportion between 1000 and 999, the measure of each element will be given from the difference of its terms. Then, this being done, the measure of any one of the 1000 proportions will be composed of as many measures of its element, as there are of those elements in the said divided proportion. And all these measures, for all the proportions, will be sufficiently exact for the nicest calculations. All these sections and measures of proportions are performed in the manner of that described at postulate 2, and the operation is abundantly explained by numerical calculations. 18 Prop. The proportion of any number, to the first term 1000, being known; there will also be known the proportion of the rest of the numbers in the same continued proportion, to the said first term. So, from the known proportion between 1000 and 900, there is also known the prop, of 1000 to 810, and to 729; And from 1000 to 800, also 1000 to 640, and to 512 ; And from 1000 to 700, also 1000 to 490, and to 343 ; And from 1000 to 600, also 1000 to 360, and to 216 ; And from 1000 to 500, also 1000 to 250, and to 125. Corol. Hence arises the precept for squaring, cubing, &c ; as also for extracting the square root, cube root, &,c. For it will be, as the greatest number of the chiliad, as a dcnomi- TRACT 21. LOGARITHMS. 8$1 nator, is to the number proposed as a numerator, so is this fraction to the square of it, and so is this square to the cube of it. 1 9 Prop. The proportion of a number to the first, or 1000, being known ; if there be two other numbers in the samepro- portion to each other, then the proportion of one of these lo 1000 being known, there will also be known the proportion of the other to the same 1000. Corol. 1. Hence, from the 15 proportions mentioned in prop. 18, will be known 120 others below J 000, to the same 1000. For so many are the proportions, equal to some one or other of the said 15, that are among the other integer numbers which are less than 1000. Corol. 2. Hence arises the method of treating the Rule-of* Three, when 1000 is one of the given terms. For this is effected by adding to, or subtracting from, each other, the measures of the two proportions of 1000 to each of the other two given numbers, accord- ing as 1000 is, or is not, the first term in the Rule-of-Threc. 20 Prop. When four numbers ai-e proportional, the first to the second as the third to the fourth, and the proportions of 1000 to each of the three former are known, there will also be known the proportion of 1000 to the fourth number. Corol. 1. By this means other chiliads are added to the former. Cerol. 2. Hence arises the method of performing the Rule- of-Threc, when 1000 is not one of the terms. Namely from the sum of the measures of the proportions of 1000 to the second and third, take that of 1000 to the first, and the re- mainder is the measure of the proportion of 1000 to the fourth term. Defimti'on. The measure of the proportion between 1000 and any less number, as before described, and expressed by a number, is set opposite to that less number in the chiliad, and is called its logarithm, that is, the number (apt$iMf) indicating the proportion (Aoyov) which 1000 bears to that number, to which the logarithm is annexed. 21 Prop. If the first or greatest number be made the radius 558 CONSTRUCTION OF TRACT 21. of a circle, or smis totiis; every less number, considered as the coaine of some arc, has a logarithm greater than the versed sine of that arc, but less than the difference between the ra- dius and secant of the arc; except only in the term next after thef^dius, or greatest term, the logarithm of which, by the |iypothesis, is made equal to the versed sine. (iTbat is, if CD be made the logarithm of AC, or the mea- * sure of the proportion of AC to AD ; then the measure of the proportion of AB to AD, that is the logarithm of AB, will be greater than BD, but less than EF. And this is the same as Napier's first rule in page 345. A BCD 22 Prop. The same things being supposed ; the sum of the versed sine and excess of the secant over the radius, is greater than double the logarithm of the cosine of an arc. Corol. The losr. cosine is less than the arithmetical mean between the versed sine and the excess of the secant. Precept 1. Any sine being found in the canon of sines, and its defect below radius to the excess of the secant above ra- dius, then shall the logarithm of the sine be less than half that sum, but greater than the said defect or coversed sine. Let there be the sine 99970.1490 of an arc : Its defect below radius is 29.8510 the covers- and less than the loc:. slue : Add the excess of the secant 29-8599 Sum 59.7109 its half or 29.8555 greater than the lo^'arithni. Therefore the log. is between 29.8510 and 29.8555 Precept 2. The logarithm of th(^ sine being ftnnid, there will also be found nearly the logarithm of the round or inte- ger number, which is next less than the sine with a fraction, by adding that fractional excess to the logarithm of the said sine. Thus, the logarithm of the sine 99970.149 is found to be about 29.854; ifnou tlie logarithm of the round number 99970.000 be required, add I i9, the fractional part of the sine, to its logarithm, observing tlie point, thus, 29.854 149 the sum 30.003 is the log. of llic round number 99970.000 nearly. TKACT 21. LOGARITHMS. 359 23 Prop. Of three equidifferent quantities, the measure of the proportion between the two greater terms, with the mea- sure of the proportion between the two less terms, will con- stitute a proportion, which Avill be greater than the proportion of the two greater terms, but less than the proportion of the two least. Thus if AB, AC, AD be three quantities having the equal differences BC, CD ; and if the measure of the ^ g ^ ^ proportion of AD, AC, he, cd, and that of AC, AB be Ic; then the proportion of cd to cb will be greater than the i i i proportion of AC to AD, but less than the proportion bed of AB to AC. 24 Prop. The said proportion between the two measures is less than half the proportion between the extreme terms. That is, the proportion between be, cd, is less than half the proportion between ab, ad. Corol. Since therefore the arithmetical mean divides the proportion into unequal parts, of which the one is greater, and the other less, than half the whole; if it be inquired what proportion is between these proportions, the answer is, that it is a little less than the said half. An Example of finding nearlii the limits, greater and less, to the measure of any proposed proportion. It being known that the measure of the proportion between 1000 and 900 is 10536.05, required the measure of the proportion 900 to 800, where the terms 1000, 9G0, 800, have equal dilFerences. Therefore as 9 to 10, so 10536.05 to 11706.72, which is less than 11773.30 tb.e measure of the proportion 9 to 8. Again, as the mean proportional between 8 and 10 (which is 8.9442719) is to U), so 10536.05 toll779-63, which is greater than the measureofthe proportion between 9 and 8. Axiom. Every number denotes an expressible quantity. 25 Prop. If the 1000 numbers, differing by 1, follow one another in the natural order; and there be taken any two ad- jacent numbers, as the terms of some proportion; the measure of this proportion will be to the measure of the proportion between the two greatest terms of the chiliad, in a proportion greater than that which the greatest term 1000 bears to the 360 CONSTRUCTION OF TRACT 21. greater of the two terms first taken, but less than the pro- portion of 1000 to the less of the said two selected terms. So, of the 1000 numbers, taking any two successive terms, as 501 and 500, the logarithm of the former being 69114.92, and of the latter 69314.72, the differ- ence of which is 199.80, Therefore, by the definition, the measure of the pro- portion between 501 and 500 is 199.80. In like manner, because the logarithm of the greatest term 1000 is 0, and of the next 999 is 100.05, the difference of these logarithms, and the measure of the proportion between 1000 and 999, is 100.05. Couple now the greatest term 1000 with each of the selected terms 501 and 500 ; couple also the measure 199.80 with the measure 100.05; so shall the proportion between 199. SO and 100.05, be greater than the proportion be- tween 1000 and 501, but less than the proportion between 1000 and 500. Corol. 1. Any number below the first 1000 being proposed, as also its logarithm, the differences of any logarithms ante- cedent to that proposed, towards the beginning of the chiliad, are to the first logarithm (viz. that which is assigned to 999) in a greater proportion than 1000 to the number proposed; but of those which follow towards the last logarithm, they are to the same in a less proportion. Corol. 2. By this means, the places of the chihad may easily be filled up, which have not yet had logarithms adapted to them by the former propositions. 26 Prop, The difference of two logarithms, adapted to two adjacent numbers, is to the difference of these numbers, in a proportion greater than 1000 bears to the greater of those numbers, but less than that of 1000 to the less of the two numbers. This 26th prop, is the same as Napier's second rule, at page Z^5. 27 Prop. Having given two adjacent numbers, of the 1000 natural numbers, with their logarithmic indices, or the mea- sures of the proportions which those absolute or round num- bers constitute with 1000, the greatest; the increments, or differences, of these logarithms, will be to the logarithm of tlie small element of the proportions, as the secants of the arcs whose cosines are the two absolute numbers, is to the greatest number, or the radius of the circle ; so that, however, of the said two secants, the less will have to the radius a less proportion than the proposed difference has tp the first of all. TRACT 21. LOGARITHMS. 361 but the greater will have a greater proportion, and so also will the mean proportional between the said secants have a greater proportion. Thus if BC, CD be equal, also Id the logarithm of AB, and cd the logarithm of AC ; then the proportion of be lo cd will be greater than the proportion of AG to AD, but less than that of AF to AD, and also less than that of the mean proportional between AF and AG to AD. A! WCD i c d Corol. 1. The same obtains also when the two terms differ, not only by the unit of the small element, but by another unit, which may be ten fold, a hundred fold, or a thousand fold of that. Corol. 2. Hence the differences will be obtained sufficiently exact, especially when the absolute numbers are pretty large, by taking the arithmetical mean between two small secants, or (if you will be at the labour) by taking the geometrical mean between two larger secants, and then by continually adding the differences, the logarithms will be produced. CoroL 3. Precept. Divide the radius by each term of the assigned proportion, and the arithmetical mean (or still nearer the geometrical mean) between the quotients, will be the re- quired increment; which being added to the logarithm of the greater term, Avill give the logarithm of the less term. Example. Let there be giten the logarithm of 700, viz. 35667.4948, to find the log to 699, Here radius divided by 700 gives 1428571 &c. and divided by 699 gives 1430672 &c. the arithmetical mean is 142.962 which added to 35667.4948 gives the logarithm to 699 35810.4568 Corol. 4. Precept for the logarithms of sines. The increment between the logarithms of two sines, is thus found : find the geometrical mean between the cosecants, and divide it by the difference of the sines, the quotient will be the difference of the logarithms. St>2 CONSTRUCTION OF TRACT 21. Example. r sine 2909 cosec. 543774682 O 'Z bine 5S18 coscc. 171887319 dif. 2909, geom. mean SiQS nearly. The qooticnt 80000 exceeds the required increment of the logarithms, because the secants are here so large. Appendix. Nearly in the same manner it may be sliown, that the second ditt'erences are in tlie duplicate proportion of the first, and the third in the duplicate ot" the second. Thus, for instance, in the beginning of the logarithms, the first dif- ference is IQO.OOOOO, viz. equal to the difference of the num- bers 1 00000.00000 and 99900.00000; the second, or difference of the differences, 10000; the third 20. Again, after arriving at the number of 50000.00000, the logarithms have for a dif- ference 200.00000, which is to the first difference, as the number 100000.00000 to 50000.00000; but the second dif- ference is 40000, in which 10000 is contained 4 times; and tfic thirti 328, in which 20 is contained 16 times. But since it) treating of new matters we labour under the want of proper v>-ords, therefore lest we should become too obscure, the de- monstmtion is omitted untried. 28 Prop. No number expresses exactly the measure of tlie proportion, between two of the 1000 numbers, constituted by the foregoing method. 29 Prop. If the measures of all proportions be expressed l)v numbers or logarithms ; all proportions will not have as- signed to them their due portion of measure^ to tlie utmost accuracy, 30 Prop. If to tlie number 1000, the greatest of tlie chiliatl, bci referred others that are greater than it, and tlie logarithm of iOOO be made 0, the logarithms belonging to iho>e grcat'.T numbers will be negative. This concludes tlie first or scientific part of the work, the principles of v/hich Kepler applies, in the second ))an, if) ilic actual eou.struction ol' the first 1000 logarithms, winch ccr.i- straction is pretty minnU'lv described. This pait is inUlied TRACT 21. LOGARITHMS. 363 *' A very compendious method of constructing the Chiliad of Logarithms;" and it is not improperly so called, the method being very concise and easy. The fundamental principles are briefly these : That at the beginning of the logarithms, their increments or differences are equal to those of the na- tural numbers: that the natural numbers may be considered as the decreasing cosines of increasing arcs: and that the se- cants of those arcs at the beginning have the same differences as the cosines, and therefore the same differences as the loga- rithms. Then, since the secants are the reciprocals of the cosines, by these principles and the third corollary to the 27th proposition, he establishes the following method of constitut- ing the 100 first or smallest logarithms to the 100 largest numbers, 1000, 999, 998, 997, &c, to 900. viz. Divide the radius 1000, increased with seven ciphers, by each of these numbers separately, disposing the quotients in a table, and they will be tlie secants of tliose arcs which have the divisors for their cosines; continuing the division to the 8th figure, as it is in that place only that the arithmetical and geometri- cal means differ. Then by adding successively the arithme- tical means between every two successive secants, the sums will be the series of logarithms. Or, bj^ adding continually every two secants, the successive sums will be the series of the double logarithms. Besides the 100 logarithms, thus constructed, the author constitutes two others by continual bisection, or extractions of the square root, after the manner described in the second postulate. And first he finds the logarithm wliich measures the proportion betv.-een 100000. CO and 97656.25, which latter terra is the third proportional to 1024 and 1000, each with two ciphers ; and this is effected by means of twenty-four continual extractions of the square root, determining the greatest term of each of twenty-four classes of mean proportionals; then the difference between the greatest of tijese means and the first or whole number 1000, with ciphers, being as often doubled, there arises 2371.6,">26 for the logaritlun sought, which made negative is the logarithm of 1021. Secondly, S64f CONSTRUCTION OF TRACT 21. the like process is repeated for the proportion hetween the Tiumbers 1000 and 500, from which arises 69314.7193 for the Jogarithm of 500 ; which he also calls the logarithm of dupli- cation, being the measure of the proportion of 2 to 1. Then from the foregoing he derives all the other logarithms in the chiliad, beginning with those of the prime numbers 1, 2, 3, 5, 7, &c, in the first 100. And first, since 1024', 512, 256, 128, 64, 32, 16, 8, 4, 2, 1, are all in the continued pro- portion of 1000 to 500, therefore the proportion of 1024 to 1 is decuple of the proportion of lOOO to 500, and conse- quently the logarithm of 1 would be decuple of the logarithm of 500, if were taken as the logarithm of 1024 ; but since the logarithm of 1024 is applied negatively, the logarithm of 1 must be diminished by as much: diminishing therefore 10 times the log. of 500, which is 693147.1928, by 2371.6526, the remainder 690775.5422 is the logarithm of J , or of 100.00, which is set down in the table. And because 1, 10, 100, 1000, are contitujcd proportionals, therefore the proportion of 1000 to 1 is triple of the proportion of 1000 to 100, and consequently ^ of the logarithm of 1 is to be set for the logarithm of 100, viz. 230258.5141, and this is also the logarithm of decuplication, or of the '0001 proportion of 10 to 1. And hence, multiplying this logarithm of 100 successively by 2, 3, 4, 5, 6, and 7, there arise the logarithms to the numbers in the de- cuple proportion, as in the margin Also if the logarithm of dupli- cation, or of the proportion of 2 to 1 , be taken from the logarithm of 1, there will remain the loga- rithm of 2 ; and from the logarithm of 2 taking the logarithm of 10, there remains the logarithm of the proportion of 5 to 1 ; which Nos. Logarithms, 100 230358.5141 10 460517.0282 1 690775.5422 .1 921034.0563 .01 1151292.5703 .001 1381551.0844 3001 161)809.5985 Log. of 1 of 2 to 1 log. of 2 log. of 10 of 5 to 1 log. of 5 690775.5422 69314.7193 621460.8221' 460517.0281 160943.794S 529831.7474 TRACT 21. LOGARITHMS. 36a taken from the logarithm of 1, there remains the logarithm of 5. See the margin. For the logarithms of other prime numbers he has recourse to those of some of the first or greatest century of numbers, before found, viz. of 999, 998, 997, &c. And first, taking 960, whose logarithm is 4082.2001 ; then by adding to this logarithm the logarithm of duplication, there will arise the several logai'ithms of all these numbers, which are in dupli- cate proportion continued from 960, namely 480, 240, 120, 60, 30, 15. Hence the logarithm of 30 taken from the loga- rithm of 10, leaves the logarithm of the proportion of 3 to 1 ; which taken from the logarithm of 1 , leaves the logarithm of 3, viz. 580914.3106. And the double of this diminished by the logarithm of 1, gives 47 105 3.0790 for the logarithm of 9. Next, from the logarithm of 990, or 9 x 10 x 11, which is 1005.03S1, he finds the logarithm of 11, namely, subtracting the sum of the logarithms of 9 and 10 from the sum of the logarithm of 990 and double the logarithm of 1, there remains 450986.0106 the logarithm of 11. Again, from the logarithm of 980, or 2 x 10 x 7 x 7, which is 2020.2711, he finds 496184.5228 for the logarithm of 7. And from 5129.3303 the logarithm of 950, or 5 x 10 x 19, he finds 396331.6392 for the logarithm of 19. In like manner the logarithm to 998 or 4 X 13 X 19, gives the logarithm of 13 ; 19, gives the logarithm of 17 ; 29, gives the logarithm of 29 ; 23, gives the logarithm of 23 ; 3 1 , gives the logarithm of S 1 . And so on for all the primes below 100, and for many of the primes in the other centuries up to 900. After Avhich, he directs to find the logarithms of all numbers composed of these, by the proper addition and subtraction of their loga- rithms, namely, in finding the logari nm of the product of two numbers, from the sum of the logarithms of the two factors take the logarithm of 1, the remainder is the logarithm of tiie to 969 or 3 X 17 X to 986 or 2 X 17 X to 966 or 6 X 7 X to 930 or 3 X 10 X 366 CONSTRUCTION OF TRACT 21. product. In this way lie shows that the logarithms of all numbers under 500 may be derived, except those of the fol- lowing 36 numbers, namely, 127, 149, 167, 173, 179, 211, 223, 251, 257, 263, 269, 271, 277, 281, 283, 293, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 419, 421, 431, 433, 439, 443, 449. Also, besides the composite numbers between 500 and 900, made up of the products of some numbers whose logarithms have been before determined, there will be 59 primes not composed of them; which, with the 36 above mentioned, make 95 numbers in all not composed of the products of any before them, and the logaritlims of which he directs to be derived in this manner ; namely, b}'^ considering the difl'erences of the logarithms of the numbers interspersed among them ; then by that method by which were constituted the differences of the logarithms of the smallest 100 numbers in a continued series, we are to proceed here in the discontinued scries, that is, by prop. 27, corol. 3, and especially by the appendix to it, if it be rightly used, whence those differences will be very easily supplied. This closes the second part, or the actual construction of the logarithms; after which follows the table itself, which has been before described, pa. 323. Before dismissing Kepler's work however, it may not be improper in this place to take notice of an erroneous property laid down by him in the ap- p'^ndix to the 27th prop, just now referred to ; both because it is an error in principle, tending to vitiate the practice, and because it serves to show that Kcplcu* was not acquainted with the true nature of the orders of dilFerenccs of the logarithms, notwithstanding wliat he says above with respect to the con- struction of them by means of their several orders of difTer- ences, and that consequently he has no legal claim to any share in tiie discoverv of the differential method, known at that time to Briggs, and it would seem to him alone, it being published in his logarithms in the same year, 1624, as Kepler's book, togetlier with the true nature of the logarithmic orders of differences, as we shall presently see in the following ac- count of his works. Now this error of Kej^Ier's, here alluded TRACT 21. LOGARITHMS, 3b7 to, is in that expression \vhere he says ti!t; ihlnl chiTerences are in the duplicate ratio of the second diiTcrcnces, like as the second differences are in the duplicate ratio of the first ; or, in other words, that the third differences are as the squares of the second differences, as uel! as the second differences as the squares of the first ; or tliat the third differences are as the fourth po-cners of tiie first differences : Whereas in truth the third differences are only as the cubes of the first differ- ences. Kepler seems to have been led into this error by a mistake in his numbers, viz. when he says in that ap}:>endix, that " the third difference is 328, in which 20 is contained 16 times ;" for when the numbers are accurately computed, the third difference comes out only 161, in which therefore 20 is contained only 8 times, which is the cube of 2, the number of times the one first difference contains the other. It would hence seem that Kepler had hastily drawn the above errone- ous principle from this one numerical example, or little more, false as it is: for had he made the trial in many instances, though erroneously computed, they could not easily have been so uniformly so, as to afford the same false conclusion in all cases. And therefore from hence, and what he says at the conclusion of that appendix, it may be inferred, that he either never attempted the demonstration of the property in ques- tion, or else that finding hiniseif embarrassed with it, and unable to accomplish it, he t'lcrefore dispatched it in tlie am- biguous manner in which itappciirs. But it may easily be shown, noi or.!y that tlic tliird differ- ences of the logantlnns at diii'^rent places, are as the cubes of the first differences; but, in genera!, tlsat the numbers in any one and the same order of differences, at different places, are as that power of the nuinoers in the i:;st differences, whose index is the same as tliat of the order; or that the second, third, fourth, &c differences, are as the second, tliird, fourtli, &.C powers of the first differences. For t;ie several orders of differences, when the absolute nnrnbers differ by indefinitclv small parts, are as the several orders of fiaxions of the lo^a- rithpis ; but if x be any number, then is the fluxion of Sr)S CONSTRUCTION OF TRACT 21. the logai'ithra of x, to the modulus w, and the second fluxion, or the fluxion of this fluxion, is , since Jc is constant j and the third, fourtli, &c fluxions, are " - , , '- ;; , &c; that is, the first, second, third, fourth, fifth, sixth, &c orders of fluxions, are equal to the modulus vi multiplied into each of these terms, X \x- 1.2i^ 1.2.3** 1.2.3.4xS 1.2.3.4.5x^ _ - , _ , &c; X ' x^ ' ^3 > x^ x^ ' x^ where it is evident, that the fluxion of any order is as that power of the first fluxion, whose index is the same as the number of the order. And these quantities would actually be the several terms of the differences themselves, if the dif- ferences of the numbers were indefinitely small. But they vary the more from them, as the differences of the absolute numbers differ from x, or as the said constant numerical dif- ference 1 approaches towards the value of x the number itself. However, on the whole, the several orders vary pvoportionably, so as still sensibly to preserve the same ana- logy, namely, that two nth differences are in proportion as the nth powers of their respective first differences. 