Norwich llniversitv Librai^, NortlAfleld, Vermont. Class NO. JT/O Boot^No.3fJ5^ University of California • Berkeley The Theodore P. Hill Collection of Early American Mathematics Books Digitized by the Internet Archive in 2008 with funding from IVIicrosoft Corporation http://www.archive.org/details/concisemathematiOOrobirich CONCISE MATHEMATICAL OPERATIONS; BEING A SEQUEL TO THE AUTHOR'S CLASS BOOKS WITH MUCH ADDITIONAL MATTER. A. WORK ESSENTIALLY PRACTICAL, DESIGNED TO GIVE THE LEARNER A PROPER APPRE* CIATION OF THE UTILITY OF MATHEMATICS ; EMBRACING THE GEMS OF SCIENCE FROM COMMON ARITHMETIC, THROUGH ALGEBRA, GEOMETRY, THE CALCULUS, AND ASTRONOMY. BY H. N. ROBINSON, A. M. FORMERLY PROFESSOR OF MATHEMATICS IN THE UNITED STATES NAVY J ATTTHOR OF ARITHMETIC, ALGEBRA, NATURAL PHILOSOPHY, GEOMETRY, SURVSYING, ASTRONOMY, ETC. ETC. ETC, CINCINNATI: JACOB ERNST, 112 MAIN" STREET. 1854. Entered according to Act of Congress in the year 1854, bj H. N. ROBINSON, In the Clerk's Ofl&ce of the District Court of the United States for the Northern District of New York. PREFACE. This book is not designed to teach Mathematical Principles, but to apply and enforce them. It contains collections and groups of mathematical prob- lems which show the utility of science, and place its fruits in the foreground. Let no one expect to find any close connection between the different parts of this book, or even in any one part of it. System and connection is essen- tial in every theoretical work, but it would be as. absurd to look for it here as to look for a composition in a dictionary. That there is need for such a work as this, all would be convinced who could see but a tenth part of the letters that every accommodating mathema- tician is constantly receiving, requesting the solution of problems or the exposition of principles. Indeed, much important matter to be found in this volume, has been sug- gested and brought to the immediate notice of the author by letters received requiring his aid ; and to save the trouble of answering such letters in future was one inducement to publish this work. There is a great deal of perfectly barren mathematical knowledge in this country; particularly among those who have studied, not for science, but for a diploma. Not unfrequently do we meet persons who can demonstrate many, if not all the elementary problems in common Geometry, who, at the same time, cannot make the least application of them, and who seem to be unaware that they were ever intended for any practical use. Knowledge, so confined and abstract, is of doubtful utility, even as a mental discipline. Unless we take in a broad expanse, and unite both theory and practice, we perceive nothing of the beauties of the Mathematics. De- tached propositions and abstract mathematical principles, give us no better idea of true and living science, than detached words and abstract grammar would give us of poetry and rhetoric. Small acquirements in the Mathe- matics serve only to make us timid, cautious, and distrustful of our own powers — ^but a step or two further gives us life, confidence, and power. The efforts of the great mass, who attempt the study of the Mathematics, are very inefficient and feeble, because the motive is not sufficiently pointed and pressing. They study for the discipline of mind. ]N'ow, we venture to assert, that those who study for any object so indirect and indefinite, can never be decidedly successful. And those who teach with no other view than giving discipline to the minds of their pupils, never more than half teach. The object, and the only object, should be to under- stand the subject studied, and if that understanding is attaiiied, the highest mental discipline that the subject can yield, will surely come with it. iii iv PREFACE. Let a person undertake the study of any science, Trigonometry for exam pie, with no other object than the discipline of the mind, and our word for it, the science will come to him with the utmost diflficulty ; and however long he may study, the spirit of the science will never find a lodgment with him. But let him be determined to understand it, for the purpose of being an architect, an engineer, or a navigator, and all is changed — beauties are now seen where none were discovered before, and the student is now sensible of possessing both knowledge and mental discipline. Let a person commence Astronomy, simply with a view to mental disci- pline, and when will he obtain a sound knowledge of that science ? We answer, never. But let him commence the study with a determination to understand it, and his efforts will be well directed, and science will come to him with ease, and with it will come a discipline of mind, the most pure and lasting that man can attain. There is another erroneous impression which serves, as far it goes, to ob- struct the progress of sound mathematical learning in this country. It is a vague, yet general idea, that Arithmetic, Algebra, Geometry, Trigonometry, and the Calculus, are distinct and separate sciences, and each is to be learned by itself and then carefully laid aside. The truth is, they are but diJOferent sections of the same science, and each one in turn may be used to illustrate the other ; and studied as a whole, under the direction of a philosophic t^^acher, the labor of acquisition would be very much reduced. Were we to say nothing in respect to our method of treating the square and cube roots in this volume, the mere arithmetician would undoubtedly depreciate it. He will perhaps still regard the method as unscientific, and call it a mere " cut and try" operation ; but when he finds the same thing in Geometry, and there finds lines which may represent all the different factors in any case, and sees the geometrical reason why the exact square root is always a little less than the half sum of two unequal factors, he must then admit that the cut and try method is not very unscientific after all. The truth is, in the hands of those who can take the geometrical view of it, and who can use it with judgment, this method is as scientific as any, and in many cases far more practical than the common rilles. The first principles of Geometry are, to a certain degree, abstract ; but the application of Geometry, as appears in this work, is far from being so ; and he must be a very practical mathematician who cannot find something here to amuse, to interest, or to instruct him. To the subject of finding sines and cosines, both'natural and logarithmic, for every minute of the quadrant, we call special attention — as strict attention to that subject in all its bearings, will so readily impress upon the mind of a learner, the importance of theoretical Geometry. To the practial application of Interpolation, we'also call attention. Some problems in Mensuration and Plane Trigonometry will be found very inter- esting to those who possess a taste for the Mathematics, and we have ex- tracted several different solutions from the works of others, to show how PREFACE. V diflferently diiFerent persons present the same thing. There are few mathe- matical students who could not be greatly benefitted bj a close perusal of Spherical Trigonometry and Astronomy as presented in this work. Any person who has the outlines of Astronomy and Elementary Mathematics can here have a view of all the details of a solar eclipse, in a comprehensible shape. There has been a great deal of unnecessary controversy about the Differen- tial and Integral Calculus, which we think can and ought to be wiped away. And we have here given a little foretaste of what we shall attempt if cir- cumstances prompt us to write a work on that subject. It is not for us to assume that we can make science clearer than others, but we have yet to see the works of an author who has made the least attempt to show the simple elementary nature of this science. They at once commence with the definition of constants and variables, and then direct what to do. We have yet to see the first book that expends a word in giving an idea of what the Calculus is, or what is the utility and object of the science, and we charge more than half the obscurity to this fact alone : hence we could not forbear being a little elementary when we came to that subject, and we leave it to those readers, who have fonnerly studied other works on this science, to say whether we have or can dispel any of the obscurity that has so long hovered around it. All sciences are obscure until they are applied. Even Arithmetic would be so in the abstract, and being alive to this fact we have extended the ap- plication of the Calculus to more subjects than we have hitherto observed in other works. For example, see the method of clearing lunar distances, and the use we made of the same principle in computing an eclipse. But neither in the Differential nor the Integral Calculus do we pretend to be any thing like full or perfect, even for a work of this kind. We have only thrown out a few practical remarks and problems, in our own unique manner, more to learn what is desired, and what can be appre- ciated, than for any thing else. "When we commenced, we did not intend to produce so large a volume; it grew on our hands; but we believe that this result will not be regretted by generous patrons. CONTENTS. PART I.— ARITHMETIC. SECTION I. Introduction, 13 14 The Philosophy of Multiplication and Division, 14 18 Canceling, 19 ^ Proportion, 23 ^29 Cause and Eflfect, 26 ^29 SECTION II. Exchange, -29 Compound Fellowship, 30 32 Problems in Mensuration and the Roots, 32 34 SECTION III. Powers and Roots, 35 16 Alligation Alternate, 46 48 Position, 48— —50 PART II.— ALGEBRA. SECTION I. Simple Equations, 51 59 Problems Producing Simple Equations, 59 67 Interpretation "of Negative Values, 67 68 Finding and Correcting Errors, 69 71 Pure Equations, 72 80 Questions Producing Pure Equations, 80 84 vii viii CONTENTS. SECTION II. Quadratic Equations, 84 89 Special Equations in Quadratics, 89 ^96 SECTION III. I Quadratic Equations containing more than one Unknown Quantity, 97 108 UnTfTought Examples, 109 110 SECTION IV. Problems producing Quadratic Equations containing more than one Unknown Quantity, 110 116 Problems Selected from Various Sources, 116 123 SECTION V. Problems in Proportion and Progression, 123 124 Geometrical Progression and Harmonical Proportion, 125 129 Proportion, 129 130 Additional Problems, 130 138 SECTION VI. Solution of Equations of the Higher Degrees, 138 161 H'ewton's Method of Approximatic«i, 138 140 Horner's Method, ^ 140 157 l^ew and Concise Formula to find Approximate Roots in Quadratics, 143 144 Cubic Equations, 148 152 The Combination of Roots in the Fca-mationof Coefl5cients,.158 161 Recurring Equations, 161 166 SECTION VII. Indeterminate Analysis, 166 181 Properties of Numbers, 166 170 Indeterminate Problems, , . . , 171 181 CONTENTS. k SECTION VIII. To determine the Number of Solutions that an Eom. Formerly all kinds of problems and puzzles were to be found in arithmetics ; but pure science, good taste, and the rapid ad- vancemert of the pupils, require that the works on arithmetic should bf concise and clear, and take no undue proportion of the student's ^ime and attention. Severo problems do not teach science — but science will subdue all severe problems, and we would use problems only as a means of elucidating science. Algebraic problems, and problems in geometry and mensuration, should never appear in arithmetic, but old custom will not yet tolerate their expulsion. We shall pay particular attention to the metaphysique of the science. Numbers only of the same kind can be added together or subtracted from each other. Numbers are either abstract or concrete. Abstract numbers are unapplied and are mere numerals. Concrete numbers bring to the mind the particular number of things to which they refer. Arithmetic proper, comprises the system of notation, and the operations to be performed with abstract numbers only — vnihout any reference to their application whatever. The application of arithmetic includes all kinds of numerical 13 U ROBINSON'S SEQUEL. computations, and they are therefore endless in variety and char- acter. In the application of arithmetic, there are two distinct opera- tions, the logical one and the mechanical one; the thinking and the doing. The undisciplined direct their attention more to the doing than to the thinking, when it should be the reverse; and nearly all the efforts of a good teacher are directed to make his pupils reason correctly. If a person fails in an arithmetical problem, the failure is always in the logic, for false logic directs to false operations, and true logic points out true operations. Abstract arithmetic we shall not touch, except when necessary to illustrate a point before us. With these introductory remarks we commence with the follow- ing principles : 1. Multiplication is the i^epetition of one number as many times as there are units in another. This is general, whether the numbers be large or small, whole or fractional. The mles in whole numbers and in fractions, apply to the mechanical operations only, and not to the one fundamental principle. 2. When the multiplicand and multiplier are both abstract num- bers, the product is abstract, or a mere numeral without a name. 3. iVb two things can be multiplied together. A multiplicand may have a name, as dollars, yards, men, e as the multipHcand. 4. Division is finding how many times one number can he sub- tracted from another of the same hind. There are other definitions to be found in books, which do very well in the main, but this is the only truly logical definition I can find. Division should never be considered in the light of sepa- rating a number into parts, for this is not true in all cases, and confusion often arises in fractions by this view of the subject. ARITHMETIC. 16 5. Division corresponds to multiplication conversely, when we take the product for a dividend, the multiplicand for a divisor, and the quotient for a multiplier, 6. In multiplication it is indifferent which of the two factors is called the multiplicand, the other must be an abstract multiplier. The name of the product (when known) is an infallible index to show which of the two factors is really the multiplicand. To illustrate principles three and six, we give the following EXAMPLES. 1 . What will 763 pounds of pork come to at 8 cents per pound? At first view, this example seems to conflict with principle three, for, says the pupil, we multiply the pounds by the price per pound; but it is not so. Two pounds would cost twice as many cents as one pound, and 763 pounds would cost 763 times as many cents as one pound ; therefore 763 is the abstract multiplier in the operation, and 8 cents is the true multiplicand, and the product will be cents, as required. In the act of multiplying, it is indifferent how the numbers are written. 2. Reduce 6£ 135. ^d. to pence. Here 20 and 5, as abstract numbers, must be multiplied together and 13 added, making 113 shillings, — but which of the two fac- tors 20 or 5, is the multiplicand? Here nine-tenths of those who teach arithmetic would call the 6£ the multiplicand and 20 the multiplier ; but this is not so. A multiplicand suffers no change of name by being multiplied, and as the name of the product is unquestionably shillings, 20 shil- lings is the multiplicand, and 5, as an abstract number, is the multiplier, there being 5 times as many shillings in 5£ as in 1 £. By the same logic, to reduce 113 shillings to pence, 12, the number of pence in a shiUing, is the true multiplicand, which must be repeated 113 times, and then the product must of course be pence as required. Dollars can be divided by dollars, and by nothing else. Yards can be divided by yards, and by nothing else, and so on, for any other thing tha* might be mentioned. 16 ROBINSON'S SEQUEL. This fact has not been sufficiently attended to ; indeed, it has scarcely been recognized by many teachers. It is true we can divide a number of dollars, yards, 5*> 2f'i, 2 " 18 ROBINSON'S SEQUEL. We will give but one more example to show that the divisor and dividend must be of the same name. The moon describes an arc in the heavens of 197° 38' 45" in 15 days; how great an arc will it describe in 1 day? The fifteenth part of 197° 38' 45" is obviously the sum required, but how will the number 15 measure 197 degrees? If we drop the name of degrees and say 197, then we can divide it by 15, and this is the usual way — during the operation all names are prac- tically destroyed — and after the operation is over, the proper name is given according to the logic or philosophy involved in the ques- tion, and it is in this logic or philosophy where the unthinking fail, if they fail at all. We may also solive this problem by conceiving the moon to move 15° in one day, and then dividing 197° 38' 45" by 15°, we shall obtain an abstract number, each unit of which corresponds to a day. Then changing the names between the divisor and the quotient, 16° will become an abstract number, and the quotient will be degrees, ^(Scc. as required. 15°) 197° 38' 45"(13 15 47 46 60 J5)158(JO 150 8 60 16)525(36 45 76 76 Our first divisor was 15°, second 15', third 15", but by making these abstract numbers, the quotient will become 13° 10' 35", the answer. CANCELING. 19 €ANCEL,ING. Within a few years the subject of canceling has been brought to the special notice of teachers and others, and like every other improvement, it has been opposed by some, and looked upon with distrust and indifference by others. But still, it being a real and substantial improvement, it is working its way ; and even at this day intelligent pupils are astonished that teachers should op- pose it as they sometimes do. Indeed, that teacher who would in any degree discountenance cancellation should be dismissed at once from the class room. Some few educationists had private reasons of a pecuniary nature for opposing cancellation; but the chief opposition arose from the disinclination of persons to break into old habits. Cancellation does not change the process of reasoning on a problem, iDut it requires a more general perception at a glance, and more rapidity of thought, than the old methods; hence the naturally dull did and do yet oppose it as a matter of course. The architect makes the design of a proposed building on paper, represents it inside and out, estimates the cost, suggests changes and improvements, and has it all in his mind before a stick of timber is prepared, or any serious labor commenced. It is economy to do so. An arithmetician should do the same ; he should be able to repre- sent what he proposes to do, on paper, look at it and consider it fully before he commences real labor. It is economy to do so, for then he may see counter operations that will cancel or abridge each other. Desirable as all this is, it is rarely thought of; — no sooner is an operation decided upon than the operator hastens to perform it, without thinking further until that is done. He then decides upon another step and performs it, — then another, and so on through the problem. Now as a general rule we would have each step of an operation distinctly indicated before it be performed, and then examined as a whole, the same as an engineer would examine a map, or an architect the plan of a building. We give the following exam- ples to exercise this faculty : 20 ROBINSON'S SEQUEL. 1. A merchaiU bought 526 barrels of flour at $4.50 per barrel^ and paid in cloth at $2.25 per yard. Bow many yards did it require? Ans. 1052. The following is the common method of thought and operation. We must find what the flour will amount to, and as soon as that thought is defined, the operator commences the multiplication. When that is done, then comes the thought about the cloth, and it is decided to divide the amount by $2.26 for the required re- sult, and the operation stands thus : 526 460 26300 2104 225)236700(1052 225 1170 1126 450 450 Now we would not change the direction of the thought in the least, but we would have it continued to the end, and each opera- tion indicated as we go along.* The map of the whole operation stands thus: 526-45 225 ~ Here is a fraction, the numerator consisting of two factors, the denominator of one factor which is contained twice in 460. Hence twice 526 is the required result, and the mechanical ope- ration is just nothing at all. ♦When two numbers are to be multiplied together, we write them with a point between, thus 4.6 indicates 4 multiplied by 6. If this is to be divided by any other number, say .3, we would write -1-. When two numbers are to be added, we write (+) plus between them; when one is to be subtracted, we write ( — ) minuB before that one. CANCELING. n 2. How much will 540 yards of cloth cost at 3s. 4d., in dollars at 6 shillings each? The map of the operation is thus: — —- ?-. This reduces to 90 'SI-. Multiply one factor by 3, and divide the other by 3, (which will not change the value of the product), then 30-10=300, the result. N. B. In this work we do not pretend to explain prime and composite num- bers, what numbers will cancel with each other, and what will not. These things must be learned elsewhere. 3. At \9.\ cents per pound what must he paid for four boxes of sugar, each containing 136 pounds? Map of the operation, = =63 dollars. 4. What will one hogshead, or 63 gallons of wiTie, cost at Q\ cerUs a gill? Ans. ^126. Map of the operation, =126. The multiplication indicated in the pumerator reduces the 63 gallons to gills, and as 6|- cents is one-sixteenth of a dollar, we divide by 16, which cancels the product of the fours in the nu- merator and leaves 63 to be doubled for the result. 5. At 1^' cents a gill, how many gallons of cider can be bought for $24? Ans. 50. 24* 100 Map of part of the operation =:the number of gills, or 24-200 ,, .,, * ' ' =the gills. HT r .1 1 1 .• 24-200 200 ^^ . Map of the whole operation, = =50 Ans. ^ ^ 3-4-2-4 4 6. Jf a man travel 39 miles 20 rods in a dag, how many days leill be required to traverse 25000 miles? Ans. 640. As 320 rods make a mile, the following is the m? 320-25 000 320-39-1-20 22 ROBINSON'S SEQUEL. Here numerator and denominator can be divided by 20, which , ,, ,. , 16-25000 reduces the operation to . ^ 16-39+1 Because no further reduction can be made, this last indicated operation must be performed in full. In all cases, whether reduction can be made or not, we would insist on having the operations first indicated; and in practice, nine-tenths of the operations can be reduced. There is now and then one that cannot be reduced. Even when the plan of an arithmetical operation is laid down, judgment should be used in drawing out the final result, as the following example will illus- trate: Required the value of the following expression : 4900\2 /43y /144y /4900\ \80/ \ 95 / \ 24 / This occurs on page 156 of Robinson's University Algebra, and we have seen it literally carried out as indicated, in several of the best schools in the country; no reductions being made until after the numbers were squared ; thus making a long and tedious process. The proper way is to take the square root of the expression ; then we shall have l?.iil.l?00^ 80 95 24 Reducing does not change the value of the expression ; the first obvious reduction, is to divide the numerator and denominator 43 6 490 by 10 and 24 ; then the expression will stand thus, — •■ . . A still further reduction ffives • — • — =166, nearly. ^ 2 19 1 ^ Now the square of 166.3, is the value of the required expres- sion. We square, because the square root was taken in the first step. We may do this, because we have no where changed the value of the expession, except in taking the root. PROPORTION. 23 PROPORTIOIV. This manner of expressing an operation is most efficacious and practical in proportion. We shall make no attempt to elucidate the principles of propor- tion, our attention for the present being entirely on numerical op- erations. EXAMPLES. 1. If 9,ciut. ^qr. ^Ub. of stigar, oo&t ^£ Is. 80?., what wUl Shcwt. Iqr* cost? cwt. qr. lb. cwt. qr. £, s. d. Statement. 2 3 21 : 35 i : : 6 1 8 This example is taken from an old but popular book, in which the solution covers about two pages. The sugar is reduced to pounds, and the money, to pence. The result of the proportion is then obtained in pence, which being reduced, gives 73£. We do it thus : Reduce the sugar to qrs. Then the proportion is 11| : 141 : : 6£ U, Sd. Multiplying the two first terms of this proportion by 4, which does not change the proportion, then we have 47 : 141-4 : : 6£U. M. or 1 : 3-4 : : 6£ U. M. Therefore 12 times the third term is the result, 73£. 2. If 3cwt. of sugar cost 9j2 Is., what will 4cwt. Sqr. 26lb. cost nt the same r.^5 per 1000 bricks, each brick being 9 inches long, 4 inches wide, and 2 inches thick? Ans. $336.96. Index 240'12'6'12'3'12-(3.25) 1000-9-4-2 . 7. The bung diameter of a cask is 38 inches, the head diameters inside the staves 28 inches, and the length 45 inches: how many wine gallons will it contain? Ans. 167.89-|-. N. B. The cask is conceived to be two equal /rwi-^wms of cones joined by their greater diameters. {^See Geometry.) Index to solution (5^^+28^+28-38),7854-45 3'231 Observe that the decimal 0.7854 is divisible by 231 : quotient .0034. Therefore we may have the following rule to find the num- ber of wine gallons in a cask : Rule. To the square of the head diameter add the square of the bung diameter, and the product of the two diameters : multiply that sum by ^ of the length of the cask and by the decimal .0034. 8. A man bought a grindstone which was 48 inches in diameter and 5 inches in thickness, for $10. When he had ground down 3 inches of its radius, a neighbor proposed to purchase it from him at the same proportional price, in case he would deduct 4 inches each way from the center ^ allowed to be the limit to which it could be used. What should the purchaser pay? Ans. $7.58-)-. Statement (43''— 8"^). 7854 : (42''— 8^). 7854 : : 10 : Ans. Or (48^'— .8'') : (4^^ —8'') : : 10 : Ans. Or 66-40 : 50-34 : : 10 : Ans. oca" Or 56-2 : 6«17 : : 10 : Ans.=^^. 112 9. If a mxin 6 feet in height travel round the earth, how muxk further must his head travel than his feet? Ans. 37 ^-Q feet nearly. Let D= the diameter of the eai*th in feet ; then rti>= the cir- cumference in feet. (D-\-12)= the diameter, and rti>+13rt=s the circumference traveled by the man's head. The difference =.1^(3.1416)=^««. POWERS AND ROOm 3ft « SECTION UIs PO^VERS ANO KOOTS. The common methods of operation, as taught under this head, Rre in general the besk One object in this work is to show some peculiarities which will in some instances abridge labor, awaken investig'ation, and inspire originality of thought. We give the following delinitions : 1 . Any number multiplied into itself is called the square of that number. Or we may say (he product of two equcd factors pro- duces a square^ Either factor is called the root. 2. The product of threie equal factors is a cube or third power, — ■ of four equal factors, a fourth power, and so on. One of the equal factors is a root in all cases. 3. A square number multiplied by a square number, will pro- duce a square numberv N. B. This is obvious in Algebi-a for a^ multiplied by b^ pro- duces a^b^ , obviously a square, whatever numbers may be repre* sented by a and b. 4. A square number divided by a square number will give a square number, either whole or fractional. 5. A cube number multiplied by a cube number will give a cube number. 6. If a root is a composite number, its power (square or cube as the case may be) can be separated in square or cube factors: but if the ix)ot is a prime number, the power cannot be so sepa^ rated. We will soon show the practical utility of these principles. While operating in powers and iHDots w-e should have the fol- lowing table before us : Numbers, j 1 j 2 | 3 | 4 1 5 1 6 1 7 i 8 1 9 1 10 1 Sq. or 2d poAver, | 1 | 4 ! 9 jl6 1 25 1 36 1 49 1 64 1 81 ! 100 1 Cube or 3d power,! 1 ! ^ !27 j64 il25 |216 |343 |512 |729 iiooo i Powers being obtained from roots by simple multiplication, there is no room for much artifice. 56 ROBINSON'S SEQUEL. Sometimes the application of the following properties of num- bers will be useful : The square of the difference of two numbers is equal to the sum qf the squares, less twice the product of the two numbers. Algebraically, (a—hy=a'-\-b^—2ab. The square of the sum of two numbers is equal to the sum of the squares added to twice the product of the number. Algebraically (a+5)2 =a2 +6^ +2aJ. EXAMPLES. 1. What is the square of 79? Ans. 624L (79)2 = (80—1)2 =6401—160=0241. 2. What is the square of 83? Ans. 6889, (83)2 ^(80+3)2 =64094-480=6889. 3. What is the square of 97? Ans. 9409, (97)2=(100— 3)2 = 10009— 600=9409. 4. What is the square of 971? Ans. 942841, (971)2 =(970+1)2 =940901-f 1940=942841. 5. What is the square of 29? Ans. 841. (29)2 = (30— 1)2=901— 60=841. These formulas are useful when one or all of the integers are large. We shall now turn our attention to the extraction of square root. We suppose the reader understands the common method, which as a general operation is the best. To call out thought, however, we will require the square root of 9409, on the supposition that we know nothing of the common rule, and only know thai two equal factors of the numbers 9409 are required. The first thought is, that if we divide any number by any factor ^ the quotient will be another factor . Take 100 for one factor. Divide 9409 by 100, and the other factor is 94, omitting the decimal ; but these factors are not equal. The factors sought then, or rather one of them, is more than 94, and less than 100. Hence it must be near the half sum of these two numbers ; that is, near 97. By trial we find 97 correct. POWERS AND ROOTS. 37 N.. B. The half sum of two unequal factors, is always a little greater than one of the equal factors, because the sum of two une- qual factors which form a product, is always greater than the sum of two equal factors. EXAMPLES. 1. Find the square root o/" 841 ; that is, we demand two equal f Victors, which, multiplied together, will produce 841. Assume any factor : say 25. 25)841(33, (plus a fraction, which we omit) is the corresponding factor. But these factors are not equal, and the equal factors must be near their half sum ; that is, near 29. By trial, 29 is found to be the number exactly. 2. Find the square root, or two equal factors of the number 444889.. Divide by 6. 6 )444889 ^ 74148 Here the two factors scre-veri/ uneqzcal, but we can bring them to a proximate equality, by conceiving one multiplied by 100, and the other divided by 100. The factors will then be 600 and 741, nearly. The half sum of these is 670, which must be near one ©f the equal factors sought. Now divide. 670)444889(664 4020 4288 4020 2689 2680 These factors being so nearly equal, and there being a slight remainder, the half sum of the two (667) may be relied upon as tlie true -root. 38 ROBINSON'S SEQUEL. 3. Find the square root of 3. Am. 1.7320508. The only two factors in whole numbers are 1 and 3 ;* these are so unequal that their half sum, 2, will be entirely too large. Hence I will assume one factor to be 1.7. 1.7)3. (1.7647 1.7 130 119 110 1.7647 1.7 2)3.46T7"" JQ2 1.7323 root nearly. 80 68 120 Making another trial with the assumed factor, 1.732, we find the result as stated in the answer. 4. Find the square root o/ 181. Ans. 13.45362-f.. If we allow ourselves to have some knowledge of square num- bers, we can find a factor near in value to one of the equal factors sought. Thus the square of 12 is 144, and of 13, 169 ; therefore one of the equal factors of 181 is more than 13. Assume it 13.5, the other factor is, then, 13.4074 ; the mean of these is 13.4537. Taking this as the assumed factor, wo approximate still nearer to the root by a like operation ; and thus we can approximate to any degree of accuracy required. By admitting that every figure in a root demands two places in its second power, we can come near the root at the first assumption. For example : 5. Find the square root c/ 617796. Ans. 786. Separate the power into periods, as in the common operation ; the superior period is 61 ; the square root of this is near 8, and being three periods the root is near 800. Assume 780, then di- vide by it,, thus, * In our geometrical problems we shall give a scientific and satisfactory method of reducing unequal to the equivalent equal factors. POWERS AND ROOTS. 39 780)617796(792 5460 7179 7020 1596 1560 36 The half sum of 780 and 792, is 786 ; the answer. By the last example we perceive that the square root of the product of two factors which are nearly equal, is very nearly equal to the half sum of the two factors. It is a little less. In the last example there was a small remainder, which was rejected ; had there been no remainder, 786 would have been too great for the root. The square root of the producf of two square factors is equal to the product of the square root of those factors. That is, the square root of a^6^ is the square root of a^ into the square root of b^; in short it is aX^- To apply this principle I adduce the following examples : 1. A section of government land is a square of 640 acres. What is the length, in rods, of one of its sides ? Ans. 320. This problem requires the square root of the product of the two factors, 640 and 160. The product of two factors is not affected by multiplying one and dividing the other by the same number. Now multiply the factor 160 by 10, and divide the other by 10, then 1600 "64 will be the equivalent factors ; both square factors ; their roots are 40 and 8. Hence, the value sought is 40* 8=320. Again. Take the original factors, 640, 160. Divide 640 by 2, and multiply 160 by 2, which gives 320, 320. As the factors are now equal, one of them is the root sought. 2. A man has 50 ^ acres of land in a. square form; what is the length of one of its sides ? Ans. 90 rods. Index. V50|-160 = 7ifi- 160 = ^405-20=^8100=90. 40 ROBINSON'S SEQUEL. 3. Find the square root of the product of the two factors^ 1 8 aw«?32. Equivalent factors, 9 and 64 roots 3-8=24. " Or, 36 and 16 roots 6-4=24. Again, — it — =25, which is too great for one of the equal factors by 1, because the factors are so unequal. In working square root, it is important that the teacher should be able to show to his intelligent pupils, that the square on the hypotenuse of a right angled triangle is equal to the sum of the squares on the other two sides, notwithstanding they have never been students in geometry. To give an ocular demonstration of this important truth, we present the following figure ; The line PQ separates two equal squares. The triangle a is the right angled triangle in question, its right angle at P, x and y are its two sides, and the side opposite the right angle P is called the hypotenuse. In each square are four equal right angled triangles. Let them be taken away from each square, and in one square the square ^will be left, and in the other square the two squares A and B will be left. Now, from each of the two equal squares on each side of PQy we took equal sums — which must leave equal sums. That is, A+B=H. When we operate on a right angled triangle, we may divide the two given sides by the same number, if we can do so without POWERS AND ROOT^S. 41 a remainder on either side, and then operate with the quotients as we would with the original numbers. But in conclusion we must multiply the result by the number which we divided bj. This is working on a similar reduced triangle. j:xa.mples. 1. The two sides of a right angled triangle are 312 and 416; what is the hypotenuse ? Ans. 520. Divide by 5 2)312. 416 Divide by 2) 6 8 3 4 Square 3 and 4, add those squares which make 25 ; the square root of 25 is. 5. Multiply 5- 2 -52=520. Ans. 2. A hawk, ^perched on a tree 77 feet high, was brought down by a sportsman 1 4 rods distant on a level with the base ; what distance in yards did he shoot? Ans. 81.15-1-yards. 14 rods reduced to feet is 14- 161=7 -.33. Now without reduction we shall be obliged to square 77 and 231. But we may operate thus, 7)7-33 77 11) 33 n 32-1-12 = 10. ^10=3.1622+. Ans. in feet, = 77 (3.1622). Ans. in yards, 77 (1.054). CUBE ROOT. The object of the Cube Root is to find three equal factors, ex- actly or approximately, Avhose product will give any required sum. The reason of its being called cube is because the three factors may be correctly represented by the length, breadth, and height of a geometrical cube. The product of three unequal 42 ROBINSON'S SEQUEL. factors may be represented by a geometrical solid of unequal length, breadth, and height, called a parallelopipedon. While operating for cube root it is convenient to have the cube numbers before us. Roots, 123456789 10 Cubes, 1 8 27 64 125 216 343 512 729 1000 We see by these cubes that one figure in a root may Kave three places in its corresponding power. Therefore separate the power into periods of three figures each, beginning at the units ; the number of periods will show the num- ber of figures in the root. Now as we are to have nothing to do with the common methods of extracting cube root, all we are permitted to know is the divi- sion of the power into periods, and the fact that three 'equal factors of the power are required. EXAMPLES. 1. Extract the cube root of 84604519 ; or in other woi'ds,find three equal factors whose product will produce this number. Ans. 439. Here are three periods 84'604'519, which show that there must be three figures in the root. The superior period is 84, and 84 referred to the line of cubes, its place would be between 64 and 125, whose roots are 4 and 5. Hence the root sought for is greater than 400 and less than 500 ; I should judge it not far from 440. Therefore I assume 440 as one of the factors of the number. 440)84604519(192283 440 4060 3960 1004 880 1245 880 "3661 3520 "1319 1320 CUBE ROOT. 43 Now if one factor is 440, the product of the other two is 192283, very nearly ; (not exactly, for the last figure 3 is too large by a very small fraction.) We will now operate for two equal factors of the number 192283, and if our first factor is near an equal factor, that same factor is near an equal factor in this number ; therefore try it thus, 440)192283(437 1760 1628 1320 3083 3080 Here we have three factors, 440, 440, 437, whose product will give the number 84604519, within 3 units. These factors are not all equal, and of course are not the factors required ; but they are so nearly equal that one-third of their sum will be one of the equal factors required. That is, 440 440 437 3)1317 439 Ans. 2. Find the cube root, or three equal factors of the number 32461759. Ans. 319. By the aid of the periods we perceive that the factors must be greater than 300, but nearer 300 than 400. Assume then 312 to be near one of the equal factors sought for. Divide by 312 twice, or once by the square of 312. That is, 97344)32461759(333.3 292032 325855 2 92032 "338239 292032 46207 It is now obvious that the product of the three factors, 312, 312, and 333.3, will produce very nearly the given power; but these factors are not all equal, and equal factors are required ; 44 . ROBINSON'S SEQUEL. but they are so nearly equal that } of their sum, 319.1, can be relied upon as extremely near the root required. A factor, or root, determined in this manner from unequal factors, will always he a little in excels of the true value required. Hence, in this case we will omit the one-tenth and take 319 as nearer the root sought, and on trial find it to be the root exactly. We will now give one of the most difficult examples. 3. Find the approximate cube root of 16. Ans. 2.519842. The factors of 16 are 2- 2- 4; the sum of these is 8, which divided by 3 gives 2.66 for the first approximation to equal factors, but as these factors are so unequal, 2.66 must be in excess. Therefore we assume 2.5 to be near one of the equal factors re- quired. To find the other two factors, divide twice by 2.5, or once by 6.25. Thus, 6.25)16.00(2.56 12 50 3500 3125_^ ' / ~375a 3750 Here we have three factors, 2.5, 2.5, 2.56, whose product will give 16 exactly ; they are not all equal however, but being nearly so \ of their sura, 2.52, is very nearly equal to the root sought : {it must he a very little in excess). Now if we repeat the operation with 2.52 as an assumed factor and find two other corresponding factors, ^ of the sum of the three will be the root to a high degree of approximation. 4. Find an approximate cube root of QQ. Ans. 4.041240. By the cube numbers we find that 4 must be near one of the equal factors, therefore divide by the square of 4. 16)66(4.125 64 20 16 40 32 CUBE ROOT. 46 Hence the root sought must be a very little less than ^ of the sum of 4, 4, 4.125 ; that is, a very little less than 4.0416. For a nearer approximation take 4.041 as one of the factors of 66, and find the other two, (fee. 5. Jf^ind an approximate cube root of 21. Ans. 2.758923. That is, find three factors, as near equal as possible, whose pro- duct will be 21, or very nearly 21. We know that 27 has three factors, each equal to 3 ; therefore the equal factors of 21 must be each less than 3, and as we can- not expect to find the equal factors at the first trial, we will assume 2.7 and 2.8 to be two of the factors, their product is 7.56 ; hence, the third corresponding factor is found by the following division : 7.56)21.00(2.777 15 12 5880 ' ' 5292 5880 5292 5880 5292 588 Here we have three factors nearly equal, whose product is very near 21 ; one-third of their sum is 2.759, which must be a little greater than the root required. We will therefore assume 2.75 as one of the equal factors sought, and find the other two corres- ponding factors, and one-third of their sum will be an approxi- mate cube root of 21. It is not necessary to give more examples. When it is necessary to multiply several numbers together and extract the cube root of their product, we may often evade or abridge the operation by resolving the numbers into cube factors. EXAMPLES. 1 . What is the side of a cubical mound, equal to one 288 feei long, 216 feei broad, and 48 feet high? Ans. 144. 46 ROBINSON'S SEQUEL. 288=2»12»12 216=6- 6- 6 48=4-12 Product, 288»216«48=123>63«8 Whence, y288- 216 -48=12*6*2=144. Am, S. Required t^ie cube root of the product of 448 by 392 in a brief manner. N. B. Divide by the cube number 8 ; then it will appear that 448=:8'8-7 and 392=s8-7»7 Product, 448-392=^83-73 Whence, 3^448*392^8- 7=66 Ans. 3. Find the cube root of the pn)duct of the two fuctors 192 and 1025 in as brief a manner as possible, Ans. 60. The three last examples are rare cases ; nevertheless they serve to awaken thought, afid for this purpose they were introduced. JlI.I.I«ATIOM AI>TE»]VATE. No arithmetical rule is more difficult to be comprehended by young pupils than this. The operations are generally very trifling, but the rationale is rarely discovered. For this reason we shall be a little unique in our exposition of the principle — we shall resort to an experiment in philosophy. It is clear to the comprehension of almost every one, that two bodies balanced on a fulcrum, the heavier body must be nearer the fulcrum than the lighter body. Thus two bodies bal- anced on the fulcrum F, 2 pounds at the distance of 6, will balance 6 pounds at the distance of 2. ALLIGATION ALTERNATE. 47 Or when we have the distances, we can take those distances, or their proportion for corresponding weights if we alternate them. That is, the long distance must go on the opposite side of the fulcrum, and there become weight, and so of the other distance, and there will be a balance. We shall apply this principle in the following example. 1. A grocer has two kinds of sugar , one at 9 cts., the other at 16 cts. per pound ; he wishes to make a mixture worth 1 1 cts. per pound : what proportion of the two kinds shall he take ? Here two quantities are to be balanced on tho, fulcrum 11. The difference between 9 and 11 is 2 ; place C 95 the 2 opposite 16. The difference between 11 and 11 ^ 16 is 5 ; place this opposite 9. (^16 2 The result is, that 5 pounds at 9 cts. = 45 cts. and 2 pounds at 16 cts. = 32 cts. Makes 7 pounds worth 77 cts., which is 11 cte. per pound as required. We may now expand the problem and add another kind of sugar, worth 10 cts. per pound. Then make a mixture, worth 11 cts., with sugars worth 9, 10, and 16 cts. per pound. 9\ 5 16^ 2+1 Link each price below 1 1 to the one above. Make a balance between 9 and 16 as before, then between 10 and 16, and all will be bal- anced as required. The result is. 6 pounds at 9, 5 at 10, and 3 pounds at 16 cts. That is, 13 pounds of this mixture is worth 143 cts., which is 11 cts. per pound as required. On the same principle, any number of ingredients may be re- duced to any given mean price or quality. We give but one example. Mix 6 bushels of oats, worth 20 cts. per bushel, with 8 bushels of oats worth 25 cts. per bushel, with rye at 70 cts. per bushel, and wheat at 80 cts., and sell the mixture at 75 cts. per bushel ; what proportion of rye and wheat will there be in the mixture ? Ans. Rye 14 bushels, wheat 160' bushels. 6 bushels at 20 cts. will cost 120 cts. and 8 at 25 cts. will cost 48 ROBINSON'S SEQUEL. 200 cts.; whence the 14 bushels of oats will cost 320 cts., or 22IJ cents per bushel. 76 22f 5 80^ 6+621 6 bushels of oats. 6 bushels of rye. 51} bushels of wheat. Here we have a true mixture worth 76 cts. per bushel, but the mixture contains only 6 bushels of oats : it must contain 14, there- fore multiply each of these quantities by y. Then 6* Y = 14. 57}- V = 160. Alligation is of little or no practical utility, yet it serves as well as any other arithmetical operation to discipline the mind. POSITION. SINGLE POSITION DOUBLE POSITION. Before Algebra became a popular study many algebraic prob- lems appeared in common Arithmetics, and were solved by special rules, which were drawn from the results of algebraic investiga- tions. But at the present day all such problems in Arithmetic are improper ; as much so as to travel 500 rdiles in a pri- vate carriage by the side of a railroad track. Problems in Single Position produce equations reduceable to this form: a:f—m. (1) Problems in Double Position produce equations in this form : ax-\-bz=m (2) Not knowing the value of x in equation (1) we assume some known number, x, which may not be the true one, and if it is not, the result will not be the given number m ; let it be m\ Then we shall have : ax'=m' (3) Divide equation (1) by (3), then X —-.m ^ X m' Converting this into a proportion, we have m : m '. '. x' '. V, POSITION. 40 The result of this proportion, put into words, is the rule of Single Position given in all the old Arithmetics. Rule. Assume a number and find the result of the supposition ; then say : As the result of the supposition is to the given result, so is the supposed number to the true number. We give but a single example. A and B have the same income, ^contracts an annual debt amounting to | of it : B lives on | of his income, and at the end of 10 years lends to A money enough to pay off his debts and has $160 to spare : what is the income of each ? Ans. $280. For the sake of convenience we will take some number divisible by 6 and 7 ; therefore take 35 for the supposed income of each. Then ^'s debt in one year is $5, in 10 years $60. B saves ^, or $7, in one year, in 10 years $70. B lends A 60 and has $20 left as the result of the supposition. Then, 20 : 160 : : 35 : 280. Ans. Now let us suppose the income to be 1, or unity. Then ^*s debts in 10 years amount to y . B saves in 10 years -^j", or 2. B pays A's debts ; he then has (2 — y>)=^. Whence, 4=160, or4=40, or 1=280. Ans. This manner of working by fractions some teachers call Arith- metic, but it is Algebra in disguise. Let X be the income, in place of 1, and the identity will be obvious. To show the arithmetical rule for Double Position we take the equation ax-\-b=m. (1) 1st. Suppose a; to be represented by the assumed number x', and m-\-e' the result of this supposition, e' being the excess, or error. Then, ax'+b^^m+e'. (2) Again, assume another number, say x" and e" the second error. ax"+i=m-{-e". (3) Subtract (1) from (2) and a (x' — x)=e'. (4) (1) from (3) alx'''-22= 16 (2) 70— ar— y== 16 (3) Sum, a:+y-{- 2=3*16 (4) Add (3) and (4) and we have 82=4-16 2=8. (6.) This problem is resolved in the book, by equation 7, Art. 53. (7.) Let X represent the better horse, and y the poorer a:+15= Uy+10) x+10=.my+15) Therefore, |(y+10)=||(y+15)+5 Reduced gives .,, y=s50. (8.) Let x= the price of the sherry. y= brandy. Put a=78. 2x-{- y=3a 7x-\-2y=9'a-\-9 '3x =3a-f9 X = a-\S— 81s. Am, (9^,) Let x=:A's time. y=^B's time. Then, _=th-e part that B can do in one day, 4 , 4_^^1 xy 16 4- Hence , ys=48. 36^3 J 4 (10.) 2ar _2 f+?=^ ^+7 3 '^f 5 Sx=y-\-7 5a:+10=6y. (11.) , 2y , 3ar ^3 ^4 (12.) Let .r= the greater, and (24 — x)= the less, X , 24 — X ..4.1 24 — ^ X ALGEBRA. 63 x^ : ^24— a:)2 : : 4 : 1 By evolution, x : 24 — x : : 2 : 1 (13.) Let a:= the number of persons. y= what each had to pay. Then, xy=: the amount of the bill. Put (a;+4) (y— 1)= the bill. Also-, (x—3) (y+l)= the bill. xy-\-47/ — X — 4=xy xy—Sy-{-x—S=:xy 4y — X — 4=0 — 3y-^x — 3=0 By addition, y — 7=0 (14.) 10x-\-y=4x-\-4y or, y=:2x» 10x-{'y^27=10y+x ' or, 9x +27= 9y X + 3= y=^2x, hence, «=3. (16.) Let x= the digits in the place of lOO's. y= *' in the place of lO's. z= " the units. W-^y-\'Z=\l Z=:z2x 100ic-f-10y+2+297= lOO^+lOy-f-a? 99a;+297= 99^ x-\-3=z=:2x Hence x=3. /ic \ T + a;— 40 X — 20 , a:— 10 . ^, (16.) Let , ., and _ — represent the parts. Then, f=12+f:r?2+5ld?=90 .=100. 2 ^ 3 ^ 4 (17.) Let X represent the part at 5 per cent, and (a — x) the part at 4 per cent. Then 5x 1 4a — 4x , ioo"*" 100 Hence x=:100b — 4a. (18.) To avoid high numerals, and of course a tedious opera* tion, Put a= 5000; then 2a= 10000, 3a=s 16000, —=1500, and A^^=800. ■ 10 100 64 ROBINSON'S SEQUEL. Put x=. A'& capital, and r — l=^'s rate. ar+2a= ^'s ** r=B'& rate. a;-j-3a= (7's " r+l==C"s rate. '' r X — X I 16a r x-\-2a r By conditions 100 ' 100 100 rx — X .3a_rx-{-3ar-\-x-\-Sa ~ioo ~^io Too Reducing (1), gives a:=(16 — 2r)a (1) (2) Hence, 32— 4r=27-~3r, or. a . . . .r=5. (19.) Put a=1000, X and y to represent the two parts, and r and t the rates expressed in decimals. ' x-\-y=z\3a rar= ty te=360 ry=490 Divide (3) by (4), and we have /J\/a;\ 36 W\y/ 49 Then by conditions, (1) (2) (3) (4) (6) From (2) we have x_t y r t . Substitute the value of — in equation (5), and .36 49 or By returning to equation (1) we have _iL-|-y=13a. 13y=13a*7 or y=7a. (20.) Let Xy y, and z, represent their respective ages. Then by conditions given, x — y=. z 5y-{-2z — ^a;=147 (21 .) Let X, y, and z, represent the respective property of each, and put 5= their sum. ALGEBRA. 66 fX'\-3y-\-3z=i7a a=100. Conditions, }y-\-4x-\-4z=58a (z-^5x-\-5y=63a Add 2x to the 1st equation, 3y to the 2d, and 4z to the 3d, observing that x-\'y-\-z=s ; then we shall have 3s=47a+2a; (1) 45=58a+3y . (2) 65=63a+42 (3) 3s— 47a . or, x= 2 45 — 58a 5s— 63a z=. 4 3*— 47a , 4s — 58a , bs — 63a By addition, s 2 ' 3 ' 4 Hence, 5=19a. This value of s, put in equation (1), gives a?=5a=500. (23.) Let X, y, and z represent the respective sums. .+^=<. (1) y+|=a (2) 1 ^ . (3) ac+y=2a 3y+2=3a 42+ar=:4a From the 1st or, 4z-\-12y= or, 4z-f- x= —x-\-12y= 24x-\-12y= = 12a : 4a : 8a :24a 25x =16a (24.) This problem is resolved in the work, by the 13th exam- ple, page 80, (Art. 61.) 