Presentxu bn ^eq^vcUcA f) . €8 .16 Sr- ^tr- CncpclopeUta Untaumca OR, A DICTIONARY OF ARTS, SCIENCES, AND MISCELLANEOUS LITERATURE; ENLARGED AND IMPROVED. THE FIFTH EDITION. JRtWtrateo UJitf) nearly stir fjunOceti CngcaWngs. VOL. XIII. INDOCTI DISCANT; AMENT MEMINISSE PERITI. EDINBURGH : Printed at the Encyclopaedia Press, FOR ARCHIBALD CONSTABLE AND COMPANY, AND THOMSON BONAR, EDINBURGH: GALE, CURTIS, AND FENNER, LONDON ; AND THOMAS WILSON AND SONS, YORK. 1815. ♦ ♦ **>• c KV pv X» - ^ ui ?>. 1993 Z' 0'' Encyclopedia Britannica. MAT 'Material, Btfaterial- ifts. MATERIAL, denotes fomething compofed of matter. In which fenfe the word Hands oppofed to immaterial See Matter and Meta¬ physics. 1 LRIALISTS, a feft in the ancient church, compofed of perfons who, being prepoffeffed with that maxim in the ancient philofophy, Ex nihilo nihiljit. Out of nothing nothing can arife,” had recourfe to an internal matter, on which they fuppofed God wrought in the creation ; inftead of admitting God alone as the foie caufe of the exiftence of all tilings. Tertullian vi- M A T goroufly oppofes the dodrine of the materialifls in his Material- treatife againft Hermogenes, who was one of their ifts- number. Materialists is alfo a name given to thofe who maintain that the foul of man is material $ or that the principle of perception and thought is not a fubftance diftinft from the body, but the refult of corporeal orga¬ nization : See Metaphysics. There are others, call¬ ed by this name, who have maintained that there is nothing but matter in the univerfe j and that the Deity himfelf is material. See Spinoza. MATHEMATICS. Definition TV/fA I HEMATICS is divided into two kinds, e mathe- ancj mixed. In pure mathematics magnitude is confidered in the abftraH $ and as they are founded on the. fimpleft notions of quantity, the conclulions to which they lead have the fame evidence and certainty as the elementary principles from which thefe conclu¬ lions are deduced. I his branch of mathematics com¬ prehends, i. Arithmetic, which treats of the properties of numbers. 2. Geometry, which treats of extcnfion as endowed with three dimenfions, length, breadth, and thicknefs, without confidering the phyfical qualities infeparable from bodies in their natural ftate. 3. Al¬ gebra, fometimes called univerfal arithmetic, which compares together all kinds of quantities, whatever be their value. 4. 1 he direB and inverfe method of F/ux- ions, (called on the continent, the differential and inte-' gral calculi f which confider magnitudes as divided in¬ to two kinds, eonftant and variable, the variable magni¬ tudes being generated by motion ; and which deter¬ mines the value of quantities from the velocities of the motions with which they are generated. Mixed Mathe¬ matics is the application of pure mathematics to certain elfablithed phylical principles, and comprehends all the phyfico-mathematical fciences, namely, 1. Mechanics ; 2. Hydrodynamics; 3. Optics; 4. Afronomy; 5. A- coufics ; 6. EleBricity ; and, *]. Magnetifm. The hi- Hoiy of thefe various branches of fcience having been given at full length, we {hall at prefent direft the at¬ tention of the reader to the origin and progrefs of pure mathematics. 2. In attempting to difcovcr the origin of arithmetic Vol. XIII. Part I. and geometry, it would be a fruitlefs talk to conduct the reader into thofe ages of fable which preceded the records of authentic hiftory. Our means of informa¬ tion upon this fubjeft are extremely limited and im- perfeft j and it would but ill accord with the dignity of a fcience whofe principles and conclufions are alike irrefiftible, to found its hiftory upon conjecture and fable. But notwithftanding this obfcurity in which The fci- the early hiftory of the fciences is enveloped, one thing ences origl- appears certain, that arithmetic and geometry, and fomenatec^ of the phyfical fciences, had made confiderable progrefs in Egypt, when the myfteries and the theology of that favoured kingdom were tranfplamed into Greece. It is highly probable that much natural and moral know¬ ledge was taught in the Eleufinian and Dionyfian my¬ fteries, which the Greeks borrowed from the Egyptians, and that feveral of the Grecian philofophers were in¬ duced by this circumftance to travel into Egypt, in fearch of thofe higher degrees of knowledge, which an acquaintance with the Egyptian myfteries had taught them to anticipate. We accordingly find Thales and a C. 640. Pythagoras fucceflively under the tuition of the Egyp- A. C. 590. tian pricfts, and returning into Greece loaded with the intellectual treafures of Egypt. By the eftablilhment of the Ionian fchool at Miletus, Thales inftruCled his Difcove^ countrymen in the knowledge which he had received, of Thales, and gave birth to that fpirit of inveftigation and dif- covery with which his followers were infpired. He taught them the method of afcertaining the height of the pyramids of Memphis by the length of their fhadows •, and there is reafon to believe that he was the A firft MATHEMATICS. firft wlio employed tKe circumference of a circle for the menluration of angles. That he was the author of greater difcoveries, which have been either loft or a- fcribed to others, there can be little doubt j but thefe are the only fails in the hiftory of Thales which time has fpared. Difcoveries 3. The fcience of arithmetic was one of the chief of Pythago-branches of the Pythagorean difcipline. Pythagoras -as' attached feveral myfterious virtues to certain combina¬ tions of numbers. He fwore by four, which he regard¬ ed as the chief of numbers. In the number three he fuppofed many wonderful properties to exift } and he regarded a knowledge of arithmetic as the chief good. But of all Pythagoras’s difcoveries in arithmetic, none have reached our times but his multiplication table. In geometry, however, the philofopher of Samos feems to have been more fuccefsful. The difcovery of the ce¬ lebrated propofition which forms the 47th of the firft. book of Euclid’s Elements, that in every right-angled triangle the fquare of the fide fubtending the right angle is equal to the fum of the fquares of the other two Tides, has immortalized his name j and whether we confider the inherent beauty of the propofition, or the extent of its application in the mathematical fciences, we cannot fail to clafs it among the moft important truths in geometry. From this propofition its author concluded that the diagonal of a fquare is incommen- fuiate to its fide $ and thus gave occafion to the dif¬ covery of feveral general properties of other incom- menfurate lines and numbers. 4. In the time which elapfed between the birth of Py¬ thagoras and the deftrudlion of the Alexandrian fchool, the mathematical fciences were cultivated with great ar¬ dour and fuccefs. Many of the elementary propolitions of geometry were difcovered during this period ; but hi¬ ftory does not enable us to refer each difcovery to its proper author. The method of letting fall a perpendi¬ cular upon a right line from a given point (Euclid, B. I. prop, xi.) ;—of dividing an angle into two equal parts, (Euclid, B. I. prop, ix.) *, and of making an angle equal to a given angle, (Euclid B. I. prop, xxiii.) were in- Difcovcries vented by Oenopidus of Chios. About the fame time »f Oenopi- Zenodorus, feme of whofe writings have been preferved dus andZe-ky Theon in his commentary on Ptolemy, demonftrated, r.o orus. 0pp0fitj0n to t]ie 0plnion then entertained, that ifo- perimetrical figures have equal areas. Coeval with this difcovery was the theory of regular bodies, for which we are indebted to the Pythagorean fchool. The cele- 5. About this time the celebrated problem of the du¬ brated pro- plication of the cube began to occupy the attention of blem of thethe Greek geometers. In this problem it was required of the^ube to conftruCt a cube whofe folid content fliould be propofed double that of a given cube ; and the afliftance of no and invefti-other inftrument but the rule and compafies was to be gated. employed. The origin of this problem has been a- fcribed by tradition to a demand of one of the Grecian deities.. The Athenians having offered fome affront to Apollo, were affli&ed with a dreadful peftilence 5 and upon confulting the oracle at Delos, received for an- fwer, Double the altar of Apollo. The altar alluded to happened to be cubical; and the problem, fuppofed to be of divine origin, was inveftigated with ardour by the Greek geometers, though it afterwards baffled all their acutenefs. The folution of this difficulty was attempt- A. C. 450. ed by Hippocrates of Chios. He difcovered, that if 3 two mean proportionals could be found between the fide of the given cube, and the double of that fide, the firft of thefe proportionals would be the fide of the cube fought. In order to effedl this, Plato invented an inftrument compofed of two rules, one of which moved in grooves cut in twro arms at right angles to the other, fo as always to continue parallel with it j but as this method was mechanical, and likewife fuppofed the defcription of a curve of the third order, it did not fa- tisfy the ancient geometers. The dodlrine of conic Conic-fec- fedlions, which was at this time introduced into geo-tions difco- metry by Plato, and which was fo widely extended as vered by to receive the name of the higher geometry, was fuccefs-^ q' fully employed in the problem of doubling the cube. Menechmus found that the two mean proportionals men¬ tioned by Hippocrates, might be confidered as the ordi¬ nates of two conic fedlions, which being conftrudled ac¬ cording to the conditions of the problem, would interfedl one another in two points proper for the folution of the problem. The queftion having affumed this form, gave rife to the theory of geometrical loci, of which fo many important applications have been made. In doubling the cube, therefore, we have only to employ the inftru- ments which have been invented for defcribing the conic feel ions by one continued motion. It was after¬ wards found, that inftead of employing two conic fec- tions, the problem could be folved by the interfedlion of the circle of the parabola. Succeeding geometers em¬ ployed other curves for this purpofe, fuch as the con- A., C. 2 So. choid of Nicomedes and the ciffoid of Diodes, &c. A.. C. 460. An ingenious method of finding the two mean propor¬ tionals, without the aid of the conic feftions, was after- A. D. 400. W'ards given by Pappus in his mathematical colledlions. 6. Another celebrated problem, to trifefl an angle, The trifec- was agitated in the fchool of Plato. It was found that this b011 °f an problem depended upon principles analogous to thofe of an^e' the duplication of the cube, and that it could be con- ftruded either by the interfedion of two conic fedions, or by the interfedion of a circle with a parabola. Without the aid of the conic fedions, it was reduced to this fimple propofition :—To draw a line to a femicircle from a given point, which line fhall cut its circumfer¬ ence, and the prolongation of the diameter that form* its bafe, fo that the part of the line comprehended be¬ tween the two points of interfedion fhall be equal to the radius. From this propofition feveral eafy conftruc- tions may be derived. Dinoftratus of the Platonic fchool, and the cotemporary of Menechmus, invented a curve by which the preceding problem might be folved. It had the advantage alfo of giving the multiplication of an angle, and the quadrature of the circle, from which it derived the name of quadratrix. 7. While Hippocrates of Chios was paving the way for Hippe- the method of doubling the cube, which was afterwards crates’s given by Pappus, he diftinguifhed himfelf by the qua- drature of the lunulae of the circle ; and had from this circumftance the honour of being the firft who found a curvilineal area equal to a fpace bounded by right lines. He w7as likewife the author of Elements of Geometry, a work, which, though highly approved of by his co¬ temporaries, has fhared the fame fate with fome of the moft valuable produdions of antiquity. 8. After the conic fedions had been introduced into geometry by Plato, they received many important ad¬ ditions from Euxodus, Menechmus, and Arifteus. The latter 3 MATHEMATICS. A. C. 380. latter of tliefe philofopliers wrote five books on conic fec- tions, which, unfortunately for fcience, have not reached A. C. 300. our Elements of g. About this time appeared Euclid’s Elements of Eudul. Geometry, a work which has been employed for 2000 years in teaching the principles of mathematics, and which is ftill reckoned the moll complete work upon the fubjeft. Peter Ramus has aferibed to Theon both the pro- pofitions and the demondrations in Euclid. It has been the opinion of others that the proportions belong to Euclid, and the demonftrations to Theon, while others have given to Euclid the honour of both. It fee ms moll probable, however, that Euclid merely colleded and ar¬ ranged the geometrical knowledge of the ancients, and that he fupplied many new proportions in order to form that chain of reafoning which runs through his ele¬ ments. This great work of the Greek geometer con- lifts of ftfteen books : the eleven ftrft books contain the elements of pure geometry, and the reft contain the general theory of ratios, and the leading properties of commenfurate and incommenfurate numbers. Dlfcoveries 10- Archimedes, the greateft geometer among the an- Archi- cients, flourifhed about half a century after Euclid, inetles. pje was the ftrft who found the ratio between the dia- A. C. J50. meter 0f a circle and its circumference 5 and, by a me¬ thod of approximation, he determined this ratio to be as 7 to 22. This refult was obtained by taking an arithmeti¬ cal mean between the perimeters of the inferibed and circumfcribed polygon, and is fufficiently accurate for every pra&ical purpofe. Many attempts have ftnee been made to aflign the precife ratio of the circumfer¬ ence of a circle to its diameter *, but in the prefent ftate of geometry this problem does not feem to admit of a folution. The limits of this article will not permit us to enlarge upon the difeoveries of the philofopher of Syracufe. We can only ftate, that he difeovered the fuperftcies of a fphere to be equal to the convex furface of the circumfcribed cylinder, or to the area of four of its great circles, and that the folidity of the fphere is to that of the cylinder as 3 to 2. He difeovered that the folidity of the paraboloid is one half that of the cir¬ cumfcribed cylinder, and that the area of the parabola is two thirds that of the eircumfcribed re&angle *, and he was the ftrft who pointed out the method of drawing tangents and forming fpirals. Thefe difeoveries are contained in his works on the dimenfton of the circle, on the fphere and cylinder, on conoids and fpheroids, and on fpiral lines. Archimedes was fo fond of his difeovery of the proportion between the folidity of the fphere and that of the cylinder, that he ordered to be placed upon his tomb a fphere inferibed in a cylinder, and likewife the numbers which exprefs the ratio of thefe folids. Hifcoveries u. While geometry was thus advancing with fuch . AP0ll°- rapid fteps, Apollonius Pergaeus, fo called from being A^C 20c b°rn at Perga in Pamphylia, followed in the fteps of Archimedes, and Avidely extended the boundaries of the fcience. In addition to feveral mathematical works, which are now loft, Apollonius wrote a treatife on the theory of the conic fettions, which contains all their properties with relation to their axes, their diameters, and their tangents. He demonftrated the celebrated theorem, that the parallelogram deferibed about the two conjugate diameters of an ellipfe or hyperbola is equal to the rectangle deferibed round the two axes, and that the fum or difference of the fquares of the two conjugate diameters are equal to the fum or difference of the fquares of the two axes. In his fifth book he de¬ termines the greateft and the leaft lines that can be drawn to the circumferences of the conic fe&ions from a given point, whether this point is fituated in or out of the axis. This work, which contains every where the deepeft marks of an inventive genius, procured for its author the appellation of the Great Geo?neter. 12. There is fome reafon to believe, that the Egyp-Mcnelaus tians were a little acquainted with plane trigonometry 5 and there can be no doubt that it was known to the Greeks. Spherical trigonometry, Avhich is a more difficult a. D. 55. part of geometry, does not feem to have made any pro- grefs till the time of Menelaus, an excellent geometri¬ cian and aftronomer. In his work on fpherical triangles, he gives the method of conftrufting them, and of refolv- ing moft of the cafes wffiich were neceffary in the an¬ cient aftronomy. An introduction to fpherical trigonome- Theodo- try had already been given to the world by Theodofius'ftus s ^IC" in his Treatife on Spherics, where he examines the ^ ‘c relative properties of different circles formed by cutting a fphere in all directions. 13. Though the Greeks had made great progrefs in Progrefs of the fcience of geometry, they do not feem to haveanabfts- hitherto conftdered quantity in its general or abftraCt ftate. In the writings of Plato wTe can difeover fome- thing like traces of geometrical analyfis; and in the feventh propofitien of Archimedes’s work on the fphere and the cylinder, thefe traces are more diftinftly mark¬ ed. He reafons about unknown magnitudes as if they were known, and he finally arrives at an analogy, which, ■when put into the language of algebra, gives an equa¬ tion of the third degree, which leads to the folution of the problem. 14. It was referved, however, for Diophantus to lay The analy- the foundation of the modem analyfis, by his invention of the analyfis of indeterminate problems •, for the me- probiems thod which he employed in the refolution of thefe pro- invented by blems has a ftriking analogy to the prefent mode of re- Diophan- folving equations of the iff and 2d degrees. He was^s- likewife the author of thirteen books on arithmetic, fe- ‘ •’^c‘ veral of which are now loft. The works of Diophantus were honoured with a commentary by the beautiful and learned Hypatia, the daughter of Theon. The fame a. D. 410. fanaticifm which led to the murder of this aceompliftied female was probably the caufe that her works have not defeended to pofterity. 15. Near the end of the fourth century of the Chriftian Mathema- era, Pappus of Alexandria publiffied his mathematical c<^ec* collections, a work which, befides many new propofi- tions of his own, contains the moft valuable productions of ancient geometry. Out of the eight books of which 400* this work confifted, two have been lofty the reft are oc¬ cupied with queftions in geometry, aftronomy, and me¬ chanics. 16. Diodes, whom we have already had occafion to Difcovene* mention as the inventor of the ciffoid, difeovered the folu- o!~ •^^oc^cs» tion of a problem propofed by Archimedes, viz. to cut a fphere by a plane in a given ratio. The folution of Diodes has been conveyed to us by Eutocius, who wrote commentaries on fome of the works of Archi¬ medes and Apollonius, A. D. 5 20. About the time A 2 of 4 MATHEMATICS. and Sere- of Diodes flour idled Serenus, wlio wrote two books on I1US> the cylinder and cone, which have been publilhed at the end of Halley’s edition of Apollonius. Labours of 17. Geometry was likewife indebted to Proclus, the Proelus. head of the Platonic fchool at Athens, not only for his pa- - .D. 500. tronage of men of fcience, but his commentary on the firlt book of Euclid. Mathematics were alfo cultivated by Marinus, the author of the Introduction to Euclid’s Data ",—by Ilidorus of Miletus, who was a difciple of Proclus, and by Hero the younger, whofe work, en¬ titled Geodefia, contains the method of determining the area of a triangle from its three lides. Deftrudlion I^* ^^ile the mathematical fciences were thus flou- of Liic a- rilhing in Greece, and were fo fuccefsfully cultivated by Icxandrian the philofophers of the Alexandrian fchool, their very library. exiitence was threatened by one of thofe great revolu¬ tions with which the world has been convulfed. The dreadful ravages which were committed by the fuccef- fors of Mahomet in Egypt, Perfia, and Syria, the de- ftru&ion of the Alexandrian library by the caliph Omar, and the difperfion of a number of thofe illuftrious men who had Hocked to Alexandria as the cultivators of fcience, gave a deadly blow to the progress of geo- Revival of metry. When the fanaticifm of the Mahometan reli- faence. gion, however, had fubfided, and the termination of war had turned the minds of the Arabs to the purfuits of peace, the arts and fciences engaged their affeftion, and they began to kindle thofe very intellectual lights which they had fo afliduoufly endeavoured to extinguilh. The works of the Greek geometers were ftudied with care ; and the arts and fciences reviving under the auf- pices of the Arabs, were communicated in a more ad¬ vanced condition to the other nations of the world. 19. The fyflem of arithmetical notation at prefent a- dopted in every civilised country, had its origin among A. D. 960. the Arabs. Their fyftem of arithmetic was made known to Europe by the famous Gerbert, afterwards Pope Syl- vefter II. who travelled into Spain when it was under the dominion of that nation. 20. The invention of algebra has been afcribed to the Arabs by Cardan and Wallis, from the circumftance of their ufing the words fquare, cube, quadrato-quadra- tum, &c. inflead of the 2d, 3d, 4th, &c. powers as employed by Diophantus. But whatever truth there may be in this fuppofition, it appears that they were able to refolve cubic, and even biquadratic equations, as there is in the Leyden library, an Arabic MS. en¬ titled “ The Algebra of Cubic Equations, or the Solu¬ tion of Solid Problems.” Progrefs of 21. The various works of the Greek geometers were the Arabs tranflated by the Arabs, and it is through the medium in geomc- 0p an Arabic verfion, that the Hfth and lixth books of Apollonius have defcended to our times. Mahomet Ben Mufa, the author of a work on Plane, and Spherical Figures, and Geber Ben Aphla, who wrote a commen¬ tary on Plato, gave a new form to the plane and fpheri- cal trigonometry of the ancients. By reducing the theory of triangles to a few propofitions, and by fubfti- tuting, inftead of the chords of double arcs, the fines of the arcs themfelves, they fimplified this important branch of geometry, and contributed greatly to the abridge¬ ment of aftronomical calculation. A treatife on the art of furveying was likewife Avritten by Mahomet of Bagdad. 22. After the deftrucHon of the Alexandrian fchool 1 founded by Lagus, one of the fucceffors of Alexander, the difperfed Greeks continued for a while to cultivate their favourite fciences, and exhibited fome marks of that genius which had infpired their forefathers. The Mofcho- magic fquares were invented by Mofchopulos, a difco- Pulos’s di£- very more remarkable for its ingenuity than for itscoveryof praftical ufe. The fame fubject was afterwards treated by Cornelius Agnppa in his work on occult philofo- 4 phy ; by Bachet de Meziriac, a learned algebraift, about the beginning of the 17th century, and in later times by Frenicle de Beffi, M. Poignard of Bruflels, De la Hire, and Sauveur, 23. The fcience of pure mathematics advanced with a Ai£ebraini doubtful pace during the 13th, 14th, and 15th centu-troduced ries. I he algebra of the Arabians was introduced in- ‘nt0 Italy to Italy by Leonard of Pifa, who, in the courfe of his by Leonar!! commercial fpeculations in the call, had confiderable intercourfe with the Arabs. A work on the Plani- ’ * fphere, and ten books on arithmetic, were Avritten by Jordanus Nemorarius. The Elements of Euclid were A D. 1230. tranflated by Campanus of Novara. A work on alge-A. d! 125c.* bra, entitled Sutnma de Arithmetica, Geometria, Proper- tione et Proportiona/itate, was publifhed by Lucas Pac- cioli; and about the fame time appeared Regiomonta¬ nus’s treatife on trigonometry, which contains the me¬ thod of refolving fpherical triangles in general, when a. D. 140/, the three angles or three fides are knoivn. 24. During the 16th century, algebra and geometry advanced with rapidity, and received many new difeo- veries from the Italian philofophers. The formula for A.D. 150-. the folution of equations of the third degree was dif-A.d! 1535* covered by Scipio Ferrei profeffor of mathematics at Bologna, and perhaps by Nicholas Tartalea of Brefcia; and equations of the fourth order were refolved by LeAvis Ferrari, the difciple of Hieronymus Cardan of Bononia. This laft mathematician publifhed nine books of arithmetic in 1539 > and in I545 he added a tenth, containing the do&rine of cubic equations which he had received in fecrefiy from Tartalea, but Avhich he had fo improved as to render them in fome meafure his own. The common rule for folving cubic equations flill goes by the name of Cardan’s Rule. 25. The irreducible cafe in cubic equations was fuccefs- Difcoverlte fully illuftrated by Raphael Bombelli of Bologna. He of Bombell has fiioAvn in his algebra, Avhat Avas then confidered as ah- paradox, that the parts of the formula which reprefents ■A" ^ I57^* each root in the irreducible cafe, form, Avhen taken toge¬ ther, a real refult; but the paradox vanifhed Avhen it was feen from the demonftration of Bombelli that the imaginary quantities contained in the tAVo numbers of the formula neceffarily deftroyed each other by their oppofite figns. About this time Maurolycus, a Sici-Labours of lian mathematician, difeovered the method of fumming Mauroly- up feveral feriefes of numbers, fuch as the feries 1, 2,tus* 3, 4, &c.; 1, 4, 9, 16, &c. and the feries of trian-^j^ gular numbers, x, 3, 6, 10, 15, 21, &c. 57P* 26. The fcience of analyfis is under great obligations Djfcover;cs to Francis Vieta, a native of France. He introduced otViltT^ the prefent mode of notation, called literal, by employ-Born 1540. ing the letters of the alphabet to reprefent indefiniteDied given quantities ; and we are alfo indebted to him for the method of transforming one equation into another, Avhofe roots are greater or lefs than thofe of the origi¬ nal equation by a given quantity j for the method of multiplying or dividing their roots by any given num¬ ber, MAT HE her', of depriving equations of the fecond term, and of freeing them from fra&ional coefficients. The method which he has given for refolving equations of the third and fourth degree is alfo new and ingenious, and his mode of obtaining an approximate folution of equations of every order is entitled to ftill higher praife. We are alfo indebted to Vieta for the theory of angular fe&ions, the objeft of which is to find the general ex- preffions of the chords or fines for a feries of arcs that are multiples of each other. Logarithms 27. While analyfis was making fuch progrefs on the invented by continent, Baron Napier of Merchifton in Scotland was Baron !N a- krjncrinnr to perfeftion his illuftrious difcovery of the lo- Born 1550. garithms, a fet of artificial numbers, by which the moft Died 1617. tedious operations in multiplication and divifion may be performed merely by addition and fubtraftion. This difcovery was publiffied at Edinburgh in 1614 in his work entitled Logarithmorum Canonis Defcriptio, feu Arithmetica Supputationum Mirabilis Abbreviatio. It is well known that there is fuch a correfpondence be¬ tween every arithmetical and geometrical progreffion, viz. q’ i’ \ that any terms of L 2, 4, 8, 16, 32, 64, J ^ the geometrical progreffion may be multiplied or divided by merely adding or fubtrafting the correfponding terms of the arithmetical progreffion ; thus the produft of four and eight may be found by taking the fum of the correfponding terms in the arithmetical progreffion, viz. 2 and 3, for their fum 5 points out 32 as the pro dud of 4 and 8. The numbers o, I, 2, 3, &c. are therefore the logarithms of 1, 2, 4, 8, &c. The choice of the two progreffions being altogether arbi¬ trary, Baron Napier took the arithmetical progreffion which we have given above, and made the term o cor- refpond with the unit of the geometrical progreffion,, which he regulated in fuch a manner that when its terms are reprefented by the abfciffge of an equilateral hyperbola in which the firft abfcifs and the firft ordi¬ nate are each equal to 1, the logarithms are reprefent- Sfables of ed by the hyperbolic fpaces. In confequence, however, logarithms cf t^e inconvenience of this geometrical progreffion, by Mr ^ Baron Napier, after confulting upon the fubjeft with Briggs. Henry Briggs of Greffiam College, fubftituted the de¬ cuple progreffion 1, 10, ico, 1000, of which o, 1, 2, 3, 4, &c. are the logarithms. Nothing now remained but to conftruft tables of logarithms, by finding the lo¬ garithms of the intermediate numbers between the terms A.D. 1618. of the decuple progreffion. Napier, however, died be¬ fore he was able to calculate thefe tables; but his lofs was in fome meafure fupplied by Mr Briggs, who applied himfelf with zeal to this arduous talk, and publiffied in 1618 a table of the logarithms of all numbers from 1 to 1000. In 1624 he publiffied another table containing the logarithms from 1000 to 20,000, and from 90,000 to 100,000. The defefts in Briggs’s tables were filled up by his friends Gellibrand and Hadrian Vlacq, who alfo publiffied new tables containing the logarithms of fines, tangents, &c. for 90 degrees. ofHaT"65 2^’ ^ur*n£ t^ie time when Napier and Briggs were BornT-6o honour to their country by completing the fyllem Died 1621! °f logarithms, algebra was making great progrefs in the hands of our countryman Harriot. His Artis ana¬ lytic te Praxis, which appeared in 1620, contains along with the difcoveries of its author, a complete view of the Hate of algebra. He fimplified the nutation by MAT ICS. 5 fubllituting fmall letters inftead of the capitals introdu¬ ced by Vieta ; and he was the firft who ffiowed that every equation beyond the firft degree may be confider- ed as produced by the multiplication of as many limple equations as there are units in the exponent of the high- eft power of the unknown quantity. From this he de¬ duced the relation which exifts between the roots of any equation, and the coefficients of the terms of which it confifts. 29. About the fame time, a foreign author named Fer- Fernel firft nel, phyfician to King Henry II. of France, had theS:vesThe merit of being the firft who gave the meafure of the J^earth0* earth. By reckoning the number of turns made by a coach wheel from Amiens to Paris, till the altitude of the pole ftar was increafed one degree, he eftimated the length of a degree of the meridian to be 56,746 toifes, which is wonderfully near the truth. He alfo wrote a work on mathematics, entitled De Proportionibus.— About this time it was ffiown by Peter Metius, a German jy[ctiuS mathematician, that if the diameter of a circle be 113, finds its circumference will be 355. This refult, fo very near more cor- the truth, and expreffed in fo few figures, has preferved J?um1" the name of its author. dkmSer 30. The next author, whofe labours claim our atten-and circum- tion, is the illuftrious Defcartes. We do not allude to ference of a thofe wild and ingenious fpeculations by which this phi- circle* lofopher endeavoured to explain the celeftial phenome- Difcoveries na ; but to thefe great difcoveries with which he en- Defcartes riched the kindred fciences of algebra and geometry. He introduced the prefent method of marking the powers Died 165c. of any quantity by numerical exponents. He firft ex¬ plained the ufe of negative roots in equations, and ffiow¬ ed that they are as real and ufeful as pofitive roots, the only difference between them being founded on the dif¬ ferent manner in which the correfponding quantities are confidered. He pointed out the method of finding the number of pofitive and negative roots in any equation where the roots are real; and developed the method of indeterminates which Vieta had obfcurely hinted at. 31. Though Regiomontanus, Tartalea, and Bombelli, had refolved feveral geometrical problems by means of alegbra, yet the general method of applying geometry to algebra was firft given by Vieta. It is to Defcartes,IIe extends however, that we are indebted for the beautiful and ex- tPe aPp^ca- tenfive ufe which he made of his difcovery. His me• to " thod of reprefenting the nature of curve lines by equa- geometry, tions, and of arranging them in different orders accord¬ ing to the equations which diflinguiffied them, opened a vaft. field of inquiry to fubfequent mathematicians; and his methods of conftruifting curves of double cur¬ vature, and of drawing tangents to curve lines, have contributed much to the progrefs of geometry. The ' in ‘crfe method of tangents, which it was referved for the fluxionary calculus to bring to perfection, originated at this time in a problem which Florimundus de Beaune propofed to Defcartes. It was required to conftruft a A. D. 1647. curve in which the ratio of the ordinate and fubtangent ffiould be the fame as that of a given line to the por¬ tion of the ordinate included between the curve and a line inclined at a given angle. The curve was con- ftrufted by Defcartes, and feveral of its properties de¬ tected, but he was unable to accompliffi the complete A. D. 16;I folution of the problem. Thefe difcoveries of Defcartes were ftiidied and improved by his fucceffors, among whom 6 Difeovei'ics of Pafcal. Born 1623. Died 166%. Difcoveries of Fermat. Born 1590. Died 1663. Cavaleri’s method of indivifibles. The fame fubjedf dif- cuffed by Roberval. 1634. MATHEMATICS. whom we may number the celebrated Hudde, who publifhed in Schooten’s commentary on the geometry of Defcartes, an excellent method of determining if an equation of any order contains feveral equal roots, and of difeovering the roots which it container 3 2. The celebrated Pafcal, who was equally diftinguiih- ed by his literary and his fcientific acquirements, extend¬ ed the boundaries of analyfis by the invention of his arith¬ metical triangle. By means of arbitrary numbers pla¬ ced at the vertex of the triangle, he forms all the figur- ate numbers in fucceilion, and determines the ratio be¬ tween the numbers of any two cafes, and the various fums refulting from the addition of all the numbers of one rank taken in any poffible direction. This in¬ genious invention gave rife to the calculation of proba¬ bilities in the theory of games of chance, and formed the foundation of an excellent treatife of Huygens, en¬ titled De Ratiociniis in Ludo Alex, publifhed in 1657. 33. Several curious properties of numbers were at the fame time difeovered by Fermat at Touloufe. In the theory of prime numbers, particularly, which had firit been confidered by Eratollhenes, Fermat made great difcoveries; and in the doftrine of indeterminate pro¬ blems, he feems to have been deeply verfed, having re- publifhed the arithmetic of Diophantus, and enriched it with many valuable notes of his own. He invented the method of difeovering the maxima and minima of variable quantities, which ferves to determine the tan¬ gents of geometrical curves, and paved the way for the invention of the fluxionary calculus. 34. Another ftep towards the difeovery of fluxions was at this time made by Cavaleri in his geometry of indivifibles. In this work, which was publifhed in 1635, its author fuppofes every plane furface to confift of an infinite number of planes; and he lays it down as an axiom, that thefe infinite fums of lines and furfaces have the fame ratio when compared with the unit in each cafe as the fuperficies and folids to be meafured. This ingenious method was employed by Cavaleri in the quadrature of the conic feclions, and in the curvature of folids generated by their revolution ; and in order to prove the accuracy of his theory, he deduced the fame refults from different principles. 35. Problems of a fimilar kind had been folved by Fermat and Defcartes, and now occupied the attention of Roberval. The latter of thefe mathematicians began his inveftigation of this fubjeft about a year before the publication of Cavaleri’s work, and the methods which both of them employed were fo far the fame as to be founded on the principles of indivifibles. In the mode, however, which Roberval adopted, planes and folids were confidered as compofed of an infinite number of redflangles, whofe altitudes and the thicknefs of their fe&ions were infinitely fmall.—By means of this method, Roberval determined the area of the cycloid, the cen¬ tre of gravity of this area, and the folids formed by its revolution on its axis and bafe. He alfo invented a general method for tangents, fimilar in metaphyfical principles to that of fluxions, and applicable both to mechanical and geometrical curves. By means of this, he determined the tangents of the cycloid ; but there were fome curves which refifted its application. Con- fidering every curve to be generated by the motion of a point, Roberval regarded this point as afled upon at -every inflance with two velocities afeertained from the nature of the curve. He conftru&ed a parallelogram having its lides in the fame ratio as the two velocities j and he affumes as a principle, that the diredlion of the tangent muff fall on the diagonal, the pofition of which being afeertained, gives the pofition of the tangent. 36. In 1644, folutions of the cycloidal problems for-La|30lirs 0f merly refolved by Roberval were publiihed by Torricelli Torricelli, as invented by himfelf. The demonftrations of Roberval l644* had been tranfmitted to Galileo the preceptor of Torri¬ celli, and had alfo been publiflied in 1637 in Merfen- nus’s Univerfal Harmony. The Italian philofopher was confequently accufed of plagiarifm by Roberval, and the charge fo deeply a fleeted his mind as to bring him prematurely to the grave. It is obvious, however, from the demonftrations of Torricelli, that he had never feen thofe of Roberval, and that he was far from merit¬ ing that cruel accufation which deprived fcience of one of its brighteft ornaments. 37. The cycloid having attrafled the notice of geo-Fartherdif- meters from the number and Angularity of its properties,covt r'es the celebrated Pafcal propofed to them a variety of new ^ ' problems relative to this curve, and offered prizes for their folution. Thefe problems required the area of any cycloidal fegment, the centre of gravity of that fegment, the folids, and the centres of gravity of the folids, which are generated either by a whole revolution, a half or a quarter of a revolution of this fegment round an abfeiffa or an ordinate. The refolution of thefe problems was attempted by Huygens, Sluze, Sir Chriftopher Wren, Fermat, and Roberval. Sluze difeovered an ingenious method of finding the area of the curve. Huygens fquared the fegment comprifed between the vertex, and as far as a fourth of the diameter of the generating circle 5 and Sir Chriftopher Wren afeertained the length of the cycloidal arc included between the vertex and the ordinate, the centre of gravity of this arc, and the furfaces of the folids generated during its revolution. Thefe attempts were not confidered by their authors as folutions of Pafcal’s problems, and therefore they did not lay claim to his prize. Our countryman Wallis, however, and Lallouere a Jefuit, gave in a folution of all the problems, and thought themfelves entitled to the proffered reward. In the methods employed by thefe mathematicians, Pafcal detetled feveral fources of error; and it rvas referved for that great genius to furniflr a complete folution of his own problems. Ex¬ tending his inveftigations to curtate and prolate cycloids, he proved that the length of thefe curves depends on the reftification of the ellipfe, and afligned in each cafe the axis of the ellipfe. From this method he deduced this curious theorem, that if two cycloids, the one curtate and the other prolate, be fuch, that the bafe of the one is equal to the circumference of the circle by which the other is generated, the length of thefe two cycloids will be equal. 38. While thefe difcoveries were making on the con-Lahpurs o[ tinent, the friends of fcience in Britain wrere aftively Wallis, employed in promoting its advancement. In 1655,1655. Wallis publifhed his Arithnetica Infinitorum, a work of great genius. He attempted to determine by the fum- mation of infinite feries, the quadrature of curves, andl682> the curvature of folids, fubjetfts Avhich were afterwards inveftigated in a different manner by Ifhmael Bullial- dus. By Wallis’s method, curves were fquared when their ordinates are expreffed by one term, and when their 7 MATHEMATICS. their ordinates were complex quantities raifed to en¬ tire and po-fitive powers, thefe ordinates were refolved into feries, of which each term is a monomial. Wallis attempted to extend his theory to curves whofe ordi¬ nates were complex and radical, by attempting to in¬ terpolate the feries of the farmer kind with a new fe¬ ries •, but he was unfuccefsful. Difcoveries It was left to Newton to remove thisdifficulty. He •f Newton. f()jvcc[ the problem in a more dire£l and fitriple manner by the aid of his new formula for expanding into an in¬ finite feries any power of a binomial, whether its exponent wus pofitive or negative, an integer or a fraftion. Al¬ gebra is alfo indebted to this illuftrious mathematician for a fimple and extenfive method of refolving an equation into com men fur able faftors 5 for a method of fumming up the powers of the roots of an equation, of extra£ting the roots of quantities partly commenfurable, and partly in- commenfurable, and of finding by approximation the roots of literal and numerical equations of all orders. Lord 4°- About this time, William Lord Brouncker, in at- Brouncker tempting to demonftrate an expreflion of Wallis on the difeovers magnitude of the circle, difeovered the theory of con- continued (_jnUCd fractions. When an irreducible fraction is ex- BornTfoo. Puffed by numbers too great and complicated to be Died 16S4. eafily employed by the analyft, the method of Lord Brouncker enables us to fubftitute an exprefiion much more fimple and nearly equivalent. This theory, which enables us to find a very accurate relation be¬ tween the diameter and circumference of the circle, * Opera was empl°ye(i by Huygens * in the calculation of his Pojhuma, planetary automaton, for reprefenting the motions of tom. ii./ub the folar fyftem, and was enlarged and improved by finem. other celebrated geometers. Lord Brouncker had like- wife the merit of difeovering an infinite feries to repre- fent the area of the hyperbola. The fame difeovery was made by Nicholas Mercator, who publithed it in his Logarithmotechma in 1668. 41. The fubjeft of infinite feries received confiderable JamesGre- addition from Mr James Gregory. He was the firft who g0ry> gave the tangent and fecant in terms of the arc, and, inverfely, the arc in terms of the tangent and fecant. He conftru&ed feries for finding direftly the logarithm of the tangent and fecant from the value of the arc, and the logarithm of the arc from that of the tangent and fecant; and he applied this theory of infinite feries to the reftification of the ellipfis and hyperbola. Labours of 42. The differential triangle invented by the learned Dr Barrow. Dr Barrow, for drawing tangents to curves, may be re¬ garded as another contribution towards the invention of fluxions. This triangle has for its fides the element of the curve and thofe of the abfeifs and ordinate, and thofe fides are treated as quantities infinitely fmall. Theory of 43* The do&rine of evolutes had been (lightly touched evolutesdif-upon by Apollonius. It remained, however, for the covered by illuftrious Huygens to bring it to perfection. His Huygens, theory of evolutes is contained in his Horologium Ofcil- latorium, publifhed in 1673, and may be regarded as one of the fineft difeoveries in geometry. When any curve is given, Huygens has pointed out the method of conftructing a fecond curve, by drawing a feries of per¬ pendiculars to the firft, which are tangents to the fe¬ cond •, and of finding the firft curve from the fecond. From this principle he deduces feveral theorems on the rectification of curves $ and that remarkable property of the cycloid, in which an equal and fimilar cycloid is produced by evolution. 44. In contemplating the progrefs of analyfis from Hiftory of the beginning of the 17th century, to the invention of the difcove- fluxions, we cannot fail to perceive the principles of that calculus gradually unfolding themfelves to view. The human mind feemed to advance with rapidity to¬ wards that great difeovery 5 and it is by no means un¬ likely that it would foon have arrived at the do&rine of fluxions, even if the fuperior genius of Newton had not accelerated its progrefs. In Cavalerius’s Geometna In~ divijibilium, we perceive the germ of the infinitefimal calculus j and the method of Roberval for finding the tangents of curves, bears a ftriking analogy to the me- taphyfics of the fluxionary calculus. It wras the glory of Newton, however, to invent and illuftrate the me¬ thod of fluxions •, and the obfeure hints which he re¬ ceived from preceding mathematicians, do not in the leaft detrail from the merit of our illuftrious country¬ man. 45. On the claims of Leibnitz as a fecond inventor General re¬ ef fluxions, and the illiberal violence with which they marks on have been urged by foreign mathematicians, we would wilh to fpeak with delicacy and moderation. Who that 24ewton can appreciate the difeoveries of that celebrated mathe- arid Leib- matician, or is acquainted with that penetrating genius nitz. which threw light on every department of human knowledge, would willingly ftain his memory with an ungracious imputation ? The accufation of plagiarifm is one of thofe charges which it is difficult either to fubftantiate or repel, and when direfted againft a great man, ought never, without the cleareft evidence, to be wantonly preferred or willingly received. If charitable fentiments are ever to be entertained towards others,— to what clafs of beings ffiould they be more cheerfully extended than to thofe' who have been the ornaments of human nature ? If fociety has agreed to regard as fa- cred the failings and excentricities of genius,—w'hen. ought that reverence to be more ftrongly excited than when we are paffing judgment on its mightieft efforts ? Inquiries into the motives and aftions of the learned ought never to be wantonly indulged. When the ho¬ nour of our country, or the charadler of an individual, requires fuch an inveftigation, a regard to truth, and a contempt of national prejudice,' ftiould guide the in¬ quiry.—We ftiould proceed with delicacy and forbear¬ ance.—We ftiould tread lightly even on the affies of genius. It is not uncommon to witnefs the indulgence of malicious pleafure, in detradling from the merits of a diftinguiffied charadler. The affailant raifes himfelf for a while to the level of his enemy, and acquires 1 glory by his fall. But let him remember that the lau¬ rels thus won cannot flourifti long. The fame public opinion which conferred them will tear them from his brow, and confign the accufer to that infamy from which, the brighteft abilities rvill be infuffieient to raife him. The confequences of fuch conduct have been feen in the fall of Torricelli. It was the charges of plagiarifm, preferred by Roberval, that hurried this, young and aceompliffied philbfopher to an early grave. 46. We have been led into thefe obfervations by ftudy- ing the difpute between the followers of Newton and Leibnitz. The claims of the Britifh, as well as thofe of the MATHEMATIC S. ^the foreign matliematicians, have undoubtedly been too high 5 and victory rather than truth feems to have been the object of conteit. Even the name of Newton has not efcaped from ferious imputations. The immenlity of the flake for which the different parties contended, may perhaps juftity the commencement of the difpute j and the brilliancy of the talents that were called into action, may leave us no caufe to regret its continuance : But nothing can reconcile us to thole perfonal animofi- ties in which the good fenfe and temper of philofophy are loft, and that violence of literary warfare where fei- ence can gain nothing in the combat.—In giving an account, therefore, of that interefting difpute, we fhall merely give a brief view of the fadts that relate to the difeovery of the higher calculus, and make a few ob- fervations on the conclufions to which they lead. Newton 47- ^he year 1669, a paper of Sir Ifaac Newton’s, publifhes entitled De analyst per equationes numero terminorum in- tainh* the ^”^’ WaS commun‘cated by Dr Barrow to Mr Col- principles of ^ns’ one t^e fecretaries of the Royal Society. In fluxions. this paper the author points out a new method of fquar- ing curves, both when the expreflion of the ordinate is a rational quantity, and when it contains complex ra¬ dicals, by evolving the expreffion of the ordinate into an infinite number of fimple terms by means of the bino¬ mial theorem. In a letter from Newton to Collins, dated December 10. 1672, there is contained a method of drawing tangents to curve lines, wdthout being ob- ftrudted by radicals j and in both thefe works, an ac¬ count of which was circulated on the continent by the fecretaries of the Royal Society, the principles of the fluxional calculus are plainly exhibited •, and it is the opinion of all the difputants, tffiat thofe works at leaft prove, that Newton mult have been acquainted with the method of fluxions when he compofed them. 48. Leibnitz came to London in 1673, and though there is no diredt evidence that he faw Newton’s paper De Analyst per Equationes, &c. yet it is certain that he had feen Sir Ifaac’s letter to Collins of 1672 $ and it is highly improbable that fuch a man as Leibnitz fhould have been ignorant of a paper of Newton’s which had been four years in the poffeflion of the public, and which contained difeuflions at that time intei’efting to every mathematician. Correfpond- 49. A letter from Newton to Oldenburg, one of the tweenLeib-^ecretarieS t^ie ^°ya^ Society, dated Odtober 24. nitz and I676, was communicated to Leibnitz. This letter Oldenburg, contains feveral theorems without the demonftrations, which are founded on the method of fluxions, and merely ftates that they refult from the folution of a general problem. The enunciation of this problem he exprefles in a cypher, the meaning of which was, An equation containing any number of flowing quantities being given, to find the fluxions, and inverfely. In re¬ ply to this communication, Leibnitz tranfmitted a let¬ ter to Oldenburg, dated June 2J. 1677, where he ex¬ plains the nature of the differential calculus, and af¬ firms, that he had long employed it for drawing tan¬ gents to curve lines. 50. The correfpondence between Leibnitz and Ol¬ denburg having been broken off by the death of the ©f th'^ diffc lafter> Leibnitz publifhed in the A ft a Erudit. Lipf for rcmialcai- Oftober 1684, the principles of the new analyfis, under cuius, the title of Nova Mcthodus pro maximis et minimis, itemque tangentibus, qiue nec fraftas, nec irrationales Xeibnitz pubiifhes quantitates moratur, et Jingulare pro illis calculus. This paper contains the method of differencing fimple, frac¬ tional, and radical quantities, and the application of the calculus to the folution of lome phyfical and geo¬ metrical problems. In 1685, he likewiie publilfied two fmall pamphlets on the quadrature of curves, contain¬ ing the principles of the Calculus Summatorius, or tho Inverfe Method of Fluxions ; and in 1686 there appear¬ ed another tra£t by the fame author, On the Recondite Geometry, and the Analysis of Indivijibles and Inf nites, containing the fundamental rule of the integral cal¬ culus. jr. Towards the clofe of the year 1686, Sir IfaacNewtos Newton gave to the world his illuftrious work entitled pubiifhes Phtlofophice Naturahs Principia Mathematica. Some lis Pno- of the moil difficult problems in this work are founded C1^ia* on the fluxional calculus; and it is allowed by Boflut, one of the defenders of Leibnitz, “ that mathematicians did Newton the juftice to acknowledge, that at the pe¬ riod when his Principia was publifhed, he was matter of the method of fluxions to a high degree, at leaft with refpect to that part which concerns the quadrature of curves.” The claim of Leibnitz, as a feparate inventor of the differential calculus, is evidently allowed by Newton himfelf, when he obferves, that Leibnitz had communicated to him a method fimilar to his own for drawing tangents, &c. and differing from it only in tho enunciation and notation. 52. About this time, it became faftuonable among Lepin;tz geometers to perplex each other by the propofal of new propofes i and difficult problems, a practice which powerfully the pro- contributed to the progrefs of mathematics. The dif klem of tho pute in which Leibnitz was engaged with the Carte-1t^^™nou* fians refpe6ting the meafure of active forces, which the * former fuppofed to be as the fimple velocity, while the latter afferted, that they were as the fquare of the velo¬ city, led him to propofe the problem of the ifochronous curve, or “ to find the curve which a heavy body muff deferi be equally, in order to approach or recede from a horizontal plane in equal times.” This curve was which is found by Huygens to be the fecond cubic parabola j folved by but he gave only its properties and conftruftion w ithout Huygens in the demonftrations. The fame folution, along with the l687* demonftration, was given by Leibnitz in 1689, who, at the fame time, propofed to geometers to find the paracentric ifochronal curve, or the curve in which a body would equally approach or recede from a given point in equal times. 53. It was at this time that the two brothers, James James Ber- and John Bernouilli, began to difplay thofe talents from nouilli alfo which the phyfical and mathematical fciences received *tntls t^ie r , • r • , t L • zr ilochronotts luch immenie improvements. James was born in 1054,tuive and died in 17055 and John, who was his pupil, was born in 1667, and lived to the advanced age of 68 years. In 1690, James Bernouilli gave the fame folu¬ tion of the ifochronous curve that had been given by Huygens and Leibnitz 5 and propofed the celebrated problem of the catenary curve, which had formerly perplexed the ingenuity of Galileo. In two memoirs, '69t* publiflied in 1691, he determined, by means of the in-s^l,t10" °* verfe method of fluxions, the tangents of the parabolic of t£e tatfc_ fpiral, the logarithmic fpiral, and the loxodromic curve, narian and likewife the quadratures of their areas. curve, and 54. The problem of the catenary curve having occupied 0l^er analo- the attention of geometers, was refolved by Huygens, Leibnitz, MATHE Leibnitz, and .Tolin Eernouilli. In tliefe folutions, however, the gravity of the catenary curve was fuppo- fed to be uniform j but James Bernouilli extended the filiation to cafes where the weight of the curve varies _ - 6 - vctiici from one point to another, according to a given law. From this problem he was alfo conduced to the deter¬ mination of the curvature of a bended bow, and that of an elaftic bar fixed at one extremity, and loaded at the other with a given weight. In the hopes of contribut¬ ing to the progrefs of navigation, the fame mathemati¬ cian confidered the form of a fail fwoln with the wind. \Vhen the wind after ftriking the fail, is not prevented from efcaping, the curvature of the fail is that of the common catenarian curve; but when the fail is fuppo- fed perfedtly flexible, and filled with a fluid preffing downwards on itfelf, as water preffes on the fides of a veffd, the curve which it forms is one of thofe denomi¬ nated hntearia:, which is exprefled by the fame equation as the common elaftic curve, where the extenfions are reckoned proportional to the forces applied at each point. “-The fame problem was folved in the Journal des Sfavans for 1692, by John Bernouilli; but there is fatisfaftory evidence that it was chiefly borrowed from his brother James. James Ber- ^ ^ a^en^on °f James Bernouilli was now direc- nouilli. ’ ted to t,ie theory of curves produced by the revolution of 1601 006 Curv? uPon another. He confiders one curve rolling upon a given curve, equal to the firft, and immoveable, He determines the evolute and the cauftic of the epicy¬ cloid, defcribed by a point of the moving circle, and he deduces from it other two curves, denominated the anti- evolute and pericaujlic. He found alfo that the loga¬ rithmic fpiral was its own evolute, cauftic, antievolute, and pericauftic; and that an analogous property belonoed to the cycloid. G i69i. _ 56* About this time Viviani, an Italian geometer diftinguifhed as the reftorer of Arifteus’s conic feaions’ Problem of required the folution of the following problem, that v iviam there exifted a temple of a hemifpherical form, pierced "with four equal windows, with fuch fkill that the re¬ mainder of the hemifphere might be perfedly fquared. With the aid of the new analyfis, Leibnitz and James Bernouilli immediately found a folution, while that of Viviani was founded on the ancient geometry. He proved that the problem might be folved, by placing parallel to the bafe of the hemifphere, two right cylin¬ ders, the axes of which fliould pafs through the centres of two radii, forming a diameter of the circle of the bafe and piercing the dome each way. Tfchirn- 57. Prior to fome of thefe difeuflions, the curves call- taufen on ed caujlic,zn& fometimes Tfchirnhaufenian, were difeo- vered by Tfchirnhaufen. Thefe curves are formed by the crofting of the rays of light, when reflefted from a curved furface, or refraaed through a lens fo as not to James Bcr- meet in a Angle point. With the aftiftance of the com- 3stoattheraOn1ge0T^’ * k.lnrrdiaukn difeovered, that they are Tamcfub- ePO ed,rby I^9< WaS foIved by James fochrenal B®rn(H1,lb, who took for ordinates parallel ftraight lines, kurve. and for abfciflac the chords of an infinite number of Vol. XIII. Part I. fclved. curves. M A T IC S. , concentric circles defcribed about the given point. In this way he obtained a feparate equation, conitrufted at firft by the reftification of the elaftic curve, and after- wards by the redfifieation of an algebraic curve. The fame problem was folved by John Bernouilli and Leib¬ nitz. 59. In 1694, a branch of the new analyfis, called the The expo- exponential calculus, was invented feparately by Johnncmial c;tl- Bernouilli and Leibnitz. It conlifts in differencing and Cldus ^n' integrating exponential quantities or powers with varia- Ldbnlx7 ble exponents. . To Leibnitz, the priority in point of and John invention certainly belongs; but John Bernouilli was Bernouilli. the firft who publiftied the rules and ufes of the cal¬ culus. 60. The marquis PHofpital, who, in 169 5, had folved The'Mar- the problem about the curve of equilibration in draw-f’Hof- bridges, and ftiewn it to be an epicycloid, publiftied in Hu.b' the following year his Analyfis of Infinites for the detjlanding of curve lines. In this celebrated work, infinites, the differential calculus, or the diredt method of fluxions, was fully explained and illuftrated ; and as the know¬ ledge of the higher geometry had been hitherto con¬ fined to a few, it was now deftined to enlighten the dif¬ ferent nations of Europe. 61. 1 he methods which were employed by Defcartes, Newton Fermat, &c. for finding the maxima and minima of ^n6. Fluxions, and of Infinite Scries, was publiflied by Dr Pemberton about nine years after the death of its au¬ thor •, but it does not contain any new invefligations which accelerated the progrefs of the new analyfis. 68. The mathematical fciences were at this time hi-jjabours 0f debted to the labours of Manfredi, Parent, and Saurin.Manfredi, The former of thefe geometers publiflied a very able Parent, and work, De Confiruclione Equationum differentialium primi gradus. To Parent we are indebted for the problem by 1 II MATHEMATICS. which we obtain the ratio between the velocity of the power, and the weight for finding the maximum effect of machines j but his reputation was much injured by the obfcurity of his writings. Saurin was celebrated for his theoretical and practical knowledge of watchmaking, and was the firft who elucidated the theory of tangents to the multiple points of curves. Account of 69. While the feienee of analyfis was thus advancing the difpute Tvhh rapidity, the difpute between Newton and Leib- betwcen n^z began to be agitated among the mathematicians of ev on jrurope. Thefe illuftrious rivals feemed to have been hitherto contented with fharing the honour of having in¬ vented the iluxional calculus. But as foon as the prio¬ rity of invention was attributed to Newton, the friends of Leibnitz came forward with eagernefs to fupport the claims of their mafter. 70. In a fmall work on the curve of fwiftefl dcfcent, and the folid of leafl refiftance, publithed in 1699, Ni- and Leib¬ nitz. Facio de Duillier Newton. Leibnitz defends himfelf. thcTdil'nite c^<)^as Facio de Duillier, an eminent Genoefe, attribu- iiAvour of ted to Newton the firfi: invention of Fluxions, and hint¬ ed, that Leibnitz, as the fecond inventor, had borrow¬ ed from the Englifh philofopher. Exafperated at this improper infinuation, Leibnitz came forward in his own defence, and appeals to the admiffion of Newton in his Principiu, that neither had borrowed from the other. He expreffed his conviction, that Facio de Duillier was not authorifed by Sir Ifaac, to prefer fuch a charge, and threw himfelf upon the teftimony and candour of the Englifh geometer. 71. The difcuffion relied in this fituation for feveral years, till our celebrated countryman, Dr Keill, infti- Dr Keill makes the fame charge gate(j by an attack upon Newton in the Leipfic Jour Leibnitz. 1708. nal, repeated the fame charge againft Leibnitz. The Ge/man philofopher made the fame reply as he did to his former opponent, and treated Dr Keill as a young man incapable of judging upon the fubjeft. In 1711, Dr Keill addrefled a letter to Sir Hans Sloane, feere- tary to the Royal Society, and accufed Leibnitz of ha¬ ving adopted the differential notation, in order to have it believed, that he did not borrow his calculus from the writings of Newton. 7 2. Leibnitz w7as with reafon irritated at this accufation, and called upon the Royal Society to interfere in his behalf. A committee of that learned body was accord¬ ingly appointed to invcftigate the fubjedl, and their re¬ port was publifhed in 1712, under the title of Com.ier- cium Epijfolicum de Anah/Ji promota. In this report the committee maintain that Leibnitz was not the firfi: in¬ ventor, and abfolve Dr Keill from all blame in giving the priority of invention to Newton. They were cautious, and report" however, in Hating their opinion upon that part of the charge in which Leibnitz was accufed of plagiarifm. John Bcr- 73. In anfwer to the arguments advanced in the Com- nouilli re- mercium Ep//?^//l7/w,.TohnBernouilji, the particular friend plies to 0f Leibnitz, publifhed a letter, in which he has the af- givw/bfthe durance to ftate, that the method of fluxions did not Commer- precede the differential calculus, but that it might have cium F.pif- taken its rife from it. The reafon which he afligns for this flrange affertion is, that the differential calcu¬ lus was publifh’ed before Newton had introduced an uniform algorithm into the method of fluxions. But it may as well be maintained that Newton did not dif- cover the theory of univerfal gravitation, becaufe the attractive force of mountains and of fmaller portions of t'jir. Leibnitz appeals to the Royal Society, 1712. Who rp point a committee to examine tolicum. matter was not afeertained till the time of Maikelyne and Cavendifh. The principles of fluxions are allowed to have been difeovered before thofe of the differential calculus, and yet the former originated from the latter, becaufe the fluxional notation was not given at the fame time ! 