0/ Briggs's Construction of his Logarithms. Nearly according to the methods described in p. 349, 3.^0, Mr. Briggs constructed the logarithmsof the prime numbers, asappears from his relation of this business in the "Arithmetica Logarithmica," printed in 1624, where hu details, in an ample manner, the whole construction and use of his logarithms. The work is divided into 32 chapters or sections. In the first of tlic'se, logarithms in a general sense are defined, and some pvnpertics of them illustrated. In the second chapter he re- marks, that it is most convenient to make O the logarithm of 1 ; and on that supposition he exemplifies these following jnopertics, namely, that the logarithms of all numbers are either the indices of powers, or proportional to them; that tiie sum of the logarithms of two or more factors, is the loga- rithm of their product ; and tliat the diffcrenec of the loga- TRACT 21. tOGARITHMS. 360 rithms of two numbers, is the logarithm of their quotient. la the third section he states the other assumption, Avhich is ne- cessary to limit his system of logarithms, namely, making i the logarithm of 10, as that which produces the most conve- nient form of logarithms : He hence also takes occasion to show that the powers of 10, namely 100, 1000, &c, are the only numbers which can have rational logarithms. The fourth section ti*eats of the characteristic ; by which name he distin- guishes the integral, or first part, of a logarithm towards the left hand, which expresses one less than the number of inte- ger places or figures, in the number belonging to that loga- rithm, or how far the first figure of this number is removed from the place of units ; namely, that is the characteristic of the logarithms of all numbers from 1 to 10; and 1 the characteristic of all those from 10 to 100; and 2 that of those from 100 to 1000; and so on. He begins the fifth chapter with remarking, that his loga- rithms may chiefly be constructed by the two methods which were mentioned by Napier, as above related, and for the sake of which, he here premises several lemmata, concerning the powers of numbers and their indices, and how many places of figures are in the products of numbers, observing that the product of two numbers will consist of as many figures as there are in both factors, unless perhaps the product of the first figures in each factor be expressed by one figure only, r. hich often happens, and then commonly there will be one figure in the product less than in the two factors; as also that, of any two of the terms, in a series of geometricals, the re- sults will he equal by raising each term to the power denoted bv the index of the other; or any number raised to the power denoted by the logarithm of the other, will be equal to this latter number raised to the pov/er denoted by the logarithm of the former; and consequently if the one number be 10, whose logarithm is I Avith any number of ciphers, then any number raised to the power whose index is 1000 &c, or the logarithm of 10, will be equal to 10 raised to the power whose index is the logarithm of that number ; that is, the logarithm VOL. I. B 370 CONSTRUCTION OF TRACT 21. of any number in this scale, where I is the logarithm of 10, is the index of that power of 10 which is equal to the given number. But the index of any integral power of 10, is one less than the number of places in that power ; consequently the logarithm of any other number, which is no integral power of 10, is not quite one less than the number of places in that power of the given number whose index is 1000 &.c, or the logarithm of 10. Find therefore the 10th, or 100th, or 1 000th &c, power of any number, as suppose 2, with the number of figures in such power; then shall that number of figures always exceed the logarithm of 2, though the excess will be constantly less. than 1. TRACT 21. LOGARITHMS. 37t An example of this process is here given in the margin ; where the 1st column con- tains the several powers of 2, the 2d their corresponding indices, and the 3d contains the number of places in the powers in the first column ; and of these numbers in tlie third column, such as are on the lines of those indices that consist of 1 with ciphers, are continual ap- proximations to the logarithm of 2, be- ing always too great by less than 1 in the last figure, t!i;it lo- garithm being 30102999566f!9S &c. And here, since the exact powers of 2 are not required, but onlv the number of figures they consist of, as shown by the third colunni, onlv a few of the first fi- gures of the powers in the first column are r{?tained, those being sufficient to determine the uum- Powers of 2 2 4. 16 256 1024 10486 10995 12089 12676 16069 25823 66680 10715 11481 13182 Indices. 10 20 40 80 100 200 400 800 1000 2000 4000 17377 8000 19950 39803 15843 25099 99900 99801 9960 1 99204 99006 98023 96085 92323 90498 81899 67075 44990 36S46 13577 18433 33977 10000 20000 40000 80000 100000 200000 400000 800000 1000000 2000000 4000000 8000000 No. of Places or logs. 4 log. of 2 7 log. of 4 13 log. of 16 25 loor. of 256 31 log. of 2 61 log. of 4 121 log. 16 241 loff. 256 302 log. 2 603 log. 4 1205 log. 16 2409 log. 256 3011 Jog. 2 6021 log. 4 120i2 log. 16 24083 locr. 256 30103 log. 2 60206 log. 4 120412 ^^ 240824 P 301030 602060 1204120 2408240 10000000 20000000 40000000 80000000 100000000 200000000 400000000 80U00O0OO 46 129 ! lOOOOCOOOO 3010300 6020600 12041200 24082400 30103000 60206000 120411999 240823997 301029996 dTS CONSTRUCTION OF TRACT 21, ber of places in them ; and the multiplications in raising these powers are performed in a contracted way. so as to have the fifth or last figure in them true to the nearest unit. Indeed these multiplications might be performed in the same man- ner, retaining only the first three figures, and those to the nearest unit in the third place; which would make this a very easy way indeed of finding the logarithms of a few prime numbers. It may also he remarked, that those several powers, whose indices are 1 with ciphers, are raised by thrice squaring from the former powers, and multiplying the first by the third of these squares; making also the corresponding doublings and additions of their indices : thus, the square of 2 is 4, and the square of 4 is 16, the square of 16 is 256, and 256 multiplied by 4 is 1024 ; in like manner, the double of 1 is 2, the double of 2 is 4, the double of 4 is 8, and 8 added to 2 makes 10. And the same for all the following powers and indices. The numbers in the third column, which show how many places are in the corresponding powers in the first column, arepro- tluced in the very same way as those in the second column, namely, by three duplications and one addition; only ob- serving to subtract 1 when the product of the first figures are expressed by one figure ; or when the first figures exceed tliose of tlie number or power next above them. It mnv further be observed, that, like as the first number in each quatcrnioi!, or space of four lines or numbers, in the third column, approximates to the logarithm of 2, the first number in the first quaternion of the first column ; so the second, third, and fourth terms of each quaternion in the third column, approximate to the logarithm of 4, 16, and 256, the second, third, and fourth numbers in the first qua- ternion in the first column. And further, by cutting off one, two, three, &.c, figures, as the index or integral part, from the said logarithms of 2, 4, 16, and 256, the first, second, third, and fourth numbers in the first quaternion of the first column, the remaining figures will be the decimal part of the logarithms of the corresponding first, second, third, and fourth numbers in -the following second, third, fourth ^ &.c. TRACT 21. LOGARITHMS. S78 quaternions : the reason of which is, that any number of any quaternion in the first column, is the tenth power of the cor- responding term in the next preceding quaternion. So that the third column contains the logarithms of ail the numbers in the first column : a property which, if Dr. Newton had been aware of, he could not easily have committed such gross mis- takes as are found in a table of his, similar to that above given, in Avhich most of the numbers in the latter quaternions are totally erroneous; and his confused and imperfect account of this method would induce one to believe that he did not well understand it. In the sixth chapter our illustrious author l>egins to treat of the other general method of finding the logarithms of prim numbers, which he thinks is an easier way than the former, at least when the logarithm is required to a gi*eat many places of figures. This method consists in taking a great number of continued geometrical means between 1 and the given number whose logarithm is required ; that is, first extracting the square root of the given number, then the root of the first root, the root of the second root, the root of the third root, and so on till the last root shall exceed 1 by a vevy small decimal, Q-reater or less accordine to the intended number 5 1-074G07 0-03125 &c. &c. ber, like as the logarithm of the last number is the half of the preceding logaritlim. Therefore, any one such small num- ber, with its logarithm, being once found, by the continual extractions of square roots out of a given number, as 10, and corresponding bisections of its given logarithm 1 ; the loga- rithm for any other such small number, derived by like con- tinual extractions from another given number, whose loga- rithm is sought, will be found by one single proportion: which logarithm is then to be doubled according to the number of extractions, or mul- tiplied at once by the like power of 2, for the logarithm of the number proposed. To find the first small number and its logarithm, our author be- gins with the number 10 and its logarithm 1, and extracts continually the root of the last number, and bisects its logarithm, as here registered in the margin, but to far more places of figures, till he arrives at the 53d and olth roots, with their annexed logarithms, as here below : Numbers. I Logarithms. 35 I V0OOO9,0fX)00,OO0()0,25563,829Rf;,40064,70 0OOOO,()0O0O,O00(X),inO'J,23O2i,6?5I."),654O4 54 I 1-CK)000,000':0,00000,12781:9!493,20032,35 J 000(XX),00000,CX)000,0555I,U512,312.=)7,82702 where the decimals in the natural numbers are to each other in the ratio of the logarithms, namely in the ratio of 2 to 1 : and therefore any other such small number being found, by continual extraction or otlicrwise, it will then be as 12781 ike, is to 555 1&.C, so IS that other small decimal, to the correspond- ing significant figures of its logarithm. But as every repeti- tion ot this pro[)ortion requires both a very long multiplication and division, he roduc(!s this constant ratio to another equi- valent ratio whose antecedent is 1, by which all the divisions are saved : thus, as 12781&C : 555 1 &c : : lOOO&c : 434294481903251804, that IS, ihe logarithm of 1-00000,00000,00000,1 is 0-00000,00000,00000,04342,9448 1 ,90325, 1 804 ; TRACT 21. .LOGARITHMS. 375 and therefore this last number being multiplied by any such small decimal, found as above by continual extraction, the product will be the corresponding logarithm of such last root. But as the extraction of so many roots is a very trouble- some operation, our author devises some ingenious contri- vances to abridge that labour. And first, in the 7th chapter, b}' the following device, to have fewer and easier extractions to perform : namel v, raising the powers from any given prima number, whose logarithm is sought, till a power of it be found such that its first figure on the left hand is 1, and the next to it either one or more ciphers; then, having divided this power by 1 with as many ciphers as it has figure., after the first, or supposing- all after the first to be decimals, the continual roots from this power are extracted till the decimal become suffi- ciently small, as when the first fifteen places are ciphers; and then by multiplying the decimal by 434-29 &c, he has the lo- garithm of this last root ; which logarithm multiplied by the like power of the numbers, gives the logarithm of the first number, from which the extraction was begun : to this loga- rithm prefixing a 1, or 2, or 3, &c, according as this number was found by dividing the power of the given prime number by 10, or 100, or 1000, &c; and lastly, dividing the result by the index of that power, the quotient will be the required logarithm of tiie given prime nun)ber. Thus, to find the logarithm ot 2: it is first raised to the 10th power, as m the margm, before the first figures come to be 10; then, dividing by 1000, or cutting off for decimals all the figures after the fir^t or 1, the root is continually extracted out oi the quotient 1,024, till the 47th extraction, wliich gives 1.00000,00000,00000,16851,60570,53949,77; the decimal part of winch multi. by 43429 i'vc, gives .00000,00000,00000,07318,55936,90623,9368 for its logarithm: and this being continually doubled for 47 times, gives the logarithms of all the roots up to the first number: or being at once 2 1 4 2 8 3 16 4 32 5 64 6 128 7 256 8 512 9 1024 10 ST6 CONSTRUCTION OF TRACT 21. o I 4 2 8 3 16 4 32 5 64 6 128 7 256 8 512 9 1024 10 1048576 20 1073741824 30 1099511627776 40 40737488355328 47 2 IS multiplied by thie 47th power of 2, viz. 140737488355328, which is raised as in the margin, it gives .0'01029, 99566, 3981 1,95265,27744 for the logarithm of the number 1,024, true to 17 or 18 decimals: to this prefix 3, so shall 3.0102 &c b'e the logarithm of 1024 : and lastly, because 2 is the tenth root of 1024, divide by 10, so shall 0.30102,99956,63981,1952 be the logarithm required to the given nuihber 2. The logarithms of 1, 2, and 10 being now known ; it is remarked that the logarithm of 5 becomes known ; for since 10 = 5, therefore log. 10 log. 2 = log. 5, which is 6.69897,00043,36018,8058 ; and that from the multiphcations and divisions of these three 2, 5, 10, with the corresponding additions and subtractions of their logarithms, a multitude of other numbers and their logarithms are produced ; so, from the powers of 2, are obtained 4, 8, 16, 32, 64, &c-, from the powers of 5, these, 25, 125, 625, 3125, Scc ; also the powers of 5 by those of 10 give 250, 1250, 6250, &c; and the powers of 2 by those of 10, give 20, 200, 2000, &c ; 40, 400, 80, 800, &c ; likewise by division are obtained 2|-, IJ, 12^, 6i-, Briggs then observes, that the logarithm of 3, the next prime number, will be best derived from that of 6, in this manner: 6 raised to the 9lh power becomes 10077696, which divided by 10000000, gives 1.0077696, and the root fronithLi continually extracted till the 46th, is 1, 00000,00000,00000. 1099S, 593 15, 88 155, 71866 ; the decimal part of which multiplied by 43429&C, gives 0.00000,00000,00000,04776,62814,78608,0304 for its loga- rithm; and this 46 times doubled, or multiplied by the 46th power of 2, gives 0. 00336, 12534,52792;69 for the logarithm TRACT 21. LOGARITHMS. S77 of 1 .0077696 ; to which adding 7, the logarithm of the divisor 10000000, and dividing by 9, the index of the power of 6, there results 0.77815,12503,83643,63 for the logarithm of 6; from which subtracting the logarithm of 2, there remains 0.47712,12547,19662,44 for the logarithm of 3, In the eighth chapter our ingenious author decribes an ori- ginal and easy method of constructing, by means of differ- ences, the continual mean proportionals which were before found by the extraction of roots. And this, Avith the other methods of generating logarithms by differences, in this book as well as in his " Trigonometria Britannica," are I believe the first instances that are to be found of making such use of differences, and show that he was the inventor of what may be called the '* Differential Method." He seems to have dis- covered this method in the following manner: having observed ^ that these continual means between I and any number pro- posed, found by the continual extraction of roots, approach always nearer and nearer to the halves of each preceding root, as is visible when they are placed together under each other ; and indeed it is found that as many of the significant figures of each decimal part, as there are ciphers between them and the integer 1, agree with the half of those above them; I say, having observed this evident approximation, he subtracted each of these decimal parts, which he called a, or the first differences, from half the next preceding one, and by comparing together the remainders or second differences, called B, he found that the succeeding were always nearly equal to i of the next preceding ones ; then taking the differ- ence between each second difference and ~ of the preceding one, he found that these third differences, called c, were nearly in the continual ratio of 8 to 1 ; again taking the difference between each c and |- of the next preceding, he found that these fourth differences, called d, were nearly in the continual ratio of 16 to 1 ; and so on, the 5th e, 6th f, &c, differences, being nearly in the continual ratio of 32 to 1, of 64 to 1, &c. CONSTRUCTION OF TRACT 21. These plain obser- vations being made, they very naturally and cleariy su 2, jested to him the notii^u and method of cov.struct- ing all the remaining numbers, from the dif- ferences of a few of the first, found by ex- tracting the roots in the usual way. This will evidently appear from the annexed spe- cimen of a few of tlie first numbers in the last example, for find- jno: the losrarithm of 6; where, after the *>th number, the rest are supposed to be constructed from the preceding differences of each, as here shown in the lOth and 11th, And it is evident that, in proceeding, the trouble will become always less and less, the ditrereuces gr;i{lu- ai'iV vauisiiing, till at last only the fir.^t dif- Icrenccs remain; and that generally each h'ss dillcrciice is shorter t'lati the next ^Teuler, by as many 1,0()776,H6 1 1,00387,72833,36962,45663,846.55,1 1 ,001 93,6766 1 .36946,61 675,870<22,9 3 1,00096,79146,39099.01728,89072,1) 4 1, 00048,38402, 68846, 6298,'5,4925:;, 5 A A b l,00i>24, 18908,78824,68563,80872,7 24,19201,34423,31492,74626,7 J A 292,5.i')9S,62928,95754,0 I! G 1, 00012i09.3s 1,26397, 1.3459,439 19,4 A 12,09454,5941 2,34281 ,90436,3 -|A 73,13015,20822,46516,9 B 75,13899,65732.23438,5 iE 884,449(^9,76921,5 C 7 l,O0i.'06,04672,35O55,;>O'.'68,01600,.5 A 6,04690.63 19S,567':'9,7 1959,7 i. 18,28143,25761,703.59,2 R 18,28253,80205,61629,2 iB 110,54443,91270,0 C 110,55613,72115,2 i^ 1169,80845,2 D 8 1,(10003.02331,60505,65775,96479,4 A 3,02336,17527,65484,00800,2 iA 4,57021,99708,04320,8 n 4,57035,81440,42589,8 i''' 13,81732,38269,0 c 13,81805,43908,7 ic 73,10639,7 D 73,11302,8 t|d 663,1 E T 1,00001,51 lii4, 65999,05672,95048,8 A 1,511 65, 8(252,82S87,98239,7 iA 1,14253,77215,03190,9 B Hitherto the 1,14255,49927,01080,2 4b smaller fiifferences 1.72711,97889,3 c are found hy sub- 1,72716,54783,6 l^- tractiiiL; the larger from 4,5689!-, 3 D tlie parts of the like pre- 4,56915,0 li^ eedins: ones. 20,7 20,7 3i = 35 = Here t!io greater clittVnuces 65 remain al'tcr subtracting 28555,89 li" the smaller from the parts 28555,24 u of the (lilVeieiiceof 21588,99736,16 ?' the next precedincr 21588,71180,92 c number. 28563,44303,75797,72 -1" 28563,22715,04616,80 i; 75582,32999,52836,47524,40 2' 10 1 ,01 000,75582,04436,30121,42907,60 A 2 I78i,70 1 7^54, 6s 1>, 2693.58897,62 '( 2(".98, 57112,1.4 c 7140,80678,76154,20 ^, '- 71 40,77980,190 il,-2'i K r-,779 1 : J 1 S, 1 ..uf,(),7 1 4 j.3,80 i.A ! 1 l.Ci ir(lO,n77'.'>>,".V'77,:37O.S0,.V24i;',J4 A TRACT 21. LOGARITHMS. 3*79 places as there are ciphers at the beginning of the decimal in the number to be generated from the differences. He then concludes this chapter with an ingenious, but not obvious, method of finding the differences B, c, d, e, &c, belonging to any number, as suppose the 9th, from that number itself, independent of any of the preceding 8th, 7th, 6th, 5th, &.C; and it is this : raise the dBcimai a to the 2d, 3d, 4th, 5th, &c powers; then will the 2d (b), 3d (c), 4th (d), &c differences, be as here below, viz. B=iA% C= iA^f 4^% D= |A^f|A5+-rVA^+ -^A'+ s-';fA% E= . 2|A'+ 7a*+ 10|-|a7 4- 12A'-tA'+- 1^ i^A'&c. F= . . iS^^A^t- 81|a7+ 296-,si5.A'+ 834t-V7a9&c. G= . . . 122^A^+1510-rV-5-A^ f 11475tV-jA96cc. H= . . . . 1937^V7A^ + 4'7l5lA?A''&c. 1 = . . . . . 54902 jS_?^A''&.c Thus in the 9th number of the foregoing example, omitting the ciphers at the l)eginning of the decimals, we have A = 1.51164,65999,05672,95048,8 A^= - 2,28507,54430,06381,6726 A^= - - 3,45422,65239,48546,2 A^= - - - 5,22156,97802,288 A-'= - - - - 7,89316,8205 A^= 11,93168,1 Consequentl}', ^a- = 1. 14253,77215,03190,8363 = B -^a^ 1,72711,32619,74273 -I-A-* - 65269,62225 lA' + iA-^ ^ 1,72" '11 ,97889,36498 = c |A^ 4 ,56887,35577 iA^ 6,90652 7 A*^ 5 ^-fTVA" 4 ,56894,26234 = D 2|A^ - 20,71957 7a* 83 2|a5 + 7a^' - - 20,72040 = e S80 CONSTKUCTION OF TRACT 21. which. agree M-ith the hke differences in the foregoing spe- cimen. In the 9th chapter, after observing that from the logarithms of 1, 2,3, 5, and 10, before found, are to be determined, by- addition and subtraction, the logarithms of all other numbers which can be produced from these by multiplication and division ; for finding the logarithms of other prime numbers, instead of that in the 7th chapter, our author then shows an- other ingenious method of obtaining numbers beginning with 1 and ciphers, and such as to bear a certain relation to some prime number bv means of which its logarithm may be found. The method is tliis : Find three products having the common difference 1, and such that two of them are produced from fjtctors having given logarithms, and the third produced from the prime number, whose logarithm is required, cither mul- tipHed by itself, or by some other number whose logarithm is given: then the greatest and ka;t of these thrce products being multiplied together, and the mean by itself, there arise two other products also diflfering by 1, of which the greater, divided by the less, gives for a quotient I with a small deci- mal, having several ciphers at the beginning. Then the lo- garithm of this quotient being found as before, from it v. ill be deduced the required logarithm of the given prime num- ber. Thus, if it be proposed to find the logarithm of the prime number 7 ; here 6x 8 = 48, 7x 7 = 4i), and 5x 10 = 50, will be the thrce products, of which the logarithms of 48 and 50, the 1st and 3d, will be given from those of tlieir factors C), 8, 5, 10 : also 48 x 50 = 2400, and 49 x 49 = 2401 are tliC tv.-o new products, and 2401 -^2400= 1.0004i| their (juoticnt: then the least of 44 means between 1 and this quo- tient is l.O0OO0,O00OO,OO0O0,O23G7, 98249, 04^533, G40.5, which Tnu!ti}>r!('d by 43429 &c, produces 0.00000,00000,00000,01028,40172,88387,2971/:; for its loga- ririim; which being 44 times doubled, or nmltiplied i)y 175!>218G044416, produces O.C0018,09]83, 45421, 30 for tlie logarithm of the quotient 1.000414; which being added to the logarithm ol the divisor 2400, ^nves the logarithm of the TRACT 21. LOGARITHMS. S8l dividend 2401 ; then the half of this logarithm is the loga- rithm of 49 the root of 2401, and the half of this again gives 0.84509,80400,14256,82 for the logarithm of 7, which is the root of 49. The author adds another example to illustrate this method ; and then sets down the requisite factors, pro- ducts, and quotients for finding the logarithms of all other prime numbers up to 100. The iOth chapter is emploj^ed in teaching how to find the logarithms of fractions, namely by subtracting the logarithm of the denominator from that of tlie numerator, then the lo- garithm of the fraction is the remainder; which therefore is either abundant or defective, that is positive or negative, as the fraction is greater or less than 1. In the 1 1th chapter is shown an ingenious contrivance for very accurately finding intermediate numbers to given loga- rithms, by the proportional parts. On this occasion, it is re- jnarked, that while the absolute numbers increase uniformly, the logarithms increase unequally, with a decreasing incre- ment; for which reason it happens, that either logarithms or numbers corrected by means of the proportional parts, will not be quite accurate, the logarithms so found being always too small, and the absolute numbers so found too great; but yet so however as that they approach much nesircr to accu- racy tov.'ards the end of the table, where ti;c ii.creinents or differences become much nearer to equality, than in the formci; parts of the table. And from this property our author, ever fruitful in happy expedients to obviate natural difficulties, contrives a device to throw the proportional part, to be found from the numbers and logarithms, always near the eiKi oftlu! table, in whatever part they may happen naturally to fali. And it is this; Rejecting the cliaracteristic of any given loi>a- rithm, whose number is proposed to be found, take the arith- metical complement of the decimal part, by subtracting st from l.OOOi^c, the logarithm of 10 ; then find in the table the logarithm next less than this arithmetical complement, toge- ther with its absolute number ; to this tabular logaritlim add the looarithra that was ^iven, and the sura will be a loiraririjia 382 CONSTRUCTION OF TRACT 21, necessarily falling among those near the end of the table ; find then its absolute number, corrected by means of the proportional part, which will not be very inaccurate, as fall- ing near the end of the table; this being O 3 25 4 15 ^ o 5 5 < 'cu 6 7 5 15 25 -^"2 y 35 ? 