5 6 ROBINSON'S SEQUEL. f (25.) Let aj-sthe greater, and y the less. ^x — y=0 or, ...2ir=3y. (26.) a:+Ky+^)=«=61 2y+(^+2/+2)=3a 324-(^+y+2) = 4a a;=2a— 5 (1) y=i(3a— 5) (2) 2=H^a- s) (3) s= 2a— 5-|-i(3a— s)+^(a— s) 6s=12a— 6«-|-9a — 35-f-2a— 25 175=29a or .5=29-3=87. Ifow equations (1), (2), and (3), will readily give x, y and z. (27.) Let x=A'&, y JB's, and z=C's sheep. Then by the conditions, a:_|_8— 4=y-[-g— 8 i(y+S)=x+z-S i(z+S)^x+y-S x+n= y+ z (1) y^24=2x-\-^z (2) 2-|-32=3a;+3y (3) Add (1) and (3), and we have x-\-44=3x-\-^. Double (1), and subtract (2), and we have 2x — y=2y — 2x or, 4x=3i/ y=8. (30.) This is a repetition of the 1 0th example, page 89, in- serted here by oversight. But 2a;+4y=44 4a;-f-8y=44-2 ny=44-2 or (28.) ar+l_l X __1 y 3 y+1 4 (29.) «+2_6 X _1 y 7 y+2 3 ALGEBRA. ♦ ^ (31) Let a?i=:-4's money, and y=zB's. x-5=Uy+5} (1) a?4-5s=r:3y— 15 (2) Subtract (1) from (2), and we have 1 =%— 1 5—1—- or ; . . V = 1 K 2 2 (32.) Let a:=s= the number of bushels of wheat flour. And y=as •<* '^ barley ^^ Then the cost of th« whol« will be expressed by 10a'-)-4y The sale at 11 shillings v/ill be H^c-j-lly Now by the coiiditi/HV^z::^=— 14 ROBINSON'S SEQUEL. Multiply by a and take the square root, and — an Jx+Jx—a= Jx — a tjx' — ax-^-x — a=an Jx^ — ax=^[n-\-\)a — x Drop x^ , and divide by a, and — x=^(n-\-\Ya — ^nx — 2a? l+2» (32.) Resolved in the work. (Art. 90.) (4.) Observe that 180=9-20 189=9-21. Put a=9. a:2y-|-a:y2_20a •^ja.j.ya—.gla Multiply the first equation by 3, and add it to the second, and a;3+3a;2y4-32:2/2_|_y3_8i«=a3 cube root, a:-|-y=a=:9 The rest of the operation is obvious. (6.) Divide the first equation by (x-\-y) and x^—xy-\-y^=xy x^ — 2a;3/-|-y^ =:0 or x — y=^^. Hence a;=2 y=2. (6.) x-\-y : a; : : 7 : 5 xy-\-y^^\tQ, 5x-{-5y=7x 5y=2x or ir=fy. Put this value of x in the second equation, and ^y^+y^ = 126 7y2 = 126-2 ya = 18.2=36 y=d=6. (7.) From the first equation we have 5x — 5y=s4y 5x=9y ALGEBRA. 7i 181y2 = l81 -25 or y»=25. (8.) From the proportion we have 5jj/=3jx or ^5y=9x. The rest of the operation is obvious. (9.) Extract square root and ia:+i=3 or a;=7|. (10.) From the first proportion x-\-y=3x — 3y or 4y=2x Hence 8y^=a;=*. 8y3_y3_56 y=*=8 y=2. (11), (12) and (13) resolved in the work. (14.) f^::J^_6 or «-H=6 From the second, .xy=5. (15.) Divide the first equation by (^-\-y), and x^ — xy-\-y^=2x7/ x^—<^yJ^y^=xyz=\Q (1) Add 4x y =64 a;2+2a^+y2= 80=16-5 Square root, x-\'y=4j5 Square root of (1) a; — y=4 2x =4^5+4. (19.) Double the 2d equation, and add and subtract it from the 1st, then a;^ -\^2xy-\-y' =a+2 J x' — ^y-\-y^ =a — 25 x+y=:Ja-\-2b 76 ROBINSON'S SEQUEL. (Art. 92.) (6.) Add the two equations and extract square root, and we have x^-\-y ^ = ±4 ( 1 ) Separate the first member of the first equation into factors, and we have x^(x^+]/^) = \2 (2) Divide (2) by (1) and x^=z±zS x=9, (6.) Is of the same form and resolved the same as (5.) (7.) Add the two equations, and extract square root, and we 3 3 have X* -{-y * = Ja-\-b 3 / JJ 3. \ But a;*,U*+yV=a rr=- «' or r ( "' (u+6y \(a+b^. (8.) Resolved in (Art. 90,) of this Key. (9.) Square the first equation, and x+2x^y^+y=^5 (1) Difference, ^.x^y"" =12 (2) Subtract (2) from (1) and By evolution, x^ — y^ =±1 But, x^ +y^ = 5 2ic2 =6 or 4. The following are not in Rol)inson's Algebra. They are mostly from Bland's Problems. We shall number them in order. 18 (1.) Given x^-\-'3x — 7=a;+2+— to find the values of x. Ans. +3, —3, —2. ALGEBRA. 77 Reducing x'-\-2x=9+— X Factoring x'(x-{-2)=9(x-\-2) This equation will be verified by putting x-\-2=0 Then will a;=— 2 Again dividing both members by (ir-j-2) and x^=9 Whence x=zt:3. As a general thing we shall not give all the roots to equations. Imaginary roots we shall not pretend to give, except in rare cases, or unless we have an ulterior object in view. (2.) Given ^^^-7^+^^-?-==^^ to find a;. ^ ' axe Let P=z^a-\-x thenP^=a-f-^ ^^^ the equation become* — ~h — =-^^ ax c Or Px-\-Pa __Jx ax c That ia P(a:+a)=^-ar^ c Whence P^:=z-'x^ c By taking the cube root we shall have By squaring and replacing the value of P^ we shall have Let the known co-efficient ( ^ V be represented by m. Then a-\-x^=mx or, a;= ^ m — 1. (3.) Giren (gjf I^+ (''+») •'=? to find x. n ROBINSOlS^S SEQUEL. This 5s the same as example 2, in case «=2, therefore we may jump to the conclusion at once. Thus, a+x=^(^y^x. (4. ) Given i«^' +y ' = i^+y)^!/) ^ find the values of x and y. Dividing the first equation by («+y) and the second by ay we obtain «^— ^+2/^— a;y (1) ^+y=4 (2) Transposing ccy in (1) from the second member to the first gives x^ — ^y-^y^ =0 Whence x — y^^O or, x^ssy These values substituted in (2) give ics=2 ys=2 In this example we divided one of the equations by {x^y)^ therefore (x-^-y) must contain a root of that equation. (See theory of equations.) That is, ar-j-yz^O, or, xss: — y, and — y substitu- ted for X in either equation will verify it (5 ) Given i^^'+^/M'^yC^+S^) =^8) to find the values of ^ '^ ix^J^y^— Sx^—Sy^ =^12} X and y. Ans. xssi or 2. y=2 or 4. Multiply the first equation by 3 and to the product add thti second ; then x^'^Sxy{x+y)'^y^tsti2l6 (1) Cube root a:+y=6 (2) By squaring and transposing (2) becomes x^-^y^z=^36—2xy (3) By the aid of (2) and (3) we perceive that the first equation is equivalent to 36— 2a:y+62y=68 or xy=Q (4) From (2) and (4) we find x aiid^. (6.) Given ALGEBRA. 1 S y ~Q X 7 79 To find the values ' of X and y. Put And shall have And 7(x—yy_7(x—yy^l 4 y 4 X 9 Ans. a:s=| y=i-. P^(x+y)^ F^^(x-^y) Q=(x—y)^ Q'''==(x—y) Divide the first equation by f , and the second by I, then we P , P^64 ~y x^ 63 y X 63 Equation (1) reduced, becomes (^+y)P^64 xy 63 F*_64 xy 63 xy 63 Bivide (3) by (4) and we shall have JL_=16 Whence F^2Q And P3^8$3 That is a;+y=8a: — 8y or 9y='7x Ix^ From this last, xy=s * which being substituted in both (3) y That is Also (2) becomes (1) (8) (3) (*) ftnd (4) gives Whence 9P' Ix^ .64 '63 or, 9P'»^64 2 i>»==|, Thatis(*+y)^=^ (6)- Substituting ~ for y in equation (6) and we have y 3 O) ROBINSON'S SEQUEL. Cubing (•+i)"=i;- Tkatis ••('+D'4:- Or, 2232 83 = — X 92 93 4=:-x or a;= Ans The following solutions refer to problems in Robinson's Alge- bra, Chapter V., Art. 93. They number from (5) to (13). QUESTIONS PRODUCING PURE EQUATIONS. (5.) Let ic-|-2/= the greater number. And X — y= the less. Difference 2y=4 Sum =2x 2x{4xy)=1600 Hence ic=10. (6.) Let x-\-y= the greater number. X — 1/= the less. 2y : x—ij : : 4 : 3 or ic=|y- ( a;2_y2)( a;_y)=504 (Vy^— y")(|y— y)=504 (Vy')(iy) =604 Hence y=4. (7.) Let 8a:= the length of the field, and 5x= its breadth. Then = — = the acres. 1^0 4 Ix^ X 8a:= the whole cost. ^6x= the rods around the field, 13X26a:= the whole cost. Hence 2a;3 = 13 -262; or a;=13. 8ar=104. An9, (8.) Let 5x= the length of the stack. 4x= the breadth. Then, —= the height. ALGEBRA. 5x*ix'-^'4x= the cost in cents. 2 Also, 5x'Ax'9,^A= the cost in cents. Hence, 5x'4x*ll'4x=5x'4.X'2M 7x'2x=224 or 81 a;=4. (9.) Put a;*— 7= the number. Then, x-^Jx^-\-9=9 Jx''-{-9=9—x or a?=4. (10.) Let X represent ^'s eggs ; then 100 — ar= ^'s eggs. 18 At ' o Tt, . ■■As price. _=^'s price. 100— re X Hence, _i^=?(100— ;r) 100— a; x^ ^ 9a;2=4(100— a:)2 3a;=2(100— ^^ = wine in 2d drawing. a Then, (a — x) — ^ ^ =^ ^ = wine left after the second a drawing. a Agam, a : ^^ L : : x : ^ — —^ — = the third a a* drawing. Whence, (^rf)'_(^=?I"*=(f=f)!= the wine left after a a^ . a^ the third drawing. Whence we conclude that ' ^^ ^ would be the wine left after n drawings. After four drawings, ^ L =81 by conditions. ALGEBRA. 83 (a— fl;)*=81-256-a2 Square root, (a— )2=9'16o=9- 16-266 Square root again, a — x=3 • 4 • 1 6 That is, 16-16— a;=12- 16 Or, 16-16— 12- 16=4- 16=64=^. (c) A and B have two rectangular tracts of land, their tenths heing as 7. to 6, and the difference between the areas is 150 acres ; B's being the greater. Jifbw had A's been as broad as B's, it would have been 672 rods long ; bid had B's been as broad as A's, 'it would have been 900 rods long. How many acres ivere there in each ? Ans. A's 2100 acres, Ws 2250 acres. Let 7a:= the length of ^'s lot in rods, And ?/= the breadth of the same, Then, lxy=^ tlie square rods in u4's tract. Again, let Qx=^ the length of i>'s tract in rods, And, v= the breadth of the same. Then, 6ya^= the square rods in ^*s tract. By the given conditions, 6?;a;— 7a:y=150-160 (1) Now had ^'s been v in breadth, it Avould have been 672 rods long, therefore Ql^v=lxy (2) Also, dOQy=Qvx (3) by the last given condition. By multiplying (2) and (3), omitting common factors, • 112-900=7^-2 Or, 16-900=a;2 Whence, a-=4-30=120 Substituting 900y for 6y.c in (1) and 120 for x, we shall hav€ 900y— 840y=150-160 Or, 60y=150-160 y=400 Lastly, IlL2^:12?=2100 .I's acres. 160 84 ROBINSOIf^S SEQUEL. (d) A and B engaged to work for a certain number of days. At the end of the time, A, who had been absent 4 days, received $18.75, while B, who had been absent 7 days, received only $12. Now, had B been absent 4 and A 7 days, each would have been entitled to the sam£ sum. ^ How many days were they engaged, and at what rate ? Ans. They were engaged for 19 days, A at $1.25, B a^ $1 per day. Let ^= the time or number of days. a:= the daily compensation of A. y= the " ** B. Then by the given conditions (^— 4)a:=18a (1) (/— 7)y=12 (2) {t--l)x={t-4)y (3) From (3) ar=' ly. This value put in (1) gives {t:^y=\^ (4) t—1 ^ ^ Dividing (4) by (2) gives (^— 4)2^18|_ 75 ^25 {t—lf !¥ 12^ 4^ Square root =_ or t=\9 t—1 4 The value of t put in (1) and (2) gives. .ar=1.25, y=I, SECTION II. QUADRATIC EQUATIONS. The following are but hints to the solutions of Equations in Robinson's Algebra, University Edition., commencing at example 10, page 167. (10.) Put (ar— 4)2=y. Then, ?=1+1? y y^ y'— 8y-|-16=0 or. ..y— 4=0, ALGEBRA, 85 (11.) Multiply by 16. Rule 2. Then, 64x^-^16x^-{-l=39- 16+1=625 8^e-|-l=25 a;«=3 a:=729. (12.) Add 5 to each member. Then (x^—Zx-\-5)-{-(x''—^x-\-5)^==16 By substitution, y2_|_gy^9_25 .y=2 or — 8. Hence x^ — 2x-\-5=4r x=i. (13.) By (Art. 99) we have . 361 19 ' -^=_-^ t=—6 t^=3Q 19 !9 361 19 ' -^— ^=db2.,. ar=152 or 76. 19 (14) Observe that 81^^ and — s^e both squares, and if these x^ are taken for the first and last terms of a binomial square, the middle term must be 9a;_.2=18. X This indicates to add one to each member. Then extract the square root 9x-|--=±10. Hence, x=l or — 1 X (15.) The first member of (15) is the same as (14.) Hence, add unity to each member and extract square root ; I 29 we then have 9ar-4--=— +4 X X 9x^-^ix=2Q Put x=-. 9 «2_4^^28- 9=252 «— 2= ±16 x=1. y^—y=\o'2. y=12or— 11. x^—x=\^ or— 11. If we take — 11, the value of x will become imaginary. 12 gives a:=4 or — 3. (7.) This equation may be put into this form : (if—cy^ )-^(^f —cy)=c'^ from which the reduction is easy. ALGEBRA. 07- (Art. 107.) (3.) Taken from the work we have {a-|-l )x^ — a'^x=a^ Or, (a-\-l)x^={x+l)aK Both members are of exactly the same form, and of course the equation could not be verified unless xz=:a. EXAMPLES. (1.) x^'+Ux^SO. Multiply by 4, &c. 4a;2-(-^+l 12 =329+121=441 2x+U= ±21 x=5 or —16. (2.) Drop 2a; from each member, and divide by 3 ; then x—l x—2 X — = x—3 ^ 2 Clearing of fractions and 2x^ —Sx—2x-{-2=x^ —3x—2x-{-6 ir2_3a:=4. Put 2a=3. Hence, (Art. 106) x=4 or —1. (3.) Multiply the equation by 6x ; then fi'y2 J^^-L.Gx-\-6=13x x+\^ ^ x+1^ 6x''+6x-{-6=7x''+7x Hence x^-\-x=e ' x=2 or —3. (4.) Clearing of fractions 70a;— 21a;2+72a;=500— 150a; 21a;2— 292a;=— 500. Or, 21a;2— 42a;=250a;— 600. or 21a;(a;— 2)=250(a;— 2). (6.) Put ('?+y)=a;. Then \y / a;2-|-a;=30. Or, a;=5 or — 6. Now, (5+y^=5or— 6 fr. ROBINSON'S SEQUEL. y»— 6y=— 6, or y^-\-ey= — 6 2y— 6= del y=3 or «. (6.) Put a:^=y ; Then y^-\-'7y=44 4y3_|_^^49=226 2y+7=±15 y-=4or-.ll. a;=(4)^ or (—11)* (7) a;2+a;=42. Hence a;= 6 or —7. That is y* +11=36 or 49 y=5 or ^38. (8.) 11— f+!^=^ ^ ^ ic— 7 3 33a;— 23 1— 3a:— 21 =a;*— 7a; a;2— 37a;=— 252 4x'^A+3V =1369—1008=361 2ar— 37= ±19 a?=28 or 9. (9.) 3a;2— 9a:=84 12 36a;2— .^+81 = 12-84+81 = 1089 6a;— 9= ±33. (10.) Clearing of fractions we have 2a:+27a;=16— to find the values of x. ^ X X X Ans. x= 1 or — V Multiply by x, and then we shall have '(•Vi)-H(■^/i)— •• Placing the value of x under the radical signs, then . ya;2+3/a;2=3— A That is a;^+a:^=3— ir^ 2a;3+a;3=3. Whence, a;3=il or — f. Whence, a;3_ior— |. Or, ar=:l or— ¥• Ans. (4.) Given ^ar— 1)'+ A— Ay=^ to find a;. Put F=(x—-y and ^ = A— i) ' ; then P+Q=x (1) Multiply this last equation by (P — Q), then ButP2_g2— (^_1). therefore, a; (P— §)=(«— 1) Or, P-^=l-i (2) X Add ( 1 ) and (2), then 2P= (^— ^) +^ That is, 2P=P2+1 Or, P2_2P+1=0, or P— 1=0 Whence, a?— 1=1, or x=^{l±:j5). ALGEBRA. (5.) Given (x^- ->)*+ (••-: values of x. 91 We observe that this equation is in the same form as the pre- ceding, and would be identical if we changed x^ to x, aMo 1. Therefore the value of ar^ in this equation will be of the same form as the value of x in the last example, except it will contain the factor a^, because the square root has been once extracted : That is x^=—{lzh^5), =-(^)' But this conclusion is too summary to satisfy the young algebraist ; therefore it is proper to take some of the intermediate steps. then the equation becomes P-\-Q=~^ (2) xMultiply (2) by \P — Q), then we have a But the value of {P^—Q^) drawn from (1), is (a:^— a^); therefore ^ (P— ^) = x^—a^ a or i>_§=a-?l (3) X'' By adding equations (2) and (3), we find 2i>=«-^+?! . (4) ■ x^ a Multiply this equation by a, then 2aP= a''——+x^ x^ that is, 2aP=a^+P^ or 0=a2— 2aP+P2. 92 ROBINSON'S SEQUEL. Square root, 0= a — P, or P = a From the first of equations (1), we find a « .x^ — — ,2 From this equation, we find x=^-=^a(. ^ -\ . (6.) Given x^{\+l-y^{^x^-\-x)=10 , to find the val- ues of X. Observe that (^x^+x) = ^x^{\-\-—). Put (14-J_)=y ; ^x 3a; then the given equation becomes x^y^ — 3a;2y=70. — 9 289 Completing the square, x^y^ — 3a;^y-|--= x^y ^=db — "^ 2 2 ic2y=10, or — 7 That is a;2 4-^=10, or —7, ^3 Whence x=d, or — V» or 1(1=^^—251). /7^ Given 5^-^J x_ ^^{x-2jx) W-2,x+4. ^ *^ ^+27^ 6+V^ (^+2V^) (6+V:r) to find the value* of x. Multiply by (6+ ^a:), then x-\-2jx ^ ^ ^^ x+SLjx Multiply by (x-{-2jx), and we shall have 9(S6—x)=23(x^—4x)-\-7x^—3x+4 Reducing, 15x^—4Sx=160. Whence a;= 5, or — ff. (8.) Given x^ -^^^4-15=^^^1^^, to find the values ^ ^ 2^ 16 x^ of ALGEBRA. 93 By transposition, a;^-|-15-f- — = + — ■ Add 1 to each member, (see Robinson's Algebra, Art. 99,) then .>+16+^=?^^-+^+l ^ ^a;2 16 '2^ By evolution, x+- = ± (~+l\ X \ 4 / O n» Taking the plus sign, _■' = — 1-1 . ( 1 ) Taking the minus sign, - = — — — 1. (2) X 4 From (1) a;*+4ar=32. Whence a;=4 or —8. From (2) 92;2-)-4ic = — 32, and x^^'Z^^y.IzI} y ( 9 . ) Given {x^^sy — 4a;2 = 1 60, to find the values of x. Subtract 20 from both sides, then ' (a;24-5)2_4(a;24-5)=140. Whence, x^ +5—2=: ±12. Therefore, x= ±3 or ±=V*): ab ab Whence, JM^|s„ j-ws^js ALGEBRA. 97 SECTION III. QUADRATIC EQUATIONS CONTAINING MORE THAN ONE UNKNOWN QUANTITY. We commence by showing the outlines of the solution of the (3), (4), (6), (6), (7), and (8) equations in Robinson's Alge- bra, Art. (Ill), page 182. (3.) Futx^=F, and y^=Q. Then the equations become F+Q=Q (1) Square (1) and we have P^-\-2FQ-\-Q''=64^ (3) Subtract (3) from (2), and we have P^ Q''—2FQ=195, Hence, P$=15or— 13. Now we have P-\-Q=8, and PQ=15, whence P=5 or 3, and ^=3 or 5. That is, x^=5 or 3, &c. (4.) Puta;^=P, and y^ = Q; then the equations become P^ + Q''+P+Q=26, andP$=8 2PQ =16 (>+^)2+(P+$)=42. Hence, P+Q=6. (5.) Put — =:u: then u^-\-4:U= — w==_ or — — . The remaining operation is obvious. (6.) Given y^ — 8a;^y=64, and y — 2a; ^y^ =4, to find a: and y. To both members of the first equation add 16x, and to the second add x, to complete the squares ; then extract square root, and we have y— 4a;2=4(a?+4)^ and y^—x^= (a;+4)^ Four times the last equation subtracted from the preceding, gives y — 4y2=0. Or, y=16. •8 ROBINSON'S SEQUEL. (7.) Multiply in the first equation as indicated, and subtract the second equation ; we then have «+y+2^'y'=25 or x^+y^=:6 But from the second equation we have (a;2_(_y2)a;2y2=3o. Hence, x^y^ = 6 3. 2^ X JL 9 (8.) Divide the first equation by y^, and x^=2y^, or y^=^x'^ This put in the second equation gives X z f a;T_16a;3+64=64— 28=36. We continue this section by adding other and more severe equations, commencing with number one. (1.) Given -I oI'^'^^^^^Ta I \ to find the values of ^ ^ ( 28 — y = x-\'^»Jx J X and y. By adding the two equations, omitting 16 on both sides, gives Squaring, 144 — Mjy-{-y—\Qx (1) Multiply the first equation by 16, and substitute the value of — 16a; from (1), then we shall have 16y— 16^^=266— 144+247y—y Whence, 17y— 40^^=112 >/y=f^±V(H|-^+lff) = n±H = 4or-^. Therefore, y = 1 6 or |f ^ . These values of y put in the first equation, give x=Jy=A, or my. (2.) Given y J(^x-\-y) 17 1 to find the (x+yy'^ y ^^^Jix+y) [values of x=y^^2 J X and y. ALGEBRA. 99 Mnltiply the first equation by >/(^+y)» *^^^ y _|_^+y^^7 ^+y y 4 Clearing of fractions, and Reducing, 4a;2 ^^^xy-^-^y^ Adding, to both sides, (Robinson's Algebra, Art. 99) and 4 4 By evolution, :?^=±(— +3^/) Whence a;=3y or — fy. These values of a; put in the second equation, readily give «=6, or 3, or 9T3V(-n9) 32 , y=8,orl,or-^-^V(-"9_) 8 (3.) Given j a:+4^^+4y=21+8Vy+4V(a:y) \ ^ ^^ and t Jx-^Jy=Q ) the values of x and y. From the first, x--'^J(xy)-\-^y—2\-\^^Jy—Ajx That is {^Jy—JxY =21+4(2^y— 7«) Let P=^Jy — ^a: ; then .^ P^— 4P=21 ^^^■L -P=2 ± 726=7 or —3. tR^ Zjy—Jx=l or —3. But Jx^Q—Jy, Therefore Sjy-^=7 or —3. 7y= V or 1. y = -If ^ or 1. (4.) Given i Sx+ljxy^+9x^y==.(x^x)y 1 and ( 6x+y : y : : x+5 : 3 J- to fand the values of x and y. From the first 9a;rf-2 Jny^-^9x^y=:3xy—y 100 ROBINSON'S SEQUEL. That is (y^9x)+2jxy {7/+9x)^' =^3xy Add xy to both members and extract square root, then Ji/+9x+J^=2jxy (1) Whence y-\-9x=xy (2) From the second — 2y-j-18a;=ary (3) By subtraction, 3y — 9a; =0 Or, y=3x This value of y put in (2) gives 12^=3a;^. Or, x=4. Whence y=12. By taking the minus sign to the second member of (1), other values of x and y can be found. (6.) Given \ x-^-y^-J "7"^ — i to find the values and Ix^+rJiT'-' ^~^) °f^aidy- The first cleared of fractions is x^ — y^ — Jx^ — y^=iQ Whence, - Jx^—y^=3, or —2 a?=±5, or =1=371 (6.) Given ( J^x+xYJ^y-+J{\-xY+y-=4 ) ^ and ( (4_;p2-)2^j8_4y3 f the values of x and y. From the first Squaring, \^2x-{-x^-\-y''=\Q—^J(\—xY-\^^+\—^-\-x^-\-y'' Reducing, a;= 4— 2 /(I— arj^+y^ Transposing 4 and squaring, gives a;2_8:y_|-lC=4( 1— 2a:+a;2 +y2 ) Reducing, 12=3a;2+4y2 (l) That is 4— a;2=^y!., U—x^Y=:]^ Comparing this last result with the second equation, we per- ceive that ALGEBRA. 101 l^'.+4y'=16 (2) Add I to both members, (Art. 99, Algebra,) then 9 ^ ^ ^4 4 T, 1 .• 4y2 3 9 By evolution, _^_-|-_==±- 3*22 Whence 4y2=9, or — 18 y=dbf, or ±fV— 2 The value of 4y^, that is 9, put in (1), gives x=\. .2 .,2 (7.) Given {^+^'^I^=^l-t-r^- ^ x—Jx^'—y^ 4 x^Jx^'—y^ > to find and ( ^2 ^xy=52—Jx^+xy-\-4 ) the values of x and y. Add 4 to both members of the last equation, and transpose the radical, then (x^-^xy+4)+(x^+xy+4)^=56 This is a quadratic, and Jx^ +xy+4+ ^ = ±V^F = ±V Whence, Jx"-\-xy-\-4=7, or — 8 a;2-|-ary=45, or 60. (1) Now take the first equation, and multiply numerator and denominator of each of the literal fractions by its numerator, then Expanding and uniting, and we have 4 16a;2=25y2 4a; Ax — ±: 5y, or y= ± — 5 10« ROBINSON'S SEQUEL. This value of y put in (1), gives ar2-|-!^'_=45, or 60. (2) 43*2 Also, a;2— Zf„=45, or 60, (3) 6 From (2), 9a;2=9*5-5. Or, a;= ±5. Or, 9a-2=25-12=25-4-3 3ar=5-273. Or, a:=dbl0^i From (3), a;2=9-6-5. Or, a;== dbl5 Or, ir2=300. Or, x= ±10^3. Here we have 8 different values of x, each of which being 4a; substituted in y=rfc — , will give 8 different values to y. 5 (8.) Given f h+V^ \ -^ -^^ /_4^ 1 to find and I V_^<5^=^=y+i [andy. Clearing the first of fractions, gives x-\-y^+^Jx=:^xy^ (1) In the second equation, multiply the numerator and denomina- tor of the fraction by the numerator ; then Multiply by (y+1), then extract square root, and we shall have Jx+Jx—y—\^y+\ (2) Or, Jx—y—\={y+\)-^Jx By squaring, x—y—\=y^-\-9.y-\'\ — '^Jx{y-\-\)-\'X B^duced, (i=^(y''+y)+^y-\'^—^Jx{y-\-\) Dividing by (y+1), 0=y+2— 2V^ (3) As we can divide by the binomial (y+1) without a remainder, it follows, by the theory of equations that (y+1) contains a root, that is y-|- 1=0. y= — 1. Corresponding with this value of y, equation (3) or (2) will give the value of :r. 2jx=l, ar = ^. ALGEBRA. 108 To find other values, we must continue the solutions. Return to equation (1) and extract the square root of both members, and we shall have Jx-{-y= ztyjx (4) From (3), 2jx=y+2 (6) Double (4), and 2jx-\-2y= :±S.Jx{y) (6) That is, y+2+23/ =y2 +2y Or, y^ — y=2' Whence, y=2, or — 1. The value — 1 we found before ; which shows two roots equal to — 1. The other value 2, put in (6), gives a;=4. If we take the Djinus sign in (6), we shall have y+2+2y=— 2/2— 2y Or, 2/2+5y=— 2 Whence, y= — f =h^ ^17 (9.) Given and 2a;2 X _1 I ues of a; and y. The first equation can be put in this form The solution of this quadratic gives or Whence, ^=16, or ??-. y=16a;. Jy=^Jx. X \Q Substituting the values of y and Jy, in the second equation, we find x__^ X J_l 8 127^ 3 iH ROBINSON'S SEQUEL. The double is ^-Jjx=- Add 3^ to both members to complete the square, (Art. 99, Algebra,) then By evolution, ^ Jx — ^=±f Whence, x=4, or V > but y=16a;=64, or ^S■. If in the, second equation we write ^j^x for the value of y, and V J^ for the value of Jy, we shall find •*' 64 > ^^ 144' values of x and y. Transposing 2a:^y^ in the the first equation, and we have By evolution, x^ — ^y^ = ±(l-|-^y) (1) The second equation can be put in this form, (23,='+l )(*+!) (2) Taking the plus sign in (1), we can put it into this form x^—xy+y^=2y^+l (3) By the help of (3) we perceive the equal factors in (2). Sup- press them, and (2) becomes x-^y=x-\-l. Or y=l. This value of y put in (1), gives a?=2, or — 1. (11.) Given and £J^-40y^=136-y»J^'-!^ I u> find ^ "3' Itheval- yyy ' y^ y J andy. It is obvious that the first equation can be put into this form a;2y2_8()y 2 =272— 2^^0:^—272 By transposition ALGEBRA. 106 (a;2y2__272)+2y7a:2y2_272=80y2 By adding y^ to both members, and extracting square root we have (a;2y 2_272) 2+y= ±9y Whence, x^'i/'^—272=e4y\ or lOOy^ (i) Clearing the second of the given equations of fractions, and reducing, we have x^y'^—SBxy^Se (2) Put 2a=35, (see Art. 106, Robinson's Algebra.) Then x^y^ — 2cui;y=2a-\-l Adding a^, and taking square root, gives xy — a=ztz(a-\-l) Whence, xy=z(2a-\-l)=36, or —1 (3) These values of xy put in (1), give 64y2 =36 -36—272, or 6V = _271 64y2 = i024, or 8y=±32. y=4, or —-4. These values of y put in (3), give x=9, or — 9. Again, by observing (1), we perceive that we may put 100y2 = i024, or 10y= ±32. 2/=3.2, or — 3.2. (12.) Given and 2y^—^Jx + 2jy-—\Jx^ ^Jx I V^+V8(2/->/^)— 4=y+l to find the val- ues of« and y. Put 7^2. Then [^x=P, in the first equation. ^^ I 2P — ^v^ Jx^ 2 4F^-]-iJx'F=3x By adding x to both members to complete the square, we have 4F^-\.4jxrF+x=4x 2F+Jx=zt:2jx Or, 2F=^x, orSjx Restoring the value of P, we find 1^ ROBINSON'S SEQUEL. Whence, Ay^ — 16jx=x, or 9a; (1) From the second of the given equations, we have >/8(y-V^)-4=(y-V^)+l Squaring, Q(y-.Jx)—4=(y—Jxy+2(y—Jx)+l Whence, (v^J^) '— 6(y— V^) = —5 And y—Jx—3=±2 (2) Taking the plus sign y — 6= Jx (3) Taking the minus sign y — 1= Jx (4) Substituting the values of Jx and x taken from (3) in (1), we have 4y2_-16y+80=y2_i0y+25; or, V— 90y+225 Whence, y=^l2±l,EI^^ or y^^^l^J^ ^/3 5 Taking the values of the same from (4), and substituting, as before, we have 4y2— •16y+16=y2__2y+l, and 9y^—lQy-\-9 Whence, . y=3, or :?, and y=l±>/^ 3 6 Substituting the values of y in (3) and (4), we have the val- ues of X. (13.) Given and ^x^—Uy—U x^ ^fA 5y+ ^ y-^^ to find the »• values of x and y. Multiply the first equation by 3, transpose, &c., and we have x^ ,2x Ix^.x^ y lSyY~^3y~4~^ V^!^1^14=(^-^_15y-14)-94 Put P=Jx^ — 16y— 14 ; then we shall have Whence, P= ^^g. db H = ^^ , or — 9^ '»4ji. ALGEBRA. 107 That is, a;2—15y— 14=100, or V//. Or, x^ = 15y+U4; and x^ = \5y+^^U^. (1) The second equation may be written thus, ■Mn)-^i.-*i 8y ' \ 3 ' 2/ >* 3y Uniting the fractions, and x^,/4x+3y\_^ I4x+3y Sy"* \ 6 / ^l 12y Dividing every term by 2y, and we have x^ / 4x-\-3y \ _x / ^rg-f3y y T62^~^\ 12y / 2y\ 12y / For the sake of perspicuity, put P= i — X-Jl j , then 16y2 2y ^ By evolution, -^— P=0 4y Whence, ^L=P^=i^±?^ 16y2 12y Clearing of fractions, Zx^ = \Qxy-\-\'2.y^ Whence, Qar^— 48a;y=36y2 Add 64y2 to both members, to complete the squares, then 9a;2__48a;y+64y2 = 100^2 By evolution, 3a; — 8y= ±10y Whence, a:=6y, and a;= — fy (2) Substituting the first of these values of x in equation (1), we have 36y2_i5y=ii4 By adding || to both members, (Art. 99, Algebra,) we shall have 36y2_-16y+?.| = 114+f| = Hi^ By evolution, Qy — f = =h V Whence, y=2, or — 1|. 108 ROBINSON'S SEQUEL. These values put in the first of equations (2), give a:=12, or — y. By taking the second set of equations in (1) and (2), we shall find other values of x and y. (14.) Given j ^^y — \z=^x^y — \y^ / to find the values and (a;2_3=a:2y 2 (x^_y\>^ ) oi x and y. Ans. ar=l. y=4. Put a;^=P, and y^= Q, and we have P3_3 ^PQ(^p_^Q) (2) Now put P=tj Q, and equation ( 1 ) becomes (471^+1) §«—16w(2' = 16. Conceive w to be a known quantity, then the last equation is quadratic, and a solution gives ^3_4(2/i^ +2^+l) _ 4^ ^__ 4 But from (2), ^^= - = - Put the two values of Q^ equal, and put n^ — n-\-l=E, (3) Then 1_ =t-^ . Whence, 2w = ^^ZI? . (4) But from (3) resolved as a quadratic, 2w=l=fc7(4i?— 3) (6) From (4) and (5), 2E ±2i2^(4J!?— 3)=6i2— 3 Or, ±:2EJ (4E—3) = 4JS— 3 Put J(4E—3)=S. . Then ' S^:^2BS = 0. Or, S(Szt2E) = 0. This last equation may be verified by taking either factor equal to zero ; and as the first factor only gives a rational quantity, we take that which gives i2=f. By retracing, we easily find x and y. ALGEBRA. 109 We now add a few unwrought examples for the benefit of those who may wish to test their own unaided powers in these difficult operations. None of these that follow are as severe as many of the prece- ding. (15.) Given (. /J-^~"^-f ./-^^— =2 ) to find the values and ( a:2_i8=;c(4y— 9) ) o^ ^ and y. Ans. x=6, or 3. y==3, or f . (16.) Given (,+,)_^(,rz,T) ^- and ( (a;2-f-3/)2_|_(.^_y) = 2x(x^-^y)-{-50e to find the values of x and y. Atis. x=5, or — ^/. y=3, or ~|^. (17.) Given ^^+^^^^^24 1 _ ^^ . ^ ' ! a; ^y x-\-y 6 I ^^ ^^"- ^"® values ^ 4ar2 \ and V^— y+^=9^(^_^-) of ar and y. ^W5. ic=3, or Y J or t\» or jf . y=2, or — ^P; or I, or — '^4^. (18.) Given W6V^+6Vy+W^=9-Wy I ^ fi„d the and ( a; — y=l2 J values of x and y. ^Tis. a;=16, or sjyuLP. y=4, or ^eV-'*, (19.) Given (x+JSy^—n+^x=7+22j-y^] to find the , i 73^71:^X7-^+2^ y values of and j^ V "^^ ^+^""^1::^ j X and y. ^?w. ir=4. y=2. 110 ROBINSON'S SEQUEL. (20.) Given j a;^— 2/^=3 ) and ( {x^+y*y+x'y''{x''—^'')^+x^-^^=32Q \ to find the values of x and y. Ans. a;=s= ±2, or ±J(—1). y=r ±1, or ±2V(— 1). (21.) Given f ^+J^+J_ Jx-^^ ^Q9_ ^ ,^ g^^ the val^ and I 2 r~2_-.l^ [ ^^s of iT and y. ^«*. ar=9, or J^f^, or ^-f », or 16. y=4, or — V> or — V» or i. SECTION IV. PROBLEMS PRODUCING QUADRATIC EQUATIONS CONTAINING MORE THAN ONE UNKNOWN QUANTITY. The following outlines of operations, refer to problems in Rob- inson's Algebra, Chapter III, page 183. We pass on to the sixth problem, page 186, and only include those which serve to illus- trate brevity and elegance in operation. The figures in parenthesis refer to the number of the problem in the book. (6.) Let t =s the time (hours) he traveled, and r= his rate per hour ; then r^=36 (1) But if r becomes (r+l), t must become (t — 3), and then {r+l){t-S)^36 (2) Or, r/_|_jf_3r— 3=36 ri ac36 ^3(r+l) Hence, ?-^+rs=12, and r=3. (7.) Let x= the number of children, and y= the original share of each. Then a:y=4680O (1) ALGEBRA. Ill (x—2) (y+1950)=46800 (2) ajy+1950a;—2y— 2- 1950=46800 1950(a;— 2) = 2y Or, 975(x — 2)z=xi/=i46809 By division, x^ — 2a;=48 x=B» (8.) Let x= the number of pieces. Then = the cost of each piece. X 48a:— 51^=675 X 4Qx^—675x=675 16x^—225x=225. (9.) Let xz= the purchase money. Then 12^= the cost, and 390—^^^'^= his whole gain. 100 100 ^ Then 12^ : 390-12^ : : 100 : ^ 100 100 12 Product of extremes and means, ^^^=39000— 104a? 300 — =3000— 8a: 300 Put a = 300 and divide by 2 ; then — =i5a — 4a: a x^-\-4ax=5a^ a:2_|_4aa:+4a2=9a2 a:-|-2a=3a a;=a=»300. (10.) Put a;+y= the greater part, and X — y= the less part. Then 2a:=60, a:=30, and ar^—y 2 = 704. (11.) Let a;= the cost ; then 89 — a:= the whole gain. X : 39— a; : : 100 : x. Ans. ar=10. 112 ROBINSON'S SEQUEL. (12.) Let (x — 20) = the number of persons relieved by A. Then x-\-20 = the number of persons relieved by B. 1200 , c 1200 +5=. a;-}-20 X — 20 Divide by 5, and put a=240 ; then ' « fl: « a:+20 a;— 20 aa;— 20a+a;2— 400=aa?4-20a a:2=40a-i-400=40(a+10)=40-260 Or, a;2=400-25 a?=20-5=100. Hence 80 is ^'s number, and 120 A's. (13.) Let x= the price of a dozen sherry and y= the price of a dozen claret. 7a:+12y=50 (1) — = the number of dozen of sherry for 10£. X n -= the number of dozen of claret for 6j£. (2) By substitution, _Z2^4-12y=50 70y-|-36y2 -f72y = 1 50y+60 • 6 36y2_8y=300 92/2— 2y=75. Hence, y=3. (14.) Let 19a:= the whole journey. Then x= £'s days, also his rate per day. Or x^ = £'s distance. Also, 7a;-j|-32= A's distance. a:2_|_7a:_j.32=19a; x^—12x=—S2. Hence, a?=8 or 4. And 19ir=162 or 76. y Then 12=3+? X y Or, ^ 10 \Qy 36 3y+6 y ALGEBRA. 113 If we put X for the whole journey, we shall obtain the 13th equation, (Art. 104.) (16.) Leta?= the bushels of wheat, and ar-4-16= the bushels of barley. 24__ 24, 1 ~x .r+16'^4 24a;+16 • 24=24a:+^J±l?.^ a;2-[-l6a;=16- 96=16- 16-6 Put 2o=16. Then 2a-2a-6=24a2 ar+a= ±:ba .a;=4a=32. (16.) A put in 4 horses, and B put in x horses. 18 Then — = the rate per head. X Hence, 4»18 X 4»20 x+2' 4-18 4-20 |-18= the price of the pasture. j-20= the price of the pasture. X 36^ X x-\-2 ' :J^+1. x+2^ x=6. ;» (17.) Let 4x= the price per yard, and 9x= the number of yards. 36;r2^324 x=3. (18.) Let 10x-\-y= the number. Then And From (1), From (2), By division, 8 xy 10a:+y4-27=10y+a; 10a:=(2a; — \)y x+3=y i^=2^-l a:+3 (1) (2) / ^ 114 ' ROBINSON'S SEQUEL. 2a;3_-5a;=3 «=3- (19.) Let (a;— y), x, and (a;+y— 6), represent the numbers. Then 3a?— 6=33, or x=lS. (ic— y)2=a;3__2a?y+y2 x^=x^ (x^y—6y== x^ +2xy+y^-'12x^l2y+ S6 3a;2_|^22/2— 12^— 122/+36=441 By subtracting the value of 3a;2— 12a;+36, we have 2y2 — 122/=64. Hence, ir*^. (26.) Let a;+y= the greater, and x—y= the less. Then (a;2— y2)(2a:2+22/2)=1248 (1) Or, , a;4_3^4^624 Also, 4xy=:20 (2) ^ 6 . 626 Whence, y=-. y*=-i- a:*-^-=624 a;* a;8— 624a;4 =626. Put 2a=624. Then x'^—2ax''+a^=a^+2a-\-l x'—a= ±(a+l) Whence, a;4=2a+l=625. a;2=d=26. a;=6, or — 6. From (2), y=l- (27.) Let x= A's stock. a=1000. Then a-^= B's stock. Observe that 780= the whole gain. Then 9a;+(6a— 6a;)=6a+3a; : 9x : : 780 : 1140— a:. Or, 2a+x : Sx : : 780 : 1140— a:. This proportion will produce a laborious equation to work through. Therefore we will try 2x to represent ^'s stock ; then 9-2a;=18ic. (a— 2a;)6=6a— 12a?. 18a;+(6a-12ar)=6a+6a; : 18a? : : 780 : 1140— 2a?. Reducing, gives us a^x : 3a? : : 390 : 670— a?. 670a— aa?+670a?—a?2 = l 170a:. ALGEBRA. 116 Whence, a?2+1600a^= 570000. a;24-1600:c+(800)2 = 1210000. a:4-800=1100. a;=300. 2^=600, ^'s stock. When the Algebra was first published, the 6 months in the problem was printed 8 months, by mistake. How could we dis- cover that mistake ? We look at the answer and see that the numbers 600 and 400, make the stated sum 1000 ; therefore we will assume that these three numbers are correct. We will now take m to represent 9, and n to represent ^'s time. Then the preceding proportion becomes ^mx—^nx-\-7ia : ^mx : : 780 : 1140—22:. Also, ^mx — 9,nx-\-na : na—^nx : : 780 : 640-(-2a; — a. Now give to X its value 300, and to a its value 1000, and these proportions will give m=9, and ?^=6. (28.) Let «2_. half the number in the first Then SLx^is=^ the number in the first. And 4a;-|-4= the number in the second/ 3(2a;2-j-4a:-[-4)= the number in the third, 3(.'c2-[-2a!-f 2)-|-10=s the number in the fourth. Sum, ll(*'2-|-2a'+2)+10=!121, the given sum. Whence, a; ^-j- 2^+2= 101. Or, a-'2+2.r-f-l = 100. By evolution, x-^\= ±10, or .r=9, for the minus sign will not apply. Then 22'2 = 162, the number in the first, (31.) Let a:= the greater of the two numbers, and y= the less. Then per conditions, xy-s^^x"^ — y^ (1) And a-'2-|-2/2_^3_^3 ^gj From ( 1 ), x^ — ^y=y^ • Conceive y a known quantity and complete the square thus ; 4a;2_4y.a:_j-y2_5^2 2.r— 2/= ztj5'7/ 116 ROBINSON'S SEQUEL. Or, 22:=(lrhV5)y. Let (l±V5)=a. Then x=^ (3) 2 . Let this value of x be substituted in (2), and we have 4 ^^ 8 ^ Dividing by y* and clearing of fractions, and 2a2+8=(a3— 8)y Whence, y= ^— Buta2=6±275. aS^^iedrS^S. Therefore, ^ 8±8V5 2\1±V5/ ^ This last operation may not be obvious to some ; it will be seen by multiplying (1±V^)' ^7 V^' ^^^* ^^' ^^^^ numerator in paren- thesis is ^5 times the denominator. To find X we must simply multiply y by |a, see (3) ; that is, ar=Kl±V5)>/5=i(V5±5). The following are not in Robinson''s Algebra, but selected from every source, — mostly from Bland^s Problems. (1.) The swn of two nwmhers is 2, and the sum of their fifth powers is 32. What are the numbers.^ Let x= one number, and y= the other. Then x+z/=2 (1) And a:5+y'=32 (2) As the 5th power of 2. is 32, therefore (x^y)^=x^-\-y^ That is, x^-^-dx'y+lOx^^j^+lOx'^y^-^-Bxy^+y^^x^-^-y^ Or, 5x*y+10x^y''+10x''y^+5xy^=0 ' By division, x^ -\-2x^ y-\-2xy^ -]-y^ =0 That is, x^-\-y^-{-2xy(x-\-y)=0 Dividing by {x-\-y), and we have g.2_^_^y2_^2xy=0 ALGEBRA, 117 Or, x'+x7/+y^=0 From (1), x^+^xi/+y^ =4 By subtraction, a;y =4 (3) Multiplying (1) by «, gives x'^-\-xy=^^ That is, x^—2x=z—4 Whence, x=l±^—3 Then y=lqr^— 3 Here we have obtained two expressions, the sum of whose 5th powers is 32, but not two numbers. We have not so clear an idea of (1=1=^ — ^)> ^^ ^^ ^^^^ ^^ ^ itself. If we compare equations (1) and (3), we shall perceive an impossibility/; for two numbers whose sum is only 2, can never make a product of 4. In the same manner when a sum is but 2, the sum of the 5th powers of any two of its parts, can never make 32. To test our quantities, we will verify (2) with them. To save trouble, we will put a=jj — 3, then a^ = — 3, a'^=9. y5 = (l— a)5 = l— 5a4-10a^ — 10a ^ +5a^— gs aj5-|-y5 =2-f-20a2_j_l0a* =2— 60+90=32. (2.) The fore wheels of a carriage make 6 revolziiions mx>re than the hind wheels in going 120 yards ; but if the periphery of each wheel be increased by one yard, then the fore-wheels will make only 4 revolutions more than the hind wheels, in running over the same dis- tance. Required the circumference of each wheel ? Ans. Fore wheels, 4, hind wheels 5 yards. Let ir= the yards in the circumference of the larger wheels, and y= the jrards in the circumference of the smaller. ' Put a=120. Then per question, -=- — 6. ' (1) X y And _JL.=_^_4 (2) ^+1 y-f 1 ^ ^ Clearing of fractions, ay=iax — ^xy. (3) ay-\-a-=ax-\-a — ^xy — ^x — Ay — 4. (4) ria ROBINSON'S SEQUEL. Suppressing a in both members of (4), and then subtracting it from (3), we have 0=— 2xy+4;r+4y+4. (6) From (3), :r= -^= J^^. = ^^ a—Qy 12U— 6y 20— y From (6), 0:=?^:? Therefore, Hd=J^ y— 2 20— y 20y— y 2 _j_2o_y = 1 Oj/2 _20y Whence, ll2/2_39y=20. If we work out this quadratic, we shall find y = 4 ; but the operation would be a little troublesome, because the numbers are prime to each other. In cases like these, when a practical operator is only in pursuit of results, he looks at the absolute term, (in this example, 20), and observes its factors, 2, 10, 4, 5, and conceives y to represent one of them ; and if it verifies the equation, then y is really that factor. I will now conceive y to be 4, and divide the first member by y, the second by 4 ; then ' lly— 39=5 Or, lly=44, or 2/=.4. Therefore, as this supposition verifies the equation, the suppo-' sition itself is truth. Now let. us suppose y to be 6 ; then operate as before, and lly— 39=4 ny=43 Now as y does not come out equal to 5, the supposition was not true. ■c 1, 2O2/ 20-4 - l*or X, we have x=. ^-= =5. 20— y 16 (3.) A and B engaged to reap afield for ^24 ; and as A could reap it alone in 9 days, they promised to complete it in 5 days. Finding, however, that they were unable to finish it, they called in to assist them the last two days, in consequence of which, B received ^ ALGEBRA. 119 $1 less than he otherwise would have done. In what time could B (yr C alone have reaped the field? Let x= the number of days in which B ffould reap the field, and y= the number of days in which C could reap it. As A could do it in 9 days, for one day's work he should have I of the money ; and as B could do it in x days, for one day's work he should have - of the money. A and B then working X together one day would do --[-- o^ the work, and in 5 days they y X would do s/'l+i^ : - : : 24 : ?1?1= the number of dollars B \9 x/ X rc+9 would have received had (7 not been called in. But as B can reap the field in x days, for one day's work he 24 6*24 should have — dollars, and for five day's work, dollars, the X X sum he did receive. Therefore, — =:1 2^+9 X Whence, x''—'^lx=:^\m<^. Here, as we are only in pursuit of results, we try Inspection. We perceive that 10 for the value of x would not be large enough, and 20, too large ; and as 1080 terminates in a cipher, we will try dividing by 1 5 ; then «— 87=— 72. Whence, x=\b. Am, Also, a;=72 ; but this will not apply to the problem. Again, as A could do the work in 9 days, for one day's work he should have — dollars, and for 5 day's work, — dollars. 9 "^9 5*24 2*24 B should have dollars, and C, dollars, and the sum 15 ' ' ^ to the three is 24 ; therefore, 5-24 , 5-24 , 2-24 =24 9 • 15 ' ^ Or, |-j_^_|_?=i. Whence, y=18, Ans, ^ 120 ROBINSON'S SEQUEL. (4.) Bacchus caught Silenus asleep hy the side of a full cash, and seized the opportunity of drinking, which he continued, for two-, thirds of the time Silenus would have taken to empty the whole cask. After that, Silenus awoke and drank what Bacchus left. Had they both drank together, it would have been emptied two hours sooner, and Bacchus would have drank only half what he left Silenus. Required the time in which each would have emptied the cask sepa/rately. Ans. Bacchus in 6 hours, and Silenus in 3 hours. Let a= the volume of the cask. a;= the time Bacchus would require to drink it alone. y= the time Silenus would require to drink it alone. Then -= the volume Bacchus drank per hour. And _= the volume Silenus drank per hour. y - . -^= the volume Bacchus drank ; then X 3 (a — -^jz= the quantity left to Silenus ; and this quantity divid- ed by the volume Silenus drank per hour, will give the hours he employed in drinking. That is (a—?^\l , or (y^?^\ = the time Silenus drank. Had they both drank together, (J^) would express the \x-\-y/ I time. Now by the given conditions. 3^* 3x »+y (1) And (" ''VY-_''y \2 Sx/a «+y (2) Reducing (2). and l ^-/^ Or, 3x'—2y'=5xy 9a»— l&ry=6y» . 2 Whence, x=2y. This vahie put in (1), gives ALGEBRA. Ill 3~^ 3 3~ (6.) A Banker has two hinds of money ; it takes a pieces of the Jirst to make a crown, and b pieces of the second to make the same sum. Some one offers him a crown for c pieces : how many of each kind shall he take ? Ans. Of the first kind l ^J^ of the second, i 4-- (6— a) (a— 6) This problem is more of a puzzle than most others, yet it is a fair scientific question. Let a;= the number of pieces of the kind a, and y= the number of pieces of the kind b. Then x-\-y=c. (1) As a pieces are worth 1 crowiji, one piece is worth -, and x a pieces are worth — a By a parity of reasoning, y pieces of the second are worth - and the worth of both together is just 1 crown ; therefore, ?+|=l (2) a Whence, hx-\-ay=^ah From (1), hx-\-by=bc By subtraction, (a — b)y=(a — c)b. y=S^^ZZL. a — 5 In 'like manner we find x=A ^'. Itt ROBINSON'S SEQUEL. (6.) A and B traveled on the same road, arid at the same time, from Huntington to London. At the bOth mile stone from London, A overtook a drove of geese which were proceeding at the rate of 3 miles in 2 hours; and iwo hours afterwards, met a stage wagon, which was moving at the rate of 9 miles in 4 hours. B overtook the same drove of geese at the Abth mile stone, and met the same stage wagon exactly forty minutes before he came to the ^\st mile stone. Where was B when A reached London ? Ans. 25 miles from London. Let x=. the rate which A and B traveled per hour. Then 50 — 9,x= the distance from London where A met the stage. m h m 3 : 2 : : 5 : y = ^li® hours required for the geese to travel 6 miles. Then when B was 45 miles from London, A must have been 50 — ) miles from the same place, and the distance between 3 / ( the two travelers must have been i — 5 \ miles, and the val- ue of this expressson is the answer demanded. Now let t be the hours elapsed between the times that A and B met the stage. The motion per hour for the stage was f miles. B met the stage / 314-— ) miles from London ; but A met it before, nearer to London by — miles. That is, A met the stage (31 -4-— — . J miles from London. We have before determined ^3 4/ that A met the stage (50 — 2x) miles from London; therefore, 31+?^— -5^=50— 2a: (1) ^3 4 ^ ' Now after A met the stage he traveled in one direction, and the stage in another for t hours, before the stage met B, Then their distance asunder must have been l-^-\-tx\. But the distance ALGEBRA. 123 the two travelers are asunder, has been expressed by ( — 5 ) Therefore, ^+to=i— — 5 (2) From (1), /= .32.-228 j.^^^ ^^ ^_40.-60 27 ' ' 27+ 12a; Therefore, 3ar— 228_40a;— 60 27 27+12a; Or, 8a;— 57__10a;— 15 • 9 9+4aj Clearing of fractions, 32a;2_228if+72a;— 513=90a;— 135 Or, 16a;2— 1232^=189 V Here the obvious whole number factors of 189 are 3, 9, and 21 ; and as we are only in pursuit of results, we will try one or two of them. 21 we perceive at once is too large, therefore, try 9 ; then 16a;— 123=21 16a;=144, or a;=9, a true result. Now because a;=9, — f — 6=25, the answer to the question. SECTION v. PROBLEMS IN" PROPORTION", AND IN" ARITHMETICAL, GEOMETRICAL AND HARMONICAL PROGRESSION". The problems contained in Robinson's Algebra are not written out ; they are only referred to by article, and number of the prob- lem, and a mere outline of the solution indicated. We commence with Art. 117, example 3. (3.) Let X — 3y, x — yy a;+y, and x-\-%y represent the numbers ; then 2y=4. 124 ROBINSON'S SEQUEL. The product of the 1st and 4th, is a;2_9y2 . of the 2d and 3d, is (x^—y*). .a;4_9^22^2 jc4_ioa:2y2^9y* = 176985 9y*= 144 a;4_40a;3 =176841 (4.) The same notation a^ in the last example. 2;r=8. x=4. x^—y^z=15. (6.) Let n= the number of days. Then L=\+{n—\)\=n. S=^{\-\-n) ^n=i the whole distance. Also, (n — 6)15 = the whole distance. «2__29^^_I30 w=9or20. 9—6=3. 20—6=14. (6.) The first day he must pay l+^; i representing the in- terest of one dollar for one day. First day, \-\- i. 2d day, 1+ 2e. 3d day, l^- 3i. Last day, 1+60J. (2-|-6U)30= the whole sum to be paid ; but as this sum is to be paid in 60 equal payments, each payment must be ^^ 1^ ^-= Ans. 81 and | of a cent, nearly. (7.) Let X — 3y, x—y, x-\-y, and x-\-^y represent the numbers ; then 2;r2+18y2=50 2a;2-|- 2 y^=34 16y2 = i6 y=l. ALGEBRA. t26 GEOMETRICAL PROGRESSION- AIND HARMONICAL PROPORTION. (Art. 124.) (1.) Let X represent the mean sought. / 18 (2.) Let x= the number sought. Then, by harmonical pro- portion 234 : X : :. 90 : 144— x 90a;=234-144— 234x 324a:=234- 144. Hence, ar=104. (3.) Let x= the number sought. Then 24 : a? : : 8 : 4—x Or, • , 3 : a; : : 1 : 4—x x=3. (4.) Let x= the second. Then 16 : 2 : : 16— a: : 1 a;=8, (6.) Let x= the first number, and y= the ratio. Then x+xy-\-xy^ =nO (1) xy^—xz=i90 (2) By subtraction,, 2a;4-^y=120, or .t= 90 From (2), we hare x= -— =— L , or 4y2— 3y=10 y=9L (5.) Let X, xy, xy^ , and xy^ represent the numbers. Then ^y3 __ y2 _4 xy-^-xy"" 1+y 3 From this equation we perceive at once that y=2 ; then a:+2a;+4a:-f 8ar = 1 5ar=30 ar!= 2. t28 KOBINSON^S SEQUEL. (6.) Let X, xy, xy^, and xy^ represent the numbers^ a?+a-2/2 = i48 (1) a'y+3ry3=888 (2) Or, /{^)+y=-14 (1) ^And a;2-}-a;2/+y2 ^34 (2) ' Put a;-|-3/=5, and J{xy)=^p; Then a;2-|^5^-|-y^="^'^' — i^^» and equations (1) and (2) become s-|-jys=i4 (3) s2_jo2^84 (4) Divide (4) by (3), and we have s^—psnQ (6) Add (3) to (5), and divide by 2, and 5fc=10. Hence, . . . ^ » » v . , . . j9s=4. (8.) Let X, xy, xy^ , and a-y^ represent the numbers ; then xy^-^xy=^9.A xy^+x : xy^-{-xy : : 7 : 3 Or, y^-\-l : e/^+y : : 7 : 3 Divide the first couplet by (y-^l), and we have y'—y+l : y : : 7 : 3 3y2_32^_|_3_.7y^ ^31- 32/2— .10y=t— 3. From this equation we have y=3, the ratio. (9.) Let X, xy, xy^ , and xy^ represent the numbers \ Then ar(l+y+y2+y3)=:y-|-l And «:=rV- Put (y+l)=^. Then j\{A+Ay^)=A A-^-Ay^^lOA. Ay^==9A, or » . . . .y=3. Hence, xV> tV> <^<^- ^^^ ^^^^^ numbers. ALGEBRA. tf7 (10.) Let X, — —y and y represent the numbers ; then ^+^+y=26 (1) And a?y=72 Put a:-j-y=* ; then equation ( 1 ) becomes 144. ^ s+ilZ=26, or s2_26s=— 144 s=.tl8. s (11.) Let a?, a;y, and xy^ represent the numbers ; Then x^y^^2\Q (0 '^k a;2_f-a;2y4^328 (2) ^ From (1) /^+y=i3^ (1) {x+y)Jxy=^Q_ (2) a;+y=13— 7a;y • (3) _ 30 _ 13 — Jxy=—=r- Hence J xy =3. Jxy (13.) Let X, — ^, and y represent the numbers ; then ^+y=i8 (1) ?^!=.676 (2) 18 ^ ' ^1=M. xy^l^ (3) o From (1) and (3), we find x and y. 128 ROBINSON'S SEQUEL. (14.) Let X, xy, and xy^ represent the numbers ; then (^xy^ — xy) (xy — x) are the first diflferences, and xy^ — 2xy-\-x= 6 xy'-\- xy-\-x=42 Difference, Sxy =36 xy=l2 (15.) Let a;, ^ , and y represent the numbers. If y is x+y supposed greater than x, then ( y — ^^ j i ^^ — x j are the 1st differences, and y — . — ^-Ua;=2, the 2d differences.- x+y ajy=72. Put {x+y)=s; Then . s— l_if=2 s s2_25+l=289 «— 1=17. s=X'\-y=lS. (17.) Let a;^, xy, and y^ represent the numbers ; then a;a_|_a;y+y2=:3i, and ic 2+2/2 =26. (18.) Let X, xy, xy^ , xy^, xy^^ and xy^^ represent the num- bers. Then, by the conditions, we have x-\-xy-\-xy^-\-xy^-\-xy'^-\-xy^ = lB9=a (1) And xy-\-xy'^=54=:b (2) But equation ( 1 ) may be put into this form ( 1+y+y ^ )^+( 1 +y+y' >y^ =« Or, x+xy^= -^— Multiply this last equation by y, and its first member will be the same as the first member of equation (2). Therefore, ^^ = 6 : a quadratic from which we obtain y, the ratio. (19.) Take the same notation as for (18) ; then we have (a:-fary)+(ir+a:2/)y^ = 189— 36=:153=a. (1) ALGEBRA. And (x+xy)y''=36=b. Divide (1) by (2), and we have 1+2^^^153^51^ Hence. y' 36 12 (2) 129 r=2. CHAPTER III.— PROPORTION. (5.) Let X and y represent the numbers ; then x—y : x-{-y x-\-y : xy From the first, 2a; : 2y ISy . Il2^ 7 * 7 y : lly^ 2 18 11 18 1 9 77 7, or x=yy, 77 77. y=7. (6.) Let a: and y represent the numbers. a;+4 : y+4 : : 3 : 4 (1) ar_4 : y— 4 : : 1 : 4 (2) From (2) we have 4x — 16=y — 4, or y=4x — 12. This value of y put in (1), gives a;+4 : 4x—S : : 3 : 4 ar-|-4 : x—2 : : 3 : 1 a;_^4=3ar— 6 a;=6. (7.) Let X and y represent the numbers ; Then x-\-y=l6 And xy : ar^+y^ : : 15 : 34 Double the first and third terms, then add and subtract, ( The- orem 4), and 2a?y x^+y' \ : 30 : 34 x'+^y+y' : x^—^y+y^ : 64 4 ^+y x—y : 8 : 2 16 x—y : 4 1 Or. «— y=4. 180 ROBINSON'S SEQUEL. (8.) Let x= the gallons of rum. And j/= the gallons of brandy, x—1/ : y : : 100 : X B X — y : X : : 4 : y Product, (x—yY \ xy \ \ 400 : xy Dividing the second and fourth by xy, and (x—yY : 1 : : 400 : 1 x—y : 1 : : 20 : 1, or x—y^10. (9.) Let x-\-y=i the greaternumber, And X — y=. the less. Then a;2_y2=:=320. (ar+y ) 3 =;^3_|.3^2y^3^y2 _j.y 3 (ar_y)3 _^3_3^2y_j_3^2/2--3/^ (1) 6a;2y-|-2y3== diflf. of the cubes. 2y= difference. Cube of (2y)=8y3 6a;2y-|-2y3 : Sy^ : : 61 : 1 Sar'-f-y^— 244y2. 3^.2^2432^2^ This value of x'^ put in equation (1), gives 80y2=320, or y=2. We now give additional problems. (1.) The sum of four whole numbers in arithmetical progression is 20, and the sum of their reciprocals is |f-. What are the numbers? Let X — 3y, x — y, ar-f-y, and x-\-3y be the numbers ; Then 4a;=20, and x=5. Affam + -4- + = — ^ x—Sy x—y ' x-\-y x-\-3y 24 Uniting the 1st and 4th, and the 2d and 3d, we have ALGEBRA. 131 2x , 2a: 25 Dividing by x=5, and then clearing of fractions, reduces the equation to 2x''—'2y''+2x^—lQi/^z=^^(x^—9y'') {x^'—y''). 96^2_43o^2^5^4_50a;2y2_j_45y4 Putting the value of a;^ in the 2d member, we have 96^-2_480^- == 1 25;i;2 — 1 230y2 ^46y4 Whence, 0=292;-— 770y--|-45y^ Dividing by 5, 0=29a?— 154y2_j_9y4 Or, 0=145— 154y2_j_9y4 Here the sum of the coefficients is the same in both members ; therefore one of the values of the unknown quantity is 1 ; and as y=l answers the conditions of the problem, we are not required to find the other roots. Hence the numbers are 2, 4, 6, and 8. (2.) Tke sum of six numbers in arUhm^tical 2>rog7'ession is 33, und the sum of their squares is 199. What are tke numbers ? Ans. 3, 4, 6, 6, 7, and 8. Let [x — y) represent the third term, and {x-\-y) the fourth term ; then 2y will be the common difference, and {x—Sy), (^•— 3y), {x—y), {x-\-y), {x-\-2>y), and {x+oy) will represent the numbers. Then 6.r=33, or 2.i = ll. (1) And 6a:2-|-7Q?/2zz=199. (2) From (1), 4a.'2 = 121, or 12^-2c=363, Double (2), and write 363 for \9^x^ , then we have 363+140^2=398 140y2=:35; y=J-. (3.) Find four numbers in proportion such tk^U their sum shall ie 20, the sum of' their squares 1 30, and the sum of their cities ^80. Ans. 6, 9, 2, and 3. Let w, X, y, and s represent the numbers 132 ROBINSON^S SEQUEL. Then because they are in proportion, wz=^xy ( I ) By conditions, ?tf-|-a;-)-y+2;=20=a. (2) And w^+x^-^-y^'+z^^^lZO, (3) And ^3_(_^3^y3_|.23^98o. (4) From (2) we have w-^-z^a — {x-\-y) (5) w^-^^xoz+z^ =a^—2a{x-\-y)-{-x^-]-2xy-^y^ Suppressing 2wz in the first member, and its equal 2xy in the second member, and adding (x^-\-y^) to both members, we have w^^x^^y''-\-z''=a''—2a(x+y)-\-2x^-\-2y-' That is 130=400— 2a(a:+y)+2a:2+2j^2 Whence, a{x+y) = l35-\-x^+y^ (6) By cubing (5), we have w^-\-32vz(w-^z)+z^=za^—3a^(x-\-y)-\'3a(x'\'yy —{x^-\-Sxy{x-\-y)-\-y^) By transposing, and observing that Swz equal 3xy, we have (w^-\-x^-\-y^-\-z^)-\~3xy{w-^x-\-y-\-z)=a^ — 3a^(a?+y) ^ -^Sa(x-{-yy That is, 9Q0-{-Saxy=za^—3a''(x-\-y)-j-3a(x-\-yy Dividing by 20, or by a, which is the same thing, and we have 49+3x7j=a'—Sa{x+7/)-\.3{x+yy Or, 3xy=35\—3a{x-^y)-{-3{x+yy Dividing by 3, and expanding the last term, gives xy=z 1 1 7 — a{x-\-7/) ^x^ -{-2xy-\-y^ Or, a{x+y) = \\7-^x--\-xy-{^/ (7) By comparing (6) and (7), we perceive that iry+1 17=135 Or, xy=lS (8) By the aid of (8), (6) becomes «2+2a:y+y2-|_135=20(ar+y)+3d Or, (x-{-7jy—20(x+y) = —99' Whence, (ar+y)— 10=±1 Therefore, x-\-y=\l, or 9 ALGEBRA. 13»' From this last equation and equation (8), we find x=9, or 6, and y=2 or 3. (4.) The sum of Jive numbers in geometrical progression is 31, and the sum of their squares, 341. What are the numbers? Am. 1, 2, 4, 8, and 16. Let a:, xy, xy^, xy^^ xy* represent the numbers. ^ ; Put a=31 lla=341 Then x-\-xy-\-xy^ -\-^y ^ -\-^y ^ =^ ( 1 ) And a;2-|-ar2y2_|.a2^4_|_^2y6_|_^2y8_iia ^2) By the formula for the sum of a geometrical series, we have -X y-1 (3) And ^''^'°~— =11«. (4) Dividing (4) by the square of (3), gives / ylO— 1 \ / y_l y_ll Factoring the first fraction, (y^+i)(y^-i)(y-i)(y-i) _n (y+i)(y-i)(y'-i)(y'— 1) « Suppressing common factors. Divide the first fraction, numerator and denominator, by (y-f-1) and the second fraction, numerator and denominator, by {y — 1), and we then have j^'+y^+y'+y+i 31 Clearing of fractions and reducing, Here we observe the same coefficients whether we begin to the right or the left of the expression. In such cases, divide by half the power of the unknown quantity. In this example, divide by y^ , ■«v.l««^*i.'