74. Notwithftanding the ridiculous affertion of John Remarkt o» Bernouilli, it has been admitted by all the foreign ma- ^VLntro' thematicians that Newton was the firil inventor of the method of fluxions. The point at iffue therefore is merely this :—did Leibnitz fee any of the w-ritings of Newton that contained the principles of fluxions before he publilhed in 1684 his Nova Methodus pro maximis et minimis? The friends of Leibnitz have adduced fom© prefumptive proofs that he had never feen the treatife of Newton, de Ana/yji, nor the letter to Collins, in both of which the principles of the new calculus were to be found; and in order to ftrengthen their argument, they have not fcrupled to alfert, that the writings al¬ ready mentioned contained but a vague and obfeure indication of the method of fluxions, and that Leibnitz might have perufed them without having difeovered it. This fubfidiary argument, however, rells upon the opi¬ nion of individuals ; and the only way of repelling it is to give the opinion of an impartial judge. M. Montu- cla, the celebrated hiltorian of the mathematics, who being a Frenchman, cannot be fufpected of partiality to the Englifh, has admitted that Newton in his trea¬ tife de Analijji “ has difclofed in a very concife and obfeure manner his principles of fluxions,” and “ that the fufpicion of Leibnitz having feen this work is not deftitute of probability, for Leibnitz admitted, that in his interview with Collins he had feen a part of the epiftolary correfpondence between Newton and that gentleman.” It is evident therefore that Leibnitz had opportunities of being acquainted with the doftrine of fluxions, before he had thought of the differential cal¬ culus ; and as he was in London, where Newton’s trea¬ tife was publifhed, and in company with the very men to whom the new analyfis had been communicated, it is very likely that he then acquired fome knowledge of the fubje£l. In favour of Leibnitz, however, it is but juf- tice to fay, that the tranfition from the method of tan¬ gents by Dr Barrow to the differential calculus is fo Ample, that Leibnitz might very eafily have perceived it ; and that the notation of his analyfis, the numerous applications which he made of it, and the perfeflion to which he carried the integral calculus, are coniiderable proofs that he was innocent of the charge which the Englifh have attempted to fix upon his memory. 7 In T708, Remond de Montmort publifhed a cu-Works on rious work, entitled the Analyfis of Games of Chance, doc- in which the common algebra was applied to the com-1 putation of probabilities, and the eftimation of chances, Though this work did not contain any great difeovery, yet it gave extent to the theory of feries, and admir¬ ably illuftrated the dodlrine of combinations. The fame fubjedl was afterwards difeuffed by M. de Moivre, a French proteftant refiding in England, in a fmall treatife entitled Menfura Sortis, in which are given the r elements of the theory of recurrent feries, and fome very ingenious applications of it. Another edition was pub- liffed in Englifti in 1738, under the title of the Dodlrine of Chances. ® 2 A MATHEMATICS. projl'fa to ,, ' r Ayl0tt t!me bofor,e h!s deft1'’ Leibnitz propofed to the Englifli * le ■t‘ngntn geometers the celebrated problem of ortho- the pro- gonaJ trajectories, which was to find the curve that cuts blem of a feries of given curves at a conftant angle, or at an an£le varying according to a given law. This pro- cs' blem was put into the bands of Sir Ifaac Newton when he returned to dinner greatly fatigued, and he brought it to an equation before he went to reft. Leibnitz being recently dead, John Bernouilli aflumed his place, and maintained, that nothing was eafier than to bring the problem to an equation, and that the folution of the problem was not complete till the differential equation of the trajeftory was refolved. Nicholas Bernouilli, the fon of John refolved the particular cafe in which the interfeCted curves are hyperbolas with the fame centre and the fame vertex. James Hermann and Nicholas Bernouilli, the nephew of John, treated the fubjedt by more general methods, which applied to the cafes in which the interfered curves were geometrical. The moft complete folution, however, was given by Dr Taylor in the Philofophical TranfaCtions for 1717, though it was not fufficiently general, and could not apply to feme cafes capable of refolution. This defedl was fupplied by John Bernouilli, who in the Leipfic TranfaCHons for 1718, publifhed a very fimple folution, embracing all the geometrical curves, and a great num¬ ber of the mechanical ones. 77. During thefe difeuftions, feveral difficult problems on the integration of rational fradions were propofed by Dr Taylor, and folved by John Bernouilli. This fub- jed, however, had been firft difeufied by Roger Cotes, profeffor of mathematics at Cambridge, who died in 17x0. In his pofthumous work entitled Hannonia Menfurarum, publiffied in 1716, he gave general and convenient formulae for the integration of rational frac- Cotes, born t;ons . ancj we are indebted to this young geometer for his method of eftimating errors in mixed mathematics, for his remarks on the differential method of Newton, and for his celebrated theorem for refolving certain equa¬ tions. 78. In X715, Dr Taylor publifhed his learned work invents the entitled MetJiodus incrementorum direct a et inverfa. In calculus of t^s work ^ doftor gives the name of increments or finite dif- decrements of variable quantities to the differences, ferecces. whether finite or infinitely fmall, of two confecutive -terms in a feries formed after a given law. When the differences are infinitely fmall, their calculus belongs to fluxions •, but when they are finite, the method of find¬ ing their relation to the quantities by which they are produced forms a new calculus, called the integral cal¬ culus of finite differences. In confequence of this work, Dr Taylor was attacked anonymoufly by John Bernouilli, who laviffied upon the Englilh geometer all that dull abufe, and angry ridicule, which he had formerly heaped upon his brother. Problem of 79. The problem of reciprocal trajectories was at this reciprocal time propofed by the Bernouillis. This problem re- trajedlories. qU;rect the curves which, being conftru6ted in two op- 1716 pofite directions in one axis, given^in polition, and then moving parallel to one another with unequal velocities, Itefolved ffiould perpetually interfe£t each other at a given angle, by Euler, was long difeuffed between John Bernouilli and an died 178?’ anonymous writer, who proved to be Dr Pemberton. 1717. 1718. Integration of rational fractions. I7I9* Labours of Eoger Cotes, 1676. Dr Taylor 1728. It was by an elegant folution of this problem that the celebrated Euler began to be diftinguiftied among mathematicians. He was the pupil of John Berncuilli, and continued through the whole of his life, the friend and rival of his fon Daniel. The great object of his labours was to extend the boundaries of analyfis ; and before he had reached his 21 ft year, he publiihed a new and general method of refolving differential equations of the fecund order, fubjecled to certain conditions. 80. The common algebra had been applied by Leibnitz Labours of and John Bernouilli to determine ares of the parabola, C°unt laS" the difference of which is an algebraic quantity, ima-11^ gining that fuch problems in the cafe of the ellipfe and hyperbola refifted the application of the new analyfis. The Count de Fagnani, however, applied the integral calculus to the arcs of the ellipfis and hyperbola, and had the honour of explaining this new branch of geo¬ metry. 81. In the various problems depending on the analyfis Problem of of infinites, the great difficulty is to refolve the differen- c;°unt Ric- tial equation to which the problems are reduced. Countcatu James Riceati having been puzzled with a differential 1725.. equation of the firft order, with two variable quantities, propofed it to mathematicians in the Leipfic A6ls for 1725. This queftion baffled the Ikill of the moft cele¬ brated analyfts, w'ho were merely able to point out a number of cafes in which the indeterminate can be fe- parated, and the equation refolved by the quadrature of curves. 82. Another problem fuggefted by that of Viviani was Problem of propofed in 1718 by Erneft von Offenburg. It was re-Offcnburg. quired to pierce a hemifpherical vault with any number of elliptical windows, fo that their circumferences ffiould be expreffed by algebraic quantities ;—or in other words, to determine on the furface of a fphere, curves algebraically re&ifiable. In a paper on the rec¬ tification of fpherical epicycloids, Herman * imagined * reterf- that thefe curves were algebraically re&ifiable, and burgh J therefore fatisfied the queftion of Offenburg • but John Tranfac- Bernouilli (Mem. Acad. Par. 1732) demonftrated, that*™^. as the re&ification of thefe curves depended on the qua- 1 ‘ drature of the hyperbola, they were only redlifiable in Refolved by certain cafes, and gave the general method of determin-•,ohl1 ?er- ing the curves that are algebraically re&ifiable on the n-°ulili- furface of a fphere. 83. The fame fubjett was alfo difeuffed by Nicole and Labours of Clairaut, (Mem. Acad. 1734). The latter of thefe Clairaut. mathematicians had already acquired fame by bis Re- cherches fur les Courbes a double Cour bur e, publiflied in 1730, before he was 2x years of age ; but his repu¬ tation was extended by a method of finding curves whofe property confifts in a certain relation between thefe branches expreffed by a given equation. In this refearch, Clairaut pointed out a fpecies of paradox in the integral calculus, which led to the celebrated theory of particular integrals which was afterwards fully illuftrat- ed by Euler and other geometers. 84. The celebrated problem of ifochronous curves be-Problem of gan at this time to be reagilated among mathematicians, ifochronous The objeft of this problem is to find fuch a curve that acurves* heavy body defeending along its concavity fiiall always reach the loweft point in the fame time, from what¬ ever point of the curve it begins to defeend. Huygens had already ffiewn that the cycloid was the ifochronous curve in vacuo. Newton had demonftrated the fame curve to be ifochronous when the defeending body ex¬ periences from the air a reliftance proportional to its ve¬ locity j M ATHE * Mcmoirs-'mcXty ; and Euler * and John Bernouilli f, had fepa- oj Peterf ra|;eiy found the ifcchronous curve when the relittance was as ^1C ^luare °f the velocity. Thefe three cafes, ^Mem. aud even a fourth in which the refiftance was as the Par. 1730. fquare of the velocity added to the produid of the velo¬ city by a conftant coefficient^ were all refolved by Fon- Solved by taine, by means of an ingenious and original method ; Fontaine, and it is very remarkable that the ifochronous curve is the fame in the third and fourth cafes.—The method of Fontaine was illuflrated by Euler, who folved a fifth cafe, including all the other four, when the refiftance is compofed of three terms, the fquare of the velocity, the produft of the velocity by a given coefficient, and a conftant quantity. He found alfo an expreffion of the time which the body employs to defeend through any arc of the curve. Algebra of 85. The application of analytical formulae to the phy- fines and fico-mathematical fciences was much facilitated by the sofmes. algebra of fines and cofines with which Frederick Chriftian Mayer, and Euler, enriched geometry. By the combination of arcs, fines, and cofines, formuke are obtained which frequently yield to the method of refo- lution, and enable us to folve a number of problems which the ordinary ufe of arcs, fines, and cofines, would render tedious and complicated. Improve- 86. About this time a great difeovery in the theory ment in theGf differential equations of the firft order tvas made fe- ofdiffers Paral;cty hy Euler, Fontaine, and Clairaut. Hitherto tial equa- geometers had no direct method of afeertaining if any tions. differential equation were refolvable in the ftate in which it was prefented, or if it required feme prepara¬ tion prior to its refolution. For every differential equa¬ tion a particular method was employed, and their refo¬ lution was often effefted by a kind of tentative procefs, which difplayed the ingenuity of its author, without be¬ ing applicable to other equations. The conditions un¬ der which differential equations of the firft order are re- folvable were difeovered by the three mathematicians whom we have mentioned. Euler made the difeovery in 1736, but did not publiftx it till 1740. Fontaine and Clairaut lighted upon it in 1739. Euler after¬ wards extended the difeovery to equations of higher orders. Difeovery 87. The firft traces of the integral calculus with pat- of tho inte-tial differences appeared in a paper of Euler’s in the j^u- Peterlburgh '1'ranfactions for 17345 but d’Alembert, partial dif- ^"ls work Sur les Vents, has given clearer notions of fcrcnces. it, and was the firft who employed it in folution of the problem of vibrating cords propofed by Dr Taylor, and inveftigated by Euler and Daniel Bernouilli. The ob- jedt of this calculus is to find a fundlion of fcveral vari¬ able quantities, when we have the relation of the coef¬ ficients which affedf the differentials of the variable quantities of which this fundlion is compofed. Euler exhibited it in various points of view, and (hewed its application to a number of pbyfical problems 5 and he | Peterf- afterwards, in his paper entitled Invfiigatio FunBio- Tranfac- num cx ^ata ^Werent^a^um conditioned., completely ex- tions, 1762. plained the nature, and gave the algorithm of the cal¬ culus. d'Tefof " ^ie analyfis of infinites was making fuch ra- fliixions Progrefs on the continent, it was attacked in England attacked by the celebrated Dr Berkeley, biftiop of Cloyne, in a by Dr work rafted ihe \nalijjl. or a difeourfe addre/fed to an In- Berkcley, jijei Mathematician, wherein it is examined whether the i734* •A MAT ICS. objeEl, principles, and inferences of the modern analyfis, are more di/linffly conceived than Religious Myjleries and Points of Faith. In this work the ciodlor .admits the truth of the conclulions, but maintains that the principles of fluxions are not founded upon reafpning ftridlly logical and conclufive. 'i his attack called forth Robins and Maclaurin. The former proven that the principles of fluxions were confident with the ftricteft rcafoning-, while M aclaurin, in his Treatife of Fluxions, gave a fynthetical demonftration of the principles of the calculus after the manner of the ancient geometricians, and eftablifties it with fuch clearnefs and fatisfaclion that no intelligent man could' refufe his affent. The differential calculus had been attacked at an earlier period by Nieuwentiet and Rol/e, but the weapons wielded by thefe adverfaries were contemptible when compared with the ingenuity of Dr Berkeley. 89. Notwithftanding this attack upon the principles of Works of the new analyfis, the fcience of geometry made rapid thorn as advances in England in the hands of Thomas Simplon, :^1InPlon• Landen, and Waring. In 174Q, Mr Simpfon publiffi-1740. ed his Treatife on Fluxions, which, befides many origi¬ nal refearches, contains a convenient method of refolv- ing differential equations by approximation, and various means of haftening the convergency of flowly conver¬ ging feries. We are indebted to the fame geometer for feveral general theorems for fumming different feries, whether they are fufceptible of an abfolute or an ap¬ proximate fummation. His Mathematical Differtations, I743.!> publifhed in 1743, his Efihijs on feveral Subjects in Mathematics, publiihed in 1740, and his Sclell Exer- cifes for Toung Proficients in the Mathematics, publifh¬ ed in 1752, contain ingenious and original rel’earches which contributed to the progrefs of geometry. 90. In \res> Mathematical Lucubrations, publiflied in The refidu- 1755, Mr Landen has given feveral ingenious theoremsal analyfis for the fummation of feries 5 and the Philofophieal Tranf- invcnted a61 ions for 1775 contain ,his curious difeovery of the jjK,q'^enj re6Iification of a hyperbolic arc, by means of two arcs of 1777. an ellipfis, which w;as afterwards, more fimply demon- ftrated by Legendre. His invention of a new calculus, called the ref dual analysis ,2.\\di in fome rcfpe6!s fubfi- diary to the method of fluxions, has immortalized his name. It was announced and explained in a fmall pamphlet publiflied in 1715, entitled a Difeourfe con* cerning the Rcfidual Analysis. 91. The progrefs of geometry in England was acce- La])0urs Gj lerated by the labours of Mr Edward Waring, proteffor Warning, of mathematics at Cambridge. His two Works entitled Phil. Tranf Meditationes An a lytic ce, publifhed in 1769, and Medita- 17s4» anc* tiones Algebraicce, and his papers in the Philofophical Tranfadlions on the fummation of forces, are filled with original and profound refearches into various branches of the common algebra, and the higher analyfis. 92. It was from the genius of Lagrange, however,.Difcoveri(j3 that the higher calculus has received the moft brilliant c-r La- improvements. This great man was born in Piedmont, granges He afterwards removed to Berlin, and hence to Paris, where he ftill refides. In addition to many improvements upon the integral analyfis, he has enriched geometry with a new calculus called \\w method of variations. The object jjjs of this calculus is, when there is given an expreffion orof varia- funftion of two or more variable quantities whole relation tions. is expreffed by a certain law, to find what this function becomes when that law fuffers any variation infinitely ffliall. 14 MATHEMATICS. Hts theory of analyti¬ cal func¬ tions. Labours of La Place. * Tom. 6.7. Works of Coufin, La¬ croix, Bof- fut, and Legendre. Agnefi’s analytical inftitutions. 17S4. Mafcheroni on the circle. fmall, occafioncd by the variation of one or more of the terms which exprefs it. This calculus is as much fu- perior to the integral calculus, as the integral calculus is above the common algebra. It is the only means by which we can refolve an immenfe number of prob¬ lems dc maximis et minimis, and is neceffary for the fo- lution of the moft interefting problems in mechanics. His theory of analytical functions is one of the moft brilliant fpecimens of human genius. In the Memoirs of Berlin for 1772 he had touched upon this intereft- ing fubjeft, but the theory was completely developed in 1797 in his work entitled Theorie des fonEtions ana- hjtiques, contenant les principes du calcul differentiei, degagees de toute conjideration d'itfniments petits, ou evanouifj'ements, ou des limites, ou des fluxions ; et re- duit a I'anahjfe algebrique des quantiles flmes. In a great number of memoirs which are to be found in the Me¬ moirs of the Academy of Paris, in thofe of the Acade¬ my of Berlin, and in thofe of the French Academy, La¬ grange has thrown light on every branch both of the common algebra and the new analyfis. 93. The new geometry has likewife been much indebt¬ ed to the celebrated Laplace. His various papers in the Memoires des Spavans Etrangers *, and the Memoirs of the French Academy, have added greatly to the higher calculi, while his application of analyfis to the celefti'al phenomena, as exhibited in the Mechanique Celefle, and his various difeoveries in phyfical aftronomy, entitle him to a high rank among the promoters of fcience. 94. Among the celebrated French mathematicians of the laft and prefent century, we cannot omit the names of Coufin, Lacroix, and Boffut} all of whom have writ¬ ten large works on the differential and integral calculi, and illuftrated the new analyfis by their difeoveries. The Elemens de Geometric by Legendre is one of the beft and moft original works upon elementary geometry, and his papers in the Memoirs of the Academy contain feveral improvements upon the new analyfis. 95. In Italy the mathematical fciences were deftined to be improved and explained by a celebrated female. Donna Maria Gaetana Agnefi was profeffor of mathe¬ matics in the univerfity of Bologna, and publilhed a learned work entitled Analytical Inflitutions, contain¬ ing tire common analyfis, and the differential and in¬ tegral calculi. It has been tranflated into Englifh by Profeffor Colfon, and was publiihed at the expence of Baron Maferes. A few years ago feveral curious pro¬ perties of the circle have been difeovered by Mafche¬ roni, another Italian mathematician, who has publiihed them in his interefting work fur le Geometric du Cam¬ pus. 96. In England the mathematical fciences have beenEnglifli fuccefsfully cultivated by Emerfon, Baron Maferes, Dr matiioma. M. Young, Dr Hutton, Profeffor Vince, and Profeffor UciaijS‘ Robertfon of Oxford. The Doctrine of Fluxions by -merfon. Emerfon, and his Method of Increments, are good in¬ troductions to the higher geometry. The Scriptures Logarithmici of Baron Maferes } his TraBs on the Re- Baron M*« folution of Equations ; his Principles of Life Annuities, fews. and his other mathematical papers, do the higheft ho¬ nour to his talents as a mathematician 5 while his zeal for the promotion of the mathematical fciences, and his generous attention to thofe who cultivate them, entitle him to the noble appellation of the friend and patron of genius. Dr Matthew Young, bilhop of Clonfert, has Dr M. given a fynthetical demonftration of Newton’s rule for Young, the quadrature of fimple curves j and has written on the extraction of cubic and other roots. Dr Hutton Dr Huttost and Dr Vince have each publifhed feveral elementary an[l Dr treatifes on mathematics, and have invented ingeni-Vince’ ous methods for the fummation of feries. Mr Robert- Mr Robert, fon of Oxford is the author of an excellent treatife onlou- conic feCtions. 97. The ancient geometry was afliduoufly cultivated Scottifii in Scotland by Dr Robert Simpfon and Dr Matthew Stewart. Dr Simpfon’s edition of Euclid and his treatife on conic feCtions have been much admired. The 2V’cC?.r Pr SimP- Phyfical and Mathematical of Dr Matthew Stewart,lon‘ and his Propofltiones Geometricce more veterum demon- flratce, contain fine fpecimens of mathematical genius. In Dr M. the prefent day the names of Profeffor Playfair and Pro- Stewart, feffor Leflie of the univerfity of Edinburgh, Mr Wal¬ lace and Mr Ivory now of the Royal Military College at Great Marlow, are well known to mathematicians. Mr Playfair’s Elements of Geometry, and his papers on Mr Play- the Arithmetic of Impoflible Quantities and on Porfms, are tUr‘ proofs of his great talents as a mathematician and a phi- lofopher. Mr Leflie, well known for his great difeo- Mr Leflie, veries on heat, has found a very fimple principle, capable of extenfive application, by which the complicated ex- preflions in the folution of indeterminate problems may be eafily refolved. Mr Wallace’s papers on GeometricalMr Wal- Porifms in the 4th vol. of the Edinburgh TranfaCtions,,ace- difplay much genius 5 and Mr Ivory’s Treatifes in the Mr Ivory, laft vol. of Baron Maferes’s Scriptures Logarithmici, and his paper on A New Series for the ReBiflcation of the Ellipfls, Edin. Tranf. vol. 4th. entitle him to a high rank among modern mathematicians. I Mathema¬ tical, Matiock. mat MATHEMATICAL, any thing belonging to the fcience of mathematics. MATHEMATICAL Inflruments, fuch inftruments as are ufually employed by mathematicians, as compaffes, feales, quadrants, &c. Machine for dividing MATHEMAT1CAL Inflruments. See R a v s den’s Machine. MATLOCK, a town or village of Derbylhire, near % MAT Wickfwmrth, fituated on the very edge of the Der- Matlock, •went 5 noted for its bath, the water of which is milk--y— warm } and remarkable for the huge rocks in its envi¬ rons, particularly thofe called the Torr, which is 140 yards liigh. It is an extenfive ftraggling village, built in a very romantic ftyle, on the fteep fide of a moun¬ tain, and containing, in 18cI, above 2000 inhabitants. Near the bath are feveral fmall houfes, whofe fituation MAT [i k is on the little natural horizontal parts of the mountain, a few yards above the road, and in fome places the roofs of fome almoft touch the floors of others. There are ex¬ cellent accommodations for company who refort to the bath 5 and the poorer inhabitants are fupported by the fale of petrifaftions, cryftals, &c. and notwithftanding the rock-inefs of the foil, the cliffs produce an immenfe number of trees, whofe foliage adds greatly to the beau¬ ty of the place. MATRASS, Cucurbit, or Bolthead, among chemifls. See Chemistry, Explanation of Plates. MATRICARIA, Feverfew 5 a genus of plants, belonging to the fyngenefia clafs j and in the natural method ranking under the 49th order, Compojitce. See Botany Index. M ATRICE, or Matrix. See Matrix. Matrice, or matrix, in Dyeing, is applied to the five Ample colours, whence all the reft: are derived or compofed. Thefe are, the black, white, blue, red, and yellow or root colour. Matrice, or matrices, ufed by the letter-founders, are thofe little pieces of copper or brafs, at one end whereof are engraven, dentwife, or en creux, the feve- ral charafters ufed in the compoling of books. Each charafter, virgula, and even each point in a difcourfe, has its feveraf matrix ; and of confequence its feveral puncheon to ftrike it. They are the engravers on me¬ tal that cut or grave the matrices. When types are to be caft, the matrice is fattened to the end of a mould, fo difpofed, as that when the metal is poured on it, it may fall into the creux or ca¬ vity of the matrice, and take the figure and impreflion thereof. See Letter FOUNDER T. Matrices, ufed in coining, are pieces of fteel in form of dies, whereon are engraven the feveral figures, arms, charadlers, legends, &c. wherewith the fpecies art; to be ftamped. The engraving is performed with feveral puncheons, which being formed in relievo, or prominent, when ftruck on the metal, make an indent¬ ed impreflion, which the French call en creux. MATRICULA, a regifter kept of the admiflion of officers and perfons entered into any body or fociety whereof a lift is made. Hence thofe who are admitted into our univerfities are faid to be matriculated. A- mong ecclefiaftical authors, we find mention made of two kinds of matricule ) the one containing a lift of the ecclefiaftics, called matricula clericorum : the other of the poor fubfifted at the expence of the church, call¬ ed matricula pauperum. Matricula was alfo applied to a kind of alms- houfe, where the poor were provided for. It had cer¬ tain revenues appropriated to it, and was ufually built near the church, whence the name was alfo frequently given to the church itfelf. MATRIMONY. See Marriage. MATRIX, in Anatomy, the womb, or that part of the female of any kind, wherein the foetus is conceived and nouriflied till the time of its delivery. See Ana¬ tomy, N0 108. Matrix is alfo applied to places proper for the ge¬ neration of vegetables, minerals, and metals. Thus the earth is the matrix wherein feeds fprout ; and marcafites are by many confidered as the matrices of metals. The matrix of ores is the earthy and ftony fubftan- 5 ] MAT ces in which thefe metallic matters are enveloped : Matrix thefe are various, as lime and heavy fpar, quartz, fluors, 8tc. _ y— MATRON, an elderly married woman. Jury of Matrons. When a widow feigns herfelf with child in order to exclude the next heir, and a fuppofititious birth is fufpe&ed to be intended, then,., upon the writ de ventre infpiciendo, a jury of women is to be impannelled to try the queftion whether the woman is with child or not- So, if a woman is convidted of a capital offence, and, being condemned to fuffer death, pleads in ftay of execution, that Ihe is pregnant, a jury of matrons is impannelled to in¬ quire into the truth of the allegation and, if they find it true, the convict is refpited till after her deli¬ very. MATRON A, in Ancient Geography, a river fepa- rating Gallia Celtica from the Belgica (Cefar). Now the Marne j which, rifing in Champagne near Langres, runs north-weft, and then weft, and pafling by Meaux falls into the Seine at Charenton, two leagues to the eaft of Paris. MATRONALIA, a Roman feftival inftituted by Romulus, and celebrated on the kalends of March, in honour of Mars. It was kept by matrons in particular, and bachelors were entirely excluded from any Ihare in the folemnity. The men during this feaft fent prefents to the women, for which a return was made by them at the Saturnalia : And the women gave the fame indul¬ gence to their fervants now which the men gave to theirs at the feaft of Saturn, ferring them at table, and treating them as fuperiors. MATROSSES, are foldiers in the train of artil¬ lery, who are next to the gunners, and aflift them in loading, firing, and fpunging the great guns. They carry firelocks, and march along with the ftore wag¬ gons, both as a guard, and to give their afliftance in cafe a waggon fhould break down. MATSYS, Quintin, painter of hiftory and por¬ traits, Avas born at Antwerp in 1460, and for feveral years folloAved the trade of a blackfmith or farrier, at leaft till he was in his 20th year. Authors vary in their accounts of the caufe of his quitting his firft occupa¬ tion, and attaching himfelf to the art of painting. Some affirm, that the firft unfolding of his genius Avas occafioned by the fight of a print Avhich accidentally Avas fiioAvn to him by a friend Avho came to pay him a vifit Avhile he Avas in a declining ftate of health from the labour of his former employment, and that by his copying the print Avith fome degree of fuccefs, be Avas animated Avith a defire to learn the art of painting. Others fay, he fell in love Avith a young Avoman of great beauty, the daughter of a painter, and they al¬ lege that love alone wrought the miracle, as he could have no profpeft of obtaining her except by a diftin- guiftted merit in the profeffion of painting : for which reafon he applied himfelf with inceffant labour to ftudy and praftife the art, till he became fo eminent as to be entitled to demand her in marriage, and he fueceed- ed. Whatever truth may be in either of thefe ac¬ counts, it is certain that he appeared to have an un¬ common genius ; his manner Avas Angular, not refem- bling the manner of any other matter •, and his pidures Tvere ftrongly coloured and carefully finiflied, but yet they are fomewhat dry and hard. By many compe¬ tent Matfys , II Matthew. MAT C i tent judges it was believed, when they obferved the ftrength of ex predion in feme of his compofitions, that if he had Ifudied in Italy to acquire fome knowledge of the antiques and the great mafters of the Roman fchoel, he would have proved one of the moil eminent painters Oi the Low Countries. But he only imitated ordinary life j and feemed more inclined, or at lead more quali¬ fied, to imitate the defers than the beauties of nature. Some hiiforical compofitions of this matter deferve cotn- xnendation ; particularly a Defcent from the Crofs, which is in the cathedral at Antwerp ; and it is juttly admired for the fpirit, Ikill, and delicacy of the whole. But the moft remarkable and bed known pi£ture of Matfys, is that ol the Two Mifers in the gallery at Windfor. He died in 1529. M A R I, in a drip, is a name given to rope-yarn, junk, &c. beat dat and interwoven uled in order to preferve the yards from galling or rubbing, in hoidine- or lowering them. MATTER, in common language, is a word of the lame import with body, and denotes that which is tan¬ gible, vifible, and extended ; but among philofophers it fignifies that fubdance of which all bodies are compo- fed ; and in this fenfe it is fynonymous with the word Element. It is only by the fenfes that we have any communi¬ cation with the external world ^ but the immediate ob¬ jects of fenfe, philofophers have in general agreed to term qualities, which they conceive as inhering in fomething which is called their fuhjeB or fuhjlratwn. It is this fubdratum of fenfible qualities which, in the language of. philofophy, is denominated matter; fo that matter is not that which we immediately fee - or handle, but the concealed' fubjeSl or fupport of vifible and tangible qualities. What the moderns term quali¬ ties, was by Aridotle and his followers called form; but fo far as the two do&rines are intelligible, there ap¬ pears to be no effential difference between them. From the moderns we learn, that body confids of matter and qualities ; and the Peripatetics taught the fame thing, when they faid that body is compofed of matter and form. How philofophers were led to analyze body into mat¬ ter and form, or, to ufe modern language, into matter and qualities} what kind of exidence they attribute to each } and whether matter mud be conceived as felf- • exident or created—are quedions which fiiall be confi- dered afterwards (See Metaphysics). It is fufficient here to have defined the term. . MATTHEW, or Gofpel of St MATTHEW, a cano¬ nical book of the New Tedament. St MATTHEW wrote his gofpel in Judea, at the re- qued of thofe he had converted ; and it is thought he began in the year 41, eight years after Chrid’s refur- retf-ion. It was written, according to the teilimony of all the ancients, in the Hebrew or Syriac language ; but the Greek verfion, which now paffes for the origi¬ nal, is as old as the apodolical times. 6V Matthew the Evany elf's Day, a fedival of the Chridian church, obferved on September 21 d. St MATTHEW, the fon of Alpbeus, was alfo called Levi. He was of a Jewifh original, as both his names difeover, and probably Galilean. Before his call to the apodolate, he was a publican or toll-gatherer to tbe Romans j an office of bad repute among the 6 ] . MAT Jews, on account of the covetoufnefs and exaflion of Matthew, thofe who managed it $ St Matthew’s office particular- ly confiding in gathering the cudoms of all merchan- ' dile that came by the lea of Galilee, and the tribute that paffengers were to pay who went by water. And here it was that Matthew fat at the receipt of cudoms, when our Saviour called him to be a difciple. It is probable, that, living at Capernaum, the place of Chrid s ufual xefidencc, he might have fome know¬ ledge of him before he was called. Matthew imme¬ diately expreffed his fatisfa&ion in being called to this high dignity, by entertaining our Saviour and his dif- ciples at a great dinner at his own houfe, whither he invited all his friends, efpecially thofe of his own pro- feffion, hoping, probably, that they might be influenced by the company and converfation* of Chrid. St Mat¬ thew continued with the red of the apodles till after our Lord’s afeenfion. For the fird eight years after¬ wards, he preached in Judea. Then he betook himfelf to propagating the gofpel among the Gentiles, and chofe Ethiopia as the feene of his apodolical minidry • where it is faid he differed martyrdom, but by what kind of death is altogether uncertain. It is pretended, but . without any foundation, that Hyrtacus, king of Ethiopia, defiring to marry Iphigenia, the daughter of his brother and predeceffor iEglippus, and the apof- tle having reprefented to him that he could not law¬ fully. do it, the enraged prince ordered his head im¬ mediately to be cut off. Baronins tells us, the body of St Matthew was tranfported from Ethiopia to Bi- thynia, and from thence was carried to Salernum in the kingdom of Naples in the year 954, where it was found in 1080, and where Duke Robert built a church bear¬ ing his name. St MATTHEW, a town of Spain, in the kingdom of Arragon, feated in a pleafant plain, and in a very fer- tive country watered with many fprings. W. Long, o. 15. N. Lat. 40. 22. Matthew of Paris. See Paris. MATTHEW of Wefminfer, a Benedictine monk and accompliflied fcholar, who wrote a hiftory from the beginning of the world to the end of the reign of Ed¬ ward I. under the title of Flores Hi/loriarum; which was aftenvards continued by other hands. He died in I3^°* St MATTHIAS, an apoffle, was chofen inftead of Judas. He preached in Judea and part of Ethi¬ opia, and fuffered martyrdom. See the A&s of the Apojlles, chap. i. There w'as a gofpel publiffied un¬ der Matthias’s name, but reje&ed as fpurious ; as likewife fome traditions, which met with the fame fate. St Matthias's Day, a feftival of the Chriftian church, obferved on the 24th of February. St Mat¬ thias was an apoftle of Jefus Chrift, but not of the number of the twelve chofen by Chrift himfelf. He obtained this high honour upon a vacancy made in the college of the apoftles by the treafon and death of Judas Ifcariot. The choice fell on Matthias by lot ; his competitor being Jofeph called Barfabas, and furnamed Jufiis. Matthias was qualified for the apoftlefhip, by having been a conftant attendant upon our Saviour all the time of his miniftry. He was, probably, one of the 70 difciples. After our Lord’s refurredlion, he preached the gofpel firft in Judea. Afterwards M A T [ I? 1 M A U Maty. .Matthias Afterwards it is probable he travelled eaflwards, his relidcnee being principally near the irruption of the river Apfarus and the liaven Hyfi'us. The barbarous people treated him with great rudenefs and inhumani¬ ty •, and, after many labours and fufferings in convert¬ ing great numbers to Chriitianity, he obtained the crown of martyrdom ; but by what kind ot death, is uncertain.—T hey pretend to thow the relics of St Mat¬ thias at Rome; and the famous abbey of St Mat¬ thias near Treves boalts of the fame advantage : but doubtlefs both without any foundation. There was a gofpel afcribed to St Matthias 5 but it was univerfally rejected as fpurious. MATTlACAi, Aqu^, or Mattiaci Foktes, in Ancient Geography, now Wifbaden, oppofite to Mentz, in Weteravia. E. Long. 8. N. Lat. 50. 6. MATTIACUM, or Mattium, in Ancient Geogra¬ phy, a town of the Mattiaci, a branch of the Catti in Germany. Now Marpurg in Hefl’e. E. Long. 8. 40. N. Lat. 50. 40. MATTINS, the firft canonical hour, or the firft part of the daily fervice in the Romifh church. MATTHIOLUS, Peter Andrew, an eminent phyfician in the 16th century, born at Sienna, was well fkilled in the Greek and Latin tongues. Fie wrote learned commentaries on Diofcorides, and other works which are efteemed and died in 1577. MxTTURANTS, in Pharmacy, medicines which promote the fuppuration of tumors. MATY, Matthew, M. D. an eminent phyfician and polite writer, was born in Holland in the year 1718. He was the fon of a clergyman, and was ori¬ ginally intended for the church ; but in confequence of fome mortifications his father met with from the fynod, on account of the peculiar fentiments he en¬ tertained about the dodlrine of the Trinity, turned his thoughts to phyfic. He took his degree of M. D. at Leyden; and in 1740 came to fettle in England, his father having determined to quit Holland for ever. In order to make himfelf known, he began in 1749 to publifli in French an account of the productions of the Englifh prefs, printed at the Hague under the name of the Journal Britannique. This journal, which conti¬ nues to hold its rank amongft the befi; of thofe which have appeared fince the time of Bayle, anfwered the chief end he intended by it, and introduced him to the acquaintance of fome of the molt refpectable lite¬ rary characters of the country he had made his own. It was to their aCtive and uninterrupted friendfhip he owed the places he afterwards poffefled. In 1758 he was chofen fellow, and in 1765, on the refignation of Dr Birch, who died a few months after, and had made him his executor, fecretary to the Royal Society. He had been appointed one of the under librarians of the Britifii mufeum at its firft inflitution in 1753, an<^ ^e' came principal librarian at the death of Dr Knight in 1772. Ufeful in all thefe fituations, he promifed to be eminently fo in the laft, when he was feized with a languifhing diforder, which in 1776 put an end to a life which had been uniformly devoted to the purfuit of fcience and the offices of humanity. He was an early and aClive advocate for inoculation ; and when there was a doubt entertained that one might have the fmallpox this way a fecond time, tried it upon him- lelf unknown to his family. He was a member of Vol. XIII. Part I. the medical club (with the Drs Parlous, Templeman, Maty, Fothefgill, Watfon and others), which met every fortnight in St Paul’s Churchyard. He was twice v married, viz. the firft time to Mrs Elizabeth Boifra- gon ; and the fecond to Mrs Mary Deners. Pie kit a fon and three daughters. He had nearly finilhed the Memoirs of the earl of Chefterfield ; which were completed by his fon-in law Mr Juftamond, and pre¬ fixed to that nobleman’s Mifeellaneous Works, 1777, 2 vols. qto. Maty, Paul Henry, M. A. F. R. S. fon of the former, was born in 1745, and was educated at Weft- minfter and Trinity college, Cambridge, and had their travelling fellowlhip for three years. He was afterwards chaplain to Lord Stormont at Paris, and foon after va¬ cated his next fellowlhip by marrying one of the three daughters of Jofeph Clerk, Efq. and filler of Captain Charles Clerk (who fucceeded to the command on the death of Captain Cook). On his father’s death in i 776, he was appointed to the office of one of the under libra¬ rians of the Britifti Mufeum, and was afterwards prefer¬ red to a fuperior department, having the care of the antiquities, for which he was eminently qualified. In 1776 he alfo-fucceeded his father in the office of fecretary to the Royal Society. On the difputes re- fpedling the reinftatement of Dr Hutton in the depart¬ ment of fecretary for foreign correfpondence in 1784, Mr Maty took a warm and diftinguilhed part, and re- ligned the office of fecretary ; after which he under¬ took to affift gentlemen or ladies in perfedling then- knowledge of the Greek, Latin, French, and Italian dallies. Mr Maty was a thinking confcientious man ; and having conceived fome doubts about the articles he had lubferibed in early life, he never could be pre¬ vailed upon to place himfelf in the way of ecclefiaflicai preferment, though his connexions were amongft thofe who could have ferved him effentially in this point; and foon after his father’s death he withdrew himfelf from miniftering in the eftabliihed church, his reafons for which he publilhed in the 47th volume of the Gent. Magazine, p, 466. His whole life was thenceforv'ards taken up in literary purfuits. He received look from the duke of Marlborough, with a copy of that beauti¬ ful work, the Gemmee Marlburienfes, of which only 100 copies were worked off for prefents ; and of which Mr Maty wrote the French account, as Mr Bryant did, the Latin. In January 1782 he fet on foot a Review of publications, principally foreign, which he carried on, with great credit to himfelf and fatisfa&ion to the public, for near five years, when he was obliged to difeontinue it from ill health. He had long laboured under an afthmatic complaint, which at times made great ravages in his conftitution, and at laft put a pe¬ riod to his life in Jan. 1787, at the age of 42 ; leaving behind him one fon.—Mr Maty was eminently ac¬ quainted with ancient and modern literature, and parti¬ cularly converfant in critical refearches. The purity and probity of his nature were unqueftionable; and his humanity was as exquifite as it would have been exten- five, had it been feconded by his fortune. MAUBEUGE, a town of the Netherlands, in Hai- nault, with an illuftrious abbey of canoneffes, who muft be noble both by the father and mother’s fide. This place was ceded to France in 1678 ; and fortified after the manner of Vauban. In September 1793, the C Auftrianl M A U [ MauWe Auftrians formed the blockade of this place, but were Maupertuis.^n/en ^1(,m t*ie^r Poillion in the following month. It feated on the river bambre, in E. Long. 4. 2. N. Lat. 50. 16. LLlUCAUCO, Macaco, or JUafa, a genus of quadrupeds belonging to the order Primates. See Mam¬ malia Index. MAVIS, a fpecies of turdus. See Ornithology Index. MAUNCH, in Heraldry, the figure of an ancient ■seat lleeve, born in many gentlemen’s efcutcheons. MAUNDY THURSDAY, is the Thurfday in paf- fion week 5 which was called Maunduy or Mandate Thurfday, from the command which our Saviour gave his apoilles to commemorate him in the Lord’s fupper, which he this day inftituted 5 or from the new com¬ mandment which he gave them to love one another after he had walked their feet as a token of his love lo them. MAUPERTUIS, Peter Louis Morceau de, a celebrated French academician, was born at St Malo in 1698 ; and w'as there privately educated till he arriv¬ ed at his 16th year, when he was placed under the ce¬ lebrated profeffor of philofophy M. le Blond, in the college of La Marche, at Paris He foon difcovered a paffion for mathematical ftudies, and particularly for geometry. He like wife praftifed inftrumental mufic in his early years with great fuccefs, but fixed on no profeffion till he rvas 20, when he entered into the ar¬ my. He firll ferved in the Grey mufqueteers ; but in the year 1 y 20, his father purchafed for him a company of cavalry in the regiment of La Rocheguyon. He re¬ mained but five years in the army, during which time he pui lued his mathematical lludies with great vigour j and it was foon remarked by M. Freret and other aca¬ demicians, that nothing but geometry could fatisfy his a£Hve foul and unbounded thirft for knowledge. In the year 1723, he was received into the Royal Academy ol Sciences, and read his firft performance, which was a memoir upon the conftruftion and form ff mufical inftruments, November 15. 1724. During the firll years of his admifiion, he did not wholly con¬ fine his attention to mathematics ; he dipt into natu¬ ral philofophy, and difcovered great knowledge and dexterity in obfervations and experiments upon animals. If the cuftom of travelling into remote climates, like the fages of antiquity, in order to be initiated into the learned myfteries of thofe times, had dill fubfided, no one would have conformed to it with greater eager- nefs than M. de Maupertuis. His fird gratification of this paffion was to vifit the country which had ven birth to Newton : and during his refidence at ondon he became as jealous an admirer and fol¬ lower of that philofopher as any one of his own countrymen. His next excurfion was to Bafil in Switzerland, where he formed a friendlhip with the fa¬ mous John Bernouilli and his family, which continued to his death. At his return to Paris, he applied him- felf to his favourite dudies with greater zeal than ever: -—And how well he fulfilled the duties of an academi¬ cian, may be gathered by running over the memoirs of the academy from the year 1724 to 1736 5 where it ap¬ pears that he was neither idle nor occupied by objetfs »f fmall importance. Fhe mod fublime quedions in geomstry and the relative feiences received from his » a.z 8 ] M A U hands that eregance, clearnefs, and precifion, fo re¬ markable in all his writings. In the year 1736, he was fent by the king of France to the polar circle’, to meafure a degree, in order to afcertain the figure of the earth, accompanied by Mefl'rs Clairault, Camus Le Monnier, 1’Abbe Outhier, and Celfius the cele¬ brated profeffor of adronomy at Upfal. This didinc- tion rendered him fo famous, that at his return he was admitted a member of almod every academy in Eu¬ rope. In .the year 1740 Maupertuis had an invitation from the king of Pruflia to go to Berlin 5 which was too flattering to be refufed- His rank among men of let¬ ters had not wholly effaced his love for his firfl pro¬ feffion, namely, that ol arms. He followed his Pruf- fi?n majefly into the field, and was a witnefs of the difpofitions and operations that preceded the battle of Molwitz ; but was deprived of the glory of being pre- fent, when viftory declared in favour of his royal pa¬ tron, by a lingular kind of adventure. His horfe, dur¬ ing the heat of the aftion, running away with him, he fell into the hands of the enemy 5 and w'as at firft but roughly treated by the Auflrian foldiers, to whom he could not make himfelf known for want of lan¬ guage j but being carried prifoner to Vienna, he re¬ ceived fuch honours from their Imperial majefties as were never effaced from his memory. From Vienna he returned to Berlin ; but as the reform of the aca¬ demy which the king of Pruffia then meditated was not yet mature, he went again to Paris, where his af¬ fairs called him, and was chofen in 1742 diredlor of the Academy of Sciences. In 1743 he was received in¬ to the French academy ; which was the firft in ft an ce of the fame perfon being a member of both the aca¬ demies at Paris at the fame time. M. de Maupertuis again affumed the foldier at the liege of Fribourg, and was pitched upon by Marihal Cogny and the Count d’Argenfon to carry the news to the French king of the furrender of that citadel. He returned to Berlin in the year 1744, when ti marriage was negotiated and brought about by the good offices of the queen-mother, between our author and Mademoifelle de Borck, a lady of great beauty and merit, and nearly related to M. de Borck, at that time minifter of ftate. This determined him to fettle at Berlin, as he was extremely attached to his new fpoufe, and regarded this alliance as the molt fortunate cir- cumftance of his life. In the year 1746, M. de Maupertuis was declared by his Pi-uffian majelly prefiderit of the Royal Academy of Sciences at Berlin, and foon after by the fame prince was honoured with the order of Merit : However, all thefe accumulated honours and advantages, fo far from leffening his ardour for the fciences, feemed to furnilh new allurements to labour and application. Not a day paffed but he produced fome new projeil or effay for the advancement of knowledge. Nor did he confine himfelf to mathematical ftudies only : -metaphyfics, chemiftry, botany, polite literature, all fliared his at¬ tention, and contributed to his fame. At the fame time, he had, it feems, a ftrange inquietude of fpi- rit, with a morofe temper, which rendered him miferable amidft honours and pleafures.— Such a temperament did not promife a very pacific life, and he was engaged in fevctal quarrels. He had A lyTaupcrtuis' M A U [ a quarrel with Koenig the profeffor of philofophy at F'-aneker, and another more terrible with Voltaire. Maupertub had inferted into the volume of Memoirs of the Academy of Berlin for 1746, a difcourfe upon the laws of motion ; which Koenig was not content with attacking, but attributed to Leibnitz. Mauper- tuis, ftung with the imputation of plagiarifm, engaged the academy of Berlin to call upon him for his proof 5 which Koenig failing to produce, he was ftruck out of the academy, of which he was a member. Several pamphlets were the confequence of this j and Voltaire, lor fome reafon or other, engaged againft Maupertuis. We fay, for fome reafon or other j becaufe Maupertuis and Voltaire were apparently upon the moll amicable terms •, and the latter refpe&ed the former as his mailer in the mathematics. Voltaire, however, exerted all his wit and fatire againil him ; and on the whole was fo much tranfported beyond what was thought right, that he found it expedient in 1753 to quit the court of Pruflia. Our philofopher’s conflitution had long been con- fiderably impaired by the great fatigues of various kinds in w'hich his aftive mind had involved him } though from the amazing hardihips he had undergone in his northern expedition, molt of his future bodily fuffer- ings may be traced. The intenfe Iharpnefs of the air could only be fupported by means of ftrong liquors, which ferved to increafe his diforder, and bring on a fpitting of blood, which began at leaft 12 years before he died. Yet ftill his mind feemed to enjoy the greateft vigour; for the beft of his writings were pro¬ duced, and moft fublime ideas developed, during the time of his confinement by ficknefs, when he was un¬ able to occupy his prefidial chair at the academy. He took feveral journeys to St Malo, during the laft years of his life, for the recovery of his health : And though he always received benefit by breathing his native air, yet ftill, upon his return to Berlin, his diforder like- wife returned with greater violence.—His lafl: journey into France was undertaken in the year 1757 j when he was obliged, foon after his arrival there, to quit his favourite retreat at St Malo, on account of the danger and confufion which that town was thrown into by the arrival of the Engliih in its neighbourhood. From thence he went to Bourdeaux, hoping there to meet with a neutral {hip to carry him to Hamburgh, in his way back to Berlin •, but being difappointed in that hope, he went to Thouloufe, where he remained feven months. He had then thoughts of going to Italy, in hopes a milder climate would reftore him to health : but finding himfelf grow vrorfe, he rather inclined to¬ wards Germany, and went to Neufchatel, where for three months he enjoyed the converfation of Lord Marifchal, with whom he had formerly been much connected. At length he arrived at Bafil, Odlober 16. 1758, where he was received by his friend Ber¬ noulli and his family with the utmofl: tendernefs and affe&ion. He at firft found himfelf much better here than he had been at Neufchatel : but this amendment was of fhort duration 5 for as the winter approached, his diforder returned, accompanied by new and more alarming fymptoms. He languilhed here many months, during which he was attended by M. de la Condamine j and died in 17 qq. He wrote in French, 1. The figure of the earth de- 19 ] M A U termined, 2. The meafure ot a degree of the meridian. 3. A difcourfe on the parallax of the moon. 4. A dif¬ courfe on the figure of the ftars. 5. The elements of geography. 6. Nautical aftronomy. 7. Elements cf aftronomy. 8. A phyfical dilfertation on a white inha¬ bitant of Africa. 9. An effay on cofmography. 10. Re¬ flexions on the origin of languages. 11. An efiay on moral philofophy. 12. A letter on the progrch <1 the fciences. . 13. An eflav on the formation of bodies. 14. An eulogium on M. de Montelquieu. 15. Let¬ ters, and other works. MAUR, St, was a celebrated difciple of St Bene- diX. If avc can believe a life of St Maur afcribed to Fauftus his companion, he tvas fent by BentdiX on a million to France. But this life is confidered as apo¬ cryphal. In rejeXing it, however, as well as the cir- cumftanc.es of the million, we muft beware of denying the miflion itfelf. It is certain that it Avas believed in France as early as the 9th century •, and notAvithftand- ing the filence of Bede, Gregory of Tours, and others, there are feveral documents Avhich prove this, or at leaft render it extremely probable. A celebrated fo- ciety of BenediXines, took the name of St Maur in the beginning of the laft century, and received the fanXion of Pope Gregory XV. in 1621. This fo- ciety was early diftinguifhed by the virtue and the knoAvledge of its members, and it ftill fupports the charaXer. There are, perhaps, feAver eminent men in it than formerly ; but this may be afcribed to the levi¬ ty of the age, and partly to the little encouragement for the refearches of learned men. The chief perfons of ingenuity Avhich this fociety has produced are, the Fathers Menard, d’Acheri, Mabillon, Ruinart, Ger¬ main, Lami, Montfaucon, Martin, Vaiflette, le Nourri, Martianay, Martenne, Mafluet, &c. &c. See L’Hif- toire Litteraire de la Congregation de St Maur, publilb- ed at Paris under the title of BruJJels, in qto, 1770, by Dom. Tallin. MAUR ICE AU, Francis, a French furgeon, who applied himfelf Avith great fuccefs and reputation to the theory and praXice of his art for feveral years at Paris. Afterrvards he confined himfelf to the diforders of preg¬ nant and lying-in-Avomen, and Avas at the head of all the operators in this way. His Obfervations fur la groffejfe and fur Caccouchement des femmes, fur leurs maladies, et celles des enfans nouveaux, 1694, in 410, is reckoned an excellent Avork, and has been tranflated into feveral languages, German, Flemifli, Italian, Eng- lifh : and the author himfelf tranflated it into Latin. It is illuftrated Avith cuts. He publifhed another piece or tAvo, by Avay of fupplement, on the fame fubjeX 3 and died at Paris in 1709. MAURICE, St, commander of the Theban le¬ gion, Avas a Chriftian, together Avith the officers and foldiers of that legion, amounting to 6600 men.-— This legion received its name from the city Thebes in Egypt, Avhere it was raifed. It Avas fent by Dio- clefian to check the Bagaudse, Avho had excited fome difturbances in Gaul. Maurice having carried his troops over the Alps, the emperor Maximinian com¬ manded him to employ his utmoft exertions to extir¬ pate Chriftianity. This propofal Avas received Avith horror both by the commander and by the foldiers. The emperor, enraged at their oppofition, command¬ ed the legion to be decimated 3 and w hen they ftill C 2 declared Maupertuis II. Maurice. M A U declared that they would fooner die than do any thing prejudicial to the Chrhtian faith, every tenth man of thofe who remained was put to death. Their perfe- verance excited the emperor to ftill greater cruelty j for when he faw that nothing could make them relinquifh their religion, he commanded his troops to furround them, and cut them to pieces. Mau¬ rice, the commander of thefe Chriitian heroes, and Exuperus and Candidus, officers of the legion, who had chiefly infligated the foldiers to this noble re- fiftance, fignalized themfelves by their patience and their attachment to the doctrines of the Chriitian re¬ ligion. They were maffacred, it is believed, at A- gaune, in Chablais, the 22d of September 286.— Notwithfianding many proofs which fupport this tranf- aftion, Dubordier, Hettinger, Moyle, Burnet, and Mofheim, are difpofed to deny the fa£i. It is de¬ fended, on the other hand, by Hickes an Englifh writer, and by Dom Jofeph de Lille a Benedidtine monk de la congregation de Saint Vnnnes. in a work of his, entitled Defence de la Vente du Marti/) e de la Le¬ gion Thebenne, 1737. In defence of the fame faCi, the reader may eonfult Hijloria de S. Mauritie, by P. Rof- fignole a Jefuit, and the Acta San&orum for the month of September. The martyrdom of this legion, written by St Eucherius bifhop of Lyons, was tranfmitted to pofterity in a very imperfeft manner by Surius. P. Chif- flet a Jefuit, difeovered, and gave to the public, an ex- a& copy of this work. Don Ruinart maintains, that it has every mark of authenticity. St Maurice is the pa¬ tron of a celebrated order in the king of Sardinia’s do¬ minions, created by Emanuel Philibert duke of Savoy, to reward military merit, and approved bv Gregory XIII. in 1572. The commander of the Theban le¬ gion muft not be confounded with another St Maurice, mentioned by Theodoret, who futfered martyrdom at Apamea in Syria. Maurice, (Mauritius Tiberius'), was born at Ara- biffus in Cappadocia, A. D. 539. He was defeend- ed from an ancient and honourable Roman family.— After he had filled feveral offices in the court of Tibe¬ rius Conftandne, he obtained the command of his ar¬ mies againft the Perfians. His gallantry was fo con- fpicuous that the emperor gave him his daughter Conftantina in marriage, and inveffed him rvith the purple the 13th Auguft 582. The Perfians ftill continued to make inroads on the Roman territo¬ ries, and Maurice fent Philippicus, his brother-in-law, againft them. This general condudted the war with various fuccefs. At firfl he gained feveral fplendid vi&ories, but he did not continue to have a decided fuperiority. As there was a great ufe for foldiers in thefe unfortunate times, the emperor iffued a man¬ date in $92, fbrbidding any foldier to become a monk till he had accomplifhed the term of his military fer- vice. Maurice acquired much glory in reftoring Chof- roes II. king of Perfia, to the throne, after he had been depofed by his fubjedts. The empire was in his reign haraffed by the frequent inroads of the Arabian tribes. He purebafed peace from them, by granting them a penfion nearly equal to 100,000 crowns •, but thefe barbarians took frequent opportunities to renew the war. In different engagements the Romans de- ifroyed 50,000, and took 17,000 prifoners. 7'hefe were reifored, on condition that the king of the Abarx M a u fhould return all the Roman captives in his dominions. Regardlefs of his promife, he demanded a raniom of 10,000 crowns. Maurice, full of indignation, refufed the fum : and the barbarian, equally enraged, put the captives to the fword. W hile the emperor, to revenge this cruelty, was making preparations againft the A- ban, Phocas, who from the rank of centurion had attained the higheft military preferment, affumed the purple, and was declared emperor. He purfued Mau¬ rice to Chalceden, took him prifoner, and condemned him to die. The five Ions of this unfortunate prince were maflacred before his eyes, and Maurice, humbling himfelf under the hand of God, wTas heard to ex¬ claim, Thou art juf, 0 Lord, and thy judgments are ’without partiality. He was beheaded on the 26th No¬ vember 602, in the 63d year of his age and 20th of his reign. Many writers have eftimated the charac¬ ter of this prince by his misfortunes inftead of his adfions. They believed him guilty without evidence, and condemned him without reafon. It cannot be de¬ nied, however, that he allowed Italy to be haraffed ; but he was a father to the reft of the empire. He re- ftored the military difeipline, humbled the pride of his enemies, fupported the Chriltian religion by his laws,, and piety by his example. He loved the fciences, and was the patron of learned men. Maurice, eledlor of Saxony, fon of Henry le Pieux, was born A. D. 1521. He was early remark¬ able for his courage, and during his whole life he vras engaged in warlike purfuits. He ferved under the emperor Charles V. in the campaign of 1544 againft France •, and in the year following againft the league of Smalkalde j with which, although a Proteftant, he would have no manner of connexion. The emperor, as a reward for his fervices, in the year 1547, made him eledfor of Saxony, having deprived his coufin John Frederick of that electorate. Ambition had led him to fecond the views of Charles, in the hope of being eleCtor, and ambition again detached him from that prince. In 151:1 he entered into a league agamfl; the emperor, together with the eleftor of Branden- burgh, the Count Palatine, the duke of Wirtem- burg, and many other princes. This league, encou¬ raged by the young and enterprifing Henry II. of France, was more dangerous than that of Smalkalde. The pretext for the aflbeiation was the deliverance of the landgrave of Hefle, w hom the emperor kept pri¬ foner. Maurice and the confederates marched, ia 1552, to the defiles of Tyrol, and put to flight the Imperial troops w'ho guarded them. The emperor and his brother Firdinand narrowly efcaped, and fled from the conquerors in great diforder. Charles hav¬ ing retired into Paffau, where he had collefted an army, brought the princes of the league to terms of accommodation. By the famous peace of Paffau, which was finally ratified the 12th of Auguft 1552, the emperor granted an amnefty without exception to all thofe who had carried arms againft him from the year 1546. The Proteftants not only obtained the free exercife of their religion, but they rvere admit¬ ted into the imperiaEchamher, from svhieh they had been excluded fince the victory of Mulberg.