10 45 3 ?i 384 CONSTRUCTION CF TRACT 2i, whose distance from A is expressed by x, will be thus, 7.=A-^xa-\-x . ~b f .r . ^ . ^\- + &c. And if now, ^vith our author, v.c make the 2d ditferences equal, t>ien c, -A &c, we shall have the annexed series of terms with their dif- ferences. Where it is to be observed, that our author had reduced the dilTerences from the 1st to the 2d form, as he thought it easier to multiply by 5 than to divide by 2. Also all the last terms [x . ~^b) are set down positive, because in the logarithms b is negative. If the two 2d differences be only nearly equal, take a;i arithmetical mean between them, and procceil with it the same as above with one of the equal 2d , and teaches their several uses; one of which is, that every other number, taken in the diirigonal lines, furnishes the coefficients of the terms of the general equation, by which the sines and chords of multiple arcs are expressed, which be amply illu- strates; and another, that the same diagonal numbers consti- tute the coefficients of the terms of any power of a binomial; which property was also mentioned by Vieta in his Angu- Jares Sectiones, theor. 6, 7; and, before him, pretty fully treated of by Stifelius, in his Arithmetica Integra, fol. 44 et seq. ; where he inserts and makes the like use of such a table of figurate numbers, in extracting the roots of all powers whatever. But it was perhaps known much earlier, as ap- pears by the treatise on figurate numbers by Nicomachus, (see Malcolm's History, p. xviii). Though indeed Cardan seems to ascribe this discovery to Stifelius. See his Opus Novum de Proportionibus Numerorum, where he quotes it, and extracts the table and its use from Stifel's book. Cardan, in p. 135 &.C, of the same work, makes use of a like table to find the number of variations, or conjugations, as he calls them. Stevinus too makes use of the same coefficients and method of roots as Stifelius. See his Arith. page 25. And even Lucas de Burgo extracts the cube root by the same co- efficients, about the year 1470: but he does not go to any higher roots. And this is the first mention I have seen made of this law of the coefficients of the powers of a binomial, commonly called Sir I. Newton"'s binomial theorem, though it is very evident that Sir Isaac was not the first inventor of it : the part of it properly belonging to him seems to be, only the extending it to fractional indices, which was indeed an immediate effect of the general method of denoting all roots like powers, with fractional exponents, the theorem being not at all altered. However, it appears that our author Briggs was the first who taught the rule for generating the coefficients of the terms, successively one from another, of any power of a binomial, independent of those of any other power. For having shown, in his *' Abacus Uxyx^rof^ (which 393 CONSTRUCTION Or TRACT 21. he so calls on account of its frequent and excellent use, and of which a small specimen is here annejjed), that the nucjibers ABACUS nArXPKZTOS . H G F E D c B A - -0 + +0 - -o +0 1 1 1 1 1 1 1 1 2 9 8 7 6 5 4 3 36 28 21 15 10 6 3 b4 56 35 20 10 4 126 70 35 15 5 12G 56 21 6 84 28 36 7 8 9 in the diagonal directions, ascending from right to left, are the coefficients of the powers of binomials, the indices being the figures in the first perpendicular column a, which are also the coefficients of the 2d terms of eacii power (those of the first terms, being 1, are here omitted) ; and that anv onq of these diagonal numbers is in proj)ortion to the next higiier in the diagonal, as the vertical of the former is to the mar- ginal of the latter, that is, as the u[)permost number in the column of the former is to the first or right-hand number in the line of the latter; having shown these things, I say, he thereby teaches the generation of the coefficients of any power, independently of all other powers, by the very same law or rule which we now use in the binomial theorem. Thus, for the 9th power ; 9 being the coefficient of the 2d term, and 1 always that of the first, to find the 3d coefficient, Ave have 2 : 8 : : 9 : 36 ; for the 4th term, 3 : 7 : : 36 : 84; for the 5th term, 4 : 6 : : 84 : 126 ; and so on for the rest. That is to say, the coefficients of the terms in any power 7n, are inversely as the vertical numbers or first line 1, 2, 3, 4, . . w, and directly as the ascending numbers 7n, vi I, m 2, ?7i 3, . . . 1, in the first column A ; and that consequently TRACT 21. LOGARITHMS. SdS those coefficients are found by the continual multiplication of these tractions , , - , -, , which is the very theorem as it stands at this day, and as applied by Newton to roots or fractional exponents, as it had before been used for integral powers. This theorem then being thus plainly taught by Briggs about the year 1600, it is surprizing how a man of such general reading as Dr. Wallis was, could be quite ignorant of it, as he plainly appears to be by the 85tli chapter of his algebra, where he fully ascribes the invention to Newton, and adds, that he himself had formerly sought for such a rule, but without success: Or how Mr. John Bernoulli, in the I.Sth century, could himself first dispute the inven- tion of this theorem with Newton, and tiien give the discovery of it to Pascal, who was not born till long after it had been tang'it by Briggs. See Bernoulli's Works, vol. 4, page 173. But it is not to be wondered that Briggs's remark was un- known to Newton, who owed almost every thing to genius, and deep iiicditation, but very little to reading: and there can be no doubt that he made the discovery himself, without any light from Briggs ; and that he thought it was new for all powers in general, as it was indeed for roots and quanti- ties with fractional and irrational exponents. When the above table of the sums of figurate numbers is used by our author, in determining the coefficients of the terms of the equation, whose root is the chord of any sub- multiple of an arc, as when the section is expressed by any uneven number, he remarks, that the powers of that chord or root will be the 1st, 3d, 5th, 7th, &c, in the alternate uneven columns, a, c, e, g, &r, with their signs + or as marked to the powers, continued till the highest power be equal to the index of the section; and that the coefficients of those powers are the sums of two continuous numbers in the same column with the powers, beginning with 1 at the highest power, and gradually descending one line obliquely to the right at each lower power : so, for a trisection, the numbers are 1 in c, and 1 -{- 2 = 3 in a ; and tlierefore the terms are 1(3) + 3(1): for a quinquisection, the numbers are 1 in e, 394 COKSTRUCTION OF TRACT il. 1 + 4 = 5 in c, 2 -{- 3 = 5 in A ; so that the terms are 1(5) 5(3) 4" 5(1) : for a septisection, the numbers are 1 in o, 1 4- 6 = 7 in E, 4 4- 10 = 14 in c, and 3 + 4 = 7 in A ; and hence the terms are 1 (7) + ^ (5) 14 (3) + 7 (1) : and so on; the sum of all these terms being always equal to the chord of the whole or multiple arc. But when the section is deno- minated by an even number, the squares of the chords enter the equation, instead of the first powers as before, and the dimensions of all the powers are doubled, the coefficients being found as before, and therefore the powers and numbers will be those in tlie 2d, 4th, 6th, &.c, columns : and the uneven sections may also be expressed the same way ; hence, for a bisection the terms will be 1 (4) + 4 (2) ; for a trisection J (6) 6 (4) 4- 9 (2) ; for the quadrisection - 1 (8) 4 8 (6> - 20(4) 4 16 (2) ; for thequinquisection 1 (10) 10(8) + 35 (6) - 50 (4) + 25 (2) ; and so on. Our author subjoins another table, a small specimen of which is here annexed, in which tlie first column consists of the uneven numbers 1, 3, 5, &.c, the rest being found by ad- dition as before, and the alternate diagonal numbers them- selves are the coefficients. F li D C B A 4(6) + (5) - (i) -(3) + (2) (1) 1 1 1 1 1 1 7 6 5 4 3 20 14 Q 5 30 16 25 7 9 11 The method is quite different from that of Vieta, who gives an- other table for the like purpose, a small part of which is here an- nexed, which is formed by adding, from the number 2, downv.ards obliquely inwards the right; and the coeffirietits of the terms stand iipon the horizoiitu! line. 1st Vieta's Table. 2 3 2d 4 2 5 5 3d J 6 9 2 7 14 7 4th 8 20 16 2 9 27 30 9 5th 10 .'-15 50 25 2 6 th TRACT 21, lOGARITHMS. 395 These angular sections were afterwards further discussed by Ought red and Wallis. And the same theorems of Vieta and Briggs have been since given in a different form, by Herman and the Bernoullis, in the Leipsic Acts, and the Me:7ioir3 of the Royal Academy of Sciences. These theo- rems they expressed by the alternate terms of the power of a binomial, whose exponent is that of the multiple angle or section. And De Lagny, in the same Memoirs, first showed, that the tangents and secants of multiple angles are also ex- pressed by the terms of a binomial, in the form of a fraction, of which some of those terms form the numerator, and others the denominator. Thus, if r express the radius, s the sine, c the cosine, t the tangent, and s the secant, of the angle a ; then the sine, cosine, tangent, and secant of n times the angle, are expressed thus, viz. r _, , 1 _ n .n I __ n.fi-1.7i-2.n-^ n-l ^ 1.2 1.2.3.4 ^ r lang.?iA=rx - 77 H/j-l ,, A.n-l.-2.-3_ ,. ] . '2 ^ 1.2.3-4 e s^ or j2 + t^ Sec.nA = rX P ^ ^ : r'--"-^f-^t'+"-ll-'.:^^r^-^t^ &c. 1.2 1.2.3.4 where it is evident, that the series in the sine of n A, con.sists of the even terms of the power of the binomial (c -f- s)", and the series in the cosine of the uneven terms of the same power; also the series in the numerator of the tangent, consists of the even terms of the power (r + t)", and the denominator, both of the tangent and secant, consists of the uneven terms of the satne power (r 4- t)". And if the diameter, chord, and chord of the supplement, be substituted for the radius, sine and co- sine, in the expressions for the multiple sine and cosine, tha result will give the chord, and chord of the supplement, of times the arc or angle a. Thee, and various other expres- 396 CONSTRUCTION OF TRACT 21. sions, for multiple and submultiple arcs, witVi other improve- ments in trigonometry, have also been given by Euler, and other eminent writers on the same subject. 'The before mentioned De Lagny offered a project for sub- stituting, instead of the common logarithms, a binary arith- metic, which he called the natural logarithms^ and which he and Leibnitz seem to have both invented about the same time, independently of each other: but the project came to nothing. De Lagny ako published, in several Memoirs of the Royal Academy , a new method of determining the angles of figures, which he called Goniometry, It consists in measuring, with a pair of compasses, the arc which subtends the angle in ques- tion : however, this arc is not measured in the usual way, by applying its extent to any preconstructed scale ; but by ex- amining what part it is of half the circumference of the same circle, in this manner: from the proposed angular point as a centre, with a sufficiently large radius, a semicircle being described, a part of which is the arc intercepted by the sides of the proposed angle, the extent of this arc is taken with a pair of fine compasses, and applied continually upon the arc of the semicircle, by which he finds how often it is contained in the semicircle, with usually a small arc remaining; in the same manner he measures how often this remaining arc is contained in the first arc ; and what remains again is applied continually to the first remainder ; and so the 3 1 remainder to the 2d, the 4th to the 3.d, and so on till there be no remainder, cr else till it become insensibly small. By this process he ob- tains a series of quotients, or fractional parts, one of another, "which being properl}- reduced into one fraction, give the ra- tio of the first arc to the semicircumference, or of the pro- posed angle, to two right angles or 180 degrees, and conse- quently that angle in degrees, minutes, &c, if required, and that commonly, he says, to a degree of accuracy far exceed- ing the calculation of the same by means of any tables of sires, tangents or secants, nocwithstanding the apparent })aT ladox in this expression at first sight. Thus, if the 1st arc be 4 times contained in the semicircle, the reuuiiiKler once TRACT 21. LOGARITHMS. 397 contained in the first arc, the next 5 times in the second, and finally the fourth 2 times in the third : Here the quotients are 4, 1, 5, 2 ; consequently the fourth or last arc was i the 3d; therefore the 3d was - or t^t of the 2d, and the 2d was -r~ or -fi of the 1st, and the first, or arc sought, was or ^ of the semicircle ; and consequently it contains Slj- degrees, or 37 8' 34"y. Hence it is evident, that this method is in fact nothing more than an example of continued fractions, the first instance of which was given by lord Brouncker. But to return from this long digression ; Mr. Briggs next treats of interpolation by difi^erences, and chiefly of quinqui- ection, after the manner used in the 13th chapter of his con- struction of logarithms, before described. He here proves that curious property of the sines and their several orders of differences, before mentioned, namely, that, of equidifferent arcs, the sines, with the 2d, 4th, 6th, &c difl'erences, are con- tinued proportionals ; as also the cosines of the means between those arcs, and the 1st, 3d, 5th, &c differences. And to this treatise on interpolation by differences, he adds a marginal note, complaining that this 13th chapter of his " Arithmetica Logarithmica" had been omitted b}' Viacq in his edition of it; as if he were afraid of an intention to deprive him of the honour of the invention of interpolation by successive differ- ences. The note is this : " Modus correctionis a me traditus est Arithmeticae Logarithmicse capite 13, in editione Londi- nensis ; Istud autem caput una cum sequenti in editione Ba- tava me inconsulto et inscio omissum fuit: nee in omnibus, fcditionis illius author, vir alioqui industrius et non indoctus, meam mentem videtur assequutus; Ideoque, ne qaicquam desit cuiquam, qui integrum canonem conficere cupiat,qu8e- dam maxime necessaria illinc hue transferenda censui." A large specimen of quinquisection by differences is then given, and he shows how it is to be applied to the construc- tion of the whole canon of sines, both for 100th and 1 000th parts of degrees ; namely, for centesms, divide the quadrant first into 72 equal parts, and find their sines by the primary S98 CONSTRUCTION OF TRACT 21. methods; then these quinquisected give 360 parts, a second quinquisection gives 1800 parts, and a third gives 9000 parts, or centesms of degrees: but for miJlesms, divide the quadrant into 144 equal parts; then one quinquisection gives 720, 9 second gives 3600, a tliird 18000, and a fourth gives 90000 parts, or millesras. He next proceeds to the natural tangents and secants, which he directs to be raised in the same manner, by interpolations from a few [)riiiiary ones, constructed from the kno-^n pr^'s methods, and in the manner which this ingenious man had pointed out in his two volumes. Thus, the 70 chiliads of logarithms, from 20000 to 90000, computed by Vlacq, and published in 1628, being extended onlv to 10 places, yield no more than two orders of mean differences, "which are also the correct differences, in quinquisection, and therefore will be made out thus, namely, one-fifth of them by the mere addition of the constant logarithm of 5 ; and the other four-fifths of them by two easy additions of very small num- bers, namely, of the 1st and 2d differences, according to the directions given in Briggs's Arith. Log.c. 13, p. 31. And as to Vlacq's logarithmic sines and tangents to every 10 seconds, they were easily computed thus*, the sines for half the qua- drant were found by taking the logarithms to the natural sines in Rheticus's canon ; and then from these the logarith- mic sines to the other half quadrant were found by mere addition and subtraction; and from these all the tangents by one single subtraction. So that all these operations migli-t easily be performed by one person, as quickly as a printer could set up the types; and thus the computation and printing might both be carried on together. And hence it appears that there is no reason for admiration at the expedition with which these tables were said to have been brought out. Of certain curves related to Logarithms. About this time the mathematicians of Europe began to consider some curves which have properties analogous to logarithms. Edmund Gunter, it has been said, first gave the idea of r curve, whose abscisses are in arithmetical progres- sion, while the corresponding ordinates arc in geometrical progression, or whose abscisses are the logarithms of their ordinates; but I cannot find it noticed in any ])art of his writings. The same curve was afterwards considered by ra^ACT 21. LOGARITHMS. 401 Others, and named the Logarithmic or Logistic curve by. Huygens, in his " Dissertatio de Causa Gravitatis," where he enumerates all the principal properties of this curve, showing its analogy to logarithms. Many other learned men have also treated of its properties ; particularly Le Seur and Jacquier, in their commentary on Newton's Principia ; by Dr. John Keill, in the elegant little tract on logarithms, subjoined to his edition of Euclid's Elements ; and by Francis INIaseres, Esq. cursitor baron of the exchequer, in his ingenious trea- tise on Trigonometry; in which books the doctrine of loga- rithms is copiously and learnedly treated, and their analogy to the logarithmic curve &c fully displayed. It is indeed rather extraordinary that this curve was not sooner announc- ed to the public ; since it results immediately from baron Napier's manner of conceiving the generation of logarithms, by only supposing the lines which represent the natural num- bers to be placed at right angles to that upon which the logarithms are taken. This curve greatly facilitates the con- ception of logarithms to the imagination, and affords an almost intuitive proof of the very important property of their fluxions, or very small increments, to wit, that the fluxion of the number is to the fluxion of the logarithm, as the number is to the snbtangent ; as also of this property, that, if three numbers be taken very nearly equal, so that their ratios to each other mav differ but a little from a ratio of equality, as for exam, the three numbers 10000000, 10000001, 10000002, their diflerenccs will be very nearly proportional to the loga- rithms of the ratios of those numbers to each other; all which follows from tho logarithmic arcs being very little different fi'om tiieir chords, \vhcn they are taken very small. And the con- stant snbtangent of this curve is what was afterwards by Cotes called the Modulus of the system of loganthujs : and since, by the former of the two properties above-mentioned, this sub- tangent is a 4t!i proportional to the fluxion of the number, the fluxion of the logarithm, and the number itself; this pro- perty afforded occasion to Mr. Baron Maseres to give the fol- lowing definition of the modulus, wjiich is the same jn effect VOL. I. D D 402 CONSTRUCTIOX OF TilACT 21. as Cotes's, but more cleiirly expressed, namely, that it is the limit of the magnitude of a -l-th proportional to these three quantities, to wit, tlie difference of any two natural numbers that are nearly equal to each other, either of the said num- bers, and the logarithm or measure of the ratio they have to each other. Or we may define the modulus to be the natural number at that part of the system of logarithms, where the fluxion of the number is equal to the fluxion of the logarithm, or where the numbers and logarithms have equal differences. And hence it follows, that the logarithms of equal numbers, or of ecjual ratios, in different systems, are to one another as the moduli of those systems. Further, the ratio whose mea- sure or logarithm is equal to the modulus, and thence by Cotes called the j^atio viodularis^ is by calculation found tobfi theratio of 2-718281828459 ;kc to 1 , or of I to 36787y41-l 171 &c } the calculation of which number may be seen at full length in Mr. Baron Maseres's treatise on tlie Principles of Life Antmities, pa. 274 and 275. The hyperbolic curve also afforded another source for de- veloping and illustrating the properties and construction of logarithms. For the hyperbolic areas lying between tlie curve and one asymptote, when they are bounded b}' ordinates pa- rallel to the other asymptote, are analogous to the logarithms of their abscisses, or parts of the asymptote. And so also are the hyperbolic sectors; any sector bounded bv an arc of the hyperbola antl two radii, being cfjual to the cpuidrilateral space bounded by the same i.rc, the two ordinates to either asymptote from the extreuiities of the arc, and the piirt of the asymptote intercepted between them. And though Na- pier's logarithms are commouly said to be the same as hyper- bolic logarithms, it is not to be understood that hyperbolas exhibit Napier's logarithms only, but indeed all other possible systems of logarithms whatever. For, like as the right-angled hyperbola, the side of whose scjuare inscribed iit the vertex is 1, gives Napier's logarithms ; so any other system of loga- rithms IS e x])ressed by the hyperbola whose asymptotes form a certain oblique angle, the side of the rhombus inscribed at TRACT 21. LOGARITHMS. 403 the vertex of the hyperbola in this case also being still l,the same as the side of the square in the right-angled hyperbola. But the areas of the square and rhombus, and consequently the logarithms of any one and the same number or ratio, dif- fering according to the sine of the angle of the asymptotes. And the area of the square or rhombus, or any inscribed pa- rallelogram, is also the same thing as what was by Cotes called the modulus of the s^'stem of logarithms ; which mo- dulus will therefore be expressed by the numerical measure of the sine of the angle formed b}' the asymptotes, to the radius 1 ; as that is the same with the number expressing the area of the said square or rhombus, the side being 1 : which is another definition of the modulus to be added to those we remarked above, in treating of the logaritlimic curve. And the evident reason of this is, that in the beginning of the generation of these areas, from the vertex of the hyperbola, the nascent increment of the abscisse drawn into the altitude 1, is to the increment of the area, as radius is to the sine of the angle of the ordinate and abscisse, or of the asymptotes ; and at the beginning of the logarithms, the nascent increment of the natural numbers is to the increment of the logarithms, as 1 is to the modulus of the system. Hence we easily dis- cover that the angle formed by the asympttttes of the hyper- bola exhibiting Briggs's system of logarithms, will be 25 deg. 44 miiuites, 25 { seconds, this being the angle whose sine is 0-4S42944819 itc, the modulus of this system. Or indeed any one hyperbola will expfess all possible sys- tems of logarithms whatever, namely, if the square or rhom- bus inscribed at the vertex, or, which is the same thing, any parallelogram inscribed between the asymptotes and the curve at any other point, be expounded by the modulus of the system ; or, which is the same, by expounding the area, in- tercepted between two ordi nates which are to each other iti the ratio of 10 to 1, by the logarithm of that ratio in the proposed system. As to the first remarks on the analogv between logarithms and the hyperboUc spaces; it having been shown by Gregory D D 2 404 CONSTRUCTIUN OF TRACT 21. St. Vincent, in his Quadratura Circuli et Sectionuni Coni, published at Antwi-r]) in 1647, that if one asymptote be divided into parts, in geometrical progression, and from the points of division ordinates be drawn parallel to the other asymptote, they will divide the space between the asymptote and curve into eijual |>ortions ; from hence it was shown by Mersenne, that, by taking the continual sums of those parts, there would be obtained areas in arithmetical progression, adapted to abscisses in geometrical progression, and which therefore were analogous to a system of logarithms. And the same analogy was remarked and illustrated soon after, by Huygens and many others, who showed how to square the hyperbolic spaces by means of the logarithms. There are also innumerable other geometrical figures hav- ing properties analogous to logarithms ; such as the equian- gular spiral, the 6guresof the tangents and secants, &c ; uhich k is not to our purpose to distinguish more particularly. Of Gregory' s Computation of Logarithms. On the other hand, Mr. James Gregory, in his Vera Circuli et Hyperbolae Quadratura, first printed at Patavi, or Padua, in the year 1667, having approximated to the hyper- bolic asymplotic spaces by means of a series of inscribed and circumscribed polygons, thence shows how to compute the logarithms, Avhich are analogous to those areas: and thus the quadrature of the hy^jcibolic spaces became the same thing as the computation of the logarithnis. He here also lays down various methods to abridge the computation, with the assistance of some properties of numbers themselves, by which we are enabled to compose the logarithms of all prime numbers under 1000, each by one multiplication, two divi- sions, and the extraction of the square root. And the same subject is further pursued in his Excrcitationes Geonietritic, to be dc^^cribed hereafter. Mr. Gregory was born at Aberdeen in Scotland 1638, where he was educated. He was professor of mathematics in the college of St. Andrews, and afterwards in that of Edniburgh. TRACT 21. LOGARITHMS. 