.,.j»«» 1^ ROBINSON'S SEQUEL. Then lOy^— 2I2/+IO— il+i^=0 (6) y y"" Now put 21y+-=P (6) y Then 2/+^=^ y 21 Squaring, y2^2+l=^ 10_ 10P» 2^ 441 Comparing (5), (6), ?tnd (7), we perceive that 441 10P2— 441P=4410 F^—^-=b. By putting 6=441, Tlien P^-lp 4-il=1^4^= W±00>. 10 ' 400 400 ' 400 10y^+20+^=J^_ (7) By evolution, F — — =J '' 91) M 441-84} . 21-29 20 ^l 400 20 p__21 -21+21 •29 __21 • 60_105 20 20' ~Y' Or, P=II?L.^=_1^ 20 5 Now from (6), y4--=-> <^r — - w2__%=^__i. y^2, or 1 2 . 2 To find a?, we must return to equation (3), and in place of y, put in its value 2, and x=\. Hence, the numbers are 1, 2, 4, 8, and 16. (5.) There are six numbers in geometrical progression ; the sum of the extremes is 99, and the sum of the four means is 90. Whai are the numbers .^ Ans. 3, 6, 12, 24, 48, 96. Let X, xg, xy^ ^ &c. represent the numbers. ALGEBRA. 136 Then xi/^+x=99 (1) And ary^ -^-xy^+xy^ -[-a:y=90 (2) Dividing (1) by (2), gives Dividing numerator and denominator by (y+l)» then 'y^+y 10 Clearing of fractions, 10y4_iOy3_|_iOy2_iOy_(-io=lly3_|_iiy Or, 10y4— 21y3_|-.l03/2_2iy_|_io=0 This is the same equation as (5), in the preceding example ; therefore, as in that example, y=2. Now from equation (1), we have 33a:=99, or a; =3, the first number. (6.) The number of deaths in a besieged garrison atnounted to 6 daily; and allowing for this diminution, their stock of provisions was sufficient to last 8 days. But on the evening of the sixth day 100 men were killed in a sally, and afterwards, the mortality increas- ed to 10 daily. Supposing their stock of provisions unconsumed at the end of the sixth day, to support 6 men for 61 days ; it is requir- ed to find haw long it would support the garrison, and the number of men alive when the provisions were exhausted. Ans. 6 days, and 26 men aUve when the provisions were exhausted. Let x^= the number of men at first, and ji?= the amount of provisions consumed by each per day. By the question, we have a decreasing arithmetical series, whose common difference is 6*, and number of terms 8. For the amount of provisions at first in store, we have px for the first term, and {px — 42ji9) for the last term. Then the sum of the terms must be {^px — 168p). Hence, 8pa; — 168p= the amount of provisions at first. ^px — 90p=: the provisions consumed in 6 days. Whence, 2pa; — 78p=6jo-61, the provisions left. 136 ROBINSON'S SEQUEL. During the 6 days, 36 men died, and 100 men were killed ; therefore, at the end of the sixth day but 86 men were alive. Now the mortality increased to 10 daily, and at the end of n days their provisions were exhausted. Here we have another decreasing arithmetical series, the first term 86, the common difference 10, and the number of terms n. The last term is, therefore, 86— 10(7J— 1) First term 86 Sum of the terms (2-86— lOw+10)?? This number of men would require (Q6n — 5n^-]-5n)p amount of provisions. Therefore, (91nr-5n'')p=6p'61 Whence, Sw^— 91w=— 366 100»2—^+(91)2=(91)2— 366-20=8281— 7320=961 By evolution, 10/1—91= ±31 Whence, w=6, or 12^; but the last number cannot apply to the problem. (7.) Out of a vessel cmdaining 24 gallons of pure mne, a vintner drew off at three successive times a certain number of gallons, which farmed an increasing arithmetical progression, in which the diff'er- en,ce hetwem the squares of the extremes was equal ^o 16 times the •mean, and filled up the vessel with water after each draught, till he fmind what he last drew off, reduced to one-sixth of its original strength. Required the number of gallons of pure udne drawn off each time. Ans. 12, 8, and 3^. Let X — y= the number of gallons first drawn. X = the number at second drawing. and a;+y= the number at the third drawing. By the first given condition Axy==\Qx. y=4. (a? — y), or (x — 4)= the pure wine at the first drawing. He then filled the cask with water, making 24 gallons of liquid, which .contained (24— a;+4), or (28— a;) gallons of wine. ALGEBRA. 137 Liquid. Wine. Liquid. ^28— -a; W Now by proportion, 24 : 28 — x : : ar : ^^ 1—=. the wine taken out at the second drawing. Therefore, ("28 — x) — i — ZZz)—z= the wine left after the second ^ ^ 24 drawing, which is i Li — ZI_Z, by reduction. For the wine taken out at the third drawing, we have the fol- lowing proportion ; 24 • (24-a;) (28-a;) . . ^ , 4 . (24-a:)(28--a^)(a:+4) 24 ' 24-24 The number of gallons of liquid at the third drawing was (ar-[-4j^ but only one-sixth of it was wine ; therefore, (24— a ;) (28— a;) (a;-f-4) _ (a;+4) 24-24 6 Whence, (jj^) (24-.+4)^^ 4-24 Put (24— a;)=P; then P(P-|-4)=96 F+SL^ ±10 P=8 or— 12. That is, 24 — x=^^, or x=\Q ; the minus sign is not appUcable. Hence the wine at the first drawing was x — 4, or 12 ; at the 2d. (28— a;)a; ^^ g ^^^^ ^^ ^^^ 2^ ^+4 ^^ ^^ ^^^ answer.. 24 6 ' (8.) There are four numbers in geometrical progression such that the sum of the extremes is 56, and the sum of the means 24. What are the numbers r' Ans. 2, 6, 18, 64. Let X, xy, xy^ , and xy^ represent the numbers. Then by the conditions, a;y_|_a;2/2 _24=3a . (1) X J^xy^=5Q=:la {%) Dividing (1) by (2), and y(i+y)-3 i+y3 7 \ i 13a ROBINSON'S SEQUEL. Dividing numerator and denominator by (1+y), and we have y _3 1— y+y^ 7 Or, 3y2_i0y-|-3=0 If y is a whole number it is a factor of 3 ; that is 1 or 3. It is obviously not 1 . Try 3 ; then 3y-- 10+1=0, ory=3. This value put in (1), gives \2x=24. x=2. Another Solution. Let X, and y be the means, and ~, and ?L the extremes. y X Then ^+?^=7a (1) y X And a:+y=3a (2) From (1) a;3-f.y3_7(^^y ^3^ (2) cubed, x^+y^-\-Sxy(x+y)=27a^ (4) That is, 7axy-\-9axy=27a^ 16xy=:27a^ (6) Square of (2) gives x^ -\-2xy-\-y^ =: 9a^ 27a2 4xy= — -- By subtraction, x^—2xy+y''= rr_ 4 a;— y=-|a (6) From (2) and (6), we readily find x and y. SECTION VI. SOLUTIONS OF EQUATIONS OF THE HIGHER DEGREES. We shall take equations and solve them. The most difficult equations in the common popular books will be selected, begin- ning with newton's method of approximation. (1.) Given x^-\-2x^ — 23ar=70, to find one value of x. ALGEBRA. ia9 By trial we find that one value of x is between 5 and 6, nearer 6 than 6 ; therefore, let a=5 and y=: the remaining part of the root. Then a:=a-|-y. Expand, neglecting all the terms- containing the powers of y after the first, and we shall have a:3=: a3-|-3a2y-|-&c. 2a;2=2a2-j-4ay -\-kQ. —23x =— 23a— 23y By addition, In this last equation we observe Ihat a has the same powers and coefficients as x, and the coefficients to y may be found by the following Rule. Multij)ly each coefficient of x by its exponent, diminish each exponent hy unity, and change x to a. Now y= — it — .^ ? ~~— . Givinof a its value, 5, we have ^ 3a^^4a—23 ° y=|| = .l-[-. Now make a=5.1, and substitute again in the preceding formula, we have a new value of y. 2*629 Thus y= = .03. Now make a=5. 13, and substitute ^ 75-43 again, aad our new value of y will be .0045784-- Hence, a-j-y ora.=:5.134578+. (2.) Given a;4—3a;2-|-75ar= 10000, to find one value of x. By trial we find x must be near ten. Hence, put a=10 and x=a-^y. Then by the preceding rule Now make a=10 — .11=9.89. If we have the patience to substitute this value for a in the equation, we shall have a new value to y, true to 6 or 7 places of decimals, and of course a value to X to the same degree of exactness. ^ (3.) Given 3a;*— 35a;3— 1 la;^— 14a;+30=0, to find one value of X. By trial we find that x mu;st be near 12. Let a=12, and x:=a-\-y. Then by the rule * \ 140 ROBINSON'S SEQUEL. ^ 12a3— 106a2_22a— 14 5338 Hence, a;=12— .00112=11.99888. (4.) Given 5x^—3x^—2x=\560, to find x. We find by trial that one value of x is more tlian 7. Put «r=a-|-y, and a=7. Then by the rule 156Q+2a+3a--5a3 _ ^_ ^ ^^^ 15a^—Sa—2 689 ' Hence, ar=7.00867+ We give Newton's method on account of its simplicity in prin- ciple; it is easily understood, and can long be retained ; but its numerical application is laborious and tedious. A more modern, delicate, scientific, and practical method is Homer's, of Bath, England, first given to the world in 1819. The principle is that of transforming one equation into another whose roots shall be less in value by a given quantity, and again trans- forming that equation into another whose roots maybe still less, &c. The theory is fully explained in Robinson's Algebra, last two chapters. We give the following examples, commencing with quadratics. Some of the equations here solved are in the author's class book, and are numbered as in that work, pages 313 — 333. (6.) Given a;2-|-7a:=l 194. to find the values of x by Homer's method. We must first find an approximate value of x by trial ; but the inexperienced might be at a loss how to make the trial. We suggest this method ; separate the first member into factors thus : Here two factors of 1194 diflFer by 7. If the factors were equal, each one would be the square root of 1194. Now one of the fac tors is a little less than the square root of 1194, and the other a little greater ; but we want the less factor. In short, the square root of 1194 is a liUte below the arithmetical mean between x and (x-\-l). ALGEBRA. 141 This principle will do for a guide, when the coefficient of x is small in relation to the absolute term. The square root of 1194 is 34*5 ; from this we will subtract the half of 7, 3.5, and the ap- proximate value of X will be 31 ; therefore, r=31. a=7. T si r+s 38 31.2 1194(31.231 1178 1600 a+2r+5 s+t 69.2 23 1384 21600 a+2r+2s+^ 69.43 31 69461 20829 77100 The operation may thus now be 69461) carried on as in simple division ; 77100(111 69461 ,i^ 76390 69461 69290 and the figures thus obtained annexed to the portion of the root already obtained. Hence, a:=31. 231111. As the sum of the two roots is equal to — 7, the other root will be— 38.231111. (7.) Given ic^— -2 l;r=2 1459 1760730, to find theA^alues of x. Conceive — 21a; not to exist ; then the value of x will be the square root of the absolute term ; but this term has six periods of two figures each, and the superior period is 21, the greatest square in this is 16, root 4 ; hence, x must be over 400000 ; take r=400000. a=— 21. -^+r =399979 r+5 460000 — a-|-2r-f-5 s-{-t 214591760730(400000=r 1599916 60000=5 859979 63000 5460016 5159874 3000= t 200=u 14S ROBINSON'S SEQUEL. -^+2r+25+i: 922979 3200 3001420 2768937 60±=t) l=w 926179 250 2324837 1852368 926429 51 4724793 4632145 ar=46325L 926480 926480 926480 A.S the algebraic sum of the two roots must make 21, the other i-oot must be —463230. We can find the negative root directly as well as indirectly, by taking r minus ; then s, t, u, v, &c., will be minus. The divisors and quotients both being minus, their products will be plus. The following example is in direct contrast to the preceding. {a) Given x^ — 32141a;=131, to find the values of x. Here the coefficient of the first power of the unknown 'quantity is large, and the absolute term comparatively very small. The factors X and {x — 32141) are so very unequal, that a resort to the square root of the absolute term for an approximate value of x, as in the preceding equation, would be useless. In this, and in simi- lar cases, we can obtain an approximate value, by conceiving the absolute term to diminish to zero. Then «2-^32141a;=0. This equation will be verified by putting a;=0, and ar=32141 ; and from this consideration we conclude that one value of x in our equation must be very small, and the other, over 30000. Hence, put riti30000, and the solution is as follows. 131(30000=r -.^ —2141 —64230000 r-H 32000 +64230131 2000=* — a+2/--[-5 29859 59718000 .+/ 2100 4512131 100=< — the true value is 2. (A) Given a;^ — %^^= — 1, to find the approximate values of x. Ans. a:=2j^ — 2^6=^ nearly ; 5 is the true value. The other value is ^. When a and b are equal, or nearly equal, in the equation x^dzax=±.b, I it is most difficult to find the value of r, or the approximate val- ue of x\ Having now sufficiently explained the means of finding r in the different cases, we resume the appUcation of Horner's method of operation. (8.) Given Ix^ — 3a;=376, to find one value of ar. ALGEBRA. 145 Or x^—^x=^^. TxLtx==}y. (Art. 166.) Then ^—^=?IA. Or, y2—3y=376X 7=2625. 49 49 7 In this equation we perceive that y must be more than the square root of 2625, that is, more than 50. Hence, put r=50. rs t T+S 47 52 2625(62.766+ 235 99 2.7 275 198 1017 7700 75 7119 10246 68100 51225 6875 Hence x= — '- Zt=7.+ 7 ^ (10.) Given x^ — y3ya;=8, to find x true to seven places of decimals. Put a;=-^ ; then a;^=-^-^, and the given equation is trans- 11 121 ^ ^ formed into ^ — _^=8. 121 121 Or, y2_3y_9g8. .,-.% It is obvious that y must be between 30 and 40 ; therefore, r=30. -«+r 27 32 968 ( 30. 810 _-a+2r+s s+t b9 2.6 158(2.64883626 118 61.6 64 4000 3696 6224 48 30400 24896 10 146 ROBINSON'S SEQUEL. 62288 660400 88 498304 62|2|9|6|8 5209600 4983744 226856 186890 38966 37376 1590 1245 346 After the 4th decimal, the operation was carried on by con- tracted division, giving ^=32.64883625. But x= ^r of y ; therefore, ic=2.96807602. Put ar=i- ; then «* =-t— — , and the equation becomes ^R (36)2 ^ (11.) Given 4a;^-j-^ar=i, to find one value of x. This equation is the same as x^-\-:J^x=^-^. ; thena;*=J''' 36 (36)2~(36)2 20 Or, y^+7y=G4.8 It is obvious that the value of y is between 6 and 6 ; therefore r=s6. a=7. «+r 12 64.8 ( 5.277812946 r+s 5.2 60 a+2r+s 17.2 480- 8+t 2.7 544 17.47 13600 77 12229 17647 137100 78 122829 176648 1427100 1404384 ALGEBRA. 227160 175548 147 17|6|614 51612 35108 16504 15788 816 702 114 Whence, ^=6.277812*946, or —12.277812946, And a;=0. 146605915, or —0,341050359, (12.) Given iX^-]--5X=^j, to find one value of x. Put x=l 5 4y_28 2 , 4a: 28 ' 5 33 Then r_+ 25 ' 25 33 Or, 5^2_j_4y_7_oj)^2l,21212121 r=:3. a=4. u+r 7. 21.21212121, &c, (3.021186235, r+s 3 21 a+2r+8 10.0 2121 .+t 02 2002 1002 11921 21 10041 10041 188021 11 100421 100421 8760021 18 8033824 10042|2|8 626197 602536 23661 20084 3577 148 ROBINSON'S SEQUEL. y=3.021 186235. x =0.60 4237 257. (13.) Given 116 — 3x^ — 7.c=0, to find one value of x. ^3 3 3 Then. 3^+!3?=ll^ 9 9 3 y2_j_7y=:345. r=16. a=7. a+r 22 345(15.40158. r-{s 15.4 330 a-{-2r-^-s 37 A 1500 s+t 40 1496 aJ^2r+2s-\-t 37|801 40000 37801 2199 1890 309 302 ^=15.40158. ar=5.13386. We will now apply Horner's method to cubic and the higher equations. For the theory, we must go to the class books. CUBIC EQUATIONS. The first example here, is the third on pag,e 319 of the author's class book. Hence, (3.) Given x^^2x^ — 23a;=70, to find one value of x. By trial we find x must be a little over 5 ; therefore, r=5, A=2, JB=— 23, iV=70 B —23 r(r+A) 35^ r 8t 1st Divisor 12 ^ 70 ( 6.134 r» 26j 60 ALGEBRA. "lit B' 72 10000 5(5+3r+^) ATi\ 7371 2d Divisor 7371 \ 2629000 »2 1 J 2276697 B" 7543 352303000 *(^Q-\-t)t ...4599 305649104 3d Divisor 758899 46653896 e 9 B'" 763507 615 76 4tli Divisor 76412276 16^ Common division will give three or four more figures to perfect accuracy. (4.) Given x^ — 17a;^-|-42a;=185, to find one value of x. Here ^=—17, ^=42, iV^=185, and we find by trial that x must be between 15 and 16 ; therefore, r=15. B 42 r{r+A) —30] r st 1st Divisor ~12 ^ 185 ( 15.02 H 225 J • 18Q B' 207 5000000 *(s+37-+^) __0] 4 154008 2d Divisor 207 \ 2Q11) "845992 (407 «2 oj 8298 J5" 207 16192 t{^Q+t) 7004 1 4539 3d Divisor 2077004 1653.0 Hence, ir=15.02407+ (5.) Given x^-\'X^ =500, to find one value of x. Here A=\, B=0, r=7. *Q represents the root as far as previously determined tSO ROBINSON'S SEQUEL. B r(r~{-A) 56^ 1st Divisior 56 [ 600 (7.61 r« 49J ' 392 ^ 161 108 (Sr-\-s-\'A)s 1356 104736 2(1 Divisor 17466 3264 8' 36 1887181 18848 ) 1376819 238 1 Sd Divisor 1887181 Continue by common division. 1 1889563 (6.) Given x^-\-10x^-{-5x=2600, to find one value of ar. Here ^=10, B=5, r=ll. B 5 2600 ( U . 006 r(r+A) ^3J ^ 2596 1st Divisor 236 \ 4 r^ j 2^ J 3529188216 B' 588 470811784 (3E-\-u)u 198036 Continue by common 4th Divisor .588198036 division. (6.) Find one value of a: from 5x^ — 6x^-{-3x= — 85. As the result is negative, we will change the second and every alternate sign of the equation, (Art. 178), and find a value of x from the equation 5x^ -\-6x^ -\-3x=85. Use the formula of (Art. 194). c=5, A=6, -5=3, and by trial we find r=2. r s B 3 85(2.1 {cr+A)r..,. .32^ 70 1st Divisor 35 f 16 or^ 20 J 9.066 "87 5.935 {^+C8+A)s ,_ZSb Continuing this, we shall find 2d Divisor 90.65 the value of x to be 2.1 6399+, ^* 5 and its sign changed will be the 94.35 value of x in the original equa. ALGEBRA. '161 (7.) Find x from the equation \2x^-\-x^ — 6a;=330. Here c=12, ^=1, jB=— 6, r=3. B ....—6 (cr+A)r .ml »• «^ 1st Divisor 106 f 330(3.036 cr^ 108 J 31£ 5'. 326 ir , (3ci2+cO^ 11208 97836 24 \, 3d Divisor 3261208 2216376 c^ 108 Continue thus. 3272624 In the same manner perform 8 and 9. (Art. 196.) Page 323. (3.) Extract the cube root of l-352-606-460'694'688 For the sake of brevity, take r=ll, in place of 1. 1st Divisor 121 rst' B'=^r^ 363 1-352-606-460-694-688 ( 110692. {pB+t)t 16525 1331 2d Divisor. . . .3646526 21 605 460 25 1 8 232 625 3663075 3 372 835 594 {ZB-\-u)u,,: . 298431 3 299 453 379 3d Divisor...: 366605931 73 382 215 688 81 73 382 215 688 366904443 (3i2+2;> 663544 36691107844 (4.) By the table of cubes which run to 8000, we perceive at once that r in this example is 17. 1st Divisor 289 r s t 3r2 =B' 867 6382674 ( 175.2 i {^RJ^s)s 2576 4913 2d Divisor 89276 469674 26 446376 IdS! ROBINSON'S SEQUEL. 91876 (SB+ty 10604 3d Divisor 9198004 4 9208612 23299000 18396008 4902992 Complete another divisor, then continue as in simple division. (6.) Find x from the equation a;3 = 16926.972604. For the sake of brevity, let r represent the value of the two superior digits. That is, let r=26. Ist Divisor.... 626 £'=3r^ 1876 (3r+s> 761 2d Divisor... 188261 8^ 1 B" 189003 (SM+ty 462 16 3d Divisor.... 18946616 36 18990768 7648 4th Divisor 1899084348 16926.972604 ( 26.16002649 16626 301 972 1 88 261 113 721604 113 673096 48408 000 000 Common division. 189.91 ) 48408 ( 2649 37982 10426 9496 931 769 172 It is not important to show a solution to the remaining exam- ples under this article. From Robinson's Algebra, page 324. (1.) Given x^+x^+x^ —X =600, to find one vah By trial, we find r=4. 1 1 1 —1 = =600 ( 4. 4 20 84 332 6 21 83 168 4 se 228 . ALGEBRA. 1 9 67 311 4 13 17 62 109 A new transformed equation is 5^+17* 3^109s^+31l5=168. IBS '' 3 11 — ^ 17. 109. 311. =168.(0.4 .4 6.96 46.384 142.9536 17.4 115.96 357.384 25.0464 4 7.12 49.232 17.8 123.08 406.616 4 7.28 18.2 130.36 4 18.6 The next transformed equation is t* +18.6^3 _|_130.36^2 _|_406.616^ =25.0464(0.06 .06 1.12 7.888 24.8602 .1862 18.66 .06 131.48 1.12 414.504 7.956 18.72 .06 132.60 1.13 422.460 18.78 .06 133.73 18.84 Several other decimal places of the root may be found by the following division, — the powers of u above the first being consid- ered valueless, 423. ) 0.1862 ( 0.00044019 1692 i 1700 1692 800 Hence the root is 4.46044019. 164 ROBINSON'S SEQUEL. (2.) Given x^—5x^-}-0x''-^9x=2.Q, to find one value of x, r=0.3, found by trial. —6. 0.3 —4.7 +0. —1.41 —1.41 +9. —.423 8.677 r =2.8 ( 0.3 2.6731 .2269 .3 —1.32 —.819 —4.4 —2.73 7-768 .3 —1.23 —4.1 .3 —3.96 —3.8 54_3.8s3_3.96s2-|-7.768«=0.2269. -3.8 —3.96 +7.758 =0.2269(0.02 .02 —.0766 —0.081 0.16364 —3.78 —4.0366 7.677 .07336 .02 .076 .082 —3.76 —4.11 7.696 The remaining figures may be found by division, thus : 7.6 ) .07336 ( 0.00978 676 686 625 610 Hence the approximate value of x is 0.32978. (3.) Given x* — 9x^ — 1 la:*— 20a;= — 4, to find one value of x. r 1 —9. .1 —11. —.89 —20. —1.189 =-4(.l —2.1189 —8.9 .1 8.8 .1 —11.89 —.88 —12.77 —.87 —21.189 — 1.277 —1.8811 —22.466 —8.7 .1 —13.64 1 —8.6 —13.64 —22.466 —1.8811 (.07 &c. ALGEBRA. 166 -8.6 —13.64 —22.466 =—1.8811 ( .07 .07 —.597 —.9966 —1.642382 —8.63 —14.237 —23.4626 — .238718 .07 —.592 —1.0380 —8.46 —14.829 —24.6006 .07 —.587 -8.39 —16.416 .07 -8.32 1 —8.32 —15.416 —24.6 =—0.238718(0.009 .009 —.0748 —.139 — .2217647 ■8.311 —15.4908 —24.639 — .0169633 9 —.0747 —.140 —8.302 —15M55 —24.779 9 —.0746 —8.293 —16.6401 9 -8.284 —24.78 )— .0169633 ( 0.900684 14868 20953 19824 11290 9912 13780 Whence, a;=0. 179684, nearly. (4.) Given x^=z5000, to find one value of a?, or we may say find the fifth root of 5000. Here all the coefficients are zero, except the first, and r=5. 1^ ROBINSON'S SEQUEL. r 1 =6000 ( 6 6 26 125 625 3126 5 26 126 625 1876 5 60 376 2600 10 76 500 3126 6 75 750 16 150 1250 6 100 20 250 5 26 1 26. 250. 1250. 3125. =1875. ( .4 .4 10.16 104.06 541.624 1466.6496 26.4 260.16 1354.06 3666.624 408.3504 .4 10.32 108.19 584.900 25.8 270.48 1462.25 4251.524 .4 10.48 112.38 26.2 280.96 1574.63 .4 10.64 26.6 291.60 .4 27.0 1 27. 291.6 1674.6 4251.52 =408.3509(.09 .09 22.38 28.26 144.26 395.6202 27.09 313.98 1602.86 4395.78 12.7307 .09 24.46 30.46 146.998 27.18 338.44 1633.32 4542.778 Now by common division, 4642.7|78 ) 12.7307 ( .0028, nearly. 9.0866 3.64i52 Whence, a;= 5.4928, nearly. ALGEBRA. 157 It is practically useless to solve such equations as the preced- ing, because solutions are so simple and direct by logarithms. (5.) Given x^ = (-^) ' , or a:^ = ^! , to find one value of X, +1/ x^-{-2x^ + l a;5-|-2a;3+a;=64. r=2 2 1 =64 ( 2. 2 4 12 24 50 2 6 12 25 14 2 _8 28 80 4 14 40 105 2 12 52 6 26 92 2 16 8 42 _2 10 10. 42. 92. 105. =14(0.1 .1 1.01 4.3 9.63 11.463 10.1 43.01 96.3 114.63 2.537 .1 1.02 4.403 10.07 10.2 .1 10.3 .1 44.03 1.03 45.06 1.04 10.4 .1 46.1 10.5 10.5 .02 46.1 .21 10.52 .02 46.31 .21 100.703 124.70 4.506 105.209 105.21 124.7 =2.537(0.02 .926 2.1227 2.536454 106.136 126.8227 .000546 .93 2.141 10.54 46.52 107.066 128.9637 128.96 ) .00054600 ( 0.000004+ 51584 Whence, a?=2. 120004, nearly. By an exact solution, the last figure would be 3> in place of 4. 158 ROBINSON'S SEQUEL. Observation. When we observe that the sum of the coeffi- cients in any equation is zero, we may be sure that unity is one root of the equation. Then we can depress the equation one degree by division. For example, we are sure that the equation has a root =1, because 1 — 7-|-7 — 1=0 ; and 1 put for x will neither increase nor decrease any of the terms. The equation x^-^-'ix^ — Ix^ — 8a?-}-12=0, also has one root = 1, for the same reason. lii Bland's problems I find the following problem, (page 426). One root of the equation x'^ — hx^ — ^a;-|-6=0, is 5; determine all the roots. Here we perceive that another root is 1 ; therefore, the equation is divisible by (x — 5) (if — 1), or by x^ — 6ar-|-6 ; thus x^^QxJ^b ) a;4— 5a;'— ar+5 ( aj^+ar-f-l a:4_6a;3_|_5a;2 x^ — ^x^-\-bx x^—Qx-\-b a;2_6a?-f5 • Whence a:2-|-a;+l=0, and x— i(±V^^— 1)- To solve some of the following problems, it may be necessary to see how the roots combine to form the coefl&cients. We shall consider all the roots as positive; and represent them by a, 5, c, c?, e, &c. Then an equation of the second degree will be represented by a;2__^^_^j_^0. (1) An equation of the third degree, by x^-^-a —5 x^-\-ab 4-ac x-^abc^O> (2) --cb An equation of the fourth degree, by x^ ALGEBRA. rc-|-a5cc?=0. 169 —a ;r3- -aS x^ — abc —5 -(m -^bd — c _ -ad — acd — rf - -cb -—cbd _ -cd H -bd (3) Now^ let A=a+b-\-^+d. £=:a(b+c+d)-[^(c+d)-\-cd. C=a(bc+bd-{-cd)-{-cbd. J)=abcd. Then the equation of the fourth degree will become x*—Ax^-\-Bx^^Cx-]-D=0. (4) This equation multipled by (x — U), gives the representative of an equation of the fifth degree, as follows : -£JD=0 (5) x^—A —E a;4+ B \-EA ^3_ C\x^J^ J) —EB\ -\-ED In these equations we observe that the coefficient of the high- est power of a: is 1 ; and that the coefficient of the next inferior power is the sum of all the roots, with their signs changed ; — ^the absolute term is the product of all the roots. EXAMPLES. (1.) The roots of the equation 6a:*— 43a;3-j-107a;2— 108a;+36=0, are of the form a,b,~, ~ , find them. a b Divide the equation by 6, so that it may compare with the fundamental equations (3) or (4), then Now if the roots are of the form a, 5, _, -, we may take these a b symbols to represent the roots. Then a ^+^!+^=13, and a6=6. ^ ab 6 That is, 6(«+5)-l-a2+52 =43. By adding 2a5=:12 to the last equation, we have 160 ROBINSON'S SEQUEL. Whence, a-\-b=5 ; but ab=6. a=2. b=3. And the roots are 2, 3, f, |. (2.) The roots of the equation a;4 _i0a;3 +352:2— 50a;+24=0, are of the form (a+1), (a—1), (6+1), (5—1), find them. Here 2a+26=10. (a^— 1) (6^— 1)=24. That is, a+b=5. a^S^—a^— 62+1=24. But 2a6+a2+62==25. By addition, a'b''-\-2ab-Jf-l=49. a6+l=7, or a6=6. Buta+6=5; hence, a=2, 6=3, and the roots are 1, 2, 3, and 4. The roots of the following equations are in arithmetical pro- gression ; find them. 1." a;3_6a;2— 4ar+24=0. 2. x^—9x^+^3x—ie=0: 3. x^—'6x^+llx—6=0. 4. a;4— 8a;3+14a;2+8a;— 15=0. 5. a;^+a;3— lla:2+9ar+18=0. We work out the fifth and last example ; it being the only difficult one. Let (a— 36), (a — 6), (a+6), and (a+36), represent the roots ; then 4a=— 1, and (a''—b^)(a^—9b^)=lS. That is, a^— 10a2&2_|_96^ = 18. Buta2=-i.V, «* = 2-k- Therefore, -1 1^+96^ = 18. 266 16 ' _i___1062+14464=288. Or, 1446t— 1062=288— yV- Add j''^^ to both members to complete the square. Then 1446^--1062+yV_=288yV4=^Hf^ By evolution, 1262— y5_=2_o_3^fi|&oi Whence, 62=1.449209 ALGEBRA. 161 And 5=1.20383. But a=— 0.25. Therefore, a— 35=— 3.86149, log. 0.586761. a— 5=— 1.45383, log. 0.162515. a+ 5= +0.95383, log. —1.979458. a+35= +3.36149, log. 0.526510. Log. 18, 1.255244, nearly. The sum of these numbers is — 1, as it ought to be, and the product of the roots is 18. Negative numbers have no logarithms, because there are in fact no such numbers. The product of several numbers is numerically the same, whether the numbers be positive or neg- ative ; therefore, we took the logarithm of each root as though it were positive. The product in every case will be positive or negative, according as the number of minus factors is even or odd. The roots of the following equations are in geometrical progres- sion : find them. 1. a;3— 7:r2+14;r— 8=0. 2. a;3— 13a;2_|_39_^_27^0. 3. a;3— 14ic2+56a:— 64=0. 4. a;3—26a:2+156a;— 216=0. Let a, ar, and ar^ represent the three roots. Then a-\-ar-\-ar^=2Qy and a^r^=:9,\Q. From these equations we find a=2 and r=3, and the roots sought are 2, 6, and 18. Problems like the preceding are impractical, because there is no natural method of finding the form of roots, a priori, and to give the form, is nearly equivalent to giving the roots themselves. RECURRING EQUATIONS. A recnrrrng equation is one in which there is a symmetry among the coefficients ; — the terms which are equally distant from the extremes, have the same numerical coefficient. For example, 11 162 ROBINSON'S SEQUEL. is a recurring equation, for the coefficients are 1, — 11, +17, which recur in the inverse order 17, — 11, 1. Here it is obvious that x= — 1 will satisfy the equation ; and if we change the second and every alternate sign, then x= 1 will- satisfy the equation ; that is, 1 is a root of the equation x^+nx'-\-17x^^l7x^—Ux—l=0. Here the sum of the coefficients is 0, and consequently x=1 must satisfy the equation. Now we arrive, at this general truth : A recurring equation of an odd decree will have either — 1 or +1 /or one of its roots. It will have — 1, if the corresponding coefficients have like signs — and -|-1, if they have unlike signs. Hence, every recurring equation of an odd degree will be di- visible by (x-{-\) or (x — 1), and can thus be reduced to an equa- tion of an even degree, and one degree lower than the original equation. Every binomial eqtcation is also a recurring equation. Every recurring equation of an even degree above the second, can be depressed one half by the following artifice : Take the equation, x^+5x^-\-2x^+5x+l=z0. Divide every term by the square root of the highest power, in this example by ic^ ; then Then • (.»+^)+5(.+l)+2=0. Put x+l=y; thenx^+24-—=y\ X x^ Whence, y^-^oyz=0, an equation of only half the degree of the given equation. This can be verified by y=0, or y= — 5.. ALGEBRA. }6S Therefore, ar-|-_=0, or x4-^ = — 5. X X Whence, a?=±V— 1, or x=\{—b^J1\) Find all the roots of the equation x'= — 1=0, One is obviously one root, therefore divide by x — 1=0. Then x'-^x^-\-x''-\-x^\=^. Divide by a:2. x x^ Put :c+l=y; then x^ +2+^^=1/^ X x'^ Whence, y^-\-y — 1=0. y=-i+iV5, or y=-i_i75. ^ Put 2a=_i-(-i^5; then — (l+2a)=— ^— i^5. Now x-\--,=2a, and x-{- = — (l+2a) X X From the first, x=a-\-jJa^ — 1, or x=a-^Ja^ — 1 That is, x=\{J^—l+J^^^^l0^2j5) Or, ^=J(^5— 1— VZTc^-SVS) Taking the second. Or, ^•=— KV^+i+V—i^+VsJ Given (^'^+1) (.^'- + l) (^+l)=30.r^ to find the values' of x. Multiply as indicated, the product is a; B -(-a;5 +;c ^ +22- 3 -[-a;2 -[-.r-{- 1 =30^-3 Dividing by x^, and X x^ x^ 0. (.3+^.)+(.+^)V(.+l)=30 Put (-+;)=y. Then (.3+^)+3(.+»)=,» 164 ROBINSON'S SEQUEL. Whence, y3_3^_|_y2_|_y_3Q Or, y3_|_y2_2y^30. The first attempt at solving this by Horner's method shows us that r is 3 exactly ; that is, y=3. Then ar+i =3, and rr=i(3db75.) The other values of y are imaginary. Given {x-\-y) {xy-{-\)=^\^xy (1) and (a:2-j-y^)(a;2^2_(_ij_.2O0^2y2 ^g^ to find the values of x and y. Multiply as indicated, then x''y-\-xy^-^x^y=nxy (3) a:4?/2_j_^2^4_j_^2_|_^2^2082;2y» (4) Divide (3) by xy, and (4) by x^y^ , then ^+2/+-+-= 18 (5) y X And a;2-|-y2+_L-|-Jl.=208 (6) Now put P=^a;+1Y and q=(vJ^\. Then P_[-§=18 (7) And P2_|_^2^212' (8) From (7), P2_|.2P^_j.^2^324 (9) 2P^=112 (10) (10) from (8) gives P~Q=±zlO Whence, P=14 or 4, and ^=4 or 14. That is, a;4-l=14, or 4. y+-=4, or 14. X y Whence, x=(7dt4jS) or (2±V3). 7j=(7^z4j3) or (2^:173). We conclude this subject by the following equations : Given 6a;5+ll«*— 88a:3— 88a;2+lla:-|*6=0, to find n^ j^qq^^^ This equation necessarily has one root equal to — 1 ; therefore we divide by (x-\-\), and we find ALGEBRA. 165 Now 5x^^6x—94+--\-^=0. X x^ Or. 6(.= +^)+6(.+l)=94. If we put x-\--=7/, then x^-{-—=y^^2. X x^ And 5(y2— 2)+6y=94. ^^5 5 2 1 6y 104 y+i=±V- y=4, or— V a;+l=4. Whence x=2zizjS. X 1 26 Or, x-\--=: — , whence x= ^, or — 5 ; and the five roots X 5 are —1,-5, i, (2-j-^3), and (2— ^S.) Given -S ('^6i"ll^"^Ql^''M^^' 1- to find at least one of the* values of x and y. By division, the equations may be reduced to ^='+^=2/+- (1) And 2,3+J_9(,+i) (2) Now put a;+-=P, and y4-_=^. X y Then a;3+J_=P3_3p^ and 2/'+— =^='— 3$. Substituting these results, and (1) and (2) become P^—3F=Q (3) Q3_sQ=9P (4) Assume F=nQ ; then (3) and (4) become 164 ROBINSON'S SEQUEL. And g3_3^^9^^ Dividing by Q^ And ^2_3^9„ , (g^ From (5), n^Q^=i^n+\ (7) From (6), w^ ^2^(3^+1)3^3 (8) Dividing (8) by (7), gives 3n3 = l. Whence, 9?^=3(9)3 ; and substituting this in (6), and we Lave ^2^3(9)3-1-3. Or, ^=\/3( 9)^+3. Having the value of n and Q, we have F=^nQ. The values of ^ and P will give us x and y. ^7^5. a:=^(V3+3)^'+(V3— 1)^ SECTION VII. mDETERMIIf ATE ANALYSIS. Preliminary to this subject it is proper to call to mind a few facts in the theory of numbers ; for the Indeterminate and Pio- phantine analysis is but an application of that theory. (1.) The sum of any number of even numbers is even. (2.) The sum of any even number of odd numbers is even* (3.) The sum of an even and an odd number is odd. (4.) The product of any number of factors, one of which is even, will be an even number ; but the product of any odd num- ber is odd ; hence, (5.) Every power of an even number is even, and every power of an odd number is an odd number ; hence, (6.) The sum and difference of any power and its root is an even number. For the power and its root will be both even, or both odd, and the sum or difference of either, in either case, is an even number. ALGEBRA. 187 PROPOSITIOJS-S. 1 . If an odd number divide an even number, it will divide the half of it. Every number is either odd or even — even numbers are in the form 2w, and odd numbers are in the form 9.n'-\-\. Now by- hypothesis, let -=0 and q a whole number. Then 27z=^(2//+l) It is obvious that one factor in the second member is odd, therefore the other factor q, must be even, otherwise the product 2w could not be even ; hence, q may be expressed by 2q\ and 2n=2g'(2n4-l). Then —~ ==q\ and the odd number ^ ^ ^ ^ 2n+l ^ (2n'-\-l) divides half the even number 2n, which was to be dem- onstrated. 2. ff a numher p divide each of two numbers a and b, it mil divide their sum and difference. a b , _z=q, _=g . F P That is q and q\ the quotients, are whole numbers by hypoth- esis. Now by addition we have ^ ' =q-\'q', and by subtrac- P .. a — b , tion, — —z=q — q. P But the sum of two whole numbers is a whole number; there- fore, ( ^i- j is a whole number : and it is obvious that^ ) is also a whole number. Q. E. D. 3. If two members are prime to each other, their sum is prime to each of them. Let a and b be the two numbers prime to each other, (a+^) their sum, is prime to each of them. 168 KOBINSON'S SEQUEL. For by the last proposition if («+&) and a have a common divisor, their difference b must have the same divisor ; but b is not divisible by a by the supposition ; therefore, if two numbers, &c. Q. E. D. Corollary. In the same manner it may be demonstrated that if a and b be prime to each other, their difference (a — b) will be prime to each of them. , '-^^L 4. If two numbers arej^rime to each other y their sum and difference will have the common measure 2, hut no other, or their sum and dif- ference mil be prime to each other. Let a and b be prime to each other, their sum is (a-|-6), and difference, (a — b) ; and if these numbers have a common divisor, their sum 2a and difference 25 will have the same ; but the only common divisor to 2a and 26 is 2. If one of these numbers, a or b, be even, and the other odd, then (a-j-5) and (a — b) are both odd and prime to each other. For if («+5) and (a — b) are not prime, let them have the com- mon measure n. Then by proposition 2, n will be the common measure of their sum and difference ; that is, the common meas- ure of 2a and 25 ; but the only common measure of these num- bers is 2 ; therefore, (a-|-5) and (a — b) are prime to each other, or have the common measure 2. Q. E. D. 5. If two numbers a and b be prime to each other, b being the greater, then b may always be represented by the formula |)=;aq-|-r, in which r is less than &, and prime to it. The formula arises from the actual division of b by a, the re- sult is q, the integer quotient, and the remainder, r ; that is, a a Multiplying this equation by a, gives the formula ; r is neces- sarily less than a, if we suppose q to be the greatest quotient. We are i\ow to sbow that r and a are prime to each other. If they are not prime to each other, they have a common measure. ALGEBRA. 169 Let us suppose a common measure and reduce the fraction - by a it, givmg - ; then the equation becomes a a a a' But by hypothesis the fraction - is irrec?wa6^^. Yet admittinor a ° a common measure to -, we have the reduced fraction ^~r^_ which is absurd ; therefore, a commom measure to - is inadmis- a sible, and r and a are prime to each other. 6, If ayiy niimher he prime to each of two others, it will be prime to their product. Let c be prime to both a and b ; then we are to show that c is also prime to ab. * By the hypothesis _ is an irreducible fraction ; multiply this fraction by h, and we shall have — .m Now if this last fraction is reducible, some common measure must exist between b and c ; but by hypothesis there is none ; therefore, ab and c are prime to each other. Q. E. D. Corollary 1. If a^=b, then -1-= — ; and if c is prime to a, c c it is prime to a^ , a^ , and any power of a. Corollary 2. If c is prime to any number of factors as a, b, d, e, &c. it will be prime to their product. 7. If two numbers, a and b, be prime to each other, then mb divided by a, and m'b divided by a, will have different remainders for all values of m less than a. Let us admit that the two operations in division will produce % 170' ROBINSON'S SEQUEL. the same remainder, — then by performing the division we shall have mh . r — =9'+- * a a And — = q-\-- a a By subtraction, -( m — vi j=Q — g' •Or, J= g-g' am — m But by hypothesis, . is irreducible, at the same time (m — mf) a is less than a, which is absurd ; therefore, the two divisions can- not have the same remainder. 8. If two numbers, a and b, be prime to each other, the equation ax — by= ±1, is always jwssible in integers. That is, positive in- tegral values of x and j mag be found which will satisfy it. By the preceding proposition — =q-\-- , and as r may be of a a any value less than a, according to the magnitude of m, r=a — 1 is possible. Whence, nih=aq-\-a — 1 Or, mb-\-\ = {q-^\)a Now let y=w, and a?=(5'-|-l) ; then - ^ by-\-\=ax Or, ax — hyz=\ But w is a whole number, therefore its equal y is a whole number, and ($'+1) is a whole number, and consequently its equal a? is a whole number ; that is, ax — by= 1 is possible, x and y being whole numbers. ALGEBRA. 171 We now come more directly to the indeterminate analysis. For a perfect and definite solution of a problem, there must bo as many independent equations as unknown quantities to be de- termined ; and when this is not the case, the problem is said to be indeterminate. For instance, a;-f-y=20. Here x may have any value what- ever, and ^he equation will give the corresponding value toy, and the number of solutions may be infinite; but if we restrict the values of x and y to whole numbers, then only 19 different solutions can be found; for x may be equal to 1, or 2, or 3, &c., to 19, and y will equal the remaining part of 20. In some cases, the number of solutions is unlimited or infinite. ax — hy=c, represents a general equation of the kind, and a solu- tion gives x= ' "^ in which y may be any whole number what- a ever that will make (c-(-5y)=a, or any multiple of a ; but num- berless such values of y may be found, and consequently number- less values of x. N". B. Such equations are generally restricted to the least values of X and y. Equations in the form ax-\-by=c, may be very limited in the number of their solutions, — may have only one solution, or a solution may be impossible, when x and y are restricted to whole numbers. A solution gives x=-^ ^ . a Now if c is very large, and b and a small, y may take a great number of integral values, before the numerator becomes so small as not to be divisible by a. If we make y=l, and then find that i \ is a proper frac- tion, a solution is impossible in integers. The equation ax-\-hy=^c is always possible in integers, when e is greater than {ab — a — b), and a and b prime to each other. The equation ax-\-by=c, is possible sometimes, or rather with tome numbers, when c is less than (a5 — a — b). For example, 7a:-{-13y=71, is impossible in integers, because (7-13 — 20) is 172 ROBINSON'S SEQUEL. not greater than 71. But 7x-|-13y=27, is possible, giving rr=2, and y=l. ' That is, 7x-\-13i/ iput equal to any whole number greater than 71, will admit of a solution in integers; and put equal some numbers less than 71, will admit of a solution. In all these equations a and b are supposed to be prime to each other. If they have a common divisor, that same divisor must divide c, .or a solution is impossible in integer numbers. For if «ir4-Jy=-, it is obvious that ax is a whole number, also n by is a whole number, and the sum of two whole numbers can never be equal to an irreducible fraction, as the preceding in- dicates. For a particular example, 6a;-|-9y=32, is impossible in whole numbers. Dividing by 3, and 2.c-l~%= V- ^^ ^ ^^ ^^ ^^ ^ whole number, 2a; must be a whole number, and 3y must also be a whole number ; but it is impossible for any two whole numbers to be equal to \^ . In cases where solutions are possible, our rules of operation rest entirely on the following facts : 1st. A whole number added to a whole numtber ^ the sum is a whole number, 2d. A whole number taken from a whole number, the remainder is a whole nurriher. 3d. Multiply a whole number by a whole number, and the product is a whole number. EXAMPLES. ( 1 .) Given 3a:-j-5y=35, to find the values of x and y in whole numbers. 35— 5y 3 Because x must be a whole number, the fractional form ^ 3 must be a whole number, or (11— y)+ — ~- must be a whole number. But (11 — y) is obviously a whole number ; therefore, the other part, ( -ZIJ^. ) must be a whole number also. ALGEBRA. 173 Again, as y is a whole number, -Z is in fact a wliole number, o which added to { -Zl-^ ), and the sum is -^-, a whole number. V 3 / 3 In this last expression the coefficient of y is reduced to 1 under- stood, and the operation had that end in view. Put this last expression equal n ; that is any wliole number, or rather some whole number. ^+l=n, 3 Or, y=3^^— 2. In this last equation we can take n=^0, 1, 2, 3, 5x — 24y=68, to find the least values of x and y in whole numbers. We require the least values, because an unlimited number of solutions may be found. From the equation, x=^^'Ay=.l+^l±l^ . 33 • ' 35 Hence, '" -= some whole number: but — •^= some whole 35 35 number ; therefore,, by taking the difference of these two whole numbers we have — ^ — = some whole number. 35 Three times a whole number is a whole number ; therefore, 33y— 99 33y— 29 ^ , , , \ — = — — 2= some whole number.* 35 35 — t. =iwh. Also, — ^-=iwh. 35 35 *Subsequently we shall put wh to represent the phrase, some whole number. 174 ROBINSON'S SEQUEL. Whence, ^^y^^lyi=??.^^y+^=wh. 35 35 35 \ 35 / ^ "^ 35 Having thus deprived y of its numerical coefficient, we may put ^-"t^-?=w. Whence, y=35w— 32. Taking w=l. y=3, and x= — 3" =4, the least possible 35 values of x and y in integers, as was required. The next values are ic=28, and y=38. (3.) A man proposed to lay out $500 for cows and sheep ; the cows at the rate of '^ 17 per head, and the sheep at ^2. How iimny of each could he purchase ? Let a;= the number of cows, and y= the number of sheep. Then 17a:-|-2y=500, is the only equation that can be obtained, and X and y must be whole numbers by the nature of the prob- lem — he could not purchase a part of a cow, or a part of a sheep. To find the least number of cows, transpose 17a;. Then y=250— 8a;— ^. Now as X and y must be whole numbers, _ must be a whole ^ 2 number, and the least number corresponding to x must be 2 ; corresponding to this, y=233. Under all suppositions, the number of cows must be divisible by 2. Now if the object was to purchase as many cows and as few sheep as possible, we would transpose the other term thus : :.=522=:?^=:29+ZlI?^ 17 ^ 17 Whence, !=I?^=im,'A. Or, ^Zl^l^wh ; to this add 17 17 III, and ^^==3+^±^ 17 17 ^ 17 ALGEBRA. 176 Therefore JA^=zn, or y=\ln — 6. , 17 Put w=l, then y=12, the smallest number of sheep. The corresponding value of a:=28. The number of cows may be any one of the even numbers from 2 to 28. (4.) A man wished to spend 100 dollars in cows, sheep, and geese ; cows a^ 10 dollars a piece, sheep at 2 dollars, and geese at 25 cents, and the aggregate number of animals to he 100. How many must he purchase of each ? Let x=. the number of cows, y the sheep, and z the geese. Then 10a;+2y+-=100. (1) And a:+y4-2=100. (2) Clear equation (1) of fractions, and 40a;4-8y-(-s=400. x-\. ^+0=100. 39a;+72/ =300. ir= £l=7-[- — ^ = a whole number. 39 ' 39 Or, 6( — IZIJl )= H — ^= a whole number; add — ^ and \ 39 / 39 39 4H-135 ^^ 40y+1350 H:24^ ^^^^^^ ^^^^^^^ 39 39 '^ ~ 39 Therefore, y+24_^ ^^^ y_39p_24_i5, 09 r This value of y, gives ar=5. Hence, 2=80. If we take ^=2, we shall have y=z5A\ then a: will come a minus quantity, an inadmissible circumstance in any problem like this. Therefore, 5 cows, 15 sheep, and 80 geese, is the only solution. (5.) A person spent 28 shillings in ducks and geese ; for the geese he paid 4s. Ad. a piece, and for the ducks, 25. Qd. a piece. What number had he of each ? Let a;= the number of geese, and y the number of ducks. % 176 ROBINSON'S SEQUEL. 28-12. Then 52ar+302/=28-12. Or, 26x-\-l52/=:168. , 3— lire 15 \ Whence, =wh. But =wh. 16 15 By addition, C-^+^)4=wk. l^Hd?=.+f+15 ^ \ 15 / 15 ^15 _JL — z=n. x=15n — 12. a;=3, when w=l. 15 Then ?/=8— 2=6. When n=2, x=lQ, and y=—20. But this is inadmissible; therefore, x=3 and y=6, is the only possible solution. (6.) Divide the number 100 into two such parts that one of them may be divisible by 7, the other by 11, Let 7x= one part, and lly the other part. Then 7a:+lly=100. _100— 7a;__ 1— 7a; ^ IT"" """Tr" 11 ^11 11 U ' 11 ^+?=?i. x=lln—3. 11 Now put ?z=l, then a;=8 and y=4. 56 and 44 are, therefore, the required parts. (7.) Mnd the least number which being divided by 6 shall leave the remainder 3, and being divided by 13 shall leave the remainder 2. Let iV= the number sought, and x and y the quotients arising from the divisions. •^V— 3 . JV—2 Then =x, and =y 6 13 ^ Whence ^^=6x+3, and iVi= 13y+2. Consequently, 6z-\-l = lSy. 6 6 ALGEBRA. HT Then ^~-=n, or y=z6n-\-l For the least values of y we must take w=0, then y= 1 and x=2. ButiV=6;r+3=15. V^ We may determine iV" more directly without the aid of x and ft y, thus : . = some whole number ; also, =wh. 6 13 As iV does not contain a coefficient to be worked off, we may 7\r 3 put =^, and iV^=6p-["2 i^ which any integral value put for 6 J) will satisfy the first whole number, but the other must be sat- isfied also ; then put the value of N in ; that is, ^ 13 ^^ I-= some whole number. 13 13 13 13 13 13 Whence, p=1i3q-\-2. Assume 2'=0, then ^=2 and JV=6i?+3=15, as before. (8.) What number is that which being divided by 11, leaves a remainder of 3, divided by 1 9, leaves a remainder of 5, and divided by 29, shall leave a remainder o/" 10 ? Let ^y be the required number, and x, y, and z the several quotients, and of course they must be whole numbers. Then llar+3=iV; and 19y4-5=iV, and 292+10=iV. Hence, ir=??!±I, and x='^?li^. 19^=29^+5 29^+5^ . lOH-5 19 ~ 19 * 200+10 , 0+10 ! — =zwh, or -J — =n. z=19n — 10. 19 19 Any integer written in place of n will give integer values to z and y, but x, or JLL must be a whole number also. Hence such a value of n must be found as will make 29(19w— 10)+7 , m ROBINSON'S SEQUEL. Or, 561^290+7^ 651.^-283 ^^^^ n-8^^^ 11 11 ^ 11 ^ o Whence, . =p. w= 11^+8. For the least value of n put^=0 ; then w=8. But 2=19n—10=19'8— 10=162— 10=142. And iV^=29- 142+10=4128, the number required. (9.) Required the least nuv}her that can he divided by each of the nine digits, without remainders. Let xz=i the number. Then -, -, -, -, -, -, -, -, must all be whole numbers. 2 3 4 6 6 7 8 9 Now if we make - a whole number, _ and _ , the double and 8 4 2 C|[uadruple, will be whole numbers of course. Also, if - is a whole number, — will he a whole number. 3 Therefore, we have only to find such a value of x as will make _, _, _, _, -whole numbers. -, may also be cast out, on 9 8 7 6 5 6 -^ consideration that 6=2-3, and 2*3 are factors, one of 9, the other of 8, in the preceding expressions. Hence we have only to find such a value of x as will make each of the fractions -, _, -, and -whole numbers; and as these 9 8 7 6 denominators contain no common factors, their product is the least number that will answer the condition. Whence, ir=72 • 36=2520. (10.) A market woman has some eggs, which when counted out by twos, threes, fours, and fives, still left one ; hut when counted hy sevens, there was none left. Wliat was the least possible number of eggs she could have had ? Ans. 301. X——\ X 1 X 1 X 1 1 X This problem requires , , , , and _ to be whole numbers. ALGEBRA. 1^9 j is a whole number, / j , its double, must be a whole number of course ; hence, we have only to make , .JUL- , , and -- whole numbers. 3 4 5 7 Put the first expression equal to the whole number n ; then a:=3w-[-l. This will satisfy the first expression ; but the others must be satisfied also; therefore, substitute (3^-|-l) for x in them. Then — , — , and — X-, must be whole numbers. 4 5 7 VS/i 7 6w . n ^ n ■, 5 5 ^5 5 That is, n=bp ; and substituting this value of n in the other two expressions, we have — ^ , and ^~^ , to be made whole numbers. 4 7 ]^P+lz=2pJf-Pj^ =:wh. Whence, put ^±1=^. p=lq--\. 15/? . Finally, this value of p put in — ^ , gives 4 im , which must be made a whole number 4 before we can be sure of a result which will satisfy all the con- ditions, l^^~^=(26^-3)+?=?=...A. Whence, 'Zl~-?=/. y=:4«:+3. For the least value of q, put /=0 ; then $-=3. Butjt?=7^— 1=20. w=5p. Then?*=100. .i-=3;i-f 1=301. (11.) Required the year of the Christian Era in which the solar cycle was, or will be 15, the lunar cycle 12, and the Roman Indiction 12. N. B. The operator must of course know the exact import of these terms, and the facts in the case, before he can be required to solve the problem. 180 ROBINSON'S SEQUEL. The solar cycle is a period of 28 years, at the expiration of which, the days of the week return to the same days of the month, (provided a common centurial year has not intervened.) The first year of the Christian Era was the tenth of this cycle ; therefore, we must add 9 to the year and divide the sum by 28, and the remainder will be the number of the cycle. The lunar cycle, or Golden number, as it is sometimes called, is a period of 19 years, after which the eclipses return in the same order as in the previous 19 years. The first year of the. Christ- ian Era was the second of this period ; therefore, we must add 1 to the year and divide by 19, and the remainder is the year of the lunar cycle. The Roman Indiction is not astronomic. It is a period o€ 15 years, the first of our Era being the 4th of the Indiction ; there- fore, add 3 to the year and divide by 15, and the remainder is the Indiction. The reader is now prepared to solve this or any other similar problem. Let X represent the required year ; then the problem requires that a;4-9— 15 a;-fl— 12 a:-f-3— 12 28 ' 19 ' 15 /^ g ^ ] \ /J 9 should be whole numbers ; that is, , , , must be 28 19 15 whole numbers. The first expression will be satisfied by putting it equal to any whole number n\ then a;=28w-f-6. But the other two expressions must be satisfied also. Therefore ^^MltU and ^Mlt:^ 19 15 Or, '^''^n—b ^^^ 28?i— g ^^^^ ^^ ^j^^j^ numbers. 19 15 Or. ^^~^ and ^^^~^ must be whole numbers. 19 15 157^_13n-3_27^+3_^^,;^ 7(2n+3)_^^^ 15 15 15 15 15n_ 14n+2 1_^^,;^ n-21 16 16 * 16 ALGEBRA. 181 Whence, n=15p-\-21. Every expression is now satisfied, I — 19" except ^ ; to satisfy this, write in the value of n ; then 9(15;.