—Mau¬ rice foon after united himfelf with the emperor againft the margrave of Brandenburg, who laid ivafte the German provinces. He engaged him in 1553, g“in~ ed 1 20 1 Mat M A U [2 Maurice, cd the battle of Siverfhaufen, and died of the wounds 1 jie jiacl received in the engagement two days after. He was one of the greateft proteftors of the Luther¬ ans in Germany, and a prince equally brave and po¬ litic. After he had profited by the fpoils of John Frederick, the chief of the Proteftants, he became himfelf the leader of the party, and by thefe means maintained the balance of power againft the emperor in Germany. Maurice de NaJJhu, prince of Orange, fucceeded to the government of the Low Countries after the death of his father William, who was killed in 1584 by the fanatic Gerard. The young prince was then only eighteen years of age, but his courage and abi¬ lities were above his years. He was appointed cap¬ tain general of the United Provinces, and he reared that edifice of liberty of which his father had laid the foundation. Breda fubmitted to him in 1590', Zut- phen, Deventer, Hulft, Nimeguen, in 1591. He gained feveral important advantages in 1592, and in the year following he made himfelf mafter of Gertru- denburg. When he had performed thefe fplendid fervices, he returned to the Low Countries by the way of Zealand. His fleet was attacked by a dreadful tempeft, in which he loft forty veffels, and he him¬ felf had very nearly periflied. His death would have been confidered by the Hollanders as a much greater calamity than the lofs of their Veffels. They watched over his fafety with exceeding care. In 1594, one of his guards was accufed of an intention to take away his life ; and it was generally believed that he was bribed to this fervice by the enemies of the republic. He fell a facrifice at Bruges, either to his own fanaticifm or to the jealous anxiety of the friends of Maurice. The prince of Orange, increafing in reputation, de¬ feated the troops of the archduke Albert in I597> an(^ drove the Spaniards entirely out of Holland. In 1600 he was obliged to raife the fiege of Dunkirk } but he took ample vengeance on Albert, whom he again de¬ feated in a pitched battle near Nieuport. Before the a£Hon, this great general fent back the fhips which had brought his troops into Flanders : Mij brethren (faid he to his armv), we rnuji conquer the enemy or drink up the waters of the fea. Determine for yourfehes ; I have determined 1 (hall either conquer by your bravery, or I (hall never furvive the difyrace of being conquered by men in every refpici our inferiors. This fpeech elevated the foldiers to the higheft pitch of enthufiafm, and the victory was complete. Rhinberg, Grave, and F.clufe, cities in Flanders, fubmitted to the conqueror the fol¬ lowing year. Maurice, however, not only laboured for the commonwealth, but alfo for himfelf. He co¬ veted the fovereignty of Holland, and was oppofed in the profecution of his defign by the penfioner Barne veldt. The zeal and activity of this wife republican coft him his life. He was an Arminian ; and at this time Maurice defended Gomar againft Arminius.— Taking advantage of the general odium under ivhich the Arminians lay, he found means to get Barneveldt condemned in 1619. His death, wholly owing to the cruel ambition of the prince of Orange, made a deep impreflion on the minds of the Hollanders. The truce with Spain being expired, Spinola laid fiege to Breda in 16 24, and in fix months, by the proper di¬ rection of his great talents, though with great ilaugh- ]• M A U ter of his troops, he took the place. The prince oi Maurice, Orange, unfuccefsful in every attempt to raife the iv~aur't"J1t‘t,'’ fiege, died of vexation in 1625, aKec* 55 years, with the reputation of the greateft warrior of his time.— “ The life of this ftadtholder (fays the abbe Ray- nal) Avas almoft an uninterrupted feries of battles, of fieges, and of victories. Of moderate abilities in every thing elfe, he (hone confpicuous in his military capacity. His camp was the ichool of Europe j and thofe who received their military education in his ar¬ mies augmented, perhaps, the glory of their mafter.— Like Montecuculi, he difeovered inimitable Ikiil in his marches and encampments ; like Vauban, he pof- feffed the talent of fortifying places, and of rendering them impregnable j like Eugene, the addrefs of find¬ ing fubliftence for great armies in countries barren by nature, or ravaged by Avar j like Vendome, the happy talent of calling forth, in the moment they became neceffary, greater exertions from his foldiers than could reafonably be expeCted 5 like Conde, that infallible quicknefs of eye Avhich decides the fortune of battles j like Charles XII. the art of rendering his troops al¬ moft invincible to cold, hunger, and fatigue 5 like Turenne, the fecret of making war Avith the leaft pof- fible expence of human blood.” The Chevalier Folard maintains, that Maurice Avas the greateft commander of infantry fince the time of the Romans. He ftudied the military art of the ancients, and applied their rules with great exaftnefs in the various occurrences of war. He not only took advantage of the inventions of others, but he enriched the fcience of Avar with feveral im¬ provements. Telefcopes Avere firft ufed by him for a military purpofe $ and, befides a kind of gallery in condufting a fiege, and the plan of blockading a ftrong place, Avhieh Avere of his iiwention, he greatly improved the Avhole art by his method of pulhing an attack with great vigour, and of defending, for the greateft length of time, and in the beft manner^ a place befieged. In ftiort, the many ufeful things Avhich he praftifed or invented, placed him in the higheft rank among men of a military charafter. On one occafion, a lady of quality alked him, Who was the firjl general of the age? Spinola (replied he) is the fecond. It Avas his conftant pra&ice, during fleep, to have two guards placed by his bedfide, not only to de¬ fend him in cafe of danger, but to awake him if there fliould be the leaft occafion. The Avar betAvixt Spain and Holland was never carried on Avith greater keen- nefs and animofity than during his adminiftration.— The Grand Signior, hearing of the vaft torrents of blood ftied in this conteft, thought that a great em¬ pire muft depend on the decifion. J he object of fo manv battles was pointed out to him on a map, and he faid coldly, If it were my bujinefs, I would fend my pioneers, and order them to cafl this little corner of earth into the fea. Maurice, like many great men, Avas im¬ patient under contradiftion, and too much devoted to Avomen. He was fucceeded by Frederick Henry his brother. MAURITANIA, an ancient kingdom of Africa, bounded on the weft by the Atlantic ocean, on the fouth by Getulia or Libya Interior, and on the north by the Mediterranean \ comprehending the greater part of the kingdoms of Fez and Morocco.— Its an¬ cient limits are not exa&ly mentioned by any hiftprian j neither M A U [2 Mauritania, neither can they now be afcertained by any modern ob- ””"'v fervations, tliele kingdoms being but little known to Europeans. This country was originally inhabited by a people called Mauri, concerning the etymology of which name authors are not agreed. It is probable, however, that this country, or at leall a great part of it, was firft called Phut, iince it appears from Pliny, Ptolemy, and St Jerome, that a river and territory not far from Mount Atlas went by that name. From the Jerufalem Tar- gum it likewife appears, that part of the Mauri may be deemed the offspring of Lud the fon of Mifraim, lince his defendants, mentioned Genefis x. are there called 'NU-no, Mauri, or Mauritani. It is certain, that this region, as well as the others to the eaftward of it, had many colonies planted in it by the Phoeni¬ cians. Procopius tells us, that in his time two pillars of white Hone were to be feen there, with the follow¬ ing infcription in the Phoenician language and charac¬ ter upon them : “ We are the Canaanites, that fled from Jojhua the fon of Nun, that notorious robber.” Ibnu Rachic, or Ibnu Raquig, an African Avriter cited by Leo, together rvith Evagrius and Nicephorus Cal- liltus, affert the fame thing. . The Mauritanians, according to Ptolemy, Avere di¬ vided into feveral cantons or tribes. The Metagonitce were feated near the itraits of Hercules, now thofe of Gibraltar. The Saccojii, or Cocojii, occupied the coaft of the Iberian fea. Under thefe tAvo petty na¬ tions the Majtces, Verues, and Verbicce or Vervicce, were fettled. The Salifce or Salinfce, Avere fituated lower, toAvards the ocean; and, ftill more to the fouth, the Vo/ubiham. The Maurenjii and Herpiditani poffeffed the eaflern part of this country, Avhich Avas terminated by the Mulucha. The Angaucani, or Jangacaucani, NeSiiberes, Zagrenjii, Baniubee, and Va¬ cant*, extended themfelves from the fouthern foot of Ptolemy’s Atlas Minor to his Atlas Major. Pliny mentions the Baniurcc, Avhom Father Hardouin takes to be Ptolemy’s Baniubae *, and Mela the Atlantes, Avhom he reprefents as poffeffed of the Aveftern parts of this diftridt. The earlieft prince of Mauritania mentioned in hiftory is Neptune *, and next to him Avere Atlas and Antaeus his tAvo fons, both famous in the Grecian fables on account of their Avars with Hercules. An¬ taeus, in his contention Avith that hero, feems to have behaved Avith great bravery and refolution. Having received large reinforcements of Libyan troops, he cut off great numbers of Hercules’s men. But that celebrated commander, having at laft intercepted a ftrong body of Libyans fent to the relief of Antceus, gave him a total overthrow, wherein both he and the belt part of his forces were put to the fword. This deciiive action put Hercules in poffeffion of Libya and Mauritania, and confequently of the riches of all thefe kingdoms. Hence came the fable, that Her¬ cules, finding Antseus, a giant of an enormous fize with Avhom he was engaged in Angle combat, to re¬ ceive freih ftrength as often as he touched his mother earth Arhen thrown upon her, at laft lifted him up in the air and fqueezed him to death. Hence likeAvife may be deduced the fable intimating that Hercules took the globe from Atlas upon his orvn (boulders, overcame the dragon that guarded the orchards of the ] m a u Hefperides, and made liimfelt mailer of all the oold- Mlumtma en inut there. Bocbart thinks that the fable alluded ' v—* chiefly to naval engagements, wherein Hercules, for the moft part, was vi&orious j thougli Animus from time to time received fuccours by fea. But at laft Hercules, coming up Avith one of his fquadrons which had a ftrong reinforcement on board, made himfelf mailer of it, and thus rendered Antaeus incapable for the future of making head againft him. The fame author likewife infinuates, that the notion of Antaeus’s gigantic ftature prevailing for fo many centuries a- mongft the Tingitanians, pointed out the fize of th® veffels of which his fleets and fquadrons Avere com- pofed. As for the golden apples fo frequently men¬ tioned by the old mythologifts, they were the trea¬ sures that fell into Hercules’s hands upon the defeat of Antaeus; the Greeks giving the oriental word hxn- riches, the fignification affixed to their oivn term p.YiXct, apples. With regard to the age in which Atlas and An¬ taeus lived, the moft probable fuppofition feems to be that of Sir Ifaac NeAvton. According to that illufi- trious. author, Ammon the father of' Sefac Avas the firft king of Libya, or that vaft tradt extending from the borders of Egypt to the Atlantic ocean ; the con- queft of Avhich country Avas effedfed by Sefac in his father’s lifetime. Neptune afterwards excited the Li¬ byans to a rebellion againft Sefac, and Acav him ; and then invaded Egypt under the command of Atlas or Antfeus, the fon of Neptune, Sefac’s brother and ad¬ miral. Not long after, Hercules, the general of The- bais and Ethiopia for the gods or great men of Egypt, reduced a fecond time the Avhole continent of Libya, having overthroAvn and flain Antaeus near a town in Thebais, from that event called Antcea or Antceopolis : this, we fay, is the notion advanced by Sir Ifaac NeAv¬ ton, who endeavours to prove, that the firft redudtion of Libya, by Sefac, happened a little above a thou- fand years before the birth of Chrift, as the laft, by Hercules, did fome few years after. Noav, though we do not pretend to adopt every particular circum- ftance of Sir Ifaac NeAvton’s fyftem, yet Ave cannot forbear obferving, that it appears undeniably plain from Scripture, that neither the weftern extremity of Libya, nor even the other parts of that region, could poffibly have been fo Avell peopled before the time of David or Solomon, as to have fent a numerous army to invade Egypt. For Egypt and Phoenicia, from avhence the greateft part of the anceftors of the Li¬ byans came, and Avhich Avere much nearer the place from whence the firft difperfion of mankind Avas made, could not themfelves have been greatly overftoeked with inhabitants any confiderable time before the reign of Saul. And that fuch an invafion happened in the reign of Neptune, or at leaf! of his fon Antseus, has been moft fully evinced by this moft excellent chrono- logcr. from the defeat of Antfeus, nothing remarkable occurs in the hiftory of Mauritania till the times of the Romans, who at laft brought the whole kingdom under their jurifdiftion; for which fee the article Rome. . I. With regard to the cuftoms, &c. of this people, it wmuld feem, from what Hyginus infinuates, that they fought only with clubs, till one Belus, the fon of Neptune, as that author calls him, taught them M A U [ 23 ] M A U Mauritania, them the ufe of the fword. Sir Ifaac Newton makes —Y—-J this Belus to have been the fame perfon with Sefoftris king of Egypt, who overran a great part of the then known world. 2. All perfons of diitindlion in Mau¬ ritania went richly attired, wearing much gold and lilver in their clothes. They took great pains in cleanl- ing their teeth, and curled their hair in a curious and elegant manner. They combed their beards, which were very long, and always had their nails pared ex¬ tremely clofe. When they walked out in any num¬ bers, they never touched one another, for fear of dif- eoncerting the curls into which their hair had been formed. 3. The Mauritanian infantry, in time of ac¬ tion, ufed Ihields made of elephants ikins, being clad in thofe of lions, leopards, and bears, which they kept on both night and day. 4. The cavalry of this nation was armed with broad Ihort lances, and carried targets or bucklers, made likewife of the Ikins of -wild beafts. They ufed no faddles. Their horfes were fmall and fwift, had wooden collars about their necks, and were fo much under the command of their riders, that they would follow them like dogs. The habit of thefe horfemen was not much dift'erent from that of the foot above mentioned, they conftantly wearing a large tunic of the fkins of wild beads. The Phutad, of whom the Mauritanians were a branch, were emi¬ nent for their Ihields, and the excellent ufe they made of them, as we learn from Homer, Xenophon, Hero¬ dotus, and Scripture. Nay, Herodotus feems to inti¬ mate, that the Ihield and helmet came from them to the Greeks. 5. Notwithftanding the fertility of their foil, the poorer fort of the Mauritanians never took care to manure the ground, being ftrangers to the art of hufbandry ; but roved about the country in a wild lavage manner, like the ancient Scythians or Arabes Scenitae. They had tents, or mapalia, fo extremely fmall, that they could fcarce breathe in them. Their food was corn, herbage, &c. which they frequently did eat green, without any manner of preparation, be¬ ing deftitute of wine, oil, and all the elegancies as well as many neceffaries of life. Their habit was the fame both in fummer and winter, eonliiling chiefly of an old tattered, though thick garment, and over it a coarfe rough tunic 5 which anfwered probably to that of their neighbours the Numidians. Mod of them lay every night upon the bare ground; though fome of them drewed their garments thereon, not unlike the prefent African Kabyles and Arabs, who, according to Dr Shaw, ufe their hykes for a bed and covering in the night. 6. If the mod approved reading of Ho¬ race may be admitted, the Mauritanians diot poifoned arrows } which clearly intimates, that they had fome dull in the art of preparing poiibns, and were excellent dartmen. This lad obfervation is countenanced by Herodian and iElian, who entirely come into it, affirm¬ ing them to have been in fuch continual danger of be¬ ing devoured by wild beads, that they durd not dir out of their tents or mapalia without their darts. Such perpetual exercife mud render them exceedingly fkil- ful in hurling that weapon. 7. The Mauritanians fa- crificed human viftims to their deities, as the Phoenici¬ ans, Carthaginians, &c. did. The country people were extremely rude and bar¬ barous \ but thofe inhabiting cities mud undoubtedly have had at lead fomc fmattering in the literature of the feveral nations they deduced their origin from. That Mauritania the Mauritanians had fome knowledge in naval affairs, II. . feems probable, not only from the intercourfe they had with the Phoenicians and Carthaginians, as well as the fituation of their country j but likewife from Orpheus, or Onomacritus, who afferts them to have made a fettlement at the entrance into Colchis, to which place they came by fea. Magic, forcery, divination, &c. they appear to have applied themfelves to in very early times. Cicero and Pliny fay, that Atlas was the inventor of adrology, and the dottrine of the fphere, i. e. he fird introduced them into Mauritania. This, according to Diodorus Siculus, gave rife to the fable of Atlas’s bearing the heavens upon his fhoulders. The fame author relates, that Atlas indruefed Hercules in the doctrine of the fphere and adrology, or rather adronomy, who afterwards brought thofe fciences into Greece. MAURITIA, the Ginkgo, or Maidenhair Tree; a genus of plants belonging to the natural order of palmae. See Botany Index. MAURITIUS, or Maurice, an ifland of Africa, about 400 miles ead of Madagafcar, lying in the la¬ titude of 20 and 21 degrees fouth. It is about 150 miles in circumference. In the beginning of the 16th century it wras difeovered by the Portuguefe, who knowing that Pliny and other ancient writers had mentioned the ifland of Cerne in thefe feas, took it for granted that this mud be it j and accordingly we find it dyled Cerne or Sirnc, in their maps : but, not- withftanding this, they did not think fit to fettle it $ and indeed their force was fo fmall, in comparifon of the vad dominions they grafped, that it was very ex- eufable. However, according to their laudable cu- dom, they put fome hogs, goats, and other cattle, up¬ on it, that in cafe any of their fliips either going to the Indies or returning to Portugal Ihould be obliged to touch there, they might meet with refrelhments. The Dutch, in the fecond voyage they made to the Ead Indies under their admiral James Cornelius Van- neck, came together with five fliips on the 13th of September 1368 j anchored in a commodious port, to which they gave the name of Warwick Haven ; and gave a very good account of the place in their jour¬ nals. Captain Samuel Caltleton, in the Pearl, an Englilh Ead India fliip, arrived there on the 27th of March 16125 and taking it to be an ifland undifeover- ed before, beffowed upon it the name of England's Fo- reft, though others of his crew called it Pearl IJland; and in the account of their voyage, written by John Tatton the in alter of the Ihip, celebrated it as a place very convenient for Hupping, either outward or home¬ ward bound, to refrefli at. This they femetimes ac¬ cordingly did, and brought fome cargoes of ebony, and rich wood from thence, but without fixing any / fettlement. , At length, in 1638, the Dutch feated themfelves here : and it is highly remarkable, that at the very time they were employed in making their fird fettle¬ ment, the French fent a veffel to take poffeflion of it, who found the Dutch beforehand with them, and re¬ filled the afliftance of an Englilh Indiaman, wooding and watering in another port of the ifland, who very frankly offered it, to drive the Dutch from their half- fettled pods. They continued for fome time in quiet; poffeflion M A U [ 24. ] Mauritius, pofleflion of the places they fortified in this illand, to defence v which they gave the name of Mauritius, in honour of Prince Maurice their ftadthdder. But having en¬ gaged the French, who were fettled on Madagafcar, to fteal 50 of the natives, and fell them for Haves, for the improvement of the Dutch fettlements here, this proved the ruin of both colonies 5 for the negroes furprifed and maffacred the French in Madagafcar j and the Haves in Mauritius fled into the centre of the ifland } from whence they fo much and fo inceflantly molefted thofe who had been formerly their mailers, that they chofe to quit a country where they could no longer remain , in any tolerable degree of fafety. The Fail India Company, however, from motives of con- veniency, and a very imperfect notion of its value, dif- approved this meafure, and therefore ordered it to be refettled j which was accordingly done, and three forts erefted at the principal havens. Things now went on fomewhat better than they did before •, but they were Hill very much difturbed by the revolted negroes in the heart of the ifle, whom they could never fubdue. One principal ufe that the company made of this place, was to fend thither ftate prifoners, who, as they were not men of the bell morals, quickly cor¬ rupted the reft of the inhabitants, and rendered them fuch a race of outrageous fmugglers, the fituation of the place concurring with their bad difpofition, that, after various ineffe£lual attempts made to reform them, orders were at length given to abandon Mauritius a fecond time, which, after fome delays, were put in execution in the year 1710. Two years after this, the French took pofleffion of it, and named it the ijie de France. This name has obtained among themlelves, but the Europeans in ge¬ neral continue to call it Mauritius. It lies in S. Lat. 20. 15. E. Long. 6. 15. The inconveniences arifing from the want of a port at the ifland of Bour¬ bon, induced the French to take pofleflion of Mauri¬ tius, it having two very good harbours, to fortify which no expence has been fpared. That on the north-weft is called Port Louis, that on the fouth-eaft fide of the ifland is called Port Bourbon. The trade-wind from the fouth-eaft in thefe latitudes blows all the year round, excepting for a few days at the fummer fol- ftice, when it is interrupted by hard gales and hurri¬ canes from the north. The eafe with which this wind enables {hips to enter the port of Bourbon, caufed the French, when they firft took pofleflion of this fpot, to efteem it the beft port in the ifland *, but experience pointing out to them, that the fame wind often ren¬ dered the paffage out of the harbour fo difficult, that a fhip was fometimes obliged to wait a confiderable time before the weather admitted of her putting to fea, this harbour is in a great meafure abandoned, and the prin¬ cipal town and feat of government is now fixed at Port Louis, which is nearly in the middle of the north fide of the ifland, and its entrance is through a channel formed by two ftioals, which advance about two miles .into the fea. When a ftiip arrives oppofite to this channel, the fouth-eaft wind hinders her from entering the port under fail, and the muft either warp in with cables or be towed in with boats. The neceflity of this operation, joined to the extreme narrownefs of the channel, which does not admit of two (hips abreaft of each other entering at the fame time, is one of the beft M A U the harbour has againft an attack by fea ; for, Mauritius,j from thefe obftacles, an enemy would find it a matter of the greateft difficulty to force the port 5 and in ad¬ dition to this natural ftrength, they have built two forts and as many batteries, which are mounted with heavy cannon, and entirely command the approach to the harbour, fliould ffiips prefume to force an entry under fail. This port is capable of containing 100 fail of ftiips, and is well provided with every requifite for repairing and even building of {hips. This port has proved of the greateft advantage to France in the fe- veral wars which have been carried on between Great Britain and her j and has proved of great utility to the French Eaft India Company’s commerce •, for here their ffiips and crews were fure to meet with all necef- fary refreffiment after a long voyage. The port of Bourbon is alfo fortified j and an army landed here would find it an extremely difficult talk to pafs the mountains to the different parts of the ifland. There are feveral places between the north-eaft extremity and Port Louis where boats may land, but all thefe are de¬ fended by batteries ; and the country behind them is a continued thicket: The reft of the coaft is inaccefli- ble. In the north-eaftern quarter is a plain extending about 10 miles from eaft to weft, and in fome places five miles inland from the northern coaft. All the reft of the ifland is full of high and fteep mountains, lying fo near to one another, and the intervals between them fo narrow, that, inftead of valleys, they rather refemble the beds of torrents j and thefe are choked with huge fragments of rocks which have fallen from the fteep fides of the impending mountains. On the fummits of the mountains ice is frequently to be found, and they are covered with forefts of ebony and other large trees. The ground they ffiade produces herbage, Ihrubs, and plants of various forts, from the common grafs to the ftrongeft thorn, and that in fuch profufion, that they form a thicket fo clofely interwoven, that no progrefs can be made but by means of a hatchet. Notwithftanding thefe difficulties, plantations have been formed on thefe mountains, and very confiderable pro¬ grefs has been made in the plains; but the productions, although moftly of the fame kind, are not only in lefr quantity, but of an inferior quality to thofe produced at Bourbon ifland. In a courfe of years, however, the fettlement coft fo much, and was eonfidered in every light worth fo little, that it had been more than once under deliberation, whether, after the example of the Dutch, they fliould not leave it again to its old negro inhabitants; which fooner or later in all likelihood would have been its fate, if, in 1735, the famous M. de la Bourdonnais had not been fent thither with the title of governor-gene¬ ral of the French iflands. He found this ifle in the worft ftate poffible, thinly inhabited by a fet of lazy people, who equally hated induftry and peace, and who were continually flatter¬ ing this man to his face, and belying him wherever and as far as they durft. He gave himfelf no trouble about this, having once found the means to make him¬ felf obeyed ; he faw the vaft importance of the ifland ; he conceived that it might be fettled to great advan¬ tage ; and, without fo much as expecting the thanks of thofe for whom he laboured, he began to execute tills great defign. His firft ftep was to bring over black M A U r 25 ] MAX Mauritius, black boys from Madagafcar, whom he carefully trained — y.—, 1 Up in good principles, and in continual exercife •, by which he rendered them fo good foldiers, that he very quickly obliged the Marones, or wild negroes, either to fubmit or to quit the illand : he taught the planters to cultivate their lands to advantage •, he, by an aqueduft, brought frefli water to the fea fide; and whereas they had not fo much as a boat at his coming thither, he made a very fine dock, where he not only built Hoops and large veflels, but even a (hip of the burden of 500 tons. However incredible it may feem, yet it is cer¬ tainly fatt, that in the fpace of five years he converted this country into a paradife, that had been a mere wildernefs for 5000 ; and this in fpite of the inhabitants, and of the company, who being originally prejudiced by them, behaved ill to him at his return. He foon made the cardinal de Fleury, however, fenfible of the true ftate of things; and compelled the company to acknowledge, though they did not reward, his ferviees. He afterwards returned into the Indies, and perfedfed the work he had begun, and to him it is owing that the ille of France was rendered one of the fineit and moll important fpots upon the globe. Here no coffee is raifed ; but by the indefatigable induftry of M. de Bourdonnais, fugar, indigo, pepper, and cotton (which are not at Bourbon), came to be cultivated with fuccefs. Since the departure of that moll excellent governor, the plantations have been neglected, and are fallen off; but if a proper fpirit of activity was raifed among the inhabitants, they might foon be made to refume their fiourifhing appearance. Mines of iron have been difeovered in the mountains near the great plain, in the north-eaft part of the ifland ; and thefe mountains affording in great abundance the neceffary fuel, forges have been eredted : but the iron produced is of a very inferior quality, it being brittle, and only fit for making cannon-balls and bomb-fhells. Black cattle, fheep, and goats, are preserved with difficulty ; the firll generally die before they have been a year in the illand, and this' occafions frequent importations of them from Madagafcar and other parts. Common domeftic poultry breed in great plenty; and, with filh and turtle, furnilh a great part of the food of the European inhabitants. The approach to the illand is extremely dangerous, it being furrounded with ledges of rocks, and many of them covered by the fea. The ffiore abounds with coral and ffiells. This illand is faid to contain 60 ri¬ vers : fome are confiderable Itreams, and molt of them have their fources from lakes, of which there are feve- ral in the middle part of the ifland. The rivers afford plenty of various kinds of filh, particularly eels. Thefe are of an enormous fize, fome having been found that were fix feet long, and fix inches in circumference, and fo extremely voracious, that it is dangerous to bathe in thofe parts of the river where they lie, as they will feize a man without fear, and have ftrength fuffieient to keep him under water till, he is drowned. Here is a great variety of birds, and bats as large as a young kitten: the inhabitants efteem them a deli¬ cate morfel. The air is both hot and moift, but not unwholefome. 'The place abounds with in feeds, which are very troublefome ; but there are no ferpents. It has been difeovered, that off Port Louis the fouth- VOL. XIII. Part I. call wind generally blows with leal! ftrength about Mauritius funrife; and it alfo happens, on four or five days, intervals, in the courfe of a month, that early in the , ~ \ morning the wind ceafes in the northern part of the iftand for an hour or two, when a breeze rifes, although but faintly, from the north-weft ; during which, a ihip ftationed at the entrance of the channel to avail herfelf of this breeze, may enter the harbour and attack the forts. This iftand, during the period of the French revo¬ lution, did not entirely efcape from the ftorm which then agitated the parent country. In the year i/99» a conspiracy was formed, and broke out, for the pur- pofe of refilling the government which had been efta- blilhed under the authority of the republic. It was, however, foon fuppreffed by the adfivity of the munici¬ pality and governor-general, fupported by the majority of the inhabitants, and order and tranquillity were again re ft or cd. The population of this iftand in 1799 amounted to 65,000, viz. 55,000 flaves, and 10,000 .whites and mulattoes. The following is a ftate of the produce of this ifland in 1800 : viz, coffee, 6000 bales, of 100 lbs, French ; indigo, 300,0Oolbs. from 2S. to 8s. per lb.; cotton, 2000 bales, of 250 lbs.; raw fugar, 20,000,000 lbs.; cloves, 20,000 lbs. The iftand of Mauritius, as w-ell as the other French iflands in the Indian ocean, were taken by the Britifti in 1811. MAURUA, one of the Society iflands in the South fea. It is a fmall iftand, entirely furrounded with a ridge of rocks, and without any harbour for ftiipping. It is inhabited; and its productions are the fame with thofe of the neighbouring iflands. A high round hill rifes in the middle of it, which may be feen at the diftance of 10 or 12 leagues. W. Long. 152. 32, S. Lat. :6. 25. MAUSOLEUM, a magnificent tomb or fepulehral monument. The word is derived from Maufolus king of Caria, to whom Artemifia his widow ereCted a moft ftately monument, efteemed one of the vronders of the world, and called it, from his own name, Mavfoleum. St MAWES, a town of Cornwall, in England, feated on the eaft fide of Falmouth haven, in W. Long. 4. 56. N. Lat. 50. 6. Though but a hamlet of the parifti of St Juft, two miles off, without a minifter, or either church, chapel, or meeting-houfe, it has fent members to parliament ever fince 1562, who are re¬ turned by its mayor or portreve. It confifts but of one ftreet, under a hill, and fronting the fea, and its inhabitants fubftft purely by fitiling. K. Henry VIII. built a caftle here, oppofite to Pendennis, for the better fecurity of Falmouth haven. It has a governor, a deputy, and two gunners, with a platform of guns. Here is a fair the Friday after St Luke’s day. MAXENTTUS, Marcus Aurelius Valerius, a fon of the emperor Maximianus Hercules, was, by the voluntary abdication of Dioclefian, and of his father, raifed to the empire A. D. 306. He afterwards in¬ cited his father to reaffmne his imperial authority; and in a perfidious manner deftroyed Severus, who had delivered himfelf into his hands, and relied upon his honour for the fafety of his life. His viefories and fucceffes were impeded by Galerius Maximianus, who oppofed him with a powerful force. The defeat D and MAX [2 Maxcntius and voluntary death of Galerius foon reftored peace Maximus tw Maxentius pafled into Africa, where v——y / he rendered himfelf odious by his cruelty and oppref- lion. He foon after returned to Rome, and was in¬ formed that Conftantine was come to dethrone him. He gave his adverfary battle near Rome, and, after he had loft the victory, he fled back to the city. The bridge over which he croiTed the Tiber was in a de¬ cayed fituation, and he fell into the river, and was drowned, A. D. 312. The cowardice and luxuries of Maxentius were as confpicuous as his cruelties. He oppreiied his fubjedts with heavy taxes, to gratify the cravings of his pleafores, or the avarice of his fa¬ vourites. He was debauched in. his manners, and nei¬ ther virtue nor innocence w'ere fafe whenever he was inclined to voluptuous purfuits. His body wTas de¬ formed and unwieldy. To vilit a pleafure ground, or to exercife himfclf under a marble portico, or walk on a ftiady terrace, was to him a Herculean labour, which required the greateft exertions of ftrength and refolution. MAXILLA, the Jaw. See Anatomy, N°2o~ 26. MAXIM,, an eftabliftied propofttion or principle ; in which fenfe it denotes much the fame with axiom. Mz^XIMILIAN I. emperor of Germany, figna- lized himfelf againft the French while he was king of the Romans, and after he w7as emperor entered into the army of Henry VIII. of England as a volunteer againft that nation : lie was a protedlor of learned men, and aboliftied an iniquitous tribunal, ftyled Ju¬ dicium occultum Wejlphalice ; he compofed fome poems, and the memoirs of his own life. He died in 1 cio, aged 60. MAXIMUM, in Mathematics, denotes the greateft quantity attainable in any given cafe. If a quantity conceived to be generated by motion increafes or decreafes till it arrives at a certain magni¬ tude or pofition, and then, on the contrary, growrs greater or leflcr, and it be required to determine the faid magnitude or pofition, the queftion is called a pro¬ blem de maximis et minimis. MAXIMUS, a celebrated Cynic philofopher, and magician, of Ephefus. He inftrucfted the emperor Julian in magic j and, according to the opinion of fome hiftorians, it was in the converfation and com¬ pany of Maximus that the apoftafy of Julian originat¬ ed. The emperor not only vifited the philofopher, but he even fubmitted his writings to his infpetftion and cenfure. Maximus refufed to live in the court of Julian ; and the emperor, not diflatisfied with the re- fufal, appointed him high pontiff in the province of Lydia, an office which he difcharged with the great¬ eft moderation and juftice. When Julian went into the eaft, the philofopher promifed him fuccefs, and even faid that his conquefts would be more numerous and extcnfive than thofe of the fon of Philip. He perfuad- ed his imperial pupil, that, according to the dodlrine of metempfychofis, his body was animated by the foul which once animated the hero whofe greatnefs and vic¬ tories he was going to eclipfe. After the death of Julian, Maximus was almoft facrificed to the fury of the foldiers 5 but the interpofition of his friends faved his life, and he retired to Conftantinople. He was foon after accufed of magical pradtices, before the em- 6 ] MAY peror Valens, and beheaded at Ephefus, A. D. 366. Maximw He wrote fome philofophical and rhetorical treatifes, H fome of which were dedicated to Julian. They are all MaJr* now loft. t —-y—■ .H Maximis of Tyre, a Platonic philofopher, wrent to Rome in 146, and acquired fuch reputation there, that the emperor Marcus Aurelius became his foholar" and gave him frequent proofs of his efteem. T his phi¬ lofopher is thought to have lived till the reign of the emperor Commodus. There are ftill extant 41 of his differtations j a good edition of which was printed by Daniel Heinfius, in 1624, in Greek and Latin, with notes. Maximus Marius. See Marius. MAT , the fiftn month in the year, reckoning from our firft, or January ; and the third, counting the year to begin with March, as the Romans anciently did. It was called Maius by Romulus, in refpedt to the fenators and nobles of his city, who were named majores ; as the following month was called Junius, in honour of the youth of Rome, in honorem junior urn, who ferved him in the war j though fome will have it to have been thus called xrom Main, the mother of Mer- cury, to whom they offered facrifice on the firft day of it 5 and Papius derives it from Madius, eo quod tunc ter¬ ra madeat. In this month the fun enters Gemini, and the plants of the earth in general begin to flower.— T he month of May lias ever been efteemed favour¬ able to love $ and yet the ancients, as well as many of the moderns, look on it as an unhappy month for marriage. I he original reafon may perhaps be re¬ ferred to the feaft of the Lemures, which was held in it. Ovid alludes to this in the fifth of his Fafti, when he fays, Nec viduez tcedis eadem, nec virginis apta Tempora ; qiue nupjit, non diuturna fuit ; Hac quoque de caufaff te proverbia tangunt, Menfe malum Maio nubere vulgus ait. MAT-dew. See Dew. MAT-duke, a fpecies of cherry. See Prunus, Bo¬ tany Index. .May, IJle of, a fmall ifland at the mouth of the frith of Forth, in Scotland, about a mile and a half in circumference, and feven miles from the coaft of Fife, almoft oppofite to the rock called the Bafs. It formerly belonged to the priory of Pittenweem ; and wras dedicated to St Adrian, fuppofed to have been martyred in this place by the Danes ; and hither, in times of Popiflr fuperftition, barren women ufed to come and worflnp at his flirine, in hopes of being cured of their fterility. Here is a tow7er and light- houfe built by Mr Cunningham of Barns, to v’hom King Charles I. granted the ifland in fee, wdth power to exaft twopence per ton from every ffiip that paffes, for the maintenance of a lighthoufe. In the middle of it there is a frelh-water fpring, and a fmall lake The foil produces pafturage for 100 flieep and 20 black cattle. On the weft fide the fteep rocks render it inacceffible ; but to the eaft there are four landing places and good riding. It was here that the French fquadron, having the chevalier de St George on board, anchored in the year 1708, when the vigilance of Sir George Byng obliged him to relinquiffi his de- fign, and bear away for Dunkirk. The fhores all round MAY I 27 1 MAY Mav* Mayernc. round the ifland abound with fifti, and the cliffs with water fowl. ( May, Thomas, an eminent Englifh poet and hifto- rian in the 17th century, was born of an ancient but decayed family in Suffex, educated at Cambridge, and afterwards removed to London, where he contracted a friendfhip with feveral eminent perfons, and particu¬ larly with Endymion Porter, Efq, one of the gentle¬ men of the bedchamber to King Charles I. While he refided at court, he wrote the five plays now extant under his name. In 1622, he publifhed a trantlation of Virgil’s Georgies, with annotations j and in 1635 a poem on King Edward III. and a tranflation of Lu- ■can’s Pharfalia *, which poem he continued -down to the death of Julius Caefar, both in Latin and Englifh verfe. Upon the breaking out of the civil wars he adhered to the parliament; and in 1647, he publifh¬ ed, “ The hiftory of the parliament of England, which began November the third, MDCXL. With a fhort and accelfary view of fome precedent years.” In 1649, he pablifhed, Hijiorus Parliamenli Anglue Bi'cviarium, in three parts ; which he afterwards tranf- lated into Englifh. He wrote the Hiftory of Hen¬ ry II. in Englifh verfe. He died in 1642. He went ■well to reft over night, after a cheerful bottle as ufual, and died in his fleep before morning: upon which his death was imputed to his tying his nightcap too clofe under his fat cheeks and chin, which caufed his fuffocation ; but the facetious Andrew Marvel has written a poem of 1C0 lines, to make him a martyr of Bacchus, and die by the force of good wine. He was interred near Camden in Weftminfter Abbey ; which caufed Dr Fuller to fay, that “ if he were a biafled and partial writer, yet he lieth buried near a good and true hiftorian indeed.” Soon after the reiteration, his body, with thofe of feveral others, was dug up, and buried in a pit in St Margaret’s churchyard ; and his monument, which was erefted by the appointment of parliament, w'as taken down and thrown afide. MAYER, Tobias, one of the greateft aftronomers ■and mechanics the 18th century produced, was born at Mafpach, in the duchy of Wirtemberg 1723. He taught himfelf mathematics, and at the age of four¬ teen defigned machines and inftruments with the great- eft dexterity and juftnefs. Thefe purfuits did not hin¬ der him from cultivating the belles lettres. He ac¬ quired the Latin tongue, and wrote it with elegance. In 1750, the univerfity of Gottingen chofe him for their mathematical profeflbr ; and every year of his fhort life was thenceforward marked with fome confi- derable difeoveries in geometry and aftronomy. He publifhed feveral works in this way, which are all rec¬ koned excellent ; and fome are inferted in the feeond volume of the Memoirs of the univerfity of Got¬ tingen.” His labours feem to have exhaufted him; for he died worn out in 1762. MAYERNE, Sir Theodore de, baron of Aul- bone, was the fon of Lewis de Mayerne, the celebrated author of the General Hiftory of Spain, and of the Monarchic arijlo-democratique, dedicated to the ftates- general. He was born in 1 C73, and had for his god¬ father Theodore Beza. He ftudied phyfic at Mont¬ pelier, and was made phylician in ordinary to Hen¬ ry IV. who promifed to do great things for him, pro¬ vided he would change his religion. James I. of Eng¬ land invited him over, and made him firft phyfi- cian to himfelf and his queen, in which office he fer- ved the whole royal family to the time of his death in 1655. His works were printed at London in 1700, and make a large folio, divided into two books ; the firft containing his Confilia, EpiJIoU, ct Ohfervationes ; the fecond his Pharmacopoeia variaque medicamentorum formulce. MAYHEM. See Maim. MAYNE, Jasper, an eminent Engliffi poet and divine in the 17th century, who was bred at Oxford, and entered into holy orders. While his majefty re- fided at Oxford, he w-as one of the divines appointed to preach before him. Fie publiihed in 1647 a piece entitled OXAOMAXIA, or The people's war ex¬ amined according to the principles of reafon and ferip- ture, by Jafper Mayne. In 1648 he was deprived of his ftudentftiip at Chrift church, and two livings he had ; but wras reftored with the king, who made him his chaplain and archdeacon of Chichefter ; all which he held till he died. Dr Mayne was held in very high efteem both for his natural parts and his acquired accompli(liments. He was an orthodox preacher, and a man of fevere virtue and exemplary behaviour ; yet of a ready and facetious wit, and a very Angu¬ lar turn of humour. From fome ftories that are related of him, he feems to have borne fome degree of re- femblance in his manner to the celebrated Dr Swift ; but if he did not poffefs thofe very brilliant parts that diftinguiftied the Dean, he probably was lefs fubjecl - to that capricious and thofe unaccountable whimfies which at times fo greatly eclipfed the abi¬ lities of the latter. Yet there is one anecdote re¬ lated of him, which, although it reflecls no great ho¬ nour on his memory, as it feems to carry fome degree of cruelty with it, yet it is a ftrong mark of his re- femblance to the Dean, and a proof that his propen- fity for drollery and joke did not quit him even in his lateft moments. The ftory is this : The Doflor had an old fervant, wrho had lived with him fome years, to whom he had bequeathed an old trunk, in which he told him he would find fomething that would make him drink after his death. The fervant, full of ex¬ pectation that his mafter, under this familiar expref- fion, had left him fomewhat that would be a reward for the affiduity of his part fervices, as fcon as decency would permit, flew to the trunk ; when, behold, to his great difappointment, the boafted legacy proved to be a red herring. The doctor, however, bequeathed many legacies by will to pious ufes ; particularly 50 pounds towards the rebuilding of St Paul’s cathedral, and 200 pounds to be diftributed to the poor of the parifties of Caffington and Pyrton, near Wattington, of both which places he had been vicar. In his younger years he had an attachment to poetry ; and wrote two plays, the latter of which may be feen in the tenth volume of Dodftey’s Collection, viz. 1. Amorous war, a tragi¬ comedy. 2. The city-match, a comedy. He publifhed a poem upon the naval victory by the duke of York over the Dutch, printed in 1665. He alfo tranflated into Englifh -from the Greek, part of Lucian’s Dia¬ logues. D 2 Mryerria iVa-. ne. MAYNOOTH, may [28 Mayr.ooth MAYNOOTH, or Manooth a poft town in the Vi. o. coualy ni Kildare, and province of JLeiniter, in Ire- >""" *' land; 12 njlies from Idublin. Xhough not very large, it is regularly laid out, and conlills of good houles. Here is a charter fchool, which wras opened 27th July 17 $9. MAYN WARING, Arthur, an eminent political wliter in the beginning of the 18th century, Raid fe- veral years at Oxford, and then went to Chelhire, where he lived feme time with his uncle Mr Francis Chol- mondeley, a very honefl gentleihan, but extremely averfe to the government of King William III. to whom he refufed. the oaths. Here he profecuted his lludies in polite literature with great vigour 5 and com¬ ing up to London, applied to the Rudy of the law. Fie was hitherto very zealous in anti-revolutional prin¬ ciples, and wrote feveral pieces in favour of King James II. j but upon being introduced to the duke of Somerfet and the earls of Dorfet and Burlington, be¬ gan to entertain very different notions in politfes. ’ His father left him an eflate of near 800I. a-year, but fo encumbered, that the intereR money amounted to al- nioR as much as the revenue. Upon the concluRon of the peace he went to Paris, where he became acquaint¬ ed with Mr Boileau. After his return he was made one of the commiflioners of the cufloms, in which poR he diflinguiflied himfelf by his fkill and induflry. He was a member .of the Kit-cat club, and w^as looked upon as one of the chief fupports of it by his pleaf- entry and wit. In the beginning of Queen Anne’s reign, the lord treafurer Godolphin engaged Mr Donne to quit the office of auditor of the impreRs, and made Maynwaring a prefent of a patent for that office worth about 2000I. a-year in a time of bufinefs. He had a confiderable Riare in the Medley, and was author of fe¬ veral other pieces. The Examiner, his antagonifl in politics, allowed that he wrote with tolerable fpirit, and in a maffierly Ryle. Sir Richard Steele dedicated the firfl volume of the Tatler to him. MAYO, one of the Cape de Verd iflands, lying in the Atlantic ocean, near 300 miles from Cape Verd in Africa, about 17 miles in circumference. The foil in general is very barren, and water fcarce ; however, they have fome corn, yams, potatoes, and plantains, with plenty of beeves, goats, and affes. What trees there are, grow on the fides of the hills, and they have fome figs and water melons. The fea round about the ifland abounds with fiffi. The chief commodity is fait, with which many Engliffi ffiips are loaded in the fummer time. The principal town is Pinofa, inhabited by ne¬ groes, who fpeak the Portuguefe language, and are Rout, lufly, and fleffiy. They are not above 200 in number, and many of them go quite naked. W. Long. 23. 5. N. Lat. 15. 10. Mayo, a county of Ireland, in the province of Con¬ naught, having Sligo and the fea on the north, Rof- common on the fouth, Leitrim and Rofcommon on the eafl, and the Atlantic ocean on the wefl. It contains 724,640 Irifii plantation acres, 68 pariffies, and 140,000 inhabitants. It gives title of earl to the family of Bourke. This county takes its name from an ancient city, built in 664 ; the ruins of the cathedral, and fome traces of the Rone v,Tills which en- compaffed the city, yet remain on the plains of Mayo. It was a univerfity, founded for the education of fuch 4 ] MAY of the Saxon youths as were converted to the Chriftiafi faith : it was fituated a little to the fouth of Lough Conn 5 and is to this day frequently called Mayo of the Saxons, being celebrated for giving education to Al¬ fred the Great, king of England. As this town has gone to decay, Balinroke is reckoned the chief town. 1 he county by the fea is mountainous j but inland has good paflures, lakes, and rivers. It is about 6 2 miles long, and 32 broad. CaRlebar is the afiizes town. Mayo was iormerly a bilhop’s fee, which is now united to Tuam. hlAY OR, the chief magifirate of a city or town chofen annually out of the aldermen. The word, an¬ ciently wrote meyr, comes from the Britifii mirct, i. e. cujiodire, or from the old Englifli mater, viz. pote/ias, and not from the Latin major. King Richard I. in 1189, change^ the bailiff of London into a mayor, and from that example King John made the bailiff of King’s Lynn a mayor anno 1204: Though the fa¬ mous city of Norwich obtained not this title for its chief magiflrates till the feventh year of King Hen¬ ry V. anno 1419; fince which there are few towns of note but have had a mayor appointed for govern¬ ment. Mayors of corporations are juflices of peace pro tempore, and they are mentioned in feveral flatutes j but no perfon fliall bear any office of magifiracy con¬ cerning the government of any town, corporation, &c. who hath not received the facrament according to the church of England within one year before his elec¬ tion, and who Riall not take the oaths of fuprcmaey, &c. If any perfon intrudes into the office of mayor, a quo warranto lies againR him, upon which he fliall not only be oufled, but fined. And no mayor, or perfon holding an annual office in a corporation for one year, is to be elected into the lame office for the next 5 in this cafe, perfons obffru&ing the choice of a fucceffor are lubject to 100I. penalty. Where the mayor of a corporation is not chofen on the day ap¬ pointed by charter, the next officer in place fliall the day after hold a court and deft one j and if there be a default or omiffion that way, the eleftors may be compelled to choofe a mayor, by a writ of mandamus out of the king’s bench. Mayors, or other magiflrates of a corporation, who ffiall voluntarily abfent them- felves on the day of cleft ion, are liable to be imprifon- ed, and difqualificd Irom holding any office in the cor¬ poration. Mayor's Courts. To the lord mayor and city of London belong feveral courts of judicature. The higheft and molt ancient is that called the hujlings, de- flined to fecure the laws, rights, franchifes, and cufloms of the city. The fecond is a court of requejf, or of confcience ; of which before. The third is the court of the lord mayor and aldermen, where alfo the flieriffs fit j to which may be added two courts of flieriffs, and the court of the city orphans, whereof the lord may¬ or and aldermen have the cuflody. Alfo the court of common council, which is a court or afl’embly, wherein are made all by-lavrs which bind the citizens of London. It confiRs, like the parliament, of two houfes : an upper, confifling of the lord mayor and aldermen ; and a lower, of a number of common council men, chofe by the feveral wards as reprefen- tatives Mayo, Mayor. Mayer I! Mazan'ne. M A Z tstivcs of tliG body of the citizens, common council are made laws for the advancement of , trade, and committees yearly appointed, &c. But adds made by them are to have the aiTcnt of the lord mayor and aldermen, by flat, 11 Geo. 1. Alfo the chamberlain’s court, where every thing relating to the rents and re¬ venues of the city, as alfo the affairs of fervants, &c. are tranfaded. Laftly, To the lord mayor belong the courts of coroner and of efeheator } another court for the confervation of the river Thames ; another of gaol- delivery, held ufually eight times a year, at the Old Bailey, for the trial of criminals, whereof the lord mayor is himfelf the chief judge. h here are other courts called wardmotes or meetings of the wards 5 and courts of halymote or affemblies of the feveral guilds and fraternities. MAZA, among the Athenians, was a fort of cake made of flour boiled with water and oil, and fet as the common fare, before fuch as were entertained at the public expence in the common hall or Prytaneurn. MAZ AG AN, a ftrong place of Africa in the king¬ dom of Morocco, and on the frontiers of the province of Duguela. It was fortified by the Portuguefe, and befieged by the king of Morocco with 200,000 men in 1 1562, but to no purpofe. It is fituated near the fea. W. Long. 8. 1 v N. Lat 3^. 12. MAZARA. an ancient town of Sicily, and capi¬ tal of a confiderable valley of the fame name, which is very fertile, and watered with feveral rivers. The town is a bi(hop’s fee, and has a good harbour is feated on the fea coaft, in E. Long. 12. 30. N. Lat. 37. 53. MAZARINE, Julius, a famous cardinal and prime miniffer of France, was born at Pxfcina in the province of Abruzzo, in Naples, in 1602. ALer having finifhed his Undies in Italy and Spain, he en¬ tered into the (ervice of Cardinal Sachetts, and became well (killed in politics, and in the interefts of the princes at war in Italy, by which paeans he was enabled to bring affairs to an accommodation, and the peace of Queiras was (hortly concluded. Cardinal Richlieu being taken with his conduct, did from thenceforward highly efteem him j as did alfo Cardinal Antonio, and Louis XIII. who procured him a car¬ dinal’s hat in 1641. Richlieu made him one of the executors of his will; and during the minority of Louis XIV. he had the charge of aftairs. At laft he became the envy of the nobility, which occafioned a civil war 5 whereupon Mazarine was forced to re¬ tire, a price was fet on his head, and his library fold. Notwjthftanding, he afterwards returned to the court in more glory than ever ; concluded a peace with Spain, and a marriage treaty betwixt the king and the in¬ fanta. This treaty of peace paffes for the mafter- piece of Cardinal de Mazarine’s politics, and procured him the French king’s mod intimate confidence : but at laft his continual application to bufinefs threw him into a difeafe, of which he died at Vinciennes in j66i.—Cardinal Mazarine was of a mild and affable temper. One of his greateft talents was his knowing mankind, and his being able to adapt himfelf, and to affume a ebaraffer conformable to the circum(lances of affairs. He poffeffed at one and the fame time the bifhopric of Metz, and the abbeys of St Arnauld, St Clement, and St Vincent, in the fame city } that of [ 29 } M E A In the court of St Dennis, Clugny, and Viaor, of Marfeilles; of St Michel at Soifl'ons, and a great number ot others. Fie founded Mazarine college at Paris •, which is alfo i. called the college of the four nations. There has been publilhed a colleftion of his letters, the moil copi¬ ous edition of which is that of 1745* 2 vo's' duc3~ decimo. MAZZUOLI. See Parmigiano. MEAD, a wholefome, agreeable liquor, prepared with honey and water. One of the bell methods of preparing mead is as fol¬ lows : Into twelve gallons of water put the whites of fix eggs } mixing thefe well together, and to the mix¬ ture adding twenty pounds of honey. Let the liquor boil an hour, and when boiled, add cinnamon, ginger, cloves, mace, and rofemary. As foon as it is cold, put a fpoonful of yeft to it, and turn it up, keep¬ ing the veffel filled as it works \ when it has done working, Hop it up clofej and, when fine, bottle it off for ufe. - Thorley fays, that mead not inferior to the bed of foreign wines may be made in the following manner : Put three pounds of the fined honey to one gallon of water, and two lemon peels to each gallon j boil it half an hour, well feummed ; then put in, while boilirtg, lemon peel: work it with yeft ; then put it in your veffel with the peel, to Hand five or fix months, and bot¬ tle it off for ufe. If it is to be kept for feveral years, put four pounds to a gallon of water. The author of the Dictionary of Chemiftry diredls to choofe the whited, pureft, and bed tafted honey, and to put it into a kettle with more than its weight of water : a part of this liquor muft be evaporated by boiling, and the liquor feummed, till its confiftence is fuch, that a frefti egg (hall be fupported on its furface without finking more than half its thicknefs into the liquor ; then the liquor is to be drained, and poured through a funnel into a barrel; this barrel, which ought to be nearly full, muft be expofed to a heat as equable as poffxble, from 20 to 27 or 28 degrees of Mr Reaumur’s thermometer, taking care that the bung- hole be (lightly covered, but not clofed. The pheno¬ mena of the fpirituous fermentation will appear in this liquor, and ivill fubfift during two or three months, according to the degree of heat ■, after which they will diminifti and ceafe. During this fermentation, the barrel muft be filled up occafionally with more of the fame kind of liquor of honey, feme of which ought to be kept apart, on purpofe to replace the liquor which flows out of the barrel in froth. When the fermenta¬ tion ceafes, and the liquor has become very vinous, the barrel is then to be put into a cellar, and well clofed •, a year afterwards the mead will be fit to be put into bottles. Mead is a liquor of very ancient ufe in Britain. See- Feast. Mead, Dr Richard, a celebrated Englifh phyfi- cian, was born at Stepney near London, where his father, the Reverend Mr Matthew Mead, had been one of the two minifters of that parifh ; but in 1662 was ejedled for nonconformity, but continued to preach at Stepney till his death. As Mr Mead had a handfome fortune, he beftowed a liberal education upon 13 children, of whom Richard was the eleventh j and for that purpofe kept a private tutor in his houfe, ■who.- RIazarine-> II , Mead. ' Mead. M E A [ 30 ■who tauglit him the Latin tongue. At 16 years of ^ age luchard was fent to Utrecht, where he ftudied three years under the famous Gra;vius; and then choofing the profeflion of phytic, he went to Leyden -where he attended the leftures of the famous Pitcairn on the theory and practice of medicine, and Her¬ man's. botanical courfes. Having alfo fpent three years m theie ftudies, he went with his brother and two other gentlemen to vifit Italy, and at Padua took his degree of doftor of philofophy and phyfic in 1695. Afterwards he fpent.feme time at Naples and at Rome ; and returning home the next year, fettled at Stepney, where he married, and praaifed’phylic w nh a fuccefs that laid the foundation of his future greatnefs. In 1703, Dr Mead having communicated to the Royal Society an analyfis of Dr Bonomo’s difcoveries relating to the cutaneous worms that generate the itch, which, they .inferted in the Philofophical Tranf- aaions; this, with his account of poifons, procured him a place in the Royal Society, of which Sir Ifaac Newton was then prefident. The fame year he was elected phyfician of St Thomas’s hofpital, and was alfo employed by the furgeons to read anatomical lec¬ tures in their hall, which obliged him to remove into the city. In 1707 his Paduan diploma for do&or of phyfic was confirmed by the univerfity of Oxford ; and being patronized by Dr Radcliffe, on the death of that famous phyfician he fucceeded him in his houfe at Bloomfbury-fquare, and in the greateft part of his bufmefs. In 1727 he was made phyfician to Kino- George II. whom he had alfo ferved in that capacity while he was prince of Wales ; and he had afterwards the pleafure of feeing his two fons-in-law, Dr Nichols and Dr Wilmot, his coadjutors in that eminent Ra¬ tion. . Dr Mead was not more to be admired for the qua¬ lities of the head than he was to be loved for thofe of his. heart. Though he was himfelf a hearty wLig, yet, uninfluenced by party principles, he was a friend to all men of merit, by whatever denomination they might happen to be diltingftifhed. T. hus he was intimate with Garth, with Arbuthnot, and with Freind j and long kept up a conflant correfpondence with the great Boerhaave, who had been his fellow Undent at Ley¬ den : they communicated to each other their obferva- tions and projefts, and never loved each other the lefs for being of different fentiments. In the mean time, intent as Dr Mead was on the duties of his profeflion, be had a greatnefs of mind that extended itfelf to all kinds of literature, which he fpared neither pains nor money to promote. He caufed the beautiful and fplen- did edition of Thuanus’s hiftory to be publilhed in 1713, in feven volumes folio: and by his interpofition and afliduity, Mr Sutton’s invention of drawing foul air from (hips and other clofe places was carried into execution, and all the fhips in his majeftv’s navy pro¬ vided with this uieful machine. Nothing pleafed him more than to call hidden talents into light; to give encouragement to the greateff projefts, and to' fee them ^executed under his own eye. During almoft half a century he was at the head" of his bufinefs, which brought him one year above feven thoufand pounds, and for feveral years between five and fix thoufand 5 yet clergymen, and in general all men of learning, 5 1 M E A were welcome to his advice. His library confifled of. I0>°0° volumes, of which his Latin, Greek, and oriental manuferipts, made no inconfiderable part. He . had a. gallery for his pidures and antiquities, which colt him great fums. His reputation, not only as a phyfician but as a fcholar, was fo univerfally efta- bhlhed, that he correfponded with all the principal li¬ terati in Europe : even the king of Naples fent to de¬ fire a complete colledion of his works 5 and in return made him a prefent of the two firft volumes of Signior Rajardi, which may be confidered as an introduction to the collection of the antiquities of Herculaneum. At the fame time that prince invited him to his pa¬ lace, that he might have an opportunity of Ihowing him thofe valuable monuments of antiquity 5 and no¬ thing but his great age prevented his undertaking a journey fo fuited to his tafte. No foreigner of learn¬ ing ever came to London without being introduced to Dr Mead ; and on thefe occafions his table was always open, and the magnificence of princes was united with the pleafures of philofophers. It was principally to him that the feveral counties of England and our co¬ lonies abroad applied for the choice of their phyfi- cians, and he was likewife confulted by foreion phy- fieians from Ruflia, Pruflia, Denmark, &c. He wrote, befides the above works, 1. A Treatife on the Scurvy! 2. De variolis et morbillis dijfertatio. 