405 He died of a fever in December 1615, being only 36 years of age. Of Mercatof s Logarithmotechnia. Nicholas Mercator, a learned mathematician, and an ino;e- wious member of the Royal Society, was a rBtive of Holstein in Germany, but spent most of his time in England, where- he died in the year 1690, at about 50 years of age. He was the author of many works in geometry, geography, astro- nomy, astrology, &c. In 1668, Mercator published his Logarithmotechnia, sive- methodus construendi Logarithmos nova, accurata,et faciiis; in which he deHvers a new and ingenious method of comput- ing the logarithms, on principles purely arithmetical ; which, ' being curious and very accurately performed, I shall here give, a rather full and particular account of that little tract, as well- as of the small specimen of the quadrature of curves by infir nite series, subjoined to it ; and the more especially as this work gave occasion to the public communication of some o Sir Isaac Newton's earliest pieces, to evnnce that he had ncrfr borrowed them from this publication. So. that it appears: these two ingenious men had, independent of each other, in- some instances fallen upon the same things. ' Mercator begins this work bv reuiarknig that the word logarithm is composed of the words ratio and number, being as nuich as to sav thennmber of ratios; which he observes is- quite agreeable to the nature of them, for that a log.irithm is' nothing else but tiie nnniber of ratiiniculce contained in the ratio which anv number bears to unitv. lie then makes a learned and critical dissertation on the nature ot ratios, their* magnitude and muisure, conveying a clearer idea of the na-. ture of logarithnis than had been given by either Napier or Brlorgs, or anv other writfr except Kepler, in his work before described ; tliough those otiier writers seem indeed to have had in their oun minds the same ideas on the subject as Kepler and Mercutor, butwitiiout having expre.ssed them so clearly. Our author indeed [)retty closely follows Kepicr in 406 CONSTRUCTION OP TRACT 21. his modes of thinking and expression, and after him in plain and express terms calls logarithms the measures of ratios ; and, in order to the right understanding that definition of them, he explains what he means by the magnitude of a ratio. This he does pretty fully, but not too fully, considering the nicety and subtlety of the subject of ratios, and their magni- tude, with their addition to, and subtraction from, each other, which have been misconceived by very learned mathemati- cians, who have thence been led into considerable mistakes. Witness theoversight of Gregory St. Vincent, which Huygcns animadverted on in the E^eraa-is Cyclometria Gregorii a Sancto Vincentio, and which arose from not understanding, or not adverting to, the nature of ratios, and their proportions to one another. And many other similar mistakes might here be adduced of other eminent writers. From all which wc must commend tlie propriety of our author's attention, in so judiciously discriminating betw-een the magnitude of a ratio, as of a to b, and the fraction -f-, or quotient arising from the division of one term of the ratio by the other ; which latter method of consideration is always attended with danger of eiTors and confusion on the subject ; though in the 5th defi- nition of the 6th book of Euclid this quotient is accounted the quantity of the ratio; but this definition is ])robably not genuine, and tlierefore verv properly omitted by professor Simson in his edition of the Elements. And in those ideas on the subject of logaritlims, Kepler and Mercator have been followed by Hal ley, Cotes, and most of the other eminent writers since that time. Purely from the above idea of logaritlims, namely, as being the measures of ratios, and as expressing the number of raii- unculic contained in any ratio, or into which it may be divided, the number of the like ecjual ratiuncuUv contained in some one ratio, as of 10 to 1, bemg supposed given, our author shows liow the logarithm or measure of any other ratio may be found. But this however only by-the-bve, as not being the principal method he intends to teach, as his last and best, and which we arrive not at till near x\\q. end of the book, as we shall see TRACT 21. LOGAHITHMS. 407 below. Having shown then, that these logarithms, or num- bers of small ratios, or measures of ratios, may be all properly represented by numbers, and that of 1 , or the ratio of equa- lity, the logarithm or measure being always 0, the logarithm of 10, or the measure of the ratio 10 to 1, inmost conveniently represented by 1 with any number of ciphers; he then pro- ceeds to show how tlie measures of all other ratios may be found from this last supposition. And he explains the prin- ciples by the two following examples. First, to find the logarithm of 100-5*, or to find how many ratiunculte are contained in the ratio of 100*5 to 1 , the number oi ratiuncuUe in the decuple ratio, or ratio of 10 to 1, being 1 .0000000. The given ratio lOO'o to 1 , he first divides into its parts, namely, 100*5 to 100, 100 to 10, and 10 to 1 ; the last two of which being decuples, it follows that the characteristic will be "2, and it only remains to find how many parts of the next decuple belong to the first ratio of 100*5 to 100. Now if each term of this ratio be multiplied by itself, the products will be in the duplicate ratio of the first terms, or this last ratio will contain a double number of parts; and if these be multiplied by the first terms again, the ratio of the last products will contain three times the number of parts; and soon, the num- ber of times of the first parts contained in the ratio of any like powers of the first terms, being always denoted by the expo- nent of the power. If therefore the first terms, lOO'o and 100, be continually multiplied till the same powers of them have to each other a ratio whose measure is known, as suppose the de- cuple ratio lOtol, whose meabure is 1.0000000; then the ex- ponent of that power shows what mul.this measure 1.0000000, of the decuple ratio, is of the required measure of the first ratio 100-5 to 100; and consequently dividing 1.0000000 by that exponent, the quotient is the measure of the ratio 100*5 to 100 sought. The operation for finding this, he sets down as here follows; where the several multiplications are all performed in Merfator distinguislics his decimals from integers thus 10Q[5, or ]0(>'5. 408 CONSTRUC5TION OF TRACT 2t. the contracted way, by inverting the figures of the multiple and retaining only the first number of decimals in each pro- duct. 100-5000 . power . 1 5001 . . 1 1005000 5025 1010025 5200101 1010025 10100 20 A 1020150 0510201 1020160 20403 102 51 1040106 6010401 108:5068 8603801 in3035 5303711 Tsieoii 1106731 1893406 6043981 3584985 5894853 12852116 16 16 32 32 64 64 128 128 256 256 512 This power being greater than the decuple of the like power oi 1 00, which must always be 1 with ciphers, re- sume therefore the 256th power, and multiply it, not by itself, but by the next before, viz. bv the 128th, thus 3584985 6043981 6787831 1106731 9340130 5303711 10956299 256 128 384 64 448 32 480 This being again too much, ir^^te<^d of tWe 16' '.dra.v it utotlu' 8tli, or next prect'diug, thus 9340130 6070401 This power again exceeding the same power of 100 more than 10nmes,there- fore draw the same 448tli, not into the 32d, but the next preceding, thus 9340130 860ii801 10115994 448 16 464 9720329 0520201 9916193 5200101 448 8 i56 4 460 2 462 power 10015603 Which again exceeds the limit; theref. draw the 460th into the 1st, thus 9916193 . 460 5001 . 1 S965774 461 S'Hce therefore the 462d power of 100*5 is greater, and the 46 1st power is less, than the de- cuple of the same power of 100, the ratio of 100*5 to 100 is contained in the decuple more than 461 times, but less than 462 times. Again, Since y Ji \ power \ 99161931 and the differences 9965774 > 49581 J nearly 1001 5603 3 49829 ( equal ; TRACT 21. LOGARITHMS. 409 therefore the proportional part which the exact power, or 10000000, exceeds the next less 9965174, will be easily and accurately found by the Golden Rule, thus : The just power . . 10000000 and the next less . . 9965774 the difference . 34226; then ^ As 49829 the dif. between the next less and greater^ : To 34226 the dif. between the next less and just, : : So is JOOOO : to 6868, the decimal parts ; and therefore the ratio of 100*5 to 100, is 461*6868 times contained in the decuple or ratio of 10 to 1. Dividing now 1. 0000000, the measure of the decuple ratio, by 461*6868, the quotient 00216597 is the measure of the ratio of 1005 to 100; which being added to 2 the measure of 100 to 1 , the sum 2.00216597 is the measure of the ratio of 1005 to 1, that is, the log. of 100*5 is 2.00216597. In the same manner he next investi- gates the log. of 99*5, and finds it to be 1.99782307. A few observaiions are then added, calculated to generalize the consideration of ratios, their magnitude and affections. It is here remarked, that be considers the magnitude of the ratio between two quantities as the same, whether the antecedent be the greater or the less of the two terms: so, the magnitude of the ratio of 8 to 5, is the same as of 5 to 8; that is, by the magnitude of the ratio of either to the other, is meant the number of I'atiunatia between them, which will evidently be the same, whether tiie greater or less term be the antecedent. And he further remarks that, of different ratios, when we di- vide the greater term of each ratio by the less, that ratio is of the greater mass or magnitude, which produces the greater quotient, et vice versa; tiiough those quotients are not pro- portional to the masses or magnitudes of the ratios. But when he considers the ratio of a greater term to a less, or of a less to a greater, that is to say, the ratio of greater or less inequality, as abstracted fron the ma^; iiitude of the ratio, he distinguishes it by the word ajfcction, as much as to say, greater or less affection, something in the manner of positive and negative quantities, or such as are affected with the signs 410 CONSTRUCTION OF TRACT 21. -f- and . The remainder of this work he delivers in several propositions, as follows. Prop. 1. In subtracting from each other, two quantities of the same alFection, to wit, both positive, or both negative ; if the remainder be of the same aflFection with the two given, then is the quantity subtracted the less of the two, or expressed by the less number; but if the contrary, it is the greater. _ ^ - ... a a+b a + 26 - Prop. 2. In any continued ratios, as r, , . &c, (by which is meant the ratios of a to a + b, a + b to a -\ 2b, a 4- 2^ to a 4- 3i, h<:,) of equidifferent terms, the antecedent of each ratio being equal to the consequent of the next pre- ceding one, and proceeding from less terms to greater ; the measure of each ratio will be expressed by a greater quantity than that of the next following ; and the same through all their orders of differences, namely, the 1st, 2d, 3d, &c, dif- ierences; but the contrary, when the terms of the ratios decrease from greater to less. Prop. 3. In any continued ratios of equidifferent terms, if the 1st or least be c, the difference between the 1st and 2d /, and c, d, e, &c, the respec- tive first term of their 2d, 1st term a 3d, 4th, &c, differences: 2d term a -\- b . then shall the several quan- 3d term a \- 2b -\- c tities themselves be as in 4th term a + 3^ -f 3f -f- d theannexedscheme; where 5th term a + 4/* -f 6c' \- Ad -\- c each term is composed of the first term, together with as many of the dif- ifiencHs as it is distant iVom tlu; first term, and to those diflerences joiriing. lor coefficients, , the num- bers in the sloping or ob- 1 hquc linosconlaincd in the annexed table of figurate 1 a umbers, in - the same &c. &c. 1 1 1 1 1 1 1 1 .1' 2 3 4 5 , 6 7 8 9 3 G 10 15 21 28 36 4 10 20 55 56 84 5 15 35 70 126 > 6 21 56 126 7 28 84 8 iG < TRACT 21. LOGARITHMS. 411 manner, he observes, 1st term a as the same figurate 2d term a h numbers complete the 3J term a 26 + c powers raised from a bi- 4th term a 3A + 3f d nomial root, as had lonf 5th term a 46 + 6c 4c? f e before been taught by &c. &c. others. He also re- marks, that this rule not only gives any one term, but also the sum of any number of successive terms from the begin- ning, making the 2d coefficient the first, the 3d the 2d, and so on; thus, the sum of the first 5 terms is Sa -f 106 4- 10c ^r od \- e. In the 4th /7/v>p. it is shown, that if the terms decrease, proceeding from the greater to the less, the same theorems hold good, bv only changing the sign of every other term, as in the margin. Prop. 5 shows how to find any multiple nearly of a given ratio. To do this, take the difference of the terms of the ratio, which multiply by the index of the multiple, from the product subtract the same difference ; add half the remain- der to the greater term of the ratio, and subtract the same lialf from the less term, which give two terms expressing the required multiple a little less than the truth Thus, to qua- druj)le the ratio |^ : the difference of the terms 3 multi[)lied by 4 makes 12, from which 3 deducted leaves 9, its half 4 i- added to the greater term 28 makes 324^, and taken from the less term 25, leave 20^ ; then 20|; and :i2l are the terms nearly of the quadruple sought, or reduced to whole num- bers gives -|i, a little less than the truth. Prop. 6 and 7 treat of the approximate multiplication and division of ratios, or, which is the same thing, the finding nearly anv powers or any roots of a given fraction, in an easy manner. The theorem for raisuig any power, v/hen reduced to a simpler form, is this, the m power ot , or (-^)'", i^=;q_ ^ nearly, v.here .? is = a -f b, and dzz a ^j b, the sum and dif- 412 CONSTRUCTION OF TRACT 21. ference of the two numbers, and the upper or under signs take place according as is a yjroper or an improper frac- tion, that is, according as a is less or-greater than b. And the tn.torextractiiigthe?;?th root oF , is ^ r or ()"' = -i_-, nearly ; which latter rule is also the same as the former, as will be evident by substituting: - instead of m in the first theorem. So that universally (-^)r, is='^^^^^ nearly. These theorems however are nearly true only in some certain cases, namelv when 4- and - do not diOlr greatly from unity. And in the 7th prop, the author shows how to find nearly the error of the tiieorems. In the 8th prop, it is shown, (hat the measures of ratios of equidifferent terms, are ncarlv reciprocally as the arithmeti- cal means between tlie terms of each ratio. 8o, of the ratios ^|j 3.3^ jo^ the mean between the terms of the first ratio is l7, of the 2d 34, of the 3(1 51, and the measure of the ratios are nearly as ^'., y^, -J-^. From this property he proceeds, in the 9 ih prop, to find the measure of any ratio less than t^I'.\, which has an equal dif- ference, 1, of terms. In the two examples, mentioned near the beginning, our author found t!ie logarithm, or measure of the ratio, of ^l'-^ , to l)e 21169 ^'c, and that of -j-'-%^ to be 21659/ff ; therefore the sum 434'29 is the logarithm of -p-gv^A, or ^1-0- X Toot; or the logarithm oH ^,X^ is nearer 43430, us found by other more accurate computations. Now to find the logarithm of |ot> liaving tite same difFerence of terms, 1, with the former; it will he, hv proj). 8, as lOO'o (the mean between 101 and 100): 100 (the nie^m between 99'5 and 100-.->) : : 43430 : 43213 the logarithm of |^, or the dilference be- tween tile logarithms ot 100 and ioi. But the log. of 100 is 2 ; therefore t!)e iogaritrmi of 101 is 2.0043213. Again, to find the logarithm of 102, \\ c mu.si first find the logarithm of 4-o4 > the mean between it^ terms being 101'3, therefore as }01'5 : 100 : : 43430 : 42788 the logarithm of -j" ^ , or the dif- TRACT 21. LOGARITHMS. 413 ference of the logarithms of 101 and 102. But the logarithru of 101 was found above to be 2.004'3213 ; therefore the log, of 102 is 2.0086001. So that, dividing continually 86859 (the double of 434298 the logarithm of tI-o-:| or 4-14) by each number of the series 201, 203, 205, 207, &c, then add 2 to the 1st quotient, to the sum add the 2d quotient, and so on, adding always the next quotient to the last sum, the several sums will be the respective logarithms of the numbers in this series 101, 102, 103, 104, &c. The next, or prop. 10, shows that, of two pair of conti- nued ratios, whose terms have equal differences, the difference of the measures of the first two ratios, is to the difference of the measures of the other two, as tiie square of the common term in the two latter, is to that in the former, nearly. Thus, .-i c , ^ a + ha + 3ba + ^b , ^ in tlie four ratios -. -, , -: as the measure or - ~ (the difference of the first two, or the quotient of the c \ 1 ~aa.+ ?inl'-\-\5hb , , ... two tractions 1 : is to the measure or ; - : : so (a + 4^) : is to {a-\-hy-^ nearly. In prop. 11 the author shows that similar properties take place among two sets of ratios consisting each of 3 or 4 &c continued numbers. Prop. 12 shows that, of the powers of numbers in arith- metical progression, the orders of diflerences which become equal, are t';e 2d differences in ti)e squares, the 3d differences in tiie cubes, tiie 4th utfterences in the 4th powers, &.c. And hence it is shown, how to construct all those powers bv the contniMal addition of their differences; as had been long before move luliy explained by Briggs. In the iii'xt, or lyth prop, our author explains his compen- dious metl)(xi of raismg the tables of logarithms; showing how to construcL tiie logaritbnis by addition only, from the pro- perties contained ni tlie 8th, 9th, and 12th props. For this purpose, he makes use of the quantity - , which by division he resolves into this innnite series -f - 4- -- -f- - ^c (in V Lu i.'3 t^ infin.). Putting then = iOO, the arithmetical mean between 41* CONSTRUCTION OF TRACT 21. the terms of the ratio t-^-otj ^ = 1 00000, and c successively equal to 0*5, i'5, 2'5. &c, that so b c may be respectively equal to 99999*5, 99998-5, 99997*5, &c, the corresponding means between the terms of the ratios r^lUh lllfl, Irl^, &c, it is evident that r will be the quotient of the 2d term divided by the 1st, in the proportions mentioned in the 8th and 9th propositions ; and when all of these quotients are found, it remains then only to multiply them by the constant 3d term 4S429, or rather 43429'8, of the proportion, to pro- duce the logarithms of the ratios ^lUU-, ^||f, |^\^ &c, till ||%T- ; then adding these continually to 4, the logarithm of 10000, the least number, or subtracting them from 5, the logarithm of the highest term 100000, there will result the logarithms of all the absolute numbers from 10000 to 100000. Now when c = 5, then r=-001,- =:-00000On05,- =-0()0000000000025, ='000()00000000000000125 l> tb L3 b* ft O. (IC oc^ Sec; therefore - = + + &c, is =-0O10O0005000O250001'25, b c b bb fc3 In like manner, if c = 1-5, then r ^^'H be =-001000015000225005375, bc and if c=2-5, then ^ will be =-OOlO0nO25O00625015625; b c &.C. But instead of constructing all the values of - in the ~ bc usual way of raising the powers, he directs them to be tound by addition only, as in the last proposition. Having thus found all the values of --^, the author then shows, that thev may be drawn into the constant loga- rithm 43429 by addition only, by the help of the annexed table of its first 9 products. The author thejn distinguishes which of the logarithms it may be proper to find in this way, and which from their component parts. Of these, the logarithms of all even numbers need not be thus computed, being composed from the number 2 ; which cuts off one-half of the numbers: neither are those numbers to be coni])utcd which end in 5, because 5 is one of thtir factors-. 1 43429 2 86858 3 130287 4 173716 5 217145 6 2fJ0574 7 301003 8 34~432 9 390861 TRACT 21. LOGARITHMS. 415 these last are ,% of the numbers ; and the two together -^ + tV make 4 of the whole : and of the other |, the ^ of them, or -j?7 of the whole, are composed of 3 ; and hence -| -\- ^, or 4-1 of the numbers, are made up of such as are composed of 2, 3, and 3. As to the other numbers which may be com- posed of 7, of 1 1 , &c; he recommends to find thei)' logarithms in the general way, the same as if they were incomposites, as it is not worth while to separate them in so easy a mode of calculation. So that of the 90 chiliads of numbers, from 10000 to 100000, only 24 chiliads are to be computed. Neither indeed are all of these to be calculated from the fore- going series for t , but only a few of them in that way, and the rest by the proportion in the 3th proposition. Thus, having computed the logarithms of 1000,'] and 10013, omit- ting 10023, as being divisible by 3, estimate the logarithms of 10033 and 10043, which are the 30th numbers from 10003 and 10013; and again omitting 10053, a nmltiple of 3, tlnd the logarithms of 10063 and 10073. Then by prop. 8, As 10048, the arithmetical mean between 10033 and 100G3, to 10018, the aritlimetical mean between 10003 and 10033, so 13006, the dif. between the logs, of J 0003 iind 10033, to 12967, the dif. between the logs, of 10033 and 10063. C129C7 That is, 1st, As -| 10078V : 10018 : : 13006 : < &c. C 12953 Attain, As ^ 10088 r : 10028 : : 12992 : < kc. 101183 t C 10068^ ri2940 And 3dly, As } 10098 > : 10033 : : 12979 : ^ &:c. And with this our author concludes his compendium for con- structing the tables of logarithms. He afterwards shows some applications and relations of the doctrine of logarithms to geometrical figures: in order to which, in prop. 14, he proves algebraically that, in the right- angled hyperbola, if from the vertex, and from any other 416 CONSTRUCT ION Of TRACT 21. point, there be drawn bi, fh perpendicular \ to the asymptote ah, or jiarallel to the other i!^^ , asymptote: then will ah : Ai : : bi : fh. And, \^}\'' hxprop. 15, it ai = bi= 1, and ui=a; then ___f\/\ will FH = - = i-a + a^-a^ + a*-a^ kc, ^1 /" in mfimfutn, by a continned algebraic divi- sion, the process of which he describes, step bv step, as a thing that was new or uncommon. But that method of divi- sion had been taught before, by Dr. Walhs in his Opus Arithmeticum. Prop. 16 is this: Anv given number being supposed to be divided into innumerable small equal parts, it is required to assign the sum of any powers of the continual sums of those innumerable [)arts. For which purpose he lays down this rule ; if the next higher power of the given number, above that power whose sum is sought, be divided by its exponent, the quotient will be the sum of the powers sought. That is, if N be the given number, and a one of its innumerable equal parts, then will a" + {2ay + (S^)" + (4r/)'' &c . . . . n" be = p^: which theorem he demonstrates by a method of induction. And this, it is evident, is the finding the sum of any powers of an infinite number of arithmeticals, of which the greatest term is a giv -n (juantity, and the least indefinitely small. It is also remarkable, th.it the above expression is similar to the rule for tinding the fluent to the given fluxion of a power, as afterwards taught by Sir I. Newton. Mercator then applies this rule, \n prop. 17, to the qua- drature of the iiyperbola. Thus, putting ai = i, conceive tliC asymptote to be divided from i into innumerable equaj parts, nau)ely ip = pq =z qr = a ; then, by the I4thand 15th 7;.s = 1 - + a^ a^ ^c \ qt = l- 2a f. 4^^ - 8^ &c f ^^^^ ^^^ ^'"^^ ^"'" ^^ = '^^^ tni^ 1 - Za \- 9a^ - 2la^ &c)'"'" ps + qt\-ru, which is =: 3 firt 4- 14a^ 3(ifl' &c, that is, equal to the number of terms contained in the line i?