+21)-5 _,^ 19 19 ^^ ^ 19 Whence, ^+^i=t.A., and ^5H^?2=^+6+^-H6 19 19 ^ ^ 19 19 ^ ^ ^ We canhot take q less than 1 . The least value of p will then be 3. But ?2=15p+21=66. a;=28?i-|-6=28-66+6 = 1854. If no interruption was to be made by the centurial years, the coincidence of these cycles would not occur again until the year 9834, which we find by making q=2. SECTION VIII. TO DETERMINE THE NUMBER 'OF SOLUTIONS AN EQUATION IN THE FORM AX-{-BY=C WILL ADMIT OF. An equation in the form ax — hy=c, will admit of an unlimited number of solutions, because x and y can increase together; but not so with equations in the form ax-[-by=c ; for in them an in- crease of X will cause a decrease of y, and an increase of y, a decrease of x; but neither x nor y are permitted to fall below 1. If c is very large in relation to a and b, the equation may have a great number of solutions, and we are now about to show a summary method of determining the solutions in any given case. The equation ax' — hy'=\ is always possible in wjiole num- bers, (Prop. 8, sec. vii;) therefore, c times the equation is also possible ; that is, acx' — cby'z=c, is possible. But ax-\-by=^c, is a general equation. Put these two values of c equal to each other, then 182 ROBINSON'S SEQUEL. (xx-{-byz=zacz — cby' ( 1 ) From (1), x=cx—(^yjh\b. (2) For the sake of perspicuit}-, put ^ "1"^=^. a Then (2), becomes x^cx' — hn (3) Multiply (3) by a, and substitute the value of ax in (1), Then acx — abm-\-by=acx' — cby' Whence, yz=iam — cy' (4) From (3), we find 7^=^:-— _ (6) h b From (4), we find m=^]^+l (6) a a Equations (5) and (6) show that m must be greater than ^ a and at the same time less than — b Therefore, the limits to m are found. Now let us observe equations (3) and (4) ; rr must be a whole number, and as c, x, and b are whole numbers, m must be taken in whole numbers, and it may be any whole number between a b The number of solutions will, therefore, correspond with the difference between the integral parts of the fractions — and ^ h a except when ^ is a whole number, in that case x becomes b, and - / b b must be considered a fraction, and rejected. If, however, we in- tend to include among^ the integral values, this precaution need not be observed. EXAMPLES. (1.) Required the number of integral solutions of the equation, 7a;+9y=100. ' First find the least value of x' and y' in the equation 7a;'— 9/= 1. ALGEBRA. 18S The result will be x'—4. y'=3. Then -'=l?2:!=44l 'l='^^,=42l b 9 9 a 7 7 Disregarding the fractions, the difference of the integral parts is 2, showing two integral solutions to the equation. If we had taken the difference between ^^^ and ^^, thus ; B|ofli__2 7.oo_i|_i^ ^e might have come to the conclusion that the equation would admit of only one integral solution. The integral difference in this case is not the difference of the integrals. When the fractional part of — is not less than the fractional b part of —, but equal or greater than it, we can find the number a of solutions by taking the difference, thus — — ^=^i^f yj. b a ab But ax' — by'=\ ; therefore, — — Jl.= — b a ab Whence we conclude that — will in this case show the num- ab ber of solutions. — In all cases it will be the number, or one less. (2.) What number of integral solutions will the equation 9;5+13y=2000 admit of? Am. 17. 9-13=117)2000(17. That is, we are sure of 17 different solutions, and there may be 18. The equation bx-\-9y^=40 admits of no solution in whole num- bers, c, 40, is not divisible by 5* 9=45, that is, no unit in the quotient. Yet the equation 5a;-|-9y=:37, admits of one solution. The auxiliary equation 5x — 9y'=\, gives a;'=2, and y'=l. Therefore, ^^J^=^. ^=?Z=7i ■2. b 9 ^ a 5 '* Here the difference of the integrals is 1, indicating one solu- tion. In fact x=2, y=S. How many solutions will the equation 2x-\-5g=40 admit of? 184 ROBINSON'S SEQUEL. The auxiliary equation, 2a;' — 6/=l, gives x'=S, y'=l. —=24. ^=20. Or, 4 solutions. ^ b a But observe that — in this case, is a complete integral, 24 ; b according to previous considerations, we must deduct one, and the number of solutions are 3, as follows : x=5. 10. 16. y=6. 4. 2, and no other solution can be found. (3.) What number of solutions in whole numbers can be found for the equation 3x-\-5y-\-7z=100. As X and y each cannot be less than one, z cannot be greater than ^^^^^~^=13}. That is, z cannot be greater than 13, in whole numbers. Now suppose 0=1, and the equation becomes 3a:-|-5y=93. The number of solutions for this equation, found as previously- directed, is 6. That is j x = 26. 21. 16. 11. 6. 1. ( y=z 3. 6. 9. 12. 15. 18. Now X and y can make these six changes, and z be constantly equal to 1, and satisfy the primitive equation. We may observe here that x diminishes from one solution to another by the coefficient of y, and y increases by the coefficient of X, but this is not a general principle. Take 2=2, and the equation becomes 3a;-|-5y=86. This equation has also 6 solutions, z being through all the changes of x and y equal to 2. Now take s=3, then the original equation is 3x-\-5y=^lQ, This equation has five solutions. Now take 0=4, then 3a;-|-5y=72. This equation has four solutions. Take 0=5, then 3a;-[-5y=66. This equation has four solutions. Take 0=6, then 3ar-}-6y=58. This equation has four solutions. In this manner, by taking equal to all the integers up to 12 in succession, we find 41 solutions to the primitive equation. (4.) Required some of the integral solutions to the equation 14a;+19y+2l0=252. ALGEBRA. 186 Here we observe that 14 and 21 and 252, are all divisible by 7, therefore y must be 7, or one of its multiples ; suppose it 7, and divide the whole by 7. Then ' 2^+19-1-32=36. Or, 2^+30=17 Whence, x may equal 1, and 2=5, or ^=7 and 2=1. Or, fl;=4 and 2=3. ix—1. 4. 1. > Hence, \ y=7. 7. 7. (2 = 1. 3. 5. As these examples are of little practical utility, we give no more of thep. SECTION IX. DIOPHAI^TINE ANALYSIS. Diophantus was a Greek mathematician, who flourished in the early days of science : and the analysis that bears his name, mostly refers to squares and cubes. The object of this analysis is to assign such values to the un- known quantities in any algebraic expression, as to render the whole a square or a cube, as may be required. The first principles of this branch of science are very simple, but in their application, they expand into the region of impossi- bility. To Euler and Lagrange, we are indebted for most that has ap- peared on this subject. Case 1st. The most simple expression to be made a square, is of this form : '- ax-\-h. All we have to do, is to put this expression equal to smne square, say n^ ; then a;= , and n, a, and &may be taken at pleasure. a The result will give a value of x which will render (aa;+5), a square as required. .*.. • * . ^w 186 ROBINSON'S SEQUEL. I EXAMPLES. (1.) Three times a certain number increased by 10, is a square. What is the number ? Here a=3, ^=10, and fl;= — III — o If we put w=l, then x^= — 3, and ax-\-b^= — 9-}- 10=1, a square. If we put w=2, then ic= — 2, and aar-(-5=4, a square; and by taking different values for n, we can find as many squares as we please. (2.) Find such values of x as will render the following expres- sions squares : (9a:+9), (7;r+2), (3;r— 5), (2^^— i). All these expressions correspond to the general expression, [ax-^b.) Case 2d. To advance another step, we require such values of X as will render any expression in the form {ax^-\-bx) a square. Because a: is a factor in every term of the power, we will make it a factor in the root : that is, put the root equal nx ; then ax^ -\-bx=^n^ x^ . ax-\-b=n'^x. x=-A- n^ — a ^ * EXAMPLES. (3.) Six tim£s the square of a certain number, added to five times the same number, is a square. What is the number ? Ans. x= , that is, the number is expressed by with the liberty to give any value to n that we please. If we make n=l, then a;= — 1, and (ax^-{-bx) will become 1, a square as required. If we make w=2, then a;=_-|= — f, and ax^ -\-bx=6.%^ — ^^=2by a square as required, and many other results may be obtained. ALGEBRA. 187 (4.) Find what values of x will render the following expressions squares : (5a?2— 3a;), {lx^—\bx), {\2x^—yx), {yx'^—^x.) (5.) Find such a number that if its square he divided by 12, and one-third of the number be taken from the quotient, the remainder will he a square numher. Ans. 16 is one number, and there are many other numbers that will correspond to the conditions. The practical utility of this analysis may be exemplified in forming examples in quadratic equations. Thus ax^ — hx=zN, is a quadratic, and we would assign such values to N, as will make the values of x commensurable quan- tities. h b^ h^ Completing the square, gives x^ — -x-\- =iV'4- . a 4a ^ 4a^ To make the values of x -commensurable, it is necessary to / 5^ \ make the expression t A^-|- ) a perfect square, and this we can do by putting it equal to some definite square, (by case 1st a and b being known quantities,) and deducting the values of N. Case 3d. Expressions in the form [x^d=.ax-^h), can be made perfect squares, by putting them equal to the square of [x — n.) This hypothesis assumes [x — n) to be the square root of {x^diziax-\-b), and as this expression may be any number between zero and infinity, we now enquire whether it be possible that [x — n) can always represent the root, whatever it may be. We reply, it can. If X is large and n small, [x — n) will be large. If x=in, then [x — n) will be zero. If n is numerically greater than x, then (x — n) will be negative ; but the square of a negative quantity is positive ; therefore, n can be so assumed that [x — nY can equal to any positive quantity whatever. - ::^ That is, x'^±:ax-{-b=x'^—2nx-\-n^. "* Or, x:= 2w±a An expression in which n may be taken of any value whatever, and we shall have a corresponding value of x. _ ^ '*•* i^ % ♦ ^^^*v, > 188 ROBINSON'S SEQUEL. Case 4th. An expression in the form (ax^ zhhx-\^c^ ) , can be made a complete square, by assuming its square root to be (c — 71X.) Because c^ is in the power, c must be in the root, and it is obvi- ous that X must be a factor in the other part of the root. ^' Whence, ax^ztbx-\-c'' =c'^—2cnx-{-n^x'^. •■^^ axd3=—2cn-\-n^x. 2cn±:b EXAMPLE. What value shall be given to x to make 8x^-[-17x-|-4 a perfect square ? Here a=8, 5=17, c=2. Whence, :.= ^^+^^. If 7^=l, then x——'2>, and '8;c2-|-17;c+4=25, a square. If n=3, then 2;=29, and the value of the expression is 7225, the square of 85. Case 5th. An expression in the form {^ax"^ -\-bx-\-c) , in which neither the first nor the last term is a square, neither branch of the root can be taken, and the expression cannot be made a square, unless we can separate it into two rational factors, or unless we can find some square to subtract from it, which will leave a re- mainder susceptible of being separated into two rational factors. By placing the expression [ax^ -\-bx-\-c) equal to zero, and solving the quadratic, we shall have two factors of the expression, but whether rational or commensurable factors or not, is the subject of enquiry. , To find the factors which make the product ax^ -\-bx-\-c, i^Mt this expression equal to 0, and work out the values of x thus, ax^-^-bx-^c^O. Or, ax^-\-bx^= — c. Complete the square, and 4:a^x^-{-Aabx-\-b^=b'^—^ac. Or, ^ax-\-b=±J{b^—'Aac.) Or ^=±iV(*'— 4ac)— A. 2a^^ ^ 2a / ALGEBRA. '* 189 We now perceive that the values of x must be rational, previ- ded J{b^ — 4ac) is a complete square. If it be so, let J_ /(P_4ac) ^=m, and —}_J{b^—4ac)—^=n. Then the two values of a; are x=m and x=n, and (x — m)(x — n), are the factors which will give the expression ax^-^x-^-c. EXAMPLES. (1.) Find such a value ofxas. mil make 6x^-|-13x-|-6 a square. Here a=6, 5=13, c=z6. h^ = l69, 4ac=144, P—4ac=25, and J(b^—4ac)=5. Now 12a:+13=db5. a:=— |. Or, x=—^. Or, 3;i:+2=0, and 2a;+3=0. That is, (Sx-\-2) (2a:-|-3), will produce the expression 6a;2+13:r-f-6. Now to find such values of x, as will make the expression a square, put (3a:+2) (2x-{-3)=n^3x-\-2y . That is, take either factor of the expression for a factor in its square root. Then 2x-{'S:=n^3x-{-2.) 2^2—3 x= 2—3n^ Take n=l, then x=l, and the expression becomes , % 6-|- 134-6=25, a square. Take n=2, then x= — i, and the expression becomes 1, a square. Take n=S, then x= — f, and the expression becomes -^j, a square. (2.) Find such a value of x as shall render the expression (\Sx^-{-15x-{-7), a square. Here as neither the first nor the last term is square, nor (b^ — 4ac) SL square, we cannot find the required values of x, unless we can find a square, which subtracted from the expression, will leave a remainder divisible into rational factors. But in this case, 4ac is 190 * . ROBINSON'S SEQUEL. greater than b^, we must therefore subtract such a square as to diminish a and c, and increase 6. To accomplish this object, we will subtract the square of (a? — 1), and not the square of (x-\-\.) That is, from 13x^+15x-{-7y Subtract a;^— 2x-{-\. Difference, 12x''-{-\7x-{-e. In this last expression, a=\2, 5=17, and c=^6. ^ Hence, b'^-^4ac=2S9—2QS=l, a square. We are now sure the difference is divisible into rational factors, and to obtain the factors, we put \2x^-\-\lx-{-Q=0. A solution of the quadratic, gives x= — f , or x= — f . Whence, (3i;-|-2)=0, and (4.2;-}-3)=0, and our original ex- pression becomes (^_l)3_^(3ar+2) (4a?+3.) It is obvious that [x — 1 ) must be in the root, and one of the factors may be in tlie other branch of the root ; that is, put (a;— 1)2_{-(3^+2) (4;r+3) = [ [x—\)-\-n{3x-\-2) y. By reduction, 4x-\-3=:2n{x—\)-\-n^ (3x^2,) Or. x=^"+3--^-. Take w=l, then x=3, and \3x^ -\-\ 5X'\-1 ^\Q^ , a square. (3.) Find such a value of x as will render 14x^-|-5x — 39, a square. After a few trials this expression is found to be the same as (2,r— l)2+(5.c— 8) (2x-\-b.) Assuming its root to be 2a:— 1+ p(5x — 8), then by squaring the root, making it equal to its power, and reducinor, we find x=-^-^^^~^ . 5p^-\-ip—2 Assuming p= 1 , x= y , and the expression equals 36, a square. Other values can be found, by assuming different values to jo. (4.) Find such a value of x as will make 2x2-f"2^^~l~28, — 1 Assume ^=4, then x=4, and the original expression is 144, a square. If (x-\-4y-{-(x+l) (x+n) = [ (x+4)—p(x-\-12)y, we shall find x=: ±- ±- If we take p=l, x=%. If we take »=| then a;=8, and we might find many other values of x that would answer the required condition. Case 6th. Expressions in the form a^x'^-\-hx^-\-cx^-\-dx-\'€y can be rendered square, provided we can extract three terms of their square roots. Assume such terms as the whole root, making its square equal to the given expression, and the resulting value of x will make the whole expression a square. EXAMPLE. Find such a value of x as will make 4x''-|-4x^-(-4x^-|"2^ — ^'^ square. We commence by extracting the square root as far as three terms, and find them to be (2a;^-|-ar-|-|.) Therefore, 4x^ ^4x'^ ■\-4x^ -\-'lLx—%^{'ix'' -^-x-^-lY . Expanding and reducing, 2a; — 6= |4:-[-t6- And ' ;r=13|. Essentially the same method must be performed in other ex- amples under this form. ^ Case 7th. Find such a value of x as will make ax^-\'C, a square. Expressions in this form, where /;=0, and where neither a nor c are squares, nor {h^ — 4ac) a square, present impossible cases : unless we can first find by inspection, some simple value of x that will answer the condition. m Idt « ROBIKSON'S SEQUEL. EXAMPLE. Find such a value of x as will make 2x^ -|-2, a square. It is obvious that if x=\, the expression is a square. Now, having found that 1 will make the expression a square, we can find other values as follows : Leta:=l-[-y; then x^ =^\-\-2y-\-y^ , And 2x''+2=4+4y-\-2yK Here the original expression is transformed into another expres- sion, having a square for its first term. Now we must find such a value of y as shall make 4-|-4y-|-2y^, a square. Assume 4-[-42/-|-22/"=(2 — my)^=4 — 4my-\^m^y^. Or, 4-|-2y= — 4m4-m^y. Hence, y=-^ — —-, ^ may be any num- m^ — 2 ber greater than one. Put m=2. Then y=6, and a;=l-f-y=7, and the original expression, 2a?2-|-2=98-|-2=:100, a square. N. B. It often occurs incidentally in the solution of problems, that we must make a square of two other squares. This can be done thus: Let it be required to make a:^-|-y^, a square. Assume x=p^ — q^, and y=2j)q. Then x''=2)^—2p^q^-\-q\ And y^= 4p^q^, Add, and x^ -\-y^ ^^p'^ -\'2p^ q"^ -\-q^ y which is evidently a square, whatever be the values of p and q. We can, therefore, assume j9 and q at pleasure, provided^ be greater than q. % SECTION X. DOUBLE AND TRIPLE EQUALITIES. We have thus far confined our attention to finding a value of x that would render a single expression a square. Now we propose finding a value of x that will render several expressions squares at the same time. ALGEBRA. 193 Case 1st. As a general expression for double equality, let it be required to find such a value of x, that will make (ax-\-b) and (ca:-[-c?), squares at the same time. X:=: . a ^2 ^ Whence, -^ =- , or d^ — cb=ap^ — ad. a c Transposing cb, and multiplying by c, gives c^t-= acp ^ -\-c 2 d — acd. As the first member of this equation is square for all values of c and t, it is only requisite to find such a value of p, as to make the second member a square ; which can be done by some of the artifices heretofore explained. To illustrate, we give the following definite problem : The double of a certain number increased by 4, makes a square ; and Jive times the same number increased by. 1, makes a square. What is that number? Let X be the number ; then 2ar+4=^2 Whience, \ ^ Then 5t''—^0=2p''—^. And 25/2 = 10^2+90. The first member of this equation is a square, whatever be the value of f ; and all the conditions will be satisfied, provided we can find such a value of p as to make the second member a square. This expression corresponds to case 7, and we cannot proceed, unless we find by trial, by intuition as it were, some simple value of ^ that will make 10p2_|_9Q^ a, square; and we do perceive that jt?=l will make the whole expression 100, a square. Now if jt>=l will give a definite and positive value to ar, which will answer the required conditions, the problem is solved. If not, we must find other values of p. 13 sions fH ROBINSON'S SEQUEL. Here x=^ /and i(p=l, x=0, and the expressions, 2a:-|-4 6 and 5a:-|-I, become 4 and 1. Squares, it is true, which answer the technicalities, but not the spirit of the problem. To find another value of jt?, put jO= !-[-?• Then 1 0^2 _^90= 1 00+20jt>+2^^ To make this a square, assume \00-{-20q-{-2q'' = (l0— nqy = 100— ^Onq+n^g'*. By reduction, q= — l_ir_''. Now n must be so taken, that n' n^ — 10 will be greater than 10; take w=5 and §'=8, p=Q, then x=l6, and the original expressions, 2^-j-4=36, a square, and 5x-\-l=z 81, a square. Case 2d. A double equality in the forpi ox'^-\-bx=Ci* and cx^-\-dx=0, may be resolved by making a:=--, then the expres- y will become -^(a-j~*y) ^^^ — (^+^y)» which must be made squares. But if we multiply a square hy a square^ or divide a square hy a square, the product or quotient will he square. Now as each of the preceding expressions are to be squares, and as they obviously have a square factor — , it is only necessary y^ to make a-\-by, and c-\-dy, squares, as in the first case. We may also take another course and assume ox^-\-bx=p^x^ , which gives x= , whic^h value put in the other expression, p^ — a and we have c( ) -\-d( ) = D . \p^—a/ ' \p^—a/ Multiplying this by the square [p^ — aY , and the expression becomes ch^ — dhd-\-abp^= some square, from which the value of p can be found, and afterwards x. EXAMPLE. Find a numher whose square increased by the number itself, and *Thi8 symbol is read a square. ALGEBRA. 196 whose square diminished by the number itself, the sum and difference shall be squares. Let ar= the number ; then by the conditions, a;2-|-a:=n, and x^ — a;= some other square. Assume x= - ; then — ( 1 -f-y ) = D . The first members of these equations are obviously squares, provided the factors (l-(-y) and (1 — y) are squares. To make these factors squares, put l-|-y=jt?2, and 1 — y=^q^ • Whence, y==p^ — l,andt/=l — g*. p^z=<2.—q^. All the conditions will be satisfied, when we discover what value must be given to q to make (2 — q^), a square ; and q=\ satisfies that condition. This value oi q makes j9=l, and y=0. But ic=:_=_= infinity. y If X is infinite, x^ can neither be increased or diminished by adding and subtractings;; ihereioxQ x^-\-x=i\2, vm^ x^ — ^•=n, because x^ is obviously a square. But practically, we say that this value of q will not answer the conditions ; therefore, we will find other values as follows : Put q=\-\-t. Then 2— y^ ^i__2t—i'' =^(\~ntY = \—ZrU+n'' f" . Or, /=?l!^i. Take n=2, n^+\ Then^=|. ^=1+1=1. y=i_||===_|4. 1 25 But x=-= — -- , the number souofht. y 24' Those who desire a positive number, can take n negative. Case 3d. To resolve a triple equality. Equations in the form cx-\-hy=:it'^ , ax-\-dy—u^, eX'\-fy='8^ , can be resolved thus : J96 BOBINSON'S SEQUEL. By eliminating x, we find y= By eliminating y, we find x au ad — be ad — be Substituting these values of a; and y in the equation &c-[^y=5^, we shall have (af-be)u-+ { de-ef )e_^, . ad — be Assume e«=r±:^2;; then u^=^t^z^, and put this value in the above, and divide by t^ , then {af—bc)z^J^[de—cf )_ s2 ad— be '¥' As the second member of this equation is a perfect square, all the conditions will be satisfied when we find a value of z that will render the first member a square. This, when possible^ can be done by case 7, of section viii. After z is found, t can be assumed of any covenient value whatever. When u and i are known, x and y will be known. We are now through with theory, — not that we have presented the whole, there are some cases in practice that no general rules will meet, and the operator must depend on his own judgment and penetration. Much, very much, will depend on the skill and foresight displayed at the commencement of a problem, by assuming convenient ex- pressions to satisfy one or two conditions at once, and the remain- ing conditions can be satisfied by some one of the preceding rules. EXAMPLES. (1.) It is required to find three nwmbers in arithmetical progres- sion, such that the sum of every two of them may be a square. Let X, x-\-y, and ar-|-2y, represent the numbers. Then by the general formula, ^x^y=t^, 2x-\-2y=:u^, 2x-\-Sy—s^, ^ _ ■ /2 y 21 2 2,11 By extermmating x, we have -= ^ . ALGEBRA. 197 Continuing thus according to the general equations, we must go through a long and troublesome process, and in conclusion we shall find the numbers to be 482, 3362, and 6242. Another Sdviion. Observing the remark immediately preceding the example, we put — — y, — , and — -|-y to represent the numbers. Ai ^ At Then {x^ — y), (^^+y), and x^ must be the squares. But x^ is a square for all values of x ; therefore, we have only to make squares of (x^ — y) and {x'^-^-y-) Let y=2x-\-l ; then x^ -\-y=x'^ -\-2x-\-l , a square for all values of X. Hence, all we have to do is to find a value of x that will make a square of the expression x^ — y, or x^ — 2x — 1. Assume the square root to be (x — n) ; then x^ — 2x — l=x'^ — 2nx-\-n^ • x=.^L+l_ 2(n—l) Take w=ll, then a;=:6.l, and -1^:2 = 18.605, y=13.2, and this numbers are 5.405, 18.605, and 31.805. Various other numbers may be found, by giving different values to n. (2.) Find two numbers such that if to each, as also to their sum^ a given square el^ be added, the three sums shall all be squares. Let x^ — a^, and y- — a^ represent the numbers ; then the first conditions are satisfied. It now remains to make x^-{-y^ — 2a^-\-a^ a square, or, x^-\' y 2 — Qj2 __ [-] ^ Assume y^ — a^ ==:2ax-\-a ^ . This assumption will make the expression a square, whatever be the values of either a; or a. But the assumed equation gives y^ =^2ax-\-2a^ , a.nd as y^ is a square, we must find such values of x and a, as shall make 2aa:-|-2a2, a square. Put x=-na. Then 2wa2-|-2a^ = n, or, a^ (2w+2)= D . Hence it is sufficient that we put 2w-|-2= some square. Therefore, assume 2w-|-2=l6. Hence, w=7 and ir=7a. Now take a equal to any number whatever. If a=l, a:=7, y=4, and 48 and 15 are the numbers, add 1 to each, and we have 49 and 16, squares ; sum, 63-|-l=64, a square. 198 ROBINSON'S SEQUEL. (3.) Find three square nunilers whose sum shall he a square. Letir*+y2_f-22_Q^ Assume y'^='^z. Then xf -{-2xz-^z^ is a square. But 2a:s=D. Let x=uz, then ^uz^ = {j, or 2u= □ = 16, u=S, x=Qz, z—\, x—^, y=4. Therefore 64+16+1=81=92. (4.) Fhui three square numbers in arithmetical progression. Let x^ — y, x^, and x^-\-y represent the numbers. Assume a.2__y2_|_i^ then the first and last will be squares, and it only re- mains to make (y^+i), a square. Therefore, put 3/2+1 =(y—j9) 2. Whence, y=£- !. 2p Take^=l, then 2/=|, and y2+|=f|=a;2. Consequently, -g^, ||, and ^\ are the numbers ; but we can multiply them all by the same square number, 64, without chang- ing their arithmetical relation^ and their products will still be squares, 1, 25, and 49. Multiplying these numbers by any square number, will give other numbers that will answer the condition. (5.) Find two whole numbers, such that the sum and difference of their squares when diminished by unity, shall be a square. Let a:+l=: one number, and y= the other. Then by the conditions, x'^-\-y^-\-^x=.{2 (1) And a:2— y2+2ar=n (2) Assume 2a:=a2, and 3/^=2cu;; then (1) and (2) become (a;2+2aa;+a2) and (a:^— 2aa;+a2), obvious squares whatever may be the values of x and a. But the equations 9,x=ia^ , and y^ =2ax, must be satisfied. Take a=4, then x=S, a;+l=9, and 9 and 8 are the numbers required. (6.) FtTid three whole numbers, such that if to the squares of each, the product of the other two be added, the three sums shall be square*. Let a?, xy, and xv be the numbers. Then by the conditions, x^ -\-x^vy= □ . x^y^-{-x'^v=0. And x^v^^x^yz=[2. ALGEBRA. 199 Omitting the common square factor x^, it will be sufficient to make squares of the following expressions : \-\-vy=U. Assuming y=4v-\-4 will make the first and third expressions square. Substituting the value of y^ in the second expression, we shall have 16y^-f-33y-j-16, which must be made a square. Whence, \ev^+^3v-\-16={4—pvy. Reduced, gives v= — X-^. Take ^=5. p'^ — 16 Then v=\^. Now take x=9, and 9, 73, 328, will be the re- quired numbers. (7.) Find two whole numbers whose sum shall be an integral cube, and the sum of their squares increased by thrice their sum, shall he an integral square. Let x-^y=^n^, that is, some cube. Then x^-^y^-\-Zn^=z\2' Put 2;ry=37i3, then x^ -\-9,xy-^y^ is a square, whatever may be the values of x and y. But x and y must conform to the equations x-\-y,=n^, and 2xy=Sn^. Work out the value of x from thesQ equations, on the supposition that n is known, and we shall find 2x=n^-{-J{n^—6n^}. Now a; will be rational, provided we can find such a value of n as shall render n^ — 6n^ a square, but if we add 9 to this, we perceive it must be a square, and we have two squares, which difi"er by 9. Therefore one must be 16, the other 25, as these are the only two integral squares which differ by 9. Hence, ;i6_6^3_|_9^25. Or, ^3—3=5. n^ = ^, n=2, and x=6, y=2. (8.) Find three numbers such that their sums, and also the sum of every two of them, may all he squares. Let x^ — Ax= the first, 4x= second, and 2:r-}-l= third. By this notation, all the conditions will be satisfied, except the sum of the last two. That is 62"-|-l must be a square, but to have three different whole numbers, no square will answer under 1 2 1 ,.the square 200 ROBINSON'S SEQUEL. of 11. Hence, put 6a;-[-l = 121. Or, a;=20. And the numbers will be 320, 80, and 41. (9.) Find two numbers such thai their difference may he equal to the difference of their squares, and the sum of their squares shall bf a square number. Let X and y be the numbers. Then x — yz=x^ — y^ . Divide by X — y, and l=x-\-y. Hence a?=l — y, and x^-\-y^ = \ — 22/-(-2y^. Which last expression, 1 — 2y-|-2y^, must be made a square. For this purpose put 1—22/4-2^2 =(1 — nyY . Hence, y=5i^IIl i. n^ — 2 Take n any value to render y less than one, in order to make X positive. Take w=3, then y=y, and a;=f , the answer. The following are not difficult, and we leave them as a pleas- ant exercise for learners. (10.) Find three numbers in geometrical progression, such that if the rman be added to each of the extremes, the sums in both cases shall be squares. Ans. 6, 20, and 80. (11.) Find three numbers, such that their product increased by unity shall be a square, also the product of any two increased by unity, shall be a square. Ans. 1, 3, and 8. Assume 1 for the first number, and x and y for the other two. (12.) Find two numbers, such that if the square of each be added to their product, the sums shall be both squares. Ans. 9 and 16. (13.) Find three integral square numbers in harmonical propor- tion. Ans. 25, 49, and 1235. (14.) Find two numbers in the proportion of S to 15, and such that the sum of their squares shall be a square number. Ans. 156 and 255. Bonnycastle's answer is 476 and 1080. (15.) Find two numbers such that if each of them be added to their product, the sums shall be both square. Ans. ^ and |. We have given as much on this topic as will be profitable, save the following remote and partial application. ALGEBRA. 201 EXAMPLES. (1.) Given -j ^2_ — 7 [ ^^ ^^^ *^® values of x and y. x" ={l—y.) y2^(74-a;.) Here (7— y) and {1+x), must be squares. x=2, and y=3, will evidently answer the conditions ; and as these values will verify the given equations, the solution is accomplished. (2.) Given | l^^Zu'^y^'TLixy'^d \ ^^ ^°^ ^^^^^^ ^^ ^ ^^^ ^' As 4 and 9 are squares, the first members are square in fact, though not in form. But we can make the first members square in form, by assuming 2.c2 — 2,xy—0, and Sy^— 2.ry=0. Then y^=4 and a;2=9, or y=2 and a;=3 ; values which ver- ify all the equations, (3.) Find such integral values of x, y, and z, as will verify the equations x^+y^+xy=^l. And a;2-j-s^+^2=49. If we add xy to the first equation, and xz to the second, the first members will be square ; and, of course, the second mem- bers will be square in fact, though not in form. We have then to make ^l-\-xy, and ^^-\-xz, squares. To accomplish this, put 37-|-a^=49, or xy=\^ (1) And 49+^2=64, or ii:s= 15 (2) From (1), 2:=-; from (2), a:=— . y z Hence, 122=15?/, or 2=—. Take y=4, then 2=5, and a;=3 ; values which will verify the given equations. (4.) Find such integral values of y and z that will verify the equation y'^ -\-z^ -\-yz=Q\ . Add yz to both members, then put Q\-\-ys=^n^ . Now if we assume w=8, yz=^2>. But yz=i'^ will give y-|-2=8, and these two equations will not give integral values to y and z. Therefore, take ?z=9, then w=^=81, y2=20, y-|-^=9. Hence, 2=4 or 5, and y=5 or 4. 202 ROBINSON'S SEQUEL. (5.) Given ■! a. ^ajx^ZL r f *^ ^°^ ^^ values of x and y. Put xy=^py transpose, &c. Then 4a;2 = 12+2/>. Ay^ = \Q—Qp. Now if we find such a value of j9 as will make (IS-f-Sp) and (16 — Qp), squares at the same time, it is highly probable that such a value will verify the original equations. It is obvious that ^=2, will make the expressions squares ; then 4a:^ = 16, a;=2 and 2/=l, and these values will verify all the equations. ,^\ n- \ 6a;24-2v^ =5^4-12 ) to fin( (6.) Given j 3^,:[:2^^_3^?1 3 ^ ^^^^^ d one value of x This problem is under (Art. 1 10, alg.) Add the equations together, and reduce, and we have 9.r2=y2_|_3a:y+9. The first member of this equation is a square ; therefore the second member is a square, but to make it a square in form, as well as in fact, we perceive it is only necessary to make a;=2. Then dx^ =:y^ -^-Qy-^-^ , and 3.r=y-|-3 ; whence y=3, and these values verify the given equations. This method of operation must be used with great caution, and taken for just what it is worth. 2 2 (7.) Given ar-|-y=35, and x^ — y^^=5, to find the values of x and y. Put x^=F, and y^=Q. Then P^+Q^=35, and F^—Q^=5. Or, P^=35—Q\ and F^=5+QK The equations can all be verified, provided we find can such a val- ue of Q that will make (35 — Q^) a cube, and (5-\-Q^), a square. We will try the next less integral cube below 35. That is, we will assume 35—^3^27. Then Q=2, and (5-{-Q^)=9, a square. Then P=3, and x^==3, or a;=27, and y=8. This problem was given in the first editions of Robinson's Algebra, page 147, under the head of pure equations, but it was out of place and is now changed. PART THIRD SECTION I. OEOMrETRY. D right Thirty-one of the following problems will be found in Robinson's Geometry, commencing on page 100. (1.) From two given points, draw two equal straight lines which shall meet in the same point in a line given in position. Let A and B be the two given points, taken at pleasure, and MO the line given in position. Join AB and bisect it in J). Draw J)U perpendicular to AB, to meet the line IIO in U. Join AH and BJEJ, the lines required. Be- cause AJ)=^DB, and DE com- mon to the two A's ADE, BDE angles, therefore AE=^BE. Q. E, N. B. For simple and obvious demonstrations, we shall not go through the steps in full, but refer to Robinson's Geometry for the proposition that applies. (2.) From two given points on the same side of a line given in position, to draw two lines which shall meet in that line and make equal angles with it. Let A and B be the two given points, and HO the line given in position. From one of the given points as B, let fall the perpendicular B 0, to the given line, and produce it to D, making 0D=B0. Then join AD : this line will neces- sarily cut the hne HO in some point E. Join EB, and AE and EB are the re- quired lines. jLBEO=I^DEO, (Book 1, Th. 13.) L.AEH= L,DEO, (Th.3, Bookl.) Whence, L.BE 0=1^ A EH. Q.E.D. 203 204 ROBINSON'S SEQUEL. (3.) If from any jiolnt without a circle, two straight lines he dravm to the concave part of the circumference, making equal angles with the line joining the same point and the center ; the parts of these lines which are intercejjted loithin the circle, are equal. Let A be the point without the circle. Join A C and draw any other hne to cut the ciroie as AD ; then draw AB so that the angle CAB=^ CAD. Then we are to show that FB^ED. The two A's, ABC and ADC, having two sides AC, CB, of the one, equal to A C, CD, of the other, and their re- spective angles at A equal, the two A's are equal. That is, AB=AD. For the same rea- son the two A's ACF, ACE are equal, and AF=AE. Whence, An—AF=zAD~AE, or BF=DE. Q. E. D. (4.) If a circle be described on the radius of another circle, any straight line drawn from the point where they meet, to the outer cir- cumference, is bisected by the interior one. Let A C be the radius of one circle and the di- ameter of another, as represented in the figure. From the pointof contact A, of the two circles, draw any line, as All; this line is bisected in D. Join DC and ffB. Then ADChemg in a semicircle, is a right angle; also, AJIB is a right angle, for the same reason: therefore, DC and HB are parallel. Whence, AD : Aff : : AC : AB GEOMETRY. 205 But as AB is double of A C, therefore All is double of AD, or ^^is bisected in I). Q. E. D. (5.) JF'rom two given points on the sante side of a line given in position, to draw two straight lin£S which shall contain a given angle, and be terminated in that line. Let A and B be the two given points and j0"(9 the line given in posi- tion. For the sake of perspicuity, we will re- quire two lines drawn from the two points, A and B, to meet in HO, and make an angle of 50°. Subtract 50 from 1 80, and divide the re- mainder by 2, this pro- duces 65°. At A make the angle BAC=Qb°, and at B make the angle ABC=^Qb° ; these two lines will meet in C, making an angle of 50°. About the A ABC describe a circle, cutting HO in //and 0. Join AH, BH AHB is equal ACB, (th. 9, b. iii, scholium,) the angle required. Lines drawn from A and B, to meet the line in 0, would also answer the conditions, N. B. When the given angle is not sufficiently small to cause the angle C to fall below the line HO, the problem is impossible. (6.) If from amj point without a circle, lines he drawn touching it, the angle contained hy the tangents is double of the angle contained by the line joining the points of contact, and the diameter drawn through one of them. This problem requires no figure. Imagine a point without a circle, a line drawn from that point to the center of the circle, and lines drawn to touch the circle on each side. Join the points 206 ROBINSON'S SEQUEL. of contact and the center of the circle. Thus we have two equal right angled triangles, having the same hypotenuse, the line from the given point without the circle to the center of the circle. With the correct figure in the mind, the truth of the proposition is obvious. (7.) If from any two points in the circumference of a circle, there he drawn two straight lines to a point in a tangent to thai circle, they ivillmake the greatest angle when drawn to the point of contact. Let A and B be the two points in the circle, and CD a tan- gent line. The prop- osition requires us to demonstrate that the angle A GB is greater than the angle ADB. ACB=AOB, {th.9, b. iii, sch.) But A OB is greater than ADB, (th. 11, b. i, cor. 1), therefore, ACB is also greater than ADB. Q. E. D (8.) Fro?n a given point within a given circle, to draw a straight line which shall make with the circumference an angle less than any angle made by any other line drawn from that point. Let P be the given point within the circle, and C the center. Join PC. Through P draw APB at right angles to PC. Also, through P draw any other line as P G ; then we are to show that PBt is less than PGH. From C let fall the perpendicular CD on the chord FG. PC is the hypotenuse of the right '4i GEOMETRY. 207 angled triangle PDC \ therefore, PC is greater than CD, con- sequently the chord FO is greater than the chord AB, (th. 3, b. iii.) and the arc OAF is greater than the arc BOA. The angle POHis, measured by half the arc OAF, and PBt is meas- ured by half the arc BOA ; therefore, the angle POH is greater than the angle PBt, or PBt is less than POH. Q. E. D. N. B. The angle which any chord makes with the circumfer- ence, is the same as between the chord and tangent, — because the circumference and tangent unite as they meet the chord. (9.) If two circles cut each other, the grectiest line that can he draion through the point of intersection, is that which is parallel to the line joining their centers. Let A and B be the center of two circles which intersect in 0. Through draw mn inclined to AB, — then we are to prove that mn is less than it would be if it were parallel to AB. Draw AC and BI) perpendicular to mn, then CD^=\mn. Draw (7^ paral- lel to AB, then CH=AB ; and CZT being the hypotenuse of the right angled A CDH, GH, or its equal AB, is greater than CB. Now conceive mn to revolve on the center 0, until CD becomes parallel to AB ; CD will then become equal to AB. But mn will be all the while double of CD : therefore, mn will be the greatest when parallel to AB. Q. E. D. (10.) Iffrofrti. any point within an equilateral triangle, perpendic- ulars he drawn to the sides, they are together, equal to a perpendicular drawn from any of the angles to the opp)osite side. 208 ROBINSON'S SEQUEL. Let ABC be the equilateral A, CD a perpendicular from one of the angles on the oposite side ; then the area of the A is expressed by \AB X CD. Let P be any point within the triangle, and from it let drop the three perpendiculars FG, PH, P Oy The area of the triangle APB is expressed by ^ABy^PG. The area of the A CPB is expressed by \CBxPO\ and the area of the A CPA is expressed by ^CAy^PH. By adding these three expres- sions together, (observing that CB and CA are each equal to AB,) we have for the area of the whole A ACB, \AB{PG+ PH-\-PO.) Therefore, \ ABX CD==^AB{PG+Pir+P 0.) Dividing by i^^, gives CD=PG-\-PH+P 0. Q. E. D. (11.) If the points , bisecting the sides of any triangle he joiried, the triangle so formed, will be one-fourth of the given triangle. If the points of bisection be joined, the triangle so formed will be similar to the given A, (th. 19, b. ii.) Then, the area of the given A will be to the area of the A formed by joining the bisecting points, as the square of a line is to the square of its half ; that is, 2^ to 1, or as 4 to 1. Hence the A cut off is I of the given A- Q. E. D. {\9..) The difference of the angles at the base of any triangle, is double the angle contain£d by a line drawn from the vertex perpen- dirular to the base, and another bisecting the angle at the vertex. Let ^^C be a A. Draw A3f bi- secting the vertical angle, and draw AD perpendicular to the base. The theorem requires us to prove that the diferenne between the angles B and Cis double of the angle MAD. By hypothesis, the angle CAM= MAB. That is, CAM=MAD-\-DAB, (1) GEOMETRY. 209 ( C+CAM+MAJ) By (th. 11, b. i, cor. 4.) -j j^j^j)^j^ :90°. I (2) :90°. f (3) Therefore, B+DA£=C-{-CAM-\-MAD. (4) Taking the value of CAM irom. (1), and substituting it in (4), gives B+DAB= C-\-3fAI)+DAB+MAD. Reducing, (B—C)=2MAD. Q. E. D. (13.) If from the three angles of a triangle, lines be drawn to the middle of the op-ponte sideSy these lines will intersect each other in the same point. Let ABC be a A, bisect BCmE, AC'mF. Join AU and BF, and through their point of inter- section 0, draw the line CD. JVow if ice prove AD =DB, the theorem is true. Triangles whose bases are in the same line, and vertex in the same point, are to one another as their bases ; and when the bases are equal, the triangles are equal. For this reason the A AFO=AFCO, and the A COF == A FOB. Put A AFO=a; then A FCO=a. Also, put A COF=b, as represented in the figure. Because CB is bisected in F, the A ACF is half of the whole A ABC. Because ^C is bisected in F, the A BFC is half the whole A ABC. That is, 2«+6==25+a. Whence, a=b, and the four triangles above the point are equal to each other. Let the area of the A AD be represented by x, and the area oiDOBhjy. Now taking COD as the base of the triangles, we have 2a : X : : CO : OD Also, 25= 14 2a CO OD 210 ROBINSON'S SEQUEL. Whence, Therefore, AD=DB, 2a : y. Or, .c=y. Q. E. I). (14.) The three straigJd lines which bisect the three angles of a triangle, meet in the saine point. Let ABO be the A, bisect two of the angles A and C — the bisecting lines will meet in the same point 0. Join OB; we are required to demon- strate that OB bisects the angle B. From 0, let fall the perpendiculars on to the sides. The two right angled A's A OH and A 00, are equal in all respects, be- cause they have the same hypotenuse A 0, and equal angles by construction. In the same manner we jirove that the A CGO = £\00L Whence, 00= OL But (7(9= 0//; therefore, 0H= 01. Now in the two right angled triangles OHB and OIB, we have 0H= 01, and OB common, therefore, the triangles are equal, 'dridIIBO=OBL Q. E. D. (15.) The two triangles formed by drawing straight lines from any point within a. parallelogram to the extremities of the opposite sides, are together half the parallelogram. Let ABD (7 be a parallelogram, E any point within. We are to show that the triangles AUB, CJED, are together half the parallelogram. Through the point £ draw a line par- allel to AB or CD, thus forming two parallelograms. The A AUB is half the lower para,llelogram, and the A CUD is half the upper parallelogram ; therefore, the sum of the two A's is half the whole parallelogram. Q. E. D. (16.) The figure formed by joining the points of bisection of the aides of any trapezium, is a parallelogram. • • ^ .Ar GEOMETRY. 211 Let AB CD ho a trapezium. Draw the diagonals ^ (7, JBD. Bisect the sides in a, b, c, and (/. Join abed. We are to prove that this figure is a parallelo- ABD is a A whose sides are bisected in a and b ; therefore, tlie A Aba is equiangular to the A ABD, (th. 19, b ii), and ab is parallel to BD, and by (th. 18, b. ii), ab=\BD. In the same manner we can prove that dc is parallel to BD and equal to half of it. Consequently ab and dc are parallel and equal. There- fore, by (th. 23, b. i), the figure abed is a parallelogram. Q. E.D. (17.) If squares be described on the three sides of a right angled triangle, and the extremities of the adjacent sides be joined, the triangles so formed are equal to the given triangle, and to each other. LetJJ5(7be the given right angled triangle and construct the figure as here represent- ed. It is ob- vious that the vertical right angled A ^l-^ff^ is equal to ABC. Draw AV perpendicular to BC, and call it X. We now pro})ose to show that HO =^x. BD is produced to G, the angles VBOojidi ABffare right angles, and Il2 ROBINSON'S SEQUEL. from these equals take away the common part ABG; thus showino- th^t ABV=BBG. o The two right angled triangles AB V, JIB G are equal, because they have equal angles, and the hypotenuse AB= the hypote- nuse JIB, because they are sides of the same square. Therefore, ffG=A V, and if one is in value x, the other has the same value. Now we designate any side of the square on BC by a, then twice the area of the A AB C is ax, and the double area of the triangle HBD is obviously ax. Therefore, HBD is equal in area to ABC. In the same manner we can prove that FCE:=ABC. Q.E.D. (18.) If squares he described on the hypotenuse and sides of a right anffled tnangle, and the extremities of the sides of the former, and the adjacent sides of the others he joined, the sum of the squares of Che lines joining them ivill he equal to five times the square of the hypotenuse. (See figure to the last Theorem.) In the right angled triangle HGD, we have x'-+{BG+ay={HDY (1) In the right angled triangle PFE, we have x-+{PC^y={FEY (2) Expanding (1) and (2), and observing that GB-=BV, PC= CV, we shall have x^-\-{BVy-\-2a(BV)-{-a^=(IIDy (3) And x''-{-(FCy+2a{FC)+a-=:(FFy (4) By adding (3) and (4), and observing that x- -\-(B V)" =b^ , andic2+(PC)2=c2, then (J2^^2 )^2a(i? r+ VC)-{-2a^=(IIFy+(FCy That is, a2_j_2rt(o)_j_2«2^ Or, 5a^ = (IIDy+(FCy Scholium, The sum of the squares of the sides of the last figure is 8a ^. (19.) The vei'ticol angle of an ohlique-angled triangle, inscribed in a circle, is greater or less than a right angle, hy the angle contained be- tween the base and the diameter draumfrom the extremity of tJie base. * GEOMETRY. 213 Let AE C be a A liav- ing the angle A CB grea- ter than a right angle, and describe a circle a- bout it. From one ex- tremity of the base as B draw the diameter BD. The angle DBC is a right angle, because it is in a semicircle. The ver- tical angle A CB is grea- ter than a right angle hj ACB; but ACD is equal ABD, because each is measured by half the arc AI). Therefore ACB is greater than a right angle by ABD. Next let A'CB be the A ; the angle A'CB is less than a righ^ angle by the angle DCA'=DBA\ hecause each is measured by half the arc DA'. Therefore, the vertical angle, (fee. (20.) If the base of any triangle he bisected by the diameter of its circumscribing circle, and from the extremity of that diameter, a per- pendicidar be let fall upon the longer side, it ivill divide that side into segmerds, one of which will be equal half the sum, and the other half the difference of the sides. Let .42? (7 be the A, bisect its base by the diameter of the circle drawn at right angles to AB. From the center let fall Om at right angles to A C, it will then bisect ylC From the extremity of the diameter B, draw Bfh perpendicular to ^1 C, and consequently par- allel to Om. Produce 214 ROBINSON'S SEQUEL. Hh to M and join ML. Complete and letter the figure as represented. The two triangles Aah and Hha are equiangular. The angle a is common to them, and each has a right angle by construction, therefore the angle H=^ the angle A. But equal angles at the circumference of the same circle subtend equal chords, (th. 2, b. iii ;} therefore CB^=ML. The angle HML is a right angle, be- cause it is in a semicircle, therefore ML is parallel to AC, andJfZ is bisected in n. Now Am^\A C. nL^md^ \ML=\B C. Therefore by addition, Am-\-md=\(AC-\-CB,) Or, Ad=\(AC-^CB.) Q. E. D. Cor. If Ad is the half sum of the sides, dc or Ah must be the half difference ; for the half sum and half difference make the greater of any two quantities. (21.) A straight line dv&wn frmi the vertex of an equilateral trian- gle, inscribed in a circle, to any point in the opposite circumference, is equal to the two lines together, which are dratvn from the extremities of the base to the same point. Let ^i>6'be the e- quilateral A in a cir- cle. Take I> any point in the arc between i> and C, and join A.D, BD, andi)a Designate each side of the given triangle by a. Now ABDC is a (juadrilateral in a cir- cle, AD is one diago- nal and BC iho, otlier, and by (th. 21, b. iii) \ve have u(AD)=a{BD)+a{DC,) Diyiding by a, and AD=:BD-^DC. Q. E. D. GEOMETRY. 215 (22.) The straight line bisecting any angle of a triangle inscribed in a given circle, cvis the circumference in a "point lohich is equidistant from the extremities of the sides opposite to the bisected angle, and from the center of a circle inscribed in the triangle. (See the figure to the last Theorem.) The angle BAD is measured by half the arc BD, (th. 8, b.iii) and the angle DA C is measured by half the arc D C ; therefore, if BAD=DAC, the arc BD must equal the arc DC. (23.) If from the cerder of a circle a line be drawn to any point in the chord of an arc, the square of that line, together with the rectangle contained by the segments of the chord, will be equal to the square described on the radius. (See the figure to the 21st Theorem.) From the center draw F to any point in A 0, and through the point Fdraw nm at right angles to OV, and join Om ; then Vm is a right angled triangle. Therefore, ( F)^-|-( Vmy = {Om)\ But (Vmy = (nV) (Vm)=(AV) (VC), (th. 17, b. iii.) Therefore, by substitution, { OVy-\-{AV) (VC)=( Omy . Q. E. D. (24.) If two poiMs be taken in the diameter of a circle, equidistant frmn the center, the sum of the squares of the two lines drawn frmn these points to any point in the circumference will be always the same. Let C be the cen- ter of a circle, and A any point in the cir- cumference. CA= r, the radius. Put AD=y, DO =ar, and CB and CG each =a. Then BD =(x — a), and DG =(x+u). Now in the triangle ADB we have y^-{-(x^ay=(ABy. And in the triangle ADG, y^-^(x-^ay—(AGy 216 ROBINSON'S SEQUEL. By expanding and adding, we find The triangle -4i> (7 gives 2?/^ -\-2x^ —^r'^ ; therefore, 2r^+2a^=(ABy-\-(AGy . Because the first member of this equation is the same for all values of x and y — that is, because it is invariable ; therefore the second member must also be invariable. Q. E. D. (25.) If on the diameter of a semicircle two equal circles be described^ and in ike space included by the three circumferences, a circle be in- scribed, its diameter will be two-thirds the diameter of either of the equal circles. It is sufficient to represent a portion of the figure. Let B be the cen- ter of the semicircle, and BA the diame- ter of one of the e- qual circles, and E the center of the cir- cle sought — BD be- ing at right angles to AB from the point B. Put CB=r, and DE=x. Then BD =2r, BE=:^2r-—x, and CE=r-\-x. Now in the right angled triangle BEC, we have That is. By expanding. Reducing, Whence, {CBY+{BEY=(CEY. r2J^[2r—xY={r+xy. 7.2 +4r2 ^Arx+x"" =r^ J^^rx+x^ . x=§r. Q. E. D. (26.) If a perpendicular be drawn from the vertical angle of any triangle to the base, the difference of the squares of the sides is equal to the differeyice of the squares of the seginerUs of the base. GEOMETRY. 