3. Medic a facra : five de Morbis infgnioribus, qui in Bib His memorantur, Comment anus. 4. Monita et Preecepta medico. A. Difcourfe concerning peftilential contagion, and the methods to be ufed to prevent it. The works he wrote, and publifhed in Latin were tranflated into Englifh, under the Doftor’s infpeaion, by Thomas Stack, M. D. and F. R. S. This great phyfician, naturalift, and antiquarian, died on the 16th "of Fe¬ bruary 1754. MEADOW, in its general fignification, means pafture or grafs lands, annually mown for hay : but it is more particularly applied to lands that are fo low as. to be. too moift for cattle to graze upon them in winter without fpoiling the fward. For the manage¬ ment and watering of meadows, fee Agriculture P- 435- MEAL, the flour of grain. The meal or flour of Britain is the fineft and whitefl in the world. The I rench is ufually browner, and the German browner than that. Our flour keeps well with us j but in carry¬ ing abroad it often contraCIs damp, and becomes bad. All flour is fubjeCI to breed worms ; thefe are white ia the white flour, and brown in that which is brown j they are therefore not always diflinguifliable to the eye : but when the flour feels damp, and fmells rank and mu fly, it may be conjeftured that they are there in great abundance. The colour and the weight are the two things which denote the value of meal or flour ; the whiter and the heavier it is, other things being alike, the better it always is. Pliny mentions thefe two eha- radters as the marks of good flour j and tells us, that Italy in his time produced the fineft in the world. Phis country indeed was famous before his time for this produce j and the Greeks have celebrated it 5 and Sophocles in particular fays, that no flour is fo white or fo good as that of Italy. The* corn of this coun- •>- try has, however, loft much of its reputation fince that time ; Mead I! Meal. M E A [ 3i ] M E A liJead. time j and the reafon of this feems to be, that the —* ' whole country being full of fulphur, alum, vitriol, m ircafttes, and bitumens, the air may have in time af¬ fected them fo far as to make them diifufe themfelves through the earth, and render it lefs fit for vegetation y and the taking fire of fome of thefe inflammable mi¬ nerals,. as has fometimes happened, is alone fudicient to alter the nature of all the land about the places where they are. The flour of Britain, though it pleafes by its white- nefs, yet wants fome of the other qualities valuable in Hour the bread that is made of it is brittle and does not hold together, but after keeping a few days becomes hard and dry as if made of chalk, and is full of cracks in all parts ; and this muft be a great difad- vantage in it when intended for the fervice of an army, or the like occafions, where there is no baking eeery day, but the bread of one making muft neceflarily be kept a long time. The flour of Picardy is very like that of Britain ; and after it has been kept fome time, is found improper for making into pafte or dough. The French are forced either to ufe it immediately on the grinding, or elfc to mix it with an equal quantity of the flour of Brittany, which is coarfer, but more undtuous and fat¬ ty •, but neither of thefe kinds of flour keep well. The flour of almofl any country will do for the home confumption of the place, as it may be always frefh ground } but the great care to be ufed in {elect¬ ing it, is in order to the fending it abroad, or furnifh- ing (hips for their own ufe. The faline humidity of the fea air rufts metals, and fouls every thing on board, if great care be not taken in the preferving them. This alfo makes the flour damp and mouldy, and is often the occafion of its breeding infedts, and being wholly fpoiled. The flour of fome places is conflantly found to keep better at fea than that of others ; and when that is once found out, the whole caution needs only be to carry the flour of thofe places. Thus the French find that the flour of Poitou, Normandy, and Guienne, all bear the fea carriage extremely well; and they make a confiderable advantage by carrying them to their A- merican colonies. The choice of flour for exportation being thus made, the next care is to preferve it in the fhips ; the keep¬ ing it dry is the grand confideration in regard to this ; the barrels in which it is put up ought to be made of dry and well feafoned oak, and not to be larger than to hold two hundred weight at the mofL If the wood of the barrels have any fap remaining in it, it will moi- flen and fpoil the flour 5 and no wood is fo proper as oak for this purpofe, or for making the bins and other veffels for keeping flour in at home, fince when once well dried and feafoned it will not contrail hu¬ midity afterwards. The beech wood, of which fome make their bins for flour, is never thoroughly dry, but always retains fome fap. The fir will give the flour a tafle of turpentine and the afli is always fubjeil to be eaten by worms. The oak is preferable, becaufe of its being free from thefe faults ; and when the feveral kinds of wood have been examined in a proper manner, there may be others found as fit, or poflibly more fo, than this for the purpofe. The great teft is their hav¬ ing more or lefs fap. See Flour and Wood. MEAN, in general, denotes the middle between two extremes : thus we fay the mean diftance, mean proportion, &c. , MEAN, Arithmetical, is half the fum of the two ex¬ tremes, as 4 is the arithmetical mean between 2 and 6 5 for =4. Mean MeF.fi.ire. MEAN, Geometrical, is the fquare root of the rect¬ angle, or product of the twro extremes : thus, Vi X9=V9 =3- To find two mean proportionals between two ex¬ tremes : multiply each extreme by the fquare of the other, then extract the cube root out of each produCt, and the two roots will be the mean proportionals re¬ quired. Required twro proportionals between 2 and 16,. 2 X 2 >< 16=64, and 3vC 64=4. Again, 3 ^2 X ibl=:3\/512 = 8. 4 and 8 therefore are the twro proportionals, fought. MEARNSSHIRE, a county of Scotland. See Kin¬ cardineshire. MEASLES, a cutaneous difeafe attended ivith a fever, in which there is an appearance of eruptions that do not tend to a fuppuration. See PvIedicine.. Index. MEASURE of an angle, is an arch deferibed from, the vertex in any place between, its legs. Hence, angles are diftinguifhed by. the ratio of the arches, de¬ feribed from the vertex beteveen the legs to the peri¬ pheries. Angles then are diftinguifhed by thofe arches ; and the arches are diftinguifhed by their ratio to the periphery. Thus an angle is faid to be fo many degrees as there are in the faid arch. MEASURE of a folid, is a cube whofe fide is an inch, a foot, or a yard, or any other determinate length. In geometry it is a cubic perch, divided into cubic feet, digits, &c. MEASURE of velocity, in Mechanics, is the fpace paf- fed over by a moving body in a given time. To mea- fure a velocity, therefore, the fpace muft be divided into as many equal parts as the time is conceived to be di¬ vided into ; the quantity of fpace anfwering to fuch a part of time is the meafure of the. velocity. Measure, in Geometry, denotes any quantity afTum- ed as one, or unity, to which the ratio of the other ho¬ mogeneous or fimilar quantities is expreffed. Measure, in a legal and commercial fenfe, denotes a certain quantity or proportion of any thing bought, fold, valued, or the like. It is neceflary, for the. convenience of commerce, that an uniformity fhquld be obferved in weights and meafures, and regulated by proper ftandards. A foot-. rule may be ufed as a ftandard for meafures of length, a bufhel for meafures of capacity, and a pound for weights. There Ihould be only one authentic ftan¬ dard of each kind, formed of the. moft durable ma¬ terials, and kept with all poflible. care. A fufheient number of copies, exaflly correfponding to the prin-. cipal ftandard, may be diftributed for adjufting the weights and meafures that are made for common ■ ufe. There are feveral ftandards of this kind both in England M E A [32 England and Scotland. See the article WEIGHTS and Meafures. II any one of the ftandards above mentioned be juftly preferved, it will ferve as a foundation for the others, by which they may be corrected if inaccurate, or reftored if entirely loft. For inftance, if we have a ftandard foot, wre can eafily obtain an inch, and can make a box which fliall contain a cubical inch, and may ferve as a ftandard for meafures of capacity. If it be known that a pint contains 100 cubical inches, we may make a veflel five inches fquare, and four inches deep, which will contain a pint. If the ftan¬ dard be required in any other form, we may fill this veflel with water, and regulate another to contain an equal quantity. Standards for weights may be obtain¬ ed from the fame foundation \ for if we know how many inches of water it takes to weigh a pound, we have only to meafure that quantity, and the weight which balances it may be aflumed as the ftandard of a pound. Again, If the ftandard of a pound be given, the meafure of an inch may be obtained from it; for we may weigh a cubical inch of water, and pour it into a regular veffel j and having noticed how far it is filled, ■we may make another veflel of like capacity in the form of a cube. The fide of this veflel may be af¬ lumed as the ftandard for an inch j and ftandards for a foot, a pint, or a bulhel, may be obtained from it. Water is the moft proper fubftance for regulating ftan¬ dards j for all other bodies differ in weight from others of the fame kind ; whereas it is found by experience that fpring and river water, rain, and melted fnow, and all other kinds, have the fame weight; and tins uni¬ formly holds in all countries when the water is pure, alike warm, and free from fait and minerals. Thus, any one ftandard is fufticient for reftoring all the reft. It may further be defired to hit on fome ex¬ pedient, if poflible, for reftoring the ftandards, in cafe that all of them fhould ever fall into diforder, or fhould be forgotten, through the length of time, and the vi- eiffitudes of human affairs. This feems difficult, as no words can convey a precife idea of a foot-rule, or a pound weight. Meafures, affumed from the dimen- fions of the human body, as a foot, a hand-breadth, or a pace, muft nearly be the fame in all ages, unlefs the fize of the human race undergo fome change •, and therefore, if we know howr many fquare feet a Roman acre contained, we may form fome judgment of the nature of the law which reftricled the property of a Roman citizen to feven acres ; and this is futficient to render hiftory intelligible 5 but it is too inaccurate to regulate meafufes for commercial purpofes. The fame may be faid of ftandards, deduced from the meafure of a barley-corn, or the weight of a grain of wheat. If the diftance of two mountains be accurately meafured and recorded, the nature of the meafure ufed will be preferved in a more permanent/ manner than by any ftandard •, for if ever that meafure fall into difufe, and another be fubftituted in its place, the diftance may be meafured again, and the proportion of the ftandards may be afeertained by comparing the new and ancient di fiances. But the moft accurate and unchangeable manner of eftablifiling ftandards is, by comparing them with the length of pendulums. The longer a pendulum is, it ] M E A vibrates the flower; and it muft have one precife length 'VTeafure. in order to vibrate in a fecond. The flighteft differ-y—J ence in length will occafion a difference in the time j which will become abundantly fenfible after a number of vibrations, and will be eafily obferved if the pendu¬ lum be applied to regulate the motion of a clock. The length of a pendulum which vibrates feconds in Lon¬ don is about 39-^ inches, is conftantly the fame at the fame place, but it varies a little with the latitude of the place, being fhorter as the latitude is lefs. There¬ fore, though all ftandards of weights and meafures Avere loft, the length of a fecond pendulum might be found by repeated trials : and if the pendulum be pro¬ perly divided, the juft meafure of an inch Avill be ob¬ tained 5 and from this all other ftandards may be re¬ ftored. See White hutjl on Invariable MEASURES. Meafures are various, according to the various kinds and dimenfions of the things meafured.—Hence arife lineal or longitudinal meafures, for lines or lengths j fquare meafures, for areas or fuperfices ; and folid or cubic meafures, for bodies and their capacities •, all Avhich again are very different in different countries and in different ages, and even many of them for different commodities. Whence arife other divifions of ancient and modern meafures, domeftic and foreign ones, dry meafures, liquid meafures, &c. 1. LONG Meafures, or Meafures of Application. I.] The Englifh and Scotch Standards. The Englifti lineal ftandard is the yard, containing 3 Englifti feet ; equal to 3 Paris feet 1 inch and TV of an inch, or ^ of a Paris ell. The ufe of this mea¬ fure Avas eftablifhed by Henry I. of England, and the ftandard taken from the length of his oavu arm. It is divided into 36 inches, and each inch is fuppofed equal to 3 barleycorns. When ufed for meafuring cloth, it is divided into four quarters, and each quar¬ ter fubdivided into 4 nails. The Englifh ell is equal to a yard and a quarter, or 45 inches, and is ufed in meafuring linens imported from Germany and the Loav Countries. The Scots elwand Avas eftabliftied by King David I. and divided into 37 inches. The ftandard is kept in the council chamber of Edinburgh, and being compar¬ ed Avith the Englifti yard, is found to meafure 37-jf inches •, and therefore the Scots inch and foot are lar¬ ger than the Englifh, in the proportion of 180 to 18^ but this difference being fo inconfiderable, is feldom attended to in practice. The Scots ell, though for¬ bidden by hrw, is ftill ufed for meafuring fome coarfe commodities, and is the foundation of the land meafure of Scotland. Itinerary meafure is the fame both in England and Scotland. The length of the chain is four poles, or 22 yards ; 80 chains make a mile. The old Scots com¬ puted miles Avere generally about a mile and a half each. The reel for yarn is 2^ yards, or 10 quarters, in circuit; 1 20 threads make a cut, 1 2 cuts make a hafp or hank, and 4 hanks make a fpindle. 2. ] The French ftandard Avas formerly the aune or ell, containing 3 Paris feet 7 inches 8 lines, or 1 yard 4 Englifti 5 the Paris foot royal exceeding the Englifti by parts, as in one of the folloAving tables. This ell M E A [ 33 ] M E A Mealure. ^ ell is divided two ways ; viz. into halves, thirds, fixths, The French, however, have alfo formed an entirely Meafure. ’ and twelfths j and into quarters, half-quarters, and fix- new fyftem of weights and meafures, according to the v—~“ teenths. following table. Proportions of the meafures of each fpe cies to its principal meafure or unity. ' 10,000 1,000 IOO 10 o 0.1 0.01 0.001 Firtt part of the name which indicates the proportion to the principal meafure or unity. Myria Kilo Hedlo Deca Deci Centi Mini Length. Metre. Proportion of the principal meafures j between themfelves and the length of r p a * CP* o 0.276TB- o 2.763 •§• O 4.144 I: o 16.579 ° SS*^8 I Ij.7°5i 6 3.50^ 05 O o I- 2 p r o O O^g- 0.01 O 0.04 o o £ 0.06 o o I I o o o 0.24 0.48 3*84 7.68 Meafure. Measure of Wood for Firing, is ufually the cord four feet high, and as many broad, and eight long •, this is divided into two half cords, called ways, and by the French membrures, from the pieces ituck up¬ right to bound them ; or voyes, as being fuppofed half a waggon load. Measure for Horfes, is the hand, which by ftatute contains four inches. Measure, among Botanifs. In defcribing the parts of plants, Tournefort introduced a geometrical fcale, which many of his followers have retained. They mea- fured every part of the plant; and the effence of the defcription confifted in an accurate menfuration of the whole. . ' As the parts of plants, however, are liable to va¬ riation in no circumftance fo much as that of dimen- fron, Linnaeus very rarely admits any other menfura¬ tion than that arifing from the refpective length and breadth of the parts compared together. In cafes that require aftual menfuration, the fame author re¬ commends, in lieu of. Feurnefort’s artificial fcale, the following natural fcale of the human body, which he thinks is much more convenient, and equally ac¬ curate. The fcale in queftion confrfts of 11 degrees,, which are as follow : 1. A hair’sbreadth, or the diameter of a hair, (capillus). 2. A lire, (linen), the breadth «f the crefcent or white appearance at the root of the finger (not thumb, meafured from the fkin towards the body of the nail ; a, line is equal to 12 hairbreadths, and is the 12th part of a Pariiian inch. 3. A nail (unguis), the length of a finger nail j equal to fix lines, or half a Parifian inch. 4. A thumb (po/ex), the length of the firft or outermoft joint of the thumb j equal to a Parifian inch. 5. A palm (palmus), the breadth of the palm exclufive of the thumb 5 equal to three Parifian inches. 6. A fpan (fpithama), the di- ftance between the extremity of the thumb and that of the fir ft finger when extended ; equal to feven Parifian inches. 7. A great fpan (dodrans), the diflance be¬ tween the extremity of the thumb and that of the little finger, when extended ; equal to nine inches. 8. A foot (pes), meafuring from the elbow to the bafis of the thumb 5 equal to 12 Parifian inches. 9. A cubit (cubitus), from the elbow to the extremity of the middle finger', equal to 17 inches. 10. An arm length (brachiutn), from the armpit to the extre¬ mity of the middle finger ; equal to 24 Parifian inches,, or two feet. 11.A fathom (orgi/a), the meafure of the human ftature *, the diftance between the extremities of the two middle fingers, when the arms are extend¬ ed ', equal, where greateft, to fix feet. Measure is alfo ufed to fignify the cadence and time obferved in poetry, dancing, and mufic, to render them regular and agreeable. The different meafures or metres in poetry, are the different M E A [ 43 1 M E A Meafure. clifferent manners of ordering and combining the quan- "“■"'V 1 tides, or the long and fliort fyllables. t hus, hexame¬ ter, pentameter, iambic, fapphic verfes, &c. coniift of different mea fares. In Englifh verfes, the meafures are extremely vari¬ ous and arbitrary, every poet being at liberty to intro¬ duce any new form that he pleafes. The moil ufual are the heroic, generally comifting of live long and five Ihort fyllables ; and verfes of four feet j and of three feet and a caefura, or fmgle fyllable. The ancients, by varioufly combining and tranfpof- ing their quantities, made a vaft variety of different meafures. Of words, or rather feet of two fyllables, they formed a fpondee, confifting of two long fyllables j a pyrrhic, of two Ihort fyllables *, a trochee, of a long and a Ihort fyllable 5 and an iambic, of a Ihort and a long fyllable. Of their feet of three fyllables they formed a mo- loffus, confifting of three long fyllables ; a tribrach, of three ftiort fyllables •, a dactyl, of one long and two fhort fyllables j and an anepaeft, of two fhort and one long fyllable. The Greek poets contrived 124 dif¬ ferent combinations or meafures, under as many dif¬ ferent names, from feet of two fyllables to thofe of fix. Measure, in Mujic, the interval or fpace of time which the perfon who beats time takes between the rifing and falling of his hand or foot, in order to con- du£t the movement, fometimes quicker, and fometimes flower, according to the kind of mufic, or the fubje£t that is fung or played. The meafure is that which regulates the time we are to dwell on each note. See Time. The ordinary or common meafure is one fecond, or 60th part of a minute, which is nearly the fpace be¬ tween the beats of the pulfe or heart j the fyftole, or contraftion of the heart, anfwering to the elevation of the hand •, and its diaftole, or dilatation, to the letting it fall. The meafure ufually takes up the fpace that a pendulum of two feet and a half long employs in making a fwing or vibration. The meafure is regu¬ lated according to the different quality or value of the notes in the piece *, by which the time that each note is to take up is expreffed. The femibreve, for inftance, holds one rife and one falland this is called • the meafure or whole meafure, fometimes the meafure note, or time note; the minim, one rife, or one fall; and the crotchet, half a rife, or half a fall, there being four crotchets in a full meafure. MEASURE Binary, or Double, is that wherein the rife and fall of the hand are equal. MEASURE Ternary, or Triple,h that wherein the fall is double to the rife •; or where two minims are played during a fall, and but one in the rife. To this pur- pofe, the number 3 is placed at the beginning of the lines, when the meafure is intended to be triple; and a C, when the meafure is to be common or double. This rifing and falling of the hands Avas called by the Greeks ct^o-is and St Auguftine calls it plaufus, and the Spaniards compos. See Arsis and Thesis. Powder MEASURES in Artillery, are made of copper, and contain from an ounce to 12 pounds : thefe are very convenient in a fiege, Avhen guns or mortars are loaded Avith loofe powder, efpecially in ricochet firing, &c. MEASURING, or Mensuration, is the ufing^. a certain knoAvn meafure, and determining thereby the precife extent, quantity, or capacity of any thing. Measuring, in general, includes the practical part of geometry. From the various fubjects on which it is employed, it acquires various names, and conftitutes various arts. See Geometry, Levelling, Mensu¬ ration, Trigonometry, &c. MEx\T. See Food, Diet, Drink, &c. Amongft the Jews, feveral kinds of animals Avere forbidden to be uied as food. The flefh with the blood, and the blood Avithout the fleih, Avere prohibit¬ ed •, the fat alfo of facrificed animals Avas not to be eaten. Roaft meat, boiled meat, and ragouts, Avere in ufe among the HebreAVs, but Ave meet Avith no kind of feafoning except fait, bitter herbs, and honey.— They never mingled milk in any ragout or hath, and never ate at the fame meal both meat and milk, butter, or cheefe. The daily provifion for Solomon’s table was 30 meafures of fine Avheat flour, 60 of common flour, 10 fat oxen, 20 pafture oxen, 100 (beep, be- fides venifon and Avildfowl. See Luxury. The principal and moft neceffary food among the ancient Greeks, Avas bread, AA'hich they called and produced in a Avicker bafket called *«ye«v. Their loaves Avere fometimes baked under the alhes, and fometimes in an oven. They alfo ufed a fort of bread called ma%a. Barley meal Avas ufed amongft the Greeks, Avhich they called tzycpHo*. They had a fre¬ quent difh called Avhich was a compofition of rice, cheefe, eggs, and honey, Avrapped in fig-leaves. The /u.vTTtu]oy was made of cheefe, garlic, and eggs, beaten and mixed together. Their bread and other fubftitutes for bread, Avere baked in the form of hol¬ lo av plates, into Avhich they poured a fauce. Garlic, onions, and figs, feem to have been a very common food amongft the poorer Athenians. The Greeks, efpecially in the heroieal times, ate flefti roafted ; boil¬ ed meat feldom Avas ufed. Fifti feems not to have been ufed for food in the early ages of Greece. The young people only, amongft the Lacedaemonians, ate animal food } the men and the old men Avere fupport- ed by a black foup called p.iyx £vp.os, which to people of other nations Avas ahvays a dilagreeable mefs. Grafs- hoppers and the extremities or tender (hoots of trees Avere requently eaten by the poor among the Greeks. Eels dreffed Avith beet root Avere efteemed a delicate difti, and they Avere fond of the jowl and belly of falt- fifti. Neither Avere they Avithout their fweet-meats j the deffert confided frequently of fruits, almonds, nuts, figs, peaches, &e. In every kind of food we find fait to have been ufed. The diet of the firfi: Romans confided wholly of milk, herbs, and roots, Avhich they cultivated and dreffed Avith their OAvn hands ; they alfo had a kind of gruel, or coarfe grofs pap, compofed of meal and boiling water ; this ferved for bread : And when they began to ufe bread, they had none for a great Avhile but of unmixed rye. Barley-meal was eaten by them, which they called polenta. When they began to eat animal food, it Avas efteemed a piece of luxury, and an indulgence not to be juftified but by fome particu- F 2 lar Meafure Meat. M E A Meath. ^ occafion. After animal food had grown into com- —^ ■ mon ufe> the meat which they moil frequently produced upon their tables was pork. Method of Preferving Flefh-MEAT without fpices, and with very little full. Jones, in his Mifcellanea Cunofa, gives us the following defcription of the Maorith Elcholle, which is made of beef, mutton, or camel’s flefli, but chiefly beef, which is cut in long fliees, and laid for 24 hours in a pickle-. They then remove it out of thole jars or tubs into others with water ; and when it has lain a night, they take it out, and put it on ropes in the fun and air to dry.. When it is thoroughly dried and hard, they cut it into pieces of two or three inches long, and throw it into a pan or caldron, which is ready with boiling oil and fuet fufficient to hold it, where it boils till it be very clear and red when cut. After this they take it out, and fet it to drain; and when all is thus done-it Hands to cool, and jars are prepared to put it up-in, pouring upon it the liquor in which it was fried ; and as foon as it is thoroughly cold, they Hop it up clofe. It will keep two years; will be hard, and the hardell they look upon to be the bell done. This, they dilh up cold, fometitnes fried with eggs and garlic-, fbmetimes Hewed, and lemon fqueezed on it. It is very good any way, either hot or eold. MEATH, commonly fo called, or otherwife Enf Meath, to diHinguifli it from the county called Wef Meath : A county of Ireland, in the province of LeinHer, bounded by the counties of Cavan and Louth on the north, the Irifh channel on the eafl, Kildare and Dublin on the fouth, and Weft Meath and Long¬ ford on the weft. It is a fine champaign country, abounding with corn, and well inhabited. It returns 14 members to parliament ; and gives title of earl to the family of Brabazan. It contains 326,480 Irilh plantation acres* 139 parilhes, and x 12,000 inhabi¬ tants. The chief towm is Trim. This diftrift being the moft ancient fettlement of the Belgians in Ireland, the inhabitants wmre efteemed the eldeft and moft honour¬ able tribe : from which feniority their chieftains were elefted monarchs of all the Belgae ; a dignity that was continued in the Hy-n-Faillian without intermiflion, until the arrival of the Caledonian colonies* under the name of Tuath de Danan, when Conor-Mor, chief¬ tain of thefe people, obtained, or rather ufurped, the monarchial throne, obliged Eochy Failloch, with fe- veral of his people, to crofs the Shannon, and eftablifli themfelves in the prefent county of Rofcommon, where Crothar founded the palace of Atha or Croghan, a circumftance which brought on a long and bloody war between the Belgian and Caledonian races, which was not finally terminated until- the clofe of the 4th cen¬ tury, when the Belgian line wras reftored in the perfon of O’Nial the Great, and continued until Briam Bo- romh ufurped the monarchial dignity, by depofing Malachy O’Malachlin, about the year 1001. Tuathal Tetfthomar, by a decree of the Tarah aflembly, fepa- rated certain large tracts of land from each of the four provinces, w here the borders joined together ; whence, under the notion of adopting this fpot for demefne lands to fupport the royal houfehold, he formed the county or kingdom of Meath, which afterwards be¬ came the peculiar inheritance of the monarchs of Ire¬ land. In each of the portions thus feparated from C 44- J M E A the four provinces, Tuathal caufed palaces to be crew¬ ed, which might adorn them, and commeAorate tin ^ name in which they- had been. added to the royal do¬ main; In the tract taken out of Munfter, he built the palace called Flachtaga, where the facred fire, fo called, was kindled, and wdiere all the priefts and druida annually met on the laft day of Oaober ; on the evening of which day it was enafted, that no other fire ihould be ufed throughout the kingdom, in order that all the fires might be derived from this, which being lighted up as a fire of facrifice, their fuperftition led them to believe would render all the reft propitious and holy ; and for this privilege every family was to pay three¬ pence, by way of acknowledgment to the king of Munfter. The fecond royal palace was erected in the proportion taken out of Connaught, and was built for the aflembly called the convocation of Vifneach, at which all the inhabitants-were fummoned to appeal on the 1 ft day of May, to offer facrifice to Beal, or Bef, the god of fire, in whofe honour tw o large fires being kindled, the natives ufed to drive their cattle between them, which was fuppofed to be a prefervative for them againrt accidents and diftempers, and this was called Beal-Tinne, or Bel-Tine, or. the feftival of the god of fire. The king of Connaught at this meeting claimed a horfe and arms from every lord of a manor or chieftain,' as an acknowledgement for the lands ta¬ ken from that province, to add to the territory of Meath. The third was that which Tailtean erected in the part taken from Ulfter, where the fair of that name was held, which was remarkable for this parti¬ cular circumftance, that the inhabitants brought their children thither, males and females, and eontraftei them in marriage, where the parents having agreed upon articles, the young people were joined according-- ly ; every couple contracted at this meeting paid the king of Ulfter an ounce of filver by w7ay of acknow¬ ledgement. The royal manfion of Tarah, formerly deftroyed by fire, being rebuilt by Tuathal, on the lands originally belonging to the king of Leinfter, was reckoned as the fourth of thefe palaces ; but as a fa¬ bric of that name had flood there before, we do not find that any acknowledgement was made for it to the king of Leinfter. Meath, with Clonmacnois, is a bifhop’s fee, valued in the king’s books at 373I. 7s. o-^d. fterling, by an ex¬ tent returned anno 28th Elizabeth ; but, by a former extent taken anno 30th Henry VIII. the valuation a- mounts to 373I. I2s. w-hich being the largeft and moft profitable for the king, is the meafure of the firft fruits at this day. This fee is reputed to be worth annually 3400I. There were formerly many epif- copal fees in Meath, as Clonard, Duleek, Kells, Trim, Ardbraccan, Donftiaghlin, Slaine, and Foure, befides others of lefs note ; all thefe, except Duleek and Kelts* were confolidated, and their common fee was fixed at Clonard, before the year 115 2 •* at which time the divifions of the bifhoprics in Ireland wer e made by John Paparo, cardinal prieft, entitled Cardinal of St Lawrence in Damafo, then legate from Pope Eu¬ gene III. to the Irifti. This divifion was made in a fynod held on the 6th of March in the abbey of Mel- lifont, or, as fome fay, at Kells : and the two fees of Duleek and Kells afterwards fubmitted to the fame fate. The conftitution of this diocefe is Angular, hav- insr Meath. M E C [ 45 3 M E C Meath ing no dean nor chapter, cathedral, or economy.— Me 'eras ^odor the bifhop, the archdeacon is the head officer, to ‘ ~, whom, and to the clergy in general, the conge cPehre iffued while bifhops were elective. The affairs of the dioeefe are tranfacled by a fynod, in the nature of a chapter, who have a common feal, which is annually lodged in the hands of one of the body, by the ap* pointment and vote of the majority* The diocefe is divided into twelve rural deaneries. Of CloNMACNOJS, now annexed to Meath: There is no valuation of this lee in the king’s books} but it is fuppofed to be included in tlie extent of the fee of Meath, taken anno 30th Henry VIII. The chapter ef this fee conlilted anciently of dean, chanter,- chan¬ cellor, trealurer, archdeacon, and twelve prebendaries, but mod of their poffeffions have fallen into lay hands. At prefent the deanery is the only part of the chapter which fublilts, to which the prebend of Cloghran is annexed, and he hath a feal of office, which appears to have been the ancient epifcopal feal of this fee. This fee wras founded by St Kiaran, or Ciaran, the younger, in 548 ur 549 ; and Dermod, the fon of Ceronill, king of Ireland, granted the fite on which the church was built. We/} Meath* See Westmeith.. MEATUS suditorius. See Anatomy, N° 144. MEAUX, an ancient town of France, in the de¬ partment of the Seine and Marne, with a bihop’s fee, feated in a place abounding in corn and cattle, on the river Marne, which divides it into two parts; and its trade confifts in corn, wool, and cheefe. It fuftained a liege of three months againft the Engliffi in J421. E. Long. 2. 58. N. Lat. 48. 58. MECfENAS, or Mecoenas, C. Cilnius, a cele¬ brated Roman knight, defeended from the kings of Etruria. He has rendered himfelf immortal by his liberal patronage of learned men and of letters ; and to his prudence and advice Auguftus acknowledged him¬ felf indebted for the fecurity he enjoyed. His fond- nefs for pleafure removed him from the reach of ambi¬ tion ; and he preferred dying, as he was born, a Ro¬ man knight, to all the honours and dignities which either the friendlhip of Auguftus or his own popularity could heap upon him. To the interference of Mecamas, Virgil owed the retribution of his lands; and Horace was proud to boaft that his learned friend had obtain¬ ed his forgivenefs from the emperor, for joining the caufe of Brutus at the battle of Philippi. Mecsenas ■was himfelf fond of literature : and, according to the molt received opinion, he wrote a hiftory of animals, a journal of the life of Auguftus, a treatife on the difterent natures and kinds of precious ftones, belides the two tragedies of Oclavia and Prometheus, and other things, all now loft. He died eight years be¬ fore Thrift; and on his deathbed he particularly re¬ commended his poetical friend Horace to the care and confidence of Auguftus. Seneca, who has liberally commended the genius and abilities of Mecaenas, has not withheld his cenfure from his diffipation, indolence, and effeminate luxury. From the patronage and en¬ couragement which the princes of heroic and lyric poetry among the Latins received from the favourite of Auguftus, all patrons of literature have ever fince been called Meccenates. Virgil dedicated to him his Geor¬ gies, and Horace his Odes. 5 MECCA, an ancient and very famous town of Alia-, in Arabia Felix; feated on a barren fpot, in a valley furrounded with little hills, about a day’s jour¬ ney from the Red fea. It is a place of no ftrength, having neither walls nor gates; and the buildings are very mean;. That which fupports it is the refort of a great many thoufand pilgrims annually, for the fhops are fcarcely open all the year befides. The inhabitants are poor, very thin, lean, and fwarthy. The hills about tire town are very numerous; and confift of a blackilh rock, fome of them* half a mile in circumfe¬ rence. On the top of one of them is a cave, wffiere they pretend Mahomet ufually retired to perform his devotions, and hither they7 affirm the greateft part of the Alcoran wras brought him by the angel Gabriel. The town has plenty of water, and yet little garden- fluff ; but there are feveral forts of good fruits to be had, fuch as grapes, melons, water melons, and cucum¬ bers. There are alfo plenty of ffieep brought thither to be fold to the pilgrims. It ftands in a very hot cli¬ mate ; and the inhabitants ufually fleep on the tops of their houfes for the fake of coolnefs. In order to pro- tect themfelves from the heat through the day, they carefully fliut the w indows, and wrater the ftreets to re- freffi the air. There have been inftances of perfonsfuf- foeated in the middle of the town by the burning wind, called Simoom. As a great number of the people of diftindlion in the province of Hedsjas ftay in the city, it is better built than any other in Arabia. Amongft the beauti¬ ful edifices it contains, the moft remarkable is the fa¬ mous Kaha or Caaba, “ The houfe of God,” which was held in great veneration by the Arabs even before Mahomet’s time. No Chriftian dares go to Mecca ; not that the ap* proach to it is prohibited by any exprefs law, or that the fenfible part of the Mahometans have any thing to objeft to it; but on account of the prejudices of the people, who regarding this ground as facred, think Chriftians unworthy of letting their foot on it; it would be profaned in the opinion of the fuperftiti- ous, if it was trod upon by infidels. The people even believe, that Chriftians are prevented from approach¬ ing by fome fupernatural power ; and they tell the ftory of an infidel, who having got fo far as the hills that liirround Mecca, all the dogs of the city came out, and fell upon him ; and who, being ftruck with this miracle, and the auguft appearance of the Kaba, immediately became a muffulman. It is therefore to be prefumed that all the Europeans -who deferibe Mecca as eye-w itneffes, have been renegadoes efeaped from Turkey. A recent example confirms this fuppo- fition. On the promife of being allowed to preferve his religion, a French furgeon was prevailed on to ac¬ company the Emir Hadsji to Mecca, in qualitv of phy- fician ; but at the very firft ftation, he was forced to fubmit to circumcifion, and then he was permitted to continue his journey. Although the Mahometans do not allow Europeans to go to Mecca, they do not refufe to give them de- feriptions of the Kaba, and information with regard to that building ; and there are perfons who gain their bread by making defigns and little pi£tures of the Kaba^ and felling them to pilgrims. See Caaba. The Mahometans have fo high an opinion of the fan&ity Mecca. M E C [ 40 ] M E G Mecca, fanclity of Mecca, that they extend it to the places “"~v in the neighbourhood. The territory of that city is held faered to certain diftanees, which are indicated by particular marks. Every caravan finds in its road a fimilar mark, which gives notice to the pilgrims when they are to put on the modeft garb in which they muft appear in thofe iacred regions. Every muffulman is obliged to go once in his life at leaf! to Mecca, to perform his devotions there. If that law was rigour- oufly enforced, the concourfe of pilgrims would be prodigious, and the city would never be able to con¬ tain the multitudes from all the countries where the Mahometan religion prevails. We mull therefore, fuppofe, that devotees alone perform this duty, and that the others can eafily difpenfe with it. Thofe ■whofe circumltances do not permit a long abfence, have the liberty of going to Mecca by a fubftitute.— A hired pilgrim, however, cannot go for more than one perfon at a time ; and he muft, to prevent frauds, bring an atteftation in proper form, from an Imam of Mecca, that he has performed the requifite devotions on behalf of fuch a perfon, either alive or dead 5 for, after the deceafe of a perfon who has not obeyed the law during his life, he is ftill obliged to perform the journey by proxy. The caravans, which are not numerous, when we confider the immenfe multitude of the faithful, are Oompofed of many people who do not make the jour¬ ney from purpofes of devotion. 1 hefe are merchants, who think they can tranfport their merchandifes with more fafety, and difpofe of them more eafily ; and contractors of every kind, who furnilh the pilgrims and the foldiers who efcort the caravans, with necelfa- ries. Thus it happens, that many people have gone often to Mecca, folely from views of intereft. The molt confiderable of thofe caravans is that of Syria, commanded by the pacha of Damafcus. It joins at fome diftance the fecond from Egypt, which is con¬ duced by a bey, who takes the title of Emir Hadsji. One comes from Yemen, and another, lefs numerous, from the country of Lachfa. Some fcattered pilgrims arrived by the Red fea from the Indies, and from the Arabian eftabliftunents on the coafts of Africa. The Perfians come in that which departs from Bagdad ; the place of conduCor to this laft is bellowed by the pacha, and is very lucrative, for he receives the ranfoms of the heretical Perfians. It is of confequence to a pilgrim to arrive early at the holy places. Without having been prefent from the beginning at all the ceremonies, and without hav¬ ing performed every particular aft of devotion, a man cannot acquire the title of Hadsji: this is an honour very much coveted by the Turks, for it confers real advantages, and makes thofe who attain it to be much refpefted. Its infrequency, however, in the Maho¬ metan dominions, Ihows how much the obfervation of the law commanding pilgrimages is neglefted. A fi¬ milar cuftom prevails among the Oriental Chriftians, who are alfo exceedingly emulous of the title of Hadsji, or Mokdafi, which is given to pilgrims of their com¬ munion. In order to acquire this title, it is not fuffi- cient that the perfon has made the journey to Jerufa- lem •, he muft alfo have kept the paffover in that city, and have affiHed at all the ceremonies of the holy weeks. I After all the effential ceremonies are, over, the pil- Mecca, grims next morning move to a place where they fay MtcEmi- Abraham went to offer up his fon Ifaac, Avhich is , ca*‘ about two or three miles from Mecca : here they pitch v their tents, and then throw feven fmall ftones againft a little fquare ftone building. This, as they affirm, is performed in defiance of the devil. Every one then purchafes a fheep, which is brought for that purpofe, eating fome of it themfelves, and giving the reft to the poor people who attend upon that occafion. Indeed thefe are miferable objefts, and fuch ftarved creatures, that they feem ready to devour each other. After all, one would imagine that this was a very fanftified place j and yet a renegado who went in pilgrimage thither, affirms there is as much debauchery praftifed here as in any part of the Turkifh dominions. It is 25 miles from Jodda, the fea port town of Mecca, and 220 fouth-eaft of Medina. E. Long. 40. 55. N. Lat. 21. 45. MECHANICAL, an epithet applied to whatever relates to mechanics : Thus we fay, mechanical powers, caufes, &c. See the articles Power, Cause, &c. The mechanical philofophy is the fame with what is otherwife called corpufcular philofophy, which explains the phenomena of nature, and the operations of corpo¬ real things, on the principles of mechanics1, viz. the motion, gravity, arrangement, difpofition, greatnefs or fmallnefs, of the parts which compofe natural bodies. See Corpuscular. This manner of reafoning is much ufed in medicine j and, according to Dr Quincy, is the refult of a tho¬ rough acquaintance with the ftrufture of animal bo¬ dies : for confidering an animal body as a compofition out of the fame matter from which all other bodies are formed, and to have all thofe properties which concern a phyfician’s regard, only by virtue of its peculiar con- ftruftion it naturally leads a perfon to confider the feveral parts, according to their figures, contexture, and ufe, either as wheels, pulleys, wedges, levers, ferews, cords, canals, ftrainers, &c. For which pur¬ pofe, continues he, it is frequently found helpful to x defign in diagrams, whatfoever of that kind is under confideration, as is cuftomary in geometrical demonftra- tions. For the application of this doftrine to the human body, fee the article Medicine. MechaNICAT., in mathematics, denotes a conftrucr tion of fome problem, by the afliftahee of inftruments, as the duplicature of the cube and quadrature of the cir¬ cle, in contradiftinftion to that which is done in an ac¬ curate and geometrical manner. Mechanical Curve, is a curve, according to Defcartes, which cannot be defined by any algebraic equation j and fo Hands contradiftinguiffied from algebraic or geo¬ metrical curves. Leibnitz and others call thefe mechanical curves tranfcendental, and diffent from Defcartes, in excluding them out of geometry. Leibnitz found a new kind of tranfcendental equations, whereby thefe curves are de¬ fined : but they do not continue conftantly the fame in all points of the curve, as algebraic ones do. See the article Transcendental. Mechanical Solution of a problem is either when the thing is done by repeated trials, or when lines ufed _ in M E- C f 47 ] M E C Mechani- in the folution are not truly geometrical, or by organi which are ufed for raifing greater weights, or over- Mechani¬ cs cal conltru&ion. coming greater refill ances, than could be effected by caI- Mechanical Powers, are certain limple machines, the natural llrength without them. See Mechanics. ' ~ MECHANICS. Definition, i. TiTECHANICS is the fcience which enquires into the laws of the equilibrium and motion of folid bodies ; into the forces by which bodies, whether ani¬ mate or inanimate, may be made to aft upon one ano¬ ther *, and into the means by which thefe may be in- creafed fo as to overcome fuch as are more powerful.— The term mechanics was originally applied to the doc¬ trine of equilibrium. It has by fome late writers been extended to the motion and equilibrium of all bodies, whether folid, fluid, or aeriform ; and has been employ¬ ed to comprehend the fciences of hydrodynamics and pneumatics. HISTORY. Progrefs f 2. As the fcience of mechanics is intimately con- mechamcs ne<^ed S16 arts °f life, and particularly with thofe amon^the which exift even in the rudeft ages of fociety, the con- ancients. ftruftion of machines muft have arrived at conliderable perfeftion before the theory of equilibrium, or the fimpleft properties of the mechanical powers, had en¬ gaged the attention of philofophers. We accordingly find that the lever, the pulley, the crane, the capftan, and other Ample machines, were employed by the an¬ cient architefts in elevating the materials of their buildings, long before the dawn of mechanical fcience j and the military engines of the Greeks and Romans, fuch as the catapultae and balilbse, exhibit an extenfive acquaintance with the conftruftion of compound ma¬ chinery. In the fplendid remains of Egyptian architec¬ ture, which in every age have excited the admiration of the world, we perceive the mofl: furprifing marks of mechanical genius. The elevation of immenfe maffes of ftone to the tops of their ftupendous fabrics, mufl: have required an accumulation of mechanical power which is not in the pofleflion of modern architefts. Ariftotle 3. The earlieft traces of any thing like the theory of who^f mechanics arf to be found in the writings of Ariftotle. tcniictl to ^ome los works we difcover a few erroneous and thi theory obfcure opinions, refpefting the doftrine of motion, and | of mech.t- the nature of equilibrium ; and in his 28th mechanical - queftion he has given fome vague obfervations on the • 3Z0- force of impulfe, tending to point out the difference be¬ tween impulfe and preifure. He maintained that there cannot be two circular motions oppofite to one another *, that heavy bodies defcended to the centre of the uni- verfe, and that the velocities of their defcent were pro¬ portional to their weights. Archimedes 4. The notions of Ariftotle, however, were fo con- foundation erl‘one°us, that the honour of laying the foun- of theoreti- dat‘on °f theoretical mechanics is exclufively due to the cal mecha- celebrated Archimedes, Avho, in addition to his inven- nics. tions in geometry, difeovered the general principles of C- 250- hydroftatics. In his two books, De Equiponderantibus, he has demonftrated that when a balance with unequal arms, is in equilibrio, by means of two weights in its oppofite feales, thefe weights muft be reciprocally pro¬ portional to the arms of the balance, f rom this gene¬ ral principle, all the other properties of the lever, and of machines referable to the lever, might have been deduced as corollaries $ but Archimedes did not follow the difeovery through all its confequences. In de- monftrating the leading property of the lever, he lays it down as an axiom, that if the two arms of the ba¬ lance are equal, the two weights muft alfo be equal when an equilibrium takes place ; and then ftiows that if one of the arms be increafed, and the equilibrium ftill continue, the weight appended to that arm muft be proportionally diminilhed. ft his important diicovery condufted the Syracufan philofopher to another equally ufeful in mechanics. Reflecting on the conftruftion of his balance, which moved upon a fulcrum, he perceived that the two weights exerted the fame preliure on the fulcrum as if they had both relied upon it. He then confidered the fum of thefe two weights as combined with a third, and the fum of thefe three as combined with a fourth ; and faw that in every fuch combination the fulcrum muft fupport their united weight, and there¬ fore that there is in every combination of bodies, and in every Angle body which may be conceived as made up of a number of leffer bodies, a centre of prefjure or gravity. This difeovery Archimedes applied to par¬ ticular cafes, and pointed out the method of finding the centre of gravity of plane furfaces, whether bounded by a parallelogram, a triangle, a trapezium, or a parabola. The theory of the inclined plane, 'the pulley, the axis in peritrochio, the ferew, and the wedge, which was firft publiftied in the eighth book of Pappus’s mathematical colleftions, is generally attributed to Archimedes. It appears alfo from Plutarch and other ancient authors, that a greater number of machines which have not reached our times was invented by this philofopher. J he military engines which he employed in the fiege of Syracufe againft thofe of the Roman engineer Ap- pius, are faid to have difplayed the greateft mechanical genius, and to have retarded the capture of his native city. 5- Among the various inventions which we have re-Invention ceived from antiquity, that of water mills is entitled to hft AB be the lame lever fupported by Fig. 3, the fulcra F, and let A f— FB and/F—2FB. Then if two weights C, D of one pound each be fufpended at the extremities A, B, they will be in equilibrio as be¬ fore. But fince the fulcrum f fupports a preffure of one pound (Axiom 2.), the equilibrium will ftill con¬ tinue when that fulcrum is removed and a weight of one pound made to aft in a ^contrary direftion y P at the point^ fo that the angle PyT may be equal to DBA. Now, (Axiom 1.) a weight E of one pound afting upward at f will be in equilibrio with a weight E' of one pound afting downwards at f; Fybeing equal to Yf, and therefore by removing E from the point f and fubftituting E at the point f, an equilibrium will ftill obtain. But fince Fy /~2FB a weight of one pound fufpended from f will have the fame influence in turn¬ ing the lever round F as a weight of two pounds fuf¬ pended at B (Cafe 2.). Let us remove, therefore, the Aveight E' from f, and fubftitute a Aveight G“2E/, fo as to aft at B. Then fince the equilibrium is not deftroy- ed, Ave have a Aveight C of one pound afting at the di¬ ftance FA, and the Aveights D -f-G —3 pounds afting at the diftance FB. But FA=3FB and D-J-G^C, confequently C: D-f-GrrFB :FA: That is, Avhen the diftances from the fulcrum are as 3 to I, and when an equilibrium exifts, the Aveights are reciprocally propor¬ tional to thefe diftances. 40. By making FA in fig. 2. equal to 2FB it may Fig. 2. ' be (hewn, as in Cafe 2. that the Aveights are reciprocally proportional to their diftances from the fulcrum, Avhen they aft on the fame fide of the fulcrum, and Avhen the diftances are as 3 to 1. 41. In the fame Avay the demonftration may be ex-pig. tended to any commenfurable proportion of the arms, by making EA to FB in that proportion, and keeping f A always equal to FB. Hence avc may conclude in general, that Avhen tAvo Aveights afting at equal angles upon a ftraight lever devoid of weight, are in equilibrio, they are reciprocally proportional to their diftances from the centre of motion, (y. E. D. 42. Cor. 1. If tAvo Aveights afting at equal angles Corollarifs. upon the arms of a ftraight lever devoid of Aveight are reciprocally proportional to their diftances from the ful¬ crum, they Avill be in equilibrio. For if an equilibrium does not take place, the pro¬ portion of the Aveights muft be altered to procure an equilibrium, and then, contrary to the propofition, the weights would balance each other when they Avere not reciprocally proportional to their diftances from the ful¬ crum. 43. Cor. 2. If a weight Wbe fupported by a hori-Fjg. zontal lever refting on the fulcra A, B, the preffure up¬ on A is to the preffure upon B in the inverfe ratio of their diftances from the point Avhere the Aveight is fuf¬ pended, that is, as BF to FA. For if avc fuppofe B to be the fulcrum, and if removing the MECHANIC S. 55 Theory, tlie fulcrum A, we fupport the extremity A of the lever V“ by a weight E equivalent to the weight fuftained by the fulcrum A, and acting upwards over the pulley P, then the weight E or that fuitained by A : W—JBF : BA (Prop, i.) and if we conceive A to be the ful¬ crum, and fupport the extremity B by a weight F equal to that which was fupported by the fulcrum B, we fhall have the weight F or the weight fuftained by B : W=AF : AB. Hence ex cequo the weight fuf¬ tained by A is to the weight fuftained by B as BF is to FA. Fig. 5. 44. Cor. 3. We may now call the two weights P and W, the power and the weight, as in fig. 5, and fince P : WrrFB : FA, we have (Geometry, Sedft. iv. Theor. 8.) P X FA=W X FB, when an equilibrium takes place, . . p WXFB „T PxFA coniequently Pzz FA= w- FA 7 WxFB FB FB= P PxFA W 45. Cor. 4. We have already feen (Axiom 2.) that when the power and the weight are on contrary fides of the fulcrum, the preffure upon the fulcrum is equal to P-j-W or the fum of the weights *, but it is obvious that when they act on the fame fide of the ful¬ crum, the preffure which it fupports will be P—W, or the difference of their weights. 46. Cor. 5. If a weight P be Ihifted along the arm of a lever AD, the weight W, which it is capable of balancing at A, will be proportional to FA. When the weights are in equilibrio (Cor. 3.) W : P—FA : FB, or by alternation W : FArrP : FB, and if iv be another value of W and f a another value- of FA, we (hall alfo have w : Prr:f a : FB or w : f a— P : FB, confequently (Euclid, Book v. Prop. xi. and xvi.) W : iu=FA : f a, that is, W varies as FA. Fig. 6. Cor. 6. It is obvious that the truths in the preced¬ ing propofition and corollaries, alfo hold Avhen the lever has the form reprefented in figure 6. only the ftraight lines AF, FB are in that cafe the length of the arm. Dcfcription 47* Cor. 7. Since by the laft corollary FA : f a-=z of the fteel- W : w, it follows that in the Roman Jlatera or J}eeli/ardy tera ^ ^ merely a lever with a long and ftxort arm, ha¬ ving a weight moveable upon the long one, the diftances at which the conftant weight muft be hung are as the 7- Aveights fufpended from the fhorter arm. The fteelyard is reprefented in fig. 7. Avhere AB is the leArer Avith un¬ equal arms AF, FB, and F the centre of motion. The body W, Avhofe Aveight is to be found, is fufpended at the extremity B of the lever, and the conftant Aveight P is moved along the divided arm FB till an equili¬ brium takes place. As foon as this happens, the num¬ ber placed at the point of fufpenfion D, indicates the weight of the body. If the lever is devoid of Aveight, it is obvious that the fcale EB Avill be a fcale of equal parts of which EB is the unit, and that the Aveight of the body W Avill be always equal to the conftant weight P multiplied by the number of divifions between P and F. Thus if the equilibrium takes place Avhen P is pull¬ ed out to the 12 divifion, avc (hall have W=i2 P, and pound, W~ 12 pounds. But when the gravity of the lever is conlidered, which muft be done in the Theory, real fteelyard, its arms are generally of unequal Aveight,' : and therefore the divifions of the fcale muft be alcer- tained by experiment. In order to do this, remove the weight P, and find the point C, at Avhich a Aveight P' equal to P being fufpended, will keep the unequal arms in equilibrio, C will then be the point at which the equal divifions muft commence. For Avhen W and P are placed upon the fteelyard and are in equilibrio, W balances P along with a weight Avhich, placed at D, would fupport P placed at C : Therefore W X BF— P x DF-f-P x CF j but PxE>F4-PxCF=:PxDC, confequentlyW X BF=P X DC,and(GEOMETRY, Seft. iv. Theor. 8.) W: DC=P : BF. By taking different values of the variable quantities W and DC as %v and d c, we fhall have w : dc~P : BF, confequently (Euclid, B. V. Prop. xi. and x\d.) W : w=DC : ds, that is, the weight of W varies as DC, and there¬ fore the divifions muft commence at C. If the arm BF had been heavier than FA, which, however, can fcarce- ly. happen in practice, the point C Avould have been on the other fide of F. In conftrufting fteelyards,. it might be advifable to make the unequal arms balance each other by placing a weight M at the extremity of the lighter arm, in which cafe the fcale Avill begin at F. In the Danifh and Swedifli.Danifh and' fteelyard the body to be weighed and the conftant Swedilh weight are fixed at the extremities of the fteelyard, but fteel>'ard'* the point of fufpenfion or centre of motion F moves along the lever till the equilibrium takes place. The point F then indicates the weight of the body required.. —There are fome fteelyards in which the conftant Aveight is fixed to the fhorter arm, Avhile the body to be Aveigh- ed moves upon the longer arm.. The method of divid¬ ing this and the preceding fteelyard may be feen in De la Hire’s Traite de Mecunique, Prop. 36, 37, 38. Prop. II. 48. To find the condition of equilibrium on a ftraight lever when its gravity is taken into the account. 49. Let us fuppofe the lever to be of uniform thick- nefs and denfity, as AB, fig. 7. and let it be fufpended Fig. 2. by the points c, d to another lever a b, confidered as Avithout Aveight, fo that a c~c fz= f d~d b. Then if be the centre of motion or point of fufpenfion, the cy¬ linder A B will be in equilibrio ; for the weight AB may be regarded as compofed of a number of pairs of equal Aveights, equally diftant from the centre of mo¬ tion. For the fame reafon, if we conceive the cylinder to be cut through at F the equilibrium Avill continue, c, d being uoav the points at which the Aveights AF, FB act, and their diftances cf df from the centre of motion being equal. Confequently the arms AF, FB have the fame energy in turning the lever round f as if weights equal to AF, FB Avere fufpended at the diftance of their middle points c, d from the fulcrum. Let P therefore, in fig. 5. be the poAver, W the weight, m the weight of the arm AF, and n the weight , of FB. Then Avhen there is an equilibrium Ave {hall ' ' have (Prop. I. Cor. 3.) P x AF-\.m X ^ AF=W X FB + w X sT B j and fince the Aveight m a fling at half the diftance AF is tlie fame as half the weight m, acting at the 16 MECHANICS. Lheory. the whole diftance AF? we may fubfHtute m X AF v inftead of m x 1 AF, and the equation becomes h X AF= W^fn x FB. Hence W+>xFB , B—— im AF •yjy- P“F i w X AF FB \V 1// x 2FB AF X 2 AF : FB ' in — 2P -2\V af=w+>xFB jjF ^"2 P+4otX AF 1E--w+i^— 50. Cor. If the arms of the lever are not of uniform denfity and thicknefs, inftead of the diftance of their middle points, we muft take the diftance of their centre of gravity from the fulcrum. Prop. III. 51. If two forces acting in any dire&ion, and in the fame plane, upon a lever of any form, are in equilibrio, they will be reciprocally propor¬ tional to the perpendiculars let fall from the fulcrum upon the dire&ions in which they a£t. Plate 52* ^‘et AFB be a lever of any form, F its fulcrum, eccXYII. A, B the points to which the forces, or the power P and Fijj. 1. & 2. weight W, are applied, and AE, BK the direftions in which thefe forces aft. Make AE to PK as P is to W, and they will therefore reprefent the forces applied at A and B. Draw AC perpendicular to AF and EC parallel to it, and complete the parallelogram AD EC. In the fame way form the parallelogram BGKH. Produce EA and KB towards m and n if neceflary, and let fall F m, F « perpendicular to AE, BK produced. Then P fhall be to W as F « is to F m. By the refolution of forces (Dynamics, § 140.) the force AE is equivalent to forces reprefented by AD and AC, and afting in thefe direftions. But as AD afts in the direftion of the arm AF, it can have no influ¬ ence in turning the lever round F, and therefore AC reprefents the portion of the force AD which conUi- butes to produce an angular motion round F. In the fame -way it may be {hewn that BG is the part of the force BK which tends to move the lever round F. Now fuppofe AF produced to B, FB being made equal to FB and B'G'—BG. Then by Prop. I. AC : B'G' =rFB' : FA *, but by Axiom 1. the effort of BG to turn the lever round F is equal to the effort of the equal force B'G' to turn the lever round F ; therefore AC: BG=FB : FA and AC xFA=BG xFB. Now the triangles ACE, AE m are fimilar, becaufe the angles at F and M are both right, and on account of the parallels DF, AC, MAC^ADF; therefore AC : AErrFw: FA, and AC xFA = AE xFra:. For the fame reafon in the fimilar triangles BGK, BF n we have BG : BK=F n : FB, and BK x F w=BG X FB. Hence AE X F-wrr BK X F and AE : BK or P : W Theory. —Yn : Fm. (^. E. D. 53. Cor. 1. The forces P and W are reciprocally corollaries, proportional to the fines of the angles which their di- jq t & 2 reftions make with the arms of the lever, for F m is evidently the fine of the angle FA m, and F n the fine of the angle FB n, FA, FB being made the radii ;— therefore P : WrrSin. FB « : Sin. FA m, or P : W Since FA : Fw=:Rad. : Sin. FA/« Sin. FB«* c- ir a i t- FAxSm. FA/» , „ Sin. IA m, we have t m— T— : and finer Bad. FB : FtfrzrRad.: Sin. FB«, wehave Fffl——B * p^'FBw had. but in the cafe of an equilibrium P : WrrF n : Ym, ccn- FF X Sin. FB n FA x Sin.FAwz Rad. ’ Rad. '* and fince magnitudes have the fame ratio as their equi¬ multiples, P : W nrEB X bin. EB n : FA x bin. FA#*. 54. Cor. 2. rI he energies of the forces P, W to turn the lever round the fulcrum F is the fame at what¬ ever point in the direftions w E, « K they are applied, for the perpendiculars to which thefe energies are pro¬ portional remain the fame.—T he truth of this corollary has been affumed as an axiom by feme writers on me¬ chanics, who have very readily deduced from it the. preceding propofition. But it is very obvious that the truth affumed as felf-evident is nearly equivalent to the truth which it is employed to prove. Thofe who have adopted this mode of demonftration illuftrate their axiom by the cafe of a folid body that is either pufhed in one direftion with a ftraight rod, or drawn by a cord ; in both of which cafes it is manifeft that the effeft of the force employed is the fame, at whatever part of the rod or firing it is applied : But thefe cafes are completely different from that of a body moving round a fixed centre. 55. Cor. 3. If AE and BK the direftions in which the forces P, W are exerted be produced till they meet at L; and if from the fulcrum E tire line FS be drawn parallel to the direftion AL of one force till it meets BL, the direftion of the other ; then LS, SF will reprefent the two forces. For as the fides of any triangle are as the fines of the oppofite angles LS : SF~fin. LFS : fin. FLS j but on account of the parallels FS, AL the angle LFS—FLA, and FL being radius F m is the fine of FLA or LFS, and F n the fine of FLS, there¬ fore by fubfiitution LS : SFnF : F #, that is as the force W : P. 36. Cor. 4. If feveral forces aft upon a lever, and keep it in equilibrio, the fum of the produfts of the forces and the perpendiculars from the fulcrum to the direftion of the different forces on one fide is equal to the fum of the produfts on the other. For fince the energy of each force to turn the lever is equal to the produft of the force and the perpendicular from the fulcrum on the line of its direftion ; and fince in the cafe of an equilibrium, the energy of all the forces on one fide of the fulcrum muft be equal to the energy of all the forces on the other fide, the produfts propor¬ tional to their energies muft alfo be equal. 