*, miuHs the sum of those TRACT 21. LOGARITHMS. 4n terms, plus the sum of the squares of the same, minus the sum of their cubes, plus the sum of the 4th powers, &c. Putting now lA = 1, as before, and ip = O'l the number of terms, to find the area Bips; by prop. 16 the sum of the terms will be = -005, the sum of their squares = -OOOSSSSSS, the sum of their cubes -000025, the sum of the 4th powers 000002, the sum of the 5th powers -000000166, the sum of the 6th powers '000000014, &c. Therefore the area Bips is = '1 005 + -000333333 - -000025 + -000002 - -000000166 + -000000014 &c = -100335347 -005025 166 = -0953 101 8 1 &c. Again, putting iq = "21 the number of terms, he finds in like manner the area Biqt = -21 - -02205 + -003087 - 000486202 + '000081682 - -000014294 + -000002572 - -000000472 + -OOOOOOOSS&c = -213171345- '022550984 = 190620361 &c. He then adds, hence it ajjpears that, as the ratio of ai to Ap, or 1 to 1-1, is half or subduplicate of the ratio of ai to Aq, or I to 1-21, so the area Bips is here found to be half of the area Biqt. These areas lie computes to 44 places of figures, and finds them still in the raiio of 2 to 1. The foregoing doctrine amounts to this, that if the rect- angle Bi X ir, which in this case is expressed by ir onl y, be put =: A, ai being = 1, as before; t!ien the area Bin^, or the hyperbolic logarithm of 1 -f a, or of the ratio of 1 to 1 + A, M'ill be equal to the infinite series a -^.a^ + I a^ i a* + iA^ kc\ and which therefore may be considered as Mercator's quadrature of the hyperbola, or his general expression of an hyj)C!bo!ic logarithm in an infinite series. And this method was further improved by Dr. Waliis in the Piiilos. Trans, for the year 1C68. In prop. 18 Mercator compares the hyperbolic areolce with the ratiuncuLc of equidifferent nmnbers, and observes that, the areola Bips is the measure of the ratiuncula of ai to a/>, the areola spqt is the measure of the ratiuncula of Ap to Aq, the areola tqru is the measure of the ratiun. of Aq to a?", &c. Finally, in the 19th prop, he shows how the sums of loga- rithms may be taken, after the manner of the sums of the VOL. I. E E Mdiidnta^^ 418 CONSTRUCTION OF TRACT 21. areola. And hence infers, us a coroMarv, how the continual proiliict of anv driven numbers in arithnietical progression may he obtained : for the sum of the logarithms is the lo. Trans, for 1( 68 are also given some further illustrations of this work, by the author himself. And in va- rious places also in a similar manner are logarithms and hv|)er- bolic areas treated of by Lord Brouncker, Dr. Wallis, Sir I, Newton, and many other learned j)ersons. Of Gregorys Exercitationes Geomelrkre. In the same year 1668 came out Mr. James Gregory's Exercitationes Geometrical, in which are contained the fol- lowing pieces: 1, Appendicula ad veram circuli et hyperbola? quadra- turam : 2, N. Mercatoris quadratura hyperboltc geometrice de- monstrata : 3, Analf^gia inter lineam meridianam planisphoerii nautici et tangentes artificiales geometrice demonstrata ; sen quod sccantium naturaliuin additio efficiat tangentes artificiales : 4, Item, quot tangeutium naturaliuni additio efficiat secan- tes artifici-ues : 5, Quadratura conchoidis : 6, Quadratura cissoidis : et 7, Methodns facilis et accurata componendi secantes ct tangepi ' n arti*''cia]es. The lir^t of tlie.se pieces, or the Appendicula, contains some f'.irrlier t'.\t(nision and illustration of his Vera circuli et hy- perixii.:- quadratura, occasioned by the animadversions made on tiiai work by the celebrated mathematician and piiiieso- ^)\uv llnygens. In the 'jd is demonstrated geometrically, tlie quadrature of TRACT 21. LOGARITHMS. 419 the hyperbola ; by which he finds a series similar to |^erca- tor's for the logarithm, or the hyperboUc space beyond th first ordinate (bi, fig. pa. 416). In like manner he finds an- other series for the space at an equal distance within that or- dinate. These two series having all their terms alike, but all the signs of the one plus, and those of the other alternately plus and minus, by adding the two together, every other term is cancelled, and the double of the rest denotes the sum ol both spaces. Gregory then applies these properties to the logarithms ; the conclusion from all which may be thus briefly expressed : since a iA* -|- ^a^ -^a.* &g, = the log. of -i^, and A + 4a^ -{- -l-A^ -{- iA* &c = the log. of ^-1-, theref. 2a + Ia^ -f- ^a^ -f- fA^ &c rz the log. of Ji^, or of the ratio of 1 A to 1 + A. Which may be accounted Gregory's method of making logarithms. The remainder of this little volume is chiefly employed about the nautical meridian, and the logarithmic tangents and secants. It does not appear by whom, nor by what accident, was discovered the analogy between a scale of logarithmic tangents and Wright's protraction of the nautical meridian line, which consisted of the sums of the secants. It appears however to have been first published, and introduced into the practice of navigation, by Henry Bond, who mentions this property in an edition of Norwood's Epitome of Navigation, printed about 1645 ; and he again treats of it more fully in an edition of Gunter's works, printed in 1653, where he teaches, from this property, to resolve all the cases of Mer- cator's sailing by the logarithmic tangents, independent of the table of meridional parts. This analogy had only been found to be nearly true by trials, but not demonstrated to be a mathematical property. Such demonstration seems to have been first discovered by Nicholas Mercator, who, desirous of making the most advantage of this and another concealed in- vention of his in navigation, by a paper in the Philos. Trans. E E 2 420 CONSTRUCTION OT TRACT 21. for Jime4, 1666, invites the public to enter into a wager with him, on his ability to prove the truth or falsehood of the sup- posed analogy. This mercenary proposal however seems not to have been taken up by any one, and Mercator reserved his demonstration. The proposal however excited the attention of mathematicians to the subject itself, and a demonstration was not long wanting. The first was published about two years after by Gregory, in the tract now under consideration, and from thence and otlier similar properties, here demon- strated, he shows, in the last article, how the tables of loga- rithmic tangents and secants may easily be computed, from the natural tangents and secants. The substance of which is as follows : Let Ai be the arc of a quadrant, extended in a right line, and let the figure ahi be composed of the natural tangents of every arc from the point A, erected perpendicular to AI at their respective points: let AP, po, ON, NM, &c, be the very small equal parts into which the quadrant is divided, namely, each -^, or -j^^ of ^ degree ; draw pb, oc, nd, me, &c, perpendicular to ai. Then it is manifest, from what had been demonstrated, that the figures abp, acq, &c, are the artificial secants of tlie arcs ap, ao, &.c, putting o for the artificial radius. It is also manifest, that the rectangles bo, CN, DM, &,c, will be found from the multiplication of the small part AP of the quadrant by each natural tangent. But, he [)roceeds, there is a little more difficulty in measuring the figures abp, bcx, cdv, ^c; for if the first difTcrences of the tancrents be equal, ab, bc, cd, &c, will not difl'er irom rigiit lines, and then the figures abp, bcx, cdv, &c, Avill b(; right- angled triangles, and therefore any one, as hqg, will b(' :=. i^n X QG : but if the second dilTcrences be equal, the said figures will be portions of trilineal (juadratices; for exan)ple, HOG will be a [portion of a trilincal quadratix, whose axis is parallel to qh ; ard each of the last dilTercnces being z, it will K L M JS Tract 21. logarithms. 42i he QHG =4H X QG ^-L X QG ; and if the 3d differences be equal, the said figures will be portions of trilineal cubices, and then shall aHG be equal -I-qh x aG^iy-^au x zx qg^- ttVf^^ ^ ^ct^) when the 4th differences are equal, the said figures are portions of trilineal quadrato-quadratices, and the 4th differences are equal to 24 times the 4th po\\^r of ae, divided by the cube of the latus rectum ; also when the 5th differences are equal, the said figures are portions of trihneal sursolids, and the 5th differences are equal to 120 times the sursolid of qg, divided by the 4th power of the latus rectum ; and so on m infinitU7n. What has been here said of the com- position of artificial secants from the natural tangents, it is remarked, may in like manner be imderstood of the compo- sition of artificial tangents, from the natural secants, accord- ing to what was before demonstrated. It is also observed, that the artificial tangents and secants are computed, as above, on the supposition that is the log. of 1, and lOOOOOOOOOOOO the radius, and 2302585092994045624017870 the log. of 10; but that they may be more easily computed, namely by ad- dition onlv^, by putting -^V of a degree =qg=AP=1, and the logarithm of 10 = 7915704467897819; for by this means 4qh X qg is = iQH = ghg, and |^qh x qg -j-Vz x qg = ^qh -lijZ = QHG, also |QH X QG - v^ (tV<1H X Z X QG' - TraT^* X QG^ = -JQH- v^(yV set down of its products b}^ the first 9 figures. But if ap or qg be ^^ of a degree, the artificial tangents and secants will ans ver to 13192840779829703 as the logarithm of 10, the first 9 mul- tiples of which are also placed in the table. But to represent the numbers by the artificial radius, rather than by the loga- rithm of 10, the author directs to add ciphers, &c. And so iuch for Gregory's Exercitationes Geometricae. 422 CONSTRUCTION OF TRACT 21. The same analogy between the logarithmic tangents and the meridian line, as also other similar properties, were after- wards more elegantly demonstrated by Dr. Halley in the Philos. Trans, for Feb. 1696, and various methods given for computing the same, by examining the nature of the spirals ito which the rhumbs are transformed in the stereographical projection of the sphere, on the plane of the equator : the doctrine of which was rendered still more easy and elegant by the ingenious Mr. Cotes, in his Logometria, first printed in the Philos. Trans, for 1714, and afterwards in the collec- tion of his works published in 1732, by his cousin Dr. Robert Smith, who succeeded him in the Plumian professorship of philosophy in the University of Cambridge. The learned Dr. Isaac Barrow also, in his Lectiones Geo- metrical, lect. xi. Append, first j)ublished in 1G72, delivers a similar property, namely, that the sum of all the secants of any arc is analogous to the logarithm of the ratio of r + -s to r 5, or radius plus sine to radius minus sine; or, which is the same thing, that the meridional parts answering to any degree of latitude, are as the logarithms of the ratios of the versed sines of the distances from the two poles. Mr. Gregory's method for making logarithms was further exemplified in numbers, in a small tract on this subject, printed in 1688, by one Euclid Speidell, a simple and illiterate person, and son of John Speidell, before mentioned among the first writers on logaritlims. Gregory also invented many other infinite series, and among them these here following, viz. a being an arc, t its tangent, iind s the secant, to the radius r ; then is I zz a -\ \- 4- -A &LC 3/- ' iDj't ^ 313r ' 2835r* . a'-' , 5a'' , 61a 277fl8 ' = '^~^-Tr + 24r3 + ^i^ +' sl].^ ^^- And if r and o- denote the artificial or logarithmic tangent ajid secant ot the same arc a, the whole quadrant being y, and ~~2r "*" T273 "^ 4J75 ' 2520^ "^ 28330? And if s denote the artificial secant of 45, and s + /the arti- ficial secant of any arc a, the artificial radius being 0; then is , , , /^ , 4/3 7Z4 , 14P 452^0 3 a = -iq 4- I ~ 4 &c. ^^ ^ r ^ 3r 3r3 ^ 3/4 ' 43/=' The investigation of all which series may be seen at pa. 298 et seq. vol. 1, Dr. Horsley's commentary on Sir I. Newton's works, as they were given in the Commercium Epistolicum, no. XX, without demonstriition, and where the number 2 is also wanting in the denominator of the first term of the series expressing the vahie o\' a-. Such then were the ways in which Mercator and Gregory applied these their very simple series a ^A-" -{- yA^ |a-* &c, and A-l-^A" + iA''4-! A^&c,for tlie purpose ot computing loga- rithms. But they might, as I apprehend, have applied them to this purpose in a shorter and more direct maimer, by com- puting, by their means, only a few logarithms of small ratios, \u which the terms of the series would have decreased bv the- powers of 10, or some greater number, the imm-:irators of all the terms being unity, and tiieir denominators the powers of 10 or some greater number, and t!icn employing these few logarithms, so computed, to the finding the loorarithms of other and greater ratios, by the easy operations of mere ad- dition and subtraction. This might have been done for the logarithms of the ratios of the first ten numbers, 2, 3, 4, 5, 6,7, 8, 9, 10, and 1 I, to I, in the following manner, com- municated bv iIr. Baron Maseres. In the first place, t'le logarithm of the ratio of 10 to 9, or of 1 to J'^, or of I to 1 xV J i'' cq^iil to the series _! I _J . __! I i I 1 s-c '^ "^ 2x100 ^3x1000 ' 4x10000 ' 3x100000 ' In like manner are easily found the logarithms of the ratios of 11 to 10; and then, by the same series, those of 121 to 120, and of 81 to 80, and of 2401 to 2400; in all which cases 424 CONSTRUCTION OF TRACT 21. the series would convero:e still faster than in the two fii^st cases. We may then proceed by mere addition and subtrac- tion of logarithms, as follows ; Log.y = L. U 4-L. I L. W = 21.. J r L. W = L. 121 I.T 8 1 T I 2 o = L. 121 _T To- ^' I 2 I TVoJ T 1 20 = L. 3 "i) T 9 = 2L. J r ) L. lo _ T V + L. [.. II =2L. L. if = L. n-L- 8 I 5-0 L. 4 = L. S L. 1 = L. I o L. = L. - L. Having thus got the logarithm of the ratio of 2 to 1, or, in common language, the logarithm of 2, the logarithms of all sorts of even numbers mav be derived from those of the odd numbers, which are their coefficients, with 2 or its ])owers. We may then proceed as follows : 2L. 2, f L. 4, 9 = L. I- + L. 4, 10 = L. V L. L. L. L. 3 =iL. 9, L. 100 =2L. 10, L. 8 =3L. 2, L. 24 = L. 8 + L. 3, L.2400 = L. 100 + L. 24, L.2401 = L. |A^+ L.2490, L. 7 =:^L. 2401, L. 11 = L. V 4 L. 9, L. 6 = L. 2 + L. 3. Thus we have got the logarithms of 2, 3, 4, 5, 6, 7, 8, 9, 10, and 11. And this is, upon the whole, perhaps t[)e best me- thod of computing logarithms that can be taken. There have been indeed some methods discovered by Dr. Halley, and other mathematicians, for computing tite logarithms of the ratios of j)rime numbers, to the next adjacent even numbers, wiiicli are still shorter than the application of the ibregoing series. But those methods iwe less simj:)Ic and easy to under- stand, and applv, than these series ; and tlie computation of logarithms by these scries, when their terms decrease by the powers of 10, or of some greater number, is so very short and easy (as we have seen in the foregoing computati(jns of the logiirithms of the ratios of 10 to 9, 11 to 10, 81 to 80, 121 to 120, ,^c,) tliat it is not worth wdiile to seek for anv shorter methods of com]-)uting them. Auf! this method of TRACT 21. LOGARITHMS. 425' computing logarithms is very nearly the same with that of Sir I. Newton, in his second letter to Mr. Oldenburg, dated October 1676, as will be seen in the following article. 0/ Sir Isaac Newton's Methods. The excellent Sir I. Newton greatly improved the quadra- ture of the hyperbolical-asymptotic spaces by infinite series, derived from the general quadrature of curves by his method effluxions; or rather indeed he invented that method him- self, and the construction of logarithms derived from it, in the year 1665 or 1666, before the publication of either Mcrcator's or Gregory's books, as appears by his letter to Mr. Olden- burg dated October 24, 1676, printed in p. 634 etseq. vol. 3, of Wallis's works, and elsewhere. The quadrature of the hyperbola, thence trans- lated, is to this effect. Let dvD be an hy- perbola, ^vhose centre is c, vertex f, and interposed square cafe= 1. In CA take ab and Ab on each side =. tV or 0*1 : And, ^ 1} A. erecting the perpendiculars bd, bd; half the sum of the spaces r , .,,, ^ , OdOl 00001 , 0-0000001 AD and Ad will be = 0-1 -\ ~ \- (- &c. 1 L . ir irr 0-01 0-0001 0-000001 O'OOOOOOOl and the half diff. 1- 7 -1 ^ -l z &c. 2 4 o 8 Which reduced will stand thus, l-()0O00000000OO,00O500000O000O The sum of these 0-1033605156577 is Af/, 3333333333 250000000 and the differ. 0-0953101798043 is AD. 20000000 16666H6 In like manner, putting AB and A.b 142857 12500 each = 0-2, there is obtained nil 100 Arf = 0-2231435513142, and 9 1 AD = 0-1823215567939. 0- 1 0033534773 1 0,0 005025 1 679267 Having thus the hyperbolic logarithms of the four decimal numbers 0-8, 0-9, 11, and 1-2; and since -^ X 2, and 0'8 and 0*9 are less than unity ; adding their logarithms to double the logarithm of V2, we have 0-6931471805597, the hyperbolic logarithm of 2. To the triple of this adding the log. of 0-8, because ^4rr-= ^0, we have 2-3025850929933, 26 CONSTRUCTION OF TRACT 21^ tlie logarithm of 10. Hence by one addition are found the logarithms of 9 and 1 1 : And thus the logarithms of all these prime numbers, 2, 3, 5, 11 are prepared. Further, by only depressing the numbers, above computed, lower in the deci- mal places, and adding, are obtained the logarithms of the decimals 0-98, 0-99, I'Ol, r02; as also of these 0-998, 0'999, rOOl, 1002. And hence, by addition and subtraction, will arise the logarithms of the primes 7, 13, 17, 37, &c. All which logarithms being divided by the above logarithm of 10, give the common logal-itinns to be inserted in tlie table. And again, a few pages further on, in the same letter, he resumes the construction of logaritlims, thus: Having found, as above, the hyperbolic logarithms of 10, 0'98, 0*99, TOl, 1"02, whicli may be etfected ui an hour or two, dividing the last four logaritlmis by the logarithm of 10, and addmg the index 2, wc have the tabular logarithms of 98, 99, 100, 101, 102. Then, by interpolating nine means between each of these, will be obtained the logarithms of all numbers between 980 and 1 020 ; and ajjain interpolating 9 means between every two numbers from 980 to 1000, the table, will be so far con- structed. Then iVom these will be collected the logarithms of all the primes under 100, together with those of their mul- tiples: ail wliich will require only addition and subtraction; for 988 9936 986 992 999 ^84 47IB-^^' T7x-27=23; ^,=29; - = 31; ^^37; -- =41 ^^3 4o ^'^^ A^ -'^11 ro 9971 _ 9882 ,,, 9849 -=43; ~ = 4,'; ^^_=,53;^^. = .9;^^- =61 :j^-^= 67 994 ^ 9928 ^ 9954 ^^ 996 9968 ^^ 989 f Tr= '^'s-7r7=^3; .^^^=.9; - :=.S;3; ^^^ = 39: -, = 97. This quadrature of the h3q)erbola, and its application to the construction of logarithms, arc still further explained by our celeurated author, in his treatise on Fluxions, published by Mr.Colson in 1736, where he gives all the three scries for the areas ad, At/, B(/, in general terms, tlie former the same as that pi,l)ii-,|,cd by Mercator, and the latter by Gregory; and he exj)lains the nu-auier of deriving the latter series from the former, namely by uniting together the two series lor the TRACT 21. LOGARITHMS. 421 spaces on each side of an ordinate, bounded by other ordi- nat;es at equal distances, every 2d term of each series is can- celled, and the result is a series converging much quicker than either of the former. And, in this treatise on fluxions, as well as in the letter before quoted, he recommends this as the most convenient way of raising a canon of logs, computing by the series the hyperbolic spaces answering to the prime numbers 2, 3, 5, 7, 1 1 , &c, and dividing them by 2-3025850929940457, which is the area corresponding to the number 10, or else multiplying them by its reciprocal 0-4342944319032518, for the comaion logarithms. " Then the logarithms of all the numbers in the canon which are made by the multiplication of these, are to be found by the addition of their logarithms, as is usual. And the void places are to be iiiterpolated after- wards by the help of this theorem : Let ji be a number to which a logarithm is to be adapted, x the difference between that and tlie two nearest numbers equally distant on each side, whose logarithms are already found, and \etd be half the dif- ference of the logarithms; then the required logarithm of the number n will be obtained bv adding erbolic areas, and then proceeds thus ; " The two first terms d -\- o^ this series 1 think to be ac- curate enough for the construction of a canon of logarithms, even though they were to be produced to 14 or 15 figures; provided the number whose logarithm is to be found be not less than 1000. And this can oive little trouble in the calcu- lation, because x is generally an unit, or the number 2. Yet it is not necessary to interpolate all the places by the help of this rule. For the logarithms of numbers which are produced by the multiplication or division of the number last found, maybe obtained by tiie numbers wliose logarithms were had before, by the addition or subtraction of their logarithms. Moreover, by the differences of the logarithms, and by their 2d and 3d differences, if there be occasion, the void places maybe more expeditiously supplied; the foregoing rule being to be applied only when the continuation of some full places 428 CONSTRUCTION Of TRACT 21- is wanted, in order to obtain those differences, &c." So that Sir I. Newton of himself discovered all the series for the above quadrature, which were found out, and afterwards published, partly by Mercator and partly by Gregoiy ; and these we may here exhibit in one view all together, and that in a general manner for any hyperbola, namely putting CA c, ap = 6, and AB = Ai^ = .r : then will bd = -^, and bd = : whence the aieas are as below, viz. , bx-i bti bx* , bi^ 2a ' 3u* 4ft3 5(1* J L , **' . bx'i , bx* , bx^ 5 2rt 3a 4a3 5a* Bd 2bjt; 4 -i &c. In the same letter also, above quoted, to Mr. Oldenburg, our illustrious author teaches a method of constructing the trigonometrical canon of sines, by an easier method of mul- tiple angles than that before delivered by Briggs, forthesame purpose, because that in Sir Isaac's way radius or 1 is tlie first term, and double the sine or cosine of the first given angle is the 2d term, of all the proportions, by which the several suc- cessive multiple sines or cosines are found. The substance of the method is thus : The best foundation for the construc- tion of the table of sines, is the continual addition of a given angle to itself, or to another given angle. As, if the angle a be to be added ; M_^^--f L V NY r inscribe hi, ik, kl, lm, mn, no, op, &c, each equal to the radius ab; and to the opposite sides draw the perpendiculars BE, HQ, IR, KS, LT, Mv, NX, OY, &c ; SO shall the angle a be the common difference of the angles Hici, IKH, KLi, lmk, &c; their sines Ha, ir, ks, &c ; and their cosines la, kr, ls, &c. Now let an}' one of them lmk, be given, then the icst will be thus found: Draw xa and jub perpendicular to sv and mt; TRACT 21. LOGARITHMS. 429 now because of the equiangular triangles abe, tla, km&, ALT, AMV, &c, it will be AB : AE : : KT : sa (=iLV + 4-Ls) : : LT : Ta 4=iMV + 4;KS,) and ab : be : : lt : Lfl (=4^LS- iLv) : : KT {=4km) : ^m^ (=^mv ^ks.) Hence are given the sines and cosines ks, mv, ls, lv. And the method of conti- nuing the progressions is evident. Namely, < LV : MT + lyix : : MX : NY + NY &c, as AB : 2AE : : -| ^^^ . ^^ ^ ^^ . . ^^ : OY + MV &c, ( LV : NX LT : : MX : OY MV &c, or AB : 2BE : t | ^V : MT - MX : : NX : NV - NY &c. And, on the other hand, ab : 2ae : : ls : kt + KR &c. Therefore put ab = 1, and make be x LT = La, aex KT = s in the year 17 19, at the Royal Observatory at Green- Avicii, where he died the 14th January 1742, in the 8Cth year TRACT 21. LOGARITHMS. 