217 Let ABC be any triangle. Let fall AD perpendicular to the base. Now the two right angled triangles give us (ADy-{-{BDy={ABy. And (ADy-\-(I)C)^={AC)K By subtraction, (BDy—{D Cy^jABy'—jA C) ^ . Q. E. D. By factoring, {BD-{-DC){BD—DC)={AB-\-AC){AB—AC.) By observing that (BD-\-J)C)=BC, and converting this equa- tion into a proportion, we have BC : (AB+AC) : : (AB—AC) : (BD—DQ.) (This is Prop. 6, Plane Trigonometry, page 149, Robinson's Geometry.) ScHo. This proportion is true whatever be the relation of AB to AC. It is true then when AB=AC. Making this supposition, then BD becomes equal to D C, and the proportion becomes BC : AB+AC : : : 0. Now (AB-^AC) being two sides of a triangle are greater than the third side BC ; therefore the last zero is greater than the first, an apparent absurdity. But this is no more than saying that zero divided by zero can have a positive quotient — for we can subtract zero from zero as many times as we please, and still have zero left. The proportion is obviously true, for times BC is equal to times {^AB-\-AC.) Indeed may be to 0, as a to any quantity Avhatever. ( 27.) The square described on the side of an equilateral triangle is equal to three times the square of the radius of the circumscribing circle. Let ABC be the equilateral tri- angle. Let fall the perpendicular AE on the base ; it will bisect the base. Draw BD bisecting the an- gle at B. D will be the center of the circumscribing circle, and AD or BD will be the radius. We are to prove AD=BD^ and find the value of BD in terms of AB. 218 KOBINSON'S SEQUEL. Each angle of an equilateral triangle is 60^, (^ of 180°.) If we bisect these, each division will be 30°. Hence BAD=^30°, and ABJ)=30° ; therefore, AD=BJ). Put AB=2a, then BE=a. Also put BD==x, then DE=lx* Now in the riifht an^^led trianojle BDE, we have Whence, ^a^^-dx"". But ^w" ={ABY . Therefore, {ABY^7>{BDY, Q. E. D. (28.) The sum of the sides of an isosceles triangle, is less than the sum of any other triangle on the same base, and between the same parallels. Let ABC be the isosceles tri- angle. AB=AC. Throuorh the point A draw GAH parallel to BC. Take G any other point on the line GH, and'draw i?6^and GO. We are to show that AB-\-AO are less than^C 6^-|- G C. Produce AB to D, making AJ)=AB, or AC. Then by reason of the parallels GH and B C, the angle BAH is equal to the angle ABO, and IIAC=ABC. Because AD=AC, the anole ADII= the anerle ACH. Whence the two triangles AD If and ACH, are equal in all respects, and GB is perpendicular to -DC; whence any point in the line GHis equally distant from the two points D and C. Now the straight hne BI)=BA-\-AC, and because I>G=GC, B G-\- GB= GB+ GC. But X> 6^+ GB, the two sides of a A are greater than the third side jDB ; therefore, GB-\-GC are greater than BD, that is, greater than B^A-fAC. Q. E. D. ' *This mjglit not be admitted, at tlie same time the reader would readily admit that BE was one-half AB. ABE is aright angled triangle, one angle being 30 deg. the side opposite that angle is half the hypotenuse, and this is a general truth. Now the angle DBE equals 30 deg., therefore DE is half BD. GEOMETRY. 219 GEOMETRICAL CONSTRUCTIONS. (29.) In any triangle, given one angle, a side adjacent to the given angle, and the difference of the other two sides, to construct the triangle. Let ^^ represent the giv- en side, and from one ex- tremity as Ay make the an- gle BAC= to the given angle, (prob. 5, b. iv.) Take AJ)= to the given difference of the sides, and join DB. From the point B make the angle DBO equal to the angle BDC, then CB=CD, smd AD is the given diflPerence of the sides, and ABC is the triangle required. (30.) In any triangle, given the base, the sum of the other two sides, and the angle opposite the base, to construct the triangle. Draw AC equal to the sum of the sides. From the point ^ as a center, with a radius e- qual to the given base AB, describe an arc as represented in the fig- ure. From the point C in the line A C, make the angle ACB equal to half the given angle. If the problem is possible, this line CB will cut the circular arc in two points, B and B'. From B and B' make the angles CBD and CB'D', each equal to the angle at C. Join AB, AB', and either A ABD or AB'D', fulfils the required conditions. For CD=DB, and CD'=B'D', (because they are sides of a A opposite equal angles,) therefore AD-{-DB=mA C ; also AD'-^- 220 ROBINSON'S SEQUEL. D B'-^AC. The angle ADB is double the angle C, (th. 11, b. i,) therefore it is the angle required. ' (31.) In any triangle, given the base, the angle opposite to the base, and the difference of the other two sides, to construct the triangle. Subtract the given angle from 180°, and divide the remainder by 2, designating the result by a. Draw an indefinite line as AC, (see figure to 29,) and take AD equal to the given difference of the sides. From the point D, make the angle CDB-=-a. From ji as a center, with a radius equal to the given base AB, strike an arc, cutting DB in B. At J5make the angle DBC=a; then DC==BC, and ABC will be the triangle required. (^^2.) In any triangle, given the base, the perpendicular, and the angle opposite to the base, to construct the triangle. Draw AB equal to the given base, and D C parallel to it at the given perpendicular dis- tance. On the other side of the base AB, make the angle BAG e- qual to j^art of the gi^i^en angle, and ABG equal to the reinain- ing part, thus forming the A AGB. About the A ABG, describe a circle cutting DCm the points D and C. Join A C, CB, and xiCB is the triangle required. The angle BCG=BAG, (th..9, b. iii, scho.), and the angle ACG=ABG. Therefore by addition, ACB=BAG+ABG; that is, ACB-= the given angle. The triangle ADB will also answer the conditions ; for ACB =ADB. (33.) In any triaiigle, given the base, the ratio of the two sides, a7id the line bisecting the vertical angle, to construct the triangle. GEOMETRY. 221 Draw the base EG, and bisect it in i). Draw DB at right angles to EG. Divide EO in the point /, so that ^/ shall be to IG in the ratio of EH to HG, Find IB of such a ralue that HI : GI : : EI : IB. The three first terms are given ; therefore the fourth is known. From / as a center, with the distance IB as radius, strike an arc, c^utting DB in B. Join BI and produce it to H, making ZST equal to the given distance. Join EH, HG^ and EHG is the A required. Because HIy^IB=EIy^IG, a circle which passes through the points E, B, and G, will also pass through the point H, and the angle EHI=IHG, and for that reason EH \ HG \ \ EI \ IG, as required. (See th. 25, b. ii.) (34.) To draw a straight line through any given j^oint within a triangle to meet the sides , or the sides produced, so that the given point shall bisect the line so drawn. Let ABO be the A, and /^ the given point within it. Through F it is required to dra2v the straight line gl, so thai gP shall be equal PL From P draw PH paral- lel to AB. Take gH=AH. Join gP and produce it to I, and gl is the line required. P^is parallel to Al, gH : HA : : gP : PL ^ But gH=HA ; therefore gP=Pl. ScHO. Had we taken Hg double of AH, then gP would have been double of PI, and we might have required gP to be any number of times PL Because (35.) Find the square roof of \3 or any other number, by a geO' metrical construction. 222 ROBINSON'S SEQUEL. Divide the number into any two factors, (say 2 and 6},) add the factors to=2 and DB=Q^; then the length of DE applied to the same scale will show the square root of 13. Because ABxDB={DJSy. When the two factors are very nearly equal, D will be very near the center of the circle, and DF will be very nearly the ra- dius of the circle, — always a little less, unless the factors ar«^ absolutely equal ; in that case each one is a root. On this prin- ciple toe extracted square root in the first part of this volume. Observe the A F> CE. CE is the half sum of the two factors, and DC is their half difference. Also, DE is the sine of the arc AE^ and DC is the cosine of the same arc ; therefore, ive can if we desire it, bring in the aid of a. table of natural sines and cosities. But the tables of natural sines are adapted to radius unity ; lieie the radius is 4|, therefore to have corresponding values of CD and DEy we have this proportion, ^ : \ : : ^ : .52941, The result of this proportion carried to the table of natural sine ; gives .848365 for the corresponding cosine, and this multiplied by 4J, gives 3.605551 for the square root of 13. Another Construction. (36.) Let it be required to find the square root of 250, (or any other number,) by a geometrical construction. # GEOMETRY. 223 Divide the number into two factors. Let one factor be represented by AB, the other hy AC; BC being their diflference. On the difference as a diameter, describe a circle. From the extremity A, draw AD touching the circle. AD represents the square root required. By (th. 18, b. iii, scho. 1), ABXAC={AD)K Therefore AD is the square root of the product of the two factors AC 2iTidiAB. Reinark. When the two factors are nearly equal, the circle will be very small, and AD will be very nearly Ao. But xio is the half sum of the factors AB and A C, hence we know that the square root of the product of two factors is always a little less than their half sum, unless the factors are absolutely equal. In the proposed example, we divide 250 into the two factors, 25 and 10 — their diflference is 15. Hence 7^ is the radius of the circle. Take 7|- from any scale of equal parts in the dividers, and describe a circle. Draw any diameter as B C, and produce it to A, making AB=^ 10. From A draw^i) to touch the circle ; take that distance in the dividers and apply it to the scale, and the result will be the square root of 250. The practical difficulty in this construction is to decide exactly where the point D is, therefore the first method of construction is the best. Geometrical constructions are not to be relied upon for numer- ical accuracy, but they are invaluable to impress theory, and are sure guides to numerical operations. Scho. If it were required to make a square equal to a given rectangle, either of the two preceding constructions may be applied. Let ^C be one side of the rectangle, AB the other; then AD will be a side of the required square. 224 ROBINSON'S SEQUEL. PROBLEMS. The following problems do not admit of geometrical construc- tions, in the sense of some of the preceding — they require alge- bra applied to geometry. We take the problems from Robinson's Geometry, pages 105 to 109. For theory, the reader must look elsewhere. We omit the first two problems, and number them as they are numbered in the geometry. PKOBLEM 3. In a triangle J having given the sides about the vertical angle, and the line bisecting that angle and terminating in the base, to find the base. Let ABC he the A, and let a circle be circumscribed about it. Divide the arc AliJB into two equal parts at the point jE, and join £JC. This line bisects the vertical angle, (th. 9, b. iii, scho.) Join BU. Put AD=x,' JDB=g, AC=a, CB=b, CD=c, and DjE=w. The two A's, ADC and JiJB C, are equiangular ; from which we have, w-\-c : b : : a : c. Or, cw-^c^ =ab. (1) But as JtJC and AB are two chords that intersect each other in a circle, we have, cw=xy (th. 17, b. iii.) Therefore, xg-\-c'^=ab (2) But as CD bisects the vertical angle, we have, a : b : : X : y (th. 23, b. ii.) Or, x=^-l (3) Hence, -y' -\^^ zz=ab : or, y=J( b" ' ' " \j ^ And, x=?J6»-el* b^ a Now as X and y are determined, the base is determined. N. B. Observe that equation (2) is theorem 20, book Hi. GEOMETRY. 225 PROBLEM 4. To determine a triangle, from the base, the line bisecting the ver- tical dngle, and the diameter of the circumscribing circle. Describe the circle on the given diameter AB, and divide it in two parts, in the point i>> so that ADxI>B shall be equal to the square of one-half the given base. Through D draw ED G at right angles to AB, and EG will be the given base of the A- Put AD=^n, DB:=m, AB=d, DG=b. Then, n-\-m=d, and nm=:b^ ; and these two equations will determine n and m ; and therefore, n and m we shall consider ijis known. Now suppose EHG to be the required A, and join TUB and HA. The two A's AHB, DBF, are equiani^ailar, and therefore, we have, AB : HB : : IB : DB. But BI is a given line, that we will represent by c ; and if we put IB=iv, we 8hall have IIB=c-\-w ; then the above proportion becomes, d : c-f-w : : w : m. Now w can be determined by a quadratic equation ; and there- fore IB is a known line. In the right angled A DBI, the hypotenuse IB, and the base DB, are known ; therefore, I>I is known, (th. 36, h. i); and if i)/is known. Eland IG are known. Lastly, let EH—x, IIG=y, and put EI=p, and IG=q. Then by theorem 20, book iii, pq-\-c^:=xy (1) But, X : y : : p : q (th. 25, b. ii.) ^ Or, x^^l (2) q And from equations (1) and (2) we can determine x and y, the sides of the A ; and thus the determination has been attained, carefully and easily, step by step. PROBLEM 5. Three equal circles touch each other externally, and thus inclose one acre of ground ; what is the diameter in rods of each of these circles ? 15 226 ROBINSON'S SEQUEL. Draw tliree equal circles to touch each other externally, and join the three centers, thus forming a triangle. The lines joining the centers will pass through the points of contact, (th. 7, b. iii.) Let H represent the radius of these equal circles ; then it is obvious that each side of this A is equal to 2JR. The triangle is therefore equilat- eral, and it incloses the given area, and three equal sectors. As each sector is a third of two right angle*, the three sectors are, together, equal to a semicircle ; but the area of a semicircle, whose radius is M, is expressed by - -*^-- (th. 3, b, v, and th. 1, b. v); and the area of the whole triangle must be -f"^^^ * but the area of the A is also equal to Ji multiplied by the per- pendicular altitude, which is BJS. . Therefore, Or, Hence, i22j3=---+160. 2 723(2^3— rt)=320. 320 320 2^3—3.1415926 0.3225 i?= 31.48-1- rods for the result. :992.248. PROBLEM 6. In a right angled triangle, having given the hose and the sum of the perpendicular and hypotenuse, to find these two sides. Let ABC he the A. Put CB= h, AB+AC=a, AB=x ; then AC =a — X. By (th. 36, b. i), a'—b^ Whence, 2a Now the numerical value of x being known, the triangle can be constructed geometrically. GEOMETRY. 227 PROBLEM 7. Given the base and altitude of a triangle^ to divide it into three equal parts, by lines parallel to the base. Let ^^ 6" represent the A. Conceive a perpendicular let drop from C to the base AB, and represent it by b. Put 2a=AJB. Then ab= the area of the triangle. Let X be the distance from C to FD ; then by (th. 22, b. ii), we have, x^ : b^ : : ^ab : ab Whence, x ; b : i \ \ J^, If X represents the distance from C to GE, then x^ : b^ : : f«6 : ab. Or, X : b : : Ji t J3, ^=^ We perceive by this tliat the divisions of the perpendicular are independent of the base, and that we may divide the triangle into any required number of parts, m, n, p, ==b, &c. Then \^^'+y'=^''\ By add. 5x^ +^'' =a^ ■\-b^ =din. 3^ 2 «2 a* — m. x=Ja^ — m. y==.Jb^ — m. ~T~ "~3~" PROBLEM 15. To determine a right angled triangle, having given the perimeter and the radius of its inscribed circle. Let ^^C be the A, OjE'the radius of the circle. It is obvious that AE=A£>, CF= CD. Put AE=x, CF^y, FB^T, ^p=. the perime- ter. Then by the conditions, :t+y-fr=p (1) From the right angled A ABC, we have (x+rY+(y+TY=^(x+yy By reduction, rx-\-ry-\-r^ =xy That is, (x-^y-\'r)r^=^rp^=xy Equation (4) expresses the area of the triangle. From (1), x^'-^-^xy+y' From (4), Axy = (2) (4) '2 — =o2 — 2»r-4-r^ . Apr By subtraction, x^ —2xy-\-y^ =p^ — 6pr-|-r» . GEOMETRY. fftt Whence, x — y= ± Jp ^ — 6jt?r-[-r - . But x-{-y—p — r. Therefore, x=z^{p—r)±^Jp''—Qpr+r^. PROBLEM 16 To determine a triangle , having given the base, the perpendicular, and the ratio of the two sides. Let ABO be the A- AB=b, CD=a, J)B=x. Then CB:=/a:^^+^. Let the given ratio of the sides be as fn to n ; then J(b — x)^-\-a^ : ^a^-\-x'^ : : m : n. This proportion will give the value of or, then AC and CB will be known. PROBLEM 17. To determine a, right angled triangle, having given the hypotenuse, and the side of the inscribed square. Let J[i) (7 be the A. (See last figure.) Put CI—x,IE=ia, A G=y, and A C=h. Then by proportional triangles, we have CI : lU : : JfJG : GA. That is, X : a : : a '. y. Whence, 2-5^=0^. In the right angled A's AGE, ECL we have AE^ Jf~+a^. EC= Jx^^a- . Observing that AE-\-EC=AC—b, and a~—xy, we perceive that Jx^-\-xy-\'Jy~-{-xy^h. Whence, »Jx+Jy=. A^^ . ■_ 7 3 By squaring, x-\-y-\-^Jxy—-,!—- . x+y ___ Put (a?+y)=-5, and observe that ^Jxy=^Za ; then 52 -f. 2^5=^2 _ 232 ROBINSON'S SEQUEL. Whence, «+a= zt^a^-j-i^. Now having the value of (x-^-y), and (ary) the separate values of X and y can be determined, which is a solution of the problem. PROBLEM 18. To determine the radii of three equal circles, inscribed in a given, circle to touch each other, and also to touch the circumference of the given circle. Let AD B be the given cir- cle. Di- vide the circumfe- rence 360 deg. into 3 equal parts. BD is one of those parts 120° ; then the arc ^i)=60°. A circle inscribed in the A COE, will be one of the equal circles required. Let A 0=a, AH=x, H being the center of the circle. From H, draw i/F" perpendicular to CO, then AH=:HV. Hence FIVz!=x, OH=a — i?-, and F= J- 0^, because the an- gle F^0=30°. (See prop. 1, plane trig., page 139.) Now by the right angled A VH, we have {OV)^-[-{VHY={OHy. That is, (^^i::fy+a:2=(a— ar)«. ^ Whence, a;={273— 3)a. PROBLEM 19. In a right angled triangle, hamng given the periimter, or sum of all the sides, and the perpendicular let fall from the right angle on the hypotenuse, to determine the triangle, that is, its sides. GEOMETRY. 238 Let ABC he the A, and represent its perimeter byjo. Put AI>=:a, AB:=x, A C=y. Then B C=zp—x—y, Because BA C is a right angle, x^-\-y^-=p^—2p{x-\-y)-\-x^-\.2xy+y And, a{p — x — y)=xy Reducing (1), ^p{^-\-y)='P^-\-^y Double (2), 'Hap — 2a{x-^^y) = '2.xy By subtraction, (2a-j-2p) (^+y) — ^op=p^ Whence, x-\-y _p^-\-2ap (1) (2) (3) (4) (6) (6) Because BC=p-x^, BC=p-t±^l= ^t.._ 2«+%) 2rt+2/) OjB" From (2) we observe that xy ^ ' 2a4-2p Equations (6) and (7), will readily give x and y. (7) PROBLEM 20. To detennine a right angled triangle, having given the hypotenuse and the difference of two lines, drawn from the two acute angles to the center of the inscribed circle. Let ABC be the A, the center of the inscribed circle; then A bisects the angle CAB, and CO bisects the an- gle C. The angle A OH, being the exterior angle of the trian- gle A C, it is e- qual to CAO-\- ACO,i\mi'is,AOH is equal to half the sum of the angles CAB, BCA, or to 45°. Produce CO to II; from A let fall All 234 ROBINSON'S SEQUEL. perpendicular on CA. Now in the A A OH, because ^=90°, and ^0//=45°, OAII=45'', and consequently AJI= Off. Put AC=a, AO=x, OC=x+d, Off and Aff, each equal to y. Then Cff=x-\-i/+d. In the A AffO, we have 2y2_^2 ^jj In the A ^^C', we have (x+y+d)'-\.y^=.a^ (2) Expanding, x''-{-y^-{-d'+{2x-\-2d)?/-\-2dx+y'' =a^ (3) Substituting the value of y^ and y from (1), and 2x^-{-(2x+2d)-^-+2dx=a''—d^. Or, 2x''+j2'x''-\-j2dx-{-2dv=a''^^\ Dividing by (2-[->/2), and we have 2 I J «^ — <^^ ^ 2+V2 Whence, ar = — ^±^/m4-^^ PROBLEM 21 To detei-mine a triangle, having given the base, the 2^^fp€ndicular, and the difference of the two sides. (See figure to Problem 19.) Let ABC be the A- Put BD=^x, DC^y, AC=z, AB= z+d, AD=a, BC^h. By the conditions, ar-|-y=6 (1) x^-\^^-=z'-\-2dz-{-d' (2) y'+a'=z^^ (3) By subtraction, x^—y^=2dz-\-d^ (4) Factoring, (^+y) (^ — y)=d(2z-\-^) That is, b(x—y)=d{2z+d) From this we have the proportion, b : (2z+d) : : d : (x—y) This proportion is the following rule given in trigonometry, viz : In any plane triangle, as the base is to the sum of the sides, so ii the difference of the sides to the difference of the segments of the base. GEOMETRY. t$6 We return to the solution. From ( 1 ) we have rr=a — y, whence a;^ — y'^=:ar — 2ay. From (3), z=Jy^-\-a^. These values put in (4), give a^ — 2ay=2c?7y2+^+c?2 Squaring, (^a^—d^Y—Aa{a^—d'' )2/+4a-r =^d''y''+^o''d^ Or, (a2— ^a^a_4^(„2_^2 jy_|_4(g2__^2 yf^^a^'d- 4^ 2 ^2 a^ — d^ — ^ay-\-^y^ ==- - a 2 — <;- a . , '. 2 ,o , 4«2(/2 5«2J2_^4 ar^2y=±:dj^ ^ a Whence, y=-=p-( I ^ 2^2\ a^—d^ / PROBLEM 22. To determine a triangle, having given the base, the perpendicular, and the rectangle, or product of the two sides. (See figure to Problem 19.) Let ABGhQ the A. Put BD=x, DC^y, BC=b, AD=za, and the recangle, {AB) (AC)=c. Now in the right angled triangles, ADB, ADC, we have AB=Jx^--\-aK AC^Jy^+a\ Whence, Ux-+a-){Jy^^+a^^)==c (0 And, x+y=l (2) From (1), x^y^+a^x'+y')+a'=c' (3) From (2), x^-\-y''=b''—tcy (4) This value substituted in (3), gives x^y^J^a^b^—'^a^xy-{-a^ =c* x^y^—^^xy+a^—c^—aH^ 2A2 xy-^a^ = ±:Jc^—aH xy=a''±:Jc^—a^b^ (6) 236 ROBINSON'S SEQUEL. From equations (2) and (5) the values of x and y can be de- termined. PEOBLEM 23. To determine a triangle, having given the length of the three lines dravmfrom the three angles to the middle of the opposite sides. Let ABC be the A. Bisect the sides AB in D, AC in F, CB'mE. Put AE=^a, BF=h, CD =e, AD=u, AF=x, BE Now by (th. 39, b. i), we have, x^+b^=4u^-\-4g^ (1) (2) (3) By addition, a^-\-b^-\-c^=7(x'--\-y'^+u'') Whence, ^a^^^-^-c'^^^ix^Jl^iy^+iu'- From (1), 2^2^c2 = 4a;2-f4y2 (4) By subtraction, ^a^-{'b^-\-c^)—u^—c^=4u^ Or, 4a2_[-452_j_4(.2_7^^2_7c2^28w2 4a3_J_462_3c2=:35«<2 By inference And, t^-±J^' 2+4*2- -3c2 V 35 ^-±J4«- 2+4c2- -362 rt^ 36 V=r^J'^ 2+462- -3a2 36 PROBLEM 24. In a triangle, having given the three sides, to find the radius of the inscribed circle. GEOMETRY. 237 Let ABC be the A. From the center of the cir- cle 0, let fall the perpen- diculars OG, OE, OD, on the sides. These perpendiculars are all eqiial, and each equal to the radius required. Let the side opposite to the angle A, be represent- ed by a, the side opposite B by h, and opposite C by c. Put OE, OD, (fee. equal to r. It is obvious that the double area of the A BOC is expressed by ar ; the double area of A OB by cr ; the double area of ^ C by hr; Therefore, the double area of ABC is (^a-{-h-\-c)r. From A let drop a perpendicular on BC, and call it x. Then cw= the double area of ABC. Consequently, {^a-\-b-\-c)r=iax ( 1 ) The perpendicular from A will divide the base BC into two segments, one of which is Jc^ — x'\ the other, Jb'^ — x^, and the sum of these is a ; therefore, (2) '^=a^—2aJb^—x^J^b^—x'- ^aJb^—x^=a^-]-b^—c^ J^'' -X^=z 2a Whence, Or, ar=^62 — m"^ This value of a; put in (1), gives Whence, __ajb''—m^ a-\-b-\-c the required result. PROBLEM 25. To determine a right angled triangle, having given the side of the inscribed square, and the radius of the inscribed circle. 23^ ROBINSON'S SEQUEL. Lei O be the center of a cir- cle, OH or OL=ir, the given radius, BE or ED^=a, a side of the given square. BO'i^ the diagonal of r-, and BD is the diagonal of «-, and B OD is one continuous line. The point D of the given square may be in the circle, in that case the hypotenuse touch- es the circle and the square in the same point, and that point is the middle of the hypotenuse. If the point D is not on the circumference, it must be without, as liere represented. Draw Dt to touch the circle in t, and that line produced both ways will define the hypotenuse. ^1(7 and ^C will meet if produced, and ABC \\\\\ be the tri- angle required. OB^J0=(a—r)j2, J)V=(a—r)j2-^r, Now as i) is a point without a circle, and Dt touching it, we have by (th. 18, b. iii), {Dty-==DVxDU; that is, (i)/)2 = [(a_r)72~r] [(a— >-)72-fr ]=2a=— 4ar-fr2. Whence, Dfz=jZa^ — 4ar-|-r2=:c. Because A is a point without a circle, and AH, At, lines drawn touching the circle, AH=At, (th. 18, b. iii, scho. 2.) Observe that KH=a — r=d. Put AH, At, each equal to x ; then in the A AXB we have AK=:x — d, AD==x-\-c, KD^c Whence, (A'-|-(-)^=a--j-(.tr — d)-. x^ +2CX+C'' =a2 J^x'^^Stdx-^d* 26+2^ Now the value of x being known, AB is known, and all the sides of the A AKD. But the A AKD is proportional to the triangle ABC, and gives us AK : KD ', i AB : BC GEOMETRY. 239 The first three terms of this proportion being known, the last is known, and the triangle is fully determined. PROBLEM 26. To determine^ a triangle and the radius of the inscribed circle, hav- ing given the lengths of three lines dravm from the three angles to the center of that circle. Let ABC be the A, the center of the circle. Put^O= a, OB=ic, OC=:h. AO bisects the angle A. Produce AO to D, Then because the angle A is bisected, CD : DB : : AC : AB. Put AB=x, AC=^y, and let the ratio of AB to BD be n\ then nx=BD and ny=^ CD. Now as the angle C is bisected by (70, we have AC : CD : : AO : OB That is, y : ny : : a : OD Whence, OD—na. Because AD bisects the angle A, we have, (th. 20, b. iii), Also, And, From (1), xy=^a^ {\'\-nY -^-n'^xy nx^ =c^-\-na^ ny^=b^-\-na^ xy. a'^ilJ^ny _a^(\^n) \—n 1— w2 The product of (2) and (3), gives n^x^y'^—{c^-\-na^) {h^^rui^) Squaring (4), and multiplying the result by n'^, also gives (1) (2) (3) (4) (5) »««y^ (6) 240 ROBINSON'S SEQUEL. Equating (5) and (6), gives This equation contains only one unknown quantity n, but it rises to the fourth power — hence this problem is not susceptible of a solution from this notation short of an equation of the fourth degree. In cases where a, b, and c are numerically given, the solution may be possible through an equation of the second or third degree. We perceive by the figure, that if b=:c, x must equal y. PROBLEM 27. To determine a right angled triangle^ having given the hypotenuse and the radius of the inscribed circle Let ABC be the A, £^0 the radius of the circle. AU =AI)= X, CD= CF=zy. Then AB =.r+r. BC=y-{-r. By the right an- gled triangle, ={x+yy {\) x-\-y=a (2) Reducing (1), gives xy=^rx-\-ry +r-. That is, xy=ar-^r^ (3) From (2) and (3), x and y are easily found. In numerical problems, great advantage can be taken of mul- tiple numbers, the same as we have shown in common algebra. The following example will be sufficient. The sum of the two sides of a 2)lane triangle is 1 1 55, the perpen- dicular drawn from the angle included by these sides to the base, is GEOMETRY. 241 I I \ HUB 300 ; the difference of the segments of the hose is 495. WIicU are the lengths of the three sides? Am. 945, 375, 780. Write the given numbers in order, thus, 300, 495, 1156. Di- vide them by 16, and their relation is 20, 33, 77. The two latter numbers have a common factor 1 1 , which call a. Put 5=20. Then the three given lines will be h, 3a, and 7a. Let CB=x,. then AC=7a — x. BD=y, then AD=y-\-3a. CD=.h. In the right angled A CDB, we have y2_|_j2^^2 (1) AD Q gives (y_[.3a)2+52==(7a— :r)2 (2) Expanding (1) and subtracting (2) from it, gives 6ay+9a2 ==49^,2 — i4«a; 3ay=20a2 — '^ax Divide by a and write h in the place of 20, then/ Sy=ab — Ix Squaring, 9y^=a''b''—14abx+49x^ From (1), 9y2=_962 + 9x^ By subtraction, 0=:(a^-\-9)b^—l4abx-{'40x' Divide by & (or 20), then 0=(a^-\'9)b—14ax-\-2x^ 4a;2— 28aa;=— 2a2 b—l 8b Add (49a2), 4x^-^2Qax+49a^=49a^—2an^l8b =9a2— .360=729 By evolution, 2x — 7a = ±27 2a:=77±27=50, or 104 a:=25, or 54 -» Here 25 is the number that corresponds with the problem ; therefore, jB (7=25- 15=375. We multiply by 15, because we reduced the numbers in the first place by dividing by 15. 16 242 ROBINSON'S SEQUEL. SECTION II. TRIOONOJWETRir. We shall here attempt to show the most practical method of finding the circumference of a circle to radius unity ; and of finding the sines and cosines. The trigonometrical equations that we may call into immediate use, are the following : We number them as they are numbered in Robinson's Trigo- nometry. sin.(a-|-5)=sin.a cos.6-l-cos.a sm.h (7) sin.(a — J)=sin.acos.6 — cos.asin.6 (8) cos.(a-|-6)=cos.a cos.6 — sin.a sin.6 (9) cos.(a — J)=cos.a cos.5-|-sin.a sin.5 (10) sin.2a+cos.^a=l (1) sin.2a=2sin.a cos.a (30) Or, sin.a=2sin.^a cos.^a By problem 23, book iv. of Robinson's Geometry, we learn that if we divide the radius into extreme and mean ratio, and take the" greater segment, that segment will be the chord of 36°. Let 1 be the radius of a circle, and x the greater segment re- quired ; then 1 : X : : X : 1 — x Whence, a:=—i-f ^^5=0.6180340, the chord of 36° in a circle whose radius is unity. We learn by theorem 6, book v, Robinson's Geometry, that when c represents any chord of a circle, and x a. chord of one- third of that arc, the following equation will exist : x^ — 3x= — c. Put c=0.6 18034000, and a solution of the equation gives the chord of 12°. Again, put c equal to the chord of 12°, and an- other application of the equation will give the chord of 4°, and thus by the successive application of this equation, we have found the following values : TRIGONOMETRY. 24» 1. The chord of 36°=0.6 18034000. 2. The chord of 12°=0.209056903. 3. The chord of 4°=0.069798981. 4. The chord of 80=0.023270628, By theorem 4, book v, we learn that if c represent the chord of any arc, the chord of hcdf that arc will be represented by Having the chord of 80' the preceding expression gives us the chord of 40', 20', and 10', as follows : Chord of 40'=0.01 16355131. Chord of 20'=0,0058 177679. Chord of 10'==0.0029088819. The chord of 10' so nearly coincides with the arc of 10', that for all practical purposes, we may consider the chord and arc the same ; then the semicircumference must be 1080 times 0.0029088819, or 3.141592462. A more exact determination gives 3.141592653+ for the length of 180'', when the radius is unity. The chords of all arcs under 10' cun be found from that chord, directly/ hy division. As the sine of an arc is half the chord of double the arc, therefore, we can have the natural sine of 18° by dividing the chord of 36° by 2. Having the sine of any arc, we can find its cosine by the fol- lowing equation : cos. «s=^l — sin.^a When sin.^a is a very small fraction, as it is for all arcs under 10', then ^1 — sin.^a is very nearly equal to (1 — ^sin*a). By the foregoing we find the following sims and cosines : sin. r=.000^.908881 cos. 1'=.9999999576 sin. 2'== .00058 17762. cos. 2'=. 9999998802 sin. 3'=.0008726643 cos. 3'=.9999996692 sin. 4'=:.001 1635524 cos. 4'=,9999993231 244 ROBINSON'S SEQUEL. sin. 5'=.001 4544405 cos. 5'=.9999989423 sin. 6'=.0017453286 cos. 6'=.9999984769 sin. 7'=.0020362167 cos. 7'=.9999979269 sin. 8'=. 002327 1036 cos. 8 =.9999972926 sin. 9=.0026179916 cos. 9'=.9999965731 sin. 10'=.0029088789 cos. 10'=.9999957689 sin. 20 =.0058177378 cos. 20'=. 9999830770 sin, 30'=.0087265343 cos. 30'=.9999618877 sin. 40'=.01 16352640 cos. 40'=.9999323090 From the chords of 4°, 12°, and 36°, we readily find sin. 2°=. 0348995000 cos. 2°=.9993908139 sin. 6°=. 1045284515 cos. 6°=. 994521 8389 sin. 18°=.309017000O cos. 18°= .951 06466 19 Having the foregoing sines and cosines, we can find the sines and cosines of certain other arcs as follows : Put 2a= to any arc whose sine is known, then we can obtain the sines and cosines of the half of 2a, or a, by the following general equations : cos.^a+sin.^a=l (1) 2cos. a sin. a = sin. 2a ( 2) Now if we suppose 2a=18°, we have sin. 2a=.3090 170000 ; and by substituting this value of sin. 2a, and adding and subtract- ing the equations, we shall have cos. 2 a+2cos.a sin. a+sin.^ a== 1 .3090 1 70000 ( 3) and cos.^a— 2cos.asin.a+sin.2a=0.6909830000 (4) By extracting the square root, cos.a-f-sin.a=l. 1441228508 (5) cos.a— sin.a=0.8312532699 (6) By adding (6) and (6), and dividing by 2, we find cos.a=cos.9°=. 9876880603 TRIGONOMElTlEtY: 246 Subtracting (6) from (5), and dividing by 2, gives sin.a=sin.9°=. 1 564342904 If we put 2a=6°, a like operation will give the cosine and sine of 3°. If we put 2a=2°, a like operation will give the cosine and sine of 1°, and so on. Again, we must not overlook the fact that the cosine of 2° is the same value as the sine of 88° ; therefore, if we put 2a=88°, an operation like the preceding will give us the cosine and sine of 44°. Another operation will give us the cosine and sine of 22°, and still another of 11°, and so on. If we require the cosine and sine of any particular arc that we cannot arrive at, by any of these subdivisions, we may apply th§ following equations : sin. (a-(-6) =sin.a cos.6-j-cos.a sin.5 (7) sin. (a — &)=sin.acos.6 — cos.asin.6 (8) For example, if a=6° and 5=1°, and we have the cosine and sine of 6° and 1° ; then (7) will give us the sine of 7°, and equa- tion (8) will give us the sine of 5°. Whence by equations (1), (2), and (7), (8), the sine and co- sine of every degree of the quadrant can be obtained without the trouble of fractional parts of degrees. But there is a better method to continue the table after a begin- ning has been made, which we illustrate by the following example : Suppose we have the sine and cosine of 15° and 16°, and also the sine and cosine of each of the small arcs from zero to 5° or 6°, and require the sine and cosine of 17° or of any other arc under 20°, we would operate as follows : Let the arc ^J9=15°, ^i>=2°; then ^^=17°, ^6^=17°, i>6^=17°+16° =32°. Draw the chord BD. Now because an angle at the circumference is measured by half its subtended arc, therefore, the angle 7ii?i)=16°. The chord BD is double the sine of 1°; and it is obvi- ous that we have BD and all the angles of the small right an- 246 ROBINSON'S SEQUEL. gled triangle nBD ; and if we compute Bn, and add it to DH, the sine of 15°, we shall have BJEy the sine of 17° ; and nD sub- tracted from CH\ will give the cosine of 17°. The computation is as follows : (We use the logarithmic sines and cosines, diminishing the indices by 10. to correspond with radius unity in the table of natural sines.) Log. sine 1° —2.241855 Log. 2 , .301030 Log. of ^i> 2.542885 —2.542885 sine 16°... .—1.440338 cosine . . .—1.982842 »i>.... 009621 —3.983223 %JS .033554— -2.525727 , Nat. cos. 15° .... 965 93 Nat. sin. 15° .258820 Nat. cos. 17°. . ..95631 Nat. sin. 17° .292374 Thus we can go on and compute the sine and cosine of 19°. RemarJc. When the triangle nBD is taken sufficienly small, the chord BD is confounded with the arc, and the triangle is then called the differential triangle, and figures largely in the differen- tial calculus ; and by it we can readily compute the sine and cosine of (16° r), (15° 2'), &c., having the sine and cosine of the de- gree, whatever it may be. If we were making a table of sines and cosines for every min- ute of the quadrant, it would require too much labor to use the foregoing equations for every minute, we would use them for every degree, and then fill up the sines and cosines for the inter- mediate minutes by INTERPOLATION. In the appendix to Robinson's University Algebra, standard edition, is the following formula for inserting any intermediate term of a series : In this formula a is the first term of a series consisting of a, a^, ^2, ttg, &c., terms, b is the first term of the first difference, c is the first term of the second difference, and so on. The interval TRIGONOMETRY. «47 between two given numbers in the series is always to be taken as unity, therefore, % is a fractional part of that unit. The following example will clearly illustrate. 1. Given the sines of 1°, 2°, 3°, 4°, 6°, and 6°, to find the sines of 1° 12', 2° 12', 3° 12', and 1° 24', 2° 24', or to find the sine of any arc between 1° and 3° by interpolation. (*«) Ist diff. Sddiflf. 3d diflf. sin. 1°=.0174524035 (-H) {-^) {-d) sin. 2°=.0348995000 .0174470965 sin. 3°=.O523359508 .0174364508 106457 sin. 4°=.0697664685 .0174205177 159331 52874 fiin. 5°=.0871557450 .0173992765 212412 53081 sin. 6°=.1045284515 .0173727165 265600 53188 To interpolate 12', we must put n of the formula = ||. 2 n — 1 4 n — 2 6 Whence, And, 10 2 n — 1 %= — n* : 10 10 n'. 1 n—\ 10 48 2 100 2 3 1000 The products will be positive or negative according to the rules of multiplication. For the sine of any arc between 1° and 2°, we take the first line of the column under a for the first term of the series, and the first line of the column under h for the first dif- ference, and so on. To find the sine of any arc between 2° and 3°, we must take the second line of the column under a for the first term of the series, and the second line of the column under h for the first difference, and so on. Whence, the following equations : sin. 1° 12'=.0174524034+T\(.0174470965)+^3.ir( 106457)— yf«~o( 52874) = .0209424308 sin. 2° 12'=.0348995000+y2^(.0174364508)+yfxr( 159331)— tHttC 53081)=.0383878114 248 ROBINSON'S SEQUEL. sin. 3° 12'=.0523359508+y\(.0174206177)+yf^(.212437)— i-UTr(-63213)=.0558214986 If we put w=||=,V» we can find the sine of 1° 24', 2° 24', and 3° 24', in precisely the same manner. In short, if we put n equal any number of times gV» we can find the sine of the degree and that number of minutes, but it is best to be regular, and find the sines to 1° 12', 1° 24', 1° 36', and so on, and then interpolate again between the numbers thus found. Little attention has been paid to this subject of late, because the labor when once done, is done forever ; and it has all been done in the preceding age ; our object has been to present a systematic view of the whole matter, and show the student that the task of computing a trigonometrical table is not so great as is generally We have thus far computed natural sines and cosines, but we generally use logarithmic sines and cosines. To find the logarithmic sine, we simply take the logarithm of the natural sine from a table of the logarithms of numbers, increasing the index bg 10. After a few logarithmic sines have been found at equal inter- vals of arc, then the intermediate logarithms can be found directly by interpolation. To make a table of logarithmic sines true to six places of deci- mals, we must compute with at least eight decimal places ; and to make a table true to nine places of decimals, we must compute with twelve decimal places. To show the advantage of working on a large scale, we will require the log. sines of 1°, 2°, 3°, 4°, 5°, and 6°, true to nin£ places of decimals. The natural sines we already have, and the necessary tabl-es of logarithms are in the latter part of this volume, the same as are to be found in our Surveying and Navigation. Most operators would take out the logarithm of each natural sine separately, having no connection mth eojch other, but this would re- quire much unnecessary labor, and it is to explain the artifices, that we bring forward the example. TRIGONOMETRY. 249 In the first place we will take the sine of 6°, that is, find the log. of the number .1045284515. Log. .104 —1.017033339299 Factor, 1.005 Table A, .002166071750 Log. prod., . 104520^7 01919941 1049 1.00008. . . Table C, 34742166 .104528 36160 W 019234153215 o Dividing the given number by this number, we find another factor to be 1.0000008. The log. of this factor corresponds to 8(c) in table C ; therefore, —1.019234153215 347432 Log. .1045284515= —1.019234500647, nearly. Add 10. Tabular log. sine 6°= 9.019234500647 Having the logarithm of .1045284515, and requiring that of .0871557425, we first consider whether we cannot find some con- venient divisor to the first that will produce the second for a quotient, or produce a number very near the second. To find definitely what this divisor is, represent it by D ; then •^^^^^-=0.087155. Whence, i>=1.2, nearly. Log. .1045284515 —1.019234500647 Divide by 1.2) log. 0.079181246048 Gives .087107043 log. —2.940053254599 Factors, 1.0005 1.00005 1.000008 1.0000008 217099966 Table C, 21714178 3474352 347435 Prod, nearly = .0871657425 log. nearly=— 2.940295890530 We found these factors by taking .0871557425 for a dividend, and .087107043 for a divisor; the quotient is 1.0005588, which 260 ROBINSON'S SEQUEL. we directly separate into the single factors, 1 .0005, 1 .00006, =1.000152. By a little examination, we shall find that if we multiply the sine of 1° by 3, and divide the product by 1.000406, the quotient will be the sine of 3° very nearly. To the log. —2.241 856 255 129 Add log. 3, 477 121 254 720 —2.718 976 509 849 Sub. log 1.0004 '.^ 173 690 053 —2.718 802 819 796 Also, sub. log. 1 .000006 2 605 764 (26) Log. sine of 3°= —2.718 800 214032 AddJlO^ Tabular log. sine 3°= 8.718 800 214 032 In a similar manner we can find the logarithmic sine of 4°. If it were our object to compute a table of logarithmic sines and cosines for every degree and minute of the quadrant, we would first compute each degree and half degree in natural num- bers — and take the logarithms of those numbers. Then we would interpolate for the intermediate logarithms. We now proceed to solve proUeTns m Trigonometry and Menswrc- tion. (Problems 1 and 2 are on page 167, Robinson's Geometry.) 1. Given AB 428, the angle C 49° 16', and (AC+CB), 918, to find the other parts. Let ABC represent the A. Draw^I>=918. From D draw DB so thatthe angle ADB shall be half the angle ACB, that is, 24° 38'. From ^ as a center with AB as a radius, strike an arc 252 ROBINSON'S SEQUEL. cutting BDm B. From B make the angle I)BC=2^° 38': then A CB will be the A required. In the AADB we have AB : AD : : sin. D : sin. ABD, That is, 428 : 918 : : sin. 24° 38' : sin. ^^i>. Sin. 24° 38' 9.619938 Log. 918 2.962843 12.682781 Log. 428 2.631444 Sin. 63° 22' 48" or its supplement 116° 37' 12". .9.951337 From this take DB C 24° 38' ABC= 9r°^59M2" Having now two angles of the A ABC, we have the third angle ^=38° 44' 48", and with all the angles, and the side AB, we find A (7=564.49, and conseqently 5(7=354.51. (2.) Given a side and its opposite angle, and the difference of the other two sides, to construct the triangle and find the other parts. Let ABC be the A. ^(7=126, 5= 29° 46', and AM, the diflference between AB and BC, =43. From 180° take 29° 46' and divide the remainder by 2. This gives the angle BMC or BCM. BMC taken from 180°, gives AMC. Now in the A AMC, we have the two sides AC, 126, AM, 43, and the angle AMC, to find the angle A. The computation is as follows : 180°— 29° 46'=150° 14' ; half, =75° r=BMC. 180°— 75° 7'= 104° 52,'= AMC. Now in the A AMC, we have AC : AM '. : sin. 104° 63' : &m. ACM Sin. 104° 63'=cos. 14° 63'. 126 : 43 : : cos. 14° 53' : sm. ACM Cos. 14° 63'. 9.986180 Log. 43 1.633468 11.618648 Log. 126 2.100371 fiin. ^C7Jf=sin. 19° 22' 28" 9.518277 i TRIGONOMETRY. 253 Whence, ^(7^=76° 7'+ 19° 22' 28"=94° 29' 28". Conse- quently ^=55° 51' 32". Now we have all the angles, and AC, of the A ABC. (3.) Two lines meet, making an angle of 50°. On one line are two objects, one 200, the other 500 yards from the angular point. Where abouts on the other line will these two objects appear under the greatest possible angle, and what unll that angle be? LetP^andPi) be the two lines, and A and B the two objects. Let i) be the re- quired point on the other line. Then pd=jTaxpb = J500X 200 = 316.226+ yards; but this requires demonstration. If we make PD=JPAxPB, and then pass a circle through the points A, B, and D, PD will touch the circle in the point D. (Th. 18, b. 3, scho.) And because PD is a tangent, the angle ABB at the point of contact, is greater than any other an- gle AdB, on either side of D, (see th. 7, page 101 of this volume.) Or we may prove it here. AeB=ADB, (th. 9, b. iii, scho.); but AeB is greater than AdB, therefore, ADB is greater than AdB ; that is, greater than any angle drawn from any point be- tween P and D. The same demonstration will apply on the other side of D. The computation for the angle, is as follows : From D let drop the perpendicular DH, then in the A PDH, we have As radius, 10.000000 10.000000 ^ To PD. 2.600000 2.500000 * So is sine 50°, To DH, 242.24, 9.884262 cosme 9.808067 2.384252 PH, 203.26, 2.308067 m 254 ROBINSON'S SEQUEL. From PH take PB, and we have JIB==3.26. From PA take PJI, and we have Aff=^296.S4. Now HB \ HD \ \ ; R : : tang. ABD, AH '. HD '. \ : E ; : tang. BAD. 12.384252. 12.384252 HB 0,513218 AH 2.471800 Tan. ABB 89° 14' 11.871034 tan. BAB 39° 16' 9.912462 180°— (89° 14'+39° 16') =51° SO'=:ABB, the greatest angle required. At the point G on the line PG, the objects A and B would extend the greatest possible angle, and in that case also, PG= JPAxPB7~ But the angle A GB must be of such a value that AI)B+AGs=nO'' ; therefore, ^6^^=128° 30'. (The following are on page 174, Robinson's Geometry.) (3.) From an eminence q/* 268 feet in perpendicular height, ike angle of depression of the top of a steeple which stood on the same horizontal plane, was found to he 40° 3', and of the bottom 56° 1 8'. What was the height of the steeple? Ans. 1 1 7.8 feet. Let BC hQ the eminence 268 feet, and AD the steeple. Draw CE par- allel to the horizontal AB. Then JSCD=40'' 3', £CA=CAB=56° 18'. i)(7^=56° 18'— 40° 3'=16° 15'. 2)^C=90°— 66° 18'=33° 42' In the A ABO, we have sin. Qe"" 18' : 268 : : sin. 90° : AC. ^^_ 268X-g sin. 56° 18' In the A ADC, we have the supplement to the angle ADC equal to 16° 15' added to 33° 42', or 49° 57' ; therefore. As sin. ADC : AC : : sm. DCA : AD 268Xi2 That is, sin. 49° 67' : sin. 56° 18' sin. 16° 15' : ^i> TRIGONOMETRY. 256 ^7)^ 268'.a-sm. 16° 15^ _ 2.428135+1 0. +9.446893 __ sin. 49° 67' -sin. 66° 18' 9.883836+9.920099" ~" 21.876028—19.803936=2.071093 Log. ^i>=2.071093. Whence, ^2>=1 17.78 feet. (4.) From, the top of a mountain three miles in height, the visible horizon appeared depressed 2° 13' 27". Required the diameter of the earth, and the distance of the boundary of the visible horizon. Ans. Diameter of the earth 7968 miles, distance of the hori- zon 164.64 miles. Let AB represent the mountain, and AD the visible distance. AB produced will pass through the center of the earth at C. From D draw CD perpendicular to AD. Join BD. AD (7 is a right an- gled triangle. C^i>=90°— 2° 13' 27"=87° 46' 33". ACD=9P 13' 27". ADB=^ACD= r 6' 44". ^^i>=91° 6' 44". " No-jF in the A ABD, we have sin. 1° 6' 44" : 3 : : sin. 91° 6' 44" : AD. Sin. 91° 6' 44"=cos. 1° 6' 44" 9.999919 Log 3 0.477121 10.477030 Sin. 1° 6' 44" 8.288029 Log. 164.64 2.189001 in the triangle AD C, we have sm.ACD : AD : : cos. A CD : CD Cos. -4Ci>=cos. 2° 13' 27" 9.999674 AD ^ 2.18900 1 12.188676 Sin. ACD=sm. 2° 13' 27" 8.588932 3.699743 Double 0.301030 Diameter log. 7958 miles, nearly 3.900773 256 ROBINSON'S SEQUEL. (Several of tlie following problems in Mensuration are taken from the Surveying and Navigation, page 60.) (5.) Find the length of an arc of 30°, the radius being 9. When the radius is 1, an arc of 180°=3. 141592 ; therefore, 3 141592 an arc of 30° and radius 1 must be — , and this multiplied by 9 must be the required result. „ 3.141592-3 . „,c>«oo A Hence, =4.712388, Ans. (6.) Find the area of a circular sector whose arc is 18°, and ra- dius 1^. We must first find the length of the arc, as in the last problem, then multiply its half by the radius. •^ 141 fiQ9 Whence, _LlZ:!^ixl8X ^=.3141592Xf=.235619=-» arc. 180 Therefore the area must be fX 0-2356 19=0.363427, Ans. The arc of 1° and radius unity is .0174533. Therefore, that of 9° is .0174533X9, and this multiplied by the square of the radius will give tlie true result. That is, .0174533X9X1=0.353403. (7.) Required the area of a sector whose radius is 26, and ar- 147° 29'. .0 174533Xl47.4833x625 __g^^ 3^^^ 2 (8.) What is the length of a chord which cuts off one-third of the area from a circle whose diameter is 289 ? Ans. 278.6716. Like many problems in relation to the circle, this can be solved only by approximation . As circles are in all respects pro- portional to their radii, I will ope- rate on radius unity, and in conclu- sion, multiply by 2-|^, If the segment FFD=} of the whole circle, ABDE will equal | of the whole. Because ^ — |=i. TRIGONOMETRY. 267 The space ABDE contains two equal sectors, DCB, ACE^ and the triande ^Ci). Put the arc BD=x, CB^\. Then o C6^=sin. X, OD=cos. x. The area of the two sectors together is x. The area of the triangle ECD is sin. x cos. x. Therefore, ir+sin. x cos. x=\7i. rt=3. 141 592653+ Double, 2a;-|-2sin. x cos. x=^7t. But sin. 2.2;=2sin. x cos. x. (See eq. (30), page 143, Geom.) Therefore, 2ar-[-sin. 2x^^7t (1) Here we have a correct and definite equation, but we cannot solve it, as it contains an arc and its sine, and they are not united. by any definite numerical law ; we must, therefore, resort to ajyproximation. We know that sin. 2x is not much less than 2x. Therefore, 4x=^7i is not far from the truth. Also, 2 sin. 2x=}7t is not far from the truth. — The one too small, the other too large. That is, x=z^^rt, approximately, and sin. 2a:=i7t, approximately. To find the arc BD approximately, we have this proportion : rt : j\7t : : 180° : Arc BD. Whence, ^i>= 15°. By the table of natural sines we find sin. 2ii; = sin. 31° 34' nearly. Or, ic=15° 47' nearly. }Ve nozv knoio that to make the area ABDE=}7t, the arc BD must be greater than 15° and less than 15° 47'. I will now suppose the arc BD=15° 20', and compute the area ABDE, corresponding to that supposition. For the numerical value of the arc 15° 20', we have the follow- nig proportion : 180° : 15}j° : : 3.14159265 : Mq BD Or, 540 : 46 : : 3.14159265 : Arc^i)=0.2676175. The tables will give us (sin. 15° 20') (cos. J5° 20') thus : 17 258 ROBINSON'S SEQUEL. Sin. 15° 20' 9.422318 Cos. 15° 20' 9.984259 Sum less 20=log — 1.406577=0.2550200 nearly. jBi>=a;=0.2676175 nearly. Area ABDE= 0.5226375 nearly. But the required area of ABDE is \7t— 0.5235987 nearly. Hence, 15° 20' for ^i>, gives an area too small by 0.00096 12 Now we wish to increase the area ABDE by the little narrow space EDdey and this is so narrow that Dd and Ee are in respect to practi- cal or numerical purposes, right lines, and EDcle is a trapezoid, and its par- alel sides may be taken as equal ; it is then practically a parallelogram whose area is given and its longer side equal to 2(cos. 15° 20'). Let y= the width of this parallelogram or trapezoid, (as we may call it either.) Then we shall have the following equation : 2cos.(15° 20')y=0.0009612 Or, cos.(15° 20')y=0.0004806 That is, 0.9644^=0.0004806. Whence, y=0.000498 That is, we must increase the natural sine of BJ) 15° 20', by 0.000498. The natural sine of 15° 20' is 0.264434 To which add 0.000498 N. sin. of 15° 21' 47" cor. to sum 0.264932 Thus we learn that the arc Bd corresponds to 15° 21' 47" as nearly as a table of natural sines computed to 6 decimal places will give it. Twice the cosine of 16° 21' 47", to a radius of ^(289) is the chord sought, which we compute as follows : Cos. 15° 21' 47" 9.984184 Log. 289 2.460898 Log. 278.67+ 2.445082 TRIGONOMETRY. 269 (9.) WTiat is the radius of a circle whose center being taken in the circumference of another containing an acre, sliall ciU of half of its contents /* , This problem is the same for circles of every magnitude ; therefore, we will operate on a circle of radius unitt/. Let X represent the number of degrees in the arc AB, and 180 the length of each degree; TCX then "'*' represents the length 180 ^ ^ of the arc AB. BF= sin. X. CF= cos. x. FA=\ — cQs.x. (ABy = (1 — cos.a;)2-|-sin.2^, or AB = jj2 — 2cos.iC, which equals the radius of the cutting circle. The area of the sector CBAD^ is measured by the arc AB'CA; that is, ---,' From this take the triangle CBD, or 180 6 sin. a: cos. a;, and the segment ABB will be left. That is, Segment ABFJ)=—— sin. x cos. x. (^=3.141592.) *Oiie reason for the appearance of this work is that it is required, because able mathematicians have written so obscurely. They seem to have written as I should, were I indifferent whether the reader, or rather the learner, un- derstood me or not. Do not the following extracted solutions justify this observation ? they are brief, to be sure, and no one sets a higher value on brevity than does the author of this work ; — but nothing is meritorious which is ^^anting in perspicuity. The following extracts are from the Mathematical Diary, published by James Ryan, 1825. Solution. — By Robert Adrain, LL. D. In a circle to radius unity, let 2« be the arc of which the chord is the re- quired radius, then ?r being the area of the given circle to radius unity, if we express analytically the area cut off by the radius sought and divide by 2, we obtain the transcendental equation (^^z\cos. 22+1 sin. 2^=1 260 ROBINSON'S SEQUEL. Again, as x= tlie degrees in AB, (180 — x) = the degrees in BE. Because the angle BAH is at the circumference, it is measured by \ of (180— a;), or (90— iic^. Whence, the arc BJI=90 — ^ x, measured in degrees. For the length of the arc BH, we observe that 180° of the cir- cumference would be measured by 7t^2 — 2cos.a?. for 1 then, we have -^ _1_, this multiplied by the num- ber of degrees, (90-|)will produce (?.?