57. Cor. 5. If two forces aft in a parallel direftion upon an angular lever whofe fulcrum is its angular point, fequently P : Wa= MECHANIC S. 57 Theory. F>S- 3- %• 4* f'g- 5* point, thefe forcc!i will be in equilibrio when a line drawn from the fulcrum upon the line which joins the two points where the forces are applied, and parallel to the direction of the forces, cuts it in luch a manner that the two parts are reciprocally proportional to the forces applied. Let AFB be the angular lever, whofe fulcrum is F, and let the forces P, W be applied at A and B in the parallel diredtions P ni^ W n ; then if the line FD, pa¬ rallel to P m or W «, cut AB in luch a manner that DB : DA=P : W, the forces will be in equilibrio. Draw F m perpendicular to P m, and produce it to « ; then lince A m, B n are parallel, m n will alfo be perpendicular to B », and by the propofition (Art. 51.) F n : F m — P : W. Now, if through F, there be drawn m! n' parallel to AB, the triangles F m m', F n n' will be fxmilar, and we lliall have F « : F m=.Y n': Ym', but on account of the parallels AB, ni n'; F ?i' : F m' —DB : DA, therefore DB : DA—P : W. 58. Cor. 6. Let CB be a body moveable round its centre of gravity F, and let two forces P, W act upon it at the points A, B in the plane AFB, in the directions AP, BW; then fince this body may be re¬ garded as a lever whofe fulcrum is F, the forces will be in equilibrio when P : WrrF n : F ?n the perpendicu¬ lars on the directions in which the forces adt. 59. Cor. 7. If AB be an inflexible rod moveable round F as a fulcrum, and adled upon by two forces P, W in the diredtions A m, A n, thefe forces will be in equilibrio when they are to one another as the per¬ pendiculars F «, F m.—For by cor. 2. the forces may be confidered as applied at in and «, and m F;/ may be regarded as the lever •, but by the propofition (Art. 51.) P : W=rF n : F m ; F ?;/, F n being perpendiculars upon A m, A n. Fig. 6. 60. Cor. 8. Let DE be a heavy wheel, and FG an obflacle over which it is to be moved, by a force P, adting in the diredtion AH. Join AF, and draw' Y m, F « perpendicular to CA and AH. The weight of the wheel is evidently the weight to be raifed, and may be reprefented by W acting at the point A in the vertical diredtion AC. We may now confider AF as a lever whole fulcrum is F, and by cor. 7. there will be an equilibrium when P : W—F « : F m. Since F m re- prefents the mechanical energy of the power P to turn the wheel round F, it is obvious that when FG is equal to the radius of the wheel, thfe weight P, how¬ ever great, has no power to move it over the obftacle ; for when FG—AC, F m—Q, and F ttc X P—o. Big. 7. 61. Cor. 9. If a man be placed in a pair of feales hung at the extremities of a lever, and is in equilibrio with a weight in the oppoftte fcale, then if lie prefifes againft any point in the lever, except that point from which the fcale is fufpended, the equilibrium will be deilroyed. Let CB be the lever in equilibrio, F its fulcrum, and let the feales be fufpended from A and B, AP being the fcale in which the man is placed. Then if he preffes with his hand or with a rod againfl D, a point nearer the centre than A, the fcale will take the pofition AP', and the fame effedl will be produced as. if AD were a folid mafs adling upon the lever in the diredlinn of gravity. Confequently if Vp be drawn perpendicular from the point P' to FC, Fp will be the lever with which the man in the fcale tends to turn fehe lever round the fulcrum j and as Yp is greater than VoL.Xin.PaA I. FA, the man will preponderate. In the fame way it Theory, it may be thown, that if the man in the fcale AP preffes '—--nr"—-' upwards again It a point C, more remote from the ful¬ crum than A, he will diminilh his relative weight, and the fcale W will preponderate, for in this cafe the icale affumes the pofition AP", and Fp' becomes the lever by which it afts. 62. Cor. 10. If a weight W be fupported by an Fig. &. inclined lever relting on the fulcra A, B, the preiiure upon A is to that upon B inverfely, as A/is to fb, the. feclions of a horizontal line by the vertical direction of the weight W. Remove the fulcrum A, and fupport the extremity A by a weight P, equal to the preiiure upon A j then B being the centre of motion, and m n being drawn through F perpendicular to the direction of the forces A /«, Ey, and confequently parallel to A b, we have (Art. 51.) P : WrrF n ; F m—fb : f A, that is, the preflfure upon A is to the preflure upon B inverfely as A/is to fb. Scholium. 63. Various attempts have been made by different writers on mechanics to give a complete and fatisfac- tory demonflration of the fundamental property of the lever. The firft of thefe attempts was made by Archi¬ medes, who aflumes as an axiom, that if two equal bo¬ dies be placed upon a lever, they will have the lame in¬ fluence in giving it a rotatory motion as if they were both placed in the middle part between them. This truth, however, is far from being fell’ evident, and on this account Mr Vince * has completed the demonitra- * Phil. tion by making this axiom a preliminary propolition. Tranf. The demonftration of Galileo f is both Ample and ele-D94-P-33 gant, and does not feem to have attracted much notice, though in principle it is exactly the fame as that oijttatiuncs Archimedes completed by Mr Vince. Galileo fuf- MathemaU pends a folid cylinder or prifm from a lever by feveral threads. When the lever is hung by its centre, the*5' 58, whole is in equilibrio. He then fuppofes the cylinder to be cut into two unequal parts, which from their mode of fufpenfion ftill retain their pofition, and then imagines each part of the cylinder to be fufpended by its centre from the lever. Here then we have two unequal weights hanging at unequal diflances from the centre of fufpenfion, and it follows from the conftruc- tion, that thefe weights are in the reciprocal ratio of their diftances from that centre. Mr Vince, on the other hand, employs a cylinder balanced on a fulcrum. He fuppofes this cylinder divided into unequal parts, and thus concludes from his preliminary propofltion, that thefe unequal parts have the fame effe6t in turning the lever as if the weight of thefe parts was placed in their centres ; which is done by Galileo by fufpending them from their centres. From this the fundamen¬ tal property of the . lever is eafily deduced The next demonftration was given by Huygens, who aflumes as an axiom, that if any weight placed upon a lever is removed to a greater diftance from the fulcrum, its ef¬ fort to turn the lever w ill he increafed. This axiom he might have demonil rated thus, and his demenftra- tion would have been completely fatisfactory, though it applies only to cafes where the arms of the lever p,at<_ are commenfiu-able. Let AB be a lever w ith equal f, and alfo m—n—o, fo that, if the laft formula is fuited to thefe conditions, we fhall have the formula of iEpinus. Prop. VI. 67. If a power and weight acting upon the arms of any lever be in equilibrio, and if the whole be put in motion, the velocity of the power is to the velocity of the weight as the weight is to the power. Fig. 3. Let AFB be any lever whofe fulcrum is F, and let the power P and weight W be applied to its extremi¬ ties A, B, fo as to be in equilibrio. Draw F /«, ¥ n perpendicular to AD, BE the direction of the forces P, W. Then -fuppofe an uniform angular motion to be given to the lever, fo as to make it defcribe the fmall angle AFA', the pofition of the lever will now be A'FB', and the directions of the forces, P, W will be A'D', B'E, parallel to AD, BE refpeCtively, fince the angle AEF is exceedingly fmall. Join A A', BB', and from A' and B' draw A'r, B'ss perpendicular to AD and BE. Now it is obvious, that though the point A has moved through the fpace AA' in the fame time that the point B has deferibed the fpace BE', yet A x is the fpace deferibed by A in the direction AD, and B % the fpace deferibed by B in the direction BE. For if we fuppofe a plane pafling through A at right angles to AD, snd another through P parallel to the former plane, it is manifeit that A x meafures the ap¬ proach of the point A to the plane paffing through P j and for the fame reafon B •z meafures the approach of the point B to a plane pafling through W at right angles to WB. Therefore A x, B 2 reprefent the fpaces uniformly and fimultaneoufly deferibed by the points A, B, and may therefore be taken to denote the velocities of thefe points (Dynamics, § 14.) 5 confe- quently the velocity of A : the velocity of B=r A a1 : B 2. Now, in the triangles A a A', ¥ m A, the exterior an¬ gle * AFzrrA m F-j-ra F, A (Euclid, B. I. Prop. 32.) and A'AF=A/raF, becaufe AFA' is fo exceedingly imall that A'A is fenfibly perpendicular to AF; confe- quently # AA':= AEwz; and as the angles at x and m are right, the triangles A a: A', Atn¥ are fimilar (Geometry, Theor. XX. SeCt. IV.). Therefore, A x : AA'—F m : FA, and in the fimilar triangles AFA', BFB' AA' : BB'=FA : FB, and in the fimilar triangles BB'^, BF«, BB': BzrrFB : F«, therefore by compofition we have A a? : B %zz.¥ m : ¥n. But by Propofition II. P : W=F/z : F/«, confequent- ly A.v : B izrrrW : P, that is, the velocity of the power is to the velocity of the weight as the weight is to the power. (). E. D. 68. Cor. Since Aa : B : P we have A x x P =B«xW, that is, the momenta of the power and weight are equal. SECT. II. On the Inclined Plane. 69. Definition. An inclmedplane is a plane fur- P^ate face AB, fupported at any angle ABC formed with the horizontal plane BC. The inclination of the plane S‘ is the angle which one line in the plane AB forms with another in the horizontal plane BC, both thefe lines being at right angles to the common interfeClion of the two planes.—The line BA is called the length of the plane, AC its height, and BC the length of its bafe. 70. In order to underhand how the inclined plane ads as a mechanical power, let us fuppofe it neceflary to elevate the weight D from C to A. If this weight is lifted by the arms of a man to the point A, he mull fupport the whole of the load •, but when it is rolled up the inclined plane, a confiderable part of its weight is fupported upon the plane, and therefore a much fmaller force is capable of railing it to A. Prop. I. 71. When any weight Wr is kept in equilibrio up¬ on an inclined plane by a power P, the power is to the weight as the fine of the plane’s inclina¬ tion is to the fine of the angle which the direc¬ tion of the power makes with a line at right angles to the plane. Let MN be the inclined plane, NO a horizontal Fig. 5, line, and MNO the inclination of the plane, and let the weight W be fuftained upon MN by means of the power P a£Hng in the dire&ion AE. From the point A, the centre of gravity of the weight, draw AB per¬ pendicular to the horizontal plane ND, and AF per¬ pendicular to MN 5 produce EA till it meets the plane in C, and from the point F where the body touches the plane draw Fra at right angles to AC, and F« at right angles to AB. Then, fince the whole body may be confidered as colledled in the centre of gravity A, AB will be the direction in which it tends to fall, or the di- redtion of the weight, and EA is the diredtion of the power •, but AF is a lever whofe fulcrum is F, and fince it is adled upon by two forces which are in equilibrio, we lhall have (Art. 59.) P : W—F« : Fra, that is, as the perpendiculars drawn from the fulcrum to the diredtioa in which the forces adl. Now FA being radius, F« is the fine of the angle FAB, and F ra is the fine of the angle FAC $ but FAB is equal to MNO the angle of the plane’s inclination, on account of the right angles at F and B and the vertical angles at D j and FAC is. the angle which the diredtion of the power makes with a line perpendicular to the plane \ therefore P : W H 2 as 6o MECHANICS. Theory. ag the fine of the plane’s inclination, is to the fine of the * " » angle formed by the direction of the power with a line at right angles to the plane. 72. Cor. 1. When the power a£fs parallel to the plane in the direction AE', P is to W as EA to E «, that is, as radius is to the fine of the plane’s inclination, or on account of the fimilar triangles FA/z, MNO, as the length of the plane is to its height. In this cafe the power a£ts to the greateft advantage. 73. Cor. 2. When the power a£ts in a vertical line A s, F m becomes equal to or coincides with F ?z, and we have P : Wz=F « : F «, that is, the power in this cafe fuftains the whole weight. 74. Cor. 3. When the power a£ls parallel to the bafe of the plane in the direction Ae, P : WirF n \Yf “F n : A.n. 7 Cor. 4. When the power adls in the direction AF e' perpendicular to the plane, it has no power to refill the gravity of the weight 5 for the perpendicular from the fulcrum F, to which its energy is proportional, vanilhes. 76. Cor. 5. Since the body W a£ls upon the plane in a diredlion AF perpendicular to the plane’s lurface, (for its force downwards may be refolved into two, one parallel to the plane, and the other perpendicular to it), and fince the reaction of the plane mull alfo be perpen¬ dicular to its furface (Dynamics, § iqq.)? that is, in the direction FA, then, when the direction of the power is A e parallel to the horizon, the power, the weight, and the preflure upon the plane, will be refpe&ively as the height, the bafe, and the length of the plane. The weight W is a£led upon by three forces j by its own gravity in the direction A n, by the rea£lion of the plane in the direction AF, and by the power P in the direftion AF. Therefore, fince thefe forces are in equilibrio, and fince Ay is parallel to n F, and Fy to A «, the three fides AF, Ay Fy will reprefeut the three forces (Dynamics, § J44-)* But the triangle AFyis fimilar to A n F, that is, to MNO, for it was already (hewn that the angle n AF is equal to MNO, therefore, fince in the triangle AFy AF reprefents the preffure on the plane, K f the weight of the body, and F f the energy of the power, thefe magnitudes will alfo be reprefented in the fimilar triangle MNO by the fides MN, MO, NO. Fig. 6. 77* Cor. 6. If a power P and weight W are in equilibrio upon two inclined planes AB, AC *, P : W=r AB : AC. Letp be the power, which a6ling on the weight W in a direflion parallel to the plane would keep it in equilibrio, then we have p : W~AD : AC ; but fince the firing is equally firetched at every point, the fame power p will alfo fuftain the power P, confe- quently P : AB : AD, and by compofition P : W r=AB : AC. Prop. II. 78. If a fpherical body is fupported upon two in¬ clined planes, the preffures upon thefe planes will be inverfely as the fines of their inclination, while the abfolute weight of the body is repre¬ fented by the fine of the angle formed by the two planes. Fig. 7. Let AC, BC be the two inclined planes, and F the 4 fpherical body which they fupport. The whole of its matter being fuppofed to be collected in its centre of gravity F, its tendency downwards will be in the ver¬ tical line 10. The reaction of the planes upon F is .evidently in the direction MF, NF perpendicular to the furface of thefe planes, and therefore we may con- fider the body F as influenced by three forces acting in the directions FC, FM, FN ; but thefe forces are re¬ prefented by the fides of the triangle ABC perpendicu¬ lar to their directions, (Dynamics, § 144.), confe- quently the abfolute weight of the body F, the prefllire upon the plane AC, and the preflure upon the plane BC, are refpeCtively as AB, AC, and BC, that is, as the fines of the angles ACD, ABC, BAC, for in every triangle the fides are as the fines of the oppofite angles, or, to expre.fs it in fymbols, W being the abfolute weight of the body, w the preflure on AC, and w' the preffure on BC, Theory. W : ‘m : it/—AB : AC : BC, or W : io : lo'zrfin. ACB : fin. ABC : fin. BAC. But on account of the parallels AB, DF, the angle ABCzrBCF, and BAC—ACD, therefore the pref¬ fures upon the planes are inverfely as the fines of their inclination, the abfolute weight of the body being re¬ prefented by the fine of the angle formed by the fur- faces of the two planes. 79. Cor. I. Since the two fides of a triangle are Corollaries greater than the third, the fum of the relative weights fupported by the two planes is greater than the abfolute weight of the body. 80. Cor. 2. If the inclination of each plane is 6o°, then ACB mult alfo be 6o°, and the triangle ABC equilateral, confequently the prelfure upon each plane is equal to the abfolute weight of the body. 81. CoR. 3. When the inclination of each plane increafes, the preffure which each fufiains is alfo in- creafed ; and when their inclination diminilhes till it almoft vanilhes, the preflure upon each plane is one half of the abfolute weight of the body F. Prop. III. 82. If a body is raifed with an uniform motion along an inclined plane, the velocity of the power is to the velocity of the weight as the weight is to the power. Let the weight W be drawn uniformly up the in- p;^ dined plane AB, from B to D, by a power whtife di¬ rection is parallel to DH. Upon DB defcribe the circle BFEDN, cutting BC in E, and having pro¬ duced HD to F, join FP, FB, FE, and draw DC per¬ pendicular to BD. Now the angles BFD, BED are right (Geometry, Se£t. II. Theor. 17.), and there¬ fore, though the power moves through a fpace equal to BD, yet its velocity in the direction DH is mealured by the fpace FD uniformly defcribed ; and for the fame reafon, though the weight W defcribes the fpace BD, yet its velocity in the direction in which it ads, that is, in a vertical direction, is evidently meafured by the fpace DE uniformly defcribed. Then becaufe the triangle DEE is equal to DFE, (Geometry, SeCt. II. Theor. 15.) and DBErrDCH, (Geometry, Sed. IV. Theor. 23.) and FDE=DHC, (Geometry, Sed. I. Theor. MECHANIC S. 6i Thwy. 21.) the triangles DT'E, DHC are fimilar, and (Geo- »■■ ■ v 1 ^ metry, Se£t. IV. Theor. 20.) DF : DE“DH : HC. But DH : HCrdin. DCId : fin. HDC, that is, (art. 71.) DF : DE, or the velocity of th# power to the ve¬ locity of the weight, as W : P. Q. E. D. Scholium. 83. The inclined plane, when combined with other machinery, js often of great ufe in the elevation of weights. It has been the opinion of fome writers, that the huge maffes of ftone which are found at great alti¬ tudes in the fplendid remains of Egyptian architecture, were railed upon inclined planes of earth, with the aid of. other mechanical powers. This fuppofition, how¬ ever, is not probable, as the immenfe blocks of granite which compofe the pyramids of Egypt could not pof- fibly have been railed into their prelent fituation by any combination of the mechanical powers with which we are acquainted.—The inclined plane has been very advantageoufiy employed in the duke of Bridgewater’s canal. After this canal has extended 40 miles on the fame level, it is joined to a fubterraneous navigation about 12 miles long by means of an inclined plane, and this fubterraneous portion is again connected by an inclined plane with another fubterraneous portion about 106 feet above it. This inclined plane is a lira turn of ftone which Hopes one foot in four, and is about 453 feet long. The boats are conveyed from one portion of the canal to another by means of a windlafs, fo that a loaded boat defeending along the plane turns the axis of the windlafs, and raifes an empty boat.—A pair of flairs, and a road that is not level, may be regarded as inclined planes 5 and hence it is a matter of great importance in carrying a road to the top of a hill, to ckoofe fuch a line that the declivity may be the leaft pofiible. The additional length, which, in order to effeCt this purpofe muft fonaetimes be given to the line of road is a trifling inconvenience, when compared with the advantages of a gentle declivity. Sect. III. 0,v the Rope Machine. 84. Definition. When a body fufpended by two or more ropes, is fuftained by powders which aft by the afliftance of thefe ropes, this aifemblage of ropes is called a rope machine. DE : BErrfin. DBE : fin. BDE, and on account of the parallels DE, AB, the angle BDE—ABD, con- fequently P : pzrfin. DBE : fin. BDE. 86. Cor. i. When the line joining the pulleys is horizontal, as AC, then P : 7?—EC : FA, for FC and FA are evidently the fines of the angles DBE, BDE. 87. Cor. 2. Any of the powers is to the weight, as. the fine of the angle which the other makes with the direftion of the weight, is to the fine of the angles which the power makes with one another. For fince DB reprefents the weight, and BE the power P, we have BE : BD—fin. BDE : fin. BED •, but on account of the parallels DE, AB, the angle DEB — ABC, the angle made by the direftion of the powrers, confequent- ly BE : BD, that is, p : W= fin. ABF : fin. ABC. In the fame way it may be fhown that P : W—fin. CBF : fin. ABC. Hence we have P-\-p : W— fin. CB.b ft-fin. ABF : fin. ABC, that is, the fum of the powers is to the weight, as the fum of the fines of the angles which the powers make with the direftion of the weight is to the fine of the angle which the powers make with one another. 88. Cor. 3. The two powers P, /J, arc alfo direftly proportional to the cofecants of the angles formed by the direftion of the powers with the direftion of the weight. For fince P : prrfin. DBE : fin. BDE, and by the principles of trigonometry, fin. DBE : fin. DBE —cofec. BDE : cofec. DBE, we have P : y^nrcofec. ABl1 : cofec. CBF. It is alfo obvious that P : jo as the fecants of the angles which thefe powers form with the horizon, fince the angles which they make with the horizon are the complements of the angles which they form with the direftion of the weight, and the cofe- cant of any angle is juft the fecant of its complement, therefore P : jorzfec. BAF : fee. BCF. Theory. Chap. II. On Compound Machines. 89. Definition. Compound machines are thofe which are compofed' of two or more fimple machines, either of the fame or of different kinds. The number of compound machines is unlimited, but thofe which properly belong to this chapter, are, 1. The wheel and axle ; 2. The pulley 5 3. The wedge ; 4. The ferew j and, 5. The balance. Prop. I. 85. If a weight is in equilibrium with two powers adfing on a rope machine, thefe powers are in- verfely as the fines of the angles which the ropes form with the diredfion of the weight. Let the weight W be fufpended from the point B, where the ropes AB, BC are joined, and let the powers P, p afting at the other extremities of the ropes which pafs over the pulleys A, C, keep this weight in equilibrio, we ftiall have P : fin. CBD : fin. ABD. Produce WB to F, and let BD reprefent the force exerted by W •, then by drawing DE parallel to AB, the fides of the triangle BDE will reprefent the three forces by which the point B is folicited (Dynamics, $ M4*)> f°r CB are the direftions of the forces P and p. We have therefore P : p—DE : BE 3 but Sect. I. On the Weed and Axle. 90. The ’wheel and axle, or the axis in peritrochio, Fig. i®, is reprefented in fig. 9. and confifts of a wheel AB, and cylinder CD having the fame axis, and moving upon pivots E, F, placed at the extremity of the cylinder. The power P is moft commonly applied to the circum¬ ference of the wheel, and afts in the direftion of the tangent, while the weight W is elevated by a rope which coils round the cylinder CD in a plane perpen¬ dicular to its axis.—In this machine a winch or handle EH is fometimes fubftituted inftead of the wheel, and fometimes the power is applied to the levers S, S fixed in the periphery of the wheel 3 but in all thefe forms the principle of the machine remains unaltered. I hat the wheel and axle is an affemblage of levers will be obvious, by confidering that the very fame effeft would be produced if a number of levers were to ra¬ diate 62 M E C H , 'ni('orv- diate from tlie centre C, and if a rope carrying the v power P were to pafs over their extremities, and i xtri- cate itfelf from the defeending levers when they come into a horizontal pofition. 91. Axiom. The effeft of the power to turn the-cy¬ linder round its axis, is the fame at whatever point in - the axle it is fixed. I Prop. I. 92. In the wheel and axle the power and weight will be in equilibrium, when they are to one another reciprocally as the radii of the circles t<* which they are applied, or when the power is to the weight as the radius of the axle is to the radius of the wheel. Fig, 11. Let AD be a fedtion of the wheel, and BE a fee* tion of the axle or cylinder, and let the power P and weight W a£t in the diredtions AP, WP, tangents to the circumferences of the axle and wheel in the points A, B, by means of ropes.winding round thefe circum¬ ferences. As the effedt is the fame according to the axiom, let the power and weight adt in the fame plane as they appear to do in the figure, then it is obvious that the effort of the power P and weight W will be the fame as if they were fufpended at the points A, B 5 confequently the machine may be regarded as a lever AFB, whofe centre of motion is F. But fince the di- redlions of the power and weight make equal angles with the arms of the lever, we have (Art. 36.) P : W r=:FB : FA, that is, the power is to the weight as the radius of the axle is to the radius of the wheel. ®orollaries. 93. Cor. 1. If the power and weight adl obliquely to the arms of the lever in the diredlions A />, B “u;, draw F iT* F « perpendicular to A/j and B to, and as in the cafe of the lever (Art. 51.) there will be an equili¬ brium when P : W“F n :¥ m. Hence the tangential diredlion is the moil advantageous one in which the power can be applied, for FA is always greater than F m, and the leaft advantageous diredlion in which the weight can be applied, for it then oppofes the greateff refinance to the power. 94. Cor. 2. If the plane of the wheel is inclined to the axle at any angle x, there will be an equilibrium when P : W— femidiameter of the axle : fin. x. 95. Cor. 3. When the thicknefs of the rope is of a fenfible magnitude, there will be an equilibrium when the power is to the weight as the fum of the radius of the axle, and half the thicknefs of its rope, is to the fum of the radius of the wheel and half the thick¬ nefs of its rope 5 that is, if T be the thicknefs of the rope of the wheel, and t the thicknefs of the rope of the axle, there will be an equilibrium when P : Wrr FB : FA-f iT. 96. Cor. 4. If a number of wheels and axles are fo •ombined that the periphery of the firft axle may adt on the periphery of the fecond wheel, either by means of a firing or by teeth fixed in the peripheries of each, and the periphery of the fecond axle on the periphery of the third wheel, there will be an equilibrium when the power is to the weight as the produdl of the radii •f all the axles is to the produdl of the radii of all the srheels. This corollary may be demonftrated by the I A N I C S. fame aeafoning which is ufed in Art. 63. for the com- Tfieorr, binauon of Levers. ^ " y—** 97. Cor. 5. In a combination of wheels, where tl e motion is commifnicated by means of teeth, the axle is called the pinion. Since the teeth therefore muff be nearly of the fame fize, both in the wheel and pinion, the number of teeth in each will be as their circum¬ ferences, or as their radii; and confequently in the com¬ bination mentioned in the preceding corollary, the power will be to the weight, in the cafe of an equili¬ brium, as the produdt of the number of teeth in all the pinions is to the produdl of the number of teeth in all the wheels. Prop. II. 98. In the wheel and axle the velocity of the weight is to the velocity of the power as the power is to the weight. If the power is made to rife through a fpace equal to the circumference of the wheel, the weight -will evi¬ dently deferibe a fpace equal to the circumference of the axle. Hence, calling V the velocity of the power, v that of the weight, C the circumference of the wheel, and c that of the axle, we have V : v~ C : c. But by the propofition P : W=rc : C, therefore P : : V. Scholium. 99. The conflrudlion of the main-fpring box of the On the fufee of a watch round which the chain is coiled, is a fufce a beautiful illullration of the principle of the wheel and watc*b axle. The fpring-box may be confidered as the wheel, and the fufee the axle or pinion to which the chain communicates the motion of the box. The power re- fides in the fpring wound round an axis in the centre of the box, and the weight is applied to the lower cir¬ cumference of the fufee. As the force of the fpring is greateil when it is newly wound up, and gradually de- creafes as it unwinds itfelf, it is neceffary that the fufee fliould have different radii, fo that the chain may adl upon the fmallell part of the fufee when its force is greateil, and upon the largell part of the fufee when its force is lealt, for the equable motion of the watch requires that the inequality in the adtion of the fpring fhould be counteradled fo as to produce an uniform ef- fedl. In order to accomplilh this, the general outline of the furface of the fufee mull be an Apollonian hyper¬ bola in which the ordinates are inverfely as their re- fpedlive abfeiffse. For further information on this fub- jedl, fee Recherches des Mathemat. par M. Parent. tom. ii. p. 678. 5 Traite d'Hor/ogerie, par M. Berthoud, tom. i. chap. 26. ; and Traite de Mecanique, par M. de la Hire, prop. 72. Sect. II. On the Pulley. 100. Definition.—The pulley is a machine com-On the pofed of a wheel with a groove in its circumference, pulley, and a rope which paffes round this groove. The wheel moves on an axis whofe extremities are fupported on a kind of frame called the block, to which is generally fufpended the wreight to be raifed. A fyllem of pulley* is called a muffle, which is either fixed or moveable ac¬ cording as the block which contains the pulleys is fixed or moveable. Prop. MECHANICS. Theory. Prof. I. Ioi. In a fingle pulley, or fyItem of pulleys where the different portions of the rope are parallel to each other, and where one extremity of it is fixed, there is an equilibrium when the power is to the weight as unity is to the number of the portions of the rope which fupport the weight. Pig. 12. F‘g- 13- Fig. 14. 15. 102. Case i. In the fingle fixed pulley A A let the power P and weight W be equal, and act againft each other by means of the rope PBAW, paffmg over the pulley A A ; then it is obvious that whatever force is exerted by P in the direftion PBA, the fame force muft be exerted in the oppofite direction WBA, con- fequently thefe equal and oppofite forces muft be in equilibrio •, and as the weight is fupported only by one rope, the propofition is demonftrated, for P : W— 1 : 1. 103. Case 2. In the fingle moveable pulley, where the rope, faftened at H, goes beneath the moveable pulley D and over the fixed pulley C, the weight to be raifed is fufpended from the centre of the pulley D by the block /?, and the power is applied at P in the dire£tion PE. Now it is evident that the portions CFjo, HGD of the rope fuftain the weight W, and as they are equally ftretched in every point, each muft fuftain one half of W •, but (Cafe 1.) in the fingle pulley C the rope CEP fuftains a weight equal to what the rope CFp fuftains *, that is, it fuftains one-half of W. Con- fequently P~FW, or W—2P, when there is an equi¬ librium ; and fince the weight W is fupported by two firings, we have P : W~ 1 : 2. 15. 104. Case 3. When the fame rope paffes round a number of pulleys, the ropes which fupport the weight W are evidently equally ftretched in every part, and there¬ fore each of them fuftains the fame weight. Confe- quently if there be ten ropes fupporting the weight, each fuftains -/^th part of the weight, and therefore P—to W, or W—10P, which gives us P : W— 1 : 10. —T he pulley in fig. 15. is the patent pulley invented by Mr White, in which the lateral friftion and fhaking motion is confiderably removed. Prop. II. Prop. III. Theory. 106. In a fyflem of moveable pulleys whofe num¬ ber is «, fufpended by feparate and parallel ropes, whofe extremities are fixed to the weight W, there is an equilibrium when P : W : 1 : 2” —1. In this fyftem of pulleys, the rope which fuftains the pjg lJs power P pafles over the pulley C, and is fixed to the weight at O. Another rope attached to the pulley C paftes over the pulley B and is fixed to the weight at E, and a third rope iaftened to B paifes over A and is fixed at F. Then it is manifeft that the rope CD furtains a weight equal to P j and finee the pulley C is pulled downward with a weight equal to 2P, the rope BC muft fupport a weight equal to 2P, and the rope B the fame weight j confequently the rope AB fuitains 4P. The whole weight therefore is P + 2P-f4P, and hence P : W=:P : P-j-^P-j-qP, or P : W=i : 1 4-2+4 &c. to « terms, fo that P : W = i : 2„—j. Prop. IV. 107. In the fyftem of pulleys reprefented in Fig. 1-. fig. 19. and called a bpamlh barton, in which two pulleys are fupported by one rope, there is an equilibaum when P : W = i : 4. In this combination of pulleys, the rope AB which fupports the power P paiies over the moveable pulley A, and beneath C towards H, where it is faxed. Ano¬ ther rope, attached to the pulley A, paftes over the fixed pulley B, and is faftened at E to the pulley C, which fupports the weight W. Then, finee the rope AP fup¬ ports 1 pound, the rope AC alio fupports 1 pound, and therefore the pulley A, or the rope BA, is pulled down with a force of 2 pounds. But the rope BDE is equally ftretched with BA, confequently the pulley C, to which DE is attached, is pulled upwards with a force of 2 pounds. Now the rope AC lupporting 1 pound, the rope GH muft likewile I'upport 1 pound, confequently, fince DE fuftains 2 pounds, AC 1 pound, and HG 1 pound, they will together fuftain V\ =4 pounds, and therefore P : ^ = 1 : 4, J05. In a fyftem of « moveable pulleys fufpended by feparate and parallel ropes, there is an equi¬ librium when P : W - 1 : 2* ; that is, if there are 4 pulleys «—4, and P : W= 1 : 2 x 2 X 2 x 2, or P : W=i : 16. rig. 17. This fyftem is reprefented in fig. 17. where the rope which carries the power P pafles over the fixed pulley M, and beneath the moveable pulley A, to the hook E "where it is fixed. Another rope fixed at A pafles over B and is fixed at F, and fo on with the reft. Then by Art. 103. P : the weight at A= r : 2 The weight at A : the weight at B—1 : 2 The weight at B : the weight at Cm : 2 The weight at C : the weight at D or W=x : 2; and therefore by compofition E : W=I ; 2x 2X 2x 2 or P : Wm : 16. £. E. d. Prop. V. 108. In the fyftem of pulleys reprefented in fig.F;„ m 20. called a Spaniih barton, where two pulleys are fupporteu by one rope, tnere is an equili¬ brium when P : W=:i : 5. In this fyftem the rope PB pafles over B round C and is fixed at E. Another rope attached to B paiies round AF and is fixed at i to the pulley CD, vvhich carries the weight W. i\iow the rope BP bemg itretched with a force of 1 pound, the ropes BGC, CDE are alfo ftretched with a force of 1 pound each, and the pulley CD is pulled upwards with a force of 2 pounds. But linee the three ropes BP, ED, and GC, are each ftretched with a force of 1 pound, the pulley B and the rope BA, upon which they all ad in one diredion, mull be pulled down with a force of 3 pounds. Now the mpe IT equally ftretched with BA, confequently it, will draw the pulley CD upwards with a force of 3 pounds, MECHANIC pounds, and 0nee it is drawn upwards by the ropes CG, DF, with a force of two pounds, the whole force will fuftain W= 5 pounds 5 but this force of 5 pounds is by the hypotheiis in equilibrio with P or I pound, confequently P : W=ri : 5. s. P : ^rrrad. p : tt—rad. 9r : W~rad. 2Cof. MAP 2cof. NBA 2cof. RGB, confequently P : W=rad. : 2 cof. MAP x 2 oof. NBA Xi^OtCB, or, which is the fame thing, Theory. Prop. VI. Plate eccxix. %, 1. 109. When the ropes are not parallel, and when two powers are in equilibrxo with a weight by means of a pulley, and have their dire£Uons at equal angles to the dire&ion of the weight, each of thefe powers is to the weight as the radius of the pulley is to the chord of that por¬ tion of the pulley’s circumference with which the rope is in contact. Let the weight W fufpended from C be fuftained in equilibrio by two powers P, p, which a& by a rope PCFEp paffing over the pulley CHEF, and touching the arch CFE of its circumference. Then fince the angles PWD,/;WD are equal, and the powers P, p in equilibrio, P mull be equal to p ; and making W A =WB, and drawing AI parallel to PW, and BI pa¬ rallel toPW WB, BI, WI will refpeflively repre- fent the forces P,/?, W or P : jo : W=WB : BI : WI, Dynamics, Art. 144. Now the triangles WBI, CDE having their refpe&ive fides at right angles to each other, are fimilar j confequently WB : BI : WI=CD : DE : EC, that is, P : : W=CD : DE : EC 5 but CD, DE are equal to radius, and EC is obvioully the chord of the arch CEE, therefore P : W or p : W as radius is to the chord of the arch with which the rope is in contact. , . no. Cor. t. Any of the powers is alfo to the weight as radius is to twice the eofme of the angle which either rope makes with the dire&ion of the weight. For fince CG is the cofine of DCG, and fince CE is double of CG, CE is equal to 2 cofine DCG — 2 Cof. PWD but P : W=:CD : CE, hence we have By fubftituting the preceding value of CE, P ; W=CD ©r radius : 2 Cof, PWD, Scholium. in. By means of this propofition and corollary, the proportion between the powers and the weight in the various fyftems of pulleys, reprefented in fig. 12, 13, 14, 15, f 6, 17, 18, 19, 20. when the ropes are not parallel, may be eafily found. Prop. VII. 112. In a fyftem of moveable pulleys, where each has a feparate rope, and wdtere the ropes are not parallel, there is an equilibrium when the power is to the weight as radius is to the cofines of half the angles made by the rope of each pulley, multiplied into that power of. 2 whofe exponent is the number of pulleys. F. Let the power P fuftain the weight W by means of ^ the pulleys A, B, C} let P, p, w be the different powers which fupport the pulleys A, B, C, and let MAP, NBA, RGB be the angles formed by the ropes. Then, by the laft propofition, P : W—rad. : 2X 2X 2 x cof. MAP X cof. NBAx cof. RGB. Prop. VIII. 113. In a fingle pulley, or in a combination of pulleys, the velocity of the power is to the ve¬ locity of the weight as the weight is to the power. 114. Case i. In the fingle fixed pulley, it is ob-pig, vious, that if the weight W is raifed uniformly one inch, the power D will alfo defcribe one inch, confe¬ quently velocity of P : velocity of W~W : P. 115. Case 2. in the fingle moveable pulley, when Fig. 15, the weight W is raifed one inch, the ropes become one inch ftiorter } and fince the rope has always the fame weight, the power muft defcribe two inches, therefore velocity P : velocity W~W : P. 116. Case 3. In the combination of pulleys, in Figs 14, i< figs. 14, 15, 16, when the weight rifes one inch, each 16. of the four ftrings becomes an inch Ihorter, fo that P muft defcribe four inches, as the length of the rope is invariable j confequently velocity P : velocity W =: W : P. 117. Case 4. In the fyftem exhibited in fig. 17. it Fig. is evident, that when the weight W rifes one inch, the rope DC is lengthened two inches, the rope CB four inches, the rope BA eight inches, and the rope AFP, to which the power is lulpended, 16 inches } fo that lince the power of this pulley is as 16 to j, we have velocity P : velocity W=W : P. 118. Case 5. In the combination of pulleys, repre- Fig. 18. fented in fig. 18. when the weight W rifes one inch, all the three ropes CD, BE, AF are each ihortened one inch. But while CD fhortens one inch, CP be¬ comes one inch longer; while BE Ihortens one inch, BC becomes one inch longer, and CP two inches longer (art. 110.) j and while AF fhortens one inch, AB be¬ comes one inch longer, BC two inches longer, and CP four inches longer j therefore CP is lengthened al¬ together feven inches, and as the power of the pulley is as 7 to 1, we have, as before, velocity P : velocity W=W ; P. 119. Case 6. In the fyftem of pulleys, called the Fig. Spanilh barton, fig. 19. when the weight W rifes one inch, the three ropes AC, DE, HG are each Ihorten¬ ed one inch. By the fhortening of HG, CA one inch each, the rope AP is lengthened two inches ; and by the ftiortening of DE one inch, BA is lengthened one inch, and AP two inches (art, 115.) $ confequently, fince AP is lengthened in all four inches, and fince the power of the pulleys is four, we have velocity P : velo¬ city W=W ; P. 120. Case 7. In the other Spanifti barton, in fig, 20, Fig. 2*. when the weight is elevated one inch, the three ropes DE, IF, CG are each one inch (horter. While ED, and CG fhorten one inch each, BP is lengthened two inches^ M E C H Theory, inches, and wlille IF becomes one inch fhorter, AB be- ^ 1 comes one inch longer'; but when AB is lengthened one inch, BP becomes one inch longer, and ED, CG one inch fhorter each, and by this Ihortening of ED, CG, the rope B is lengthened two inches, therefore, finee the rope BP is lengthened altogether five inches, and fince the pulleys have a power of five, we have, as formerly, velocity P ; velocity Vv zr W : P. Sect. III. On the Wedge. 121. Definition. A wedge is a machine compofed ®f two inclined planes with their bafcs in contad ; or, more properly, it is a triangular prifm, generated by the motion of a triangle, parallel to itfelf, along a ftraight line pa ding through the vertex of one of its angles. The wedge is called ifofceles, rectangular, Plate or fcalene, according as the triangle ABC by which CtCXIX the wedge is generated, is an ifofceles, a redangular 3' er a fcalene triangle. The part AB is called the head or back of the wedge, DC its altitude, and AC, BC its faces.—The wedge is generally employed for cleaving wood, or for quarrying ftones but all cutting inilruments, fuch as knives, fwords, chifels, teeth, &c. properly belong to this mechanical power, when they ad in a diredion at right angles to the cutting furface j for when they ad obliquely, in which cafe their power is inereafed, their operation refembles more the adion •f a faw. Prop. I. 122. If each of the faces of an ifofceles wedge, which are perfe&ly fmooth, meet with an equal refiflance from forces a£Hng at equal angles of inclination to their faces, and if a power ad: perpendicularly upon the back, thefe forces will be in equilibrio, when the power upon the back is to the fum of the refiltances upon the fides, as the fine of half the angle of the wedge, multiplied by the fine of the angle at which the refilling forces ad upon its faces, is to the fquare of radius. Fig. g. 'h,et ABC be the wedge, AC, BC its ading faces, and iVID,ND the diredions in which the refilling forces ad upon thefe faces, forming with them the equal angles DMA, DNB. Draw CD, DF, DE at right angles to three fides of the wedge, and join F, E meeting CD in G. On account of the equal triangles CAD, CDB (Euclid, Book i. Prop. 26.) AD=rDB ; and in the equal triangles ADM, BDN, MDrrND. In the fame way DF—DE and AF=BE, therefore CF=CE. But in the triangles CFG, CEG there are two fides FC, CG equal to EC,CG, and the angle FCGrrECG, con- fequently FG —GE, and FGC, ABC are both right angles,, therefore FE is parallel to AB.—Now the force Mi) is refolvable into DF, EM, of which FM has no elfed upon the wedge. But, as the effedive force FD is net in dired oppofition to the perpendicular force ex¬ erted on the back of the wedge,, we may refolve it into the t wo forces FG, GD, of which GD ads in dired op¬ pofition to the power, while FG ads in a diredion paral¬ lel to the back of the wedge. In the fame way it mav be Ihewn that EG GD are the only effedive forces which Vol. XIII. Part I. A N I C S. 65, refult from the force ND. But the forces FG EG be- Theory. ing equal and oppolite, deliroy each other , c, rncquent- ''■“"'v ’ ly 2GD is the force which oppoles that which is exert¬ ed upon the back of the wedge, and the wedge will be kept at reft if the force upon the back is equal to 2GD, that is, when the force upon the back is to the fum of the refinances upon the faces as 2Gi> is to MD-j-ND, or as 2GD : 2DM, or as Gd is to DM. Now DG : DF — fin. DFG : radius, or as (Euclid, vi. 8.} fin. DCF radius, and DF : MD—fin. DMF : radius; therefore by compofi* fition, DG : MD — fin. DCF x fin- DMF : rad. x rad. or rad.J1. But, DG : MD as the force upon the back is to the fum of the refiftanees, therefore the force up¬ on the back is to the fum of the refinances as fin. DCF X fin. DMF is to the fquare of the radius. I23* Cor. i. If the diredion of the refifting forces Corollaries, is perpendicular to the faces of the wedge, DMF be¬ comes a right angle, and therefore its fine is equal to radius. Confequently we have, in this cafe, the force upon the back to the fum of the refiftances, as fin. DCF X rad. is to radipsj1, that is, as fin. DCF is to radius, or as AD half the back of the wedge is to AC the length of the wedge. 124. Cor. 2. In the particular cafe in the propofi- tion, it is obvious that the forces MF, NE are not op- pofed by any other forces, and therefore the force upon the back will not fuftain the refifting forces ; but in the cafe in cor. 2. the forces MF, NE vanilh, and there¬ fore the other forces will fuftain each other. 125. Cor 3. If the refifting forces ad in a direc¬ tion perpendicular to AB, the angle DMF becomes equal to ACD, and therefore the force upon the back is to the fum of the refiftances as fin. ACDj* is to radius|*, that is, as the fquare of AD half the back of the wedge is to the fquare of AC the length of the uredge. 1 26. Cor. 4. When the diredion of the refiftances is parallel to the back of the wedge, the angle of in¬ clination DMC becomes the complement of the ferni- angle of the wedge, and therefore the force upon the back is to the fum of the refiftances as the fin. ACD X col. ACD is to the fquare of the radius, that is, as DA X DC is to AC*. But in the fimilar triangles DAF, DAC, we have DF : DArrDC : AC, and DF X ACnrDA X DC, confequently the force upon the back of the wedge is to the fum of the refiftances as DFx AC is to AC*, that is, as DF : AC. Prop. II. 127. If, on account of the fridlion of the wedge, pig or any other caufe, the refiftances are wholly effedlive, that is, if the refifting furfaces adhere to the places to which they arc applied without Aiding, there will be an equilibrium, when the force upon the back is to the lum of the refin¬ ances, as the fine of the acute angle which the diredlion of the refifting forces makes with the back of the wedge is to radius. Join MN, which will cut DC perpendicularly at the I" point 66 , M E C H , Thcory- . point H. 1'hen, lince the forces MD, ND are refolvable into MH, HD and into NH, HD, and fince MH, HN deftroy each other, the force upon the back is fuftained by 2 HD. Confequently, the force upon the back is to the fum of the refiltances as 2 HD is to 2 MD, or as HD is to MD. But the angle ADM, which the direc¬ tion of the forces makes with the back of the wedge, is equal to DMN, and HD is the fine of that angle, MD being radius, therefore the force upon the back is to the fum of the refinances as fin. ADM : radius. Q. E. D. Corollaries. 128.C0R. I . Since the angle AMD=:MDC-fMCD, the angle MDC is the difference between MCD the fe- miangle of the wedge, and AMD the angle which the direction of the refilling forces makes with the face of the wedge, and fince HD is the cofine of that angle, MD being radius, we have the force upon the back to the fum of the refiftances, as the cofine of the difference be¬ tween the femiangle of the wedge and the angle which the direction of the refilling forces makes with the face of the wedge, is to radius. • Prop. III. 129. When there is an equilibrium between three forces a£ting perpendicularly upon the Tides of a wedge of any form, the forces are to one an¬ other as the lides of the wedge. This is obvious from Dynamics, $ 144. Cor. 2. where it is fhewnthat when three forces are in equilibrio, they are proportional to the fides of a triangle, which are refpedlively perpendicular to their directions. Prop. IV. 130. When the power adling upon the back of a wedge is in equlibrio with the refiftances op- pofed to it, the velocity of the power is to the velocity of the refiftance as the refiftance is to the power. Tig. 3. Produce DM to K, and draw CK perpendicular to DK. Then, by Art. 122. the power is to the refin¬ ance as MD : DPI. Let the wedge be moved uni¬ formly from D to C, and DK is the fpace uniformly defcribed by the refilling force in the direftion in which it acts j therefore, the velocity of the poAver is to the velocity of the refiftance as DC : DK ; that is, on ac¬ count of the equiangular triangles DHM, DKC, as MD : DH 5 that is, as the refiftance is to the power. Sect. IV. On the Screw. 131. Definition. A fcrerv is a cylinder with an inclined plane AATapped round it, in fueh a manner, that the furface of the plane is oblique to the axis of the cy¬ linder, and forms the fame angle with it in every part of the cylindrical furface. When the inclined plane winds round the exterior furface of a folid cylinder, it is called a male fcrew ; but Avhen it is fixed on the in¬ terior circumference of a cylindrical tube, it is called a female fcrew. In the female lerew, the fpiral grooves formed by the inclined plane on the furface of the cy¬ lindrical tube, mull be equal in breadth to the inclined A N I C S. plane in the male ferew, in order that the one may Theory. move freely in the other. By attending to the mode ' * in which the fpiral threads are formed by the circum¬ volution of the inclined plane, it will appear, that if one complete revolution of the inclined plane is deve¬ loped, its altitude will be to its bafe as the diftance be- tiveen the threads is to the circumference of the fcreAv. 1 hus, let a b c (fig. 4.) be the inclined plane, whofe Fig. 4. bafe is a c and altitude b c, and let it be wrapped round the cylinder MN (fig. 5.) of fuch a fize that the points a, c may coincide. The furface ab of the plane (fig. 4.) will evidently form the fpiral thread a d e b (fig. 5.), and ab the diftance betAA^een the threads Avill be equal to be (fig. 4.) the altitude of the plane, and the circumfer¬ ence of the fcreAv MN Avill be equal to ac the bafe of the plane. If any body, therefore, is made to rife along the plane a deb in fig. 5. or along the fpiral thread of the fcreAv, by a force acting in a direction parallel to a deb, there Avill be the fame proportion betAveen the poAver and the refiftance as if the body afeended the plane abc (fig. 4.). 132. A male fcrew Avith triangular threads is repre-pj 6 „ fented by AB (fig. 6.), and its correfponding female fcreAv by AB (fig. 7.). A male fcrew Avith quadrangu¬ lar threads is exhibited in fig. 8. and the female fcreAv Fi g in which it works in fig. 9. The friftion is confidera- lg’ ’ 9 bly lefs in quadrangular than in triangular threads, though, when the fcrew is made of Avood, the triangular threads iliould be preferred. When the ferews are me¬ tallic and large, the threads fliould be quadrangular; but the triangular form is preferablein fmall fcreAvs. When the fcreAv is employed in practice, the porver is always applied to the extremity of a lever fixed in its head. This is ftiCAvn in fig. 10. where AB is the lever afting upon the fcrew BC, Avhich works in a female fcrew in Ir the block F, and exerts its force in bending the fpring CD ». Prop. I. 133. If the fcrew is employed to overeome any refiftance, there will be an equilibrium when the power is to the refiftance as the diftance be¬ tween two adjacent threads is to the circum¬ ference defcribed by the power. Let FAKbe a fection of the fcrew reprefented in fig. Fig. 11.. 8. perpendicular to its axis 5 CD a portion of the inclined plane Avhich forms the fpiral thread, and P the porver, Avhich, when applied at C in the plane ACF, will be in equilibrium with a weight upon the inclined plane CD. Then, in the inclined plane, Avhen the direction of the poAver is parallel to the bafe, avc have (Art. 72.) P: VV, as the altitude of the plane is to the bafe, or (Art. 131.) as the diftance betAveen two threads is to the Avhole circumference FKCF. If avcfuppofe another poA\rer PTo aft at the end of the lever AB, and deferibe the arch HBG, and that this poAcer produces the fame effeft at B as the power P did at C, then (Art. 36.), avc lurve P': P=:CA : BA, that is, as FKCF is to the circumfe¬ rence HBG ; but it Avas fheAvn before, that P : W~ as the diftance between tA\To contiguous threads is to FKCF; therefore, by compofition, P: W as the diftance betAveen two threads is to HBG or the circumference of a circle whofe radius is AB. f). E. D. j34. Cor. I. It is evident from the propofition that the S M E C H Tlieory. the power does not In the lead: depend upon the fize of ~-v cyiincier FCK, but that it incieafes with the di- ftance of that point from the centre A, to which the power is applied, and alfo with the fhortnefs of the di- llance between the threads. Therefore, if P, p be the powers applied to two different fcrews, D, d the di- itances of thefe powers from the axis, and T, t the di- ftances between the threads; their energy in overcom¬ ing a given refiftance will be diredlly as their diftances fro'm the axis, and inverfely as the diftances (»f their , . , • T2 1) d . D threads, that is, r : ,p- : -, or P vanes as Prop. II. 135. In the endlefs ferew, there will be an equi¬ librium when the power is to the weight, as the diftance of the threads multiplied by the radius of the axle, is to the diltance of the power from the axis of the ferew multiplied by the radius of the wheel. Fig. 12. The endlefs ferew, which is reprefented in fig. 12. confifts of a ferew EF, fo combined with the wheel and axle ABC, that the threads of the ferew may work in teeth fixed in the periphery of the wheel, and thus com¬ municate the power exerted at the handles or winches P, p. Let W' repreient the power produced by the ferew at the circumference of the wheel 5 then, by the laft propofition, P : W7 as the diftance between the threads is to the diftance of P from the axis of the ferew •, but (Art. 92.) in the wheel and axle W7: Was the radius of the axle is to the radius of the wheel \ therefore, by compofilion, P : W as the diftances of the threads multiplied by the radius of the axle C, is to the diftance of the poirer P from the axis multiplied by the radius of the wheel A B. Prof. Ill, 136. When there is an equilibrium in the ferew, the velocity of the weight is to the velocity of the power, as the power is to the weight. r'£- ii- It is obvious from fig. 11. that while the power de- feribes the circumference of the circle HBG uniformly, the weight uniformly rifes through a fpace equal to the diftance between two adjacent threads ; therefore, the velocity of the power is to the velocity of the weight as the diftance between the threads is to the arch deferibed by the power, that is, (by Art. 133.), as the weight is to the power. Prop. IV. 137. To explain the conftru&ion and advantages * See Phil, of Mr Hunter’s double ferew *. Tranf. vot. Ixxi. p. 5s. Let the ferew CD work in the plate of metal BA, Fig. 13. and have n threads in an inch : the cylinder CD, of which this ferew is formed, is a hollow tube, which is alfo formed into a ferew, having «-f-1 threads in an inch, and into this female ferew is introduced a male ferew DE, having, of courfe, threads in an inch. The ferew DE is prevented from moving round with CD by the frame ABGF and the crofs bar a bt but is A N I C S. 67 permitted to afeend and defeend without a motion of Theory, rotation. Then, by a revolution of the ferew CD, " ’" the other ferew DE will rife through a fpace equal to and if the circumference deferibed by the //-f-i X« lever CK be m inches, we (hall have P : W— : «+iX« m ; or P : W~i : m « X « T 1 • 138. This reafoning will be more perfpicuous by fup- pofing «, or the number of threads in CD, to be 12, and or the number of threads in DE, will confe- quently be 13. Let us fuppofe that the handle CK is turned round 12 times, the ferew CD will evidently afeend through the fpaee of an inch, and if the ferew DE is permitted to have a motion of rotation along with CD, it will alfo advance an inch. Let the ferew DE be now moved backwards by 12 revolutions, it will evidently deferibe a fpace of 4f of an inch, and the confequence of both thefe motions will be that the point E is advanced Th- of an inch. But, fince DE is prevented from moving round with CD, the fame effeft will be produced as if it had moved 12 times round with CD, and had been turned 1 2 times backwards j that is, it will in both cafes have advanced t't of an inch. Since, therefore, it has advanced Tt °f an inch in 12 turns, it will deferibe only of Th-, or ^ of an inch uniformly at one turn ; but if the length of the lever CK is 8 inches, its extremity K will deferibe, in the fame time, a fpace equal to 16 X 3-1416=: ^0.2656 inches, the circumference of the circle deferibed by K j therefore the velocity of the weight is to the velocity of the power, as of an inch is to 50.2656 inches, or as 1 is to 7841.4336, that is, (Art. 136.) P : W =ri : 7841.4336. Hence the force of this double ferew is much greater than that of the common ferew, for a common one with a lever 8 inches long muft have I 56 threads in an inch to give the fame power, which would render it too weak to overcome any confiderable refiftance. 139. Mr Hunter propofes * to conned! with his* double fcrews, a wheel and a lantern, which are put in T/a/// vol. motion by a winch or handle. The power of this com-lxxi- P- 65* pound machine is fo great, that a man, by exerting a force of 32 pounds at the winch, will produce an effect of 172100 pounds 5 and if we fuppofe y of this effedt to be deftroyed by friction, there will remain an effedt of 57600 pounds.—In fome fcrews it would be advan¬ tageous, inftead of perforating the male ferew CD, to have two cylindrical fcrews of different kinds at differ¬ ent parts of the lame axis. Scholium. 140. The ferew is of extenfive ufe as a mechanical power, when a very great preffure is required, and is very fuccefsfully employed in the printing prefs. In the prefs which is ufed for coining money, the power of the ferew is advantageoufly combined with an impulfive force, which is conveyed to the ferew by the intervention of a lever. The ferew is alfo employed for raifing water, in which form it is called the ferew of Archimedes (Hydrodynamics, § 328)5 and it has been lately employed in the flour mills in America for pufhing the flour which comes from the millftones, to the end of a long trough, from which it is conveyed to other parts I 2 of 63 MECHANICS. Theory. Plate (2CCXX. Fig. x. of the maclimery, in order to undergo the remaining precedes. In this cafe, the fpiral threads are very large in proportion to the cylinder on which they are fixed. 141. As the lever attached to the extremity of the ferew moves through a very great fpace when compared with the velocity of its other extremity, or of any body which it puts in motion •, the ferew is of iramenfe ufe in fubdividing any fpace into a great number of minute parts. Hence it is employed in the engines for dividing mathematical inftruments, and in thofe which have been recently ufed in the art of engraving. It is likewife of great ufe in the common wire micrometer, and in the divided objeft-glafs micrometer, inftruments to which the fcience of aftronomy has been under great obliga¬ tions. See Micrometer. Sect. V. On the Balance. 142. Definition. The balance, in a mathematical fenfe, is a lever of equal arms, for determining the weights of bodies.—The phyfical balance is reprefented in fig. 1. where FA, FB are the equal arms of the ba¬ lance, F its centre of motion fituated a little above the centre of gravity of the arms, FD the handle which al¬ ways retains a vertical pofition, P, W the fcaies luf- pended from the points A, B, and CF the tongue or index of the balance, which is exadlly perpendicular to the beam AB, and is continued below the centre of motion, fo that the momentum of the part below F is equal and oppofite to the momentum of that part which is above it. Since the handle FD, fufpended by the hook H, muft hang in a vertical line, the tongue CF will alfo be vertical when its polition coincides with that of FD, and confequently the beam AB, which is perpen- IV. ffl-j-L x AO X Sin.AAO-}-S X OG X Sin.^=, dicular to CF, muft be horizontal. When this happens, the weights in the fcale are evidently equal. Prop. I. Theory. 143. To determine the conditions of equilibrium Fig. 2> in a phyfical balance. Let AOB be the beam, whofe weight is S, and let P, be equal weights exprefied by the letter /x, and placed in the fcaies, whofe weights are L and /. Let O be the centre of motion, and g the centre of gravity of the whole beam, when unloaded j we fhall have in the cafe of an equilibrium, LpHh L X K.Q, — p-\-lxBC + SxCc; for fince S is the weight of the beam and g its centre of gravity, its mechanical energy in afting againft the weights /x-f L is =5 X Cc, the diftance of its centre of gravity from the vertical line palling through the centre of motion o. II. But fince ACrrEC', /xxAC—^xBC~c. Then, after tranfpofition, take this from the equation in N° I. and w7e fhall have, III. /xBC—Lx AC + SxCc; or L—1~ 8 X Cc Let placed points of fufpenfion will be no longer horizontal, but will affume an inclined pofition. Let BA/=:

,b A. the fines of the angles F B, F £ A, to which they are proportional, and alfo by taking inftead of F B the difference of the angles y'FB,yF £, and in¬ ftead of AF b, the fum of thefe angles, we fhall have P—W AFB I ang. /F £= p X Tang. ——, whence, by tranfpofition, and by Geometry, Theor. VIII. Sea. IV. P-ffW : P—W=Tang.—-— : Tang-j^F^. Hence, when the angle formed by the arms of the ba¬ lance, and the angle of aberration f¥ b or ZFL, are known, the weights may be found, and vice verfa. Chap. IV. On the Centre of Inertia, or Gravity. 154. Definition.—The centre ofinertia, or the centre of gravity, of any body or fyftem of bodies, is that point upon which the body or fyftem of bodies, when influ¬ enced only by the force of gravity, v7ill be in equilibrio in every pofition. The centre of inertia of plane fur- faces bounded by right lines, and alfo of fome folids A N I C S. may be eafily determined by the common geometry. The application of the method of fluxions, however, to this branch of mechanics is fo Ample and beautiful, that we (hall alfo avail ourfelves of its afliftance. The centre of gravity has been called, by fome writers, the centre of pofition, and by others, the centre of mean diftances. Prop. I. 155. To find the centre of inertia of any number of bodies, whatever be their pofition. Let ABCD be any number of bodies influenced by Fig. 4 the force of gravity^ Suppofe the bodies A, B connedr- ed by the inflexible line AB confidered as devoid of weight, then find a point F, fo that the weight of A : the weight of B—BF : FA. The bodies A, B will therefore be in equilibrio about the point F in every pofition (Art. 36.), and the preffure upon F will be equal to A-{-B. Join FC, and find the point f fo that A-f-B : C~Cf: fF> the bodies A, B, C, will confe- quently be in equilibrio upon the point f which will fuftain a preffure equal to A-f-B-f-C. Join Df and take the point

x XB -)-E x XC -J-Z xXD-P ExXE, then dividing by A-j-B-f C-J-D-f-E, we 70 Theory, Theory. MECHANICS. v A _A X XA+BxXB+C xXd'+I) X XD+Ex^I ^ a+b+c+d+e Now AxXAj B xXB, &c. are evidently the mo¬ menta of the bodies A, B, &c. and the divifor A-f-B -J-C-f-D-j-E is the fum of the weights of all the bodies 5 therefore the diftance of the point X from the centre of gravity

in, y, then in the fimilar triangles AjvF, ByF, we have A* : ByzrAF : BF, that is, (Art. 155.) as B : A, hence AxAv=rBxBy, that is, A X * «—A «zrB X B b—yb, or on account of the e- quality of the lines arn, F_/^B£^Ax Fy£—Aa=r B X B b F/, therefore, by multiplying and tranfpofing, we have A-j-B X^/—A X Aa-J-B X B£. In the very fame way, by drawing G z parallel to the plane, it may be (hewn that A + B-j-C X G^rzA X A«-f-B XB^+CxCc. Q.E.D. 162. Cor. By dividing by A-f B-f-C we have ^_AxAfl4-BxB/>+CxCc A+B+C Prop. IV. 163. To find the centre of inertia of a ftraight line, compofed of material particles. If we confider the ftraight line as compofed of a number of material particles of the fame fize and den- fity, it is evident that its centre of inertia will be a point in the line equidiftant from its extremities. For if we regard the line as a lever fupported upon its mid¬ dle point as a fulcrum, it will evidently be in equilibrio, in every pofition, as the number of particles or weights on each fide of the fulcrum is equal. Prop. V. 164. To find the centre of inertia of a parallelo¬ gram. Let ABCD be a parallelogram of uniform denfity, Fig. 7, bifeft AB in F, and having drawn Ff parallel to AC or BD, bifeft it in