4-31 of his age. His method was first printed in the Philosophical Transactions for the j^ear 1695, and it is entitled " A most compendious and facile method for constructing the loo-a- rithms, exemplified and demonstrated from the nature of jiumbers, without any regard to the hyperbola, with a speedy method for finding the number from the given logarithm." Instead of the more ordinary definition of logarithms, as iiumeroruvi proportionalium (equidifferentes comites, in this tract our learned author adopts this other, numeri ralionem exponentes, as being better adapted to the principle on which logarithms are here constructed, where those quantities are not considered as the logarithms of the numbers, for example, of 2, or of 3, or of 10, but as the logarithms of the ratios of 1 to 2, or 1 to 3, or 1 to 10. In this consideration he first pursues the idea of Kepler and Mercator, remarking that any such ratio is proportional to, and is measured by, the number of equal ratiunculae contained in ( dcli ; which ratiunculse are to be understood as in a continued scale of proportionals, in- finite in number, between the two terms of the ratio ; which infinite number of mean proportionals, is to that infinite number of the like and equal ratiunculge between any other two terms, as the logarithm of the one ratio, is to the loga- rithm of the other : thus, if there be supposed between 1 and 10 an infinite scale of mean proportionals, whose number is 100000 &c in infinitum ; then between 1 and 2 there will be 30102 (*^c of such proportionals; and between 1 and 3 there will be 47712 &c of them; which numbers therefore are t!ic logarithms of the ratios of 1 to 10, 1 to 2, and 1 to 3. But for the sake of Aw mode of constructing logarithms, he changes this idea of equal ratiuncula;, for that of other ratiuncula?, so constituted, as that the same infinite number of them shall be contained in the ratio of 1 to every other number whatever ; and that therefore these latter ratiunculs will be o^ unequal or different magnitudes in all the difi'erent ratios, and in such sort, that in any one ratio, the magnitude of e.xh of the ra- tiunculoe in this latter case, will be as the number of them in the former. And therefore, if between 1 and any number 452 CONSTRUCTION OF TRACT 21. proposed, there be taken any infinity of mean proportionals, the infinitely small augment or decrement of the first of those means from the first term 1, will be a ratiuncula of the ratio of 1 to the said number ; and as the number of all the ratiun- culse in these continued proportionals is the same, their sum, or the whole ratio, will be directly proportional to the mag- nitude of one of the said ratiunculse in each ratio. But it is also evident that the first of any number of means, between 1 and any number, is always equal to such root of that num- ber, whose index is expressed by the number of those pro- portionals from 1 : so, if w denote the number of proportionals from 1, then the first term after 1 will be the jth root of that number. Hence, the indefinite root of any number being ex- tracted, the differentiola of the said root from unity, shall be as the logarithm of that number. So if tlicrc be required the log. of the ratio of 1 to 1 -f y the first term after 1 will be (1 + y)""? '^nd thcref. the retjuired log. will be as (1 +q)'" 1. But, (1 f (/)'" IS rr 1 -^ n \ .-'7 F - . -r -'i; s^ &c ; or by omitting the 1 in the com[joui)d nunjerators, as infinitely small in respect of the infinite number 7n, the same scries will become 1 -\ a + - . q^ 4- -. . a^ m'- m 2m ' vi 2m 3m ' &c, or by abbreviation it is 1 -] be supposed divided into two parts, namely, into the ratio of ato \a-{- ^b or \z, and the ratio of iz to b, then will the sum of the logarithms of those two ratios be the logarithm of the ratio of a to b. Now by substituting in the foregoing series, the logarithms of those two ratios will be into 1 1 &c. and - into ~ - \- + &c: and hence the sum, 1 . ^ 2x , 2-i3 2r5 2t7 2j.' , ^ -^ ^"^ T + 5? + ^3 + ^7 + "5? + &^> will be the logarithm of the ratio of a to b. Further, if from the logarithm of the ratio of a to ^z, be taken that of iz to b, we shall have the logarithm of the ratio of ab to iz- ; and the half of this gives that of <^ab to ^z, of of the geometrical mean to the arithmetical mean. And con- sequently the logarithm of this ratio will be equal to half the difference of that of the above two ratios, and will therefore be -^ into _ + _ + _ + - + &c. The above series are similar to some that were before given by Newton and Gregory, for the same purpose, deduced from the consideration of the hyperbola. But the rule Avhich is properly our author's own, is that which follows, and is de- rived from the series above given for the logarithm of the sum of two ratios. For the ratio oiab to iz^ or ia^+ia^-f-x^^ having the difference of its terms -^^-^ab -f- ^b'^ or {ib j^zf or ^x^, which in the case of finding the logs, of prime num- bers is always 1, if we call the sum of the terms iz* + a^=j/% VOL. I. F F 434' CONSTRUCTION OF TRACT 21. t the loj^. of tliera. of ^/ab to \a-\-i^b or iz will be found to be - into h T7 + ~.o + r^. + rrrs + &c. And these r:ilcs oir learned author excniplifies by sonne cases in numbers, to show the easiest mode of application in practice. Again, by means of the same binomial theorem he resolves, with equal facility, tlie reverse of the problem, namely, from the log. given, to (ind its number or ratio: For, as the log, of the ratio of 1 to 1 -f ^ was proved to be (1 + j)"* 1, and X that of the ratio of 1 to 1 - ^ to be 1 - ( I y)"" ; hence, calling the given logarithm l, in the former case it will be (1 -{^ q)'" z^ 1 + l, and in the latter (1 q)'" r= 1 l ; and therefore 1 + y = (1 + l)"7 that is,, by the binomial and 1 ^ r= (1 l)"' 5 theorem, 1 4- 7 =: 1 4- ? L 4 Inf l" + Im' l^ + -^Vk'^ l'^ + tto"^^ l^ &c, and 1 -7 r= 1 ?;; L 4 Inr ir i??i ' l^ -f ^.V^^ ^"^ ~ t' o'"^ ^' &c, wi being any inhnite index whatever, differing according to the scale of logarithms, being JOOO^c in Napier's or the hy- perbolic logarithms, and 230258J&C in Briggs's. If one term of the ratio, of wiiich l is the logarithm, be given, the other term will be easily obtained by the same rule : For if l be Napier's logarithm, of the ratio of a the less term, to b the greater, then, according as a or b is given, we shall have, b = a into ] + l + -^l^ 4- -II^ 4- ..'-l'* -J- &c, a= b into 1 - l 4- -M/ ^L^ 4" zV^^ - <^^'-"- Hence, by help of the logarithms contained in the tables, may easilybe found the number to anviiiven I02:. to a ureat extent. For if the small diflc.rence between the given log, l and thu nearest tabvdar logiiiithm, cither greater or less, be called /, and tlie number ansut-riiig to the tabular logarithm L* Alcrt , because - = 7ldZX : 1 H :x n ' or loof, n ' nix = 0- -loi ndzx AISOj f n theref. loGf. o n + x -1 + .r3 + X* 4n* &C, n . ^ 1 .r'^ .l3 X* 0.^ and loe. = + - + TT^, + ir-, f , n nx n 2ir on' %m And by adding and subtracting any of these series, to or from one another, and multiplying or dividing their corre- sponding numbers, various other series for logarithms may be found, converging much quicker than these do. In like manner, by assuming quantities otherwise com- pounded, for the value of n, various other forms of logarith- mic series may be found by the same means. Of Mr. Cotes' s Logomdria. INIr. Roger Cotes was el^ed the first Plumian professor of :sfrouoniy and experiraerml philosophy in the university of 4S8^ CONSTRUCTION OF TRACT 21. Cambridge, January 1706, which appointment lie filled with the greatest credit, till he died the 5th of Jmie 1716, in the prime of life, having not quite completed the 34th year of his age. His early death was a great loss to the mathemati- cal world, us his genius and abilities were of the brightest order, as is manifest by the specimens of his performance given to the public. Among these is, his Logometria, first printed in number 338 of the Philosophical Transactions, ancl afterwards in his Harmonia !\Iensurarum, published in 1722, with his other works, by his relation and successor, in the Plumian professorship, Dr. Robert Smith. In this piece he fir^t treats, in a general way, of measures of ratios, which measures, he observes, are quantities of any kind, whose mag- nitudes arc analogous lo the magnitudes of the ratios, these magnitudes mutually increasing and decreasing together in the same proportion. He remarks, that the ratio of equality has no magnitude, because it produces no change by adding and subtracting ; that the ratios of greater and less inequality, are of difti^rent affections; and therefore if the measure of the one of these be considered as positive, that of the other will be negative ; and the measure of the ratio of equality nothing : That there are endless systems of these, which have all their measures of the same ratios proportional to certain given quantities, called moduli, which lie defines afterwards, and the ratio of which they are the measures, each in its peculiar sys- tem, is called the modular ratio, ratio modularis, which ratio is the same in all systems. He then adverts to logarithms, wliich lie considers as the numerical measures of ratios, and he describes the method of arranging tiiem in tables, with their uses in multiplication and division, raising of powers and extracting of roots, by means of the corresponding operations of additioii and subtraction, multiplication and division. After this introduction, wiiich is only a slight abridgment of the doctrine Jong before very amply treated of by others, and particularly by Kepler and Mercator, Ave arrive at the first proi.'Osition, which has justly been censured as obscure and impeifect, seemingly throu^'an affectation of brevityj TRACT 21. LOGARITHMS. -liSd intricacy, and originality, without sufficient room for a dis- play of tJiis quality. The reasoning in this proposition, such as it is, seems to be something between that of Kepler and the principles of fluxions, to which the quantities and expressions are nearly allied. However, as it is my duty rather to nar- rate than explain. I shall here exhibit it exactly as it stands. This proposition is, to determine the measure of any ratio, as for instance that of ac to ab, and Avhich is eft'ected in this manner: Conceive the differ- ence BC to be divided into . : r A li P fei. C innumerable very small par- ticles, as PQ, and the ratio between ac and ab into as many such very small ratios, as between Aa and ap: then if the magnitude of the ratio between aq and ap be given, by divid- ing, there will also be given that of va to ap ; and therefore, this being given, the magnitude of the ratio between aq and ap may be expounded by the given quantity ; for, ap re- maining constant, conceive the particle pq to be augmented or diminished in any proportion, and in the same proportion will the magnitude of the ratio between aq and ap be aug- mented or diminished : Also, taking any determinate quan- tity M, the same may be expounded bj- m x ; and therefore the quantity M X will be the measure of the ratio between Aci and AP. And this measure will have divers magnitudes, and be accommodated to divers systems, according to the divers magnitudes of the assumed quantity m, which therefore is called the modulus of the system. Now, like as the sum of all the ratios aq to ap is equal to the proposed ratio AC to ab, so the sum of all the measures m x , found by the known methods, will be equal to the required measure of the said proposed ratio. The general solution being thus dispatched, from the ge- neral expression, Cotes next deduces other forms of the measure, in several corollaries and scholia; as 1st, the terms AP, aq, approach the nearer to equality as the small differ- F; 44 CONSTRUCTION OF TRACT 21. ence pq is less ; so tliat either m x or m x - will be the AP AQ measure of the ratio between Aa and ap, to the modulus m. 2d, That henee the modulus m, is to the measure of the ratio between aq and ap, as either ap or aq is to their dif- ference PQ. 3d, The ratio between ac and ab being given, the sum of all the will be civcn ; and the sum of all the m X XP O AP is as M : therefore the measure of any given ratio, is as the modulus of the sj'stem from which it is taken. 4th, There- fore, in every system of measures, the modulus will always be equal to the measure of a certain determinate and immut- able ratio; which therefore he calls the modular ratio. 5th, To illustrate the solution by an example : let z be any deter- minate and permanent (juantity, .r a variable or indeterminate quantity, and x its fluxion; then, to find the measure of the ratio between z-\- x and z .v, put this ratio equal to the ratio between ?/ and 1, expounding the number j/ b}' ap, its fluxion j by pq, and 1 by ab : then the fluxion of the re- quired measure of tha ratio between y and 1 is m x . y Now, for 7/, restore its val. ^ ^, and for j the flux, of that val. -, ^,1 SO shall the flux, of the measure become 2m x "-^^-^ ('-^> _ _ _ z-aa' or 2m into -- + '- -f ^ + &c ; and therefore that measure will be 2m into ^- -f ^-j-&.c. In like manner the measure of the ratio between 1 -{ v and 1, wall be found to be - - - _ M into x; \v- -f .]t;3 ^t;'^ -}- &c. And hence, to And the number from the logarithm given, he reverts the series in this manner : If the last measure be called ?/;, we shall have -'" or q = x) -\v + \v' ^v" + ix;^ &c, therefore q'= - v^ tj^ \-Wv'' ^v^ hc^ and hicli being called 711, and = q, that ratio will be the ratio of 1 to I Q + 4G' ^a' + -^QC Sec. And hence, takinj; ?7Z = M, or Q 1, the said modular ratio will also be tiie ratio ofltol 4- + T i + ^ TTo &c. And the former of these expressions, for the modular ratio, comes out the ratio of 2.718281828459 8cc to 1, and the latter the ratio of 1 to 0.367879441171 &c, which number is the reciprocal of the former. In the 2d prop, the learned author gives directions for con- structing Briggs's canon of logarithms, namely, first bv the general series 2m into -t ^ -{- \-i -{ &.c, finding the loga- rithms of a few such ratios as that of 12G to 125, 225 to 224, 2401 to 2400, 4375 to 4374, Sec, from which the logarithm of 10 will be found to be 2.30-2585092994 Sec, when m is 1 ; but since Briggs's log. of 10 is 1, therefore as 2.302585 &c is to the mod. l,soisl (Briggs's log. of 10) to0.4342S4i8l903&c, Avhich therefore is the modulus of Briggs's logarithms. Hence he deduces the logarithms of 7, 5, 3, and 2. Iti like manner are the logarithms of other prime numbers to be found, and from them the logarithms of composite numbers by audition and subtraction only. Cotes then remarks, that the first term of the general series 2m into^ + ^ + "^ + ^^i ^""'^^^ '^^ sufficient for the loga- rithms of intermediate numbers between those in the table, or even for numbers beyond the limits of the table. Thus, to 443 consthuction of tract ll. find the logarithm answerinj^ to any intermediate number ; jet a and e be two numbers, tlie one the given number, and the other the nearest tal)ular number, a being the greater, ^nd c the less of them ; put z =. a \- e their sum, x = a e their dilJerencc, A rz the logarithm of the ratio of a to e, that is the excess of the logarithm of a above that of e: so shall the said dificrence of their logarithms be A zr 2m x very nearly. And, if there be required the number answer- ing to any given intermediate logarithm, because A is = 9m 2mi 2Mr ^ x^ y.e , = or , there!, x = or very nearly. z 'Zax 2c .fx M + ^x M 5X -^ " In the 3d prop, the ingenious author teaches how to convert the canon of logarithms into logarithms of any other system, by means of their moduli. And, in several more propositions, he exemplifies the canon of logarithms in the solution of va- rious important problems in geometry and physics ; such as the quadrature of tlie hyperbola, the description of the logi- stica, the equiangular spiral, the nautical meridian, &c, the descent of bodies in resisting mediums, the density of the atmosphere at any altitude, &c, 0,28, 0,800000000000 B = . . = 3/2- /lO <0,33 ! 1 ,024000000000 C = A -r B = 10/';;- 3/10 > 0,300 0,99035'203U29 n = B+ 9c = 93^2- 28/10 <0,30107 1 1,004336277664 E =c+ 2d = 169/2- 59/10 >0,301020 0,998959336107 F =D+ 2e = 485/2- 146/10 <0,3010309 1,000162894165 G =E+ 4f = 2136/2- 643/10 > 0,50 102996 0,999936281874 H =F+ 6g = 13301/2- 4004/10 0,3O10299951 0,99997172083(> K =H + I = 42039/2- 12655/10 <0,S0IO299959 1,000007161046 L =1 + K = 70777/2- 21306/10 > 0,301 02999562 0,999993203514 M = K+ 3l= 254370/2- 76573/10 < 0,30 102999567 1,000000364511 V) =L+ M = 325147/2- 97879/10 >0,3010299956635 0,999999764687 = M+ 18n = 6107016/2- 1838335/10 < 0,301 0299956640 comp.ar. 235315 = 364^ 110 + 235313N ='.'3l2:)S,-;^2r IS7/2 693147 100972/ in > 0,301029995663987 composed of the logarithms of 2 and 10, the numbers in ques- tion, those logarithms being denoted thus, l2 and /lO. Then multiplying the two numbers in the first column together, there is produced a third number 1,024, against which is written c, for its logarithm, expressing likewiFc by an equa- tion in what manner c is formed of the foregoing logarithms A and B. And in the same manner the calculation is conti- nued throughout; only observing this compendium, that be-.' fore multiplying the two last numbers already entered in the table, to consider what power of one of them must be used to bring the product the nearest that can be to unity. Now after having continued the table a little way, this is found by onl}' dividing the dilferences of the numbers from unity one bv the other, and taking tlie nearest quotient for the index of the 414 CONSTRUCTION OF TRACT 21, power sought. T'hiis, tlie second and third numbers in the table being 0,8 and 1,024, their differences from nnity are 0,200 and 0,024 ; hence 0,200 -^ 0,024 gives 9 for the index ; and therefore niultiplying tlie 9th po;ver of 1,054 by 0,8, produces the next number 0,l'9035203l429j whose logarithm is D = B -}- 9c. When the calculation is continued in this manner till the numbers become small enough, or near enough to J, the last logaritiim is supposed equal to nothing, which gives an equa- tion expressing the relation of the logarithms, and thence the required logarithm is dcterniined. Thus, supposing g = 0, we liave 2136/2 643/10 = 0, and lience, because the loga- rithm of 10 is 1, ve obtain l2 = --=0,30102996, too small in the last figure only; which so happens, because the num- ber corresponding to g is greater than 1. And in this manner are all the numbers in the third or last column obtained, which arc continual approximations to the logarithm of 2. There is another expedient, which renders this calculation still sliorter, and it is founded on this consideration: that when jr is small, (l+.r)' is nearly = 1 +??.r. Hence if l+,r and 1 z be the two last numbers already found in the first column of the table, the product ot their powers (1 -f- .vY'x (1 z)" will be nearly 1 j and hence the relation of in and V may be thus I'ound, (1 -\- x)" x (1 z)" is nearly = (1 -\-vix)x (1 c) =z 1 -J- ^'"^' "2 7}i77.rz r: 1 -f- mx nz nearly, which being also = 1 nearly, therefore m : n : : z : V : : l.{l-z) :/.(l+.r); whence xl.{l-z) -\-zl .{I +x) = 0. For example, let 1,024 and 0,9903.52 be the last numbers in the table, their In^s. being c and D : here we have 1,024 = 1 +.r, and 0,990352 = l-z; conseq. .v = 0,024, and ^r =:O,009648, 201 and hence the ratio in small numbers is -. So that, for X 500 ' ".nuding the logaiithms proposed, we may take 5OOd + 20Jc = 4S510/2- 1 1603/10 = 0; which gives /2=0,3010307. And in this lu.uuier arc found the numbers in the last line of the table. TRACT 21. ' LOGARITHMS. 445 Of Mr. Long's Method. In number 339 of the Philosophical Transactions, arc given a brief table and method for finding the logarithm to any num- ber, and the number to any logarithm, by Mr. John Long, B. D. Fellow of C. C. C. Oxon. This table and method are similar to those described in chap. 14, of Briggs's Arith. Log. differing only in this, that in this table, by Mr, Long, the logarithms, in each class, are in arithmetical progression, the common difference being 1 ; but in Briggs's little table, the column of natural numbers has the like common difference. 1'he table consists of eifjht classes of logarithms, and their corresponding numbers, as follow : L. Nat. Numb. Log. ,009 Nat. Numb. Log. Nat. Numb. Log. Nat. Numb. ,9 7,943232.347 1,020939484 ,00009 1,000207254 ,0000009 1,000002072 ,8 6,309573445 8 1,018591388 8 1,000184224! 8 1,000001842 ,7 5,011872336 7 1,016248694 7 1,000161194' 7 1,000001611 ,6 3,981071706 6 1,013911386 6 1,000138165' 6 1,000001381 5 3,162277660 5 1,011579454 5 1,000115136 5 1,000001151 ,4 2,511886432 4 1,009252886 4 1,000092106 4 1,000000921 ,3 1,995262315 3 1,006951669 3 1,000069086 3 1,0000006-90 ,2 1,584893193 2 1,004615794 2 1,000046053 o 1,000000460 1 '09 1,258925412 1 ,0009 1,002305238 1 1, 0000230261 . 1 1,000000230 1,230263771 1,002074475 ,000009, 1,0C0020724| ,00000009 1 ,000000207 8 1,202264455 8 1,001843766 s' 1,0000 1 842 11 8 1,000000184 7 1,174897555 7 1,001613109 7,1,000016118| 1 1,000000161 6 1,148153621 6 1,001382506 6-1.000013816' 6 l,00('0O0138 5 1,122018454 5 1,001151956 5il,000011513i 5 1,000000^15 "* 1,096478196 4 1,000921459 4 1,0000092101 4 1,000000092 S 1,071519305 31,000691015 3 1,0000069081 S 1,000000069 2 1,047128548 2 1,000460623 2 1,00(1004605! 2 1,000000046 1 1 ,023292992 1 1,000230285 1 1,000002302: 1 1.000000023 where, because the logarithms in each class ate the continual multiples 1, 2, 3, &c, of the lowest, it is evident that the na- tural numbers are so many scales of geometrical proportionals, the lowest being the common ratio, or the ascending num- bers are the 1, 2, 3, &c, powers of the lowest, as ex])ressed by the figures 1, 2, 3, See, of their corresponding logarithms. Also the last number in the first, second, third, See class, is the 10th, 100th, 1000th, ^c root of 10 ; and anv iiumbtT in 4iG CONSTRUCTION OF TRACT 21. any class, is tlie 1 0th power of the corresponduig number in tiie next following class. To find the logaritiim of any number, as suppose of 2000, by this table, Look in the first class for the number next less than the first iigure 2, and it is 1,995262315, against which is 3 for the first figure of tiic logarithm sought. Again, di- viding 2, the number proposed, by 1,995262315, the number found in the table, the quotient is 1 ,002374-4'67 ; which being looked for in the second class of the table, and finding neither its equal nor a loss, is therefore to be taken for the second figure of the logarithm ; and the same quotient l,0O2374'i67 being looked for in tlie third cLiss, the next less is there found to be 1,002305238, against which is 1 for the third figure of the logarithm; and dividing the quotient 1,002374467 by the said next less number 1,002305238, the new quotient is 1,000069070 ; which being sought in the fourth class, gives 0, but sought in the fifth class gives 2, which are the fourth and fifth figur.;s of the logarithm sought : again, dividing the last quotient by 1,000046053, the next less number in the table, the (juotient is 1,000023015, which gives 9 in the 6th class for t'.ie Gth fioure of the 'o^arithm souo'ht: and ao;ain dividing the last quotient by 1,000020724, the next less number, the quotient is 1 ,000002291, the next less than which, in the 7th class, gives 9 for the 7th figure of the logarithm : and dividing tlie last quotient by 1,000002072, the quotient is 1,000000219, which gives 9 in the 8th class for the 8th figure of the log.: and again the last quotient 1,000000219 being divided by 1,000000207, the next less, the quotient 1,000000012 gives 5 in tlK; same Sth class, when one figure is cut oiF, for the 9th figure of the logarithm sought. All which figures collected together give 3,301029995 for Briggs's log. of 2000, the index 3 being supplied ; which logarithm is true in the last figure. To find the number answering to any given logarithm, as suppose to 3,3010300 : omitting the characteristic, against the other figures 3, 0, 1, 0, 3, 0, 0, as in the first column in the margin, arc the several numbers as in the 2d column. TRACT 21. LOGARITHMS. 447 found from their respective 1st, 2d, 3d, 3 &c classes; the effective numbers of which multipHed continually together, 1 thelastproductis 2,000000019966, which, because the characteristic is 3, gives 3 2000,000019966, or 2000 only, for the required number, answering to the given G loirarithm^ 1,995262315 1,002305238 1,000069080 Of Mr. Jones's Method. In the 61st volume of the Philosophical Transactions, is a small paper on logarithms, which had been drawn up, and left unpublished, by the learned and ingenious William Jones, Esq, The method contained in this memoir, depends on an appli- cation of the doctrine of fluxions, to some properties drawn from the nature of the exponents of powers. Here all num- bers are considered as some certain powers of a constant de- terminate root: so, any number x may be considered as the 2 power of any root r, or that x = r" is a general expression for all numbers, in terms of the constant root r, and a vari- able exponent z. Now the' index z being the logarithm of the number x, therefore, to find this logarithm, is the same thing, as to find what power of the radical r is equal to the number x. From this principle, the relation between the fluxions of any number x^ and its logarithm s, is thus determined : Put r=:l +7Z ; then is .r = 7-^ = (1 + iif, and ^ + ^ = (1 -f ??)-+^=: {\-\-7iYx{i-\-nY=:xx (1-f 7z)% which by expanding (i-j-)z^ omitting the 2d, 3d, &c powers of i, and writing q for -^, becomes x + xk x (i' + 4?* + l?