/?^|^V90-?) for the linear measure of the arc BIT. This multiplied by the radius AB, or J2 — 2cos.ar, will give the area of the sector ABHD ; that is, Sector ABHD=^(^-i^^y\Uo-'^\ From this subtract the triangle ABD, which is measured by sin. a:(l — cos.a:), and we have the segment BIIDF. That is. Segment ^^i>^=. Then, by the rules of mensuration, it will be found that the two parts into which the given circle is divided are equal, each to 2 — (2cos.^i4) — 1)=-, Or by trigo., sin.^ — ^cos.4)=-, whence, ^=109° 11' 17" and the required radius =1.15874. If the contents of the given circle be one acre, then the required radius will be 206.7336 links, or about 45.4814 yards. TRIGONOMETRY. 261 The area ABHD=^—^m.x-\-—J(\—cos.x)(\^0.^-x)\ 180 ' 180\^ ^^ ^/ This reduces to ( 1— cos. x-\- \ — sin. x. If we put this expression equal to the given quantity, _ we jit cannot resolve the equation, because it would contain the linear quantity x and the transcendental quantities sin. x and cos. x. Therefore if we solve the problem at all, we must do it indirectly, by approximation, as we are obliged to do with nearly all problems pertaining to the circle. This expression is a general one, and if we assume x any num- ber of degrees, we can readily obtain the corresponding value of the expression, and if any assumption corresponds to a given value, the problem is solved ; and if it nearly corresponds, we shall have nearly the radius required, which can be increased or de- creased, as we are about to explain. For the area ABHD to contain half of the circle, it is our judgment that the arc AB should contain about 76° ; therefore, we assume x=-lb^ and the expression becomes (^ _P , 10 COS.75\7t . ^e 1 — COS.754- I — sin.75. ^ 24 / By the table of natural sine^ we find (0.741 18+0.10787)7t— 0-96693 The final result of this supposition is that the area ABHD=^ 1.701370. But the half of the circle is 1.570796 ; therefore, we have taken x too great to obtain the area of half the circle. We will now take x=10°. Then A— cos.70+!^^^V— .sin.70° will be an expression for a less area than before. By log. log. 0.79099 —1.898175 log. 3.1415926 0.497149 2.48500 0.395324 Nat. sin. 70^ 93969 Area ABHD 1.54531 Given result .1.57079 Error too small 0.02548 t62 ROBINSON'S SEQUEL. This error must be conceived to be a winding 'parallelogram, whose length is BHD. Dividing .02548 by BHBy will give the amount to be added to the radius AH. The radius AH or AB is 2sin.35°=l. 14716. The angle ^^li)=180— 70=1 10°. The Unear value of ^^i>=Qif!im^^i!il^??)li^. 180 Now the amount that the radius must be increased is expressed ^^ (.02548)18 (1.14716) (3.141592)11 By logarithms. Log. .02548 —2.406199 Log. 18 1.255273 —1.661472 Log. 1.14716 0.0596187 Log. 3.141592 0.4971499 Log. 11 1.0413927 1.5981613 1.698161 0.01 1 568 —2.06331 1 Add... 1.14716 1.158728= the required jradius AB, which will cut the circle into two equal parts. If the radius of the given circle is (a), in place of unity, then the radius of the cutting circle must be (1.15828)a To find the number «>f degrees and minutes m ABy divide 1.15828 by 2, which gives .57914 for the sine of half AB, or 35° 23' 30" or ^jB=70° 47'. The following theorems are extracted from pages 219 and 220 of Robinson's Geometry, (1.) Show geometrically, that R(R-l-.cos. A) = 2 cos.' ^ A ; and that R(R— cos. A)=2 sin.'^ A. TRIGONOMETRY. S^5 Let CB or CA represent the radius of a circle and call it E. Let the arc ^i>= Ay and draw the lines here represented. Then GD=^m.A, CG=cos.A, BG= E-]-Qos.A, OA—R — cos.^, AI=^m.\A, CI=co8.\A, jBi>=2cos.J^. From C draw CO perpendicular to JBD ; then JB 0= OD, and the two A's BOCy Bl) G, are equiangular ; therefore, BG : BD : : BO : BC. That is, jB-j-cos.^ : 2cos. ^-4 : : cos.^J^ : B. Whence, B(B+cos.A)=2cos.^ I A. Q. E. D. Again, by the similar A's AGDy ADB, we have AG : AD : : AD : AB, That is, i2— cos. A : 2sin. ^A : : 2sin.i^ : 2E. Whence, B(E—cos. A)=2sm.^A. Q, E. D. (2.) Show that R'sin. A=2sin.^ A cos.^ A. By similar triangles, we have AG I CI : : AD : DG. That is, E : cos. ^A : : 2sin. ^A : sin. A. Whence, E s'm. A=2sm.^Acos.^A. Q. E. D. (3.) Prove that tan. A-}-tan. B= ^ "^ — ^~t — I radius beinff unity. COS. A COS. B It is admitted that tan.^= — '- — , and tan. Bz By addition, tan. A-{-tsLn. B= COS.Jl COS. ^ sin A , sin.B COS. A cos.B sin, A cos. ^-f-cos. A sin. B sm.(A-\-B) n E D COS. -4 COS. ^ cos.-4cos. ^ (4.) Demonstrate geometrically y that R sec. 2 A = tan. A tan. 2 A +R2. 264 ROBINSON'S SEQUEL. Take CB radius, let the arc BD—9.A. Then ^^= tan. 2^, ^(7= sec. 2^. Draw CjE' bisecting the angle J^ OS, then BE= tan. -4. Also, i?-£'= tan. ^, because the two triangles CBE and CDS, are in all respects equal. Now by the similar A's ADE, ABO, we have this proportion, AD \ DE \ \ AB ', BO. That is, AD : tan. A : : tan. 2A : H Whence, AD • i2=tan. A • tan. 2 A By adding jK^ to both members and fac- toring, we have (AD-{-E)E= tan. A tan. 2^-|-i22 But (^i>+i^)=^(7=sec. 2^; therefore, B sec. 2^=tan. A tan. 2A-\-R^ . Q. E. D. (5.) Show that in any plane triangle, the base is to the sum of the other two sides, as the sine of half the vertical angle is to the cosine of half the difference of the angles at the base. Let ABC be the A. Call AB the base, and produce A C the shorter side so that CD=: OB and OEz= OB. Then if be taken as the center of a circle and CB radius, that circle must pass through the points E, B, and D, and the angle EBD must, therefore, be a right angle. Because A CB is the exterior angle of the A ODB, and that A isosceles, the angle A CB must equal 2D, or the angle D is half the vertical angle. Because BA is the exterior angle of the A AEB, we have BAC=AEB+ABE (1) But AEB=CBE=CAB+ABE. This value of AEB, sub- stituted in ( 1 ), gives BA 0= CBA+ABE+ABE (2) Whence, BAC^CBA=^2ABE (3) TRIGONOMETRY. ; Ml This last equation shows us that ABE is half the difference of the angles at the base. Now in the A ABD, we have AB : AD : : sm.D : sin. ^^i9. But the sin. ABD=cos. ABE^ because the sum of these two angles make 90°. Hence the preceding proportion becomes AB : {AC-\-CB) : : ^m.\ACB : gob.\{BAC--CBA). Q.E.D. ScHO. 1 . The A AEB gives us this proportion, AB : AE : : sin. E : sin. ABE. Because the angles E and D together make 90°, sin.^=cos.i>. Hence, AB : AE : : cos. D : sin. ABE. That is, in relation to the triangle ABC, and generally, The base of any 'plane triangle, is to the difference of the other two sides, as the cosine of half the angle opposite to the base, is to the sine of half the difference of the other two angles.'^ ScHO. 2. Draw AH parallel to EB, and of course perpendic- ular to DB ; then we have DA '. AE \ '. JDH \ HB. If Affhe made radius, DR is tangent to the angle DAB, and BB is tangent to the angle BAB. Because ^^is parallel to EB, the angle BAB is equal to ABE; but ABE has been demonstrated to be equal to the half difference of the angles CAB, CBA ; therefore, DAB is the half sum of the same angles, for the half sum and half difference of any two quantities make the greater of the two. Therefore, the preceding proportion becomes the following theorem : As the sum of the sides is to their difference, so is the tangent of the half sum of the angles at the base, to the tangent of half their dif- ference. This theorem is demonstrated in some form in every treatise on plane trigonometry. It is the 7th prop., page 149, Robinson's Geometry (6.) The diference of two sides of a triangle, is to the difference of the segments of a third side, made by a perpendicular from the oppo- site angle, as the sine of half the vertical angle is to the cosine of half the difference of the angles at the base ; required the proof *This is theorem v, Robinson's Geometry, page 220. , «66 ROBINSON'S SEQUEL. Let ABC he the A. On the shor- ter side CB as ra- dius, describe a circle, cutting AB in F, AC in B, and produce AC to K Draw CD perpendicular to the base, then DB is one segment of the base, AD is the other, and AF is their difference. AJI is obviously the difference of the sides. Now in the A ABF, we have Aff : AF : : sin. AFff : sm. AHF (1) This proportion demonstrates the theorem, as will appear when we show the values of these angles. Because CHFB is a quadrilateral in a circle, the angles HFB+E^im". But BFB-\-AFir=lBO''. By subtraction, F—AFJI=0, or AFJI=F. In the same manner we prove that AIIF=ABF. Substituting these equals in proportion (1), it becomes AH : AF : : sin. ^^ : sin. ABF (2) The angle F is half the angle A CB, because A CB is at the center of the circle, and F at the circumference intercepting the same arc. Also, sin. ABF=cos. ABIT, because EBE is a right angle, and the sine of an arc over 90° is equal to the cosine of the excess over 90°. Again, BCF— the sum of the angles at the base. BHC, or its equal CBH, is half BCF\ therefore, CBH = the half sum of the angles at the base of the triangle A CB, and HBA is their half difference. Whence proportion (2) becomes Aff : AF : : 8m.^(ACB) : cos.:^( ABC—A). Q.E.D. ScHO. Because -4 is a point without a circle, &c. ABxAF=AFxAff. Whence, AB : AF : : Aff : AF, TRIGONOMETRY. «67 The two A's ABE, AHF, have the common angle A, and the sides about the equal angle proportional, therefore, (th. 20, b. ii.) the two A's are similar, and AFH=E. AHF=ABE. (7.) Given the base, the difference of the other two sides, and the difference of the angles at the base, to construct the triangle. (See figure to Theorem 5.) Draw AB equal to the given base. From B on the opposite side of the base, make the angle ABE=z half the difference of the angles at the base. Take AE, the given difference of the sides, in the dividers ; put one foot on A, and strike an arc cutting BE in E. Join AE, and produce EA. Make the angle EBD=i90°. BD and EA produced will meet in D. Bisect ED in C, and join BG, and AOB will be the A required. N. B. This problem was Suggested bj the investigation of theorem v. (8.) Prove that smr\l ^ =tan.-\/-. Remark. The notation sin.~*w, signifies an arc of a circle whose radius is unity, and sine u, &c., &c. Hence the above proposition in plain English is this : The radius of a circle is unity, the sine of an arc in thai circle is ^1 Prove that the tangent of the same arc must be - /- . ^a-\-x ^a Let y= the cosine of the arc in question. Then ya+_^=l. Whence, y^^.JL-. a-\-x a-\'X But to every arc we have the following proportion : cos. : sin. : : 1 : tan. That is, J-^ : -/-^ : : 1 : tan. \a+x ^a+x Or, Ja : Jx : : I : tan.=^-. Q. E. D. 268 ROBINSON'S SEQUEL. (9.) If tan.(a-j)^,_sin^c^ ^^ tan.a tan.4= tan.^c. tan. a sin.^a To perform the reduction, multiply by tan. a, and in the last term take its equal — '— ; then cos.a tan. (a — 5)= tan. a — _; '. . sin. a cos.a That is, ^1l^J?IL*_=tan.- ^'"'"^ l-|-tan.atan.5 sin.a cos.a tan.a-toii.J=tan.a+tan.=« tan.4-C!in:!£+E5:!i^iL?i^) \ sin.a cos.a / Whence, ( l+tau.=a) tan. j^sin.'c+sin.'otan^atoij sin.a cos.a Multiplying by cos.a, and observing that (l-|-tan.^a)=sec.^a, and cos.a sec. a=l ; then , , sin.^ c . sin.^ctan.atan.6 sec. a tan.o= + ' sin.a sm.a Or, sec.a tan.6 sin.a=sin.^ c-|-sin.^ c tan.a tan.J. Take — '— for tan.a in the last term, then cos.a , , . . „ , sin. ^c sin.a tan. 5 sec.atan.6 sm.a=sm.^c4- cos.a Or, (cos.a sec.a)tan.5 sin.a=sin.2c cos.a-j-sin.^c sin.a tan.6. Observing again that (cos.a sec.a) = l, we have tan.6 sin.a=sin.^c cos.a-j-sin.^c sin.a tan.6 Divide each term by cos.a, and taking tan.a for ! — 1-, we find cos.a tan.a tan.J=sin.^c-{-sin.^ctan.a tan.^. (1 — sin.^c)tan.a tan.5=sin.^c. But (1 — sin.*c)=cos.2c, because sin.2c-|-cos.^c=l ; therefore, cos. ^ c tan.a tan.5=sin. ^ c tan.a tan.6= '- — ^=tan.^c. Q. E. D. cos.^c TRIGONOMETRY, v 269 SECTION III. PROBLEMS IN SPHERICAL TRIGONOMETRY AND ASTRONOMY Let ABC he a right angled triangle, right angled at B. a the side opposite A, h the side opposite B, and c the side opposite G. Taking the complement of the oblique an- gles A and C, calling them A\ G\ and the complement of h calling it b\ Then Napier's Circular Parts give us the following equations. We retain the same numbers for the equations as in our Geometry, page 186. (11) i2 sin.c=tan.a tan. ^' (16) i? sin.^'=tan.2>' tan.c (12) i2sin.rt=tan.ctan.(7' (17) i2sin.^'=cos.a cos.C" (13) i2sin.a=cos.5'cos.^' (1^) i2sin.5'=cos.a cos.c (14) jRsimc=cos.5'cos.(7' (19) i2 sin. C"=tan.5' tan. a (15) i2sin.5'=tan.^'tan.(7' (20) J^ sin. (7'=cos.c cos.^' These equations are written in the present form to assist the memory, the second members being the products of two cosines or two tangents ; but in practice, we often modify an equation by taking sine for cosine, and cotangent for tangent, and the reverse. For instance, in equation (18), we invariably take cos.S for sin.i', it being the same, which saves the trouble of finding the com- plement to the hypotenuse. The same may be said of other com- plements. In all spherical triangles, right angled or oblique angled, the sine of the sides are to each other as the sines of the angles opposite to them. When two sides of a spherical triangle are given, there can be but one result, that is, there can be no ambiguity about the parts required ; but when only one side is given, and one of the ob- lique angles in a spherical triangle, the conditions correspond equally to two triangles, and the answer is said to be ambiguous. For a learner fully to comprehend this, it is necessary to learn to construct his triangles as follows : We shall illustrate by examples, beginning with the 10th ex- 270 ROBINSON'S SEQUEL. ample, page 199, Robinson's Geometry, which will sufficiently illustrate several others. (1.) In the right angled triangle ABC, right angle at B, given AB 29° 12' 50" and the angle C 37° 26' 21", to find the other parts. To construct a spherical A, the operator should have a scale of chords and semitan- gents ; but he can do all with a ruler and dividers. Take 0(7 in the dividers equal to the chord of 60°, (or any distance if no scale is at hand), and from any point as a center describe the circle CHDh. Draw CD and Hh at right angles through the center. Each of the lines (7, ODy OH, as well as the curve HC, HD, &c. represent 90° on a sphere. OHC is a right angle, — that is, any line from to the circumference will make a right angle with the circumference- Now from C we propose to make the angle JICA=3'7° 26' 21". Divide the quadrant BJ) into degrees, beginning at II. Take IFF equal to 37° 26' 21" ; or if the scale is used, take the chord of 37° 26' 21" from the scale and apply it from II to F. Apply the ruler from C to P, and through the point n, where this line crosses ITO, describe the curve CnD. From If, set off IfQ=^29° 12' 50", apply the ruler between Q and C, and mark F where this line cuts HO. From the center 0, with V radius, strike the arc VA. Lastly, through A and draw BAG through the center. The A ABC or its supple- ment DBA, is the one required. The side A C is measured by the arc, but neither A C nor the angle A can be measured instru- mentally. To measure sides, they must either be on the circum- ference or on the straight lines through the center. Remark. If the angle BAC had been given, we should call the triangle ADQ the supplemental triangle, for ^6^ is the sup- TRIGONOMETRY. 271 plement to AB, AD to AC, and the angle ABO is supplemental to ADB or its equal BCA. When we have all the parts of the triangle ABC, we in effect have all the parts of the triangle DAB, also all parts of the A AD G and all parts of the triangle GA C. That is, when one spherical triangle is determined, we have three others, the whole four making up a hemisphere. For the numerical computation oi AC we take equation (14) modified thus : cir. jn c'.r. 7. -Ssin.c 19.688483 sm. A C=sm. 6= — -, — — sm.(7 9.783843 sin. 63° 24' 13" 9.904640 To find BC or a, we take a modification of (18). E cos.5 19.775374 cos.a= cos.c 9.940917 cos. 46° 56' 2" 9.834457 To find the angle A, we take a modification of equation (13). . . Bsin.a 19.863539 sm.-4= — — — sm.b 9.904640 sin. C6° 27' 60" • 9.958899 Whence, ^C=:53° 24' 13", BC=46° 55' 2", and the Z- A, 65° 27' 60". AD=126° 36' 47", ^2>=133° 4' 58", and theA BAD, 114° 32' 10" The same figure will sufficiently illustrate example 12, page 199, Robinson's Geometry. (2.) In the right angled triangle ABC, given AB, 64° 21' 36", and the angle C, 61° 2' 16", to find the other parts. (14) sin.^C7=sin.6=^^^ l^M^nS ^ ' sm. C 9.941976 sin. 68° 16' 16" 9.967940 Whence, -4(7 == 68° 15' 16", and ^i> = 111° 44' 44". The answers given in the book correspond to the triangle ADB, — • and those answers were given to exercise the judgment of the learner. The other parts are found as in the last example. 272 ROBINSON'S SEQUEL. (3.) In the right amjled spherical triangle, given AB, 100® 10' 3" and the angle BCA, 90° 14' 20", to find the other parts. Because the sines, cosines, (kc, of the tables correspond to arcs under 90° ; therefore we will operate on the sup- plemental triangle, ADE. BC^DE. 180°— .^j5=79° 49' 67"= JZ>=c. The angle ^7)^=90°— (14'20")=89°45'40". AC=h, and AB = h' in the equations. AB=^Cy AED =90°, ^i>^=:(7' = 89°46' 40". To solve this A, we use equation (20). R cos. C 17.620026 sin. A COS. c 9.246810 sin. 1° 2ri2" 8.373216 To compute AI>y we take equation (16) ; AB the supplement of AC=h. R COS. 5=cot. A cot. C. cot.^=l°21' 12" 11.626819 cot.(7=89°46'40" 7.619860 AB COS. 79° 60' 6" 9.246679 AC 100° 9' 65" To find BE, or its equal BC, we take equation (13). i2sin.a= sin.6sin.-4. sin. 79° 60' 6" 9.993128 ' sin. 1° 21' 12" 8.373216 BC, sin. 1° 19' 62" 8.366344 These examples give a sufficient key to the solution of all other examples in right angled spherical trigonometry. * TRIGONOMETRY. 273 We now turn to the application of spherical trigonometry — taking the examples from page 215, Geometry. MISCELLANEOUS ASTRONOMICAL PROBLEMS. (1.) In latitude 40° 48' north, the sun hore south 78° 16' west, at 3h. 38m. P. M., apparent time. Required his altitude and declina- tion, making no allowance for refraction. Ans. The altitude, 36° 46', and declination, 15° 32' north. Let Hh be the horizon, Z the zenith of the observer, P the north pole, and PS a meridian through the sun. PZ is the co-latitude, 49° \9.\ and PS is the co-decli- nation or polar distance, one of the arcs sought. ZS is the co-altitude or ST is the altitude of the sun at the time of observation. The angle ZPS is found by reducing 3h. 38m. into degrees at the rate of 4m. to one degree ; hence, ZPS=54° 30' Because EZS=7Q° 16', PZS=101° 44'. From Z let fall the perpendicular Z^ on PS. Then in the right angled spherical A PZQ, equation (13) gives us* P sin. Z^=sin. PZsin. P. sin. PZ=sin. 49° 12' .9.879093 sin. P =sin. 54° 30' 9.910686 sin. Z^=sin. 38° 2' 33" 9.789779 To obtain the angle PZQ, we apply equation (19), which gives P COS. PZQ=cot. PZ tan. ZQ, That is, i2cos. PZQ=t3,n. 40° 48' ta,n. 38° 2' 33". * To apply the equations ■witliout confusion, letter each right angled spher- ical triangle ABC, right angled at B, then A must be written in place of P ; nndwhen operating on ZSQ, write A in place of S, and C for the angle SZQ 18 io ^4 . ROBINSON'S SEQUEL. 9.936100 9.893464 PZQ= COS. 47° 30' 50' 9.829564 PZS= 101° 44' SZQ= 54° 14' 10" To obtain ZS or its complement, we again apply (19) (19) E COS. SZQ=cot. ZStsLYL. ZQ. That is, i2cos. 54° 14' 10"=tan. STtan.38° 2' 33". i? COS. 54° 14' 10"= 19.766744 tan. 38° 2' 33 " = 9. 893464 tan. 36° 46', nearly, 9.873280 To find PS, we take the following proportioif : sin. P : sin.Z^ : : sm. PZS : &m. PS That is, sin. 54° 30' : cos. 36° 46' : : sin. 101° 44' : sin. PS COS. 11° 44' 9.990829 cos. 36° 46' .9.903676 19.894505 sin! 54° 30' 9.910686 PS, 74° 28' sin 9.983819 Whence, the sun's distance from the equator must have been 15° 32' north. (2.) In north latitude, when the sun's declinatlo7i ?m5 14° 9.Q' north, his altitudes, at two different times on the same forenoon, were 43° 7'-}-, (ind 67° 10'-(- ; and the change of his azimuth, in the inter' vol, 45° 2'. Required the latitude. Ans. 34° 20' north. Let PK be the earth's axis, ^^^^^^^ Qq the equator, and Bh the ho- ^KUBm^^ Also, let Z be the zenith of the ^■^•^^^ observer, Sm the first altitude, Tn the second, and the angle rZ>S=45° 2'. Our first opera- tion must be on the triangle ZTS. ZT=22° 50', Z^=46° 53', and we must find TS, and the Z_ TSZ- TRIGONOMETRY. 275 From T, conceive TB let fall on ZS making two right angled A's ; and to avoid confusion in the figure, we will keep the arc TB in mind, anA not actually draw it. Then the A ZTB furnishes this proportion : R : sin. 22° 60' : : sin. 46° 2' : sin. TB=sm. 16° 66' 8" To find ZB we have the following proportion, (see p. 186 Geo.) R : COS. ZB : : cos. 16° 66' 8" : cos. 22° 60' Whence, we find Z5=16° 34' 20". Now in the right angled spherical A TBS, we have TB = 15° 56' 8", BS=46° 63'— 16° 34' 20", or ^5=30° 18' 40" ; and TS is found from the following proportion : R : COS. 15° 56' 8" : : cos. 30° 18' 40" : cos. TS This gives TS=33° 53' 16". To find the angle TSZ, we have the proportion, sin. 33° 53' 1 6" : R : : s'm. TB 15° 66' 8" : sin. TSZ. Whence, the angle TSZ=29'' 30'. The next step is to operate on the isosceles spherical A PTS, We require the angle TSF. Conceive a meridian drawn bisecting the angle at P, it will also bisect the base TS, forming two equal right angled spherical triangles. Observe that P>S^=75° 40' and ^ TS=16° 56' 38". To find the angle TSF we apply equation (19), in which a= 16° 56' 38", 5=75° 40', and the equation becomes R cos. TSP—coi. 76° 40' tan. 16° 56' 38" Whence, TSP=S5'' 31' 40", and PSZ=Q5° 31' 40"— 29° 30' =66° 1' 40". The third step is to operate on the A ZSP ; we now have its two sides ZS and SP, and the included angle. From Z conceive a perpendicular arc let fall on SP, calling it ZB ; then the right angled spherical triangle SZB, gives R : sin. ZS : : sin. Z SB : sin. ZB That is, R : sin. 46° 53' : : sin.66° 1'40" : sin.Z^=sin.37° 15'20" To find SB we have the following proportion, (see Geo. p. 185.) R : cos. SB : : cos. ZB : cos. ZS Thai is, R : cos. SB : : cos. 37° 15' 20" : cos. 46° 63' 276 ROBINSON'S SEQUEL. Whence SB=30'' 49' 40". Now from FS, 15" 40', take SB, 30° 49' 40", and the diflference must be BP, 44° 50' 20". Lastly, to obtain PZ, and consequently Z Q the latitude, we have R : COS. ZB : cos. BP : cos. ZP==sin. ZQ That is, B : cos. 37° 16' 20" : : cos. 44° 50' 20" : sin. ZQ= sin. 34° 21' north. This computation differs one mile from the given answer, but any two operators will differ about this much, unless each observe the utmost nicety. This is a modification of latitude by double altitudes, but in real double altitudes the arc ^aS^ is measured from the elapsed time between the observations, and the angle TZS is not given. (3.) In latitude 16° 4' north, when the su7i's declination is 23° 2' north. Mequired the time in the afternoon, and the sun's altitude and hearing when his azimuth neither increases nor decreases. Ans. Time, 3h. 9m. 26s. P. M., altitude, 45° 1', and bearing north 73° 16' west. Let Pp be the earth's axis, Hh the horizon, Qq the equator, QZ and Pp, each equal to 16° 4' north, and Qd, qd, each e- qual to 23° 2' ; then the dotted curve dd represents the parallel of the sun's declination. Through Z and N an infinite number of vertical circles can be drawn, one of these will touch the curve dd ; let it l^e Z OK At the point where this circle touches the curve dd will be the position of the sun at the time required, and P OZ will be a right angled spherical A, right angled at 0. The problem re- quires the complement of ZO, and the time corresponding to the angle ZPO. In the spherical A P OZ, we have R : COS. PO : : cos. ZO : cos. PZ That is, R : sin. 23° 2' : : sin. altitude : sin. 16° 4' Whence, sin. alt. = TRIGONOMETRY, i? sin. 16° 4' 277 sin. 46° r nearly'. Ans. sin, 23° 2' To find the angle at P, we have the following proportion : COS. 16° 4' : R : : cos. 46° 1' : sin. P Whence, sin. P = sin. 47° 21' 30", and ZPO = 47° 21' 30", which being changed into time, at the rate of 16° to one hour, gives 3h. 9m. 26s. To find the angle PZ 0, we have this proportion : cos. 16° 4' : R '. : cos. 23° 2' : sin. PZO = sin. 73° 16' (4.) The sunset south-west ^ sovthy when his declination was 16° 4' south. Required the latitude. Ans. 69° 1' north. Draw a circle as before. Let Hh be the horizon, Z the zenith, ^^^vn P the pole. The great circle PZH'i^ the meridian, and ZCN at right angles to it, and of coui'se east and west. Let BC ^^^S^^ff^BBi^^KKKUi be a portion of the equator, and ^^H^H|HBn^S^^HI B the arc of declination. The ^^|SHH^H|^H^SBI position on the horizon where ^^^|^8^^^H^|B^B^| the sun set is the arc 110=45° ^^^B^l^^^P^^Bs^^M —6° 37' 30"=39° 22' 30". Consequently, the arc 00=50° 37' 30". , In the right angled spherical triangle BOO, we have BC, BO given to find the angle BOO, which is the complement of the latitude, or the complemc'nt of the angle B CZ. To find the angle BOO, we apply equation (14). i? sin. ^ 0=sin. OCsin. J5(70 That is, R sin. 16° 4'=sin. 50° 37' 30" sin. BOO Rsin. 16° 4' 19.442096 sin. 50° 37' 30" 9.888184 cos. 69° 1' nearly, 9.653912 ScHO. The arc -6 (7 on the equator measures the angle -BP (7, corresponding to the time from 6 o'clock to sun rise or sun set. S78 ROBINSON'S SEQUEL. This arc is called the arc of ascensional difTerence in astronomy. The time of sun set is before six if the latitude is north and tlie declination south, as in this example, but after six, if the latitude and declination are both north or both south. To obtain this arc, the latitude and declination must be given ; that is, BO and the angle BCO, the complement of the latitude. Here we apply (12), that is, M sin. BO = tan. D tan. L an equation in which D represents the declination, and L the latitude. (5.) The altitude of the suUy when on the equator, was 14° 28'-]-, hearing east 22° 30' south. Required the latitude and time. Ans. Latitude 56° T, and time 7h. 46m. 12s. A. M. Let S be the position of the sun on the equator. (See the last figure.) Draw the arc ZS, and the right angled spherical A ZQS is the one we have to operate upon. Then ZS is the complement of the given altitude, and the an- gle QZS, is the complement of 22° 30'. The portion of the equator between Q and S, changed into time, will be the required time from noon, and the arc QZ will be the required latitude. First for the arc QS. R : sin. ZS : : sin. QZS : sin. QS That is, R : cos. 14° 28' : : cos. 22° 30' : sin. ^aS^ = 73° 27'38" But 73° 27' 38" at the rate of 4m. to one degree, corresponds to 4h. 13m. 48s. from noon, — and as the altitude was marked -|-, rising, it was before noon, or at 7h. 46m. 12s. in the morning. To find the arc QZ we have the following proportion : R : COS. 63° 27' 38" : : cos. ^Z : sin. 14° 28' Whence, cos. ^Z=cos. 66° 1' nearly, and 56° 1' is the latitude souffht. o (6.) The altitude oft/ie sun was 20° 41' at 2h. 20m. P. M. when his declination was 10° 28' south. Required his azimuth and the latitude. Ans. Azimuth south 37° 5' west, latitude 51° 58' north. TRIGONOMETRY. 279 This problem furnishes the spherical A PZ 0, in which the ^H!|^^B9I side Z is the complement of ^B|fig8Q^&^^|B| 20"" 41' or eO"" W, F0=90'' ^H^j^KKBrn -\-iO° 28' = 100° 28', and the ||yi^|^SIHBI angle ZFO is 2h. 20m., chang- ed into degrees at the rate of 15° to one hour, or ZPO=35°. Now in the triangle ZFO, we have sin. ZO : sin. ZPO : : sin.PO : sin.PZO That is, cos. 20° 41' : sin. 35° : : cos. 10° 28' : sin. BZO = cos. 37° 5'. In the right angled spherical A B OZ, we apply equation (16). (16). R cos.37° 5'=tan. 20° 41' tan. BZ. i2cos. 37° 6' 19.901872 tan. 20° 41' 9.576958 tan. ^Z=tan. 64° 40' 40" 10.324914 To find FB in the right angled A BFO, we apply the same equation, (16). R cos. 35°=tan. 10° 28' tan. FB. R cos. 35° 19.913365 tan. 10° 28' 9.266555 tan. 12° 42' 40" ..10.646810 But FB is obviously greater than 90°, therefore the point B is 12° 42' 40" below the equator, but from jB to Z is 64° 40' 40"; therefore from Z to the equator, or the latitude, is the difference between 64° 40' 40" and 12° 42' 40", or 51° 58' north. Ans. Lat. 51° 58' north. (7.) If in August 1840, Spica was observed to set 2h. 26m. 14s. hefore Arcturus, what was the latitude of the observer ? Taking no account of the height of the eye above the sea, nor of the effect of refraction. Ans. 36° 48' north. By a catalogue of the stars to be found in the author's Astron- omy, or in any copy of the English Nautical Almanac, we find the positions of these stars in 1 840 to have been as follows : ^ Spica, right ascension, 13h. 16m. 46s. Dec. 10° 19' 40" south. Arcturus, " *' 14h. 8m. 25s. Dec. 20° 1' 4" north. 280 ROBINSON'S SEQUEL. Let L = the latitude sought. Put d=10° 19' 40", and D— 20° r 4". The difference in right ascensions is 61m. 39s., and this would be about the time that Arcturus would set after Spica, provided the observer was near the equator or a little south of it ; but as the interval observed was 2h. 26m. 14s., the observer must have been a considerable distance in north latitude. In high southern latitudes Arcturus sets before Spica. When an observer is north of the equator, and the sun or star south of it, the sun or star will set within six hours after it comes to the meridian. When the observer and the object are both north of the equa- tor, the interval from the meridian to the horizon is greater than six hours. The difference between this interval and six hours, is called the ascensional difference, and it is measured in arc hj £0 in the figure to the 4th example. Now let X = the ascensional difference of Spica corresponding to the latitude £, and y = the ascensional difference correspond- ing to the same latitude ; then by the scholium to the 4th exam- ple, calhng radius unity, we shall have sin. a:=tan. L tan. d ( 1 ) sin. y^tan. L tan. J) (2) The star Spica came to the observer's meridian at a certain time that we may denote by M. Then Jlf-^/^e— — ) = the time Spica set. And M-{-51m. 39s.4-(6-|-^ )= the time Arcturus set. By subtracting the time Spica set from the time Arcturus set we shall obtain an expression equal to 2h. 26m. 14s. That is 51m. 39s. +^+i^=2h. 26m. 14s. Ot, -^+I_=lh. 34m. 36s. (3) ' 16^16 ^ ^ ^ ar+y=16(lh. 34m. 36s.) (4) Equation (3) expresses time. Equation (4) expresses arc. When we divide arc by 16 we obtain time, one degree being TRIGONOMETRY. 281 the unit for arc, and one hour the unit for time ; therefore, when we multiply time by 15 we obtain arc ; that is, Ih. multiplied by 15 gives 15° ; hence (4) becomes a;4-y=23° 39'=a x—a—y (5) That is, the arc x is equal to the difference of the arcs a and y ; but to make use of these arcs and avail ourselves of equations (1) and (2), we must take the sines oi the arcs, (see equation (8), plane trigonometry) ; then (5) becomes sin. a:=sin. a cos. y — cos. a sin. y (6) Substituting the values of sin. x and sin. y from (1) and (2), (6) becomes tan. L tan. c?=sin.a cos. y — cos. a tan. L tan. D (7) Squaring (2), sin.2y=tan.^i/tan.2i). Subtracting each member from unity, and observing that (1 — sin.^y) equals cos.^y, then cos.^2/=l — tan. ^Z tan. 2 i). Or, cos. y= ^1— tan.^i/tan.^i). This value of cos.y put in (7), gives tan. Ztan. <^=sin. aj\ — tan.^Z tan.^i> — cos. «tan. Xtan. D (8) By transposition and division, /tan.rf+cos.atan^\ tan. i;=Vl-tan.=itan.=i) \ sin. a / Squaring, (^g^^lJ+g"^- "*'"'• -"Vtan.^X==l-tan.'X tan.'i) \ sin. a J Dividing by tan.^X and observing that _- = cot.^Z we lan. j-i have /tan^+cos^tan^\ =oot.=i-tan.^/) \ sin. a / Or, cot.'X=tan.'2)+('-ggl ^''"^- " ^.g^^-gV \ sin. a / =tan.^2)+C^+'^-V Vsin. a tan. a / We must now find the numerical value of the second member. Using logarithmic sines, cosines, tangents, (fee, we must diminish the indices by 10, because the equation refers to radius unity, log. tan. Z>.==— 1.561460. tan.-i>=— 1.122920=0.132712 num. 1282 V ROBINSON'S SEQUEL. log. tan. d —1.260623 log. tan. i> —1.561460 sin. a — 1.603305 tan. a — 1.641404 0.45424 —1.657318 0.83188. . . . . . —1.920056 0.45424-1-0.83188=1.28612 (1.28612)2 = 1.654105 Whence, cot.2Z=0.132712-|^l. 654105= 1.786817 Square root, ..cot. Z= 1.33672 Taking the log. of this number, increasing its index by 10 will give the log. cot. in our tables. log. 1.33672=0.126076-f-10.=10.126076=cot. 36° 48' (8.) On the 14th of November, 1829, Merikar was observed to rise 48m. 3s. before Aldebaran : what was the latitude of the observer ? Ans. 39° 34' north. The positions of these two stars in the heavens, Nov. 1829, were as follows : Menkar, right ascension, 2h. 53m. 21s. Dec. 3° 24' 52" north. Aldebaran, " 4h. 26m. 7s. Dec. 16° 19' 31" north. Aldebaran passes the meridian Ih. 32m. 46s. after Menkar. Now let M represent the time Menkar was on the meridian, then M-\-\\i. 32m. 46s. represents the time Aldebaran was on the meridian. Also, let x= the arc of ascensional diflference corres- ponding to the latitude and the star Menkar, and y that of the star Aldebaran. Then M—(q-\-—\ — the time Menkar And Jf-l-lh. 32m. 46s — ( 6-[- - ) = the time Aldebaran rose. Subtracting the upper from the lower, the difference must be «8m. 3s. ; that is, lh.-|-32m. 46s 'L-X-—=A^m, 3s. ^ 16 ' 15 Whence, -^— !_= —44m. 43s.= —0.74527. 16 16 That is, Ih. being the unit, 44m. 43s. = 0.74527 of an hour, and multiplying by 1 5, we shall have as many degrees of arc as we have units ; therefore, a;— y=— (0.74527)15=— 11° 10' 45"=— a. x=y—a. sin. a:=sin.^ cos.o — cos.y sin.a ( 1 ) rose. TRIGONOMETRY. 283 Put c/=3° 24' 52", D=\6° 19' 31", and L= the required lat- itude. Then by scholium to the 4th example, sin. a;=tan.c? tan.Z. sin.y=tan.i> tan.Z. These values of sin.a: and sin.y, substituted in (1), give tsm.d tan.Z=cos.a tan.D tan.Z — cos.y sin.a ( 2) But sin.2y==tan.2i>tan.2i;, and 1— -sin.2y=l— tan.^i^tan.^Z. Or, cos.^y=l — ^tan.^*Z)tan.2j&. Or, cos.y=iJl — ^tan.^2>tan.^X. By substituting this value of cos.y in (2) and transposing, we find sin.a^l — tan.^i) tan.2j&=(cos.a tan.i) — tan.e?)tan.X Dividing by sin.a, and observing that -r-^= , we have "^ •' sm. a tan.a n — 1 — rm — rr /tan.i) tan.c?\ , r J\ — tan.^i) tan.2Z=l — ) tan.ii. \ tan.a sm.a/ Squaring and dividing by tan.^Z, and at the same time observ- insr that =cot.Z, and we shall have ^ tan.Z cot.«Z-tan.«i>=f-^^-*_^V \ tan.a sin. a/ We will now find the numerical values of the known quantities. Log. tan.i). . .—1.466696 Log, tan.c?. . . — 2.776685 Log. tan.a —1 .296 1 79 Log. sin.a 1 .2876 1 7 Log. 1 .48089 . . . 0.170517 Log. 0.3076 . . . —1.488068 tan.2i)=0.085778 1.48089—0.3076=1.17329 Whence, cot.^ Z— 0.085778= (1.1 7329) 2. Or, cot.2ii= 1.462293. cot.Z= 1.20925. Log. cot.i;+10i=10.082785=cot. 39^ 34'. Ans, (9.) iw latitude 16° 40' north, when the sun's declination was 23° 1 8' northy I observed him twice, in the same forenoon, bearing north 68° 30' east. Required the times of observation, and his altitude at each time. Ans. Times 6h. 15m. 40s. A. M., and lOh. 32m. 48s. A. M., altitudes 9° 69' 36", and 68° 29' 42". 2B4 ROBINSON'S SEQUEL. Let Z be the zenith, P the north pole, and the curve dd be the parallel of the sun's declina- tion along which it appears to revolve. Make the angle PTiS' equal to 68° 30' ; then the sun bii^^^^^^^^^^I was at S at the time of the first HV^^^BHHH^^^H observation, and at S' at the time IIS^^^^HH|B|HI the ^I^S^^BBB^BSfl In the spherical A PZS' there ^^^I^^H^bIB^^I is given PZ, PS' and the angle PZaS" ; also, in the A PZS' there is given PZ, PS, and the angle PZS. Observe that PSS' is an isosceles A- Describe the meridian PB bisectincr the anole S'PS, and then we have three right angled spherical triangles, BPS, BPS\ and BPZ ; taking the last, we have the following proportion : R : sin. PZ : : sin. PZB : sin. PB That is, P : cos. 16° 40' : : sin. 68° 30' : sin. P^=sin.63° 2' 30". To find ZB, we take the following proportion, (see page 185, observation 1, Robinson's Geometry) : P : coB.ZB : : cos. BP : cos. PZ That is, P : cos. Z^ : : cos. 63° 2' 30" : sin. 16° 40' P sin. 16° 40' 19.457584 cos. 63° 2' 30" 9.656411 cos. 50° 45' 48" 9.801173 To find S'B, we have P : COS. S'B : : cos. 63° 2' 30" : sin. 23° 18' P sin. 23° 18' 19.597196 COS. 63° 2' 30" 9.656411 • COS. 29° 14' 38" 9.940786 Observe that S'B=:BS ; therefore, Z;S'=50° 45' 48"+29° u' 38"=80° 0' 26", and Z>S"=50° 45' 48"— 29° 14' 38"=21° 31' 10", the complement of the altitudes. Consequently the altitude at the first observation was 9° 59' 34", and at the second, 68°28'50". » * Our results differ a little from the given answer, owing, perhaps, to our not being minute in taking out the logarithms, or finding the nearest second corresponding to a given logarithm. — Experienced men on these matters do not pi'etend to work to seconds. TRIGONOMETRY. 285 To find the time from noon at the first observation, we have the following proportion : sin. PaS^ : sin-PZ^S^ : ": s'm. ZS : sin. ZFS. That is, cos.23° 18' : sin.68°30' : : sin.80° 0' 26" : sin.ZPAS^=sin.86°5'30" Had the angle been 90°, the time would have been just 6h. but the angle 3° 54' 30" less ; this corresponds to 15m. 38s. in time. Therefore, the time was 6h. 15m. 38s. For the time at the second observation, we have cos.23°18' : sin.68°30' : : sin,21°31'10" : sin.ZPAS"=sin.21°48'40" 21° 48' 40"=lh. 31m. 14s. from noon, or lOh. 32m. 46s. ap- parent time in the morning. (10.) An observer in north latitude marked the time when the stars Megulus and Sjnca were eclipsed by a plumb line, that is, they were both in the same vertical plane passing through the zenith of the ob- server. One hour and ten minutes afterwards, Regulus was on the observer's meridian. What was the observer's latitude ? The positions of the stars in the heavens were Regulus, right ascension lOh. Om. 10s. Dec. 12° 43' north. Spica, *' " 13h. 17m. 2s. Dec. 10° 21' 20" south. Let R be the position of Regu- ' lus, S the position of Spica, P the pole, and Z the zenith. Then the side PaS'=100°21'20", PE=n° 17', and the angle BPS =3h. 16m. 52s., converted into de- grees ; that is, PPS=49'^ 13'. One hour and ten minutes re- duced to arc, give 17° 30'; but the stars revolve according to siderial, not solar time, and to reduce solar to siderial arc we must increase it by about its ^\-g th part ; this gives about 3' to add to 17° 30', making 17° 33' for the angle ZPR. Our ultimate object is to find PZ, the complement of the latitude. In the A PRS, we have the two sides PB, PS, and the in • eluded angle P, from which we must find PS and the angle SEP, and we can let a perpendicular fall from M on to the side PS and 286 ROBINSON'S SEQUEL. solve it in the usual way ; but to show that a wide field is open for a bold operator ; we will put the unknown arc IiS=x, the side opposite Ii=r, and opposite S=Sy and apply one of the equations in formula (S), page 191, Robinson's Geometry. rpr - • D cos.a; — cos.r cos.s That IS, cos.i^= sm. r sin.5 Whence, cos.P sin.r sin.s-j-cos.r cos.5=cos.a; We now apply this equation, recollecting that radius is unity, which will require us to diminish indices of the logarithms by 10. cos.P=cos. 49° 13' —1.815046 sin.r=sin.lOO° 21' 20". . .—1.992068 —cos —1.254579* sin. s =sin.77° 17' —1^89214 cos —1.342679 0.6268 —1.797123 .03956 . . .—2.597258 cos.a;=0.6268— 0.03956=. 58724. Whence, by the table of natural cosines, we find a:=54°2'20". To find the angle SEP or ZEP, we have sin. 54° 2' 20" : sin. 49° 13' : : sin. 100° 21' 20" : sin. ZBP Whence, ZEP=66° 57' 30". Let fall the perpendicular PB on PZ produced, then the right angled spherical A PPP gives this proportion : P : sin. 77° 17' : : sin. 17° 33' : sm.PB=sm. 17° 6' 22" To find PP we have P : COS. PB : : cos. 17° 6' 22" : cos. 77° 17' Whence, P^=76° 41'. Now to find the angle BPP, we have sin. 77° 17' : P : : sin. 76° 41' : sin. ^i2P=sin. 86° 1' From PPB take PPZ, and ZPB will remain ; that is. From 86° 1' take 66° 57' 30", and ZPB=19° 3' 30". By the application of equation (12), we ^nd that P sin. 17° 6' 22"=tan. BZ cot. 19° 3' 30" Whence, ^Z=5° 48' And PZ=76° 41'— 5° 48'=70° 63'. The complement of 70° 53' is 19° 7', the latitude sought. By this example we perceive that by the means of a meridian line, a good watch, and a plumb line, any person having a knowl- edge of spherical trigonometry, and having a catalogue of the stars at hand, can determine his latitude by observation. ♦Observe that r is greater than 90®, its cosine is therefore, negative in value, rendering the product cos. r cos. «, or .03956, negative. PART FOURTH. PHYSICAL. ASTRONOIWY. KEPLER'S LAWS. 1 . The orbits of the planets are ellipses, of which the sun occupies one of the foci. 2. The radius vector in each case describes areas about the focus which are proportional to the times. 3. The squares of the times of revolution are to each other as the cuhes of the mean distances from the sun. The first of these is a mere fact drawn from observation. The second is also an observed fact — but susceptible of mathematical demonstration, under strict geometrical principles, and the law of inertia. The demonstration is to be found in Robinson's Astron- omy, and in various philosophical works. The third is also susceptible of demonstration by means of the calculus — and by simple geometrical proportion, if we suppose the orbits circular. We now propose to investigate and determine the relative times of revolutions of two bodies about the sun, on the supposition that they revolve in circles, (which is not far from the truth,) and are attracted towards the center inversely proportional to the squares of their distances. Let S be the center of the sun, AS the radius vector of one planet, and SV that of another. Let m be the mass of the sun, SA=^ri and S V=E. Then ~ is the force which is exerted on the planet at A, and -— is the force exerted on the other planet at V. If we take any small interval of time, say one minute, and let AI) represent the distance the first planet falls from the tan- 288 ROBINSON'S SEQUEL. gent of its orbit in unity of time, and VH the distances the other falls in the same time, Then ^ ; ^ '. '. AD '. VH (1) That the planets may maintain themselves in their orbits, the first must run over the arc AB in the unit of time, and the sec- ond must run over the arc VF. But this interval or unit of time can be taken ever so short ; and when very short, as a minute or a second, AB and VF, may, yea must be considered straight lines, chords 'Comc'iding with the arc. But if we take any chord of an arc, as AB, and from one ex- tremity draw the diameter, and from the other let fall the perpen- dicular BD, we shall have AB : AB : : AB : 2r AB^ VF^ Whence, AI)= , and in like manner VIf= Substituting these equals in proportion (1 j, and dividing the first couplet by m, and multiplying the last couplet by 2, we have 1 1 AB^ . VF^ W r R 1 1 Or, _ : _ : : AB^ : VF^ (2) r R Because the first planet is supposed to run along the arc AB, in one minute, the number of minutes it will require to make its revolution will be found by dividing the whole circumference by AB. The circumference is expressed by 2r7t, and put t to repre- sent the time of revolution ; then t = , or AB = - — In AB t the same manner if T represents the time of revolution of the 272 Tt second planet, we must have VF=^——-. By squaring these ex- pressions and substituting the values of AB^ and VF^ in pro- portion (2), we have r R t^ T^ O 1 • JL • • l! • :?" ASTRONOMY. Multiply the first couplet by tR, then i2 : r : . l! : Or, R^ R^ 1^2 L.=.^, Whence, t^ : T" : 289 •3 : i23. This last proportion corresponds with Kepler's third law. The following propositions are to be found on page 146 of Robinson's Astronomy. The frequent requests we have received to demonstrate them, suggested the propriety of pubhshing the demonstrations in this connection. The propositions are as follows : (1.) If two comets m-ove in parabolic orbits, the areas described by them in the same time are proportional to the square roots of their perihelion distances. Conceive a comet to revolve in an ellipse, F' the position of the sun, and A'F' the perihelion distance. Let F'D=r, F'C=x, DC=y, and put t to represent the number of hours required by the comet to make a rev- olution. Now nry == the area of the ellipse. This area divided by t, will express the area described by the comet about the sun in one hour. Let that area be represented by a. Then nry _ a. Let R, x\ y\ T, and A, represent similar quantities pertaining to another orbit, and by parity of reasoning, T Whence, By squaring, ry^ t r^y^ T ' ' qi3 By Kepler's 3d law, t^ : T^ 19 : R\ or t^ = (1) 290 ROBINSON'S SEQUEL. The value of /' substituted in (1), and reduced, will give y- X y : : a^ : A^ (2) r xt By inspecting the right angled triangle F'CDy we readily per- ceive that y^=:r^ — x'=(r-\-x) (r — x). Similarly, y'^=(B-\-x') Now if we suppose the ellipse to be infinitely eccentric, (as we must when it becomes a parabola,) {r-\'x) = 2r nearly, and (r — x)^=A'F'z=p exactly, (calling p the perihelion distance of one comet, and P the perihelion distance of the other.) Similarly , {R+x')z=9,R, and (B—x')=F. Substituting these values in (2), we have r R Or, p '. P \ \ a^ \ A^ Or, Jp : JP : : a : A Q. E. D. , (2.) j^we sui^pose a planet moving in a circular orhit, whose radius is equal to the perihelion distance of a cornet moving in a parabola, the areas described by these two bodies, in the same time, will be to each other as 1 to the square root of 2. Thus are the motions of comets and planets cminected. Let S be the position of the sun, P the perihelion point of a comet revolving in an ellipse. Put SP=x, and let i=the time in which the planet would revolve in the circle, and T= the time required for the comet to revolve in the ellipse. By the first law of Kepler the same body describes equal areas in equal times ; therefore if we divide the area of the circle by the number of units in the time of revolution, we shall have the area described in one unit of time. The area of the circle is ^ar*, and this divided by <, gives ASTRONOMY. 291 = the sector described by the planet in unity of time. Also, I = the sector described by the comet in the same time. Conceive these two sectors to commence on the line SP, then 3 A Ti (sector in circle) : (sector in ellipse) : : — : -__ (1) f JL A and B are the semi-conjugate axes of the ellipse. By Kepler's third law, e : T^ : : x^ : A^ (2) Multiplying the last couplet of ( 1 ) by tT, gives (sector in circle) : (sector in ellipse) : : Tx^ : [tA)B By squaring, we have (sec.incircle)^ : (sec. in ellipse )2 :: {T''x^)x : (il^^^^^a ^3^ From (2) we find T^x^=.t^A^, and substituting the value of T^x^ in (3), we have (sec.incircle)2 : (sec.inellipse)2 :: t^'A^'x : (^M^)^^^ :\ Ax \ B^ (5) Observe the right angled triangle CSQ. SG=A, CS=A — x, OG=B. A''—(A—xy=B^ Or, 2JaJ— a;2=52 Substituting this value of B^ in (5), and dividing the last couplet, gives (sector in circle)^ : (sector in ellipse) ^ : : A : 2A — x Dividing the last couplet by A, and extracting the square root, gives (sector in circle) : (sector in ellipse) : : 1 : ^2 — — (6) When the ellipse is very eccentric, A is very great in relation , X • to X, and the fraction, — is then very insignificant in value. As an ellipse becomes more and more eccentric, its curve approaches nearer and nearer to a. parabola, and when it becomes a parabola, A is infinite in respect to x, and the fraction — is then absolutely zero, and proportion (6) becomes (sector in circle) : (sector in parabola) : : I : J2 Q. E. D. 292 ROBINSON'S SEQUEL. The following inquiry has frequently come to us. We now give it in the words of a correspondent. Mr. Robixsox : Dkar Sir. — On page 192, Art- 180 of your Astronomy, it is stated that because the mean radial force causes the moon to circulate at -I.- part greater distance from the earth than it otherwise would, its periodical revo- lution is increased by its 179th part. The question is, where does the fraction -K come from? RE PL y. The mean radial force acting in the direction of the radius vector does not prevent the moon from describing equal areas in equal times. Therefore the moon describes the same area with, as it would without this action ; but the radius is increased, and consequently the angular velocity diminished. We will now give the increase of radius, and require the cor- responding decrease of angular velocity, and we shall find the ratio of one will be double that of the other, on the condition that the increase or decrease of either, is small in relation to the whole- Let U be the angular point of two equal sectors, 7' the radius of one, and A its arc. x its angle on the radius of unity. Let {r-\-h) be the radius of the other sector, A ^ its arc and y its angle. Then by reason of the two equal sectors, rA=(r-{-h)A^ (1) From one sector, \ : x : : r : A. Or, A=rx. From the other, 1 : y : : (r-\-h) : A^ Or, A ^=(r-\-h)y. Substituting the values of A and ^, in (1), we have r'x={r-\-kyj/ Or, X : y : : (r-]-hy : r^ Or, % \ y \ \ r-^-'lrh-^-h^ : r^ Because A is a very small fraction in relation to r, h^ can be omitted ; then X \ y \ \ r^-f-2rA : r^ Or, X \ y \ '. r -j-2 h r ASTRONOMY. 293 This last proportion shows that if the radius r is increased by h, the angular velocity and consequently the periodic time must be di- minished by 2A. PROPOSITION . Given the position of the earth as seen from the sun, the position of any other planet as seen from the sun, to find the position of that planet as seen from the earth. The motion of the earth and planets being known, and the elements of their orbits, the astronomical tables give the position of the earth and any planet for any given instant of time.. The position of the planet from the earth must then be computed by plane trigonometry. But before we give a definite example, we adduce the following LEMMA. 1 . In any plane triangle the greater of two sides is to the less, as radius to the tangerd of a certain angle. 2. Radius is to the tangent of the difference between this angle and 46°, as the tangent of half the sum of the angles at the base of the tri- angle is to the tangent of half their difference. To obtain that certain angle, we must place the two sides at right angles to each other. Let CA be the greater of two sides of a A, and CE a less side placed at right angles ; then CAE is the certain angle spoken of, less than 45°, and EAB is the difference between it and 45°. From (7 as a center with the longer side as radius, describe the semicircle. Then DE = the sum of the sides, and EG their difference. Join DA, A G, and from E draw EB parallel to DA. DA G is & right angle because it is in a semicircle ; therefore, EB being parallel to DA; EBG is a right angle also. DA=zAG, and EBz=BG. Let a be the greater side of a triangle represented in magnitude but not in position, by CA, and c the shorter side, represented in magnitude by CE ; then it is obvious that a '. c '. '. R : tan. CAE ■ 294 ROBINSON'S SEQUEL. This angle taken from the table and subtracted from 46° will give the angle EAB. By proportional triangles we have BE That is, a-f-c But. AB Whence, a-\-c EG EB AB AB B R BG^EB EB tan. EAB tan. EAB By proportion 7, page 149, Robinson's Geometry, we find that a-^c : a — c : : tan.^sum ang. atbase : tan. ^ their diflf. Therefore by comparison, R : tan.