but in the fimilar triangles ADC, Aec; AD : DC= A e : e c, and in the triangles ADC, ADB, Aeb ; BD : DArz be \ e A\ hence by compofition BD : Y)C—be \ e c ; but BD and DC are equal j therefore, b e~e c ; and the line b c, fuppofed to confift of material particles, will be in equilibrio about e. In the fame way it may be (hewn that every other line /3 * will be in equilibri© about a point fituated in the line AD •, confequently the centre of gravity is in that line. For the lame rea- fon it follows, that the centre of gravity is in the line CE, that is, it will be in F, the point of interfe&ion of thefe two lines. In order to determine the relation be¬ tween FA and FD, join ED-, then, fince BErrEA, and BD=DC, BE : EA=BD : DC, and confequent¬ ly, (Geometry, Se£t. IV. Theor. 18.) ED is pa¬ rallel to AC, and the triangles BED, BAG fimilar. We have, therefore, CA : CBnDE : DB, and by al¬ ternation CA : DEnCB : DB, that is, CA : DE^ 2: 1. In the fimilar triangles CFA, DFE, AF : AC— DF : DE, and by alternation AF : DFzzAC : DE, that is, AF : DF=2 : 1, or AF=|AD. 166. Cor. 1. By Geometry, Theor. 16. Sett. IV. we have ABl+AC1=2BD*-|-2AB,( = 4^C*-f ?AF* ABl+BC*=2CC» + 2BG2 -f AC»-KCF» AC*4- BC,= 2AE1 + 2EC1 AB*+1BF*. By MECHANICS. 7i Theory. By adding thefe thre£ equations, and removing the V'"—"’ fraftions, we have AC2=:3 AF*-f-3CF* +3 BP, or in any plane triangle, the fum of the fquares of the three fides is equal to thrice the fum of the fquares of the diftances of the centre of gravity from each of the angular points. 167. Cor. 2. By refolving the three quadratic equa¬ tions in the preceding corollary, we obtain AFrrf AB*+ 2 AC*—EG*; CF^rf^/ 2 B A*-f-2 BC*— AC*-, and BF=fy/2 BC*-{-2 AC*—ABZ, formulae which exprefs the diftances of the centre of gravity from each of the angular points. Prop. VII. 168. To find the centre of inertia of a trapezium or any re£Iilineal figure. Fig. 9. Let ABCDE be the trapezium, and let it be divid¬ ed into the triangles ABC, ACE, ECD by the lines AC, EC. By the laft propofition find wz, «, 0, the centres of gravity of the triangles, and take the point F in the line m n, fo that F « : F /Tzrrtfiangle ABC : triangle ACE, then F will be the centre of gravity of thefe triangles. Join F 0, and find a point f, fo that fo : Yf~ triangle ABC-}-triangle ACE : triangle CED, then all the triangles will be in equilibrio about f that is, f is the centre of gravity of the rectilineal figure ABCDE. The fame method may be employed in finding the centre of gravity of a trapezium, what¬ ever be the number of its fides. Prop. VIII. 169. To find the centre of inertia of a pyramid with a polygonal bafe. Fig. 10. Let the pyramid be triangular, as A BCD, fig. 10. Bifeft BD in F, and join CF and FA. Make Ff— •f- of FC, and F (pzz f of FA, and draw f p. It is evi¬ dent, from Art. 159. that f is the centre of gravity of the triangular bafe BCD, and that the line AF, which joins the vertex and the point f will pafs through the centre of gravity of all the triangular laminae or fec- tions of the pyramid parallel to its bafe ABC ; for, by taking any fedtion bed, and joining c m, it may be eafily (hewn, that b m—m d, and m n—±m c, fo that n is the centre of gravity of the fedlion bed. It fol¬ lows, therefore, that A f will pafs through the centre of gravity of the pyramid. In the fame way it may be fliewn, by confidering ABD as the bafe, and D the vertex, and making F frifFA, that the centre of gra¬ vity lies in the line C. But, as the lines Af

fluent of 2 tv x y z , Weight, we mail have hB=r J . , and di- fluent of 2 7r y % .... , . fluent oi x y % vidmg by 2 5r we obtain rB— ——. fluent oiy z 176. If the body, whofe centre of inertia is to be found, be a curve line, as GBD, then it is manifefl that the fmall weights will be expreffed by the fluxion of GBD, that is, by 2 •%, fince GBDrr 2 BD=2 « ; con- fequently their momenta will be 2 x x, and we fliall fluent 2X z fluent x z fluent x z have FB=r. fluent 2 z fluent a Prop. IX. 177. To find the centre of inertia of a circular fegment. Let AE=rAr, FC=y, and AD the radius of the cir¬ cle =R, confequently ME=:2 R—EA. Then, fince by the property of the circle (Geometry, Theor. 28. Sedt. IV.) MExEA—BE*, we have, by fubftitution, BE* = 2 R x EA—EA x EA, or y* = 2R » —x* j hence y=\/2 R x—,v*. Now, by Art. 174. we have the diftance of the centre of gravity from A, that is, fluent x y x . AG= . ; but the fluent of y x or the fum of fluent y x all the weights, is equal to the area of half the feg¬ ment ABEC *, therefore AG~ fluent x y x Then, by _ 4BEC fubftituting inftead of y, in this equation, the value of it deduced from the property of the circle, we have . ^ fluent of x x Jr B x—x* . AG=r AirEC ’ or’ m or^er to GD the diftance of the centre of gravity from the cen¬ tre, we muft fubftitute inftead of x (without the A N I C S. vinculum) its value R—x, and we have GD = fluent Theory. (R—Jc)a*(2RA:—v*) . ^ e— .LABEd “• Now, in order to find the fluxion of the numerator of the preceding fradlion, af- fume 2;:=2R *—**, and %^f—J2 R 2—^*, and by tak¬ ing the fluxion, we have ssn 2 R x—2xx~2\{^2x x x j but this quantity is double tif the firft term of the nu¬ merator, therefore ^zrR—at x By fubftituting thefe values in the fraftional formula, wTe obtain GDrrfluent , • *t v/—-wll- 2 3.3 we have, by raifing both fides to the third power, , but fince y— 2B.X—xx'£ 2 R x—x x \ : therefore GD=—tt — — ^ - ? 4ABEC 4ABEC ^ 2 ~^ g 0, that is, the diftance of the centre of gravi¬ ty of a circular fegment from the centre of the circle, is equal to the twelfth part of the cube of twice the ordi¬ nate, (or the chord of the fegment) divided by the area of the fegment. 178. Cor. When the fegment becomes a femicircle > r')3 we have 2 7j—2 r ; and therefore zr GD—--4 - ■■■ — 3 ABEC (2r)* 8x?'3 r* . . 12 ABEC 12 AiiEC~ 14ABEC’ that IS’ the dlf' tance of the centre of gravity of a femicircle from the centre of the femicircle, is equal to the cube of the ra¬ dius, divided by one and a half times the area of the fegment. Prop. X. 179. To find the centre of inertia of the fedfor of a circle. Let ABDC be the fedlor of the circle. By Art. 157. find m the centre of inertia of the triangle BCD, and by the laft propofition find G the centre of inertia of the fegment *, then take a point n fo fituated between G and m, that ABEC : BCBrzwz n : G n, then the point n will be the centre of gravity of the fedlor.— By proceeding in this way, it will be found that D ny or the diftance of the centre of gravity of the fedlor from the centre of the circle, is a fourth proportional t« the femiarc, to the femichord, and to two-thirds of the radius. Prop. XL 180 To find the centre of inertia of a plane fur¬ face bounded by a parabola whofe equation is y—a x>7. Since y~a xK, multiply both terms by x xy and x fe- paratcly, and we have y x xzza x+* x, and y x—a xn x. But, by Art. 174. we have FB--Uent °f there- fluent y x fore, by fubftituting the preceding values of x y x and • f! PTlt of // /Y? 77“I"'1 V yx in the formula, we obtain FEzz— —, fluent of a xn x and M E C FI A N I C S. Theory, and by taking the fluents it becomes * ax +* Fig. 12. FB=r n 2 a i « atH-1 n-\- 2 n~hI X *• and FB= 2 a —2 a* y'n U' 2/2 + 1 2/Z + 2 X When n—\, the folid becomes a common parabo¬ loid, and we obtain FBrr| ;r. When n—\, the folid becomes a cone, and FB as in Art. 171. Prop. XIII. 182. To find the centre of gravity of a fpherical furface or zone, comprehended between two parallel planes, or of the fpherical furface of any fpherical fegment* Let BMNC be a feftion of the fpherical furface comprehended between the planes BC, MN, and let EP=a?, EC—y, DCrzR, and a rr the arc CN. Sup- pofe the abfcifla EP to inereafe by the fmall quantity E 0, draw 0 r parallel to EC, C r parallel to E 0, and C r perpendicular to DC 5 then it is evident, that in the fimilar triangles CDE, Cv r, EC : DCzrC s : C r, that is, ij : Rt=C r : C r ; but C r is the flux¬ ion of the arc NC, and C r the fluxion of the abfcif- fa PE 5 therefore y : Rzr.v : 55, and zyrzR x, and a R * XT , . fluent oi x 11 % , = . Now, by Art. 17 c. FB= ^— y fluent o{ x y % therefore, by fubftituting the preceding value of ss Vol. XIII. Part I. in this formula, we obtain FB— fluent of R x x fluent of R a; R .v .v 2; Ft AC « R v x x % If n, therefore, be equal to then y=® arid, fquaring both Tides, if—a* x, which is the equation of the common or Apollonian parabola. Hence, F.3— ^ x, that is, the diltance of the centre of gravity from the vertex is -l-ths of the axis. When n is equal to 1, then y~a x, and the para¬ bola degenerates into a triangle, in which cafe FB rzy.v, as in Art. 165. Prop. XII. 181. To find the centre of inertia of a folid, ge¬ nerated by the revolution of the preceding curve round its axis. Since y=o xn, fquare both fides, and we have y2— m2x2n ; then multiply both fides by x x, and x feparate- ly, we obtain y*xxzza2x2n-\-tx, and y2x—a2 a?1® x. But, , . , fluent of w* a; a; , . by Art. 174. we have r B— —r-; therefore, fluent of y2Af by fubftituting the preceding values of y2x x, and if x . r , , . „„ fluent of a2x2n-\'2x m that formula, we obtain rBzz ., fluent of a2x2nx by taking the fluents we ftiall have a2x2n-V2x 2/2 + 1 R y a; a r (and dividing by y 2): Rata? for By 73 Theory. y R a;* taking the fluents wre obtain FB=r 4- 'jfTT—^ x’ a which requires no correftion, as the other quantities’ vamih at the fame time with x. 183. When DP is equal to DC, the folid becomes a fpherical fegment, and EA becomes the altitude of the fegment, fo that universally the centre of gra¬ vity of the fpherical furface of a fpherical fegment is in the middle of the line which is the altitude of the fegment, or in the middle of the line which joins the centres of the two circles that bound the fpherical fegment. 184. When the fpherical fegment is a hemifpheroid, the centre of gravity of its hemifpherical furface is ob- vioufly at the diftance of one-half the radius from its centre. Prop. XIV. 185;. To find the centre of gravity of a circular arc. Let BAG be the circular arc, it is required to Fig. 13. find its centre of inertia, or the diftance of the cen¬ tre of inertia of the half arc AC from the diameter HG ; for it is evident, that the line which joins the centres of gravity of each of the femiarcs AB, AC muft be parallel to HG, and therefore the diftance of their common centre of gravity, which muft be in that line, from the line HG, will be equal to the diftance of the centre of gravity of the femiarc from the fame line. ' Make PC—DE—a; ; EC=y; DC=DA=R, and AC rrs;, then it may be ftiewn, as in the laft propofition, that y : Rzza: : z ; hence % y=R x. But, by Art. 176. . fluent of w 55 , . ... r , we have I B— , y being in this cale equal to x in the formula in Art. 176. and fubftituting the . r • . t fluent of Ra? preceding value 01 y as, it becomes rB= , R 9C and, taking the fluent, we have FB=-^—, which re¬ quires no corredtion, as the fluent of y sz vaniflies at the fame time with x. Calling I7^» t^ie reader will find no difficulty in determining the centre of inertia of other fur faces and folids, when he is acquainted with the equa¬ tion of the curves by which the furfaces are bounded, and by whofe revolution the folids are generated. A knowledge of the nature of thefe curves, how¬ ever, is not abfolutely necefl'ary for the determination of the centres of inertia of furfaces and folids. A me¬ thod of finding the centre of gravity, without employ¬ ing the equation of the bounding curves, was difcover- ed by our countryman, Mr Thomas Simfon *. It was afterwards more fully illuftrated by Mr Chapman, in his work on the Conftru&ion of Ships ; by M. Le- veque, in his tranflation of Don George Juan’s Trea- tife on the Conftru£tion and Management of Veffels ; and by M. Prony, in his Architeklure Hydraulique, tom. i. p. 93. to which we mull refer fuch readers as with to profecute the fubjedt. Scholium II. 189. As it is frequently of great ufe to know the po¬ fition of the centre of inertia in bodies of all forms, we ffiall collefl all the leading refults which might have been obtained, by the method given in the preceding propofitions. 1. The centre of inertia of a ftraight line is in its middle point. 2. The centre of inertia of a parallelogram is in the interfeftion of its diagonals. 3. The centre of inertia of a triangle is diftant from its vertex two-thirds of a line drawn from the vertex to the middle of the oppofite fide. 4. The centre of inertia of a circle, and of a regular polygon, coincides with the centres of thefe figures. 5. The centre of inertia of a parallelopiped is in the interfeftion of the diagonals joining its oppofite angles. 6. The centre of inertia of a pyramid is diftant from its vertex three-fourths of the axis. 7. The centre of inertia of a right cone is in a point in its axis whofe diftance from the vertex is three-fourths of the axis. 8. In the fegment of a circle, the centre of inertia is diftant from the centre of the circle a twelfth part of the cube of the chord of the fegment divided by the area of the fegment, or d~ |C_3 A ’ where X —, where h is the diftance 4 between the centres of the circles which contain the paraboloidal fruftum, R the radius of the greater circle, and r the radius of the lelfer circle. 3 2. In a conic fruftum or truncated cone, the dif¬ tance of the centre of inertia from the centre of the fmalleft circular end is 3RI-f-2Rr-{-r1 h X — which re- 4 R*-f-Rr-f-r* prefents the diftance between the centres of the circles which contain the fruftum, and R, r the radii of the circles. 33. The fame formula is applicable to any regular pyramid, R and r reprefenting the fides of the two polygons by which it is contained. Prop. XIV. 190. If a quantity of motion be communicated to a fyftem of bodies, the centre of gravity of the fyftem will move in the fame dire£Uon, and with the fame velocity, as if all the bodies were collected in that centre, and received the fame quantity of motion in the fame dire&ion. Let A, B, C be the bodies w'hich compofe the fyftem, and let F be the centre of gravity of the bodies B, C, andy the centre of gravity of the whole fyftem, as determined by Art. 1 55. Then if the body A re¬ ceives fuch a momentum as to make it move to a in a fecond, join F a, and take a point

at the end ot a fecond. But as the quantity of motion is equal to the produdt of the velocity of the body multiplied by its quantity of matter, the velo¬ cities are inverfely as the quantities of matter, and con- fequently the velocity of the body at f is to A’s velo¬ city as A is to A -f-B-j-C, that is, as is to A a ; therefore A a and f

• Continue h f //

X«+ jrX*=P 4-J»+?vX G^-. But PX A-f-/;X«-f-5rX« &c. make up the whole folid a D, and P+^-f tr, &c. make up the whole furface ABCD 5 therefore the folid a D is equal to the generating furface ABCD multi¬ plied by the path of its centre of gravity. (^. L. D. 207. Cor. 1. Let us fuppofe the circle B AGO to be generated by the revolution of the line DA round the Fig. iz, point D j then fince the centre of gravity of the line DA is in its middle point G, the path of this centre will be a circumference whofe radius is DG, or a line equal to half the circumference BONAB, therefore, by the theorem, the area of the circle BONB will be equal to the radius DA multiplied by the femieircumfe- rence, which coincides with the refult obtained from the principles of geometry. See Playfair’s Geometry, Supp. B. I. Prop. 5. In the fame way, by means of the preceding theorem, we may readily determine the area of any furface, or the content of any folid that is generated by motion. Scholium. 208. The centro-baryc method, which is one of the finelt inventions of geometry, was firft noticed by Pap¬ pus in the preface to the feventh book of his mathema¬ tical colle£Hons, but it is to Father Guldinus that we are indebted for a more complete difeuflion of the fub- je£t. He publilhed an account of his difeovery partly in 1635, and partly in 1640, in his work entitled De Centro Gravitatis, lib. ii. cap. 8. prop. 3. and gave an in- diretl demonftration of the theorem, by {bowing the con¬ formity of its refults with thofe which were obtained by other means. Leibnitz demonflrated the theorem in the cafe of fuperficies generated by the revolution of c urves, but concealed his demon {{.ration (Ad. Lerpf. 1695, p. 492- M E C H 493. The theorem of Leibnitz, however, as well as that of Guldinus, was demonftrated by Varignon in the Me¬ moirs of the Academy for 1714, p. 78. Leibnitz ob- ferves that the method will itill hold, even if the centre round which the revolution is performed be continually changed during the generating motion. For further information on this tubjeft, the reader is referred to Dr Wallis’s work, De Calculo Centri Gravitatis, Hut¬ ton’s Menfuration, Prony’s Architecture Hydraulique, •vol. i. p. 88. and Gregory’s Mechanics, vol. i. p. 64. Prop. XVIII. >209. To (bow the ufe of the do&rine of the centre of gravity in the explanation of fome mechanical phenomena. On the mo- In the equilibrium and motion of animals, wTe per- an‘" ceive many phenomena deducible from the properties of the centre of gravity. When we endeavour to rife from a chair, we naturally draw our feet inwards, and reft upon their extremities, in order to bring the centre of gravity directly below our feet, and we put the body into that polition in which its equilibrium is tottering, a pe¬ tition which renders the fmalleft force capable of pro¬ ducing motion, or of overturning the body. In this fituation, in order to prevent ourfelves from falling backwards, we thruft forward the upper part of the body for the purpofe of throwing the centre of gravity beyond our feet: and when the equilibrium is thus de- ftroyed, we throw out one of our feet, and gradually raife the centre of gravity till the pofttion of the body is ereft.—When we walk, the body is thrown into the poiition of tottering equilibrium by refting it on one foot ; this equilibrium is deftroyed by puthing for- ward the centre of gravity, and the body again afl’umes the poiition of tottering equilibrium by refting it on the other foot. During this alternate procefs of creating and deftroying a tottering equilibrium, the one foot is placed upon the ground, and the other is railed from it •, but in running, which is performed in exaClly the fame way, both the feet are never on the ground at the fame rirne : A t every ftep there is a Ihort interval, during which the runner does not touch the ground at all. 210. When we afeend an inclined plane the body is thrown farther forward than when we walk on a hori¬ zontal one, in order that the line of diredlion may fall without our feet; and in defeending an inclined plane, the body is thrown backward, in order to .prevent the line of direction from falling too fuddenly without the bafe. In carrying a burden, the centre of gravity is brought nearer to the burden, fo that the line of direc¬ tion would fall without our feet if we did not naturally lean towards the fide oppofite to the burden, in order to keep the line of direffion within our feet. When the burden is therefore carried on the back, we lean for¬ ward ; when it is carried in the right arm, we lean to¬ wards the left} when it is carried in the left arm, we lean towards the right •, and when it is carried before die body, we throw the head backwards. 211. When a horfe walks, he firft fets out one of his Tore feet and one of his hind feet, fuppofe the right foot; then at the fame inftant he throws out his left fore £*ot and -hi* left hind foot, fo as to be fupported only A N I C S. by the two right feet. His two right feet are then Theory, brought up at the fame inftant, and he is fupported on- y-——J ly by his two left feet.—When a horfe pulls at a load which he can fcarcely overcome, he railes both his fore feet, his hind feet become the fulcrum of a lever, and the weight of the horfe colledted in his centre of gravi¬ ty acts as a weight upon this lever, and enables him to furmount the obftacle. (See Appendix -to Fergulbn’s Le&ures, vol. ii.). 212. When a rope-dancer balances himfelf upon thei\^clhod in fore part of one foot, he preferves his equilibrium in two which a ways, either by throwing one of his arms or his elevated roFe-dancer foot, or his balancing pole, to the fide opnofite to that to-^k3.^8 wards which he is beginning to fall, or by fhifting the bni.m. point of his foot, on which he refts, to the fame fide to¬ wards which he is apt to fall; for it amounts to the fame thing whether he brings the centre of gravity diredfly above the point of fupport, or brings the point of fupport diredfly below the centre of gravity. For this purpofe the convex form of the foot is of great ufe, for if it had been perfedfly flat, the point of lup- port could not have admitted of fmall variations -in its pofition *. * See Dr 213. We have already feen ( Art. 197.) that any body T. Toung't is more eafily overturned in proportion to the height of its centre of gravity. Hence it is a matter of great importance that the centre of gravity of all-carriages °'1'F 4* fhould be placed as low as poflible. This may often be effedled by a judicious difpolition of the load, of which the heavieft materials fliould always have the loweft place. The prefent conftrubfion of our mail and poll coaches is therefore adverfe to every principle con- of fcience, and the caufe of many of thole accidents in which the lives of individuals have been loft. Thees erronc_ * elevated pofition of the guard, the driver, and theous. outfide paflengers, and the two boots which contain the baggage, raifes the centre of gravity of the load¬ ed vehicle to a very great height, and renders it much more eafily overturned than it would otherwife have been. When any accident of this kind is likely to hap¬ pen, the paffengers ftiould bend as low as poflible, and endeavour to throw themfelves to the elevated fide of the carriage.—In two wheeled carriages where the horfe bears part of the load upon its back, the elevation of the centre of gravity renders the draught more difficult, by throwing a greater proportion of the load upon the horfe’s back when he is going dowm hill, and when he has the leaft occafion for it ; and taking the load from the back of the horfe when he is going up hill, and requires to be preffed to the ground. 214. A knowledge of the laws of the centre of gra-Fig. 3^. vity enables us to explain the experiment reprefented in fig. 24. where the vefiel of wrater CG is -fufpended on a rod A B, palling below its handle, and refting on the end E of the beam DE. The extremity B of the rod AB is fupported by another rod BF, which bears a- gainft the bottom of the vefiel; fo that the veflfel and the two rods become, as it were, one body, which, by Art. 199. will be in equilibrio when their common centre of gravity C is in the fame vertical line with the ^ |oa(Ld point of fupport E. may* be I 215. The cylinder G may be made to afeend the in-mfc;ie t0 dined plane ABC by putting a piece of lead or any cend an is, heavy fubftance on one fide of its axis, fo that the cen-1'1™ "ne tre of gravity may be moved from G towards g. Hence^•tk°wm MECHANICS- A double cone may be made to af< end an inclined plane by its own tveight. Fig. 26. Fig. 27. Theory it is obvious, that the centre of gravity y will defeend, and by its defeent the body will rife towards A. The inclination of the plane, however, mull be fuch, that before the motion commences, the angles formed by a vertical line drawn from g with a line drawn from G perpendicularly to AB, muft be lefs than the angle of inclination ABC, or, which is the fame thing, when the vertical line drawn from g does not cut the line which lies between the point of contact and the centre of the cylinder. When the vertical line, let fall from g, meets the perpendicular line .drawn from G to the plane in the point of contaft, the cylinder will be in equilibrio on the inclined plane. 2 i 6. Upon the fame principle, a double fealene cone may be made to afeend an inclined plane without being loaded with a weight. In fig. 26. let ABC be the feftion of a double inclined plane, AB, BC being fec- tions of its furfaces perpendicular to the line in which the double fealene cone ADEFC moves. Then, fince the centre of gravity of a cone is in the line joining the vertex and the centre of its bafe, and fince the axis of a fealene cone is not perpendicular to its bafe, the line which, joins the centres of both the cones, when in the pofition reprefented in the figure, will be above the line which joins the centres of their bafes. If the circle, therefore, in fig 27. reprefents the bafe of one of the cones, and C its centre, the line which joins the cen¬ tres of gravity of the two cones will terminate in fome point G at a diftance from the centre, and therefore the double cone will afeend the plane upon the fame prin¬ ciples, and under the fame conditions, as thofe men¬ tioned in the laft paragraph. Chap. V. On the Motion of Bodies along inclined Planes and Curves, on the. Curve of fwiftejl defeent, aitd on the Ofcillutions of Pendulums. Prop. I. 217. When a body moves along an inclined plane, the force which accelerates or retards its motion, is to the whole force of gravity as the height of the plane is to its length, or as the fine of its inclination is to radius. Let ABC be the inclined plane, A the place of the body, and let AB reprefent the whole force of gravi¬ ty. The force AB is equivalent to the two forces AD, DB or AE, AD, of which AD is the force that accelerates the motion of the body down the plane, while AE is deftroyed by the refiftance or re-a61ion of the plane. The part of the force of gravity, therefore, which makes the body arrive at C is reprefented by AD, while the whole force of gravity is reprefented by AB j but the triangle ABD is equiangular to ABC, and AD: AB=AB : AC, that is, the accelerating force which makes the body defeend the inclined plane, is to the whole force of gravity as the height of the plane is to its length, or as the fine of the plane’s in¬ clination is to radius ; for when AC is radius, AB be¬ comes the fine of the angle ACB. 218. Cor. 1. Since the force of gravity, Avhich is uniform, has a given ratio to the accelerating force, the accelerating force is alfo uniform ; confequently the laws of accelerated and retarded motions, as exhibited in the article Dynamics, are alfo true when the bodies 5 Plate ceexxr. fig. I. move along inclined planes. If H, therefore, repre¬ fent the height AB of the plane, L its length AC, g the force of gravity, and A the accelerating force, we (hall have, by the propofition, L : Hrry : A, hence AzryX-j-^ or, fince : A=radius : fin. ACB, and A — g X fin. ACB. Now, from the principles of Dynamics, s — i-g t*,. v — gt: \l 2 g s, and / — —— , E'v=c, C'B'=™, B'A'=//. Then fince F'E' may be confidered as a ftraight line, and fince B'C'rrF'i;, we have (Euclid, B» I. Prop. 47.) F'E'— Vz^+c'2, and fince F'x;r=:E'«,E'D'— Vrf+c** Now the velocities at F' and E' vary as */(t~and and F'E', E'Dr are the elementary fpaces deferibed with thefe velocities j but the times are directly as the Iquare root of the fpaces, and inverfely as the velocities, therefore the time of deferibing F'E'is ~^rH ~j~—,and V a the time of deferibing E'D' is —confequently, Vb the time of deferibing FD muft be — ~^~C ^ 4.— tj 1 a b b-%- But the propofition requires that this time fhould be the leaft poflible or a minimum, therefore taking its fluxion «nd making it equal to 0, we have 2mm ^ 2 mi 2JaXmmj-c* 2/%//'X««Xc,a VOL. XIII. Part I. But fince CA is invariable m-\-n is invariable, and therefore its fluxion or m=z—« and mf therefore by tranfpofing the fecond member of the preceding equation, and fubftituting thefe values of m and n, it becomes j- - =: — — Let us now call the variable abfeifs yC'—at, the or¬ dinate C'F'zzy, and the arc yF'srss, then m and n. are fluxions of *, and F'E' is the increment of F or « when ij is equal to «, and E'D' the increment of q F or 55, when y is equal to b, therefore by fubfti¬ tuting thefe values in the preceding equation, we ob- tain ^ which ftiews that this quantity is conftant, and gives us the following analogy, 55' : V — ^ y* Now m the cycloid y is always the chord of the generating circle when the dia¬ meter is y (for by Euclid, Book. I. Prop. 47, Book. II. Prop. 8. and Book III. Prop. 35.) AF— y'ADX AO ^ ^ and fmee AOzri and ADrry, we have AF=:y'y! But fince the arc of the cycloid at F is perpendicular to the chord AF, the elementary triangle FE® is fimi¬ lar to hDO, (for BE is parallel to AO) and conle- quently to AFO (Euclid. B. VI. Prop. 8.), therefore, we have FE : E'a'rrAO : AF; but FE=r', E^=y;, AOrzi and AF— y'y, confequently : x'— 1 : y'y, which coincides with the analogy already obtained, and being the property of the cycloid fhews that the curve of quickeft defcent is an inverted cycloidal arc. Properties of the Cycloid. Definition.—If a circle NOP be fo placed as Fig. 3. to be in contact with the line AD, and be made to Properties roll along that line from D towards A, till the fameoft.he CP* point D of the circle touches the other extremity A clold’ the point D will deferibe a curve DBA, called a cycloid. The line AD is called the bafe of the cycloid; the line CB, which bifefts AD at right angles and meets the curve in B, is called the axis, and B the vertex. The circle NOP is called the generating circle. I* 232. 1. The 82 MECHANICS. heory. 232. I. The bafe AD is equal to the circumference of the generating circle, and AC is equal to half that circumference. 2. The axis CB is equal to the diameter of the ge¬ nerating circle. 3. If from any point G of the cycloid, there be drawn a flraight line GM parallel to AD, and meet¬ ing the circle BLC in L, the circular arc BL is equal to the line GL. 4. If the points L, B be joined, and a tangent drawn to the cycloid at the point G, the tangent will be pa¬ rallel to the chord LB, and the tangent is found by joining G, E, for GE is parallel to LB. 5. The arc BG of the cycloid is double of the chord BL, and the arc BA or BD is equal to twice the axis BC. 6. If the two portions AB, DB of the cycloid in fig. 3. be placed in the. inverted pofition AB, DB (fig. f/ig, 4. 4.), and if a firing BP equal in length to BA be made to coincide with B A, and then be evolved from it, its extremity P will defcribe a femicycloid AF, fimilar and equal to BA. In the fame way the femicycloid DF, produced by the evolution of the firing BP from the femicycloid BD, is equal and fimilar to BD and'to AF. Therefore, if BP be a pendulum or weight at¬ tached to the extremity of a flexible line BP, which vibrates between the cycloidal cheeks BA, BD, its ex¬ tremity D will defcribe a cycloid AFD, equal to that which is compofed of the two halves BA, BD. 7. The chord CN is parallel to MP, and MP is per¬ pendicular to the cycloid AFD, at the point P. 8. If P p be an infinitely fmall,arc, the perpendicu¬ lar to the curve drawn from the points P/> will meet at M, and P p may be regarded as a circular arc, whofe radius is MP. An infinitely fmall cycloidal arc at F may likewife be confidered as a circular arc whofe ra¬ dius is BF. • As thefe properties of the cycloid are demonftrated in almoft every treatife on mechanics, and as their de- monftrations more properly belong to geometry than to mechanics, they are purpofely omitted to make room for more important matter. 233. Definition.—If a body defcend from any point of a curve, and afcend in the fame curve till its velocity is deftroyed, the body is faid to oicillate in that curve, and the time in which this defcent and ai- cent are performed is called the time of an ofcillation or vibration. 234. Definition.—A cycloidal pendulum is a pen¬ dulum which ofcillates or vibrates in the arch of a cy¬ cloid. 235. Definition,—Ofcillations which are perform¬ ed in equal times are faid to be ifochronous. Prop. V. Hg. 4. 236. The velocity of a cycloidal pendulum BP at the point F, varies as the arch which it de- feribes. The velocity of the pendulum at F is that which it •would have acquired by falling through EF (Prop. 2. and Cor. 3. Prop. 2.), and the velocity of availing bo¬ dy is as the fquare root of the fpace which it deferibes (Dynamics, § 37.% therefore the velocity of the pen¬ dulum P, when it reaches F, varies as ^/EF. But (Geometry, Sedt. IV.Theor. 23. and 8.) FE varies as FN* ——, and fince FC is a conftant quantity, FE will vary t C Theory. as FNJ varies, or, to adopt the notation ufed in the article Dynamics, FEziiFN2, or V'FEirFN, but the ve¬ locity acquired by falling through EF varies as \/FE, therefore the velocity of the pendulum at F varies as FN, that is, as FP, for (Art. 232. N° 5.) FN is equal to half FP. Q. E. D. Prop. VI. 237. If the pendulum begins its ofcillation from the point P, the velocity of the pendulum at any point R varies as the fine of a circular arc whofe radius is FP, and whofe verfed fine is PR. Through F draw p F 7 parallel to AD, and with a Fig. 4. radius equal to the cycloidal arc FP, defcribe the femi- circle p 0 q. Make p r equal to the arc PR of the cy¬ cloid, and through r draw r m perpendicular to p F. Through the points P, R draw PE, R1 parallel to AD, and cutting the generating circle CNF in the points N, S.—By Prop. 4. the velocity at R varies as Vet, that is, as VEF—TF, or fince CF is conftant, as VCF x EF—CF X TF, that is, as VFNV—FS\ For, (Playfair’s Euclid, Book I. Prop. 47, Book II. Prop. 7. and Book III. Prop. 35O hNI”Cll xEh, and FS=CFxTF), that is, as VpTP—TFS% that is (Art. 232. N° 5.) as VFP*—FR*. But or F m was made equal to FP, and, p r being made * equal to PR, the remainder F r muft be equal to FR, therefore the velocity at R varies as \/y mz—F r*, but (Euclid, 47. 1.) rm ~ s/¥m1—Fr1, and rm is by con- ftru£Hon equal to the fine of a circular arc, whofe ra¬ dius is FP, and verfed fine PR, confequently, the velo¬ city at R varies as the fine of that arc. C,L E. D. 238. Corollary. The velocity of the pendulum at F is to the velocity of the pendulum at R, as F m : r m, for the verfed fine is in this cafe equal to radius, and therefore the correfponding arc muft be a quadrant whofe fine is alfo equal to radius or F m. Prop. VII. 239. The time in which the pendulum performs Fig. 4, one complete ofcillation from P to O, is equal to the time in which a body would defcribe the femicircle p ° uniformly with the velocity > which the pendulum acquires at the point F. Take any infinitely fmall arc RV, and making r v equal to it, draw v 0 parallel to r m, and mn r v. Now, by the laft propofition, and by Dynamics, Art. 28. ; the velocity with which RV is deferibed is to the velocity with which w 0 is deferibed as r m is to F zw, J that. Theory. Fig. 4. , . RV that is as r m m 0 m n ——, or as — F m r m til 0 F /«’ for m n~r RV. But in the fimilar triangles Y mr, m n o,Y m : r m~m 0 • m n, confequently = —therefore the velocity with which RV is defcribed is equal to the velocity with which m 0 is defcribcd, and the times in which thefe equal fpaces are defcribed muft likewife be equal. The fame thing may be demonftrated of all the other correfponding arcs of the cycloid and circle, and there¬ fore it follows that the time in which the pendulum performs one complete ofcillation is equal to the time in which the femicircle poqu uniformly defcribed with the velocity acquired at !. Prop. VIII. 240. The time in which a cycloidal pendulum performs a complete ofcillation is to the time in which a body would fall freely through the axis of the cycloid, as the circumference of a circle is to its diameter. Since FP — 2FN, and fince the velocity acquired by falling down NF is equal to the velocity acquired by falling down PF, the body, if it continued to move uniformly with this velocity, would deferibe a fpace equal to 2PF (Dynamics, § 37. N° 6.) in the fame time that it would defeend NF or CF (Art. 219.). Calling T therefore the time of an ofcillation, and t the time of defeent along the axis, we have, by the preced¬ ing proportion, Trz time along p 0 q, with the velocity at F, and by the preceding paragraph, /=time along Y p, with the fame velocity j therefore T :: /rrtime along po q with velocity at V : time along Fp with the fame velocity ; that is, T : t—poq : Yp ~lpoq\ 2 Yp— the circumference of a circle : its di¬ ameter. 241. Cor. 1. The ofcillations in a cycloid are ifochronous, that is, they are performed in equal times whatever be the fize of the arc which the pendulum deferibes. For the time of an ofcillation has a conftant ratio to the time of defeent along the axis, and is there¬ fore an invariable quantity. 242. Cor. 2. The ofcillations in a fmall circular arc whofe radius is BF, and in an equal arc of the cycloid, being ifochronous (Art. 23 2. N° 8.), the time of an ofcil¬ lation in a fmall circular arc will alfo be to the time of defeent along the axis, as the circumference of a circle is to its diameter. 243. Cor. 3. Since the length BF of the pendulum is double of the axis CF, the time of an ofcillation in a cycloid or fmall circular arc varies as the time of de- feending along CF, half the length of the pendulum, the force of gravity being conftant. But the time of defeent along CF varies as \fCF, therefore the time of an ofcillation in a fmall circular or cycloidal arc varies as the fquare root of half the length of the pen¬ dulum, or as the fquare root of its whole length. If T, t therefore be the times of ofeillations of two pendulums, MECHANICS. and L, / their refpe&ive lengths, we have by this co¬ rollary T : /= x/L : and T X v/L j hence y ty. yL y/ TXy/ _t x y/. t— — ) w 'x yR and L = yL 7'~V T , from which we may find the time in which a pendulum of any length will vibrate •, a pendulum of 39.2 inches vibrating in one fecond. 244. Cor. 4. When the force of gravity varies, which it does in going from the poles to the equator, the time of an ofcillation is direftly as the fquare root of the length of the pendulum, and inverfely as the fquare root of the force of gravity. T he time of an ofcillation varies as the time of defeent along half the length of the pendulum, and the time of defeent through any fpace varies as where j is the fpace de¬ fcribed and g the force of gravity 5 but in the prefent cafe s — — therefore, by fubftitution, the time of defeent along half the length of the pendulum, time of an ofcillation, varies as ^ 1 Hence T : yL y/ Vg or the yL Vg' from which it is eafy to de- Vg' Vg’ duce equations fimilar to thofe given in the preceding corollary. 245. Cor. 5. Since 4/^ xTrr yL j and if ' Vg the time of ofcillation is 1 fecond, we have i/ g^\/h, or ^rzL, that is, the force of gravity in different lati¬ tudes varies as the length of a pendulum that vibrates feconds. 246. Cor. 6. The number of ofcillations which a pendulum makes in a given time, and in a given la¬ titude, are in the inverfe fubduplicate ratio of its length. The number of ofcillations n made in a given time are evidently in the inverfe ratio of /, the time of each ofcillation ; that is n±i ~ ; but by Corollary 3. tzz y/, therefore nzz —jj, and /~ t ' \ l n from which it is eafy to find the length of a pendulum which will vibrate any number of times in a given time, or the number of vibrations which a pendulum of a given length will per¬ form in a given time. Prop. IX. 247. To find the fpace through which a heavy body will fall in one fecond by the force of gravity. Since by Propofition 8. the time of an ofcillation is to the time along half the length of the pendulum as 3.1411^9 is to 1, and fince the {paces are as the fquares of the times, the fpaees defcribed by a heavy body in the time of an ofcillation will be to half the length of the pendulum as 3.14159* is to 1. Now it app ars from the experiments of Mr Whitehurft, that the length of a pendulum which vibrates feconds at London at 113 feet above the level of the fea, in a temperature of L 2 6s4 84 MECHANICS. Theory, go0 of Fahrenheit, and when the barometer is 30 inches, is 39.1196 inches j hence i* : 3.14159I* — — - : 19.5598 X3.i4i59|,= i6.o87 feet the fpace required. The methods of determining the centre of ofcillation, gyration, and percuflion, properly belong to this chap¬ ter, but they have been already given in the article Rotation, to which we mull refer the reader who willies to profecute the fubjeft. have 2B'xy=B'X2VrorB': zB'rrV : 2V* but 2V is the velocity of B', and V is the velocity of 2 B', therefore when one body is double of the other, they will remain at reft when the maffes of the bodies are in- verfely as their velocities. In the fame way the propolition may be demonftrated when the bodies are to one another in any commen- furable proportion. Theory. Prop. II. Chap. VI. On the Collijion or Imfmljion of Bodies. 248. Def. I. When a body moving with a cer¬ tain velocity ftrikes another body, either at reft or in motion, the one is faid to impinge againft, or to im- pell the other. This effeft has been diltinguilhed by the names collilion, impullion or impulfe, percuffion, and Impafl. 249. Def. 2. The collilion or impullion of two bodies is faid to be direti when the bodies move in the fame ftraigbt line, or when the point in w'hieh they ftrike each other is in the ftraight line which joins their centres of gravity. When this is not the cafe, the im¬ pulfe is faid to be oblique. 250. Def. 3. A hard body is one which is not fuf- ceptible of compreflion by any finite force An elafic body is one fufceptible of compreflion, which recovers its figure with a force equal to that which comprelfes it. A foft body is one which does not recover its form after compreffion. There does not exift in nature any body which is either perfectly hard, perfeftly elaftic or perfectly foft. Every body with which vTe are ac¬ quainted polfelfes elafticity in fome degree or other. Diamond, cryftal, agate, &c. though among the hard- eft bodies, are highly elaftic 5 and even clay itfelf will in fome degree recover its figure after compreflion. It is neceflary, however, to confider bodies as hard, foft or elaftic, in order to obtain the limits between which the required refults mult be contained. 251. Def. 4. The mafs of a body is the fum of the material particles of which it is compofed •, and the momentum, or moving force, or quantity of motion of any body, is the produdl arifing from multiplying its mafs by its velocity. Prop. I. 252. Two hard bodies B, B' with velocities V, V' (Iriking each other perpendicularly, will be at reft after impulfe, if their velocities are inverfe- ly as their mafles. 1. When the two bodies are equal, their velocities muft be equal in the cafe of an equilibrium after im¬ pulfe, and therefore B : B'—V' : V, or BY—B'V'5 for if they are not at reft after impulfe, the one muft carry the other along with it: But as their mafles and velocities are equal, there can be no reafon why the one fhould carry the other along with it. 2. If the one body is double of the other, or B=:2B', we ftiould have V'r: 2V'. Now inftead of B we may fub- ftitute two bodies equal to B', and inftead of V; we may fubftitute two velocities equal to V, with which the bodies B' may be conceived to move j confequently we - 3 253. To find the common velocity v of two hard bodies B, B' whofe velocities are V, V', after ftriking each other perpendicularly. If the bodies have not equal quantities of motion they cannot be in equilibrio after impulfe. The one will carry the other along with it, and in confequence of their hardnefs, they will remain in contact, and move with a common velocity v. 1. In order to find this, let us firft fuppofe B' to be at reft and to be ftruck by B in motion. The quantity of motion which exifts in B before impulfe is BY, and as this is divided between the two bodies after impulfe, it muft be equal to the quantity of motion after impulfe. But X B -f- B' is the quantity of motion after impulfe, RV therefore vx B + B'zrB V, and v zz———. B+B' 2. Let us now fuppofe that both the bodies are in motion in the fame direction that B follows B'. In order that B may impel B', we muft have V greater than V'. Now we may conceive both the bodies pla¬ ced upon a plane moving with the velocity V'. The body B', therefore, whofe velocity is V; equal to that of the plane, will be at reft upon the plane, while the velocity of B with regard to B' or the plane, will be V—V'J confequently, the bodies are in the fame eir- cumftances as if B' were at reft, and B moving with the velocity V—V'. Therefore, by the laft cafe, we have the common velocity of the bodies in the move- B V—BY" able plane - ; and by adding to this V', the velocity of the plane, we ftiall have v, or the abfolute velocity of the bodies after impulfe, v zz — Hence the quantity of motion, after impadl, is equal to the fum of the quantities of motion before impaft. 3. If the impinging bodies mutually approach each other, we may conceive, as before, that the body B' is at reft upon a plane which moves with a velocity V' in an oppofite direfti'on to V, and that B moves on this plane with the velocity V-J-V'. Then, by Cafe 1. BY 4-BY' . —— will be the common velocity upon the plane after impulfe ; and adding to this V', or the velocity of the plane, we fhall have v, or the abfolute velocity BY—-B'V' of the bodies after impaft, v zz —5———. Hence the quantity of motion after impa£t is equal to the differ¬ ence of the quantities of motion before impadt. It is obvious that v is politive or negative, according as BY is greater or lefs than B'V', fo that when BY is great¬ er than B'V', the bodies will move in the diredlion of B’s. M E C H Theory. motion ; and ivlien. BV is lels than the bo- dies will move in the direftion of A’s motion. 254. All the three formulae which we have given, may be comprehended in the following general formu- BVrtrB'V' la, v=————•, for when B' is at reft, V'=0, and B-f-B the formula afiumes the form which it has in Cafe 1. 255. Cor. I. If BrrB', and the bodies mutually approach each other, the equation in Cafe 3. becomes V V' . . . . . zi=: , or the bodies will move in the direction 2 of the quickeft body, with a velocity equal to one half of the difference of their velocities. 256. Cor. 2. If V=V, and the bodies move in the fame direction, the laft formula will become vzz R 1 "R/ V X ———, or V •, for in this cafe there can be no B-|-B impulfion, the one body merely following the other in contaft with it. When the bodies mutually ap¬ proach each other, and when V—V', we have DzrV B—B' X B+B' 257. Cor. 3. When the bodies move in the fame di- ^ r BV-fBV' reftion, we have, by Cale 2. vzz velocity gained by B' is evidently v- BV—BW B-f-B' -V',orBV+B'V' Now the hence B + B': B=V—V': B+B' BV- -V -BV' B+B' ’ r“ ’ ' B + B' but this laft term is the velocity gained by B, and V—V' is the relative velocity of the two bodies. Therefore, in the impaft of two hard bodies moving in the fame dir eel ion, B+B' : B as the relative velocity of the two bodies is to the velocity gained by B'. It is obvious alfo that the velocity loft bv B is V—vzz. BV+B'V' B'V—B'V' , ' , „ ■ hence B+B V—V' B+B' B'V- B'= -B'V' B + B B+B' j but this laft term is the velocity loft by B, and V—V' is the relative velocity of the bo¬ dies, therefore in the impaB of two hard bodies B + B' : B' as their relative velocity is to the velocity lofl by B. The fame thing may be {hewn when the bodies move in oppoftte direftions, in which cafe their relative velocity is V+V'. Prop. III. 258. To determine the velocities of two elaftic bodies after impulfe. If an elaftic body ftrikes a hard and immoveable plane, it will, at the inftant of collifion, be compreffed at the place of contaft. But as the elaftic body in- ftantaneoufly endeavours to recover its figure, and as this force of reftitution is equal and oppofite to the force of compreflion, it will move backwards from the plane in the fame direftion in which it advanced.—If two elaftic bodies, with equal momenta, impinge a- gainft each other, the effeft of their mutual compref- iion is to deftroy their relative velocity, and make them move with a common velocity, as in the cafe of A N I C S. 85 hard bodies. But by the force of reftitution, equal Theory, to that of compreflion, the bodies begin to recover ^ their figure,—the parts in contaft ferve mutually as points of fupport, and the bodies recede from each other. Now, before the force of reftitution began to exert itfelf, the bodies had a tendency to move in one direftion with a common momentum ; therefore, the body whofe effort to recover its figure was in the fame direftion with that of the common momentum, will move on in that direftion, with a momentum or moving force equal to the fum of the force of reftitution and the common momentum ; while the other body, whofe effort to recover from compreflion is in a direftion op¬ pofite to that of the common momentum, will move with a momentum equal to the difference between its force of reftitution and the common momentum, and in the direftion of the greateft of thefe momenta : Af¬ ter impulfe, therefore, it either moves in the direftion oppofite to that of the common momentum, or its mo¬ tion in the fame direftion as that of the common mo¬ mentum is diminilhed, or it is flopped altogether, ac¬ cording as the force of reftitution is greater, lefs, or equal to the common momentum. 259. In order to apply thefe preliminary obferva- tions, let us adopt the notation in the two preceding propofitions, and let v be the common velocity which the bodies would have received after impulfe, if they had been hard, and v', v" the velocities which the elaf¬ tic bodies B, B' receive after impaft. 260.. 1. If B follows B', then V is greater than V', and when B has reached B', they are both compreffed at the point of impaft. Hence, fince v is the common velocity with which they Avould advance if the force of reftitution were not exerted, we have V—i;=the velocity loft by B,and v—V'crthe velocity gained by B' in confe- quence of compreflion.—But, when the bodies ftrive to recover their form by the force of reftitution, the body B will move backwards in confequence of this force, while B' will move onward in its former direftion with an acce¬ lerated velocity. Hence, from the force of reftitution, B will again lofe the velocity V—v, and B' w ill, a fecond time, gain the velocity v—V' $ confequently, the whole velocity loft by B is 2 V—2iq and* the whole velocity gained by B' is 2 v—2 V'. Now, fubtrafting this lofs from the original velocity of B, we have V—2 V—2 v> for the velocity of B after impaft, and adding the ve¬ locity gained by B to its original velocity, we have V'+I^=I V' for. the velocity of B' after impaft j hence we have v'zz V 2 V 2VZZ2V—V ^"=:V'+ 2 v—2 V'= 2 v—V. Now, fubftituting in thefe equations, the value of v as found in Cafe 2. Prop. 2. we obtain , BV—B'V+2 B'V' v"=z B+B' BV'—B'V'+2BV B + B' 261. 2. When the bodies move in oppofite direftions or mutually approach each other, the body B is in pre- cifely the fame circumftances as in the preceding cafe j but M E C H but the body B' lofes a part of its velocity equal to 2 "y -p 2 V'—V'. Hence we have, by the fame reafoning that was employed in the preceding cafe, v' —2 v—V' v”— 2 “y-l-V', and by fubftituting inftead of v its value, as determined in Cafe 3. Prop. 2. or by merely changing the fign of V' in the two laft equations in the preceding corollary, wre obtain the two following equations, which will an- fwer for both cafes, by ufing the upper fign when the bodies move in the fame direction, and the under fign when they move in oppofite dir eft ions. B V—B'V=t2 B'V' v'~ b7+!5 „ rdnBV'rtrB'V' + aBV V) — . E + B' From the preceding equation the following corol¬ laries may be deduced. 262. Cor. 1. The velocity gained by the body that is ftruck, and the velocity loft by the impinging body, are twice as great in elaftic as they are in hard bodies for in hard bodies the velocities gained and loft were v—V', and V—v ; whereas in elaflic bodies the velo¬ cities gained and loft were 2V—2 V', and 2 V—2v. 263. Cor. If one of the bodies, fuppofe B', is at ref, its velocity \'—o, and the preceding equation be¬ comes A N I C S. and V—V', both the bodies will recoil or move back¬ wards after impact with the fame velocities which they had before impaft. For in the formulae in Cafe 2. with the inferior figns, when BrrB' and V—V', we have vf——V and 270. CoR. 9. If the bodies move in oppofite direc- VB—VB' v— ■. v—; 2VB B-ftB' 7 ~ B-fB' 264. Cor. 3. If one of the bodies B' is at reft, and their mafles equal, we have BrrB' and \'—o, by fub¬ ftituting which in the preceding formulae, we obtain ■y'—0, and v"—'V ; that is, the impinging body B re¬ mains at reft after impaft, and the body B' that is itruck when at reft moves on with the velocity of the body B that ftruck it, fo that there is a complete tranf- fer of B’s velocity to B'. 265. Cor. 4. If B' is at reft and B greater than B', both the bodies will move forward in the direclion of B’s motion } for it is obvious from the equations in Cor. 2. that when B is greater than B', v', and v" are both pofitive. 266. Cor. 5. If B' is at reft, and B lefs than B', the impinging body B will return backwards, and the body B' which is ftruck will move forward in the direc¬ tion in which B moved before the ftroke. For it is evi¬ dent that when B is lefs than B', v' is negative, and v" pofitive. 267. Cor. 6. If both the bodies move in the fame di¬ reclion, the body B' that is ftruck will after impadt move with greater velocity than it had before it. This is obvious from the formula in Cafe 1. of this propofi- tion. 268. CoR. 7. If the bodies move in the fame direc¬ tion, and if B=:B', there will at the moment of impadt be a mutual transfer of velocities, that is, B will move on with B'’s velocity, and B' will move on with B’s velocity. For in the formula; in Cafe 1. when B“B, vre have v'~\T' and v"—'V. 269. CoR. 8. When the bodies move in oppofte di¬ rections, or mutually approach other, and when B=B' tions, and VrrV'', we have v' 3B—B' B—3 B' 'VX B-f-B' ’ and vn =VX Hence it is obvious, that if B=3 B', B-fB' or if one of the impinging bodies is thrice as great as the other, the greateft will be Hopped, and the fmalleft will recoil with a velocity double oi that which it had before impadt. For fince B—3 B', by fubftituting this value of B in the preceding equations, we obtain ‘t/=0, and v"—2 V. 271. Cor. 10. If the impinging bodies move in op¬ pofite diredtions, and if B=zB', they will both recoil after a mutual exchange of velocities. For when B~B', we have v'——V', and y' r= V. 272. Cor. 11. When the bodies move in oppofite diredtions, the body which is ftruck, and the body which ftrikes it, will flop, continue their motion, or return backwards, according as BV—B'V is equal to, or greater or lefs than 2 B'V'. 273. Cor. 12. The relative velocity of the bodies after impadt, is equal to their relative velocity before impadt, or, which is the fame thing, at equal inftants be¬ fore and after impadt, the diftance of the bodies from each other is the fame. For in the different cafes we have v'—2v—V j v"—2 vz+zV. But the relative ve¬ locity before impadt is in the different cafes VrqpV', and the relative'velocity after impadt is y'—■y'—\ r+rV'. 274. Cor. 13. By reafoning fimilar to that which was employed in Prop. 2. Cor. 3. it may be (hewn that B+B' : 2 B as their relative velocity before impadt is to the velocity gained by B' in the diredtion of B’s mo¬ tion •, and B-f-B': 2 B' as their relative velocity before impadt is to the velocity loft by B in the diredtion of A’s motion. 275. Cor. 14. The vis viva, or the fum of the pro- dudts of each body multiplied by the fquare of its ve¬ locity, is the fame before and after impadt, that is, B y^-j-B'y',1=:BV,-f-B,^r,1< From the formulae at the end of Cafe 2. we obtain By* B—B'i* x BV14- B'V'* , : — - ■■ ■ and B+B'l1 B' 4BB'X oV2 * B'V'2 , hence their fum By'* xBV' B+B'j* BUB7!* X I/V7* -f-4 BB'x BV2 +177 : B+B',* Theory, BV*-l-b'V'2xB-B'2-KEB' _-By3 B,v/J> B+B'* 276. Cor. 14. If feveral equal elaftie bodies B, B", B'", B"", &c. are in contadt, and placed in the fame ftraight line, and if another elaftic body /S of the (ame magnitude impinges againft B, they will remain at reft, except the laft body B"", which will move on with the velocity of /3. By Art. 264. B will transfer MECHANIC S. »7 Theory, to B" all its velocity, and therefore B will be at reft, ' in the fame way B" will transfer to Bw all its velocity, and B" will remain at reft, and fo on with the reft ; but when the laft body B"" is fet in motion, there is no other body to which its velocity can be transferred, and therefore it will move on with the velocity which it received from B"', that is, with the velocity of /3. 277. Cor. 15. If the bodies decreafe in lize from B to B,w, they will all move in the dire&ion of the impinging body /3, and the velocity communicated to each body will be greater than that which is communi¬ cated to the preceding body. 278. Cor. 16. If the bodies increafe in magnitude, they will all recoil, or move in a direftion oppoftte to that of /3, excepting the laft, and the velocity commu¬ nicated to each body will be lefs than that which is communicated to the preceding body. Prop. IV. 279. To determine the velocities of two imperfect¬ ly elaftic bodies after impulfe, the force of com- preflion being in a given ratio to the force of reftitution or elafticity. Let B, B' be the two bodies, V, V' their velocities before impadt, v', v" their velocities after impaCt, and l : ;; as the force of compreflion is to that of reftitu¬ tion. It is evident from Cafe 1. Prop. 8. that in con- fequence of the force of compreflion alone we have, V-^=velocity loft by B j from reffion. v—V—velocity gamed by ±i 3 But the velocity which B lofes and TV gains by the force of compreflion will be to the velocity which B lofes and B' gains by the force of reftitution or elaftici¬ ty as 1 : n ; hence I : «—V—v : n\T—n v, the velocity loft by B1 from ela- 1: n—v—V: n v-n V' the velocity gained by B j fticity. therefore by adding together the two portions of ve¬ locity loft by B, and alio thofe gained by B', we ob¬ tain I 4-« V—1 -\-n v, the whole velocity loft by B, x _j_7/ <11—1 _}-7;V', the whole velocity gained by B. Hence by fubtrafting the velocity loft by B in confe- quence of collifton from its velocity before impadt, we fliall have 1/ or the velocity of B after iinpadt, and by adding the velocity gained by B' after collifion to its velocity before impact, we ftiall find v" or the velocity of B' after impaft, thus •y'rrV— 1 -j-7/ V —\ -\-nv the velocity of B after impafl. t;"—V'-j-1 -\-nv—1—nV the velocity of B after impaff. Now by fubftituting in the place of v its value as de¬ termined in Cafe 2. Prop. 2. we obtain , I-MXB'V—FV7 v=V B + B, • ,,-y, 1 TPxBV-BV' ^ B+B 280. Cor. 1. Hence by converting the preceding equation into analogies, B + B: i+«xBas the relative t r^ . i velocity of the bodies before ixnpaft is to the velocity gained by B' in the direction of B’s motion j and B+B': 1 +« X B' as the relative velocity of the bodies before impadt is to the velocity loft by B. 281. Cor. 2. The relative velocity before impadt is to the relative velocity after impadt as the force of com¬ preflion is to the force of reftitution, or as 1 : The relative velocity after impact is v"—v', or tak¬ ing the preceding values of thefe quantities v"—v'zzTV1 , r+//xBV^BV'l 1X B^V^-W _v,_ + ^ + ^' B + B' y+ 1 dividing by B + B' we have v"—v’=. V'-V+V—V + n x V-V'=«XV_V' — the relative velocity after impadt. But the relative velocity before impadt is V—V', and V—V' : n X V V'—1 : n. Q. E. D. The quantity V' has evident¬ ly the negative fign when the bodies move in oppoiite diredtions. 282. Cor. 3. Hence from the velocities before and after impadt we may determine the force of reftitution or elafticity. Prop. V. 283. To find the velocity of a body, and the di- redtion in which it moves after impinging upon a hard and immoveable plane. 284. Case i. When the impinging body is perfectly Wlien the hard. Let AB be the hard and immoveable jplane, and let the impinging body move towards AB in the j ^ § diredtion CD, and with a velocity reprefen ted by CD. Then the velocity CD may be refolved into the two velocities CM, MD, or MD, FD ; CM DF being a parallelogram. But the part of the velocity FD, which carries the body in a line perpendicular to the plane, is completely deftroyed by impadt, while the other part of the velocity MD, which carries the body- in a line parallel to the plane, will not be affedted by the collifion, therefore the body will, after impadt, move along the plane with the velocity MD. Now, CD : MD —radius : cof.^lCDM, therefore fince MD =rCF the fine of the angle of incidence CDF, the ve¬ locity before imp add is to the velocity after impadt, as radius is to the fine of the angle of incidence ; and fince AM —CD—MD, the velocity before impadi is to the velocity lojt by impadt, as radius is to the verfedfne of the complement of the angle of incidence. 285. Case 2. When the impinging body is perfedtly^Ten the clafic. Let the body move in the diredtion CD with ,1Sel^“ a velocity reprefented by CD, which, as formerly, may tiCt be refolved to MD, FD. The part of the velocity MD remains after impadt, and tends to carry the body parallel to the plane. The other part of the velocity FD is deftroyed by compreflion ; but the force of refti¬ tution or elafticity will generate a velocity equal to FD, but in the oppofite diredtion DF. Confequently the impinging body after impadt is folicited by two ve¬ locities, one of which would carry it uniformly from D to F in the fame time that the other would carry it uni¬ formly from M to D, or from D to N j the body will, therefore SB MECHANICS. When the body is im¬ perfectly elaftic. Theory, therefore, move along DE, the diagonal of the paral- —-y——; ie]0grjim DFEN, which is equal to the parallelogram DFCM. Hence the angle CDF is equal to the angle EDF, therefore, when an elajlic body impinges oblique¬ ly againjl an immoveable plane, it will be reflected from the plane, fo that the angle of reflexion is equal to the angle of incidence. Since CD, DE are equal fpaces deferibed in equal times, the velocity of the body after impadt will be equal to its velocity before impaft. 286. Case 3. When the impinging body is imperfeElly elaflic. In DF take a point m, fo that DF is to D m as the force of compreflion is to the force of reftitution or elafticity, and having drawn me parallel to DB, and meeting NE in e, join D e; then, if the impinging bo¬ dy approach the plane in the direction CD, with a ve¬ locity reprefented by CD, D e will be the direflion in which it will move after impaft. Immediately after compreflion, the velocity DF is deftroyed as in the laft cafe, while the velocity MD tends to carry the body parallel to the plane. But, by the force of reftitution, the body would be carried uniformly along D m, per¬ pendicular to the plane, while, by the velocity MDzr: DNzr/w e, it would be carried in the fame time along m e, confequently, by means of thefe two velocities, the body will deferibe D e, the diagonal of the parallelo¬ gram D * N. The velocity, therefore, before impaft is to the velocity after impaft as DC : D e, or as DE : D ?, or as fin. D £ E, fin. DE e, or as fin. T) emx fin. DE while the energy of the moving force, or the fum of all the moving poAvers, is equal t* 5°°. 300. Def. 2.—The impelled point of a machine is that point to Avhich the moving poAver is applied, if there is- only one poAver, or that point to Avhich all the moving powers are reduced, or at which the moving force is fup- pofed to ail. The working point of a machine is that point at Avhich the refiftance ails if it is fingle, or that point to Avhich all the refiftances are reduced, and at Avhich they are fuppofed to ail when combined. Thus in fig. 1. G is the impelled point of the machine, andpj 2 D the working point. Had a fingle force w been ap- 0 * plied at the point B to raife a fingle weight u, ailing at MECHANICS. 9* Theory, at the point A, then B would have been the impelled v""1"' point, and the working point of the machine. In the wheel and axle, the point of the wheel at which the rope touches its circumference is the impelled point, while the working point is that point in the circumfe¬ rence of the axle where the rope which carries the Weight is in contact with it. 301. Def. 3.—The velocity of the moving power, and the velocity of the reliftance, are refpedtively the fame as the velocity of the impelled point, and the velocity of the working point. 302. Def. 4.—The effett of a machine, or the luorh performed, is equal to the refiltance multiplied by the ve¬ locity of the working point; for when any machine raifes a mafs of matter to a given height in a certain time, the effeft produced is meafured by the produdl of the mafs, and the height through which it rifes, that is, by the product of the mafs by the velocity with which it moves. 303. Def. 5.—The momentum of impulfe is equal to the moving force multiplied by the velocity of the im¬ pelled point. Esplana- 304* In any machine that has a motion of rotation, *,0\°F l6* x I56 ^e velocity of the impelled point, and y the ym 0 ' velocity of the working point. When the machine is a lever, x, y will exprefs the perpendiculars let fall from the centre of motion upon the line of direction in which the forces a6t j and if the machine is’ a wheel and axle, x, y will reprefent the diameters of the wrheel and the axle refpeflively. In compound machines, which may be regarded as compofed of levers, (Art. 90.) x will reprefent the fum of all the levers by which the power a6ts, and y the fum of all the levers by which the refiftance adds. 305. Let P be the real preflure which the moving power exerts at the impelled point of the machine, and R the actual preflure which the mere reliftance of the work to be performed exerts at the working point, or which it dire&ly oppofes to the exertion of the power. Let a be the inertia of the power P, or the mafs of matter which the power P muft move with the veloci¬ ty of the impelled point, in order that P may exert its prefliire at the impelled point; and let b be the inertia of the reliftance R, or the mafs of matter which muft be moved with the velocity of the working point in the performance of the work. 306. Since the refiftance ariling from the fri&ion of the communicating parts is an uniformly retarding force, it may be meafured by a weight