^ + i^* + <^c) ; therefore Je = axz, putting a for the series q -{- iq^ + iQ^ &c, orfx=xxy putting/ = K Now when 7- = 1 +72 = 10, as in the common logarithms of Briggs's form; then 7? = 9, q = ,9, and the series g + ig^+jq^ &c, gives a=2,302585Scc, andthevv-f. itsrecip./=:,434294&c. But if a=l=/, the form will be that of Napier's logarithms. 448 CONSTRUCTION OF TRACT 21. From the above form xz=fx, or z = --^j are then deduced many curious and <;eneral properties of logaritlims, Avith the several series hoetofore given by Gregor}-, Mercator, Wallis, Newton, and Ilallev. But of ail these series, that one which our autlior selects for conitructing the logarithms, is this, puttinfr N = ' ^ the losrarithm of is = 2/ x : n -1- '-n^ + -}^N^ + ^k' \ 6\.c, in the case in Avhich 7' p is = I, and con- senuentlv in that case n = or : which series will tlicn converge very fast. Hence, having- given any numl)crs, p, q, r, &c, and as many ratios a, b^ c, &c, composed of them, the difference between the two terms of each ratio b^infj 1 : as also the logarithms a, b, c, &.c, of those ratios given : to find the logarithms p, o., R, &c, of those numbers; sup[)Osingy= 1. For instance, if p = 2, q = o, r :=z 5 ; and a = '- z= ^ b zz =: --, c :=. ~- =r . Now the loparithms A, B, c, of ).-) i>\) 2-i 3-23 => 7 7 7 these ratios a, h, c, being fouiul by the above series, from the nature of j)owers we have tliesc three ef(nations, A = 20. 3p \ j> = 4p _ q _ R>- which equations reduced give c == 2u Q 3p3 p = 3a 1- 4b !- 2c = log. of 2. Q = 5a + 6b -\- 3c = log. of 3. n 7a 1- i)B -f 5c = log. of 5. And hence p 1- r = 10a 4- 13b \- Ic is = the logarithm of 2 X 5 or 10. An elegur.t tract on logarithms, as a comment on Dr. Ilal- ley's method, was also given by Mr. Jones, in his Synopsis :\i!m.;vi()nnn Matheseos, published in the year 1706. And, :n t')e Fhilo iophical Transactions, he ctnnmunicated various mi.rovenients in gonionictrical properties, and the series re- iating to the circle and to trigonometrv. Tiic nietnoir above described was delivered to the Royal .Society by their then librarian, Mr. John Robertson, a wor- thv,ingen'!(;us, and industrious man, who also communicated TRACT 21. ' LOGARITHMS. 449 to the Society several little tracts of his own relating to loga- rithmical subjects ; he was also the author of an excellent treatise on the Elements of Navigation in two volumes ; and he was successively mathematical master to Christ's hospital in London ; head master to the royal naval academy at Ports- mouth ; and librarian, clerk, and housekeeper, to the Royal Society; at whose house, in Crane Court, Fleet-street, he died in 1776, aged 64 years. And among the papers of Mr. Robertson, I have, since his death, found one containing the following particulars relat- ing to Mr. Jones, which I here insert, as I know of no other account of his life, &c, and as any true anecdotes of such ex- traordinary men must alw^ays be acceptable to the learned. This paper is not in Mr. Robertson's hand writing, but in a kind of running law-hand, and is signed R. M. 12 Sept. 1771. *' William Jones, Esquire, F. R. S. was born at the foot of Bodavon mountain [Mynydd Bodafon], in the parish of Llan- lihangel tre'r Bardd, in the isle of Anglesey, North Wales, in the year 1675. His father John George* was a farmer, of a good family, being descended from Hwfa ap Cynddelw, one of the 15 tribes of North Wales. He gave his two sons the common school education of the country, reading, writing, and accounts, in English, and the latin grammar. Harry his second soon took to the farming business ; but William the eldest, having an extraordinary turn for mathematical studies, determined to try his fortune abroad from a place where the same was but of little service to him ; he accordip.gly came to London, accompanied by a young man, Rowland Williams, afterwards an eminent perfumer in Wych-street. The report in the country is, that Mr. Jones soon got into a merchant's counting-house, and so gained the esteem of his master, that he gave him the command of a ship for a West-India voyage ; and that upon his return he set up a mathematical school, " It is tlie custom in several parts of Wales for the name of the father to become, the surname of his children. John George the father was commonl}' called Sion Siors of Llambado, to which parish he moved, and where his children were brought up." VOL. I. G G 4f50 CONSTlttJCil'IOij t)F TRACT 21. and published his book of uavii^jittttn* ; and that upon the death of the merchant he mahied his widow : that I ord Mac^ clesfield's sOti being his pupil, he was made secretary to the chancellor, and one of the D. tellers of the exchequer and thev have a stor}' of an Italian wedding which caused great disturbance in Lord Macclesfield's family, but comjoromised by Mr. Jones ; which gave rise to a saying, that Macclesfield Avas the making of Jone<, and Jones the making of Maccles- field." Mr. Jones died July 3, 1749, being vice-president of the Roval Society; and left one daughter, and a young son,. Avho was the late Sir William Jones, oneof the judges in India, and highly esteemed for his great abilities^ extensive learning, and eminent patriotism. Of Mr. Andrew Rtid and Others. Andrew Reid, Esq. published in 1767 a quarto tract, under tiie title of An Essay on Logarithms, in which he also shows the computation of logarithms, from principles depending on the binomial theorem and the nature of the exponents of powers, the logarithms of numbers being here considered as the exponents of tiie powers of 10. He hence brings out the "Usual scries for logarithms, and largely exemplifies Dr. Llal- lev's most simple construction. l^esides the authors whose methods liave been here parti- cularly described, many others have treated on the subject of logarithms, and of the sines, tangents, secants, &cj among the ])rincipal of whom arc Leibnitz, Euler, Maclaurin, Wol- fius, atid prof(''ssor Simson, in an elegant geometrical tract on logarithms, contained in his posthumous works, printed in ito at Glasgow, in the year 1T7G, at tlie expense of the very learned Earl Stanhope, and by his Lordship disposed of in * This tract on navigation, intitlcd, " A New Conipcndium of the wfiolc- Ait of Pracfina! Navigation," was ijuhlished in 1702, and dedicated " to the rtveiend and learned Mr. Jolm Harris, M. A. and F.K.S." the author, I apprehend, of the " Universal Dctioniu y of .Art.sand Sciences," under whose roof Mr. Jones says he composed the said treatise oa Navigation. TRACT 21. - Jf-OGARITHMS. 444. presents among gentleilien most eminent for mathematical learning. Of Mr. Dodsovi' s Anti-logarithmic Canon. The only remaining considerable work of this kind pub- lished, that I know of, is the Anti-logarithmic Canon of Mr. James Dodson, an ingenious mathematician, which work he published in folio in the year 1742 ; a very great performance^ containing all the logs, under 100000, and their correspond- ing natural numbers to 11 places of figureSj with all their differences and the proportional parts ; the whole arranged in the order contrary to that used in the common tables of numbers and logarithms, the exact logarithms being here placed first, and increasing continually by 1 , from 1 to 100000, with their corresponding nearest numbers in the columns op- posite to them ; and, by means of the differences and pro- portional parts, the logarithm to any number, or the number to any logarithm, each to 1 1 places of figures, is readily found. This work contains alsoj besides the construction of the na- tural numbers to the given logarithms, ** precepts and ex- amples, sho\Ving some of the uses of logarithms, in facilitating the most difficult operations in common aritlimetic, cases of interest, annuities, mensuration, &c; to which is prefixed an introduction, containing a short account of logarithms, and of the most considerable improvements made, since their in- vention, in the manner of constructing them." The manner in which these numbers were constructed, consists chiefly in imitations of som.e of the methods before described by Briggs, and is nothing more than generating a scale of 100000 geometrical proportionals, from 1 the least term, to 10 the greatest, each continued to 1 1 places of figures; and the means of effecting this, are such as easily flow from the nature of a series of proportionals, and are briefly as folloAv. First, between 1 and 10 are interposed 9 mean proportionals ; then between each of these 11 terms there are interposed 9 other means, making in all 101 terms; then between each of these a 3d set of 9 means, making in G G 2 452 CONSTRUCTION OF TRACT 21. all 1001 terms; again between each of these a 4th set of 9 means, making in all 10001 terms ; and lastly, between each two of these terms, a 5th set of 9 means, making iti all lOOOOl terms, including both tlie 1 and the 10. The first four of these 5 sets of means, are found each by one extraction ot the 10th root of the greater of the two given terms, which root is the least mean, and then multi[)lying it continually by it- self, according to the number of terms in the section or set; and the 5th or last section is mailc by interposing each of the 9 means by help of the method of diflerenccs before taught. Namely, putting 10, the greatest term, = A, a"^'^ = b, b"^' = c, c"^"^ =: D, D^ = K, and r7 = f ; now extracting the lOth root of a or 10, it gives 1,25892.54118 = B=a"^, for the least of the 1st set of nieans ; and then multi- plying it continually by itself, we have K, B", B% B*, &c, to b'" r=A,for all the 10 terms: 2dly,thc 10th root of 1,2589254118 sives 1,0232921)92.3 = c = B^" = a"^", for the least of the 2d class of means; whicli being continually multiplied gives C, C-, c\ &c, to c^ = b' = A, for all the 2d class of 100 terms : 3dly, the 10th root of 1,02:52929923 gives 1,00230.5238 1 = D =: c'"= b"^^''' = a"'"^'", for the least of the 3d class of means ; which being contiinially multiplied, gives D, D^, D', C=cc, to d'"""" c''^" b' = A, for the 3d class of 1000 terms : 4thly, the 10th root of 1,0023052381 gives 1,0002302850 = B := d'^'^' c^-" b'^"''"" = a'^'^'''''\ for the least of the 4th class of means, which being continually nnaltiplicd, gives e, E-, vJ, &.C, to k'^= = 1)'"= z= c'=B' =: A, for the 4th class of 1 0000 terms. Now these 4 classes of terms, thus })rodu- ced, recjuire no less than 11110 multiplications of the least uicans by themselves; which however are nmch facilitated by making a small table of tlui first 10, or even 100 products, of ilu' cionstaiit multiplier, and from it only taking out the pro- per lines, and adding tht-m together: and these 4 classes of numbers iilwuys prove thiMiiselves at every 10th term, which jnu>t ulw.iv;-, agrcH' v.ith t!i<; corre^.ponding sucoi'^sive terms TRACT 21. LOGARITHMS. 453 of the preceding class. The remaining 5th class is constructed by means of differences, being much easier than the method of continual multiplication, the 1st ami 2d differences only being used, as the 3d difference is too small to enter the com- putation of the sets of 9 means, between each two terms of the 4th class. And the several 2d differences, for each of tliese sets of means, are found from the properties of a set of proportionals, 1, ?', r^, r^, ike, as disposed in the 1st column of the annexed table, and their several orders of differences as in the other columns of the table ; where it is evident that Terms. 1st dif. 2d dif. 3d dif. &x. I X (r-l)x (r-l)^x {r-iyx 1 7' 7'- i r 1 r &c. I r ,.3 &c. &c. eacli coluinn, both that of the given terms of the progression, and those of their orders of differences, forms a scale of pro- portionals, having the same common ratio r; and that each horizontal line, or row, forms a geometrical progression, Isaving all the same coimuon ratio r 1, which is also the 1st (lifferi-'iice of c.ich set of means : so, (r l)- is tlie 1st of the '2(1 differei:ces, and which is constantly the same, as the 3d (lifferiMices become too small in the required terms of our pro- gression to be regarded, at least near the beginning of the table: lieiice, like as I, r 1, and (/' 1)" are the 1st term, with its Ist and 2d differences ; so ?% r" . {r 1), and r" . (;'!)-, are any other term with its 1st and 2d differences. And by tttis rule the 1st aiul 2d differences arc to be found, for evcrv set of 9 means, viz, multiplying the 1st term of any class (which will be the several terms of the series E, E', e^, Kc, or every lOtli term of the series r, F% F% &.c) ^y r I, or F 1, for the 1st difference, and this multiplied by f~ J 454 CONSTRUCTION OF LOGARITHMS. TRACT 21 . again for the true 2d clirTerence, at the beginning of that class. Thus, the 10th root of 1,0002302850, or e, gives 1,0000230261 16 for f, or the 1st mean of the lotvcsl class, therefore f - 1 = r - 1 = ,0000230261 16, is its 1st differ- ence, and the square of it is (r- 1)' = , 0000000005 302 its 2d diff. ; then is ,000023026 11 ep'^" or ,0000230261 16e', the 1st difference, and ,0000000005302?^" or ,0000000005 302 e" is the 2d difference, at the beginning of the ;zth class of decades. And this 2d difference is used as the constant 2d difference through all the 10 terms, except to^vardsthe end of the table, "where the differences increase fat enough to require a small correctioh of the 2d difference, which Mr. Dodson effects by taking a mean 2d difference among all the 2J differences, in this manner; having found the series of 1st differences (f-1).e% (f- !)."+', {f-1).e"*', &c, he takes the differ- ences of these, and ji of them gives the mean 2d differences to be used, namely, -~ (e""'-e"), ^ (e + 2 _ e"*'), &c, are the mean 2d differences. And this is not only the more exact, but also the easier way. The common 2d difference, and the successive 1st differences, are then continually added, through the whole decade, to give the successive terms of the required progression. , TRACT XXII. SOME PROPERTIES OF THE POWERS OF NUMBERS. 1. Of any two square numbers, at any distance from each other in the natural series of the squares 1%'2-, 3% 4^, &c, the mean proportional between the two squares, is equal to the less s(]uare plus its root multiplied by the difference of the roots, that is, by the distance in the series between the two square numbers, or by 1 more than the number of squares between them. The same mean proportional, is .also equal to the greater of the two squares, minus its root the same TRACT 22. PpWEBS OF NUMBERS. 4^5 number of times taken. That is, mi = mm + dm = im dn^ where d \s =. n in, the distance between the two squares 7/2% n*. For, since n=m-\-d; multiph' by m, then mn = mvi-i- vid, which is the first part of the proposition. Again, 7w = ?i f/j multiply this by n, then mn = nn flf> which is the latter part. 2. An arithmetical oican between the two squares mm and nn, exceeds their geometrical mean, by half the square of the difference of their roots, or of their distance in the series. For, by the first section, wm :=. mm^ dm, and also inn=:nn d?i; add these two together, and the sums are 2m7i =: mvi -\-nn ^ d (n m) =: mm + mi My divide by 2, then nn ~ ^mn + ^nn Uld. 3. Of three adjacent squares in the series, the geometrical mean between the extremes, is less by 1 than the middle square. For, let the three squares be m-, (m+ 1)^, (74-2)'; then the mean between the extremes, m{m-\-2) =:Jimi + 2m is = {m + ])- - ]. In like niauDer, the mean between the extremes, of any three squares, whose common distance or difTerence of their roots is d, is less than the middle square by the square of the distance dd. 4. The difference between the two adjacent squai'es wzm, nn, or nn mm, is (m + 0* ^'^^ 2wi -\- I. In like man- ner, the difference between 7r and the next following square p^, or p^ 11-, is 2n -{- 1; and so on. Hence, the difference of these differences, or the 2d difference of the squares, is 2(n w) = 2, which is constant, because n 7n zz 1. And thus, the 2d diff^^rences being constantly the number 2, all the first differences will be found by the continual addition of this number 2 ; and then the whole series of squares them- selves will be found by the continual addition of the first differences. Thus, the 2d difs. 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, &c. 1st difs. 1, 3, 5, 1, 9, 11, 13, 15, 17, 19, &c. squares, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, &c. 456 POWERS OF NUMBERS. TRACT 22. 5. Again, if w% n^, p', be three adjacent cubes ; then n-m3 = 3m* + 3m + M ^^d the differences of these first differences is 3(' - V2^) f 3(n- wz) = 6(m + 1), the 2d differ- ence. In like manner, the next 2d difference will be 6(n+ l). Then the dif. oi these 2d dilferenccs is 6(w m) = 6 the 3d difference, which therefore is constant. Now, supposing the series of cubes to begin from 0, the first of each of the several orders of differi;nccs will be foimd bv making m =0, in the general ex[)ression for each order: tliuf:, 6(ni-\- \) becomes 6 for the first of the 2d differences; and 3m^+3m+ 1 becomes 1 for the first of the 1st differences. A:id hence is found all the others, as in this tabic. 3d difs. 6, 6, 6, 6, 6, 6, 6, 6, G, &c, 2d difs, 6, 12, l'^, 24, SO, 36, 42, 48, 54, &c. 1st difs. 1, 7,19,37,61, 91, 127, 169, 217, See. cubes 0, I, 8, 27, 64, 1^5, 216, 343, 512, 8cc. And thus may all the powers of the S(j:r:es of natural num- bers 1,2, 3, 4, 5, 8cc, be found, by addition only, adding continually the numbers throughout thest'\:;rai orders of dif- ferences. Ai:d here it is remarkable, that tiie number of the orders of differences, will be the same as the index of the powers to be formed ; that is, in the series of scjuarc'^, there are two orders of differences; in the cubes, three; in the 4th powers, f(Mir, &c: or, which is the same thing, of the squares, the 2d differences are equal to each other; of the cubes, the 3d differences are eq\ial ; of the 4th power, the 4th diffs. are equal ; Sec Further, the 2d diffs. in the squares are 1.2 = 2 ; the 3d diffs. in the cubes 1.2.8 = 6; the 4th diffs. in the 4tli powers 1.2.3.4=24; and so on. And from these properties were found, by continual additions only, all the series of squares and cubes in the table at the end of this volume, and in my large Table of the Products and I'owers of iSumbers, published in 1781, by the Board of Longitude. ( 457 ) TRACT XXIII. A NEW AND EASY METHOD FOR THE SGUARE ROOTS OF NUMBERS. FROM MY MATHEMATICAL MISCEL. P. 323. Problem. Having given any nonquadrate number n; it is required to find a simple vulgar fraction , the value of which shall be within any degree of nearness to ^^n, the surd root of n. Investigation. Since y'N is nearl}^, or ?*n = n* nearly ; let ^'n be zr n^n. Then, since 72, d, and n, are all inte- gers by the supposition, d must also be an integer ; and the smaller that integer is, the nearer will the value of - be to -v/n, as is evident: therefore let d zz 1 the smallest integer; then IS cf"N :zi w- 1 , or = f/'N + 1 suppose this to be = (dx I )- = c/^r^ 2dx -\- 1 , where x is evidently some near value of Vn ; from this equation we have t/=:-^-, and con- sequently nz=.^id^^ - 1) = T~~. ' lience theref. y/^n = is =: 2x nearly. Thus then the function ' is an approximate value of y/N, where .r is to be assumed of any value whatever; but the nearer it is taken to \^n, tlie nearer will the value of the fraction be tOy/N required. And since is always nearer to // N than what .r is, therefore assume any integer, or ra- tional fraction, for .r, but tiie nearer to ^/N the more conve- nient, and write that assumed value of it in this expression, instead of it, so shall we have a nearer approximate rational value of ^/n; then use this last found value of ^/^ instead of .r, in the same expression, and there will result a still nearer rational value of ^/n ; and thus, by always substituting the 45S saUARE ROOTS. TRACT 23. ];irst found value for x, in the fraction , or i.r -\- - , the result will be a still nearer value. And thus we may proceed to any degree of proximity required. But a theorem somewhat easier for this continual substitu- tion, maybe tluis raised: being any one approximate va- lue of V Nj ^^'vite it instead of r, in the general function , tnen v.c hav" - tor t!ie general ypproxunation. That is, having assumed or found any one approximation , the numerator of the next nearer approximation v ill be equal to the sum of the square of the numerator n an! n times the square of the denominator of this one, ai d the de- nominator of tlie new one will b:; double the product of the numerator and denominator ot this. Ur. a still easier continual approxunai ion is -, ^vhich is equal to the former, because n^ is = fl!^N + 1. Example I. To find near rational values of the square root of the number 2. Here n =: 2. T;dce l\ or | for the first value of x, as being nearly equal to ,cn ; o quickly obtained. For, the larger the given number n is, the quicker \\\\\ the theorem approxi- mate. Thus, taking for Example 2. To find the root of the number 920. Here :< = f,'20, and .r zz. 30 nearly. Now we must first use the rule ^ , becaiiiC x is taken = 30, below the true value. Hence TRACT 24. ROOTS AND RECIPROCALS. 459 then 900+920 60 1S20 60 91 i^920. Next make- = 4 3 d 304- the second value of 1 16561 o3_l _ 2x91^ _ ^ ~Jdn~ ~~ ^T7^nr3 ~ "546 30-33150183, differing from the truth hut by 6 in the tenth place of figures, the true number being 30*33150177. And in this way may the square roots, in the tabic at the end of this volume, be easily found. TRACT XXIV. TO CONSTRUCT THE SQUARE AND CUBE ROOTS AND THE RECIPROCALS OF THE SERIES OF THE NATURAL NUMBERS. 1. F'or the Square Iioots. Since the square root of a' + n is a -\- - r-r+r^ 2cc; therefore the series of the square roots of a^, a"' + 1 > * + 2, *+3, &c, and their 1st, 2d, 3d, 4th, &.c diiVerenoes, will be as below : Nos. Square Roots. 1st Diffs. te a 1 1 1 2tl Diffs. T . . 1 1 1 3d Diir!= '+ 1 a 4- 1- 2a 8a3 ' iii^s 2a~83 ' 16a^ 1 3,7 1 o 4fl3 S/jO 5 a^+2 ,2 4,8 --- + - 1 J 4_ 19 _1 ^ 8-^ 6cc 3 aH3 ,3 9 , 27 ,4 16 , 64 1 7 37 _1__^ fl^ 1-4 a 4- -- 4--;- Where, the columns of fractions having in each of them the same denominator, after the first line, in each class, a dot is written in the place of the denominator?, to save the too fre- quent repetition of the same quantities. Now it is evident that, in every class, both of roots and of every set of differ- ences, the .first terms are all alike; and therefore, by the subtractions, it happens that every class of differences con- 460 ROOTS AND RECIPROCALS TRACT 24.. tains one term fewer than the one immediately preceding it. These differences are to be employed in constructing tables of square roots ; and the extent to which the orders of differ- ences are to be continued, must be regulated by the number of decimal figures to which the roots in the table are to be carried. In the above specimen the differences are continued as far as the 3d order, where the common first term is -, which may be sufficiently small for constructing all the preceding orders of differences, and then the series of roots themselves, as f.ir as to 7 places of decimals in each, when we conmience with the nutnber 1024, for the first square a% the root of which is 32. After this, the squares 1025, 1026, 1027, &c, conti- ntuiiiv increasing, their roots 32 + , &-c, proceed increasing- also; but the series of numbers, in every order of differences, arc ail in a decreasing progression ; so that the followiu'/ orders arc; ail found by taking each latter difference from the one ininicdiatclv above it. Then, to construct the table of roots, having found the first term of each order of diiferences, as far as necessary, su])pose to the 3d order ; subtract that continually from the iu'st of the 2d differences, which will complete the series of this order of differences. Then these bei'.ig taken each from the first difference, the successive re- mainders will form the whole series of first differences. Lastlv, these first differences added continually with the first squan? root, will form the whole series of roots, from the first rational root, suppose 32, the root of the square num- ber 1024, to he continued to tiie next r;itional root :'i3, or root of t'lc next s(]u;ire number 10S9. T len begin again, from this last square number, in l;ke manner, with a new series o( roots and differences, whic h are to be continued to the tliifd s()u:'re number 1156, the root of which is tlie next r;ition;;l root ?jV. 'T'lien tiie llki; process is to be re- ])e;ited again, ami continued from the 3d to the 4th scjuire nuiulicr. ;\;id so on, eo'.itiini!i)g from e.ich Mircessiv(> ,qn;re iiiiu. !)<".-. \n tlie nevt f')!!owin;_!; one, ;i'> lar as n<'ec:ss.irv ; the l.i^t '^i (-..ch s ";. of iv>"'^ and djirereuces ahvays verii}'in904I5S 2-802039 23 52p 121 67 0434783 4-7Q5b315 2-843867 24 576 13824 04)6666 4-sg8g7g5 2-884499 25 625 15625 04 5-OCOOOOO 2-9240 1 8 26 6/0 17576 038-1615 5-0990195 2-962496 27 729 I9OS3 0370370 5-1961524 3-OCOOOO 28 784 21952 0357143 5-2915026 o-0365Sg 29 841 24389 0341828 5-3851 648 3-072317 30 900 27000 0333333 5-4772256 3-107232 31 961 20791 03 225 81 5-56776A4 3-14)381 32 1024 32768 03 125 5-656S542 3-1 74802 33 10!S9 o5ir~i7 0303030 5-7445626 3-207534 34 1 1 56 39^04 G29-1 1 1 8 5-8309519 3-239612 35 1225 4 2875 0285714 5-9160798 3-271066 3(j 1296 46656 0277777 6-0000000 3-301927 37 1369 5C653 0270270 6-08276. 