-£'^-5 :: tan. | sum ang. at base : tan.^ their difF. Q.E.D. The application of this proposition is very advantageous when the logarithms of the two sides of a triangle are given and not the sides themselves. It obviates the necessity of finding the numerical values of the sides. This proposition is almost solely used in Astronomy, and we give the following example as an illustration. In the Nautical Almanac for 1864, Ifirid that on the first day of April at noon, mean time at Greenwich, the sun's longitude is 11° 26' 28", and the logarithm of the radius vector of the earth is 0.0000224. At the same time the heliocerUric longitude of Jupiter is 283° 46' 7", soitth latitude 6' 41", and logarithm of its radius vector 0. 71 45152. Required the geoceyitric latitude and longitude of Jupiter, and the logarithm of its true distance from the earth. Let S be the sun, 'Y'=a= the line made by Aries and Libra in the plane of the ecliptic, /y^ , and let this distance be made radius ; then IS will be the cotangent of the heliocentric latitude, and IE the cotangent of the geocentric latitude. Denote the geocentric latitude by x ; then D \ R \ \ IS \ cot. 6' 41" And D : E '. '. IE : cot. a; 7-ET Whence, IS : cot 6' 41" : : IE : cot. ar=lr cot. 6' 41" That is, Erom the log. of the ijlaneCs distance from the earth, svhtract the log. of its distance from the sun, and to the difference add the log. cotangent of the heliocentric latitude, and the sum is the log. cot. of the planet's geocentric latitude. To apply this equation with accuracy, requires some little tact in using logarithms. Observe that cot. 6' 41" is the tan. of 6' 41", subtracted from 20.0000. To find the tan. of 6' 41" or 401", first find the tangent of 1", fhen add the log. of 401. tan. l'=60" *. . .6.463726 sub. log.60 1.778151 tan. 1" 4.685575 log. 401 2.603144 tan. 6' 41" 7.288719 Log. -£^/— log. //S'=0.7257234— 0.7145150=0.0112084. The log. cot. must be increased by this quantity, therefore the log. tan. must be diminished by the same ; hence 7.2887190 0.0112084 ASTRONOMY. ' 29^7 Log. tan. of geocentric latitude is 7.2775106 Subtract log. tan. of 1" 4.685575 Log. of 390"8, or 6' 31" nearly, 2.5919356 Thus we find the geocentric latitude of Jupiter to be 6' 31" south at this particular time. Having the planet's latitude and longitude, we can compute its corresponding right ascension and declination, and the following results will be obtained : Right ascension, 19h. 46m. 9s. South declination 21° 19' 41". SOLAR ECLIPSES. We will now show the computation to determine the times of beginning and end, and other circumstances attending a solar eclipse as seen from any assumed locality on the earth. No person can do this with any safety, depending on the rules of another, he must understand the nature and scope of the problem for him- self. It requires a general knowledge of astronomy and philoso- phy, and a familiar knowledge of both plane and spherical trigonometry. The mathematical philosophy of the subject is explained gen- erally on page 214 of Robinson's Geometry, and here we will illustrate it by an example. As near as we can determine by some rough projections, the eclipse of May 26, 1854, will* be nearly central and annular as seen from Burlington in Vermont. Curiosity has, therefore, led us to make minute calculations for that place. We take the elements from the English Nautical Almanac. — Let the reader observe that the elements here correspond to the mean time of conjunction in rif/ht ascension. The elements in Robinson's Astronomy correspond to conjunction in longitude, the difference is 8m. 37s. in time. 1854, May, 26. Greenwich mean time (/ in R. A 8h. 55m. 43.8s. Sun and moon's R. A * 4h. 13m. 7.41s. Moon's Dechnation North, 21° 33' 3r'8 * This was written in April, 1853, and therefore spoken of in the future tense. «98 ROBINSON'S SEQUEL. Sun's Declination North, 21° 11' 16"8 Moon's Horary motion in R. A 31' 18"9 Sun's Horary motion in R. A 2' 31"8 Moon's Horary motion in Declination N 8' 7"3 Sun's Horary motion in Declination N 25"9 Moon's Equatorial Horizontal parallax 54' 32"6 Sun's ** ** ** 8"5 Moon's semidiameter 14' 53"5. Sun's S. D. 15' 48"9. Lat. of Burlington 44° 28' N. West Long. 73° 14'=4h. 52m. 56s Greenwich mean time of q^ ^^' 55ra. 44s. Long, in time 4h. 52m. 66s. Mean time of (/ at Burlington . Equation of time, add 4h. 2m. 48s. P.M. 3m. 15s. Conjunction at B., apparent time. . . .4h. 6m. 3s.=61° 30' 45". As the earth is not a perfect sphere, (the equatorial diameter being the largest,) the equatorial horizontal parallax requires re- duction for other latitudes, and latitude itself requires a reduction at all points, except at the equator and the poles. The horizontal semidiameter of the moon requires .augmenta- tion, as the moon rises in altitude, for the nearer the moon is to the zenith of the observer, the nearer it is in absolute distance. The following tables correct the elements in these particulars* Reduction of the Parallax and also of the Latitude. Lat. Red. of par. Red. of Lat. Lat. Red. of par. Red. of Lat. Lat. Red. of par. Red. of Lat. o " / // o " 1 / II o /' / II 0.0 0.0 3 0.0 1 11.8 33 3.3 10 28.3 63 8.8 9 18.3 6 0.1 2 22.7 36 3.8 10 64.3 66 9.2 8 32.9 9 0.3 3 32.1 39 4.4 11 13.2 69 9.7 7 42.0 12 0.5 4 39.3, 42 4.9 11 24.7 72 10.0 6 45.9 16 0.7 6 43.4 46 6.5 1128.7 75 10.3 6 46.4 18 1.0 6 43.7 48 6.1 1125.2 78 10.6 4 41.0 21 1.4 7 39.7 51 6.7 11 14.1 81 10.8 3 33.5 24 1.8 8 30.7 54 7.2 10 65.7 84 11.0 2 23.7 27 2.3 9 16.1 67 7.8 10 30.0 87 11.1 1 12.3 30 2.7 9 55.4 60 8.3 9 57.4 90 11.1 0.0 ASTRONOMY. 299 Atigmentaiion of the Mooti's Semi- diameter. Horizon. Semi-diameter. Alt. Horizon. Semi-diameter. Alt. f4'30" If 16' 16' It 17' 14'30" 15' 16' It \r — OT H o // // 2 0.6 0.6 0.7 0.8 42 9.2 9.8 11.2 12.6 4 1.0 1.1 1.3 1.5 45 9.7 10.4 11.8 13.3 6 1.6 1.6 1.9 2.1 48 10.2 10.9 12.4 14.0 8 2.0 2.1 2.4 2.7 51 10.6 11.4 13.0 14.7 JO 2.4 2.6 3.0 3.4 54 11.1 11.8 13.5 15.2 12 2.9 3.1 3.6 4.0 57 11.5 12.3 14.0 15.8 14 3.4 3.6 4.1 4.7 60 11.8 12.7 14.4 16.3 16 3.8 4.1 4.7 5.3 63 12.2 13.0 14.9 16.8 18 4.3 4.6 5.2 5.9 =sin. 68° 26' 28". .. ..9.968853 cos 9.562944 0.31787 —1.502252 .255192 —1.406861 Whence, the natural sine of the moon's true altitude, or cos. Zm=0.31787+0.25519=.57306. * As the moon is at m, and the sun at And i? : 2694.6 : : cos. 50° 10' 8" : q mn 3.430466 3.430466 sin. 60° 10' 8" . 9.885322 cos 9.806537 p 2069.2 3.315788 q 1726.8. . .3.237003~ Observe that p is the effect of parallax perpendicular to the lunar meridian at that time; and q is the parallax in declination. 302 ROBINSON'S SEQUEL. Moon^s true declination north of the sun 1336" Moon's parallax in declination south 1725"8 Moon's apparent dec. south of the sun 390^8 The apparent distance between sun and moon is, therefore, V(2069.2)2+(390.8)2=2105"7 Moon's S. D., 14' 53"5. Augmentation for altitude 8". Sun's S. D. 16'48"9. Sum = 30' 50"4=1860"4. But the distance (apparent) from center to center, we have just determined to be 2106"7 ; therefore the distance from limb to limb must be 255"3, and the eclipse has not yet commenced, and cannot commence, until the moon gains 256" on the sun's motion, which will require more .than ten minutes of time. We now require the apparent distance between the centers of the sun and moon, ten or twelve minutes later, so as to get the ap- parent rate of approach. The rate ^^^ is continually changing, but du- ^Qf^ ring any short interval of ten or twelve minutes, it may be consid- ered uniform, without any sensible error. If we vary the time, the angle ZPS will vary 1° to 4 minutes, but in that variable time the moon will move from c to m, and the angle ZPm will vary, but not quite so much as ZPS. The question now is. If we make a small difference in the angle ZPm, what corresponding difference will it make in the arc Zm ; and this is a question in the differential calculus,'^ although we can work it out at large by spherical trigonometry. We will take the interval of 12m., then the angle ZPS will increase 3°. But in one hour the moon's motion in right ascen- sion exceeds, that of the sun 28' 47" ; this in 12m. will be 5' 45"4, therefore the angle ZPm varies in that interval of time, 2° 64' 14"6 =2.90406, taking one degree as the unit. * The differential calculus is the science of minute variations, or of corres- ponding small differences — a science which owes its birth to the varying dfeements of astronomy. ASTRONOMY. 30S The equation as before is sin. ^=cos. Zw=cos. P cos. L sin. D-\-m!L, L cos. D But we have caused P to vary, while L and D remain constant. What variation will this give to the altitude A ? Taking the differential of the equation, we find * jM sin. P COS. L sin. B-dP cos. ^ But we have assumed c?P=2. 90405, while P, Z, B, and -4, in this equation, have the same values as before. That is, />=61° 30' 45", i;=44° 16' 33", i>=68° 26' 28", and ^=34° 58' 10". log. 2.90405 0.463000 sin.P —1.943954 (radius unity.) COS. L — 1.854910 sin. B —1.968853 COS. complement A 0.086445 rf^=2.0755 0.317162 Thus we find that the moon changes its altitude at this time, in the interval of 12 minutes, .2° 4' 32", and because the second member of the last equation is minus, the altitude has diminished. Moon's altitude was 34° 58' 10" Variation 2° 4' 32" Moon's altitude at this time 32° 53' 38" The angle ZPm=61° 30' 45"+2° 54' 15"=64° 25'. cos. 32° 53' 38" : sin. 64° 25' : : cos. 44° 16' 33" : sin. ZmP=sin. 60° 16' 30". To find mn, or the parallax in altitude. To log. of the horizontal parallax , 3.513044 Add COS. 32° 53' 38" 9.924100 Approximate value of mn 45' 36"=2736" 3.437144 From the moon's true alt. 32° 53' 38" Subtract apparent paralla x 45' 36" 3.513044 Moon's appa. alt. nearly 32° 8' 8" cos 9.927877 True value of mn 2760" 3.440921 As before, E : 2760 : : sin. 50° 16' 30" : p R : 2760 : : cos. 60° 16' 30" : q 4 ?i04 ROBmSON'S SEQUEL. 3.440921 3.440921 sin. 50° 1 6' 30" ..9.885 996 cos 9.805420 J^2123" 3.326917 q 1763"5.. . .3.246341 During the 12 minutes the moon moves over the oblique stkuiU. arc C7n, (in relation to the sun, as conceived to be stationary,) which is 5' 45"4 in right ascension, or the difference between the two meridians PS and Pm on the equator, is 5' 45"4, or 345"4. The perpendicular distance at the point m is therefore found by multiplying 345"4 by the cosine of the moon's declination to radius unity. Therefore, Log. 345"4 2.538322 Moon's dec. 21° 35' nearly, cos 9.968429 Perpendicular dis. between PS and Pm 32r'2. ..2.506751 During one hour the moon's relative motion in declinatian is 7' 41"4. During 12 minutes it is therefore 92"2, which added to 22' 15" or 1335" makes 1427"2 for the distance represented by ma. But q 1763"5 is the effect of parallax on the line or the ef- fect in declination, and it being greater than 1427"2, their differ- ence, 336"2 is the apparent distance in declination of n below S, or of the center of the moon below the center of the sun. Again, p is the parallax in right ascension, projecting the moon 2123" west of its true place, while it is 321 "2 east of the sun ; therefore the apparent right ascension of the moon is 1801 "8 west of the sun. Consequently the apparent distance of the two centers is V(1801"8p+(336"2)2 = 1 833"2. But the semidiameter of the sun and the augmented semidiam- eter of the moon at this time amount to 1850"4, differing only 17"2. The distance between the centers being less than the sum of the semidiameters, shows that the eclipse has already com- menced Twelve minutes before this time, the distance between the centers was ■* 2105"7 Now it is 1833"2 Moon's apparent motion in 12 minutes . . .'. 272"5 or 22^7 in one minute. Then 22"7 : 17"2 : : 60s. : 46.4 seconds. ASTRONOMY. 306 That is, the eclipse commences 11m. 14.6 sec. after the appa- rent time of conjunction at Burlington, or at 4h. 17m. 17.6 sec. If 6 seconds be taken from the sum of the semidiameters for irradiation and inflection, as most astronomers recommend, the eclipse will commence at 4h. 17m. 30s. THE POINT OF FIRST CONTACT. The point n, the apparent place of the center of the moon is nearly west of S and the angle ZmP=50°. Therefore the point of first contact, from the sun's vertex, must be (50"-}~^0°)» ^40° towards the right, but if viewed through an inverting telescope, the appearance will be directly opposite. GREATEST OBSCURATION. The time of greatest obscuration will take place not far from Ih. 20m. after conjunction at Burlington, or not far from 5h. 26m. 3s. ; we will therefore compute the apparent distance between the two centers for this time. We could compute it by proportion, provided the apparent motion of th,e moon was uniform, and in a straight hne ; but that motion being neither uniform nor in a straight line, we are compelled to compute it by points to obtain any thing like accuracy. Using the last figure, the angle ZP^=5h. 26m. 3s. =81^ 30' 45" ; but during Ih. 20m. the moon will gain 38' 22" in right as- cension ; therefore the angle ZPw=80° 62' 23". In Ih. 20m. the moon will increase her declination 10' 49", making it 21° 44' 21", or Pm=68° 15' 39", and am is now 32' 30"= 1950". As before, sin. -4=cos. Z7»=cos. P cos. L sin. i)-|-sin. L cos. D. COS. P=cos. 80° 52' 23" 9.200404 cos. Z=cos. 44° 16' 33" 9.854910 sin 9.843917 sin. i>=sin. 68° 1 6' 39" .9.967959 cos 9.568656 0.10555. —1.023273 .25856 —1.412571 Nat. sin. ^=0. 10555+0.25856=.3641 1 . Whence, A, moon's true alt.=21° 21' 12". Zm=68° 38' 48". 20 ' % 306 ROBINSON'S SEQUEL. sin. 68° 38' 48" : sin. 80° 52' 23" ; : cos. 44° 16' 33" : sin. ZwP= sin. 49° 22' 45". To find mn. Moon's horizontal par. log 3.513044 COS. 21° 21' 12" 9.969114 Approximate value of mn 50' 35"=3035" 3.482168 From moon's true alt. . .21° 21' 12" Take 50' 35" 3.513044 Moon's appa. alt. nearly 20° 30' 37" cos 9.970630 True value of mn 3046" 3.483674 R : 3045" : : sin. 49° 22' 45" : p B : 3045" : : cos. 49° 22' 45" : q 3.483674 3.483674 sin. 49° 22' 45" 9.880265 cos 9.813620 i?=2311"8 3.363939 ^=1982"6... .3.297294 In the Ih. and 20m. which elapses after conjunction, the moon gains 38' 22" or 2302" in right ascension on the sun ; but this is arc on the equator, it is not perpendicular distance, the two me- ridians PS and Pm, drawn from m ; but that distance is required and it is found thus : Log. 2302 3.362105 Add COS. of moon's declination 21° 44' 21" .9.9679 59 wic 2138"5 3.330064 p 231 1"8 Moon apparently west . . 173"3 Moon's declination north of sun am 1960" Moon's parallax in declination q 1982"6 Moon apparently south of the sun 32"6 Distance between centers= ^(iTS^p +(32"6) ^ = 1 76"3. We know by comparing this result with the last, that the gi-eatest obscuration or nearest approach of the centers, must take place about 7 minutes after this time. We will, therefore, differentiate for 10 minutes. In 10 minutes the sun's polar angle will increase from the meridian 2° 30' For the 3'^ motion in R. A. sub. 4' 47" The angle ZPm will increase 2° 26' 13 '=2°.4202. ASTRONOMY. 307 As before, . _ sin. P COS. L sin. D ( g.4202) COS. -4 An equation in which ^=21° 21' 12", P= 80° 62' 23", Z= 44° 16' 33", and i>=68° 16' 39". sin. P —1.994465 (radius 1) COS. L —1.864910 sin. i> —1.967959 log. 2.4202 , 0.383861 COS. complement A 0.030886 (^-4=1.7062 0.232071 The minus sign before the second member shows that this must be subtracted from A. A 21° 21' 12" 1.7062= 1° 42' 22" Moon's true altitude at this time, 19° 38' 60" Log. Horizontal parallax 3.513044 cos. 19° 38' 60" 9.973950 51' 9" 3.486994 Moon's app. alt. nearly 18° 47' 41" 3.613044 cos. 18° 47' 41" 9.976154 True value of mn 3084"5 3.489198 cos. 19° 38' 60" : sin. 83° 17' 36" : : cos. 44° 16' 33" : mi.ZmP =sin. 49° 1' 40". R : 3084"6 : : sin. 49° 1' 40" : ^ R : 3084"6 :: cos. 49° 1' 40" : g 3.489198 3.489198 8in.49° 1^ 40" 9.877978 cos . 9.816700 p 2329". . . . 3.367176 g 2022"7 3.306898 At the last point, am was 1950" which has increased 77" by the moon's motion ; therefore it is now 2027". - At the last point, Sa was 2302" of arc which has increased 287", making 2589", which must be reduced to the arc of a great circle as before. f !► ^ ROBINSON'S SEQUEL. Log. 2589 3.413132 Moon's declination 21° 46' cos 9.967927 Moon east of sun 2404"6 3.381069 Parallax in R. A. p., .2329" Moon east of sun, apparently 76"6 Moon north of sun 2027" Parallax in declination, q., ,, 2022"7 Moon north of sun, apparently .... 4"3 Distance between centers = ^(76"6)2-|- (4"3)=75''6, appa- rently. Now to find the nearest approach of the centers, the time of forming the ring, its continuance, &c., we have a very delicate and simple problem in plane geometry. Let S be the center of the sun. Take SV = 173"3, F"u4=:32"6. Then ^aS'=176"3. Also take then /Sf^=75"6, BD= 173"3+75"5=248"8. i>^ =37" nearly ; then ^J5 the moon's apparent motion on the face of the sun during 10 minutes must be V(248"8)2-f(3rp =251 "5. Therefore the apparent motion per minute is 25"15. We must now find Sm, the distance between the two centers at the time of their nearest approach. In the triangle ABS, we have all the sides, therefore by (Prop. 6, page 149, Geom.), we have AB : AS-\-SB : : AS—SB : Am-^B That is, 26r'6 : 26r'9 : : 100.7 : 100.86 ^wi+7w^=251.6 Am—mB=100M 2mJ5= 150.64 m.B= 76.32. Whence, Am=n&'lB. In the right angled triangle BmS, we have Sm= J(75"6y—{75"32y =6"5 Moon's semidiameter from giv^a elements 14' 63"6 Augmentation for alt. 18° <"" (see table) 4^7 ASTRONOMY. 309 Augmented semidiameter 14' 58"2 Sun's semidiameter from given elements 15' 48"9 Diflference, 50"7 * It is obvious that the ring will form when the distance between the two centers comes within the difference of the semidiameters. Suppose it to form when the moon's center passes n ; then in the right angled triangle Smn, Sn=50"7. Sm—6"5. And mw=7(50"7)2— (6"6)2=50"28. An=Am — mw=126"9, and at the rate of 25"16 per minute, this will be passed over in 5m. 2.7 seconds, nm in 2 minutes very nearly, and an equal line on the other side of m in 2 minutes more. The appa. time the Q)'s center arrives at A is 5h. 26m. 3s. To which add 5m. 2.7s. Ring forms at 5h. 31m. 5.7s. Time of nearest approach 5h. 33m. 5.7s. Rupture of the ring 5h. 35m. 5.3s. At the time of nearest approach the breadth of the ring on the north limb of the sun will be ST'O, and on the south limb 18"8 ; but if the customary allowance be made for irradiation and inflec- tion, these quantities reduce to 31 "3 and 18"3, and the duration of the ring must be reduced from 3m. 59.6s. to 3m. 55.2s. THE END OF THE ECLIPSE. We know by the moon's apparent motion (25''15 per minute, which is continually increasing) that more than an hour will be required from the time of nearest approach, for the eclipse to pass ofif. We will therefore compute the apparent distance between the two centers, one hour and ten minutes after the moon passes A^ of the last figure. By referring back we shall find that the point A corresponds with 6h. 26m. 3s. or 81° 30' 45" for the angle ZP5. One hour and. ten minutes later will be 6h. 36m. 3s. and will correspond to 99° 0' 45" for the angle ZPS. (See next figure.) * Astronomers recommend a diminution of 3" for the sun's semidiameter for irradiation, and a diminution of 2" of the moon's semidiameter for in/lio^ Hon, this would make 49"7 for their difference instead of 50"7. • ' 310 ROBINSON'S SEQUEL. Butthe difference between the right ascensions of the sun and moon is now 1° 21' 66" ; therefore the angle ZFm=97° 38' 49". At 5h. 26m. 3s. the value of ma was 1950", in one hour and ten minutes it increased 540", it is now 2490". Sa in arc=l° 21' 66"= 49 16" which we reduce to distance. Log. 4916 3.691612 Sun's dec. cos. 21° 12' 9.969667 Sa 4583" 3.661179 As before, sin. ^=cos. Zm=cos. P cos. L sin. D-j-sin. L cos. D cos. P=cos. 97° 38' 49". .—1.124088 cos. L =cos. 44° 16' 33", .—1.854910 sin.. .—1.843917 sin. D =sin. 68° 7' 13". .— 1.967531 cos. .—1.671327 —0.088416* —2.946529 0.26015-1.415244 sin. A or cos. Zm=0.26015 — 0.08842=. 17173 Whence, ^=9° 53' 21". Zw=80° 6' 39". sin. 80° 6' 39" : sin. 97° 38' 49" : : cos. 44° 16' 33" : sin.Zwi'= sin. 46° 6'. Log. Horizontal parallax 3.613044 cos. 9° 63' 21" 9.993500 63' 30" 3.506544 Moon's app. alt. 8° 69' 61" cos 9.994618 3.613044 mn 3.607662 sin. 46° 5' : p cos. 46° 6' : g 3.607662 cos 9.841116 p 2918.4 3.465192 q 2232.4 3.348778 Sa 4583. 2490 ~1664.6 267.6 H : 3218 : B : 3218 : 3.507662 '5'... ..9.857530 Distance between the centers = 7(1 664.6 )2-f-(267.6)» = 1674"5. * This number must be minus because cos. 97°^ is minus, the cosine of any arc over 90°, as far as 270°, is minus. m ASTROHOMY. Sll The sum of the semi- diameters is now 1 843" ; therefore the sun is still eclipsed, and will be un- til the apparent motion of the moon passes over ^V^^^^^^^^HB^ 169", which will require a little over 6 minutes of time, we will therefore compute the apparent distance of the cen- ters for 8 minutes later. In 8m. the angle ZPS will increase 2°, and the angle ZFm will increase 2°— (3'+50") or 1^ 66' 10". At the last operation ZFm was 97° 38' 49", now it must be 99° 34' 59", say 99° 35'. If we take D as constant in the last equation, we shall find that all our logarithms will be the same except cos. F, and all we have to do is to add to log. — 2.946529 the difference between COS. F in the last operation and the cosine of 99° 35', or the sine of 9° 35' for the log. of the first number composing the natural sine of^. That is, to —2.946529 Add .097279 ' Number, —0.110604 —1.043808* sin. u4=0.26015— 0.1 10604=. 149546. Whence, ^=8° 36'.t COS. 8° 36' : sin. 99° 35' : : cos. 44° 16' 33" : sin. ZwP=sin. 45° 33' 50". Log. Horizontal parallax 3.513044 COS. ^ 8° 36' 9.995089 Approx, val. of mn 53' 42" 3.508133 Q)'s appa. alt. nearly 7° 42' 18" cos 9.996064 3.513044 True value of rm 3229"5 3.509108 JR : 39.2d"5 : : sin. 45° 33' 50" : p R : 3229"5 : : cos. 45° 33' 50" : q ♦Artifice should be employed to take out the number corresponding to this logarithm, such as is taught in the author's Surveying and Navigation. t Here we have jumped from one result to another, and did not obtain the difference between one result and another, as we do by the differential method. 312 ROBINSON'S SEQUEL. 3.509108 3.509108 sin. 46° 33' 50\ ... 9.853717 cos 9.845168 ^ 2902"6 3.462825 q 2260.6 3.364276 Before Sa was in arc 4916", increase in 8m. 230"; therefore it is now 6146" which must be reduced as before. Log. 6146. 3.711470 cos. 21° 12'. . . ..9.969667 Sa in space, . . .4797"6. . .3.681037 p.... .2902"6 ma before was 2490 Paral. in R. A — 1896 Licrease in 8m 62 2662 q 2260.6 Moon apparently north of sun 291 .6 Distance between the centers=^(1895)2-|-(291.5)2= 1917"3. This being greater than 1843" shows that the eclipse has passed oior. The distance between the centers now is 191 7"3 Eight minutes ago the distance was 167^45 Apparent motion of the moon in 8 minutes 242"8 Corresponding motion for 1 minute 30"35. From the sum of the semidiameters 1843", subtract 1674"5, and we obtain 168"5 for the moon to pass over before the end of the eclipse. This at the rate of 30"36 per minute requires 5m. 33s. Hence, to 6h. 36m. 3s. appa. time. Add 5m. 33s. Eclipse ends 6h. 41m. 36s. appa. time. But if we reduce the semidiameters for irradiation and inflection 6", then we must diminish the time of ending 10 seconds. We may now observe that the moon's apparent motion across the sun was at the rate of 22"7 per minute at the beginning of the eclipse, 25"15 at the time of nearest approach, and 30"35 per minute at the end. This variability of the apparent motion is owing to the varying effects of the moon's parallax correspond- ing to the different altitudes, and this makes the problem tedious, and throws over it an air of complexity. Since six o'clock the rate of the moon's motion from the sun ASTRONOMY. ^» increased very much, and any one can see the rationale of this by inspecting the projection on page 293 of Robinson's Astronomy. Along the mid- day hours the sun and moon have an apparent motion together, but with diflPerent velocities. As the time from noon increases, the sun's motion along the ellipse is slower, and the moon appears to run over it faster and faster. After 6 the sun's apparent motion is no longer with the moon's, hence a rapid increase in the moon's apparent motion. We have made the problem much longer than we should have done, had we simply been in pursuit of results. Our object has been to explain and illustrate the problem to a learner, through each consecutive step, and we have found the following SUMMARY. Appa. time Burlington, Vt. Mean time. ^ Beginning of the eclipse, 4h. 17m. 17.6s. 4h. 14m. 2.5b. Formation of the ring, 5h. 31m. 5.7s. 5h. 27m. 50.6s. Time of nearest appr. of cen. 5h. 33m. 5.7s. 5h. 29m. 50.6s. Rupture of the ring, 5h. 35m. 5.3s. 5h. 31m. 50.2s. End of the eclipse, 6h. 41m. 36 s. 6h. 38m. 21 s. Duration of the ring 4 minutes nearly ; duration of the eclipse 2h. 24m. 19s. When the ring is most perfect, its breadth on the north limb will be 31", and on the south limb 18". Not long since the author received the following request : we extract from the letter. ** One request more. In your Astronomy, page 191, near the bottom, you say, (speaking of the radial force,) 'and the diminu- tion in the one case is double the amount of increase in the other, and by the application of the differential calculus, we learn the mean result for the entire revolution, is a diminution whose analytical rS expression is ; an expression which holds a very prominent place in the lunar theory.* r Sf Now my enquiry is, how can we obtain the expression for 314 ROBINSON'S SEQUEL. the mean result ? What operation in the calculus shall we go through ? Yours, &c., Wm. T " To this we returned the following reply : On page 193 you will find the following expression, 4^ rS ( S COS.' X— I) for the radial force corresponding to any angle x from the syzigies. We already know the value of this force at the syzigies and quadratures, and at these points the result has the same general form ; therefore the result for the entire quadrant, that is, the mean result for the whole quadrant, will be found by taking the an- gle ar=45°, and as the mean result for each quadrant is the same, this will be the mean result for the entire revolution. Whence, a;=46°, sin. a;=cosic, and 2cos.^a;=l, or cos.^a:=|. Or, (Scos.^ar — 1)=^ ; whence the above general expression becomes ~-. 2a3 To this was returned the following observation : *'I understand your explanation, it is very simple ; but why did you not make this explanation in the book, — and more than all, why do you call it an application of the differential calculus ? /can see no calculus in it. Yours, &c., Wm. T." To this we rejoined as follows : If the operation I sent you is not calculation, I know not what it is — it may therefore be called calculus ; ahd if in any operation small quantities may be omitted on account of their insignificance in relation to larger quantities, the small difference so omitted constitutes the differential calculus, and to obtain that geneial expression, you will see, by looking on page 193 of the Astronomy, that the powers of r above the first were omitted. CALCULUS. 316 THE €AjL€UI.irS. DIFFERENTIAL CALCULUS. The differential calculus is a branch of Analytical Geometry. It k a science for computing the ratio of small diflferences. For example, the side of a square is increased by a very small quantity, what will be the corresponding increase of the square itself ? The side of a cube is increased or diminished by a quantity very small in relation to the side itself, — ^how much will this in- crease or diminish the cube ? The arc of a circle is increased or diminished by a quantity very small in comparison with itself, what effect will this have on the sine and cosine of the arc ? The sun's longitude increases a certain distance in 10 minutes, what is its corresponding change in declination ? Or, find the law of these corresponding changes, or differences — called differ- etUicds, These questions explain in part the object of the calculus. The calls of astronomy gave birth to this science, as we have before remarked. For the development of this science, see the various works upon it. We confine ourselves in this book to a few difficult or curious operations. We presume the reader is acquainted with all the rules of operation. EXAMFLSS. (1.) Differentiate the expression jY—x^. Ans. Jl—x' Put u=Jl—x^. Square, u^=:l—x^ ; then xdx 2udu= — 2xdXf or du^=-d* J\ — x^=- Jl—x' (2.) Find the differential of the equation X u= 516 ROBINSON'S SEQUEL. By the rule for diflferentiating a fraction, we have x'dx dx(x-\- J 1 — x^) — xdx- {x+j\-x-y J\-X-+-^^:^-_ dx (^x+JU^"")^ By multiplying numerator and denominator of the second member by ^1 — x^ y then multiplying the equation by dx, we have J dx du= {x+J\-^x^Y J\—x' (3.) Q\yenu^=la-\-Jb — —\ to find the differential of u. Put y=J^^ — — and extract the 4th root of the original equa- tion ; then I tt*~ du=dy du:=^dy{a-\-yy (1) cdx But y^ = h — ~ Whence, yc?y= — _. dy 2 *- x^ " ^Ah-^— Substituting the values of y and £?y in (1), we have v- CALCULUS. 317 /I \ X \ J\ X C4.) Given u=~ ■ ^ to find the differential of u* ^ ^ Jl-\-x—Jl—x Reduce the second member by multiplying numerator and de- nominator of the fraction by the numerator ; then X Apply the rule to differentiate a fraction, and to differentiate the numerator, simply make use of the first example. — x^dx du J^Z^-d<^+J^-^') X' Dividing both members by dx, and changing signs, and ~dx Jl-x- '^ X- ^^^ X^J\—X^ X^'Jl- n-\-J\—x^\ Whence, du= — I )dx \ x^J\—°*-i(log.a;)°->. dx (10.) mffereniiateu=^'^^"^^^Mf)+'^ As before, let 0=log.a; ; then the equation becomes _j:^z^ x*ZjX^ ^ 4~ ~8"^32 Then ^./ = ^^,,2^^xx'zdz_ x^zdx_ x^ dz.x^dx '2 2 8^8 But dz= — , and z = log.a; ; substituting these values, the pre- X ceding equation becomes du=xH\09r ^y^^ji ^''(^^g'^)d x_ x^(\og.x)dx ___x^dx,x^dx V e- y -r 2 ^ 8^8 Whence, du=x^{\og.xydx. 320 ROBINSON'S SEQUEL. (11.) Differentiate u=\og.^x : that is to say, the logarithm of the logarithm of x. Put log. x=:z. Then w=log. z. And du=—. But dz=^. z as That is, du=—= -— zx a;(log.a:) (12.) Differentiate u=\og.^x. '^ By the aid of the previous example we learn that log.5a;=log.(log.''a;). Whence, du='iJ^?L^ (1) . log. 4:?; d ( log.*a;)=-i — 5.' — L, this substituted in (1) reduces it to log.^a; du= ^(l^g-^^) (log.'»a;)(log.3^) Another step gives du= _i— ^l_^i (I0g.^..)(l0g.=^2r)(l0g^a:) Using the result of the previous example, we finally have , dx du= (log.'*a:)(log.^a;)(log.^a;)(log.a;)a; (13.) Differentiate u=e (x — 1 ) . Here we must take the logarithm of each member, observing that the log. of e^ is simply x, because e is the base of the Na- perian system of logarithms, the system always used in such examples. log. w=a;-|-log.(a: — 1). du , , dx dx — =dx-\- = u X — 1 x — 1 u _e\x-\). Whence, r:!!=_Jl_=lAlIZiZ=c\ Or, du^edx, dx X — 1 X — 1 ,(14.) Differentiate u=^e'{x'^—^x'^'\-^x—Q). log.w=^-|-log.(4;^ — 'ix^-\-^x — 6) du_. .Sx^dx — Gxdx-\-6dx w'"" x^-^x^'+ex—G du x^ udx^'x^-^Sx^-^-Qx^ CALCULUS. 321 du x^u dx x^'—^x'^'+Qx—Q (15.) Differentiate u= Or, du^e^x^dx. \—x u(\ — x)^=e'x. log.w-|-log.(l — x)-=x-\-\q^.x. dxL dx 7 [ dx du 1 . x-\-\ \-\-x — x^ u 1 — X X udx 1 — X X (1 — x)x du_ {\^x—x ^) e^x _[\J^x —x'' )e^ ^^ Jw=(i+fr:^!if!^ dx {\—x)x \ — x ' {\—xY ' (1—^)^ (16.) Dffererdiate uz^e'log.x. Put y=log. X ; then u=e^y. \og.u=x-\-\og.y. -_=cZa;-f--- u y ■r, . J dx T dy dx But dy= — , and ^ — X y X log. X ^TTj, du y . dx du x\opi;.x-\-\ Whence, — =dx4- = — ^ L_ u x\og:.x udx a;locr.a; du (x]og.x-\-'i)u (x log. x-\-\ )f log. X dx X log. X X log. X (17.) Differentiate u=^ — - — Ande'+1 = $ [ (2) Q au^^JL-^^ (3) Differentiating (1) gives e^dx^^dP. (4) And (2) gives e'dx^=dQ (5) Whence. (e^-f- 1 )e'dx= QdP And, {e''-^\)e'dx=PdQ By subtraction, ^e^dx=QdP—PdQ Whence, du= ~-. 21 M ROBINSON'S SEQUEL. CIRCULAR FUNCTIONS. For the sake of reference we will here note down the differential expressions for trigonometrical lines. Let the radius of a circle be unity. Represent an arc by x, then its differential will be dx. d sm.x=cos.xdx (1) d cos.x= — sin.x dx (2) d ver. sin.a;=sin.a;G?a; (3) d sec.a;=-?^ — (4) cos.ar d i2ing.x'=—-— (5) c^ cot.=— __^_ (6) cos.^ar ^ ^ sin.2.r ^ ^ One great difficulty which troubles and perplexes the studeiit in the calculus, arises from the fact that only the abstract theory of the science has been hitherto brought to our notice, in our ele- mentary books. All can understand how these equations, (1), (2), (3), = sin. E, or J)=iE, as it should be. Now suppose that the sun changes its longitude 10', which we may call tlie {dL), or the differential of L, what will be the cor- * Those who are naturally more nice than wise, are commonly prejudiced against this science, and such frequently say it is no science at all ; how- ever, their objections are of no consequence. 324 ROBINSON'S SEQUEL. responding change in its declination, or wliat will be the value of (dD)'! To answer this question we must take the differential of each member of the general equation, then we shall have COS. DdD=fim. U COS. LdL (2) ^ Now whatever values we may assign to L and dL, equations (1) and (2) will always give D and dD at any point. For a definite example we give the following : What will be ihe differential in declination corresponding to the differential of 10' in longitude at 35° of longitude ? In other words, what will be the change in the sun's declina- tion'while it passes from longitude 35° to 35° 10' ? From ( 1 ) we' find D thus : "^ sin. E 9.599970 sin. Z 35° 9.758591 sin. B 13° 12' 5" 9.358561 From (2), ^^^ sin. ^c o.^Zr/Z cos. 1) sin. E... 9.599970 COS. Z 35° 9.913365 c?Z= 10 log 1. COS. D, complement 0.011629 Sum (less 20) 3.343 0.524964 This is 3' 20"6 nearly, and if the sun's declination is 13° 12' 5", when its longitude is 35°, the declination must be 13° 15' 25"6 at the longitude 35° 10'. This is nc^ strictbj true, because the ratio of motion in declina- tion changes in a very slight degree between 35° and 35° 10'. But the ratio between the motion in longitude and the motion in declination, is strictly as 1 to .3343, at the beginning of the arc between 35° and 35° 10'. This ratio, is constantly changing, but still equation (2) always represents it. We now require the ratio of motion in declination when the sun's longitude is 90°, that is, Z=90° ; then cos". Z=0 ; and sub- stituting this value in (2) will cause the second member to dis- appear ; and cos. D dD=0 CALCULUS. 326 Now one or the other of these factors must be zero ; but cos. D is evidently not zero ; therefore {dD) must equal 0, showing that there is no motion in declination exactly at that point. On the contrary, we may demand the sun's longitude when the motion in declination becomes zero. In other words, what will equation (2) show when (dD)=0 ? Then, "sin. ^cos. L dL=0. One of these three factors must be zero; but (sin. JE) cannot be zero, for it is a known constant quantity ; (dL) cannot equal zero in case the least possible time elapses, for the sun's apparent motion never ceases ; then (cos. L) must be zero, and Z=90°, or 270°. To show another example of the utility of the calculus, we present the problem that appears in another shape, on page 229, of our Surveying and Navigation, namely : Under ivhat circumstances will an error in the altitude of the sun, produce the least possible error in the time deduced therefrom ; the declination and latitude being constant quantities. Let P be the polar point, Hh the horizon, S the posi- tion of the sun, and Z the zenith of the observer. Let A = the altitude of the sun, D its decHnation, and X the latitude of the observer. Then by spherical trig- onometry (see page 214, Surveying and Navigation) we have COS. P= sin. A — sin. L cos. D COS. L sin. D Tht altitude A varies, and P the polar angle, or time from apparent noon, must vary in consequence of the variations of A ; and^if A is not accurately taken, P will not be accurate. In short, a differential to A, will produce a differential in P. 326 ROBINSON'S SEQUEL. Therefore we must diflferentiate the equation, taking A and P as variables, and L and D constants. Whence, —sin. P dP=.^^^-—- ( 1 ) cos.L sin.i> But in the triangle PZS, we have cos. A ^ : sin. P : : cos. D : sin. SZP, or sin. Z f. . sin. P COS. D ,a\ Or, cos. A=^ (2) sin.Z ^ ^ Substituting this value of cos. ^ in (1), and dividing by sine P, we obtain - dP= ^?.^;A^___x^^ cos. Z sin. D sin. Z Or, —dP=^- ^.^ , (3) cos. L tan. D sin. Z Now it is obvious that the numerical value of (dP), or error in time, will be least when the denominator of the fraction in the second member is greatest, and that will be greatest when sin. Z is greatest, which is the case whenever Z=90°, which is when the sun is east or west of the observer. The sign before {dP) being minus, shows that when the altitude A increases, the angle P decreases. If we make dA^=0, equation (3) Avill give dP=0, showing that if there be no error in A, there will be none in P. We give a few more examples in circular functions. (1.) Let Z be an arc or angle whose radius is wiity and cosine (mx) : we require the differential of the arc in terms ofmx. In other words, differentiate Z=cos.~*(77iar). The rules of operation are all comprised in the following equations : d sin.a:=cos. x dx ( 1 ) d cos.a:= — sin.a; dx (2) " /lzll_, By substituting this value, proportion (a), becomes rfZ (1— y')Vi— 2y' VI— y' Or, dZ : M^-y') : : 1 : 1 Whence, dZ^ ^^ (\-.y^) J\—Oiy^ (4.) Differentiate the equation Z=sin.~^(2w^l — ^m^.) If 2w^l — w^ is the sine of an arc, the cosine of the same arc must be (1— 2«^2). By the proportion observed in the last example, dZ : ^d{\—2u^) : : 1 : ^ujv^^ dZ : ' ^udu Whence dZ=. That is, dZ : ' ^udu : : 1 : ^ujX-^w" 9,du (5.) Differentiate the equation w=cos. x sin. 2x. Regarding the second member as a product, and observing the dififerentials for sines and cosines, we have du= — sin. X sin. 2xdx-\- 2 cos.^2a; cos. xdx Whence, _if =(cos. 2ar cos. x — sin. 2a: sin. a;)-|-cos.ar cos.2;r. dx By observing equation (9), page 141, Robinson's Geometry, we perceive that the quantity in parenthesis is the same as cos. (te-\-x)y or COS. 3a; ; therefore, c?w=(cos. 3ar-}-cos 2a; COS. ar)c3?a;. CALCULUS. 389 (6.) Differentiate the equation w=(taii. «)". Put tan. x==y ; then w=2/°, and duz=^ny^^dy (a) But v=tan.a;'; therefore c/y=(^ (tan. 0?)= , see (3). cos.^a; Whence (a) becomes du=. -^ '--l- cos.^a; Sin W/j* (7.) Differentiate the equation u=--—-^ — ---. Let sin. nx^=P, and sin. x^ Q. p Then the equation becomes w= — Whence, du^^^^J^H.^^^ Dividing numerator and denominator by Q'^S and we have Because P=sin. nx, dP=n cos. nx dx, and because ^=sin. a; dQ=GOS.xdx. Now by substituting the values of P, Q, dP and dQ, (a) becomes 7 (n sin. X cos. nx — n sin. 7ix cos. x) dx ("sin. xy"*'' That is, by equation (8), page 141, Geometry^ , nsin.(nx — x)dx "^ (ibT^'^T (8.) Differentiate the equation «=log.(cos. x-\-J — 1 sin. x. The second member being a logarithm, its differential is the differential of the quantity divided by the quantity. That is, 7 — sin. X dx-\-J — 1 cos. x dx. J — 1 sin.a:-|-cos.a; Or, du — sin. x-^-J — 1 COS. a; . — r- ^^ J — 1 sin. ic-f-cos.ir Whence, du=J — \dx. (9.) Differentiate the equation z<=sin. Jl+: TT 33G ROBINSON'S SEQUEL. X must be ; and we must have If the sine of an arc is the cosine of the same arc / 1 =) ,, X That is, du : ^dx^l+^x^ : : 1 : ^x — - : : 1 : 1 (;w=- 1+x^ l-\-x^ Orv ^.. : -J^- : : 1 : 1 du= ^"^ LUKAR OBSERVATIONS. The differential calculus may be used with great facility and success in clearing lunar distances from the effects of parallax and refraction. Let S'm' = the apparent central distance between the sun and moon, or the moon and a star, SS' is the refraction of the sun or star, and it is sufficiently small to be taken as the differential of the altitude. Also, m'm is the correction for the moon's apparent altitude, and we may call it the differential of the moon's altitude. The observed triangle is ZS'm'. Let aS^ represent the altitude of the sun or star, m the altitude of the moon, and x=S'm'y the observed distance. Now by the fundamental equation of spherical trigonometry, (see Geometry, page 191), we have ^ COS. ir — sin. /S sin. m * ^^k COS. Z= -— (1) COS. O^COS. Wl We now take the differential of this equation, observing that Z is constant, and that x varies only on account of the variations of m and S, * Observing that the sine of an altitude is the same as tlie cosine of the corresponding zenith distance. CALCULUS. sfl First clear of fractions, tlien differentiate ; then we shall have — cos,Z(sin./S'cos.m dS-\-cos.Ssin.m dm)= — sin.a; dx — cos.S sin. m dS — cos.w sin.S dm. Changing all the signs, and substituting the value of cos. Z from (1), reduces to /cos.arsin. S — sin.^AS^sin.mX ^cr i /cos.arsin.m — sin.^msin.^S'^ ',\y^_./cos,x sin.m — sin.^m sin.^S'X , / \ • cos.m / \ cos. S =sin.a; t/ar-j-cos./S^sin.m dS-]-CQS.7/i sm.Sdm. Transposing and uniting the coefficients of dS and of dm, will give (cos.a; sin. S — sin. ^ S sin.m — cos. ^ S sm.m)dS Ccos.a: sin.m — sin. ^m sin. /S' — cos.^m sin./S^ )c?m . , ^ 1 — = sm.a; ax COS. m Observing that sin. 2 /S'4-cos.^/S'=l, andsin.^m-[-cos.^m=l, and changing the order of the terms, we perceive that \ cos.m / \ COS. S / Here we should observe that (dm) is an elevation of the moon's apparent altitude, and (dS) is a depression of the sun or star's apparent altitude, therefore if we take (dm) positive, (dS) must be taken negative. Therefore, ^^_/c_os^^^^i^Illn^y^_/^^ (2) \ sin. a; cos.m / \ sin. a; cos. aS / This is the final equation, (dx) representing the quantity be- tween the true and apparent distance. Sometimes (dx) is positive and sometimes negative, according as the differential coefficient, or quantities in parenthesis are posi- tive or negative. When the differential coefficient of (dS) is negative, that term becomes positive, because (dS) is negative, and the product of two negatives is positive. When the altitude of the sun or star is greater than 60°, the corresponding refraction ((iS) will be a very small quantity, which can never be augmented by its coefficient ; therefore in that case the value of (dx) will be sufficiently represented by /c_os^^n.m-sin^5y^ (3) \ sin.a: cos.m / % 332 ROBINSON'S SEQUEL. Equation (2) will solve any example that may be prepared. We will solve one or two of those found on page 227, of our Surveying and Navigation. For the first example there found, in which S=Q6° 3', m=39° 18', x==46° 45', and the mpon's horizontal parallax 53' 51", ex- pression (3) will be sufficient. We must first find (dm), which is the parallax in altitude diminished by the refraction. 53' 51 "=3231" log 3.509337 cos. m 39° 18' 9.880651 log. 2499" 3..397988 39° 18' Refraction,... 69" 2430"=(Zm For the coefficient, we operate thus : (radius unity.) sin. m.. — 1.801665 cos. 7W. .—1.888651 cos. a;.. — 1.835807 sin. a:... — 1.862353 log. 4340 ..—1.637472 —1.751004 Nat. sin. aS' 99762 [sub. the upper.] -56362* log — -1 .750975 ^—1.999971 c?m=2430 log -. 3.385606 dx=40' 29"=.2429" 3.385577 X orthe app. dis. 46° 45' True distance 46° 4' 31" The answer in the book is 46° 4' 25". Our omission of the second term in equation (2) might have produced an error of 4", not more, still making a difference of 2", but this is of no conse- quence in itself. Different operators may work the same exam- ple by the same or different methods, and they will rarely produce results within 5" of each other ; and as no observations can be relied on within that limit, such results in a practical point of view are said to agree. We now take the 6th example from page 227, Surveying and Navigation. * Because this quantity is minus, (dx) must be minus, and therefore we subtract it from the apparent distance to find the true distance. CALCULUS. 3to Given sun's app. alt. 8° 26', Q)'s app. alt. 19° 24', horizontal parallax 67' 14". Apparent central distance 120° 18' 46", to find the true distance. Ans. 120° 1' 46".. Here S=^° 26', m=19° 24', and ar=120° 18' 46". Horizontal parallax 57' 14"=3434" 3.535800 cos.m 19° 24' 9.974 614 Parallax in altitude 3239" 3.51041 4 Refraction ^^ 161" dm=: . .T:T3078" dS=:6' 10"=370". (Because x is greater than 90° its cosine will be minus, which will render the differential coefficient of dm minus.) sin.m... — 1.521349 cos.m... — 1.974614 COS. a;...— 1 .703045 sin. a: —1.936152 —.16763 —1.224394 —1..9 10766 den. Bin.S .1466 6 —.31429 log — 1.497340 num --1. 586574 c?m=3078" log 3.488269 First term of (dx) —1188" 3.074843 sin. S. . .—1.166307 cos. S . . . — 1.995278 cos. ar. . .—1.703045 sin. .r —1.936152 —.07402 —2.869352 —1.931430 sin.wi .33216 —.40618 -1.608736 —1.677306 — c?iS^=370" 2.568202 Second term of (dx) + 176" 2.245508 —1188" dx= —1012"= 16' 52" Apparent central distance, 120° 18' 46" True distance, 1 20° 1' 54" The result differs 8" from the given answer as determined by other methods, which arises from taking 3078" as a differential ; it is a large arc, rather too large to be taken for a differential arc. 334 ROBINSON'S SEQUEL. MAXIMA AND MINIMA. The differential of any quantity is a general expression for a small increase of the quantity ; but if the quantity is already a maximum, an increase is impossible, and the expression for an in- crease must be zero. A decreasing differential is a general expression for a small relative decrease of any quantity ; but when the expression is already a minimum, it can have no further decrease, and the expression for such decrease must therefore be zero. Hence, in cases of a maximum or minimum, we must put the differeniial of the quantity equal to zero, thus forming a new equation, which equation generally gives the results sought. For example, the following equation always unites the declina- nation of the sun with its longitude : sin. i)=sin. E sin. L ( 1 ) Here E is the obliquity of the ecliptic, and is a constant quan- tity. L is any longitude, and D the corresponding declination. Taking the differential of this equation, we find COS. D dl)=sin. E COS. LdL (2) If we now assume the condition that D is a maximum, it is the same as to assume that (dD)=0, which makes sin. E COS. L dL=0 Here is the product of three factors, one of which must equal 0. Sine E is of known value, not equal to zero ; therefore cos. LdL=0. If cos.Z=0, Z=90°. If dL=0, L=0. Substitu- ting these values of L in equation (1), we have sin. i>=sin. ^, or JJ=E, when D is a maximum, and sin. i>=0, or D=0, when i> is a minimum ; which are obvious results. Again, the general value of the differential of the sine of any arc whose length is x and radius unity, is cos. x dx. But if the sine is a maximum, it can have no differential, ex- cept in form. That is, cos. x=0, or dx=0 ; whence a:=90°, or x=0. Showing that when the arc is 90°, the sine is a maximum, "when 0, the sine is 0, a minimum. For another iWMsirsiiion, I propose to divide the number a into (wo suck parts that ike product of tke parts skall be the greatest pos- sible. • K CALCULUS. 335 At first I will simply get an expression for any indefinite rect- angle. — That is, if x= one part, (a — x) will equal the other part, and (ax — x^) will be an expression for a rectangle which will be larger or smaller according lo the relative values of x and a. Taking the differential of the expression, we have adx — 2xdx. Now if I assume (cix — x"^) to be a maximum, it cannot in- crease, therefore its differential must be zero, or adx — 2xdx=^0. Or, a — 2a;=0, or, x=^a, which is the value of x when the product is a minimum, and may be verified by trial. EXAMPLES. (1.) Required the greatest rectangle that can he inscribed in the quadrant of a given circle. It is evident that one extremity of the diagonal of the rectangle must be at the center of the circle, and the other extremity at some point on the arc of the quadrant. Let a = the diagonal or radius of the circle, and x = the arc from one extremity of the quadrant to the point in which the rect- angle meets the curve. Then a sin .r= one side of the rectangle : acos. ar= the adjacent side, and the area of the rectangle is a^sin. iccos. x. The problem requires that this expression should be a maxi- mum, which is the same thing as requiring that its differential expression should be zero. Hence, a'^cos. x dx cos. x — a^sin. x dx sin. a;=0. Dividing by a^dx, and cos.^a; — sin.^;i:=0. Or, COS. a;=:sin. a;. But the arc which has its cosine equal to its sine is 45*^, which shows that the diagonal of the rectangle bisects the quadrant, and the rectangle is in fact a square. . We observe* that a^ disappears in the result, and this shows that the problem is independent of the radius, and equally ap- plies to all circles. In short, constant factors in a maximum may he cast out hy divi- sion hefore we take the diferential. ' (2.) Required the greatest possible rectangle that can he inscribed in a {fiven parabola. m 336 ROBINSON'S SEQUEL. Put VD=a, VB=x, PB=.y. Then BD=a — X, 2t/(a — a:) = maximum, and y^ =: 2px, by the equation of the curve. Taking the differentials, we have dy{a^x)—ydx=0. Or, dy^^^ a — X ydy=pdx. Or, dy^=^^J^ y Whence, ^ =±. Or, y-=iap — px. a — X y That is, 2px^=ap — px. 9.x=a — x. x^^a. This result shows that from the vertex, J- of the given distance is the point through which to draw one side of the maximum rectangle. ^ (3.) Problem 3 on page 253 of this work, is a beautiful exam- ple to show the power and utility of the calcidus. In fact it was the result of the calculus that pointed out the geo- metrical construction. Let AP=a, PB=^b, PD=x, and call the angle APD=P. Then I)JI=xsm.P, PB'=xcos.P, All^a—x cos. P, HB= X COS. P±ib, according as IT falls between A and B, or between B and P. Corresponding to our figure, HB=x cos.P — b. In the triangle AHD, we have 1 : i2.ii.ADH 1 4. AnzT « — ^ COS.P 1 : isin. AI>Ii= DH : HA Or, X sin .P : a- —X cos.P Also, icsin .P : X cos.P— b 1 : tsin. ffDB= X sin.P X cos.P — b X sin. P Adding these two tangents according to the mathematical law of the sum of tangents, expressed in equation (28), page 143 of Robinson's Geometry, will give , a — b X sin.P tan. ADB= 5-^ 5 — 7\ ^ a — X cos.P /x cos.P — b \ X sin.P \ X sin.P ^ (a — b) X sin.P ar^sin.^P — (a — x cos.P) (;c cos.P — b CALCULUS. 337 t^n.ADB=—^-^^J^^''J^^ x"- sin.2P+a6— (a4-6) x cos.P-f a;^ cos.^P (a — i)a;sin.P x^-\-ab — (a-f-^) a: cos.P By the requirement of the problem, the angle ADB must be a maximum ; but the angle will be a maximum when its tangent is a maximum. Hence we must put the differential of the ex- -^ 1 equal to zero. pression x'^-\-ab — (ci-\-b) X cos.P By omitting the constant factor (a — b) sin. P, and dividing nu- merator and denominator by x, we shall have only 1 r+— — (a+i) cos. P to differentiate. — dx-\- — dx ^x^ Whence. -; ; " r-^=0. (x-{J'l-^{a-Yb) cos. pV Or, a;2 =«6. This equation directed us to make FD = J 500 -200 as was done on page 263. (4.) The difference of arc between the sun's right ascension and its longitude gives rise to one part of the equation of time. What is the sun's right ascension when this part of the equation is a maximum, and what is the maximum value ? Let Z= the sun's longitude, x=. the corresponding right ascen- sion, ^= the obliquity of the echptic ; then by spherical trigo- nometry we have I'cos. -£'=:cot. Z tan. a:= — 1- fl) tan.Z ^ ^ (See equation (16) Robinson's Geometry, page 186, also (5), page 139.) But, tan.