^-pr, ■, u-f X *- i-fPb When R—o, we have a*/42, that of the Aveight be¬ ing 1, &c. If the refiftance is very great, compared Avith the poAver, the velocity Ihould at lealt be double of that Avhich would procure an equilibrium, in order that the machine Anil produce a maximum effedt. 315. If the velocity of the poAver, or its diftance from the centre of motion, be equal to, double, triple, qua¬ druple, &c. &c. of the velocity of the Aveight or re- ffftance, a maximum effedt Avill be produced Avhen the power P is equal to R X i-f’-v/2? R X T-byW > R X l + I/ Ay ^ X t + V ^c* Avhere ft is the refiftance or weight to be raifed. If the velocity of the porver lie very large, a maximum effedt will be produced when the poAver P is, at leaft, double of that which would procure an equilibrium. It ap¬ pears alfo from Mr Lellie’s paper, that in whatever way the maximum be procured, the force which impells the MECHANICS. 94 Theory, the weight can never amount to one-fourth part of the diredt adiion of the power 5 and that in machines where the velocity of the power is great, we may difregard the momenta of the connedling parts, and confider the force which ought to be employed as double of what is barely able to maintain the equilibrium. Chap. VIII. On the Equilibrium of Arches, Piers, and Domes. 3* 316. Def. I. An arch is reprefented in fig. 3. by the affemblage of Hones ab,cd,ef &c. forming the mafs ABMN, whofe inferior furface is the portion of a curve. The parts A, B are called the fpring of the arch, the line AB the /pan of the arch, C b its altitude, b its crown, ab the keystone, the curve or lower fur- face A B the intrados, and the roadway TUV the extrados ; P£), RS, the piers when they Hand between two arches, and the abutments when they are at the extremities of the bridge. ■Fig. 4. 317. Def. 2. A catenarian curve is the curve formed by any line or cord perfedtly flexible, and fufpended by its extremities. Thus if the chain ACB be fufpend¬ ed by its extremities A, B, it will by the adtion of gra¬ vity upon all its parts aflume the form ACB, which is called the catenary or catenarian curve. 318. There are three modes of determining the con- ftrudtion of arches •, the firft of which is to confider the arch as an inverted catenary j the fecond is to eftablilh an equilibrium betweeen the vertical preflures of all the materials between the intrados and extrados ; and the third is to regard the different arch-ftones as por¬ tions of wedges without fridtion, which endeavour by their own weight to force their way through the arch. The firft of thefe methods rvas given by the ingenious Dr Hook, and is contained in the following propoli- fcion. Prop. I. 319. To determine the form of an arch by con- fidering it as an inverted catenary, when its fpan, its altitude, and the form of the roadway or extrados are given. Fig. 5. Let a, b, c, d be a number of fpheres or beads con- nedted by a ftring, and fufpended by their extremities A, B ) they will form a catenarian curve A a £ c B, and be in equilibrio by the adtion of gravity. Each fphere is adted upon by two forces •, at its lower point by the weight of tire fpheres immediately below it, and at its upper point by the weight of the fame fpheres added to that of the fphere itfelf j that is, any fphere c is in «quilibrio from the refult of two forces, one of which *? produced by the weights c d e adling at the lower j.oint of b , while the other force arifes from the weight of b c d e adling at its upper point. The equilibrium of this chain of fpheres is evidently of the liable kind, as it will immediately recover its polition when the equilibrium is difturbed. Let us now fuppofe this arch inverted, fo as to Hand in a vertical plane as in Fig. 6. fig* 6. It will Hill preferve its equilibrium. For the relative pofitions of the lines which mark the diredlions remain unchanged by inverting the curve, the force of gravity continues the fame, and therefore the refult of Theory, thefe forces will be the fame, and the arch will be in ‘—■"Y'"— equilibrio. The equilibrium, however, which the arch now poffeffes is of the tottering kind, fo that the leaft difturbing force will deftroy it, and it will confequently be unable to fupport any other weight but its own. 3 20. Let us now fuppofe that it is required to form an equilibrated arch, whofe fpan is AB, whofe altitude is D k, and which ivill fupport the materials of a road¬ way, whofe form TUV is given. It is obvious, that if the fpheres a, b, c, d increafe in denfity from k towards a, the catenarian curve will grow lefs concave at its vertex e, and more concave towards its extremities A,B. Let us then fuppofe that the denfities of the fpheres a, b, c, d, e, &c. are refpedtively as a m, b n, c 0, dp, eq, &c. the vertical diftances of their refpedlive centres from the roadway TUV, the arch will have a form different from that which it would have affumed if the fpheres were of equal denfity, and will be in equilibrio when inverted as in fig. 6. Now, in place of the Fig. & fpheres a, b, c, d, e, &c. of different denfities, let us fub- ftitute fpheres of the fame denfity, and having the fame pofition as thofe of different denfities j let us then load the fphere a with a weight which, when combined with the weight of a, will be equal to the weight of the cor- refponding fphere a, that had a greater denfity j and let us load the other fpheres b, c, d, &c. with weights proportional to bn,c o,dp, &c. Then it is obvious that the preffure of each fphere when thus loaded upon that which is contiguous to it, is precifely equal to the preffure of the fpheres of different denfities upon each other, becaufe the denfity of thefe fpheres varied as their diftances from the roadway. But the arch com- pofed of fpheres of different denfities was in equilibria when inverted, therefore fince the loaded fpheres of the fame denfity have the fame pofition and exert the fame preffures, the arch compofed of thefe fpheres and fup- porting TUVB k A compofed of homogeneous materi¬ als, will be in equilibrio. Hence a roadway of a given form, and compofed of homogeneous materials, will be fup- ported by an arch whofe form is that of a catenary, each of whofe points varies in denfty as their di/lance from the furface of the roadway; or, which is the fame thing, A roadway of a given form, and compofed of homogeneous materials, will be fupported by an arch whofe form is that of a catenary, each of whofe points is a Pled upon by forces proportional to the diflances of thefe points from the furface of the roadway. 32I. Hence we have the following praflical method of afeertaining the form of an equilibrated arch, whofe fpan is AB, and altitude D k, and which is to fupport a roadway of the form T'U'V'. Let a chain Aa b ck~&, of uniform denfity, be fufpended from the points A, B, fo that it forms a catenary whofe altitude is D k, the required height of the arch. Divide AB into any number of equal parts, fuppofe eight, and let the vertical lines 1 m, 2n, 30, drawn from thefe points, interfeft the catenary in the points a, b, c. From the points a, b, c, k, r, s, t, fufpend pieces of chain of uni¬ form denfity, and form them of fuch a length, that when the whole is in equilibrio, the extremities of the chains may lie in the line T'U'V'j then the form Avhich the catenary A £ B now affumes, will be the form of an equilibrated arch, which, when inverted like AKB, will fupport the roadway TUV, fimilar to T'U'V'. This M E C H Theory. This is obvious from the laft paragraph, for the pieces of chain a m, b n, c o, k\J, &c. are forces afting upon the points a, b, c, k of the catenary, and are proportional to a m, b n, c 0, &c. the distances of the points a, b, ct i, &c. from the roadway. 322. An arch of this conftruftion will evidently an- fwer for a bridge, in which the weight of the materials between the roadway and the arch ftones is to the weight of the arch ftones, as the weight of all the pieces of chain fufpended from a, b, c, &c. is to the weight of the chain A £ B. As the ratio, howrever, of the weight of the arch ftones to the weight of the fuperincumbent materials is not known, we may affume a convenient thicknefs for the arch ftones, and if from this aflumed thicknefs their weight be computed, and be found to have the required ratio to the weight of the incumbent mafs, the curve already found will be a proper form for the arch. But if the ratio is different from that of the weight of the whole chain to the -weight of the fuf¬ pended chains j it may be eafily computed how much muft be added to or fubtra&ed from the pieces of chain, in order to make the ratios equal. The new curve which the catenary then affumes, in confequence of the change upon the length of the fufpended chains, will be the form of an equilibrated arch, the weight of whofe arch ftones is equal to that which we affumed. Scholium. 323. In moft cafes the catenarian curve thus deter¬ mined will approach very near to a circular arc equal to 120 degrees, which fprings from the piers fo as to form an angle of 60 degrees with the horizon. The form of the arch, however, as determined in the pre¬ ceding propofition, is fuited only to thofe cafes in which the fuperincumbent materials exert^a vertical preffure. A quantity of loofe earth and gravel exerts a preffure in almoft every diredtion, and therefore tends to deftroy the equilibrium of a catenarian arch. This tendency, however, may be removed by giving the arch a greater curvature towards the piers. This will make it approach to the form of an ellipfis, and make it fpring more ver¬ tically from the piers or abutments. 324. We ftiall now proceed to deduce the form of an arch and its roadway, by eftabliftiing an equilibrium a- mong the weights of all the materials between the arch and the roadway. Ihis method was given by Emerfon in his Fluxions, publifhed in 1742, and afterwards by Dr Hutton in his excellent work on bridges. Prop. II. 325. To determine the form of the roadway or extrados, when the form of the arch or intrados is given. g. 8- Let the lines AD, DE, EB, BF, FG, GH lie in the fame plane, and let them be placed perpendicular to the horizon. From the points D, E, B, &c. draAV the vertical lines D d, Ee, B b, &c. and taking D p of any length, make E r equal to D p, &c. and complete the parallelograms pc,qr. Again, make B szzqe, and com¬ plete the parallelogram /r; in like manner make Ek^isb, and complete the parallelogram F f ; and fo on with all the other lines, making the fide of each parallelogram ©qual to that fide of the preceding parallelogram which A N I C S. is parallel to it. Let us new flippofe that the lines CD, DE, EB, &c. can move round the angular points D, E, B, F, &c. the extremities A, C being immove¬ able ; and that forces proportional to Dr/, Ee, BZ», &c. are exerted upon the points D, E, B, F, &c. and in the direflion Dr/, Ee, &c. Now, by the refolution of forces, the force Dr/ may be refolved into the forces D c, D p, the force E e into the forces E y, E r, and the force B b into the forces B j-, B /, and fo on -with the reft. The force D c produces no other effedt than to prefs the point A on the plane on which it refts, and is therefore deftroyed by the refiftance of that plane 5 but the remaining force Dp tends to bring the point D to. wards E, and to enlarge the angle ADE 5 this force, however, is deftroyed by the equal and oppofite force E q, and in the fame way the forces E r, B t, F x are deftroyed by the equal and oppofite forces Br, Ek, G v, while the remaining force G u> is deftroyed by the re¬ fiftance of the plane which fupports the point C. When the lines AD, DE, &c. therefore are adted upon by vertical forces proportional to D or dE)p \ fin. AD = £ b r Ef—B, produced in C. We may take b c to exprefs the prelTure of all that is above it, propagated in this dire&ion to the joint KL. We may alfo fuppofe the weight of the courfe HL united in b, and adling on the vertical. Let it be reprefented by b F. If we form the paral¬ lelogram ^ FGC, the diagonal b G will reprefent the direction and intenfity of the whole preffure on the joint KL. Thus it appears that this preffure is conti¬ nually changing its diredlion, and that the line, which will always coincide with it, muft be a curve concave downward. If this be precifely the curve of the dome, it will be an equilibrated vaulting •, but fo far from being the ftrongeft form, it is the weakeft, and it is the li¬ mit to an infinity of others, which are all ftronger than it. This will appear evident, if we fuppofe that b G does not coincide with the curve A B, but paffes without it. As we fuppofe the arch-ftones to be ex¬ ceedingly thin from infide to outfide, it is plain that this dome cannot ftand, and that the weight of the upper part will prefs it down, and fpring the vaulting outwards at the joint KL. But let us fuppofe, on the other hand, that b G falls within the curvilineal ele¬ ment b B. This evidently tends to pufli the arch-ftone inward, toward the axis, and would caufe it to Aide in, finee the joints are fuppofed perfectly fmooth and flip¬ ping. But finee this takes place equally in every ftone of this courfe, they muft all abut on each other in the vertical joints, fqueezing them firmly together. There¬ fore, refolving the thruft b G into two, one of which is A N I C S. '99 perpendicular to the joint KL, and the other parallel Theory, to it, we fee that this laft thruft is with flood by the 'r~~ vertical joints all around, and there remains only the thruft in the diredlion of the curve. Such a dome muft therefore be firmer than an equilibrated dome, and can¬ not be fo eafily broken by overloading the upper part. When the curve is concave upwards, as in the lower part of the figure, the line b C always falls below B bt and the point C below B. When the curve is con¬ cave downwards, as in the upper part of the figure, ’b C' paffes above, or without b B. The curvature may be fo abrupt, that even b' G' fliall pafs without 'b B', and the point G' is above B7. It is alfo evident that the force which thus binds the ftones of a horizon¬ tal courfe together, by pufhing them towards the axis, ■will be greater in flat domes than in thofe that are more convex j that it w ill be ftill greater in a cone 5 and greater ftill in a curve whofe convexity is turned inwards : for in this laft cafe the line b G will deviate moft remarkably from the curve. Such a dome will ftand (having poliftied joints) if the curve fpringe from the bafe with any elevation, however fmall} nay, fince the friction of two pieces of ftone is not lefs than half of their mutual preffure, fuch a dome wfill ftand, although the tangent to the curve at the bottom fhould be horizontal, provided that the horizontal thruft be double the weight of the dome, which may eafily be the cafe if it do not rife high. “ Thus we fee that the liability of a dome depends on very different principles from that of a common arch, and is in general much greater. It differs alfo in another very important circumftance, viz. that it may be open in the middle : for the uppermoft courfe, by tending equally in every part to Hide in toward the axis, preffes all together in the vertical joints, and atls on the next courfe like the key-ftone of a common arch. Therefore an arch of equilibration, which is the weakeft of all, may be open in the middle, and carry at top another building, fuch as a lantern, if its weight do not exceed that of the circular legment of the dome that is omitted. A greater load than this would indeed break the dome, by caufing it to fpring up in fome of the lower courfes ; but tins load may be increafed if the curve is flatter than the curve of equi¬ libration : and any load w hatever, which will not cruth the ftones to powder, may be fet on a truncate cone, or on a dome formed by a curve that is convex toward the axis ; provided always that the foundation be effec¬ tually prevented from flying out, either by a hoop or by a fufficient mafs of folid pier on which it is fet. “ We have feen that if b G, the thruft compounded of the thruft b C, exerted by all the courfes above HILK, and if the force F, or the weight of that coiuffe, be everywhere coincident with b B, the ele¬ ment of the curve, we {hall have an equilibrated dome j if it falls within it, we have a dome which w ill bear a greater load; and if it falls without it, the dome will break at the joint. We muft endeavour to get analytical expreflions of thefe conditions. Therefore draw the ordi¬ nates b £ b", BDB", C c! C'7. Let the tangents at b and b" meet the axis in M, and make MO , ?dP, each equal to b c, and complete the parallelogram MONP. and draw OQ perpendicular to the axis, and produce b F, cutting the ordinates in E and e. It is plain that MN N 2 is M E C H is to MO as the weight of the arch HA h to the thruft b c which it exerts on the joint KL (this thruft being propagated through the courfe of HILK) ; and that M(), or its equal £ or M pd r V2AM—-M* n Let PC) be a feel ion of the cylinder P in fig. 2. and let all the elements of the cylinder be projefted upon this circular feftion in 2/, d’, d". Let ACB, the primitive angle of torfion, be called A, and let this angle, after the time /, become AC b, fo that it has been diminiftied by the angle BCZ>—Mq then AC b—A—Mm the angle of torfion after the time t. Since the force of torfion is fuppofed to be pro¬ portional to the angle of torfion, the momentum of the force of torfion mull be fome multiple of that angle, or 22 X A—M, 22 being a conftant coefficient, whofe va¬ lue depends on the nature, length, and thicknefs of the metallic wire. If, therefore, we call v the velocity of any point d at the end of the time t, when the angle of torfion becomes AC b, and r~C d the diilance of the point d from the axis of rotation C, we fliall have, by £he principles of Dynamics, M ■Rut—7==-^==== renrefents an arch or angle whof® ^UV2AM—Ml 1 * t radius is A and whofe verfed fine is M, which arch vanilhes when IVI~o, and which becomes equal to 90 when M=rA. Therefore the time of a complete ofcil- lation will be •p r* t=f-n X 1800. 345. In order to compare the force of torfion with the force of gravity in a pendulum, We have for the time of a complete ofcillation of a pendulum whofe length is /, g being the force of gravity, T T= § 180®. •p A rP* J 22 Therefore, fince the time in which the cylinder ofcil- lates muft be equal to the time in which the pendulum ofcillates, we have ’ a ^ 7 i. - x i8o°=— x X 1800. S Hence dividing by 1800, and fquaring both fides, we obtain rp A 1 / J n \ g' We mull therefore find for a cylinder the value of jTp r1, or the fum of all the particles multiplied by the fquares of their diftances from the axis. Now, if we make ^=6.28318 the ratio of the circumference of a circle to its radius, a— radius of the cylinder, *= its length, d— its denfity •, then we fliall have for the area of its bafe which multiplied by A gives the folid 2 content of the cylinder , and this multiplied by Theory. i d gives —for the fum of all its particles. MECHANICS. But IO3 as this is to be multiplied by the fum of the fquares of all the diftances of the particles from the centre But the number: f { G, we (hall have fp r'1 — A d fecond from A to B', where the force of torfion, as well Theory, as the other forces, concur in diminifhing u or retarding the motion.. 347. Ex. 1. If S=rwX A—M we (hall have for the date of motion in the fir ft portion BA of particles in the cylinder, or the mafs ^ of the cy- «X2AM—M*-j- linder, is —^ , therefore fubflituting ft, inftead of 2/«XA—M^1 this value of it in the preceding equation, we have LC (l* Jp r1— ——, and, dividing both fides by «, we have fpr' ■ —, and, ext raffing the fquare root and mul- n 2 n 0 tiplying by 180, it becomes v-j-l v-f-1 ^ a* Hence, when the angle of torfion becomes equal to no-> thing, or A-—M—o, we have 2/«A,'('1 1 7+! n A1 2 m A^1 >p r v + i r,p r* = UU/,/i J a which dividing hyJ~—, becomes 2 w Ay+I n A1—J X1800 2 n Ul—- y-f-l X1800. Therefore riL J a* T- p. a 2 n T X 180, and fince^-— p rz —— rr —, and by reduffion But r tt is the mg1 J 2/ 6 weight W of the cylinder, therefore, by fubftituting W .Pa* inftead of g p, we obtain n———, a very fimple formu¬ la for determining the value of n from experiments. * If it were required to find a weight (^, which, affing at the extremity of a lever L, would have a mo¬ mentum equal to the momentum of the force of torfion when the angle of torfion is A—M, we muft make QX L—a X A—M. 346. In the preceding inveftigation we have fuppo- fed, what is conformable to experiment, that the force of torfion is proportional to the angle of torfion, which gives us n X A—M for the momentum of that force. Let us now fuppofe that this momentum is altered by any quantity S, then the momentum of the force of tor¬ fion will become n X A—M—S', and the general equa¬ tion will aflume this form n X A—M—S X "t—u : J a and by multiplying in place of / its value taking the fluent, we have « X 2 AM—M2—2/^SM— Now, in order to find the value of T or a complete ofcillation, we muft divide the ofcillation into two parts, the firft from B to A, where the force of torfion accelerates the velocity a, while the retarding force, Let us now confider the other part of the motion from A to B', and fuppofe the angle AC Z/urM', we (hall find, by calling U the velocity of the point A, n M7* m M'v+1 U*—a* rp r* ~+-,+r-=-— Then, by fubftituting inftead of U its value as lately found, and taking the fluents, we (hall have, w hen the velocity vaniflies, or when the ©fcillation is tmifhed, A—M'=- 2m A11 i_I 4-M,,’+i X - ^ «Xv + 2 A-J-lVi' and if the retarding forces are fuch, that at each ofcilla¬ tion, the amplitude is a little diminilhed, we (hall have for the approximate value of A—M? »X*+I and if the angle A—M' is fo fmall that it may be treated as a common fluxional quantity, we (hall then have for any number of ofcillations NX = X aM ■, and v—1 Mv—1 Ay where M reprefents the angle to which A becomes equal after any number of ofcillations N. Hence we obtain M—- f y 2 ffl X ^ “f" I Nx — + ^ «X» + I 1 \ 1 7 - )X A”-1 } v—1 which determines the value of M after any number of ofcillations N. 348. Ex. 2. If S=mx A—M]*1; - m'X A—M V/, m' - - . o. , and being different values of and y, we (hall obtain ari.mg trom the refiftance of the air and the imper- by following the mode of inveftigation in the laft ex- feet ion of elafticity, diminiflies the velocity u ; and the ample, . n/r 2 m A*1 1 n X A—M=r x » + A-f-M +M’ t-i + 2m' Ay+I XM11^1 +1 A-fM and if the retarding force is much lefs than the force of torfion, we (hall have for an approximate value of /*X-A—M 104 Theory. Torfion balance. Fig. 3. mechanics. « x A—M~ 2 m At . 2 m'A*' ■ + „_|-i * ^4Q. Ex. 3. In general, if S —m X A—M[ -j- X 'A=M>' + m" x A=Mr" + ^ x A-Mj"", &e. we fhall always have for an ofcillatxon when S is fmaller than the force of torlion. — 2mA* 2m!A*' 2m"A'" 2m!"A*1' ^ „xa-m=—+7;rr+-7+r+y''+i ,!Sc- 2 co. Having thus given after Coulomb, the mode of deducing formulae for the ofcillatory motion of the cy¬ linder, we fhall proceed to give an account of the refults of his experiments. ' „ In thefe experiments M. Coulomb employed the torh°H balance reprefented in fig. 2. in which he fufpended cylinders of different weights from iron and brals wires of different lengths and thickneffes ; and by obferying carefully the duration of a certain number of ofcillations, he was enabled to determine, by means of the preceding formulae, the laws of the force of torfion relative to the length, the thicknefs, and the nature of the wires em¬ ployed. If the elafticity of the metallic wires had been perfect, and if the air oppofed no refiftance to the of- cillating cylinder, it would continue to ofcillate till its motion was flopped. The diminution of the ampli¬ tudes of the ofcillations, therefore, being produced fole- ly by the imperfection of elafticity, and by the refin¬ ance of the air, M. Coulomb was enabled, by obferving the fucceffive diminution of the amplitude of the olcil- lation, and by fubflrafting the part of the change which was due to the refiftance of the air, to afcertam, with the afiiftance of the preceding formulae, according to what laws this elaftic force of torfion was changed. o c 1. From a great number of experiments it appeared, that when the angle of torfion was not very great, the ofcillations were fenfibly ifochronous $ and therefore it may be regarded as a fundamental law, l/iat Jor all metallic wires, when the angles of torfion are not very .real, the force of torfion is fenfibly proportional to the angle of torfion. Hence, as the preceding formulae are founded on this fuppofition, they may be fafely applied to the experiments. _ 352. In all the experiments, a cylinder of two pounds weight ofcillated in twice the time employed by a cy¬ linder which weighed only half a pound ; and there¬ fore the duration of the ofcillations is as the fquare root of the weights of the ofcillating cylinders.. Confequently the tenfion of the wires has no fenfible influence upon the force of torfion. If the tenfions however be very great relative to the ftrength of the metal, the force of torfion does fuffer a change 5 for when the weight of the cylinder, and confequently the tenfion of the wire, is increafed, the wire is lengthened, and as this diminilhes the diameter of the wire, the duration of the ofcillation muft evidently be affefted. 3 c 3. When the lengths of the wires are varied without changing their diameters or the weights of the cylin¬ ders, the times of the fame number of ofallations are as the fquare roots of the lengths of the wires, a relult alfo deducible from theory. 354. When the diameters of the wires are varied without changing their lengths, or the weight of the cylinders, the momentum of the force of torfion varied as the fourth power of the diameters of the wires. Now this refult is perfeftly conformable to theory 5 for if we fuppofe two wires of the fame fubftance, and of the fame length, but having their diameters as one to two, it is obvious that in the wire whofe diameter is double of the other, there are four times as many parts extended by torfion, as in the fmaller wire, and that the» mean extenfion of all thefe parts will be proportional to the diameter of a wire, the fame as the mean arm of a lever is, relative to the axis of rotation. Hence it ap¬ pears that, according to theory, the force of torfion of two wires of the fame nature and of the fame length, but of different diameters, is proportional to the fourth power of their diameter. 355. From this it follows in general, that in metallic wires the momentum of torfion is directly in the com¬ pound ratio of the angle of torfion and the fourth power of their diameter, and inverfely as the length of the wires. If a therefore be the angle of torfion, X the length of the thread, J its diameter, and F th© force of torfion, we lhall have Theory. where w is a conftant coefficient for wires of the lame metal, depending on the tenacity of the metal, and deducible from experiment. 3 56. When the angle of torfion is not great, relatives to the length of the wire, the index of the cylinder re¬ turns to the pofition which it had before the torfion took place, or, in other words, the wire untwifts itfelf by the fame quantity by wffiich it had been twilled. But when the angle of torfion is very great, the wire does not completely untwift itfelf, and therefore the centre of torfion will have advanced by a quantity equal to that which it has not untwifted.—When the angle of torfion was below 450, the decrements of the ampli¬ tudes of the ofcillations were nearly proportional to the amplitudes of the angle of torfion •, but when the angle exceeded 450, the decrements increafed in a much greater ratio.—The centre of torfion did not begin to advance or be difplaced till the angle of torfion wa* nearly a femicircle : its difplacement was very irregular till the angle was one circle and 10 degrees, but be¬ yond this angle the torfion remained nearly the fame for all angles. 357. The theory of torfion is particularly ufeful in delicate refearches, where fmall forces are to be afeer- tained with a precifion which cannot be obtained by ordinary means. It has been fuccefsfully employed by Coulomb in difeovering the laws of the forces of electricity and magnetifm, and in determining the refiftance of fluids when the velocities are very fmall. PART MECHANICS. Pratt ical Mechanics. PART II. ON THE CONSTRUCTION OF MACHINERY. i OS Pratticai Mechanics. 358. WE have already feen, when confidering the maximum effects of machines, the various eaufes which afteft their performance. It appeared from that invef- tigation, that there mull be a certain relation between the velocities of the impelled and working points of a machine, or between the power and the refiftance to be overcome, before it can produce a maximum effedt, and therefore it muft be the firft objedt of the engineer to afcertain that velocity, and to employ it in the con- ftrudlion of this machine. The performance of the ma¬ chine is alfo influenced by the fridlion and inertia of its various parts \ and as both thefe adl as refiftances, and therefore deftroy a conflderable portion of the impel¬ ling power, it becomes an objedl of great importance to attend to the Amplification of the machinery, and to af¬ certain the nature of fridlion fo as to diminifli its ef- fe£l, either by the application of unguents or by mecha¬ nical contrivances. Since the impelled and working points of a machine are generally connected by means of toothed wheels, the teeth mufl: be formed in fuch a manner, that the wheels may always adt upon each other with the fame force, otherwife the velocity of the machine will be variable, and its ftrudlure foon injured by the irregularity of its motion. The irregular mo¬ tion of machines fometimes arifes from the nature of the machinery, from an inequality in the refiftance to be overcome, and from the nature of the impelling power. In large machines, the momenta of their parts are generally fufficient to equalize thefe irregularities j but in machines of a fmall fize, and in thofe where the irregularities are confiderable, we muft employ fly¬ wheels for regulating and rendering uniform their vari¬ able movements. Thefe various fubjedls, and others intimately connedted Avith them, we ftiall noAv proceed to difcufs in their order. Chap. I. On the 'Proportion between the Velocity of the Impelled and Working point* of Machines, and be¬ tween the Power and Reftfance, in order that thetj may perform the greattjl work. 359. In the chapter on the maximum effedl of ma¬ chines avc have deduced formulae containing x and y, the velocities of the impelled and working points of the ma¬ chines,and including every circumftance which can affedt their motion. The formula Avhich exhibits the value of y, or the velocity of the working point, affumes various forms, according as vte negledt one or more of the ele¬ ments of Avhich it is compofed.— When the Avork to be performed refifts only by its inertia, Avhich is the cafe in urging round a millftone or heaAry flv, the quantity R may be negledted, and the fecond formula, (Page 92. col. 2.) fhould be employed. In fmall machines, and particularly in thofe Avhere the motion is conveyed by wheels Avith epicycloidal teeth, the fridlion is very trifling, and the element cp may be fafely omitted. In corn and faw mills, the quantity b or the inertia of the refiftance may be left out of the formula, as the mo¬ tion communicated to the flour or to the farv duft is too finall to be fubjedted to computation. In ma- Vol. XIII. Part I. chines where one heavy body is employed to raife an¬ other merely by its weight, the inertia of the power and the refiftance, viz. a,b, are proportional to P, R, the powers and refiftances themfelves, and confequently P, R may be fubftituted in the formula, in the place of a, b.—The engineer therefore muft confider, before he conftrudt his machine, Avhat elements fhould enter into the formula, and Avhat ftiould be omitted, in order that he may adapt it to the circumftances of the cafe, and obtain from his machine the greateft poflible effedl. 360. When the inertia of the poAver and that of the re- To find die fiftance are proportional to the poAver and refiftance them- relation be- felves 5 and when the inertia and ffidtion of the machine-twee? tlie „ may be omitted, the formula becomes j from which the following table is computed, which contains the values of y for different values of P; R be- a machine.. ing fuppofed =10, and m—\. TABLE containing the bef Proportions between the Ve¬ locities of the Impelled and Working Points of a Ma¬ chine, or between the Levers by which the Power and Reffance a£l. Proportional value of thi impelling power, or P. 1 2 3 4 5 6 7 8 9 10 11 12 *3 H J5 16 r7 18 *9 Value of the ve¬ locities of the working point or y ; or of the lever by which the re¬ fiftance atts, that of x being r. Proportional value of the impelling I power, or P. 0.048809 O.095445 O.140175 O.183216 O.224745 0.26491X 0-3°3^4I 0.341641 0-3784°J 0.414214 0.449x38 0.483240 o-5j6575 °-549I93 0.581139 0.612451 0.643168 0-67332° 0.702938 20 21 22 23 24 25 26 27 28 29 3° 40 5° 60 70 80 90 100 V alue of the. ve¬ locities of the working point or y ; or of the lever by which the re¬ finance atts, that of x being 1. °-73205I 0.760682 0.788854 0.816590 0.843900 0.870800 0.897300 0-92350° 0.949400 0.974800 1.000000 1.236200 1'44950° 1.645700 1.828400 2.ooooco 2.162300 2.316600 In order to explain the ufe of this table, let us fup- poft that it is required to raife one cubic foot of water in a fecond, by means of a ftream which diftharges three cubic feet of water in a fecond ; and let it be re¬ quired to find the conftruttion of a Avheel and axle for performing this work ; that is, the diameter of the axle, that of the wheel being 6. Here the poxver is evident¬ ly 3 cubic feet, while the refiftance is only one cubic foot, therefore P=3 R 3 but in the preceding fable O R=xo, io6 MECHANICS. Practical n mo, confcquently P=3 X 10=030. But it appears from Mechanic*, table that when P=30, y or the diameter of the axle T is I, upon the fuppofition that the diameter a of the wheel is I } but as x muft be = 6, we {hall have y=6. 361. Inftead of uling the preceding table, we might find the belt proportion between X and y by a kind of tentative procefs, from the formula —-—-— ---.which ^ ’ Ptf’ + Ry* ’ exprefies the work performed. This method is indeed Practical tedious; and we mention it only for the fake of Ihowing Mechanic* the conformity of the refults, and of proving that there y"" is a certain proportion between x and y which gives a maximum effedt. Let x=z6, as in the preceding para¬ graph, and let us fuppofe y to be fucceflively 5, 6, and 7, in order to fee which of thefe values is the beft* Since P=3, R=l, and x=z6, we have When y=5 When y=6 When y—7 P a? R y—R* y* 3 X 6 X i X 5—1X5X5 _ 6 5 P^ + Ry* ” 3 X6x6 + i X5X 5 P a: Ry—R*y*_3 X 6 x I X 6—I x 6x 6 72 _ P^-i-Ry* “ 3 x6x6 + i x6x6 —144 P * R y—K*ya_3 X 6x 1X 7—1 X 7 X 7_ 77 _ P «*-j-Ry* ~ 3 X 6 x 6-j-i X 7 X 7 _157 0.488 =0.500 0.49045 It appears therefore that when y=5> 6, 7? the work per¬ formed is 0.488 ; 0.5000 ; 0.49045 5 fo that the effedt is a maximum when y=6, a refult limilar to what was To find the obtained from the table. belt proper- 362. When the machine is already conftrudted, x tion be- and y cannot be varied fo as to obtain a maximum ef- tween the The fame objedt however will be gained by pro- the'refilt^ perly adjufting the power to the work when the work aace. " cannot be altered, or the work to the power when the power is determinate. The formulae in Prop. 2. Chap. 7. exhibit the values of R under many circumftances, and it depends on the judgment of the engineer to feledl fuch of them as are adapted to all the conditions of the cafe. 363. The following table is founded on the formula which anfwers to the cafe where the inertia of the impelling power is the fame with its pref- lure, and where the inertia and the fridlion of the ma¬ chine may be fafely negledted. The fecond column contains the different values of R correfponding to the values of y in the firft column. The numbers in the third column fhew the ratio of y to R, or they have the fame proportion to 1, which R has to the refiftance which will balance P. In the table it is fuppofed that P=i and *=1. Table containing the bejl proportions between the Power and the Refiftance, the inertia of the impelling power being the fame with its prejfure, and the friBion and inertia of the Machine being omitted. Values of y, or the velo¬ city of the working point; x being equal to 1. V alues of R, or the refift¬ ance to be overcome, P being — 1. I.8885 i>3928 0.8986 0.4142 0.1830 o.mi 0.0772 0.0580 0.0457 Ratio of R to the refiftance which would balance P. Values oi y, or the velo¬ city of the working point; x being equal to 1. O.4724 to I O.4639 0-4493 0.4142 0.3660 o-3333 0.3088 0.2900 0.2742 7 8 9 10 11 12 r3 14 *5 \ Values of R, or the refift- • ance to be overcome, P being — i. O.03731 O.03125 O.02669 O.O2317 O.O2037 0.01809 0.01622 O.OI466 O.OI333 Ratio of R to the refiftance which would balance P. O.26117 to I 0.25000 O.2402I O.23170 O.22407 — O.21708 0.21086 O.20524 O.I9995 364. To exemplify the ufe of the preceding table, let us fuppofe that we are to raife water by means of a fimple pulley and bucket, with a power =10, and that it is required to find the refiftance R, or the quan¬ tity of water which muft be put into the bucket, in or¬ der that the work performed may be a maximum. In the fimple pulley, at, y, the arms of the vertical le¬ vers or the velocities of the impelled and working points are equal} and fince x is fuppofed in the table to be = 1 , we have y=i, which correfponds in the table with 0.4142, the value of R, P being = 1 in the ta¬ ble : But in the prefent cafe P=ro. Therefore, 10 : 1=0.4142 : 4.142, the value of R when P=io. 365. The fame refult might be obtained in a more circuitous method by means of the formula —* P**+Ry* » which expreffes the performance of the machine. Thus, let a=i 5 y=i j P= 1 o, and let us fuppofe R fuccef- fively equal to 35 4; 4.142} 5} fo that we may de¬ termine which of thefe values gives the greateft per¬ formance. WhcR MECHANICS. Practical Mechanics. When R=3, the preceding formula becomes = — =1.61 u. io+3 J3 When R=4, the formula becomes X ^—4><4 -—^4 _I>7I . . io-f-4 14 ' When R=4.i42, the formula becomes 10 X M+Mi 10+4.142 14.142 When R=5, the formula becomes I-° —--X ^ Sr j,5555 io + j 15 107 Practical Mechanics. Hence it appears, that when R=3 ; 45 4.142 5 5 j the work performed is refpeaively — 1.6154 • 1.-140 < X’^STi 1*6666 j fo that the work performed is a maximum when R is =4.142, the fame refult which was ob¬ tained from the table. Chap. II. 0/t the Simplification of Machinei'y. 366. As the inertia of every machine adds greatly to the refiftance to be overcome, and as the friftion of the communicating parts is proportional to the preflure, it becomes a matter of great praftical importance, that the different parts of a machine fliould be proportioned to the ftrains to which they are expofed. If the beam of a fteam-engine, for example, is larger than what is neceffary, an immenfe portion of the impelling power muff be deftroyed at every Itroke of the pifton, by drag¬ ging the fuperfluous mafs from a ftate of reft into mo¬ tion *, the preffure upon the gudgeons will alfo be in- creafed, and their fri&ion in their fockets proportional¬ ly enlarged. The engineer, therefore, Ihould be well •acquainted with the ftrength of the materials of which the machine is to be conftru£ted, and fhould frame its different parts in fuch a manner that they may not be heavier than what is neceffary for refifting the forces with which they are urged.—When the motions of the machine are neceffarily irregular, and when the ma¬ chine may be expofed to accidental ftrains, the parts muft be made confiderably ftronger than what is ne- -eeffary for refifting its ordinary ftrains ; but it is not often that fuch a precaution fhould be obferved. The gudgeons of water-wheels, and of the beams of fteam- engines, ought to be made as ftiort and fmall as pofli- ble, as the friclion increafes with the rubbing furfaces. This is very feldom attended to in the conftruclion of water-wheels. I he diameter of the gudgeons is fre¬ quently thrice as large as what is neceffary for fupport- ing the weight of the wheel. 367. In the conftruftion of machinery we muft not only attend to the fimplification of the parts, but alfo to the number of thefe parts, and the mode of con- nefting them. From the nature and quantity of the work to be performed, it is eafy to afcertain the velo¬ city of the working point which is moft proper for per¬ forming it. Now this velocity may be procured in a variety of ways, either by a perplexing multiplicity of wheels, or by more fimple combinations. The choice of thefe combinations muft be left folely to the judge¬ ment of the engineer, as no general rules can be laid down to direct him. It may be ufeful, however, to remark, that the power ftiould always be applied as near as poflible to the working point of the machine, and that when one wheel drives another, the diameter of the one fliould never be great, when the diameter of the other is very fmall. The fize of wheels is often determined from the ftrains to which they are expofed. If, for example, we are obliged to give a certain velo¬ city to an axle by means -of a wheel with 120 teeth and if the force with which this wheel is urged, re¬ quires the teeth to be at leaft one inch thick in order to prevent them from breaking, we fnall be obliged to make its diameter at leaft feven feet; for fuppofino- the fpaces between the teeth to be equal to the thicknefs of the teeth, the circumference of the wheel muft at leaft be equal to 120 -f-1 20= 240 inches, the fum of the teeth and their intervals, which gives a diameter of fix feet eight inches. There are fome cafes where our choice of combination muft be direaed by the nature of the machinery. If the work to be performed is a load raifed ivith a certain velocity by means of a rope wind¬ ing round a hollow drum, and if the fimpleft combina¬ tion of mechanical powers for producing this velocity fhould give a fmall diameter to the drum, then this com¬ bination muft give way to another which correfponds with a larger fize of the drum, for, on account of the inflexibility of the ropes, a great portion of the impel¬ ling power would be wafted in winding them about the circumference of a fmall drum. o a. ue auvantages 01 nmplitying machinery are Dcfcriptiora well exemplified in the following capftane, which unites a Power* great ftrength and fimplicity. It is reprefented in fin. 4. J"1 caP- where AD is a compound barrel compofed of two cy- pfate linders of different radii. The rope DEC is fixed atCCCXXIII. the extremity of the cylinder D j and after pafling over %• 4* the pulley E, which is attached to the load by means ot the hook F, it is coiled round the other cylinder D and fixed at its upper end. The caprtane bar AB urges the compound barrel CD about its axis, fo that while the rope coils round the cylinder D it unwinds it- lelf Irom the cylinder C. Let us fuppofe that the dia¬ meter of the part D of the barrel is 21 inches, while the diameter of the part C is only 20 inches, and let the pulley E be 20 inches in diameter. When the barrel AD, therefore, has performed one complete revo¬ lution by the preffure exerted at B, 63 inches of rope equal to the circumference of the cylinder, will be fa! thered upon the cylinder D, and 60 inches will be un- winded from the cylinder C. The quantity of wound rope, therefore,, exceeds the quantity that is unwound by 63-—6o__3 inches, the difference of their refpe&ive perimeters j and the half of this quantity, or if inches, will be the fpace through which the load or pulley E moves by one turn of the bar. If a fimple capftane ot the lame dimenfions had been employed, the length ot rope coiled round the barrel would have been 6q O 2 inches j io8 MECHANICS Practical inches 5 and the fpace deferibed by the pulley, or load Aiechaiiics.^ to ioe overcome> Would have been 30 inches. Now, as the power is to the weight as the velocity of the weight is to the velocity of the power, and as the velocity of the power is the fame in both capftanes, the weights which they will raife will be as i~ to 30. If it is wifh- ed to double the power of the machine, we have only to cover the cylinder C with lathes a quarter of an inch thick, fo that the difference between the radii of each cylinder may be half as little as before ; for it is obvious that the power of the capftane increafes as the difference between the radii of the cylinders is diminilh- ed. As we increafe the power, therefore, we increafe the ftrength of our machine, wThile all other engines are proportionably enfeebled by an augmentation of power. Were we for example to increafe the power of the common capftane, we muft diminifti the barrel .in the fame proportion, fuppofing the bar A B not to ad¬ mit of being lengthened, which will not only diminifti its ftrength, but deftroy much of its power by the ad¬ ditional flexure of the rope.—This capftane may be ea- fily converted into a crane by giving the compound barrel a horizontal pofition, and fubftituting a winch inftead of the bar AB. The fuperiority of fuch a crane above the common ones does not require to be pointed out ; but it has this additional advantage, that it allows the weight to flop at any part of its progrefs, without the aid of a ratchet wheel and catch, becaufe the two parts of the rope pull on the contrary Tides of the barrel. The rope indeed which coils round the larger part of the barrel afts with a larger lever, and confequently with greater force than the other ; but as this excefs of force is not fufftcient to overcome the friftion of the machine, the weight will remain fta- tionary in any part of its path. {Appendix to Fer- gufon's LeEiures, vol. ii.). % Compound 3^9- ^'^e principle on which the preceding capftane chine Oi^" conftrU(^ed> might be applied with great advantage the fame w^en two feparate axles AC, JBD are driven by means principle. °f the winch H and the -wheels B and A. It is evi¬ dent that when the winch is turned round in one di- ffig- 5‘ reftion, the rope R is unwinded from the axle BD ; the wheel B drives the wheel A, fo that the axle AC moves in a direction oppofite to that of BD, and the rope is coiled round the axle A.C. If the wheels A, B are of the fame diameter and the fame number of teeth, the weight W will be ftationary, as the rope wunded about one axle will be always equal to what is unwind¬ ed from the other. If the wheels have different diame¬ ters, or different numbers of teeth, the quantity of rope wound round the one axle will exceed what is un¬ wound from the other, and the weight will be raifed. Chap. III. On the Nature of Fri&ion and the Method of diminijhing its cjfeEls in Machinery ; and on the Rigidity of Ropes. 370. The fridlion generated in the communicating parts of machinery, oppofes fuch a refifl ance to the impel¬ ling power, and is fo injurious to the machine itfelf, that an acquaintance with the nature and effects of this re¬ tarding force, and with the method of diminilhing its effe£ls on machinery, is of infinite importance to the practical mechanic. 371. The fubjeffc of friction has been examined at Practical great length, by Amontons, Bullinger, Parent, Euler, ^Panics, and Boffut, and has lately occupied the attention of v— a our ingenious countryman Mr Vince of Cambridge. He found that the friftion of hard bodies in mo- Refu!t 0f tion is an uniformly retarding force, and that the Vince’s ex* quantity of friction confidered as equivalent to a weight periments. drawing the body backwards is equal to M M-f-W x S where M is the moving force expreffed by its weight, W the weight of the body upon the horizontal plane, S the fpace through which the moving force or weight defeended in the time t, and 16.087 feet, the force of gravity. Mr Vince alfo found that the quantity of fridlion increafes in a lefs ratio than the quantity of matter or weight of the body, and that the friction of a body does not continue the fame when it has dif¬ ferent furfaces applied to the plane on which it moves, but that the fmalleft furfaces will have the leaft friction. 372. Notwithftanding the attempts of preceding philofophers to unfold the nature of friction, it wras referved for the celebrated Coulomb to furmount the Experi- difficulties which are infeparable from fuch an in-merits of veftigation, and to give an accurate and fatisfadtory Coulomk view of this difficult branch of mechanical philofo- phy. By employing large bodies and conducting his experiments on a large fcale, he has corrected feveral errors which arofe from the limited experiments of others j he has brought to light many new and ftrik- ing phenomena, and confirmed others which were hi¬ therto but partially eftablifhed. As it would be foreign to the nature of this work to follow this ingenious phi- lofopher through his numerous and varied experiments, we lhall only prefent the reader with the interefting re- fults to which they led. r. The friClion of homogeneous bodies, or bodies of the fame kind, moving upon one another, is generally fuppofed to be greater than that of heterogeneous bo¬ dies 5 but Coulomb has (hewn that there are exceptions to this rule. He found, for example, that the friClion of oak upon oak was equal to —^— of the force of pref- fion ; the friClion of pine againft pine 1.78’ and that of oak againft pine M * The friClion of oak againfl: copper was , and that of oak againfl: iron nearly the fame. 2. It was generally fuppofed, that in the cafe of wood, the friClion is greateft w'hen the bodies are drag¬ ged contrary to the courfe of their fibres ; but Coulomb has ftiewn that the friClion is in this cafe fometimes the fmalleft. When the bodies moved in the direction of their fibres, the friClion was of the force with 2-34 which they were preffed together ; but w hen the mo¬ tion Avas contrary to the eourfes of the fibres, the fric¬ tion was only — 3-76 3. The longer the rubbing furfaces remain in contaEl, the greater is their fri£lion.—-NN\izx\ wood was moved upon 5 MECHANICS. 109 Practical upon wood, according to tlie direftion of the fibres, the Mechanics frj.-'!joa was incrcafed by keeping the furfaces in eon- tatc). for a few feconds; and when the time was prolong¬ ed to a minute, the frittion teemed to have reached its fartheft limit. But when the motion was contrary to the courfe of the fibres, a greater time was neceffary be¬ fore the fritiion arrived at its maximum. When wood Avas moved upon metal, the friction did not attain its maximum till the furfaces continued in contact for five or fix days •, and it is very remarkable, that when wooden furfaces Avere anointed with tallow, the time requiiite for producing the greatell quantity of friction is increafed. The increafe of fridtion which is gene¬ rated by prolonging the time of conta£t is fo great, that a body weighing 1650 pounds A\’as moved with a force of 64 pounds when firfi. laid upon its correfpond- ing furface. After having remained in contact for the fpace of three feconds, it required 160 pounds to put it in motion 5 and, Avhen the time Avas prolonged to fix days, it could fcarcely be moved Avith a force of 622 pounds. When the furfaces of metallic bodies were moved upon one another, the time of producing a maxi¬ mum of fridtion Avas not changed by the interpofition of olive oil ; it was increafed, hoAvever, by employing fwine’s greafe as an unguent, and Avas prolonged to five or fix days by befmearing the furfaces with tallow. 4. FriEHon is in general proportional to the force with which the rubbing furfaces are preffed together ; and is, for the mofl part, equal to between f and ^ of that force. —In order to prove the find part of this propofition. Coulomb employed a large piece of wood, Avhofe fur- face contained three fquare feet, and loaded it fuccef- fively Avith 74 pounds, 874 pounds, and 2474 pounds. In thefe cafes the fridlion Avas fucceflively .16’ 2.21 of the force of preffion; and Avhen a lefs furface and other Aveights Avere ufed, the fridtion Avas 2.40 2.36’ 2.42’ Similar refults Avere obtained in all Coulomb’s experiments, even Avhen metallic furfaces Avere employ¬ ed. The fecond part of the propofition has alfo been ' eftablilhed by Coulomb. He found that the greatefl: fridtion is engendered Avhen oak moves upon pine,, and that it amounts to -A- 0f the force of preflion j on the contrary, Avhen iron moves upon brafs, the leaft fridtion is produced, and it amounts to of the force of prefiion. 5. FriBion is in general not increafed by augmenting the rubbing furfaces.—When a fuperficies of three feet fquare Avas employed, the fridtion, Avith different weights, Avas — at a medium : but when a fmall 2.28 furface Avas ufed, the fridtion inftead of being greater, as might have been expedted, Avas only 2-39 Fri&ion di- 6. FriElion for the mo ft part is not augmented by an min idled by mcreafe of velocity. In fome cafes, it is diminift.wd by the' veloa an auSmentat^on °f celerity.—M. Coulomb found, that Avhen wood moved upon wood in the diredtion of the fibres, the fridtion was a conftant quantity, hoAvever much the velocity was varied 3 but that when the fur¬ faces \vTere very fmall in refpedt to the force with which PmcTh 'dl they were preffed, the fritiion was diminif:ed by aug- - -e:!iai,>C9“ menting the rapidity : the fridtion, on the contrary, was increafed when the furfaces were very large when com¬ pared Avith the force of prefiion. When the wood was moved contrary to the diredtion of its fibres, the fric¬ tion in every cafe remained the fame. If Ai’ood be moved upon metals, the tridtion is greatly increafed by an increafe of velocity 3 and when metals move upon Avood befmeared Avith talloAv, the fridtion is fiill aug¬ mented, by adding to the velocity When metals move upon metals, the tridtion is ahvays a conftant quantity ; but when heterogeneous fubftances are employed which are not bedaubed Avith talloAv, the fridtion is fo increa¬ fed Avith the velocity, as to form an arithmetical pro- grefiion A\rhen the velocities form a geometrical one. 7. The friSlion of loaded cylinders rolling upon a ho¬ rizontal plane, is in the dire El ratio of their weights, and the inveife ratio of their diameters. In Coulomb’s ex¬ periments, the fridtion of cylinders of guaiacum Avood, which Avere two inches in diameter, and were loaded Avith 1000 pounds, Avas 18 pounds or of the force of prefiion. In cylinders of elm, the fridtion was greater by-f-, and was fcarcely diminifhed by the interpofition of tallow. 373. From a variety of experiments on the fridtion of the axes of pulleys, Coulomb obtained the folloAving refults.—When an iron axle moved in a brafs bufh the fridtion Avas \ of the prefiion 5 but M’hen the bufh Avas befmeared A\7ith very clean talloAv, the fridtion Avas only f 3 Avhen fwine’s greafe was interpofed, the fric¬ tion amounted to A 3 and when olive oil Avas employ- 8*5 ed as an unguent, the fridtion was never lefs than -f- 7-5 When the axis Avas of green oak, and the bufh of guaiacum Avood, the fridtion Avas f6 when talloAv Avas interpofed 3 but Avhen the talloAv Avas removed, fo that a fmall quantity only covered the furface, the fric¬ tion was increafed to xrT. When the bufh Avas made of elm, the fridtion Avas in fimilar circumftances and 2Tq, which is the leaft of all. If the axis be made of box, and the bufh of guaiacum Avood, the fridtion Avill be and circumftances being the fame as before. If the axle be of boxAvood, and the bufh of elm, the fridtion Avill be -gk and and if the axle be of iron and the bufh of elm, the fridtion Avill be -5— of the force of prefiion. 374. Having thus confidered the nature and effedts of Method of fridtion, Ave fliall noAV attend to the method of leffeningdii»iniflung the refiftance Avhich it oppofes to the motion of ma- chines. The molt efficacious mode of accomplifhing this is to convert that fpecies of fridtion which arifes from one body being dragged over another, into that which is occafioned by one body rolling upon another. As this Avill always diminifh the refiftance, it may be eafily effedted by applying Avheels or rollers to the foc- kets or bufhes which fuftain the gudgeons of large Avheels, and the axles- of wheel carriages. Cafatus feems to have been the firft Avho recommended this ap¬ paratus. It Avas afterwards mentioned by Sturmius and Friction Wolfius ; but was not ufed in pradtice till Sully applied v-hcels. it to clocks in the year 1716, and Mondran to cranes in 17 Nbtwithftahding thefe folitary attempts to introduce fridlion Avheels, they feem to have attradl- cd little notice till the celebrated Euler examined and polling power. HO M^lr ^ aU^ ('xP^a'ne^5 ills ufual accuracy, their nature and w^ niCS- advantages. The diameter of the gudgeons and pivots Ihould be made as fmall as the weight of the wheel and the impelling force will permit. The gudgeons fhould reft upon wheels as large as circumltances will alloAV, having their axes as near each other as poffible, but no thicker than what is abfolutely neceffary to fuftain the fuperincumbent weight. When thefe precautions are properly attended to, the refiftance which arifes from the fridfion of the gudgeon, &c. will be extremely trifling. Friction 37 5. The effedts of fridtion may likewife in fome may be di- meafure be removed by a judicious application of the mimfhed by impe^Jnpr power, and by proportioning the fize of the application fridfion wheels to the preflure which they feverally fuf- of the im- tain. If we fuppofe, for example, that the weight of a wheel, whofe iron gudgeons move in buflies of brafs, is 100 pounds ; then the fridfion arifing from both its gudgeons will be equivalent to 25 pounds. If we fup¬ pofe, alfo, that a force equal to 40 pounds is employed to impel the wheel, and adls in the diredtion of gravi¬ ty, as in the cafes of overfhot wheels, the preffure of the gudgeons upon their fupports will then be 140 pounds and the fridtion 35 pounds. But if the force of 40 pounds could be applied in fuch a manner as to adl in diredl oppofition to the wheel’s weight, the pref¬ fure of the gudgeons upon their fupports would be 100—40, or 60 pounds, and the fridfion only 15 pounds. It is impoflible, indeed, to make the moving force adl in diredf oppofition to the gravity of the wheel, in the cafe of water mills ; and it is often im- pradlicable for the engineer to apply the impelling power but in a given way : but there are many cafes in which the moving force may be fo exerted, as at leaf! not to increafe the fridfion which arifes from the wheel’s weight. 376. When the moving force’is not exerted in aperpen- dicular diredf ion, but obliquely as in under fhot wheels, the gudgeon will prefs with greater force on one part of the focket than on any other part. This point will evi¬ dently be on the fide of the bulh oppofite to that where the power is applied ; and its diftance from the loweft point of the focket, which is fuppofed circular and con¬ centric with the gudgeon, being called #, we fhall have H Tang. x — y-, that is, the tangent of the arch con¬ tained between the point of greateff preffure and the loweft point of the bufii, is equal to the fum of all the horizontal forces, divided by the fum of all the vertical forces and the weight of the wheel, H reprefenting the former, and V the latter quantities. The point of greateff preffure being thus determined, the gudgeon muff be fupported at that part by the largeff fridfion wheel, in order to equalize the fridfion upon their axles. The application of thefe general principles to parti¬ cular cafes is fo fimple as not to require any illuffration. To aid the conceptions, however, of the pradfical me¬ chanic, we may mention two cafes in which fridfion wheels have been fuccefsfully employed. Plate 377* Gottlieb, the conffrudfor of a new crane, CCCXXIII. has received a patent for what he calls an anti-attrition 6* axle-tree, the beneficial effedfs of which he has afcer- tained by a variety of trials. It confiffs of a ffeel roller R about four or fix inches long, which turns within a groove cut in the inferior part of the axle-tree C which runs in the nave AB of the wheel. When the wheel- M E C H A N I C S. carriages are at reft, Mr Gottlieb has given the friction Practical wheel its proper petition ; but it is evident that the Mccham«. point of greatetf prefiure will change when they are v " 'J put in motion, and will be nearer the front of the car¬ riage. This point, however, will vary with the weight of the load ; but it is fufficiently obvious that the fric¬ tion roller tliould be at a little diftance from the lowTeft point of the axle-tree. 378. Mr Garnett of Briftol has applied fridfion rollers in a different manner, which does not, like the prece¬ ding method, aveaken the axle-tree. Inftead of fixing them in the iron part of the axle, he leaves a fpace be¬ tween the nave and the axis to be filled with equal roll¬ ers almoft touching each other. A fedfion of this Fig. 7, apparatus is reprefented in fig. 7. where A BCD is the metallic ring inferted in the nave of the wheel. The axle-tree is reprefented at E, placed between the fridfion rollers I, I, I, made of metal, and having their axes inferted into a circle of brafs which paffes through their centres. The circles are rivetted together by means of bolts palling between the rollers, in order to keep them feparate and parallel. 