5 3-332222 38 1444 54S72 0263)58 6-1644140 3-361975 39 152 1 59319 0256410 6-24499bO 3-391211 ! 40 1 tioo O'-IOLX) 025 6-3245553 3-4199-52 41 lOSl 6892 1 0243902 6-4031242 3-448217 42 1704 7-iUSS 02380(15 6-4807407 3-47'>027 43 IS) 9 79^07 0232558 6-5574385 3503398 44 io:;(j 85]b4 0227272 6-63324()6 3-530348 45 202 5 9 1 1 25 0222222 0-7G820.o'9 3-556sg3 46 2116 97336 0217391 6-7823300 3-583048 47 2209 103823 0212766 6-8550546 3-608826 48 2 J 04 I 1 0592 020i>333 6'9282032 3-634241 49 2401 117649 0204082 7-ooooo(.^o 3-659306 50 2500 125000 02 7*0710678 3-684031 , TR. 25. SQUARES, CUBES, RECIPROCALS, AND ROOTS. 467 Numb. Square. Cube. Recipr. Sq. Root. C. Root, 51 2601 132651 0196078 7-14i428-+ 3-708430 52 2/04 140608 OI923O8 7-2111026 3-/325 1 1 53 2809 148877 - OI8S679 7-2801099 3-756286 54 2916 157464 0185185 7-348 -1692 ^'779762 55 3025 166375 0181818 7-4161985 3-b02953 50 3136 175616 0178i71 7-4833148 i '825862 57 3249 185193 0175439 7-5498344 3-848501 58 33(54 195112 0172414 7'6i5773i 3-870877 59 3481 205379 0109492 7 0811457 3-8(12996 GO 3060 21600b 01 06666 77459667 3-914867 til 3721 2 2698 1 0103934 7-8102497 3-936497 62 3844 238328 0161290 7-8740079 3-957892 t)j 3969 250047 0158/30 7-g3725^g 3-979057 04 4090 262144 015625 8 0000000 4-0000(30 Q5 4225 274625 0153846 8-0022577 4-020726 06 4356 287496 0151515 8-1240384 4041240 67 4489 300703 0149254 b- 1853528 4-061548 68 4024 314432 0147059 8-2462113 4-081656 69 476] 328509 0144928 8-3060239 4- [01566 70 4900 34300O 0142857 8-3666003 4-12I2S5 7i 5041 357911 01408-45 b-426l408 4-140818 72 5164 373248 0138888 8-4852814 4-160168 7- 5329 3S9OI7 0136080 8-5440037 4-179339 74 5476 405224 0135135 8-0023253 4-198336 75 5025 421875 0133333 8-66025 iO 4-217103 76 5776 438976 0131579 87177979 4-235824 77 5Q2g 456533 0129870 8-7749644 4-254321 78 6084 474552 0128205 8-83 17009 4-272059 79 6241 493039 0126582 8-888I944 4-290841 so 6400 512000 0125 8-9442719 4-S08870 SI 6561 531441 0123457 9-0000000 4-326749 82 6724 551368 0121950 9-0553851 4-344481 83 6889 1 571787 0120482 9-1104336 4-362071 8-1 7056 592704 01 1904 8 9-1651514 1 "3795 19 85 7225 014125 0117647 9-2195445 4396^30 -80 7396 630056 0110279 9-2730185 4-414005 87 756g 658503 01 14943 9-3273791 j-i-431047 88 7744 68I472 0113636 i 9-3808315 4-447960 sg 7921 704909 01 123 CO 9-433y81l 4--1 64745 90 8100 72t,000 Ollll) 1 9-4808330 4-481405 91 8281 753571 0100890 j 9-5393920 4-497942 92 8464 778688 0l0S0\j6 ' 9-5910630 4-514357 93 8649 804357 0107527 j 9-6-136508 4-530655 9-t 8806 830584 01 003 83 g-6g5..yog7 4-546836 95 9020 857375 0105263 9-7407943 4-562903 9e 9216 884736 0104166 9-7979590 4-578857 97 9409 912673 0103003 ; 9"84b8578 4-594701 98 9O04 941 J 92 0102041 j 9-89949-19 4-610436 99 98OI 97t)299 OIOIOIO i 9-9^97'i4 4-626065 100 100(0 1 bocobb 01 1 lOOCOCOCO 4-641589 468 SQUARES, CUBES, RECIPROCALS, AND ROOTS. TR. 25. Numb. Square. Cube. Recipr. Sq. Root. C. Root. 101 10201 1030301 oou;,oo9 10-0498756 4-657010 102 10404 106 1208 O0980i>9 10-0995049 4-672330 103 10609 1092727 O09708"7 10-1488916' 4-687548 104 108 16 1124864 0090' 154 10-19803CO 4-702669 105 11025 1157625 0095238 10-2469508 4717694 106 11236 1191016 0094340 10-2956301 4-732624 107 11449 1225043 0093458 10-3440804 4-747459 108 11664 1259712 0092592 10-3923048 4-762203 109 11881 ] 295029 0091743 104403065 4-776856 no J 2100 13310CO 0090909 10.4880885 4-791420 111 12321 1367631 0090090 10-5356538 4'805S96 112 12544 1404928 OO89286 10-5830052 4-820284 113 12769 14428Q7 OO88496 100301458 4-834588 114 12996 1481544 00877 J 9 i 0-6770783 4-848808 115 13225 152O875 0086957 10-7238053 4-862944 116 13456 I56O896 0086207 10-7703296 4-876999 117 13689 1601613 0085470 10-8166538 4-890973 118 13924 1643032 00847^6 IO-S627SO5 4-904 S68 119 14161 1685159 0084034 10-9087121 4-918085 120 14400 I728OOO 0083333 10-9544512 4-932424 121 14641 1/71561 0082645 1 1 0000000 4-94()088 122 14684 1815848 008 1967 11-0453610 4-g5g675 123 15129 1860867 008 1 300 ] 1-0905365 4-973190 124 15376 1906624 00SO645 11-1355287 4-98603 1 125 15625 1953125 008 11-1803399 5-000000 126 15876 20OOS76 0079365 11-2249722 5-013298 J 27 16 129 2048383 00787^10 11-2694277 5-026526 128 16384 2097152 0078125 11-3137085 5-039684 129 16641 2146689 0077519 11-3578167 5-052774 130 16900 210/000 0076923 n -401 7543 5-065797 131 17161 224b091 007C-336 11-4455231 5-078753 132 17^-24 229996"8 0075757 11-4891253 5-091643 133 1 7689 2352637 0075188 11 -5325626 5-104469 134 J 7056 2400104 0074627 1 i-575-6H6g 5-117230 135 1 8225 246OJ75 0074074 11-6189500 5-129928 136 1 8496 2515450 0073529 11-0619036 5-142563 137 )e76o 257 i 353 0072993 11-7046990 5-155137 138 19044 262S072 0072464 11-7473444 5-]07t;49 1 39 19^21 26&56i9 0071942 2\Vr25 ;;w4;-62.'; 000^966 12 04 15946 5 -253 J 88 146 2 ' 3 1 ( J :;!12i;j6 0068^93 I2()S3'; ;'o 5-2656:^7 147 2hO,.( 3i7u,'.23 ooo?027 r: ]24.-i,,57 5-277632 148 2 1 UVA ..'J41792 006756/ [:-);'-,.5251 5-^c9'.72 M.U 22201 3307; )49 OC'67n4 1. -2' :ir,55u 0-301459 K'^b 22500 3375000 00(;6<;66 1 2-24 7-; 48." 5-313293 TR. 25. SauARES, CUBES, RECIPROCALS, AND ROOTS. 469 Numb. Square. Cube. 3442951 Recipr. bq Koot. C hoot. 151 22S01 0066225 12-2882057 5-325074 152 23104 3511808 0065 7 89 12-3288280 5-336803 153 23409 3581577 0065359 12-3693169 5-348481 154 23716 3652264 0064935 12-4096736 i 5-360108 155 2-1025 3723875 0064516 12-4498996 5-371685 15(J 24336 3796416 0064103 12-4899960 5-383213 157 24649 3869893 0063694 12-5299041 5 '394690 158 2^964 3944312 006329 1 12-5698051 5-406120 i59 25281 4019679 0062893 12-6095202 5-417501 i6o 25600 4096000 00625 12-619! 106 5-428835 161 25921 41 73281 0062112 12-0885775 5-440122 l62 26244 4251528 006 1 728 12-7279221 5-451362 l63 2656g 4330747 0061350 127671453 5-^&1556 l64 26896 4410944 0060975 12-8062485 5-473703 l65 27225 4492125 OO60606 12-8452326 5-484806 l66 27556 4574296 0060241 12-8840987 5-495865 lt>7 2/889 4657463 OO5988O 1 2-9228480 5-50687 g l6d 28224 4741632 0059524 12-9614814 5-517848 169 28561 48268O9 0059172 13-0000000 5-528775 170 28900 4913000 0058824 13-0384048 5-539658 171 2924 1 5000211 0058480 13-0766968 5-550499 172 29584 5088448 0058140 13' 1148770 5-561298 173 29929 51777^7 0057803 13-1529464 5-572054 174 30276 5268024 0057471 13-1909060 5-582770 175 30625 5359375 0057143 13'2287566 5-593445 170 30976 545177() 0050818 l3-26"64992 5-604079 177 31329 5545233 0056497 13-3041347 5-614673 178 31684 563'J752 0056 160 13-3416641 5-625226 i79 32041 5735339 0055866 13-3790882 5-635741 ISO 32400 5832000 0055555 13-4164079 5-646216 181 32761 5929741 0055249 13-4536240 5-656652 182 33124 602 35 69 0054945 13-4907376 5-667051 183 33489 6128487 0054645 13-5277493 5-677411 184 33856 6229504 0054348 13-5646600 5-687734 185 34225 6331625 0054054 13-6014705 5-698019 186 34596 6434856 0053763 13-6381817 5-708267 187 34969 6539203 0053476 13-6747943 5-718479 188 35344 604 4672 0053191 13-7113092 5-728654! I89 35721 6751209 0052910 13-7477271 5-738794 100 : 36100 685CiO0O 0052632 13-7840488 5-748897 191 36-181 6c;6'787l 0052356 13-8202750 5-758965 192 i 36861 7077888 0052083 13-8564065 5-768998 193 1 37249 71 8905 7 0051813 13-8924440 5-778996 194 ; 3 7 03 6 7301384 0051546 13-9283883 5-788960 J 95 j 38025 7414S7-J 0051282 13 9642400 5-798890 yqt) i 38416 7529536 0051020 14-bcoooco 5-808786 197 1 38809 76-15373 0050761 14-0356688 5-S 18648 198 39204 7762392 0050505 14-0712473 5-828476 199 39601 7S6O599 005025 1 14-100/360 5-83272 200 40000 8000000 005 14-1421356 5-848035 4-10 SaUARES, CUBES, KECIPROCALS, AND ROOTS. TR. 25. Numb. ; Square. Cube. Recipr, Sq. Root. C. Root. 5-8577()5 201 40401 8120001 004975 1 14- 1774469 202 40804 8242408 (X)49504 14-2126704 5-86/464 203 41209 8305427 004926' 1 14-247S068 5-877130 204 4l6l6 8489664 0049020 14-282S569 5-88676.^. 205 42025 8615125 0048/80 14-3178211 5-896368 206' 42436 8/41816' 0048544 14-3527001 5-905941 207 42849 8869743 004-J309 14-387J9'6 5-915481 208 43264 8998yl2 0048077 14-4222051 5-924991 209 436s 1 9123329 0047817 14-4568323 \ 5-934473 | 210 44 1 00 9261000 0047019 14-4913767 5-943911 211 4452 1 939393 1 0017393 14-5258390 5-953341 212 44944 952s 128 0047170 14-5602198 5-96273 [ 213 45369 (jo1 O.SII7929 0024450 20-2237484 7-422914 4 10 1 OS 1 0."^ 0: 92 1 000 0024390 t.'0-2484567 7 '428958 411 10SQ2I ' O94J6531 0024331 20-2731349 7-434993 412 16' 174 4. 6iJ934528 0024272 20-2977831 7-441018 413 1705 09 ; 70444(;97 0024213 20-322^014 7*447033 414 171390 1 70951944 0024155 20-3469899 7-453039 415 171^225 71473375 0024096 iO-37 15488 7-459036 4]() l7'-i05() 71991290 0024038 20-396078 1 7-465022 417 173889 72511713 0023081 20-4205779 /47O999 418 174724 73O34032 0023923 20-4450483 7 476966 419 1755OI 735OOO59 0023860 2O-4.094895 7'482921' 420 176400 74OSSCOO 0023810 20-4939015 7-4885<72 4il 177241 74018401 00 3753 20-5 1 82845 7-4948 10 422 178064 75151448 0023097 20-5420386 7-500740 423 17S929 /SosdgOj 0023041 20-5069038 7-5o(}00o 424 179770 70225024 0023585 20-59 1 2003 7-^12571 425 1 S0625 70705625 0023529 20-6155281 7 "5 1 8473 420 1814/6 77^0ii77(i 0023474 20-0397074 7-524305 427 ] 82329 77854483 0023419 20-66397t>3 7-530248 428 183184 78-10.752 0023364 20-6881 009 7-530121 429 1840-11 7StJ535Hg 0023310 20-7123152 7-541986 430 184900 79507COO 0023256 207364414 7-547811 431 I857OI 81002991 (1023202 20-7005395 7-553O8S 432 1 b0624 80021508 0023148 20-7840097 7'559525 433 187489 811 82737 0023095 20-8086520 7'565353 434 ibSisb 81746504 0023041 20-8320607 7'57n73 435 189225 8231287.5 0022(189 20-8566536 7-570984 430" 19(;o;j6 82881850 0022930 208806130 7-58^780 437 i 909-^9 83453453 0()22o83 20-9045450 7-588579 438 91844 84027072 (X)2283 1 20-(.i284495 7'5('A3u3 439 192721 840O45 1 9 0022779 20-9523268 7-600138 44(J 193000 85184000 ((022727 20-9701770 7 600003 411 194 IS 1 85700121 OO22O76 21-0000000 7-61 J 002 442 195304 8035(,883 0022024 2 1 -0237900 7-0i74ii 443 1 ',10219 8(;93b3i,;7 00.^2573 21-0-^t75052 7-6_'3l51 4-14 ic7'->(j 87528384 0(.22522 21-0713075 7-02c?883 4-15 I9b025 b '12 1125 0022-172 21-095/231 ; -03 iOOO 4 10 10811 10 88710530 0022-122 21-1 I8712] 7 64;:32i 4-; 7 \U[)0[, 8931 402 > 0022371 2i-l i-2:-745 7-040027 44 M }o:o\ 8i,9J539i 0022321 21-100010-. 7-6517,.; 44i.} J.llCUI (,05 1 WMg 0022^72 2rt89(r20i 7-''5;4i4 f5M 1 ;() 'M.'i; 9; 125{-; 0(!222,r2 1 21-2132034 1 7-0 /3094 TR. 25. SaUARES, CUBES, RECIPROCALS, AND ROOTS. 475 Numb. Square, Cube. Recipr. i Sq Root. C. Root. 451 203401 91733851 00221 73 2 1-2307606 , 7-o6S760 452 204304 923454O8 0022124 C 1-26029 '6 7-674430 453 205209 92959677 0022075 2r2S37967 7 -61-00.^5 454 20()H6 g.if)7i5664 PO22O26 2 1-307275 ; 7-6&5732 455 207025 9419S375 002 J 978 21-3307290 ; 76 ! 1371 456 207936 94818810 002 1 930 2 i -3541565 ; 7-6970(^2 457 208S49 95443993 0021*882 21-3775583 '7702624 458 209764 96071912 00^1834 21-4009346 7-708^33 459 210081 90702579 0021786 21-4242853 , 77l38-(4 460 211600 97336000 0021739 21-4476166 7-719442 401 212521 97972 181 0021692 21-47(^106 7-72c052 462 213444 98611128 0021645 21-49+1853 ; 7730614 46.i 214369 99252847 0021598 21-5174348 ; 7-7361 87 464 215296 P9897344 0021552 21-5406592 : 7741753 465 2 i 6225 100544625 0021505 21-5638587 i 7747310 466 217156 101194696 0021459 21-5870331 ' 7-752SOO 467 2I8O89 101847563 0021413 21-6101828 ' 7-758402 46s 219024 102503232 0021308 21-6333077 ' 7-763936 469 219961 103161709 0021322 21-6564078 ! Ty()g46'i 470 220900 103823000 0021277 21-6794834 ! 7-774980 471 221841 104487111 002 1 23 1 21-7025344 7780490 4/2 'Z22784 105154048 002 1 1 86 21-7255610 7-785992 473 223729 105823817 0021142 21-7485632 7791487 474 224676 106490424 0021097 21-7715411 7-796974 475 225625 107171875 0021053 21-7944947 7-802453 470 226076 107850176 0021003 21-8"i74242 7 -807 92 5 477 227529 108531333 CO20964 21-8403297 7-813389 478 228484 109215352 0020921 21-8632111 7-818845 479 229441 109902239 0020S77 21-8860686 7-824294 480 230400 1 I0592a)0 0020833 21-9089023 7-829735 481 231361 111284641 0020790 21-9317122 7-83516S 482 232324 III98OI68 0020747 21-9544984 7-840594 483 2332S9 112678587 0020704 21-9772610 7-846013 484 234256 113379904 0020661 220000000 7-851424 485 235225 1140S4125 0020619 22-0227155 7-856828 486 236106 114791256 0020576 2 2 '0454077 7-862224 487 237169 115501303 0020534 22-0680705 .786761 3 488 236144 116214272 GO20492 ! 22-0907220 7-872994 489 23cjl21 116930169 0020450 22-1 J 33444 7-87S368 490 240100 11764900b 0020 108 22-1359430 7-883734 491 241081 11 8370771 0020367 22-I5S5193 7-8S9094 492 242C61 119095488 0020325 22- IS 10730 7-894446 493 243049 119823157 0020284 22-2036033 7-^99791 494 244036 120553784 0020243 22-2261 lOS 7-905129 495 245025 I212S7375 0020202 22-2485955 7-910100 496 246016 i 122023936 0020162 22-2710575 7-915734 497 247009 122763473 0020121 22-2934968 7-921100 498 248G04 123505992 0020080 22-3159136 7-926408 499 249001 124251499 00200-iO 22-3383079 7 931710 500 2;5000C ) 125000000 002 22-3600798 7-937^)05 176 SaUARES, CUBES, RECIPROCALS, AND ROOTS. TR. 25. Numb. 501 bquare. Z5i001 Cube. Recipr. 0()li,>jj6'0 Sq. Root. 22-3830293 C Root. 125/51.101 7 942293 502 252004 1 2050-008 0019."*20 22--^.0.3505 7'947.';73 1 503 25300() 127^63027 00 1 9881 22-4/7U015 7-952847 j 504 254010 1 >b0240*:i4 (;Ql.9f>4i 22-44 994 -13 7'958n4 505 255025 1287^70'25 0019S01 -22-472/05 1 7-963374 5C6 25u030 120554216 f301976"3 22-49-!4i38 7-968D27 507 2570-19 130323843 0019724 22-5 160605" 7-973873 508 25806-1 J 3 10905 12 0015^685 22'53ti8553 7-979112 509 25908! 131872229 0019046 22-5610283 7-984344 510 260100 132651000 001 9608 22-/ 83 1796 7-989.069 511 201 121 133432831 00i9j69 22-6053091 7-994788 512 262144 1342177^8 CO; 9531 22-6274 1 70 8 000000 513 263169 135005697 00 19493 22-6495033 8-001205 514 2641. 6 13579^744 0019455 22-1715631 8-010403 515 265225 I3659O&75 0019417 22-6930114 8-0:5595 516 266256 137388O9J 0019380 22-7150334 8-0207:9 517 26/2S9 J381884I3 00 1 9342 22-737634i> 8-025957 518 268324 138991 S32 Ott 19305 >2-;og() 34 8-Oil 129 519 269;}6l 139798359 0019268 22-78I 57 15 8-036293 520 270400 1 40608000 0019231 22-8035085 8-0414^1 521 271441 141420761 0019191 22-8254244 8 04 6603 522 272'i84 142230648 00191.57 22-8473193 8 051/48 523 27352g 143055607 O0l9<20 22-86919.33 8-056886 524 27i576 143377&21 001 9084 22-8910463 8-06201 8 525 275625 144703125 0019048 22-9128785 8-067 1 43 52(3 270U76 14r.53]576 00(9011 22 9340899 8-072262 527 277729 146363183 0018975 22 -9564805 8-077374 528 2/8784 147197952 00189.39 22-9;82506 8-052480 5JQ 279811 14803.'.S&9 0018904 23-0: '00000 8-O87579 530 280000 14fc8770^-b 0018868 23-0217289 s -09267 '2 531 281901 140721291 CK) 18832 23-0434372 8-097/58 532 /8sb24 1.0056^768 0018797 23-0051252 1 8-102838 533 284059 151419437 (XJ 18762 23 .0867928 8-107912 534 285156 152273304 OOIS72; 23-10844001 8-112980 535 286225 153130375 0018692 23-1300670 8- 1 i 804 1 I 536 2K7296 153990656 0018657 23-1510738 8-123096 537 288300 154854153 OOI8622 23-17326(35 8-128144 538 259444 I5572O872 0018.587 23-1948270 8-133J86 539 2U0521 I5659O8I9 0018553 23-2103735 8-138223 540 29 1 OGO 15740400b 0018518 23-2379001 8-143253 1 541 2u208 1 15b34042l 0018484 23 ^S' 4067 8-1 48270" 1 5-12 293704 I5922(,0SS 0018150 23-280^^035 s- 153293 1 543 2v4SJ9 i6b)o,>oo7 CKJ18416 23-3023604 8-158304 5^14 2959J6 1009S9184 0018382 23-32'^8076 8- 1 63309 545 297025 !6l87b625 0018349 23-345 ?351 8-168308 5A6 2()81 16 162771336 0018315 23 3666429 8-173302 547 21^9209 i 6361 ;7323 0018282 23 -386031 1 8- 1 78289 548 3(XJ3u4 1 64506592 0018248 23 --1093998 8-183269 549 301401 165469)49 0018215 23-43074.90 8-188244 550 3025G0 166375000 001 8 181 23-4520788 8-1.93212 TR. 25. SQUARES, CUBES, RECIPROCALS, AND ROOTS. 47T Numb. Square. Cube. Recipr. Sq. Root. C. Root. 551 ci*;36>' 167284151 00.8149 23-473 J Oy2 Fi98r7T 332 30-, 701: J 6810' 608 0018116 23 '494 0602 8-203131 55.i 305 009 logii 2:^77 0018083 23-5 159520 8' 208082 554 3 069 ) 6 j;00ol404 0018051 23 "53 7 2040 8-.> 13027 355 30Li025 i, Oj53fv5 001fc018 23'5o8j380 8-.>i7965 33Q 309 1 06 171879616 0017986 23 57i^6522 s- ^22898 557 3] 02-19 172 803090 0017953 23-600c.4:'4 8-227825 558 31130-* 173741112 0017921 23 -622023 a 8-232746 55.0 312461 174670879 0017889 23-0431808 8-237001 560 3 1 3(>00 I75616OOO 0017857 23-6043191 3 567 321489 18228.203 0017037 23-8117618 8 -2767 7 '-i 568 322024 183250432 0017606 23-8327500 8-281635 569 323761 1 84220'JOi^ 001757*5 23-85*7209 8-28649;^ 5/0 32490O 185 103-0-0 0017544 23-8746728 8-291344 571 326041 186l6r4ll OJI7513 23-8950063 8 296 190 57'^ 327184 187149248 0017483 23-9165215 8-301030 573 3283-29 188132517 0017452 23-93741 b4 8-30565 574 329^76 IS9IJ9224 0017422 239 ^297 i 8-3 i 0694 575 330025 190109375 0017391 23-9791570 8-3'55l7 576 33177O 191102970 0017361 24OOLOJOO S 320335 577 3329.-9 192 100033 0017331 240208243 8-325147 5/8 334084 193100552 CO 17301 24-041 0306 8-329954 579 335241 194104539 0017271 24 -0024 188 8-3.^4755 5cO 336400 195112000 001/241 24-083 1892 8-33955 I 581 337561 1961 22941 0017212 24-lOiQ4l6 8 344341 582 336724 197137368 0017182 24- 1246/62 8-349125 583 339889 ll;8 155287 0017153 24-1453929 8-3539')4 584 341056 199170704 OO17123 24-1660919 8-35078 585 3^2225 200201625 001,094 24-1867'7.-J2 8 63446 586 343390 201230056 0017065 24-2074369 8-308209 587 344509 202262003 0017036 24'2280>*29 S-372.i66 588 345744 203297472 0017007 24-24871.3 8-3,7718 589 346921 204336469 0016978 24-2603^22 8-3b2405 590 348100 205379000 00. 69^; 9 24 -2899 150 8-387200 591 3492s 1 2f)6-l 25071 001692b 24-3104916 8-391942 5Q2 350464 207474688 0016891 24-3310501 8-390673 93 351049 20b527857 00168*13 24-3515913 8-401398 594 3 5 283 6 209584584 0016835 24-3721 152 8-4001 18 595 354025 2 1 0644875 OOIO8O7 24-3926218 8-41 083 i 596 355216 2117O8736 0016779 24-4131112 8-415541 597 35640;) 2127761/3 0016750 .i4 -433 5834 8-4_0-'.i-5 598 35/604 21:^847192 0016722 24-4540385, 8-424944 599 3 5 8801 214921799 00 1 6694 24-4744765 8- 4-' 963 8 600 S6OUOO 2 1 ooooobb 0016006 24-4948974 8-434327 478 SQUARES, CUBES, RECIPROCALS, AND ROOTS. TR. 25. Numb. Square. Cube. Recipr. | Sq. Root. C. Root. doi 361201 2I708!8()I 00 1 0039" 24T1 TioTo" 8-439009 d02 362404 J 18 J 0/208 OOlOoll 24'53508b3 8-443687 b03 363609 2 1 925O227 0010584 ! 24-5560j83 8-44830O 604 364810 2'2O348804 0016556 24-57641 5 8-+53027 605 366025 '221445125 0016529 24-5967478 8-457689 600 30/236 222.'j45010 0016501 : 24-6170673 6-40-347 607 36.-i44(i .'2.i048543 0016474 1 24-6373700 8--i60;j99 60S 36:iJbf 2247557:2 0016447 i 24O57056O 8-471647 609 37U&t J 225b6Lc29 0016420 240779'^54 8-476289 610 372100 22098 J OUO 0016393 24-6981781 8-480926 oil 373321 228*099131 00 i 0367 1 24-7'] 84 142 8-485557 612 374544 2292.;a2y 0016340 24-7386338 8-490184 613 375761; 230346397 0016313 24-7588368 8-494SO6 614 37099:; 23 147? 5 44 0016287 1 24-7790234 8-499423 615 378225 <>320O8cJ 75 OOIO26O 1 ^47991935 8 504034 616 37:^45,. 233744B96 0010234 j 24-8J 93473 8-508041 617 380689 234885 1*13 0016207 24-8394847 8-513243 618 38 1924 -23 00290;:, 2 00 10] 81 '. 2i-8.i"u0058 8 517840 619 383 l6t 237 76059 OOIO155 24-8797100 8-522432 620 384400 238328000 001 129 24-b(, 97(^-92 8-5270 ! 8 621 385641 23948306i 0016103 24-91^5710 S-531OOO 622 3bc8b4 240ti4ib48 0016077 '^'^9399..78 8-53U1 77 623 3 8a 129 '241804307 00 U 051 '^'* 9 99-79 8-540749 624 3893/0 \'42 y/002i 0016026 2-* 97999^" 8-545317 625 3(:;Oj2) 244140025 0016 2^-ot)OX):)o S-549879 626 39I&7O 245314376 0015974 25-0] 09920 8-554437 027 393129 240491 883 0015949 25-039* 168 i 8 558990 628 394384 247073152 0015924 25o:)i,y2H2 ,N- 5 03 53 7 629 39504 ; 248S58189 00 1 589s 25-(J7gs724 8-568080 630 39O900 250047000 0015073 1 25-0 ,(8008 s-57201.:. 631 398161 25 1239^()i 0015848 2j-i"i;,7134 8-5/7152 632 399424 '252435908 00(5823 I 25 K^(i6l02 8-581680 633 40(j689 253030137 0015798 25-i5C)4913 8 -580204 634 40195*6 254840104 0015773 25-)7();5,.60 S-590723 635 403225 25O047&75 0015748 25- 19(^2003 8-595238 636 40i !9'j 25725945O 0015723 25 "2 If, 0404 8-599747 0^7 406 769 258474 853 0015699 25 23885 b9 8-604252 638 407044 259094072 001 5 074 25-'25800ly 8 -008752 639 408321 2O091 71 19 0015649 25-2784493 8-613248 640 4O:)0OO 2O2 144000 0015U25 25-^982213 8-6 177;^ 8 6.11 410bbJ 2O33 74721 OOI5OOI 25-3179778 8-022224 642 412104 264OO92S8 0015576 25 -3377 J 9 8-026700 6-13 4! 3 119 265 847/07 0015552 25-3574447 8-031183 644 414730 2O7O899S4 0015528 25-377 1551 8-035055 645 4l(H)25 208330125 00 1 5504 25-39G8502 8 040 122 6iG 41/310 2O9586136 0015480 25-4l65.iOI 8-044585 647 418009 270840023 0015456 25-43619^17 8-049043 648 4 1 9^i04 27209779'2 0015432 25-455844 1 8-053497 649 421201 273359449 0015408 25-4/54784 8-05 79*40 650 422500 1 974025000 0015385 25-4950070 8-0623O1 TR. 25. SaUARES, CUBES, RECIPROCALS, AND ROOTS. 479 Numb. 051 Square. 423801 Cube. Recipr. Sq. Root. C. Root. 27589445 1 0015361 25-5147010 8-666331 652 425 1 04 277 167 808 0015337 25-5342907 8-671266 653 426409 273445077 0015314 2D'55386ni7 S-675697 054 427716 2797'i62Q4 0015291 25-5734237 8-680123 055 429035 2810! 1375 0015267 25-5929678 8-684545 656 430336 2823004 i 6 0015244 25-6124969 8-688963 657 431649 233593393 0015221 25-6320112 8-693376 658 432964 2848903 1 2 0015198 25-6515107 8-097784 65q 43^281 286191 J 79 0015175 25-6709953 S -702 138 660 435600 237496000 0015151 25-6904(552* 8-70S587 661 4S6921 28380-^731 0015129 25-7099203 8-710932 662 438244 290117528 0015106 25-7203607 8-715373 663 439569 291434247 0015083 2^7437304 8-719759 664 440S96 292754944 0015060 25-7681975 8-724141 665 442225 294079625 0015038 25-7875939 8-728518 666 443556 2954OS296 0015015 25-3069758 8-732391 667 44 J 8 89 296740963 0014993 25-8263431 8-737260 668 446224 298O77632 0014970 25-3456960 8-741624 669 447561 299418309 0014948 25-8650343 8-745984 670 448900 300763000 004925 25-8843582 8-750340 671 45024 1 3021 11711 0014903 25-9036677 8 754691 672 451584 303464448 0014381 25-9229628 8-759038 673 452929 304821217 0014859 25-9422435 8*763380 674 454276 306] 82024 0014837 25-9615100 S-7Q7719 675 455625 307516375 0014814 25-9307621 S-772053 676 456976 303915776 0014793 26-0000000 8-776382 677 458329 3lO:8v33 0014771 26-0192237 8-780708 678 459684 311665752 0014749 26-0384331 8-735029 679 461041 313040839 0014728 26-0570284 8739346 660 462400 314432000 OO147O6 26-0763096 8-793059 631 463761 3 1 582 i 241 00146&4 26-0959767 8-797967 6b2 465 1 24 317214563 0014663 26-1151297 8-802272 6s3 466489 318611937 0014641 26-1342637 8-806572 684 467856 320013504 0014620 20-1533937 8 -S J 0368 685 469225 321419125 0014599 26-1725047 8-815159 686 470590 322828850 0014577 2fi- 191 6017 S-819447 687 471969 324242703 0014556 26-2 1 06S4B 8'823730 688 473344 325660672 0014535 26-22,1,7541 8'828009 689 474721 327O82769 0014514 26-2488095 8"832285 6yO 476100 32350900b 0014493 26-207851 1 8*836556 6qi 477481 329939371 0014472 26-28(,S78y 8-840322 692 478864 33137>^888 0014451 26-3058,y29 8*845085 693 48O249 332312557 0014430 26-324 8932 8-849344 694 431630 33^255334 0014409 26-3433797 8-853598 695 4830^5 335702375 00l43Hb 26-3623527 8-857849 6q6 4844 1 6 337153536 0014308 26-33181 19 1 8-862005 1 697 435809 333603373 0014347 26-4007576 8-866337 6y8 487204 340063392 0014327 26-4190896 8-870;75 099 488001 3415;-} i099 0014306 i6-4386061 ! 8-874809 700 l-r>00(X) 343000000 (X) 14280 26-4575131 8-8790^0 480 SQUARES, CUBES, RECIPROCALS, AND ROOTS. TR. 25. Numb. 701 Square. 491-101 Cube. Recipr. Sq. Root. 1 C. Root. 1 344472101 0014265 26-4764046 8-883266 702 492804 3459-48008 0014245 26-4952826 8-887488 703 -191209 ,347428927 0014225 26-5141472 8-891 706 704 49.5610 348913664 0014205 26-5329983 8-895920 705 497025 350402625 0014184 26-5518361 8-900130 70(i 493436 351895816 0014164 26-5706605 8-904336 707 499S49 353393243 0014144 26-5894716 8-908538 70s 501264 354894912 0014124 20-6082694 8-912736 709 5026s I 356400629 0014104 26-6270539 8-916931 710 501100 357911000 0014085 26-6458252 8-921121 711 505 321 359425431 0014065 26-6645833 8-925307 712 506944 360(^44 128 0014045 26-6833281 8-929490 713 508309 362467097 0014025 26-7020598 8-933668 714 50979^ 363994344 0014006 26-7207784 8-937843 715 511225 365525875 0013986 26-7394889 8-942014 7\0 512650 367061696 0013966 26-7581763 8-9461 80 717 514089 368601 813 0013947 26-7768557 8-950343 718 515524 370146232 0013928 26-7955220 8-954502 719 516961 371694959 OOI39O8 26-8141754 8-958658 720 5 1 S400 373248000 0013888 26-8328157 8-96280( 721 519841 374805361 0013870 26-8514432 8-g6695"7 722 521284 376367OJ8 0013S50 26-8700577 8-971100 723 522729 3779^3067 001383 1 26-8886593 S-975240 724 524176 3/9503424 0013812 26-90724 SI 8-979376 72J 525625 3810/8125 0013793 26-9258240 8-983508 726 527076 3S.i657176 0013774 26-9443872 8-987637 727 528529 38424058 i 0013755 26-9629375 8-991762 723 520984 385828352 0013736 26-9S 14751 8-9958S3 729 531441 3874204S9 0013717 27-OOOOC nnf-n ijii i.:i Form L9- .KAV 3 2004 QUARTER LOAHi [3 3^j.k ^QJ^ ^ ^ Jl^i-i^ri OA Xa 000 081418 6 3 H97t v.l I jf**-.