379. As it appears from the experiments of Coulomb, that the leaft fridtion is generated when polifhed iron moves upon brafs, the gudgeons and pivots of wheels, and the axles of fridfion rollers, fhould all be made of polifhed iron ; and the bufhes in which thefe gudgeons move, and the fridfion wheels, fhould be formed of po¬ lifhed brafs. 380. When every mechanical contrivance has been Fri&ion di. adopted for diminifhing the obftrudlion which arifes miniftied bj from the attrition of the communicating parts, it may unguents, be ffill farther removed by the judicious application of unguents. The molt proper for this purpofe are fwine’s greafe and tallow when the furfaces are made of wood, and oil wfien they are of metal. When the force with which the furfaces. are preffed together is very great, tallow will diminifh the fridtion more than fwine’s greafe. When the wooden furfaees are very fmall, un¬ guents will leffen their fridtion a little, but it •will be greatly diminifhed if wood moves upon metal greafed with tallow. If the velocities, however, are increafed, or the unguent not often enough renewed, in both thefe cafes, but particularly in the laft, the unguent will be more injurious than ufeful. The beft mode of applying it, is to cover the rubbing furfaces with as thin a ftra- tum as poffible, for the fridlion will then be a eonftant quantity, and will not be increafed by an augmentation of velocity. 381. In fmall works of wood, the interpofition of the powder of black lead has been found very ufeful in re¬ lieving the motion. The ropes of pulleys ffiould be rubbed with tallow, and whenever the ferew is ufed, the fquare threads ffiould be preferred.” Appendix to Tergufon's Lediures, vol. ii. 382. When ropes pafs over cylinders or pulleys, a On thc ri- confiderable force is neceffary to bend them into the 0 form of the circumference round which they are coiled. The force which is neceffary to overcome this refiftance is called the Jiiffnefs or rigidity of the ropes. This im¬ portant fubjedt was firft examined by Amontons, * who * Mem. contrived an ingenious apparatus for afeertaining the^^'16^* rigidity of ropes. His experiments were repeated and1 confirmed in part by fubfequent philofophers, but par¬ ticularly by M. Coulomb, who has inveftigated the fub- MECHANICS. Pra&ical jecfc with more care and fuccefs than any of his prede- Mechanics. cefforg. His experiments were made both with the ap- L » paratus of Amontons, and with one of his own inven¬ tion ; and as there was no great difcrepancy in the re- fults, he was authorifed to place more confidence in his experiments. The limits of this article will not per¬ mit us to give an account of the manner in which the. experiments were conduced, or even to give a detailed view of the various conclufions which wTere obtained. We can only prefent the reader with fome of thofe leading refults which may be ufeful in the conftrudtion of ma¬ chinery. 1. The rigidity of ropes increafes, the more that the fibres of which they are compofed are twilled. 2. The rigidity of ropes increafes in the duplicate ratio of their diameters. According to Amontons and Dcfaguliers, the rigidity increafes in the fimple ratio of the diameters of the ropes 5 but this probably arofe from the flexibility of the ropes which they employed : for Defaguliers remarks, that when he ufed a rope whofe diameter was half an inch, its rigidity was increafed in a greater proportion ; fo that it is probable that if they had employed ropes from twro to four inches in diame¬ ter, like thofe ufed by Coulomb, they would have ob¬ tained fimilar refults. (See N° 9.) 3. The rigidity of ropes is in the fimple and dire£t ratio of their tenfion. 4. The rigidity of ropes is in the inverfe ratio of the diameters of the cylinders round which they are coiled. 5. In general, the rigidity of ropes is direflly as their tenfions and the fquares of their diameters, and in- verfely as the diameters of the cylinders round which they are wnund. 6. The rigidity of ropes increafes fo little with the velocity of the machine, that it need not be taken into the account when computing the effefls of machines. 7. The rigidity of fmall ropes is diminilhed when pe¬ netrated with moilture •, but when the ropes are thick, their rigidity is increafed. 8. The rigidity of ropes is increafed and their ftrength diminilhed when they are covered with pitch j but when ropes of this kind are alternately immerfed in the fea and expofed to the air, they laft longer than when they are not pitched.—This increafe of rigidity, however, is not fo perceptible in fmall ropes as in thofe which are pretty thick. 9. The rigidity of ropes covered with pitch is a fixth part greater during froft than in'the middle of fummer, but this increafe of rigidity does not follow the ratio of their tenfions. xo. The refiftance to be overcome in bending a rope over a pulley or cylinder may be reprefented by a for- a Y)n mula compofed of two terms. The firll term is a eonftant quantity independent of the tenfion, a being a eonftant quantity determined by experiment, a power of the diameter D of the rope, aVid r the radius of the pulley or cylinder round which the rope is coil- b ed. The feoond term of the formula is Tx —, r where T is the tenfion of the rope, b a conllant quanti¬ ty, and D« and r the fame as before. Hence the com- D* _ , Practical X « + Mechanics. . .a Dw „ b D* plete formula is 1-1 X : r r The exponent n of the quantity D diminilhes with the flexibility of the rope, but is generally equal to 1.7 or 1.8 ; or, as in N° 2. the rigidity is nearly in the dupli¬ cate ratio of the diameter of the rope. When the cord is much ufed, its flexibility is increafed, and n becomes equal to 1.5 or 1.4. Chap. IV. On the Nature and Advantages of Fly Wheels. 383. A FLY, in mechanics,is a heavy wheel or cylinder ■which moves rapidly upon its axis, and is applied to machines for the purpofe of rendering uniform a deful- tory or reciprocating motion, arifing either from the nature of the machinery, from an inequality in the refiftance to be overcome, or from an irregular applica¬ tion of the impelling power. When the firlt mover is inanimate, as wind, water, and fleam, an inequality of force obviouflyarifes from a variation in the velocity of the wind, from an increafe or decreafe of water occafioned by hidden rains, or from an augmentation or diminution of the fleam in the boiler, produced by a variation in the heat of the furnace •, and accordingly various methods have been adopted for regulating the adlion of thefe variable powers. The fame inequality of force obtain? when machines are moved by horfes or men. Every animal exerts its greatefl ftrength wThen firfl fet to work» After pulling for fome time, its ftrength will be impair¬ ed *, and when the refiflance is great, it will take fre¬ quent though fhort relaxations, and then commence its labour with renovated vigour. Thefe intervals of reft and vigorous exertion mull always produce a varia¬ tion in the velocity of the machine, which ought parti¬ cularly to be avoided, as being detrimental to the com¬ municating parts as well as the performance of the ma¬ chine, and injurious to the animal which is employed to draw it. But if a fly, confifting either of crofs bars, or. a maffy circular rim, be connected with the machinery, all thefe inconveniences will be removed. As every fly wheel muft revolve with great rapidity, the momen¬ tum of its circumference muft be very confiderable, and will confequently refill every attempt either to accelerate or retard its motion. When the machine therefore has been put in motion, the fly wheel will be whirling with an uniform celerity, and with a force capable of con¬ tinuing that celerity when there is any relaxation in the impelling power. After a fhort reft the animal renews his efforts 5 but the machine is now moving with its for¬ mer velocity, and thefe frefh efforts will have a tendency to increafe that velocity. The fly, however, now a£ts as a refilling power, receives the greateft part of the fuperfluous motion, and caufes the machinery to preferve its original celerity. In this way the fly fecures to the engine an uniform motion, whether the animal takes occafional relaxations or exerts his force with redoubled ardour. 384. We have already obferved that a defultory or va¬ riable motion frequently arifes from the inequality of the refiftance, or work to be performed. This is particu¬ larly manifeft in thrafhing mills, on a fmall fcale, which arc driven by water. When the corn is laid unequally 112 MECHANIC S. Practical 0n the feeding board, fo that too much is taken in by Mechanics. t]ie fluted rollers, this increafe of refinance inftantly afFefts the machinery, and communioates a defultory or irregular motion even to the water wheel or fir ft mover. This variation in the velocity of the impelling power may be diftincfly perceived by the ear in a calm even¬ ing when the machine is at work. The bell method of correcting thefe irregularities is to employ a fly wheel, which will regulate the motion of the machine when the refiftance is either augmented or diminifhed. In machines built upon a large fcale there is no neceffity for the interpolation of a tlv, as the inertia of the ma¬ chinery fupplies its place, and refnts every change of mo¬ tion that may be generated by an unequal admiffion of the corn. 385. A variation in the velocity of engines arifes alfo from the nature of the machinery. Let us fuppofe that a weight of 1000 pounds is to be raifed from the bottom of a well 50 feet, by means of a bucket attached to an iron chain which winds rounds a barrel or cylinder, and that every foot length of this chain weighs two pen ds. It is evident that the refiftance to be overcome in the firft moment is 1000 pounds added to 50 pounds the weight of this chain, and that this refiftance diminifhes gradually as the chain coils round the cylinder, till it is only 1000 pounds when the chain is completely wound up. The refiftance therefore decreafes from 1050 to 1000 pounds ; and if the impelling power is inanimate, the velocity of the bucket will gradually increafe ; but if an animal is employed, it will generally proportion its aftion to the refifting load, and muft therefore pull with a greater or lefs force according as the bucket is near the bottom or top of the well. In this cafe, however, the afliftance of a fly may be difpenfed with, becaufe the refiftance diminifhes uniformly, and may be render¬ ed conftant by making the barrel conical, fo that the chain may wind upon the part neareft the vertex at the commencement of the motion, the diameter of the bar¬ rel gradually increafing as the weight diminifhes. In this wTay the variable refiftance will be equalized much better than by the application of a fly wheel, for the fly having no motion of its own muft neceffarily wafte the impelling power. 386. Having thus pointed out the chief caufes of variation in the velocity of machines, and the method of rendering it uniform by the intervention of fly wheels, the utility, and in fome inftances the necefli- ty, of this piece of meehanifm, may be more obvioufly illuftrated by (hewing the propriety of their application in particular cafes. ^ See In the defeription of Vaulone’s pile engine *, Part III. the reader will obferve a ftriking inftance of the Plate utility of fly wheels. The ram O is raifed between V fi thf guides bb by means of horfes adling again!! the levers S, S ; but as foon as the ram is elevated to the top of the guides, and difeharged from the fol¬ lower G, the refiftance againft which the horfes have been exerting their force is ftiddcnly removed, and tbev would inftantaneoufly tumble down, wrere it not for the fly O. This fly is connected with the drum B by means of the trundle X, and as it is moving with a very great force, it oppofes a fufBcierit refiftance to Pradtica! the adfion of the horfes, till the ram is again taken up Mechanics, by the follower. 1 s-—y— 388. When machinery is driven by a fingle-ftroke fleam engine, there is fuch an inequality in the impel- ling p°v er, that for two or three feconds it does not act at all. During this interval of inactivity the ma¬ chinery would neceffarily flop, were it not impelled by a mafly fly wheel of a great diameter, revolving wdth rapidity, till the moving power again refumes its energy. 389. If the moving power is a man afting with a handle or winch, it is fubjea to great inequalities. The greateft force is exerted when the man pulls the handle upwards from the height of his knee, and he acts with the leaf! force when the handle being in a vertical po- fition is thruft from him in a horizontal direflion. The force is again increafed when the handle is puthed , downwards by the man’s weight, and it is diminiflied when the handle being at its loweft point is pulled to¬ wards him horizontally. But when a fly is properly conneCled with -the machinery, thefe irregular exer¬ tions are equalized, the velocity becomes uniform, and the load is raifed with an equable and fteady mo¬ tion. 390. In many cafes, where the impelling force is al¬ ternately augmented and dinunifhed, the performance of the machine may be increafed by rendering the refif¬ tance unequal, and accommodating it to the inequali¬ ties of the moving power. Dr Robifon obferves that “ there are fome beautiful fpecimens of this kind of adjuftment in the mechanifm of animal bodies.” Belides the utility of fly wheels as regulators of machinery, they have been employed for accumulating or collecting power. If motion is communicated to a fly wheel by means of a fmall force, and if this force is continued till the wheel has acquired a great velocitv. fuch a quantity of motion will be accumulated in its circumference, as to overcome refiftances and produce effeCts which could never have been accomplifhed by the original force. So great is this accumulation of power ; that a force equivalent to 20 pounds applied % the fpace of 37 feconds to the circumference of a cylinder 20 feet diameter, which weighs 4713 pounds, would, at the diftance of one foot from the centre, give an impulfe to a mufleet ball equal to what it receives from a full charge of gunpowder. In the fpace of fix minutes and 10 feconds, the fame effeCt would be pro¬ duced if the cylinder was driven by a man who con- ftantly exerted a force of 20 pounds at a winch one foot long (d). 39J. This accumulation of power is finely exem¬ plified in ibr jling. When the thong which contains the ftofte is fvvung round the head of the {linger, the force of the hand is continually accumulating in the re¬ volving ftone, till it -is difeharged with a degree of rapi¬ dity which it .could never have received from the force of the hand alone. When a ftone is projefted from the hand itfelf, there is even then a certain degree of force accumulated, though the ftene only moves through the arch of a circle, if we fix the ftone in an opening at the (d) This has been demonftrated by Mr Atwood. See his Treatife on Redtilineal and Rotatory Motion. Practical extremity of a piece of wood two feet long, and dif- Mcchanics. charge it in the ufual way, there will be more force ac- cumulated than with the hand alone, for the ftone de- fcribes a larger arch in the fame time, and mult there¬ fore be projected with greater force. 392. When coins or medals are Itruck, a very con- fiderable accumulation of power is neceffary, and this is effected by means of a fly. The force is firft accumu¬ lated in weights fixed in the end of the tly. This force is communicated to two levers, by which it is farther condenfed ; and from thefe levers it is tranfmitted to a ferew, by which it fuffers a fecond condenfation. The (tamp is then impreffed on the coin or medal by means of this force, 'which was firft accumulated by the fly, and afterwards augmented by the intervention of two mechanical powers. 393. Notwithftanding the great advantage of fly wheels, both as regulators of machines and collectors of power, their utility wholly depends upon the pofition which is afligned them relative to the impelled and working points of the engine. For this purpofe no particular rules can be laid down, as their pofitions depend altogether on the nature of the machinery. We may obferve however, in general, that when fly wheels are employed to regulate machinery, they (hould be near the impelling power *, and when ufed to accumulate force in the working point they fliould not be far diftant from it. In hand mills for grinding corn, the fly is for the moft part very injudicioufly fixed on the axis to which the winch is attached ; whereas it fliould always be faftened to the upper millftone fo as to revolve with the fame rapidity. In the firft pofition indeed it muft equalize the varying efforts of the power which moves the winch *, but when it is attached to the turning mill¬ ftone, it not only does this, but contributes very effec¬ tually to the grinding of the corn. 394. A new kind of fly, called a conical pendulum has been ingernoufly employed by Mr Watt for procur¬ ing a determinate velocity at the working point of his fteam-engine. It is reprefenfed in fig. 8. where AB is a vertical axis moving upon pivots, and driven by means of a rope pafllng from the axis of the large fty over the (heave EF. The large balls M, N are fixed to the rods NG, MH, which have an angular motion round P, and are connected by joints at G and H, with the rods GK, HK attached to the extremity of the lever KL whofe centre of motion is L, and whole other extremity is connefted with the cock which admits the fteam into the cylinder. 'Hie frames CD and QR prevent the balls- from receding too far from the axis, or from ap¬ proaching too near it. Now when this conical pendu¬ lum is put in motion, the centrifugal force of the balls M, N makes them recede from the axis AB. In con- fequenee of this recefs, the points C, H, K are deprefs- ed, and the other extremity of the lever is raifed $ and the cock admits a certain quantity of fteam into the cy¬ linder. When the velocity of the fly is by any means increafed, the balls recede ftill farther from the axis, the extremity of the lever is raifed higher, and the cock clofes a little and diminilhes the fupply of fteam. From this diminution in the impelling power, the velocity of the fly and the conical pendulum decreafes, and the balls ^efume their former pofition. In this way, when there is any increafe or diminution in the velocity of the fly, Vol. XIII. Part I. MECHANICS. • f 13 the correfponding increafe or diminution in the centrifu- Practical gal force of the balls raifes or depreffes the arm of the le- M<-'chanics. ver, admits a greater 01* a lefs quantity of fteam into the W-L 'V' ’'' -• cylinder, and reftores to the engine its former velocity. DeforiptioH •f tlie co¬ nical pen¬ dulum. Rg.8. Chap. V. On the Teeth of Wheels, and the Wipers Stampers. 395. In' the conftrutlion of machines, we muft not only attend to the form and number of their parts, but alfo t© the mode by which they are to be eonnedted. It would be eafy to'lhew, did the limits of this article permit it, that, when one wheel impels another, the impelling power will fometimes adt with greater and fometimes with lefs force,- unlefs the teeth of one or both of th© wheels be parts of a curve generated after the manner of an epicycloid by the revolution of one circle along the convex or concave fide of another. It may be fufficient to fliew, that, when one wheel impels another by the ac¬ tion of epicycloidal teeth, their motion will be uniform. Let the wheel CD drive the wheel AB by means ot the epicycloidal teeth mp, n q, or, adting upon the infinite- Fig. j, ly fmall pins or fpindles a, b, c ; and let the epicycloids mp, n q, &c. be generated by the circumference of the wheel AB, rolling upon the convex circumference of the wheel CD. From the formation of the epicycloid it is obvious that the arch « £ is equal to m n, and the arch a c to mo ; for during the formation of the part n b of the epicycloid n q, every point of the arch is ap¬ plied to every point of the arch m n, and the fame hap¬ pens during the formation of the part c 0 of the epicy¬ cloid 0 r. Let us now fuppofe that the tooth m p be¬ gins to adt on the pin a, and that b, c are fucceflive pofitions of the pin a after a certain time ; then, n q7 0 r will be the pofitions of the tooth mp after the fame time 5 but a bzzm n and a c~m 0, therefore the wheels AB, CD, when the arch is driven by epicycloidal teeth, move through equal fpaces in equal times, that is, the force of the wheel CD, and the velocity of the wheel' AB, are always uniform. 396. In illuftrating the application of this property of the epicycloid, which wTas difeovered by Olaus Roe- mer the celebrated Danifti aftronomer, we (hall call the fmall wheel the pinion, and its teeth the leaves of the pinion. The line which joins the centre of the wheel and pinion is called the line of centres. There are three different ways in which the teeth of one wheel may- drive another, and each of thefe modes of adlion re¬ quires a different form for the teeth. 1. When the adlion is begun and completed after the teeth have paffed the line of centres. 2. When the adtion is begun and completed before they reach the line of centres. 3. When the adlion is carried on, on both fides of the line of centres. 397- 1 • firft of thefe modes of adlion is reprefented Firft moth in fig. 1. where B is the centre of the wheel (d), A thatof a&km. of the pinion, and AB the line of centres. It is evident rr^xiV from the figure, that the part b of the tooth a b of the C u wheel, does not adl on the leaf m of the pinion till b they arrive at the line of centres AB ; and that all the adlion is carried on after they have paffed this line, and is completed when the leaf m comes into the fituation «. When this mode of adlion is adopted, the adting faces P of $d) In figs. I, 2, 3, 4, the letter B is fuppofed t» be placed at the centre of the wheels. i U MECHANICS. Practical 0f the leaves of the pinion (liould be parts of an interior l1t' Jlnlcs'i epicycloid, generated by a circle of any diameter rolling upon the concave fuperficies of the pinion, or within the circle a d h; and the faces a b oi the teeth of the wheel ihould be portions of an exterior epicycloid formed by the fame generating circle rolling upon the convex lu- perficies o dp oi the wheel. 398. But when one circle rolls within another whofc diameter is double that of the rolling circle, the line ge¬ nerated by any point of the latter is a fraight line, tend¬ ing to the centre of the larger circle. Therefore, if the generating circle above mentioned Ihould be taken with its diameter equal to the radius of the pinion, and be made to roll upon the concave fuperficies a d h of the pinion, it will generate a ftraight line tending to the pinion’s centre, which will be the form of the faces of its leaves ; and the teeth of the wheel will be exterior epicycloids, formed by a generating circle, whofe dia¬ meter is equal to the radius of the pinion, rolling upon the convex fuperficies 0 dp of the wheel. Thisre&ili- Fig. a* neal form of the teeth is exhibited in fig. 2. and is per¬ haps the moft advantageous, as it requires lefs trouble, and may be executed with greater accuracy, than if the epieycloidal form had been employed, though the teeth are evidently weaker than thofe in fig. 1.5 it is recom¬ mended both by De la Hire and Camus as particularly advantageous in clock and watch work. Fig. 1. 399. The attentive reader will perceive from fig. 1. that in order to prevent the teeth of the wheel from act¬ ing upon the leaves of the pinion before they reach the line of centres AB; and that one tooth of the wheel may not quit the leaf of the pinion till the fucceeding tooth begins to adl upon the fucceeding leaf, there mull be a certain proportion between the number of leaves in the pinion and the number of teeth in the wheel, or between the radius of the pinion and the radius of the wheel, when the dillance of the leaves x^B is given. But in machinery the number of leaves and teeth is always known from the velocity which is re¬ quired at the working point of the machine : It be¬ comes a matter therefore of great importance to de¬ termine with accuracy the relative radii of the wheel and pinion. Relative 400< For this purpofe, let A, fig. 2. be the pinion hav- fize of the a&ing faces of its leaves ftraight lines tending to wheel anti the centre, and B the centre of the wheel. AB will be pinion. the diftance of their centres. Then as the tooth C is fup- pofed not to aft upon the leaf Am till it arrives at the line AB, it ought not to quit Am till the following tooth F has reached the line AB. But fince the tooth always afts in the direflion of a line drawn perpendicu¬ lar to the face of the leaf Am from the point of contaft, the line CH, drawn at right angles to the face of the leaf Am, will determine the extremity of the tooth CD, or the laft part of it which Ihould afl upon the leaf Am, and will alfo mark out CD for the depth of the tooth. Now, in order to find AH, HB, and CD, put a for the number of teeth in the wheel, b for the num¬ ber ot leaves in the pinion, c for the diftance of the pi¬ vots A and B, and let x be the radius of the wheel, and ?/ that of the pinion. Then, fince the circumference of the wheel is to the circumference of the pinion, as the number of teeth in the one to the number of leaves in the other, and as the circumferences of circles are pro¬ portional to their radii, we fhall have a : b—x : y, then 5 by compofition (Fuel. v. 18.) a-\-b : b~c : y {c being Practical equal to a:+y), and confequently the radius of the pinion, f'echanics-i cb , , . . , viz.ym—then by inverting the firft analogy, we have b ; a=zy : x, and confequently the radius of the' wheel, viz. x — y being now a known number. Now, in the triangle AHC, right-angled at C, the fide AH is known, and likewife ail the angles (HAC being equal to ^ ; the fide AC, therefore, may be found by plain trigonometry. Then, in the triangle ACB, the ^iCxAB, equal to HAC, is known, and alfo the fides AB, AC, ivhich contain it ; the third fide, therefore, viz. CB, may be determined \ from which DB, equal to HB, already found, being fub- ftradfed, there will remain CD for the depth of the teeth. When the adlion is carried on after the line of centres, it often happens that the teeth will not work in the hollows of the leaves. In order to pre¬ vent this, the CBH muft always be greater than half the HBP. The ^ HEP is equal to 360 degrees, divided by the number of teeth in the wheel, and CBH is eafily found by plane trigonometry. 401. If the teeth of wheels and the leaves of pinions be formed according to the directions already given, they will act upon each other, not only with uniform force, but nearly without friction. The one tooth rolls upon the other, and neither Aides nor rubs to fuch a degree as to retard the wheels, or wear their teeth. But as it is impoffible in practice to give that perfect curvature to the faces of the teeth which theory requires, a quantity of friction will remain after every precaution has been taken in the formation of the communicating parts. 402. 2. The fecond mode of action is not fo advantage-gccor^ ous as that which we have been confidering, and Ihould, mociG 0f if poflible, always be avoided. It is reprefented inadlion. fig. 3. where A is the centre of the pinion, B that of the wheel, and AB the line of centres. It is evident ^ from the figure that the tooth C of the wheel acts upon the leaf D of the pinion before they arrive at the line B A ; that it quits the leaf when they reach this line, and have affumed the pofition of E and F; and that the tooth c works deeper and deeper between the leaves of the pinion, the nearer it comes to the line of centres. From this laft circumftance a confiderable quantity of frieftion arifes, becaufe the tooth C does not, as before, roll upon the leaf D, but Aides upon it ; and from the fame caufe the pinion foon becomes foul, as the duft which lies upon the aefting faces of the leaves is pufiied into the interjacent hollows. One advantage, how¬ ever, attends this mode of aeftion : It allows us to make the teeth of the large wheel rectilineal, and thus renders the labour of the mechanic lefs, and the ac¬ curacy of his work greater, than if they had been of a curvilineal form. If the teeth C, E, therefore of the wheel BC are made rectilineal, having their fin-faces directed to the wheel’s centre, the acting faces of the leaves D, F, &c. muft be epicycloids formed by a ge¬ nerating circle, whofe diameter is equal to the radius B 0 of the circle op, rolling upon the circumference m n of the pinion A. But if the teeth of the wheel and the leaves of the pinion are made curvilineal as in the figure, the faces of the teeth of the wheel muft be portions of an interior epicycloid formed by any gene¬ rating MECHANICS. 115 Fig. 4. Practical rating circle rolling within the concave fuperficies of Mechanics. ^ cjrcle 0 and the faces of the pinion’s leaves muft be portions of an exterior epicycloid produced by roll¬ ing the fame generating circle upon the convex circum¬ ference m n of the pinion. Third mode 403. 3. The third mode of aftion, which is reprefented of action. jn fig. is a combination of the two firfl: modes, and confequently partakes of the advantages and difadvan- tages of each. It is evident from the figure that the portion eb oi the tooth a6ts upon the part b c of the leaf till they reach the line of centres AB, and that the part e d oi the tooth acts upon the portion b a oi the leaf after they have pafled this line. Hence the a£ting parts eand £ c muft be formed according to the directions given for the firft mode of adtion, and the remaining parts e d,b a, muft have that curvature which the fecond mode of aCtion requires ; confequent¬ ly e ^ fhould be part of an interior epicycloid formed by any generating circle rolling on the concave cir¬ cumference m n of the Avheel, and the correfponding part £ c of the leaf ftiould be part of an exterior epi¬ cycloid formed by the fame generating circle rolling upon b EO, the convex circumference of the pinion : the remaining part c d oi the tooth (hould be a portion of an exterior epicycloid, engendered by any genera¬ ting circle rolling upon e L, the concave fuperficies of the wheel : and the correfponding part b a oi the leaf fhould be part of an interior epicycloid defcribed by the fame generating circle, rolling along the concave fide b EO of the pinion. As it would be extremely troublefome, however, to give this double curvature to the afting faces of the teeth, it will be proper to ufe a generating circle, whofe diameter is equal to the radius of the wheel BC, for defcribing the interior epicycloid e h and the exterior one b c, and a generating circle, whofe diameter is equal to AC, the radius of the pi¬ nion, for defcribing the interior epicycloid b a, and the exterior one ed. In this cafe the two interior epi¬ cycloids e h, b a, will be ftraight lines tending to the centres B and A, and the labour of the mechanic Avill by this means be greatly abridged. 404. In order to find the relative diameters of the wheel and pinion, when the number of teeth in the one and the number of leaves in the other are given, and when the diftance of their centres is alfo given, and the ratio of ES to CS, let a be the number of teeth in the wheel, b the number of leaves in the pinion, c the diftance of the pivots A, B, and let w be to « as ES to CS, then the arch ES, or SAE, will be equal 360° and LD, or LED, will be equal to Relative diameters of the wheel and pinion to 360° LCrr/W But ES : CS=:z therefore (Eucl. . , „ LD X n , X«, and LC— j but x; confequently LD : vi. 16.) LCxot=LD LD is equal to 360 therefore by fubftitution LC= 360x4 Now, in the triangle APB, AB is known, and alfo Pradticai PB, which is the cofine of the angle ABD, PC ,‘v'ec^amcs/ being perpendicular to DB ; AP or the radius of the v pinion therefore may be found by plane trigonome¬ try. The reader will obferve that the point P marks out the parts of the tooth D and the leaf SP where they commence their a£Hon ; and the point I marks out the parts where their mutual action ceafes (e) ; AP therefore is the proper radius of the pinion, and BI the proper radius of the wheel, the parts of the tooth L without the point I, and of the leaf SP ivithout the point P, being fuperfluous. Now, to find BI, we have ES : CS=m : «, and CS= ES x « but ES Avas fheAvn to be therefore, by fubfti¬ tution, CS~-^—^ n. Noav the arch ES, or-^EAS, bm being equal to and CS, or CAS, being equal o 5q ^ to — , their difference EC, or the angle EAC. bm ^ * 360 360 X« 360° —n b m The will be equal to ~ EAC being thus found, the triangle EAB, or IAB, Avhich is almoft equal to it, is known, becaufe AB is given, and likewife AI, which is equal to the cofine of the angle IAB, AC being radius, and AIC being a right angle, confequently IB the radius of the wheel may be found by trigonometry. It was former¬ ly (hewn that AC, the radius of what is called the pri- cb r mitive pinion, Avas equal to —, and that BC the a+b' radius of the primiti\re wheel Avas equal to AC X a If then avc fubftraft AC or AS from AP, we ihall have the quantity SP which muft be added to the radius of the primitive pinion, and if we take the difference of BC (or BL) and DE, the quantity LE Avill be found, which muft: be added to the radius of the primitive Avheel. We have all along fuppofed that the Avheel drives the pinion, and have given the proper form of the teeth upon this fuppofition. But Avhen the pinion drives the wheel, the form Avhich Avas given to the teeth of the wheel in the firfl: cafe, muft in this be given to the leaves of the pinion ; and the fliape which Avas formerly given to the leaves of the pinion muft noAv be transferred to the teeth of the Avheel. 405. Another form for the teeth of Avheels, differ- Form of ent from any Avhich we have mentioned, has been re- the teeth commended by Dr Robifon. He ftiews that a perfeft according uniformity of adtion may be fecured, by making the Ro" adling faces of the teeth involutes of the wheel’s circum- 1 0n* ference, Avhich are nothing more than epicycloids, the centres of whofe generating circles are infinitely diftant. Thus, in fig. 1. let AB be a portion of the wheel on P 2 which (e) The letter L marks the interfedlion of the line BL Avith the arch em, and the letter E the interfedlion of the arch b O with the upper furface of the leaf m. The letters D and S correfpond with L and E refpedtivelv and P with I. r ^ MECHANICS. Practical which tlieiootli in to be fixed, and let A/? a tie a thread Mechanics. }appecJ round its circumference, having a loop hole at jr\s * its extremity a. In this loop hole fix the pin a, and with it defcribe the curve or involute ab c de h, by unlapping the thread gradually from the circumference A p m. This curve will be the proper ihape for the teeth ©f a wheel whofe diameter is AB. Dr Robifon obfervcs, that as this form admits of feveral teeth to be afting at the fame time (twice the number that can be admitted in M. de la Hire’s method), the preffure is divided among feveral teeth, and the quantity upon any one of them is fo diminilhed, that thofe dents and impreflions which they unavoidably make upon each other are part¬ ly prevented. He candidly allows, however, that the teeth thus formed are not completely free from Hiding and friction, though this Aide is only -^-th of an inch, tvhen a tooth three inches long fixed on a wheel ten feet in diameter drives another wheel whofe diameter is two feet. Append, to Fergufoil's LeBures. 406. On the Formation of Exterior and Interior Epi¬ cycloids, and on the Difpojition of the Teeth on the Wheel"1 s Circumference. . Nothing can be of greater importance to the prae- methoTof tical mechanic, than to have a method of drawing epi- fomiing cycloids with facility and accuracy j the following, we epicycloids, truft, is the moft Ample mechanical method that can be employed.—Take a piece of plain wood GH, fig. Fig. 6. 6. and fix upon it another piece of wood E, having its circumference mb oi the fame curvature as the circu¬ lar bafe upon which the generating circle AB is to roll. When the generating circle is large, the feg- jnent B will be fufficient: in any part of the circum¬ ference of this fegment, fix a fliarp pointed nail a, floping in fuch a manner that the diftance of its point from the centre of the cirele may be exaftly equal to its radius \ and faften to the board GH a piece of thin brafs, or copper, or tinplate, a b, diftinguilhed by the dotted lines. Place the fegment B in fuch a pofition that the point of the nail a may be upon the point b, and roll the fegment towards G, fo that the nail a may rife gradually, and the point of contadl between the two circular fegments may advance towards m; the curve a b defcribed upon the brafs plate will be an ac¬ curate exterior epicycloid. In order to prevent the fegments from Aiding, their peripheries Amuld be rub¬ bed with rofin or chalk, or a number of fmall iron points jnay be fixed on the circumference of the generating fegment. Remove, with a file, the part of the brafs on the left hand of the epicycloid, and the remaining concave arch or gage ab will be a pattern tooth, by means of which all the reft may be eafily formed. When an interior epicycloid is wanted, the concave fide of its circular bafe muft be ufed. The method of de- fcribing it is reprefented in fig. 7. where CD is the ge- F‘g- ?• nerating circle, F the concave circular bafe, MN the piece of wood on which this bafe is fixed, and c d the interior epicycloid formed upon the plate of brafs, by rolling the generating circle C, or the generating feg¬ ment D, towards the right hand. The cycloid, which is ufeful in forming the teeth of rack-woi'k, is generated precifely in the fame manner, with this difference only, that the bafe on which the generating circle rolls muft be a ftraight line. In order that the teeth may not cmbarrafs one an- ?raAieii other before their adftlon commences, and that one tooth Mechanics, may begin to a6t upon its correfponding leaf of the pi- fV - nion, before the preceding tooth lias ceafed to a£t upon0f the * U the preceding leaf, the height, breadth, and diftance of teeth, tire teeth muft be properly proportioned. For this pur- pofe the pitch-line or circumference of the wheel, which is reprefented in fig. 2. and 3. by the dotted arches, muft be divided into as many equal fpaces as the num¬ ber of teeth which the wheel is to carry. Divide each of thefe fpaces into 16 equal parts •, allow 7 of thefe for the greateft breadth of the teeth, and 9 for the dif¬ tance between each j or the diftance of the teeth may be made equal to their breadth. If the wheel drive a trundle, each fpace firould be divided into 7 equal parts, and 3 of thefe allotted for the thicknefs of the tooth, and 34 for the diameter of the cylindrical Have of the trundle. If each of the fpaces already mentioned, or if the diftance between the centres of each tooth, be divided into three equal parts, the height of the teeth muft be equal to two of thefe. Thefe diftances and heights, however, vary according to the mode of aftion which is employed. The teeth ftiould be rounded off at the extremities, and the radius of the wheel made a little larger than that which is deduced from the rules in Art. 400, 404. But when the pinion drives the wheel, a fmall addition ftiould be made to the radius of the pinion. On the Nature of Bevelled Wheels, and the method of giving an epicycloidalform to their Teeth. 407. The principle of bevelled -wheels was pointed out Ecvcllei by De la Hire, fo long ago as the end of the 17th cen-wheels, tury. It confifts in one fluted or toothed cone aiding upon another, as is reprefented in fig. 8. where the cone OD Fig. 8, drives the cone OC, conveying its motion in the direc¬ tion OC. If thefe cones be cut parallel to their bafes as at A and B, and if the two fmall cones between AB and O be removed, the remaining parts AC and BD may be confidered as two bevelled wheels, and BD will adt upon AC in the very fame manner, and with the fame effect, that the whole cone OD a died upon the whole cone OC. If the fection be made nearer the bafes of the cones, the fame effedt will be produced : this is the cafe in fig. 9. where CD and DE are but very fmall portions of the imaginary cones ACD and ADE. 408. In order to convey motion in any given diredtion, and determine the relative fize and fituation of the wheels for this purpofe, let AB, ,hg. 10. be the axis Fig. 1$. of a wheel, and CD the given diredtion in which it is required to convey the motion by means of a wheel fixed upon the axis AB, and adling upon another wheel fixed on the axis CD, and let us fuppofe that the axis CD muft have four times the velocity of AB, or muft perform four revolutions while AB performs one. Then the number of teeth in the wheel fixed upon AB muft be four times greater than the number of teeth in. the wheel fixed upon CD, and their radii muft have the fame proportion. Draw c d paraUel to CD at any convenient diftance, and draw a b parallel to AB at four times that diftance, then the lines 1 m and i n drawn perpendicular to AB and CD refpedtively, will mark the fituation and fize of the wheels required. In 4 pra&ical Mechanics. On the for¬ mation of their teeth.1 Fig. 8. ©n crown wheels. Fig. II. tills cafe the cones are O n l and O m i, and s r n i, rjj m i, are the portions of them that are employed. The formation of the teeth of bevelled wheels is more difficult than one would at firft imagine. The teeth of fuch wheels, indeed, mull be formed by the fame rules which have been given for other wheels •, but lince dif¬ ferent parts of the fame tooth are at different dirtances from the axis, thefe parts muff have the curvature of their afting furfaces proportioned to that diftance. Thus, in fig. 10. the part of the tooth at r muff be more incurya- ted than the part at /, as is evident from the infpection of fig. 9. and the epicycloid for the part i muff be formed by means of circles whole diameters are im and r/, while the epicycloid for the part r muff be gene¬ rated by circles whofe diameters are C n and D d. 409. Let us fuppofe a plane to pafs through the points O, A, D •, the lines AB, AO, will evidently be in this plane, which may be called the plane of centres. Now, when the teeth of the wheel DE, which is fuppofed to drive CD the fmalleft of the two, commence their a£Hon on the teeth of CD, when they arrive at the plane of centres, and continue their action after they have paffed this plane, the curve given to the teeth of CD at C, fliould be a portion of an interior epicycloid formed by any generating circle rolling on the concave fuperficies of a circle whofe diameter is twice C n perpendiculat to CA, and the curvature of the teeth at 1 ffiould be part of a fimilar epicycloid, formed upon a circle, whofe diameter is twice im. The curvature of the teeth of the wheel DE at D, ffiould be part of an exterior epi¬ cycloid formed by the fame generating circle rolling upon the concave circumference of a circle whofe dia¬ meter is twice D d perpendicular to DA \ and the epicycloid for the teeth at F is formed in the fame way, only inftead of twice D d, the diameter of the circle muff be twice Ff. When any other mode of action is adopted, the teeth are to be formed in the fame manner that we have pointed out for common wheels, with this difference only, that different epicycloids are ne- ceffary for the parts F and D. It may be fufficient, however, to find the form of the teeth at F, as the re¬ maining part of the tooth may be fliaped by dire&ing a ftraight ruler from different points of the epicycloid at F to the centre A, and filing the tooth till every part of its acting furface coincide with the fide of the ruler. The reafon of this operation will be obvious by attending to the ffiape of the tooth in fig. 8. When the fmall wheel CD impels the large one DE, the epicycloids which were formerly given to CD muff be given to DE, and thofe which were given to DE muff be transferred to CD. 410. The wheel reprefented in fig. 1 r. is fometimes called a crown wheel, though it is evident from the figure that it belongs to that fpecies of wheels which we have juft been confidering ; for the acting furfaces of the teeth both of the wheel MB and of the pinion EDG are dire£ted to C the common vertex of the two cones CMB, CEG. In this cafe the rules for bevelled wheels muft be adopted, in which AS is to be confider- ed as the radius of the wheel for the profile of the tooth at A, and MN as its radius for the profile of the tooth at M j and the epicycloids thus formed will be the fec- tions or profiles of the teeth in the direction MP, at right angles to MC the furfaces. of the cone. When MECHANICS. 117 the vertex C of the cone MCG approaches to N till it PraAical^ be in the fame plane with the points M, G, fome of Mcc the curves will be cycloids and others involutes, as in the cafe of rack-work, for then the cone CEG will re¬ volve upon a plane furface. Appendix to Fergufon's LeBlures. Sect. II. On the Wipers of Stampers, , F, draw the two curves E / h O, and. O tp F, which will be the proper form, that muft be given to the arm of tire lever. If the handle DE moves from E towards F, the curve EO muft be ufed, but if in the contrary, direftion, we muft employ the curve OF. It is evident, that when the extremity E of the handle DE, has run through the arch E£, or rather E/, the point / will be in k, and the point s$ in v, be¬ came x* is equal to £/, and the lever will have the po¬ fition Ab. For the fame reafon, when the extremity E of the handle has arrived at z, the point h will be in h and the point g in/, and the lever will be raifed to the pofition Art. Thus it appears, that the motion of the power and the weight are always proportional. When a roller is fixed at E, a curve parallel to EO, or OF, muft be drawn as formerly. See Appendix to Fer- gufon's Lectures. Chap. VI* On the Firjl Movers of Machinery, 418. The powers which are generally employed as the firft movers of machines are water, wind, fleam, and animal exertion. The mode of employing water as an impelling power has already been given at great leng th in the article Hydrodynamics. I he application of wind to turn machinery will be difeufled in the chapter on Windmills ; and what regards fleam will be more properly introduced into the article STEAM-Engine. At prefent, therefore, we fliall only make a few general remarks on the ftrength of men and horfes; and con¬ clude with a general view of the relative powers of the firft movers o{ machinery. The following table con¬ tains the weight which a man is able to raife through a certain height in a certain time, according to differ¬ ent authors.. Table of the Strength of Men, according to diferent authors. Number of pound raifed. IOOO 6o7 4 5 £ 1000 1000 30 29 or 30 Height to which theweight is raifed. 180 a 220 y c 11 £ 33° 225 34 2.45 feet Time in which it is raifed. 60. minutes 1 fecond 145 feconds 1 fecond 60 minutes 60 minutes I fecond 1 fecond Duration of the Work. 8 hours half an hour 10 hours Names of the authors. Euler Bernouilli Amontons Coulomb Defaguliers Smeaton Emerfon Schulze 419. . (g) In the figure we have taken the point B in a diiadvantageous pofition, cafe more diftinft. s E > becaufe the interfeftions are in this * MECHANICS. Practical 419, According to Amontons, a man weighing 133 Mechanics, pQUnc|s prench, afcended 62 feet French by tteps in 34 ForcJof 1 feconds, but was completely exhaufted. The fame au- mcn ac- thor informs us that a fawer made 200 ftrokes. of 18 cording to inches French each, with a force of 25 pounds, in 145 Amontqns, pecori(js . ijuj that he could not have continued the ex¬ ertion above three minutes. According 420. It appears from the obfervations of Defaguliers, to Defa- that an ordinary man can, for the fpace of ten hours, goiters, turn a winch with a force of 30 pounds, and with a ve¬ locity of two feet and a half per fecond j and that two men working at a windlafs with handles at right angles to each other can raife 70 pounds more eafily than one man can raife 30* rlhe reafon of this is, that when there is only one man, he exerts variable efforts at dif¬ ferent pofitions of the handle, and therefore the motion of the windlafs is irregular j whereas in the cafe of two men, with handles at right angles, the effeft of the one man is greateft when the effedt of the other is leaft, and therefore the motion of the machine is more uniform, and will perform more work. Defaguliers alio found, that a man may exert a force of 80 pounds with a fly when the motion is pretty quick, and that by means of a good common pump, he may raife a hogfhead of water 1 o feet high in a minute, and continue the ex¬ ertion during a whole day. llefults of 42I. A variety of interefting experiments upon the Coulomb s force u£ men were made by the learned M. Coulomb, ments." He found that the quantity of adtion of a man who af- fcended flairs with nothing but his own weight, was double that of a man loaded with 223 pounds avoirdu¬ pois, both of them continuing the exertion for a day. In this cafe the total or abfolute effeB of the unloaded man is the greateft poflible •, but the tfeful effeB which he produces is nothing. In the fame way, if he were loaded to fuch a degree that he was almoft incapable of moving, the ufeful effedt would be nothing. Hence there is a certain load with which the man will produce the greateft ufeful effedt. This load M. Coulomb found to be 173.8 pounds avoirdupois, upon the fuppo- fltion that the man is to afcend flairs, and continue the exertion during a whole day. When thus loaded, the quantity of adtion exerted by the labourer is equivalent to 183.66 pounds avoirdupois raifed through 3282 feet. This method of working is however attended with a lofs of tliree-fourths of the total adtion of the workman.—It appears alfo from Coulomb’s experiments, that a man go¬ ing up flairs for a day raifes 205 chiliogrammes (a chi- liogramme is equal to three ounces five drams avoirdu¬ pois) to the height of a chiliometre (a chiliometre is equal to 39571 Englifh inches) •, that a man carrying wood up ftairs raifes, together with his own weight, 109 chiliogrammes to one chiliometre 5—that a man weigh¬ ing 150 pounds French, can afcend by ftairs three feet French in a fecond, for the fpace of 15 or 20 feconds $ —that a man cultivating the ground performs as much labour as a man afcending ftairs, and that his quantity of adtion is equal to 328 pounds avoirdupois raifed through the fpace of 3282 feet j—that a man with a winch does as much as by afcending ftairs and that in a pile-engine, a man by means of a rope drawn horizontally, raifed for the fpace of five hours 55^. pounds French through one foot French in a fe- C^nd.—When men walk on a horizontal read, Cou¬ lomb found that the quantity of adtion was a maximum Pra&icai when they were loaded, and that this maximum quanti- Mechanic*, ty of adtion is to that which is exerted by a man loaded ' » —1 with 190.25 pounds avoirdupois as 7 to 4.—The weight which a man ought to carry in order that the ufeful ef¬ feB may be a maximum, is 165.3 pounds avoirdupois. When the workman, however, returns unloaded for a new burden, he muft carry 200.7 pounds avoirdu¬ pois. 422. According to Dr Robifon a feeble old man raifed feven cubic feet of water—437.5 pounds avoir¬ dupois, Ilf feet high, in one minute, for eight or ten hours a day, by walking backwards and forwards on a lever }—and a young man weighing 135 pounds, and carrying 30 pounds, raifed 9F cubic feet of water —578.1 pounds avoirdupois, 1feet high, for 10 hours a day, without being fatigued. 423. From the experiments of Mr Buchanan, it ap¬ pears that the forces exerted by a man pumping, atting at a winch, ringing and rowing, are as the numbers 1742, 2856, 3883, 4095. 424. According to Defaguliers and Smeaton, the Qn ^ power of one horfe is equal to the power of five men. ftrength ^ ’ Several French authors fuppofe a horfe equal to feven horfes. men, while M. Schulze conliders one horfe as equiva¬ lent to 14 men.—Two horfes, according to the experi¬ ment of Amontons, exerted a force of 150 pounds French, when yoked in a plough. According to De-’ faguliers, a horfe is capable of drawing, with a force of 200 pounds, two miles and a half an hour, and of con¬ tinuing this aftion eight hours in the day. When the- force is 240 pounds he can work only fix hours. It appears from Smeaton’s reports, that by means of. pumps a horfe can raife 250 hoglheads of water, 10 feet high, in an hour.—The moft difadvantageous way of employing the power of a horfe is to make him carry a load up an inclined plane, for it was obferved by De la Hire, that tliree men, with 100 pounds each, will go fafter up the inclined plane than a horfe with 300 pounds. When the horfe walks on a good road, and is loaded with about two. hundred weight, he may eafily travel 25 miles in the fpace of feven or eight hours. 425. When a horfe is employed in raifing coals by means of a wheel and axle, and moves at the rate of about two miles an hour, Mr Fenwick found that he could continue at work 12 hours each day, two and a half of which were fpent in ftiort intervals of reft, when he ^aifed a load of iaoo pounds avoirdupois, with a velocity of 13 feet per minute •,—and that he will exert a force of 75 pounds for nine hours and a half, when moving with the fame velocity. Mr Fenwick alfo found that 230 ale gallons of water delivered every minute on an overftiot water wheel, 10 feet in diameter’, that a common fleam-engine, with a cylinder eight inches in diameter, and an improved engine with a cylinder 6.1 2 inches in diameter, will do the work of one horfe, that is, will raife a weight of icoo pounds avoirdupois, through the height of 13 feet in a minute. It appears from Mr Smeaton’s experiments, that Dutch fails in their com¬ mon pofition with a radius of nine feet and a half,—that Dutch fails in their beft pofition with a radius of eight feet, and that his enlarged fails with a radius of feven feet, perform the fame work as one man j or perform OK** M E C H pra&iea one-fifth part of the work of a horfe. Upon thefe fa£b Mechanics. we have conitruiled the following table, the four firft A N I C S. 121 columns of which are taken from Mr Fenwick’s Effay* on Pra&ical Mechanics. Practical Mechanic®. TABLE Jhevoing the relativeJlrength of Overjhot Wheels, Steam Engines, Horfes, Men, and Wind-mills of v different kinds. Number of ale gallons delivered on an overfliot wheel, to feet in dia¬ meter, every minute. 230 390 528 660 79O 97O II70 1445 !584 1740 1900 2100 2300 2500 2686 2870 3°55 3240 3420 3750 4000 4460 4850 5250 Diameter of the cylinder in the com¬ mon ftcam- engine, in inches. 8. 9-5 10.5 ir-5 12.5 I4‘ I5-4 16.8 17-3 18.5 I9-4 20.2 21. 22. 23.1 23-9 24.7 25-5 26.25 27‘ 28.5 29.8 31*1 32.4 33-6 Diameter of the cylinder of the im¬ proved fteam-engine, in inches. 6.12 7.8 8.2 8.8 9-35 10- 55 11- 75 12.8 13.6 14.2 14.8 15.2 16.2 I7* 17.8 18.3 19. 19.6 20.1 20.7 22.2 23- 23-9 24.7 25-5 Number of horfes work¬ ing 12 hours per day, and moving at the rate of two miles per hour. Z 2 3 4 5 6 7 8 9 10 11 12 J3 J4 J5 16 !7 18 !9 20 22 24 26 28 3° Number of men work¬ ing 12 hours a-day. 5 10 15 20 25 30 35 40 45 5° 55 60 65 70 75 80 85 90 95 100 no 120 130 140 150 Radius of Dutch fails in their com¬ mon polition in feet. 21.24 30.04 36.80 42.48 47-50 52.03 56.90 60.09 63-73 67.17 70.46 73-59 76-59 79-49 82.27 84.97 87.07 90.13 92.60 95.00 99.64 104.06 108.32 112.20 ij6-35 Radius of Dutch fails in their beft pofition, in feet. Radius of Mr Smea- ton’s en¬ larged fails, in feet. 17.89 25-30 30.98 35-78 40.00 43-8 2 47-33 50.60 53-66 56.57 59-33 61.97 64-5 66.94 69.28 7r-55 73-32 75-9° 77.98 80.00 83.90 87.63 91.22 94.66 97.98 15-65 22.13 27.11 3*-3° 35-oo 38-34 4I-4I 44.27 46.96 49-50 5I-9I 54.22 56.43 58.57 60.62 62.61 64.16 67.41 68.23 70.00 73-42 76.68 79.81 82.82 85-73 Height to which thefe different powers will raife 1000 pounds avoir¬ dupois in a minute. 13 26 39 52 65 78 9° 104 217 130 M3 156 169 182 J9S 208 221 234 247 260 286 312 338 364 39° 426. Dutch fails are always conftrufted fo that the angle of weather may diminifh from the centre to the extremity of the fail. They are concave to the wind, and are in their common poftion when their ex¬ tremities are parallel to the plane in which they move, or perpendicular to the direftion of the wind. Dutch fails are in their hefl poftion when their extremities make an angle of feven degrees with the plane of their motion. Mr Smeaton’s enlarged fails are Dutch fails in their beft polition, but enlarged at their extremi¬ ties. 427. It appears from M. Coulomb’s experiments on Dutch wind-mills, with rectangular fails, that when the diftance between the extremities of two oppofite fails is 66 feet French, and the breadth of each fail fix feet, a wind moving at the rate of 20 feet per fecond will produce an effect equivalent to icoo pounds raifed through the fpace of 218 feet in a minute. According to Watt and Boulton, one of their ‘fleam-engines, with a cylinder 31 inches in diameter, and which makes 17 double ftrokes per minute, is equi¬ valent to 40 horfes working day and night; that is, to 101 horfes working nine hours and a half, the time of conftsmt exertion in the preceding table. When the Vol. XIII. Part I. cylinder is 19 inches in diameter, and the engine make* 25 ftrokes of four feet each per minute, its power is equivalent to twelve horfes working conftantly, or thirty horfes working nine hours and a half;—and when the cylinder is 24 inches in diameter, and the engine makes 22 ftrokes, of five feet each, in a mi¬ nute, ’ I s power is equal to that of 20 horfes working conftantly, or 50 horfes working nine hours and a half. Chap VII. On the ConfruBion of Wind-mills. 428. A WIND-MILL is reprefented in fig. 1. w-here MN Plate is the circular building that contains the machinery, E CCCXX.V. the extremity of the windlhaft, or principal axis, which I" is generally inclined from 8 to 15 degrees to the hori¬ zon } and EA, EB, EC, ED four reditangular frames upon which fails of cloth of the fame form are ftretch- ed. At the lower extremity G of the fails their fur- face is inclined to the axis 7 2° 5 and at their fartheft extremities A, D, &c. the inclination of the fail is about 83°. Now, when the fails are adjufted to the wind, which happens when the wind blows in the di- reflion of the windlhaft E, the impulfe of the wind & , 122 MECHANICS. Practical upon the oblique fails may be refolved into two Mechanics. £orces^ one 0f -which a(c£s at rJght angles to the windihaft, and is therefore employed folely in giving a motion of rotation to the fails and the axis upon which they are fixed. When the mill is ufed for grinding corn, a crown wheel, fixed to the principal axis E, gives motion to a lantern or trundle, whofe axis carries the moveable millltone. Methods of 429. That the wind may a£l with the greateft efficacy fells ufthe6 uPon t'16 ^le v.'indlhaft mull; have the fame direc- wind ^on as ^v^n(a* ^ut as direction is perpetually changing, fome apparatus is neceffary for bringing the windfhaft and fails into their proper petition. This is fometimes effefted by fupporting the machinery on a ftrong vertical axis, whofe pivot moves in a brafs focket firmly fixed into the ground, fo that the whole ma¬ chine, by means of a lever, may be made to revolve upon this axis, and be properly adjufted to the direc¬ tion of the wind. Moll wind-mills, however, are furnifhed with a moveable roof which revolves upon friffion rollers inferted in the fixed kerb of the mill j and the adjuflment is effected by the affiftance of a fimple lever. As both thefe methods of adjuftment require the aflifiance of men, it would be very defirable that the fame effeft fliould be produced folely by the aftion of the wind. This may be done by fixing a large wooden vane or weather-cock at the extremity of a long horizontal arm which lies in the fame vertical plane with the windfhaft. By this means when the lurface of the vane, and its diftance from the centre of motion, are fufficiently great, a very gentle breeze will exert a fufficient force upon the vane to turn the ma¬ chinery, and will always bring the fails and windfhaft to their proper pofition. This weather-cock, it is evi¬ dent, may be applied either to machines which have a moveable roof, or which revolve upon a vertical arbor. On the Form and Pojiiion of Wind-mill Sails. 430. It appears from the inveftigations of Parent, that a maximum effeft will be- produced when the fails are inclined 54^- degrees to the axis of rotation, or when The inch- the angle of weather is 35^ (g) degrees. In obtain- nation af- ing this conclufion, however, M. Parent has affirmed Parentbe data which are inadmiffible, and has neglefted feveral roneous. circumftances which muff materially affeft the refult of his invefligations. The angle of inclination affigned by Parent is certainly the moft efficacious for giving motion to the fails from a date of reft, and for prevent¬ ing them from flopping when in motion ; but he has not confidered that the adlion of the wind upon a fail at reft is different from its aftion upon a fail in motion: for fince the extremities of the fails move with greater rapidity than the parts nearer the centre, the angle of weather fhould be greater toward? the centre than at the extremity, and ftiould vary with the velocity of each part of the fail. The reafon of this is very ob¬ vious. It has been demonftrated by Boffut, and efta- PnAica! blilhed by experience, that when any fluid a&s upon Mechanic*, a plain furface, the force of impulfion is always exerted v moft advantageoufly when the impelled furface is in a ftate of reft, and that this force diminiflies as the velo¬ city of the furface increafes. Now, let us fuppofe with Parent, that the moft advantageous angle of weather for the fails of wind-mills is 35-i- degrees for that part of the fail which is neareft the centre of rotation, and that the fail has every where this angle of weather j then, fince the extremity of the fail moves with the greateft velocity, it will in a manner withdraw itfelf from the aftion of the wind, or, to fpeak more proper¬ ly, it will not receive the impulfe of the wind fo ad¬ vantageoufly as thofe parts of the fail which have a lefs degree of velocity. In order therefore to counteract this diminution of force, we muft make the wind a.£t more perpendicularly upon the fail, by diminifliing its obliquity or its angle of weather. But fince the velo¬ city of every part of the fail is proportional to its di¬ ftance from the. centre of motion, every elementary por¬ tion of it muft have a different angle of weather dimi¬ nifliing from the centre to the extremity of the fail. The law or rate of diminution, however, is ftill to be difeovered, and we are fortunately in poffeflion of a the- ’ orem of Euler’s, afterwards given by Maclaurin, which determines this law of variation. Let 0 reprefent the Euler’s velocity of the wind, and c the velocity of any given theorem, part of the fail 5 then the effort of the wind upon that part of the fail will be greateft when the tangent of the angle of the xvind’s incidence, or of the fail’s inclina- • •• • / QC C 3 C tion to the axis, is to radius, as ^ 2-4--—-p — t° 4aa 2a Fig 431. In order to apply this theorem, let us fuppofe that hxplana- the radius or whip ED of the fail /3 2 y, is divided in- ^ alld aP' to fix equal parts; that the point n is equidiftant from E and D, and is the point of the fail which has the lame rcm> 1 " velocity as the wind ; then, in the preceding theorem, we fliall have cz=:a, when the fail is loaded to a maxi¬ mum ; and therefore the tangent of the angle, which the furface of the fail at n makes Avith the axis, Avhen a= 1, Avill be 2 3-56i =: tangent of 74° 19', Avhich gives 150 41'for the angle of Aveather at the point n. Since, at 4 of the radius c—0, and fince c is proportional to the diftance of the correfponding part of the fail from the centre, Ave Avill have, at ■£ of the radius sm, c=.~-, at of the radius, c—— ; at -g, 3 3 40,50, . . , at an