(Z-^) = A^"^i^:i (2) ^ ^ 1+tan.Ztan.a; ^ ^ (See page 143, equation (29), Geometry.) The problem requires that the differential of {L — x)^ or of tan. (Z — x) should be put equal to zero. 22 338 ROBINSON'S SEQUEL. Substituting the value of tan. L taken from (1) in equation (2), we have tan. « . — tan. a; cos.^ (1— cos.^)tan.a; tan. li/— a;)= -: — ^ =^^ j I tan.^a; cos. ^+tan.2a? COS. ^ 1^, tan. X . 1 Whence, =maximum, or COS. jE'-j-tan.^a; cos. -£'cot.a;-J-tan.a; = maximum. But this fraction is obviously a maximum when its denomii).ator is a minimum ; therefore, cos. ^ cot. a:-|-tan. a;=minimum. Taking the differential, we find — COS. JEJdx, dx ^ sin.^ar j-, -j- =0 — — = cos. -A sin.^a: cos.^a; cos.^a; tan. a:=^cos. ^ (3) » This value of tan. x put in ( 1 ) gives tan.Z = >/^^ (4) cos. ^ Equations (3) and (4) correspond to radius unity, but we can use the logarithmic table if we add 10 to the index of cos. ^ be- fore we take the root, thus : jE'=23° 27' 30" cos. +10 19. 962535 ( 2 tan. re 43° 45' 50" 9.981267 9.962535 tan. Z 46° 14" 10" 10.018732 Diff. of arc 2° 28' 20"=9m. 54.6s. (5.) What must he the inclination of the roof of a huUding that the water will run off in tlve least possible time ? Ans. 45°. Let a= the base of a roof, x= its perpendicular altitude : then Ja^+x^ will equal its length, and J^ will equal the time re- quired for water to fall through the height. But the time down an inclined plane is to the time through its perpendicular, as the length of the plane is to its height. Let <= the time down the plane ; then CALCULUS, 339 4 Or, ^=. J9^ But the problem requires that ^ should be a minimum ; its square must therefore be a minimum ; hence, — ! = a mmimum. X It will be observed that ^ is the force of gravity, and being a constant factor in the last expression, it was omitted. Taking the differential, we have 'ix^ dx—dx{a^ J^x"" )_^ x^=:a^^ -or a;=a. The perpendicular being equal to the base, shovrs that the in- clination required must be 45°. (6.) Within a triangle is a given point P, the distance to the nearest mngle A is given, and the line AP divides the angle A into two an- gles m and n, of which m is greater than n. It is required to find the line EF drawn through the point P, so that the triangle AEF shall he the least p>ossiMe, Let AP==a, AF=x, AE=y. The angle PAF=^m, FAE=^n. The area of the A ABF=ax sin. m. The area of the A APF=ag sin. n. By the conditions, ax sin. m-|-ay sija. ?*= minimum* ^ Also, by the conditions, arysin. (m-\-n)= minimum. From tlie first minimum, sin. mdx-^sm. ndg:=^0 From the second, xdg-^-gdx^O From(l), (1) (2) From {2}, sin.m dx=:= — -dy y 340 ROBINSON'S SEQUEL. Whence, y sin. n = x sin. m. Or, ay sin. n=.ax sin. m. This last equation shows that the line EPF mustjbe so drawn that the triangle APF will be equal to the triangle APE. We presume that the foregoing examples are sufficient to illus- trate the power and utility of the calculus in respect to maxima and minima. INTEGRAL CALCULUS. The integral calculus is the converse of the differential. In the differential cakulus we give the integral and require the dif- ferential. In the integral calculus we give the differential quan- tity, and require the integral from which the dijBferential was derived. Hence all our rules of operation msust have reference to the differential rules inversed. We cannot investigate the rules of operation in this work, but suppose the reader already acquainted with them. Many persons can operate ta some extent in the integral cal- culus without any distinct idea of what the integral calculus is, and this vagueness can never be fully driven away, except by close attention to the application and utility of the science. For example. — If we have the differential of a circular arc, by integration, we shall have the arc itself. If we have the differential o-f a circular segment, by integration we shall have the segment itself. If we have the differential quantity of a cone, by integrating that quantity we shall have the solidity of the cone itself. If we have the differential of a logarithm, by integration we shall have the logarithm itself. Thus we might go through the chapter. Because the integral is the opposite of the differential, the inex- perienced might conclude that one operation would be obvious from the other. In some instances the converse is obvious, but not generally so. We must not conclude that it is as easy to move in one direc- tion as in the opposite. — It is not as easy to ascend as to descend a plane — not so easy for a vessel to move against the stream as CALCULUS. 341 with it. It is not so easy to find the cube root of a number as it is to cube the root when found. The sign for the differential is (d). The sign for integration is ( f). Hence, J'du=u. The two signs destroy one another, and give the quantity u. If we take the differential of a;* we shall have 4x^dx. There- fore if we must integrate 4x^dx, we must frame a rule of opera- tion that will give x^^y which is the following : Add unity to the index, divide hj the index so iTicreased, and take away dx. The differential of - is ^ conversely then the the integral of is . But to integrate this by the above rule appears at first sight impossible, nevertheless it can be accomplished by the following artifice : Put 1 — x^=y. Then xdx=^ — \dy. A J xdx dy 1 -1-7 And, _=_-_JL=_iy Uy To the second member of this equation the rule will apply. Hence, J — ^ _=y~2__ — ______ . ^^2^^ The differential of xy is {xdy-\-ydx) ; therefore, J {xdy-\-ydx)=^xy . But the inquiry is, how we shall effect the integration under the rule. Here y is equal to, or greater, or less than x. Therefore we may assume that y=ax, a being a constant quantity. Whence, dy=:-adx, xdy=axdx. And, f (xdy-\-ydx)= C (axdx-\-axdx)=: f^axdx==^ax^. But ax^=ax'X, and because y=ax, ax^ =xy. Ans. Then we perceive that the actual integration was performed by the rule ; but the reader must not infer that the rule will apply to all cases — -far from it.* * The subject of integration requires the keenest algebraical talent, and fe"W persons are skilful algebraists in the highest sense of that term, who have aot been severely disciplined iu integration. » f 342 ROBINSON'S SEQUEL. We give a few examples under this rule. (1.) Given the diferenilal {^x^ydy -\-^^^xdx), to find its i$Uegr(d. Ans. x^y^ . Here we may assume y=£ax ; then dyz^etdx, ydy=.a^xdx. Whence, J ( ^"^ ydy-\-^y^ xdx) =J' ( 2a-x^dx-{-2a''x^dx) = C Aa^x^dx. To the second member the rule applies ; that is, C Aa^x^dx=a^x'^=za^x^ ^x^z=iy^x^ . Ans. (2.) Inte^e^te (J!+y'i^+(^!+^--M Ans. JV+^ Ja^'^^. Assume j¥^+^=P, and Ja^+P==Q. Then ydy=FdF, and xdx=QdQ. Substituting these values, and the given expression becomes F ^QdQ - j-PQ^dP That is, PdQ-\-QdP. ' But, J{PdQ-{-QdP)=PQ. That is, J^^+y^ X Ja'^+x'' . Ans. (3.) Inte^mte _— 3^y+3c/^ ^^^ (a— y+0)^ Put (a_^-|^s)*=zP ; then a— .y+2=/'^ And —dy+dz=4P^dP. Whence. __3^+3^ =3P^ JP. 4(a-y+z)^ And, r_Z±.^i'±?^=P3=(«-y+2)'. ^«*. Observatton. The differential of - is VJ^^J . therefore. y y2 . X the integral of this last expression is — But how shall we inte- y grate this, provided we did not know the integral ? The numerator would be the differential of the product xy, if the sign between the terms in the numerator were plus. CALCULUS. 843 Let us put y^=ax. Then ydx=:^axdx, and xdi/=axdXf and the yclx — xdv , ax dx — ax dx j /. expression f-— ^ becomes or , and our ef- ^ y^ a^x^ a^x^ fort fails. Now let us examine the cause of the failure. The product xy can represent any magnitude whatever, and if we put y=ax, then xy becomes ax^ ; and because x is variable, ax^ is still capable of representing any magnitude whatever. But iu the fractional expression -, if y=-ax, and a be regarded as con- y V X \ X stant, _ becomes — , or _, and in that case _ can only represent y ax a y * 1 X the ever constant fraction - ; but - must be capable of repre- a y senting any fraction whatever ; therefore we cannot put y = ax, unless we regard a as variable. Therefore to integrate the expression tL_^ ?L , put y = tx ; both t and x being variable. Then ydx=ixdx, dy:=tdx-\-xdt. xdy=txdx-\-x ^ dt. Whence, ydx — xdy=. — x^dt J ,, . r x^dt dt and the expression becomes — or— — t X z That is, jyl^HpL^J^t-^dt^t-^ hythe rule. Whence, the required integral is - ; but y ^=ix ; therefore, t y This branch of the subject may be treated as follows, provided the operator is cautious, and does not assume too much : When we differentiate a product as xy, we assume x as con- stant and y variable ; and then y constant and x variable, and thus we get two partial diflferentials. Now either one of these integrated on the supposition thai the letter which is affixed to (^d) is the variable one, and all others con- 344 ROBINSON'S SEQUEL. stant, will give the true integral. Thus the diflferential of xy is xdy-\-ydx. Now integrate xdy on the supposition that x is constant and y variable, and we have xy for the integral. ^Iso, Cydx=xy. It would therefore appear that ^xy is the whole integral, provided we did not know, a priori, that xy is the integral. ffence, when we integrate two differential expressions, on the sup- position that the letter not affected with the differential sign (d), is constant, and find two equal integrals, we must take but one of them. The same principle holds good in relation to the three or more letters. The diflferential of xyz is xydz-\-xzdy-\-yzdx. Now if we integrate this expression on the supposition that xy is constant in the first term, xz constant in the second, and yz constant in the third, we shall have xyz-\-xyz-\-xyz. Here are three equal integrals, but we must take but one of these for the whole integral, because the differential was eflfected by three distinct suppositions. The diflferential of - is yJf^Z^l^J^—xy-^dy, y y"" y Integrating each of these expressions on the supposition that y is constant in the first, and x constant in the second, we have x.x y y but we must only take one of these for the integral, for the same reason as before. EXAMPLES. (1.) Integrate (Sxy^y'^ )dx-\-(3x^ —'2xy)dy Ans. Sx^y — y^x. We integrate the first part on the supposition that y is constant, and the second on the supposition that x is constant, and we obtain 3x^ij—y^x-{-3x^y—y^x, and because we make two distinct suppositions, we divide by 2. Then test the result by taking the diflferential. CALCULUS. 346 (2.) Integrate {'iy''x-\'^^)dx-{-{^x^y-\-^xy'^'\'^y^)dy. Integrating each term, we obtain y^x^-{-Sy^x-\-x^y^^^xy^-\-^y\ Here we find two terms equal to aj^y^, and two terms equal to ^xy^f and one term Sy* ; hence I will take for the integral sought — which I find to be true by taking the differential. To integrate the varied expressions in the form ' x'^{a-\-bx'^ydx, we must resort to the established formulas, explained in elaborate works, which of course we cannot touch upon in a work like this. Because c? log. ar= Therefore, f =.\og.x-\-c {a) X ^ X ** d sin. x=cos.xdx. " f cos. xdx=sm. x-\-c (b) ** d cos. a;= — sin. xdx. " C — sin.icc?ar=cos.a;-j-c (c) '* .<^tan.x=_^-. » ** r_^_=tan.a;+c {d) cos.^a; *^ cos.^a; Each of these formula is a fundamental rule for integration. It is not necessary for us to explain the constard c. APPLICATION" OF THE INTEGRAL CALCULUS. We shall explain the application and utility of this science by examples. If x represents an arc of a circle whose radius is unity, and y the sine of the same arc ; then ^1 — y'^ will represent the cosine, and equation {b) above will become The integral of the first member will give the arc^ but it will be numerically indefinite, unless we can integrate the second member, and know the value of y corresponding to some definite value of X. 346 ROBINSON'S SEQUEL. We cannot integrate the second member in finite terms, there- fore we must develop it in a series, and integrate term by term, and if the series is suflficiently converging, the value of x can be known to any required degree of approximation. ' — L :=(1 — y^) ^dy. The binomial, expanded by the bi- /I- nomial theorem, produces Multiplying each term by dy, and integrating, we obtain , l-y3 , 1 3 ys , 1 3 5 y^ , 1 3 5 7 v% « , '2-3 245'2467'24689' ' This equation is true for all values of x. It is true then when a;=0 ; but if we make a:=0, y must equal at the same time. Therefore, if we make the supposition that x=0, the last equa- tion will become 0=0-|-c, or c=0. By some such special con- sideration, the value of c can be determined in almost every problem, although it is indeterminate in the abstract. Now the value of y is known to be \ when x, the arc, equals 30°; therefore. The arc of 30==l+iA+llll+M-l-l+^Li:^lI^_ 2 ' 3.2* ' 4.5.2« ' 4.6.7.28 ' 4.6.8.9.2»'> + &C. Multiplying by 6 and taking ten terms of the series, we shall have the value of a semicircle to radius unity. That is, we shall have 7t=3.1415926, which is true to the last figure. Thus we perceive that one operation in the integral calculus brings a result requiring many operations in common geometry. SURFACES AND SOLIDS. In general terms, aydx will represent the differential of any plane surface, and if so, Caydx-\-c, will represent any surface ; I CALCULUS. 347 but we can find the integral only when we know some relation between x and y. Also, ay^dx will represent the differential of any solid ; there- fore Cay^dx-\-c will represent the solid itself ; but we can find this integral only when we have some relation between x and y. EXAMPLES, Suppose x to represent the perpendicular of any triangle, and y its base ; then if x increases downwards by dx, ydx will be the differential of the triangle, the angles remaining constant. Therefore, Cydx will be the area of the triangle itself. This integral will require no correction, for when x=^0, y=0. The area being a triangle, we have a relation between x and y, for no triangle can exist without this numerical relation. Suppose we measure one unit down the base, and through that point draw a line parallel to the base, and find the length of this to be a units. Then whatever be the magnitudes of x and y, this relation will be constant, and a will be greater or less according to the angles of the triangle. That is, X : y : : 1 : a. Or, y=ax. Consequently, Cydx^zfcixdx^ =— .ar=:-iL That is, the area of any triangle is half the product of its base and altitude. Let VCI be any area. VC= x, CI=y, CD = dx, then the space CDRI=i ydx, the differential of the area. If VCI represents a portion of a parabola, 1. I then y^=z Then (r — x)'^-\-y'^=r^. y^z=2rx — ar^. Then J'Tti/^dx=ftJ^{2rx^x^)dx=:7i(rx^—^—\-\-c. This integral requires no correction, because when x=0, y=0 and then the area equals 0, and c=0. This integral represents the true value of any segment corres- ponding to any assumed yalue of x between x=0 and x=2r. If x=2r the segment will comprise the whole sphere. Then n{ rx^ — — )==7t{ 4r^ — )= \ 3/ \ 3/3 This corresponds to theorem 17, book vii. Geometry. (4.) The differential of is conversely. Iniegi'cUe . Ans. . Put n-{'l=m, then n=m — 1, and n — l=m — 2. With these substitutions the expression to be integrated is (m—\)x''-^dx ^^ (m—\) fx'^^dx(\-^)-'^. But (l+.)-"=l-..+..('!!±i).^-..(-t^) (^^).3+ Multiply the second member by x'^'^dx, then it becomes -f- (fee. ■ Now integrate each term separately, and the result will be ^::ii^x-jL.^x -^-^f !!H:i^ V"''-^+ &c. d«ced instead of the O's through the re to indicate that from thence the corre irst column stands in the next lower I to O's, points or dots are now .st of the line, to catch Ihe eye, spending natural numbers in ine, and its annexed first two figUJ 'es «f the Logarithms in the second co .ttmn LOGARITHMS OF NUMBERS. 3 N. I 2 3 4 5 6 7 8 9 100 000000 0434 0868 1301 1734 2166 2698 3029 3461 3891 101 4321 4750 5181 5609 6038 6466 6894 7321 7748 8174 102 8600 9026 9461 9876 .300 .724 1147 1570 1993 2415 103 012837 3259 3680: 4100 4521 4940 5360 5779 6197 6616 104 7033 7461 7868 8284 8700 9116 9632 9947 .361 .775 105 021189 1603 2016 3428 2841 3252 3664 4075 4486 4896 108 5306 5715 6125 6633 6942 7350 7767 8164 8671 8978 107 9384 9789 .195 .600 1004 1408 1812 2216 2619 3021 108 033424 3826 4227 4628 502» 5430 5830 6230 6629 7028 109 7426 7825 8223 8620 9017 9414 9811 .207 .602 .998 110 041393 1787 2182 2576 2969 3362 3765 4148 4540 4932 111 5323 5714 6106 6496 6886 7276 7664 8053 8442 8830 112 9218 9606 9993 .380 .766 1163 1638 1924 2309 2694 113 053078 3463 3846 4230 4613 4996 5378 5760 6142 6524 114 6905 7286 1075 7666" 8046 8426 8805 9186 9563 9942 .320 115 060698 1452 1829 2206 2582 2958 3333 3709 4083 116 4458 4832 6208 5580 6963 6326 6699 7071 7443 7815 117 • sise 8557 8328 92S8 S6C8 ..38 .407 .776 1146 1614 118 071882 2250 2617 2985 3362 3718 4086 4451 4816 6182 119 6647 5912 6276 6640 7004 7368 7731 8094 8457 8819 120 9181 9543 9904 .266 .626 .987 1347 1707 2067 2426 121 082V85 3144 3503 3861 4219 4676 4934 5291 5647 6004 122 6360 6716. 7071 7426 7781 8136 8490 8846 9198 9562 123 9905 .268 .611 .963 1315 1667 2018 2370 2721 3071 124 093422 3772 4122 4471 4820 5169 5618 5866 6216 6662 125 6910 7257 7604 7951 8298 8644 8990 9335 9681 0026 126 100371 071& 1059 1403 1747 2091 2434 2777 3119 3462 127 3804 4146 4487 4828 6169 5510 5851 6191 6631 6871 128 7210 7549 7888 8227 8565 8903 9241 9679 9916 .263 129 110590 0926 1263 1599 1934 2270 2606 2940 3275 3609 130 3943 4277 4611 4944 5278 5611 5943 6276 6608 6940 131 7271 7603 7934 8265 8596 8926 9256 9586 9915 0246 132 120574 0903 1231 1560 1888 2216 2544 2871 3198 3525 133 3852 4178 4604 4830 6166 6481 5806 6131 6466 6781 134 7105 7429 7753 8076 8399 8722 9045 9368 9690 ..12 135 130334 0655 0977 1298 1619 1939 2260 2580 2900 3219 136 3539 3858 4177 4496 4814 5133 5451 5769 6086 6403 137 6721 7037 7354 7671 7987 8303 8618 8934 9249 9564 138 9879 .194 .508 .822 1136 1450 1763 2076 2389 2702 139 143015 3327 3639 3951 4263 4574 4886 5196 6607 6818 140 6128 6438 6748 7068 7367 7676 7985 .8294 8603 8911 141 9219 9527 9836 .142 .449 ,756 1063 1370 1676 1982 142 152288 2594 2900 3206 2610 3815 4120 4424 4728 6032 143 5336 5640 5943 6246 6549 6852 7164 7457 7769 8061 144 8362 8664 8965 9266 9567 9868 .168 .469 .769 1068 145 161368 1667 1967 2266 2664 2863 3161 3460 3758 4055 146 4353 4650 4947 5244 5641 5838 6134 6430 6726 7022 147 7317 7613 7908 8203 8497 8792 9086 9380 9674 9968 148 170262 0565 0848 1141 1434 1726 2019 2311 2603 2895 149 3186 3478 3769 4060 4351 4641 4932 5222 6612 6802 4 LOGARITHMS N. 1 2 3 4 5 6 7 8 9 150 176091 6381 6670 6959 7248 7536 7825 8113 8401 8689 151 8977 9264 9552 9839 .126 .413 .699 .985 1272 1568 152 181844 2129 2415 2700 2985 3270 3565 3839 4123 4407 153 4691 4975 5259 5642 5825 6108 6391 6674 6956 7239 164 7621 7803 8084 8366 8647 8928 9209 9490 9771 ..51 155 190332 0612 0892 1171 1451 1730 2010 2289 2567 2846 156 3125 3403 3681 3959 4237 4614 4792 5069 5346 6623 157 5899 6176 6453 6729 7005 7281 7556 7832 8107 8382 158 8657 8932 9206 9481 9765 ..29 .303 .677 .860 1124 159 201397 1670 1943 2216 2488 2761 3033 3305 3677 3848 160 4120 4391 4663 4934 5204 5476 5746 6016 6286 6566 161 6826 7096 7365 7634 7904 8173 8441 8710 8979 9247 162 9515 9783 ..51 .319 .586 .853 1121 1388 1654 1921 163 212188 2454 2720 2986 3252 3518 3783 4049 4314 4679 . 164 4844 5109 5373 5638 5902 6166 6430 6694 6957 7221 165 7484 7747 8010 8273 8536 8798 9060 9323 9585 9846 166 220108 0370 0631 0892 1163 1414 1676 1936 2196 2456 167 2716 29/6 3236 3496 3765 4016 4274 4633 4792 5051 168 5309 5568 5S26 6084 6342 6600 6858 7115 7372 7630 169 7887 8144 8400 8667 8913 9170 9426 9682 9938 .193 170 230449 0704 0960 1215 1470 1724 1979 2234 2488 2742 171 2996 3250 8504 3767 4011 4264 4517 4770 6023 5276 172 5528 5781 6033 6286 6537 6789 7041 7292 7544 7795 173 8046 8297 8548 8799 9049 9299 9550 9800 ..50 .300 174 240549 0799 1048 1297 1546 1795 2044 2293 2541 2790 175 3038 3283 3534 3782 4030 4277 4525 4772 5019 5266 176 5513 5759 6006 6262 6499 6746 6991 7237 7482 7728 177 7973 8219 8464 8709 8954 9198 9443 9687 9932 .176 178 250420 0664 0908 1151 1396 1638 1881 2125 2368 2610 179 2863 3096 3338 3580 3822 4064 4306 4548 4790 5031 180 5273 5514 5755 5996 6237 6477 6718 6958 7198 7439 181 7679 7918 8168 8398 8637 8877 9116 9356 9594 9833 182 260071 0310 0548 0787 1026 1263 1601 1739 1976 2214 183 2451 2688 2926 3162 3399 3686 3873 4109 4346 4582 184 4818 5054 6290 6525 5761 5996 6232 6467 6702 6937 185 7172 7406 7641 7875 8110 8344 8678 8812 9046 9279 186 9513 9746 9980 .213 .446 .679 .912 1144 1377 1609 187 271842 2074 2306 2638 2770 3001 3233 3464 3696 3927 188 4158 4389 4620 4850 5081 5311 5542 5772 6002 6232 189 6462 6692 6921 7161 7380 7609 7838 8067 8296 8526 190 8754 8982 9211 9439 9667 9895 .123 .351 .578 .806 191 281033 1261 1488 1715 1942 2169 2396 2622 2849 3075 192 3301 3527 3768 3979 A205 4431 4656 4882 5107 5332 193 5557 5782 6007 6232 6466 6681 6905 7130 7354 7578 194 7802 8026 8249 8473 8696 8920 9143 9366 9589 9812 195 290035 0257 0480 0702 0925 1147 1369 1591 1813 2034 196 2258 2478 2699 2920 3141 3363 3684 3804 4026 4246 197 4466 4687 4907 6127 5347 5667 6787 6007 6226 6446 198 6665 6884 7104 7323 7542 7761 7979 8198 8416 8636 199 8853 9071 9289 9507 9725 9943 .161 .378 .596 .813 OF NUMBERS 5 N. 1 2 3 4 g 6 7 8 9 200 301030 1247 1464 1681 1898 2114 2331 2547 2764 2980 201 3196 3412 3628 3844 4059 4275 4491 4706 4921 5136 202 5351 5566 5781 5996 6211 6425 6639 6864 7068 7282 203 7496 7710 7924 8137 8361 S564 8778 8991 9204 9417 204 9630 9843 ..56 .268 .481 .693 .906 1118 1330 1542 205 311754 1966 2177 2389 2600 2812 3023 3234 3445 3656 206 3867 4078 4289 4499 4710 4920 5130- 5340 5661 5760 207 5970 6180 6390 6599 6809 7018 7227 7436 7646 7854 208 8063 8272 8481 8689 8898 9106 9314 9522 9730 9938 209 320146 0354 0562 0769 0977 1184 1391 1698 1805 2012 210 2219 2426 2633 2839 3046 3252 3458 3665 3871 4077 211 4282 4488 4694 4899 6105 5310 5516 5721 5926 6131 212 6336 6541 6745 6950 7155 7359 7663 7767 7972 8176 213 8380 8583 8787 8991 9194 9398 9601 9805 ...8 .211 214 330414 0617 0819 1022 1225 1427 1630 1832 2034 2236 215 2438 2640 2842 3044 3246 3447 3649 3850 4051 4253 216 4454 4655 4866 5057 5257 5458 6668 6859 6059 6260 217 6460 6660 6860 7060 7260 7459 7659 7858 8058 8257 218 8456 8656 8865 9054 9253 9451 9660 9849 ..47 .246 219 340444 0642 0841 1039 1237 1435 1632 1830 2028 2225 220 2423 2620 2817 3014 3212 3409 3606 3802 3999 4196 221 4392 4589 4785 4981 5178 5374 5570 6766 5962 6167 222 6353 6549 6744 6939 7135 7330 7625 7720 7915 8110 223 8305 8500 8694 8889 9083 9278 9472 9666 9860 ..54 224 350248 0442 0636 ^829 1«23 1216 1410 1603 1796 1989 225 2183 2375 2568 2761 2954 3147 3339 3532 3724 3916 226 4108 4301 4493 4685 4876 5068 5260 5452 5643 5834 227 6026 6217 6408 6599 6790 6981 7172 7363 7654 7744 228 7935 8125 8316 8506 8696 8886 9076 9266 9466 9646 229 9835 ..25 .215 .404 .593 .783 .972 1161 1350 1539 230 361728 1917 2105 2294 2482 2671 2859 3048 3236 3424 231 3612 3800 3988 4176 4363 4561 4739 4926 5113 5301 232 5488 5676 5862 6049 6236 6423 6610 6796 6983 7169 233 7356 7542 7729 7915 8101 8287 8473 8659 8845 9030 234 9216 9401 9587 9772 9968 .143 .328 .513 .698 .883 235 371068 1253 1437 1622 1806 1991 2175 2360 2544 2728 236 2912 3096 3280 3464 3647 3831 4016 4198 4382 4565 237 4748 4932 6115 5298 5481 6664 5846 6029 6212 6394 238 6577 6759 6942 7124 7306 7488 7670 7852 8034 8216 239 8398 8580 8761 8943 9124 9306 9487 9668 9849 ..30 240 380211 0392 0573 0754 0934 1115 1296 1476 1656 1837 241 2017 2197 2377 2657 2737 2917 3097 3277 3456 3636 242 3815 3995 4174 4353 4533 4712 4891 5070 6249 1 5428 243 5606 5785 5964 6142 63^1 6499 6677 6856 7034 7212 244 7390 7568 7746 7923 8101 8279 8466 8634 8811 8989 245 9166 9343 9520 9698 9875 ..51 .228 .405 .582 .759 246 390935 1112 1288 1464 1641 1817 1993 2169 2345 2521 247 2697 2873 3048 3224 3400 3575 3751 3926 4101 4277 248 4452 4627 4802 4977 5152 5826 5601 6676 5850 ! 6025 249 6199 6374 6548 6722 6896 7071 7245 7419 7592 j 7766 6 LOGARITHMS N. 1 2 3 4 5 6 7 8 9 250 397940 8114 8287 8461 8634 8808 | 8981 9154 9328 9501 261 9674 9847 ..20 ,192 .366 ,538 .711 .883 1056 1228 252 401401 1573 1745 1917 2089 2261 2433 2605 2777 2949 253 3121 3292 3464 8636 3807 3978 4149 4320 4492 4663 254 4834 5006 5176 6346 5517 5688 5858 6029 6199 6370 255 6540 6710 6881 7061 7221 7391 7561 7731 7901 8070 266 8240 8410 8679 8749 8918 9087 9257 9426 9595 9764 267 9933 .102 .271 .440 .609 .777 .946 1114 1283 1451 258 411620 1788 1956 2124 2293 2461 2629 2796 2964 3132 250 3300 3467 3635 3803 8970 4137 4305 4472 4639 4806 260 4973 5140 5307 5474 5641 6808 5974 6141 6308 6474 261 6641 6807 6973 7189 7308 7472 7638 7804 7970 8135 262 8301 H467 8633 8798 8964 9129 9295 9460 9625 9791 263 9956 .121 .286 .451 .616 ,781 .946 1110 1275 1439 264 421604 1788 1933 2097 2261 2426 2590 2764 2918 3082 265 3246 3410 3574 3737 3901 4065 4228 4392 4555 4718 266 4882 5045 5208 f-371 C634 5697 5860 6023 6186 6349 267 6511 6874 G83o 6999 7161 7324 7486 7648 7811 7973 268 8135 8297 8459 8621 8783 8944 9106 9268 9429 9591 269 9762 9914 ..75 ; .236 .398 .559 .720 .881 1042 1203 270 431364 1525 1685 1846 2007 2167 2328 2488 2649 2809 271 2969 3130 3290 3450 3610 3770 3930 4090 4249 4409 272 4569 4729 4888 5048 5207 5367 5526 5685 5844 6004 273 6163 6322 6481 6640 6800 6957 7116 7275 7433 7592 274 7751 7909 8067 8226 ,8384 8542 8701 8859 9017 9175 5i75 9333 9491 9648 9805 9964 .122 .279 .437 .594 .752 276 440909 1066 1224 1381 1538 1695 1862 2009 2166 2323 277 2480 2637 2793 2950 3106 3263 3419 3676 3732 3889 278 4045 4201 4357 4513 4669 4825 4981 5137 5293 5449 279 5604. 6760 5915 6071 6226 6382 6637 6692 6848 7003 280 7158 7313 7468 7623 7778 7933 8088 8242 8897 8552 281 8706 8861 9015 9170 9324 9478 9633 9787 9941 ..95 282 450249 0403 0557 0711 0865 1018 1172 1326 1479 1633 283 1786 1940 2093 2247 2400 2553 2706 2859 3012 3165 284 3318 3471 3624 3777 3930 4082 4286 4387 4540 4693 285 4845 4997 5150 5302 5454 5603 6758 5910 6062 6214 286 6366 6618 6670 6821 6973 7125 7276 7428 7679 7731 287 7882 8033 8184 8336 8487 8638 8789 8-940 9091 9242 288 9392 9543 9694 9845 9995 .146 .296 .447 .597 .748 289 460898 1048 1198 1348 1499 1649 1799 1948 2098 2248 290 2398 2548 2697 2847 2997 3146 3296 3445 3594 3744 291 3893 4042 4191 4340 4490 4639 4788 4936 5085 5234 292 5383 5532 5680 6829 5977 6126 6274 6423 6671 6719 293 6868 7016 7164 7312 7460 7608 7756 7904 8052 8200 294 8347 8495 8643 8790 8938 9085 9283 9380 9627 9676 295 9822 9969 .116 .263 .410 .567 .704 .851 .998 1145 296 471292 1438 1585 1732 1878 2025 2171 2318 £464 2610 297 2756 2903 3049 3195 3341 3487 3633 8779 8925 4071 298 4216 4362 4508 4(>53 4799 4944 5090 5235 5381 6526 299 5G71 6816 6962 6107 6252 6397 6542 6687 6832 6976 OF NUMBERS. 7 N. I 2 3 4 6 6 7 8 9 300 477121 7266 7411 7555 7700 7844 7989 8133 8278 8422 301 8566 8711 8855 8999 9143 9287 9481 9575 9719 9863 302 480i)07 0151 0294 0438 0582 0725 0869 1012 1156 1299 303 1443 1686 1729 1872 2016 2159 2302 2445 2588 2731 304 2874 3016 3159 3302 3445 3587 8730 3872 4015 4167 305 4300 4442 4585 4727 4869 5011 5153 6295 5437 5579 306 6721 5863 6005 6147 6289 6430 6572 6714 6856 6997 307 7138 7280 7421 7563 7701 7845 7986 8127 8269 8410 308 8551 8692 8833 8974 9114 9255 9396 9537 9667 9818 309 9959 ..99 .239 .380 .520 .661 .801 .941 1081 1222 310 491362 1502 1642 1782 1922 2062 2201 2341 2481 2621 311 2760 2900 3040 3179 3319 3458 3697 3737 3876 4015 312 4163 4294 4433 4572 4711 4850 4989 5128 5267 6406 313 6544 5683 5822 5960 6099 6238 6376 6515 6653 6791 314 6930 7068 7206 7344 7483 7621 7759 7897 8035 8173 315 8311 8448 8586 8724 8862 8999 9137 C275 9412 8560 316 9687 9824 9902 ..99 .236 .374 .511 .648 .785 .922 317 601059 1196 1333 1470 1607 1744 1880 5017 2154 2291 318 2427 2564 2700 2837 2973 3109 324^i 3382 3518 3655 319 3791 3927 4063 4199 4335 4471 4607 4743 1878 6014 320 5150 5283 5421 5557 5093 5828 5964 6093 C234 6370 321 6505 6640 6776 6911 7046 7181 7316 7451 7583 7721 322 7856 7991 8126 8260 8395 8530 8664 8799 8934 9008 323 9203 9337 9471 9606 9740 9874 ...9 .143 .277 .411 324 510546 0679 0813 0947 1081 1215 1349 1482 1616 1750 325 1883 2017 2151 2284 2418 2551 2684 2818 2951 3034 326 3218 3351 3484 3617 3750 3883 4016 4149 4282 4414 327 4548 4681 4813 4946 6079 5211 5344 5476 5609 5741 328 5874 6006 6139 6271 6403 t635 6668 6800 6932 7064 329 7196 7328 7460 7692 7724 '.855 7987 8119 8251 8382 330 8514 8646 8777 8909 9040 9171 9303 9434 9566 9697 331 9828 9959 ..90 .221 .353 .484 .615 .745 .876 1007 332 521138 1269 1400 1630 1661 1792 1922 2053 2183 2314 333 2444 2575 2705 2835 2966 3096 3226 3356 3486 3616 334 3746 3876 4006 4136 4266 4396 4626 4656 4785 4916 335 5045 5174 5304 5434 5563 5693 5822 5951 6081 6210 336 6339 6469 6598 6727 6856 6985 7114 7243 7372 7501 337 7630 7759 7888 8016 8146 8274 8402 8531 8660 8788 338 8917 9045 9174 9302 9430 9559 9687 9815 9943 ..72 339 530200 0328 0466 0584 0712 0840 0968 1096 1223 1L61 340 1479 1607 1734 1862 1960 2117 2245 2372 2500 2f27 341 2754 2882 3009 3136 3264 3391 3518 3645 3772 3899 342 4026 4163 4280 4407 4534 4661 4787 4914 6041 5167 343 6294 5421 5547 5674 5800 5927 6053 6180 6306 6432 344 6658 6685 6811 6937 7060 7189 7316 7441 7667 7693 345 7819 7945 8071 8197 8322 8448 8574 8699 8825 8951 346 9076 9202 9327 9452 9678 9703 9829 9954 ..79 .204 347 640329 0465 0580 0705 0830 0956 1080 1205 1330 1454 348 1579 1704 1829 1953 2078 2203 2327 2462 2576 2701 349 2825 2960 3074 3199 3323 3447 3571 3696 3820 3944 8 LOGARITHMS N. 1 2 3 4 5 6 7 8 9 350 544068 4192 4316 4440 4664 4688 4812 4936 5060 5183 351 5307 5431 5555 5678 5805 5925 6049 6172 6296 6419 352 6543 6666 6789 6913 7036 7159 7282 7405 7529 7652 363 7775 7898 8021 8144 8267 8389 8612 8636 8758 8881 354 9003 9126 9249 9371 9494 9616 9739 9861 9984 .196 355 550228 0351 0473 0595 0717 0840 0962 1084 1206 1328 356 1450 1572 1694 1816 1938 2060 2181 2303 2425 2547 357 2668 2790 2911 3033 3155 8276 3393 3519 8640 3762 358 3883 4004 4126 4247 4368 4489 4610 4731 4852 4973 359 5094 6216 5346 5467 5578 5699 5820 5940 6061 6182 360 6303 6423 6544 6664 6786 6905 7026 7146 7267 7387 361 7507 7G27 7748 7868 7988 8108 8228 8349 8469 8589 362 8709 8829 8948 9068 9188 9308 9428 9548 9667 9787 863 9907 . .26 .146 .266 .886 .504 .624 .743 .863 .982 364 561101 1221 1340 1459 1678 1698 1817 1986 2055 2173 365 2293 2412 2531 2650 2769 2887 3006 3125 3244 3362 366 8481 3600 3718 3837 3956 4074 4192 4311 4429 4548 367 4666 4784 4903 5021 5139 5267 6376 5494 5612 5780 368 5848 5966 6084 6202 6320 6437 6555 6673 6791 6909 369 7026 7144 7262 7879 7497 7614 7732 7849 7967 8084 370 8202 8319 8436 8554 8671 8788 8905 9023 9140 9257 371 9374 9491 9608 9725 9882 9959 ..76 .198 .309 .426 372 570543 0660 0776 0898 1010 1126 1243 1359 1476 1692 373 1709 1825 1942 2058 2174 2291 2407 2523 2639 2755 374 2872 2988 3104 3220 3836 8452 3568 3684 8800 3915 375 4031 4147 4263 4379 4494 4610 4726 4841 4957 5072 376 5188 5303 6419 5634 5660 5766 5880 5996 6111 6226 377 6341 6457 6572 6687 6802 6917 7032 7147 7262 7377 378 7492 7607 7722 7836 7951 8066 8181 8296 8410 8625 379 8639 8764 8868 8983 9097 9212 9826 9441 9555 9669 880 9784 9898 ..12 .126 .241 .355 .469 .583 .697 .811 381 580926 1039 1163 1267 1381 1496 1608 1722 1836 1950 382 2063 2177 2291 2404 2618 2631 2746 2868 2972 8085 383 3199 3312 3426 8639 3652 3766 3879 3992 4105 4218 384 4331 4444 4667 4670 4783 4896 5009 5122 5235 5348 385 5461 5574 5686 5799 5912 6024 6137 6250 6862 6475 386 6587 6700 6812 6926 7037 7149 7262 7374 7486 7599 887 7711 7823 7935 8047 8160 8272 8384 8496 8608 8720 388 8832 8944 9056 9167 9279 9391 9503 9615 9726 9834 389 9950 ..61 .178 .284 .396 .507 .619 .730 .842 .953 390 591065 1176 1287 1399 1510 1621 1732 1843 1965 2066 391 2177 2288 2399 2510 2621 2732 2843 2964 8064 3175 392 3286 3397 8508 3618 3729 3840 3950 4061 4171 4282 393 4393 4603 4614 4724 4834 4945 5055 5165 5276 5386 394 5496 5606 5717 5827 6937 6047 6157 6267 6877 6487 395 6507 6707 6817 6927 7037 7146 7256 7866 7476 7586 396 7695 7805 7914 8024 8134 8243 8353 8462 8572 8681 397 8791 8900 9009 9119 9228 9337 9446 S556 9666 9774 398 9883 9992 .101 .210 .319 .428 .537 .646 .755 .864 399 600973 1082 1191 1299 1408 1617 1626 1734 1843 1951 OF NUMBERS. 9 N. 1 2 3 4 5 6 7 8 9 400 602060 2169 2277 2386 2494 2603 2711 2819 2928 3036 401 3144 3253 3361 3469 3573 3686 3794 3902 4010 4118 402 4226 4334 4442 4550 4658 4766 4874 4982 5089 5197 403 5305 5413 5521 5628 5736 5844 5951 6059 6166 6274 404 6381 6489 6596 6704 6811 6919 7026 7133 7241 7348 405 7455 7562 7669 7777 7884 7991 8098 8205 8312 8419 406 8526 8633 8740 8847 8954 9061 9167 9274 9381 9488 407 9594 9701 9808 9914 ..21 .128 .234 .341 .447 .554 408 610660 0767 0873 0979 1086 1192 1298 1405 1511 1617 409 1723 1829 1936 2042 2148 2254 2360 2466 2572 2678 410 2784 2890 2996 3102 3207 3313 3419 3525 3630 3736 411 3842 3947 4053 4159 4264 4370 4475 4581 4686 4792 412 4897 5003 5108 5213 5319 5424 5529 5634 5740 5846 413 5950 6055 6160 6265 6370 6476 6581 6686 6790 6895 414 7000 7105 7210 7315 7420 7525 7629 7754 7839 7948 415 8048 8153 8257 8362 8466 8571 8676 8780 8884 8989 416 9293 9198 9302 9406 9511 9615 9719 9824 9928 ..32 417 620136 0140 0344 0448 0552 0656 0760 0864 0068 1072 418 1176 1280 1384 1488 1592 1695 1799 1903 2007 2110 419 2214 2318 2421 2525 2628 2732 2835 2939 3042 3146 420 3249 3353 3456 3559 3663 3766 3869 3973 4076 4179 421 4282 4385 4488 4591 4695 4798 4901 5004 6107 5210 422 5312 5415 5518 5621 6724 5827 5929 6032 6135 6238 423 6340 6443 6546 6648 6751 6853 6956 7058 7161 7263 424 7366 7468 7571 7673 7775 7878 7980 8082 8185 8287 425 8389 8491 8593 8695 8797 8900 9002 9104 9206 9308 426 9410 9512 9613 9715 9817 9919 ..21 .123 .224 .326 427 630428 0530 0631 0733 0835 0936 1038 1139 1241 1342 428 1444 1545 1647 1748 1849 1951 2052 2153 2255 2356 429 2457 2559 2660 2761 2862 2963 3064 3165 3266 3367 430 3468 3569 3670 3771 3872 3973 4074 4175 4276 4376 431 4477 4578 4679 4779 4880 4981 5081 5182 5283 5383 432 5484 5684 6685 5785 5886 5986 6087 6187 6287 6388 433 6488 6588 6688 6789 6889 6989 7089 7189 7290 7390 434 7490 7590 7690 7790 7890 7990 8090 8190 8290 8389 435 8489 8589 8689 8789 8888 8988 9088 9188 9287 9387 436 9486 9586 9686 9785 9885 9984 ..84 .183 .283 .382 437 640481 0581 0680 0779 «879 0978 1077 1177 1276 1375 438 1474 1573 1672 1771 1871 1970 2069 2168 2267 2366 439 2465 2563 2662 2761 2860 2959 3058 3156 3255 3354 440 3453 3551 3650 3749 3847 3946 4044 4143 4242 4340 441 4439 4537 4636 4734 4832 4931 5029 5127 6226 5324 442 5422 5521 5619 5717 5815 5913 6011 6110 6208 6306 443 6404 6502 6600 6698 6796 6894 6992 7089 7187 7286 444 7383 7481 7579 7676 7774 7872 7969 8067 8165 8262 445 8360 8458 8555 8653 8750 8848 8945 9043 9140 9237 446 9335 9432 9530 9627 9724 9821 9919 ..16 .113 .210 447 650308 0405 0502 0599 0696 0793 0890 0987 1084 1181 448 1278 1375 1472 1569 1666 1762 1859 1956 2053 2160 449 2246 2343 2440 2530 2633 2730 2826 2923 3019 3116 ' 10 LOGARITHMS N. 1 2 3 4 6 6 7 8 9 450 653213 3309 3405 3502 3598 3695 3791 3888 3984 4080 461 4177 4273 4369 4466 4562 4658 4754 4850 4946 6042 453 5138 6235 5331 5427 5626 6619 6715 5810 5906 6002 453 6098 6194 6290 6386 6482 6577 6673 6769 6864 6960 454 7056 7162 7247 7343 7438 7634 7629 7725 7820 7916 456 8911 8107 8202 8298 8398 &488 8584 8679 8774 8870 456 8966 9060 9156 9250 9346 9441 9536 9631 9726 9821 457 9916 . .11 ,106 .201 .296 .391 .486 .581 .676 .771 458 660866 0960 1055 1150 1245 1339 1434 1629 1623 1718 469 1813 1907 2002 2096 2191 2286 2380 2475 2569 2663 460 2768 2852 2947 3041 3135 3230 3324 3418 3512 3607 461 3701 3796 3889 3983 4078 4172 4266 4360 4454 4648 462 4642 4736 48jO 4924 6018 5112 5206 5299 6393 6487 463 6581 6676 67d9 6862 5956 6050 6143 6237 6331 6424 464 6518 6612 6705 6799 6892 6986 7079 7173 7266 7360 465 7453 7546 7640 7733 7826 7920 8013 8106 8199 8293 466 8386 8479 8672 8665 8759 8852 8946 9038 9131 9324 467 9317 9410 9503 9596 9689 9782 9876 9967 ..60 .153 468 670241 0339 0431 0524 0617 0710 0802 0895 0988 1080 469 1173 1265 1358 1451 1643 1636 1728 1821 1913 2005 470. 2098 2190 2283 2375 2467 2560 2652 2744 2836 2929 471 3021 3113 3205 3297 3390 3482 3574 3666 3758 3850 472 3942 4034 4126 4218 4310 4402 4494 4686 4677 4769 473 4861 4953 5045 6137 5228 5320 5412 5503 5695 5687 474 5778 6870 6962 6053 6146 6236 6328 6419 6611 6602 475 6694 6785 6876 6968 7059 7151 7242 7333 7424 7616 476 7607 7698 7789 7881 7972 8063 8154 8245 8336 8427 477 8518 8609 8700 8791 8882 8972 9064 9155 9246 9337 478 9428 9519 9610 9700 9791 9882 9973 ..63 .164 .245 479 680336 0426 0517 0607 0698 0789 0879 0970 1060 1151 48a 1241 1332 1422 1513 1603 1693 1784 1874 1964 2056 481 2145 2235 2326 2416 2506 2696 2686 2777 2867 2957 482 3047 3137 3227 3317 3407 3497 3587 3677 3767 3857 483 3947 4037 4127 4217 4307 4396 4486 4576 4666 4766 484 4845 4936 5026 5114 5204 5294 6383 5473 6563 6652 485 6742 6831 5921 6010 6100 6189 6279 6368 6458 6647 486 6636 6726 6816 6904 6994 7083 7172 7261 7351 7440 487 7629 7618 7707 7796 7886 7976 8064 8153 8242 8331 488 8420 8509 8598 8687 8776 8865 8953 9042 9131 9220 489 9309 9398 9486 9576 9664 9753 9841 9930 ..19 .107 490 690196 0285 0373 0362 0550 0639 0728 0816 0905 0993 491 1081 1170 1258 1347 1435 1524 1612 1700 1789 1877 492 1966 2053 2142 2230 2318 2406 2494 2583 2671 2759 493 2847 2935 3023 3111 3199 3287 3375 3463 3551 3639 494 3727 3815 3903 3991 4078 4166 4254 4342 4430 4517 495 4605 4693 4781 4868 4956 6044 5131 5210 5307 5394 496 5482 6669 6657 6744 5832 6919 6007 6094 6182 6269 497 6356 5444 6531 6618 6706 6793 6880 6968 7056 7142 498 7229 7317 7404 7491 7678 7666 7752 7889 7926 8014 499 8101 8188 8275 8362 8449 8535 8622 8709 8796 b«83 OF NUMBERS. 11 N. I 2 3 4 5 6 7 8 9 500 693970 9057 9144 9231 9317 9404 9491 9578 9664 9751 501 9838 9924 ..11 ..98 .184 .271 .358 .444 .631 .617 502 700704 0790 0877 0963 1050 1136 1222 1309 1396 1482 503 1568 1664 1741 1827 1913 1999 2086 2172 2258 2344 504 2431 2617 2603 2689 2775 2861 2947 3033 3119 3206 505 3291 3377 3463 3549 8636 3721 3807 3896 3979 4065 508 4151 4236 4322 4408 4494 4579 4666 4751 4837 4922 507 5008 5094 5179 5266 6350 6436 6522 6607 5693 5778 508 5864 5949 6035 6120 6206 6291 6376 6462 6647 6632 509 6718 6803 6888 6974 7069 7144 7229 7315 7400 7485 610 7570 7655 7740 7826 7910 7996 8081 8166 8261 8336 511 8421 8506 8591 8676 8761 8846 8931 9015 9100 9186 512 9270 9356 9440 9524 9609 9694 9779 9863 9948 ..33 513 710117 0202 6287 0371 0466 0540 0625 0710 0794 0879 514 0963 1048 1132 1217 1301 1386 1470 1554 1639 1723 515' 1807 1892 1976 2660 2144 2229 2313 2397 2481 2566 516 2650 2734 2818 2902 2986 3070 3154 3238 3326 3407 517 3491 5576 3659 3742 3826 3910 8994 4078 4162 4246 518 4330 4414 4497 4581 4666 4749 4833 4916 5000 5084 519 5167 5251 6336 5418 5602 6586 5669 5763 5836 6920 520 6003 6087 6170 6254 6337 6421 6504 6588 6671 6754 521 6838 6921 7004 7688 7171 7254 7338 7421 7504 7587 522 7671 7764 7837 7920 8003 8086 8169 8253 8336 8419 523 8502 8585 8668 8761 8834 8917 9000 9083 9165 9248 524 9331 9414 9497 9580 9663 9745 9828 9911 9994 ..77 525 720159 0242 0326 0407 0490 6573 0655 0738 0821 0903 526 0986 1068 1151 1233 1316 1398 1481 1663 1646 1728 527 1811 1893 .975 2058 2140 2222 2305 2387 2469 2552 528 2634 3716 2798 2881 2963 3046 3127 3209 8291 3374 529 3456 3538 3620 3702 3784 3866 3948 4030 4112 4194 630 4276 4358 4440 4622 4604 4686 4767 4849 4931 5013 531 5096 5176 5268 5340 6422 5503 6585 5667 5748 5830 532 5912 6993 6075 6156 6238 6320 6401 6483 6564 6646 533 6727 6809 6890 6972 7653 7134 7216 7297 7379 7460 534 7641 7623 7704 7786 7866 7948 8029 8110 8191 8273 535 8354 8435 8516 8597 867S 8759 8841 8922 9003 9084 536 9166 9246 9327 9403 9489 9570 9651 9732 9813 9893 537 9974 ..65 .136 .217 .298 .378 .469 .440 .621 .702 538 730782 0863 0944 1024 1105 1186 1266 1347 1428 1508 539 1689 1669 1750 1830 1911 1991 2072 2152 2233 2313 540 2394 2474 2656 2636 2716 2796 2876 2966 3037 3117 541 3197 3278 3368 3438 3518 3598 3679 3769 3839 3919 542 3999 4079 4160 4240 4320 4400 4480 4560 4640 4720 543 4800 4880 4960 5040 5120 5200 5279 5359 5439 5619 544 6399 6679 5759 5838 5918 5998 6078 6157 6237 6317 545 6397 6476 6656 6636 6715 6796 6874 6954 7034 7113 546 7193 7272 7352 7431 7611 7590 7670 7749 7829 7908 547 7987 8067 8146 8226 8305 8384 8463 8543 8622 8701 548 8781 8860 8939 9018 9097 9177 9266 9335 9414 •9493 549 9672 9651 9731 9810 9889 9968 ..47 .126 .205 .284 12 LOGARITHMS N. 1 2 3 4 6 6 7 8 9 650 740363 0442 0521 0560 0678 0757 0836 0915 0994 1073 551 1152 1230 1309 1388 1467 1546 1624 1703 1782 1860 552 1939 2018 2096 2175 2254 2332 2411 2489 2568 2646 553 2726 28(M 2882 2961 3039 3118 3196 3276 3353 3431 554 3510 3568 3667 3745 3823 3902 8980 4058 4136 4215 555 4293 4371 4449 4528 4606 4684 4762 4840 4919 4997 556 5075 5163 5231 5309 5387 6465 5543 6621 6699 6777 657 5855 6933 6011 6089 6167 6245 6323 6401 6479 6656 558 6634 6712 6790 6868 6946 7023 7101 7179 7256 7334 559 7412 7489 7567 7646 7722 7800 7878 7956 8033 8110 560 8188 8266 8343 8421 8498 8676 8653 8731 8808 8885 561 8963 9040 9118 9196 9272 9360 9427 9504 9582 9659 562 9736 9814 9891 9968 ..45 .123 .200 .277 .354 .431 663 750508 0586 0663 0740 0817 0894 0971 1048 1126 1202 564 1279 1356 1433 1610 1587 1664 1741 1818 1895 1972 565 2048 2125 2202 2279 2356 2433 2609 2586 2663 2740 566 2816 2893 2970 3047 3123 3200 3277 3353 3430 3606 567 3582 3660 3736 3813 3889 3966 4042 4119 4196 4272 568 4348 4425 4501 4578 4654 4730 4807 4883 4960 5036 569 5112 5189 5266 6341 5417 6494 5670 6646 5722 6799 570 5875 6951 6027 6103 6180 6256 6332 6408 6484 6560 571 6636 6712 6788 6864 6940 7016 7092 7168 7244 7320 572 7396 7472 7648 7624 7700 7775 7851 7927 8003 8079 573 8155 8230 8306 8382 8458 8533 8609 8685 8761 8836 574 8912 8988 9068 9139 9214 9290 9366 9441 9517 9692 575 9668 9743 9819 9894 9970 ..45 .121 .196 .272 .347 576 760422 0498 0573 0649 0724 0799 0875 0060 1025 1101 &77 1176 1251 1326 1402 1477 1552 1627 1702 1778 1853 578 1938 2003 2078 2163 2228 2303 2378 2453 2529 2604 579 2679 2754 2829 2904 2978 3053 3128 2203 3278 3353 580 34S8 3503 3578 3653 3727 3802 3877 3962 4027 4101 581 4176 4261 4326 4400 4476 4650 4624 4699 4774 4848 582 4923 4998 5072 5147 5221 5296 6370 5445 6520 6594 583 6669 5743 5818 6892 6956 6041 6115 6190 6264 6338 584 6413 6487 6562 6636 6710 6785 6859 6933 7007 7082 585 7156 7230 7304 7379 7453 7527 7601 7675 7749 7823 586 T.89S 7972 80i6 8120 8194 8268 8342 8416 8490 8564 687 8638 8712 8786 8860 8934 9008 9082 9166 9230 9303 588 9377 9461 9525 9699 9673 9746 9820 9894 9968 ..42 689 770115 0189 0263 0336 0410 0484 0557 0631 0705 0778 590 0852 0926 0999 1073 1146 1220 1293 1367 1440 1514 591 1687 1661 1734 1808 1881 1956 2028 2102 2175 2248 592 2322 2395 2468 3542 2615 2688 2762 2836 2908 2981 593 3055 3128 3201 3274 3348 3421 3494 3567 3640 3713 594 3786 3860 3933 4006 40?9 4152 4225 4298 4371 4444 595 4617 4590 4663 4736 4809 4882 4966 5028 5100) 6173 596 5246 5319 5392 5465 5538 6610 5683 5756 5829 5902 597 5974 6047 6120 6193 6265 6338 6411 6483 6556 6629 698 6701 6774 6846 6919 0992 7064 7137 7209 7282 7354 699 7427 7409 7572 7644 7717 7789 7862 7934 8006 8079 ^ OF NUMBERS. 13 N. 1 2 3 4 5 6 7 8 9 600 778151 8224 8296 8368 8441 8513 8585 8658 8730 8802 601 8874 8947 9019 9091 9163 9236 9308 9380 9452 9524 602 9596 6669 9741 9813 9885 9967 ..29 .101 .173 .245 603 780317 0389 0461 0533 0505 0677 0749 0821 0893 0965 604 1037 1109 1181 1253 1324 1396 1468 1540 1612 1684 605 1755 1827 1899 1971 2042 2114 2186 2258 2329 2401 606 2473 2644 2616 2688 2759 2831 2902 2974 3046 3117 607 3189 3260 3332 3403 8476 3646 3618 8689 3761 3832 608 3904 3975 4046 4118 4189 4261 4332 4403 4475 4546 609 4617 4689 4760 4831 4902 4974 5045 5116 5187 5259 610 5330 5401 5472 5543 6615 5686 5757 5828 5899 5970 611 6041 6112 6183 6254 6325 6396 6467 6538 6609 6680 612 6761 6822 6893 6964 7035 7106 7177 7248 7319 7390 613 7460 7531 7602 7673 7744 7815 7885 7956 8027 8098 614 8168 8239 8310 8381 8451 8522 8593 8663 8734 8804 615 8875 8946 9016 9087 9157 9228 9299 9369 9440 9510 616 9581 9651 9722 9792 9863 9933 ...4 ..74 .144 .216 617 790285 0356 0426 0496 0567 0637 0707 0778 0848 0918 618 0988 1059 1129 1199 1269 1340 1410 1480 1550 1620 619 1691 1761 1831 1901 1971 2041 2111 2181 2252 2322 620 2392 2462 2532 2602 2672 2742 2812 2882 2952 8022 621 8092 3162 3231 3301 8371 3441 3511 3581 8661 3721 622 3790 3860 3930 4000 4070 4139 4209 4279 4849 4418 623 4488 4558 4627 4697 4767 4836 4906 4976 5045 5115 624 5186 5254 5324 5393 6463 5532 5602 5672 5741 5811 626 5880 5949 6019 6088 6158 6227 6297 6366 6436 6505 626 6574 6644 6713 6782 6852 6921 6990 7060 7129 7198 627 7268 7337 7406 7476 7545 7614 7683 7752 7821 7890 628 7960 8029 8098 8167 8236 8305 8874 8443 8513 8582 629 8661 8720 8789 8858 8927 8996 9066 6134 9203 9272 630 9341 9409 9478 9547 9610 9685 9754 9823 9892 9961 631 800026 0098 0167 0236 0305 0373 0442 0511 0580 0848 632 0717 0786 0854 0923 0992 1061 1129 1198 1266 1335 633 1404 1472 1541 1609 1678 1747 1816 1884 1952 2021 634 2089 2158 2226 2295 2363 2432 2500 2668 2637 2705 635 2774 2842 2910 2979 3047 3116 3184 8252 3321 3389 636 3457 3526 3594 3662 3730 3798 8867 3935 4003 4071 637 4139 4208 4276 4354 4412 4480 4548 4616 4685 4753 638 4821 4889 4957 5025 6093 6161 5229 5297 5365 5433 1 639 5601 5669 5637 6706 6773 6841 5908 5976 6044 6112 j 640 6180 6248 6316 6384 6451 6519 6587 6655 6723 6790 ! 641 6858 6926 6994 7061 7129 7157 7264 7332 7400 7467 ! 642 7635 7603 7670 7738 7808 7873 7941 8008 8076 8143 1 643 8211 8279 8346 8414 8481 8549 8616 8684 8751 8818 i 644 8886 8963 9021 9088 9166 9223 9290 9358 9426 9492 645 9560 9627 9694 9762 9829 9896 9964 ..31 ..98 .165 646 810238 0300 0367 0434 0501 0596 0636 0703 0770 0837 647 0904 0971 1039 1106 1173 1240 1307 1374 1441 1508 648 1676 1642 1709 1776 1843 1910 1977 2044 2111 2178 649 2246 2312 2379 2446 2512 2o79 2648 2713 2780 2847 14 LOGARITHMS N. I 2 3 4 5 6 7 8 9 6B0 812913 2980 3047 3114 3181 3247 3314 3381 3448 3514 651 3581 3648 3714 3781 3848 3914 3981 4048 4114 4181 652 4248 4314 4381 4447 4514 4681 4647 4714 4780 4847 663 4913 4980 5046 6113 5179 5246 5312 5378 5445 5511 654 5578 5644 5711 5777 5843 5910 5976 6042 6109 6175 665 6241 6308 6374 6440 6506 6573 6639 6705 6771 6838 656 6904 6970 7036 7102 7169 7233 7301 7367 7433 7499 657 7565 7631 7698 7764 7830 7896 7962 8028 8094 8160 658 8226 8292 8368 8424 8490 8556 8622 8688 8764 8820 659 8885 8951 9017 9083 9149 9215 9281 9346 9412 9478 660 9544 9610 9676 9741 9807 9873 9939 ...4 ..70 .136 661 820201 0267 0333 0399 0464 0530 0695 0661 0727 0792 662 0858 0924 0989 1055 1120 1186 1251 1317 1382 1448 663 1514 1579 1646 1710 1775 1841 1906 1972 2037 2103 664 2168 2233 2299 2364 2430 2495 2560 2626 2691 2766 666 2822 2887 2962 3018 3083 3148 3213 3279 3344 3409 666 3474 3539 3605 3670 3736 3800 3865 3930 3996 4061 667 4126 4191 4266 4321 4386 4451 4516 4681 4646 4711 668 4776 4841 4906 4971 6036 5101 6166 5231 5296 5361 669 5426 5491 6656 5621 5686 5751 5815 5880 5945 6010 670 6075 6140 6204 6269 6334 6399 6464 6528 6593 6658 671 6723 6787 6862 6917 6981 7046 7111 7175 7240 7305 672 7369 7434 7499 7663 7628 7692 7767 7821 7886 7961 673 8015 8080 8144 8209 8273 8338 8402 8467 8531 8595 674 8660 8724 8789 8853 8918 8982 9046 9111 9175 9239 675 9304 9368 9432 9497 9661 9625 9690 9764 9818 9882 676 9947 ..11 ..75 .139 .204 .268 .332 .396 .460 .525 677 830589 0653 0717 0781 0845 0909 0973 1037 1102 1166 678 1230 1294 1358 1422 1486 1560 1614 1678 1742 1806 679 1870 1934 1998 2062 2126 2189 2253 2317 2381 2445 680 2509 2673 2637 2700 2764 2828 2892 2956 3020 3083 681 3147 3211 3276 3338 3402 3466 3530 3593 3657 3721 682 3784 3848 3912 3976 4039 4103 4166 4230 4294 4367 683 4421 4484 4548 4611 4675 4739 4802 4866 4929 4993 684 5056 5120 5183 6247 5310 5373 5437 5500 6564 5627 685 5691 6754 5817 5881 5944 6007 6071 6134 6197 6261 686 6324 6387 6461 6614 6577 6641 6704 6767 6830 6894 687 6957 7020 7083 7146 7210 7273 7336 7399 7462 7626 688 7588 7652 1 7715 7778 7841 7904 7967 8030 8093 8166 689 8219 8282 8345 8408 8471 8534 8597 8660 8723 8786 690 8849 8912 8975 9038 9109 9164 9227 9289 9352 9416 691 9478 9641 ! 9604 9667 9729 9792 9855 9918 9981 ..43 692 840106 0169 j 0232 0294 0357 0420 0482 0545 0608 0671 693 0733 0796 0859 0921 0984 1046 1109 1172 1234 1297 694 1359 1422 1 1485 1547 1610 1672 1735 1797 1860 1922 695 1985 2047 2110 2172 2235 2297 2360 2422 2484 2547 696 2609 2672 2734 2796 2869 2921 2983 3046 3108 3170 697 3233 3295 3357 3420 3482 3544 3606 3669 3731 3793 698 3855 3918 3980 4042 4104 4166 4229 4291 4363 4416 699 4477 4539 4601 4664 4726 4788 4850 4912 4974 6036 OF NUMBERS. 15 N. 1 3 3 4 5 6 7 8 9 j 700 845098 5160 6222 5284 5346 5408 6470 6532 5694 6666 701 6718 5780 6842 6904 5966 6028 6090 6151 6213 6'276 702 6337 6399 6461 6623 6585 6646 6708 6770 6832 6894 703 6955 7017 7079 7141 7202 7264 7326 7388 7449 7511 704 7673 7634 7676 7768 7819 7831 7943 8004 8066 8128 1 705 8189 8251 8312 8374 8435 8497 8559 8620 8682 8743 1 706 8805 8866 8928 8989 9051 9112 9174 9235 9297 9358 707 9419 9481 9542 9604 9665 9726 9788 9849 9911 9972 703 850033 0095 0156 0217 0279 0340 0401 0462 0524 0585 703 0646 0707 0769 0830 0891 0952 1014 1076 1136 1197 710 1258 1320 1381 1442 1503 1564 1625 1686 1747 1809 f 711 1870 1931 1992 2053 2114 2175 2236 2297 2358 2419 712 2480 2541 2602 2668 2734 2786 2846 2907 2968 3029 713 3090 3150 3211 3272 3333 3394 3455 3516 3577 3637 714 3698 3759 3820 3881 3941 4002 4063 4124 4185 4245 715 4306 4367 4428 4488 4549 4610 4670 4731 4792 4852 716 4913 4974 6034 5095 5156 6216 5277 6337 6898 6469 717 5519 5680 6640 6701 5761 5822 63S2 5943 6003 6064 718 6124 6185 6245 6306 6366 6427 6487 6548 6608 6668 719 6729 6789 6860 6910 6970 7031 T091 7152 7212 7272 720 7332 7393 7453 7513 7574 7634 7694 7755 7815 7876 721 7936 7995 8p56 8116 8176 8236 8297 8357 8417 8477 722 8537 8597 8657 8718 8778 8838 8898 8958 9018 9078 723 9138 9198 9258 9318 9379 9439 9499 95.39 9619 9679 724 9739 9799 9859 9918 9978 ..38 ..98 .168 .218 .278 725 860338 0398 0458 0518 0578 0637 0697 0757 0817 0877 72(5 0937 0995 1056 1116 1176 1236 1296 1356 1415 1476 727 1634 1594 1654 1714 1773 1833 1893 1952 2012 2072 728 2131 2191 2251 2310 2370 2430 2489 2549 2608 2668 729 2728 2787 2847 2906 2966 3026 3085 3144 3204 3263 730 3323 3382 3442 3501 3561 3620 3680 3739 3799 3868 731 3917 3977 4036 4096 4155 4214 4274 4333 4392 4452 732 4511 4570 4630 4689 4148 4808 4867 4926 4985 6046 733 5104 6163 5222 6282 5341 5400 5469 5519 5578 6687 734 5696 5756 6814 6874 5933 5992 6051 6110 6169 6228 735 6287 6346 6405 6465 6624 6583 6642 6701 6760 6819 736 6878 6937 6996 7055 7114 7173 7232 7291 7350 7409 737 7467 7526 7586 7644 7703 7762 7821 7880 7939 7998 738 8056 8115 8174 8233 8292 8350 8409 8468 8527 858<> 739 8644 8703 8762 8821 8879 8988 8997 9056 9114 9173 740 9232 9290 9349 9408 9466 9525 9684 9642 9701 9760 741 9818 9877 9936 9994 ..53 .111 .170 .228 .287 .345 742 870404 0462 0521 0679 0638 0696 0765 0813 0872 0930 743 0989 1047 1106 1164 1223 1281 1339 1398 1466 1616 744 1673 1631 1690 1748 1803 1865 1928 1981 2040 2098 746 2156 2215 2273 2331 2389 2448 2506 2564 2622 2681 746 2739 2797 2865 2913 2972 3030 8088 3146 3204 3262 747 3321 3379 8437 8495 3553 3611 3669 3727 3785 3844 748 8902 8960 4018 4076 4134 4192 4260 4308 4360 4424 749 4482 4540 4698 4656 4714 4772 4830 4888 4945 5003 24 16 LOGARITHMS N. 1 2 3 4 6 6 7 8 9 750 875061 5119 6177 6235 6293 6351 6409 5466 5524 5582 751 6640 5698 6756 5813 5871 6929 5987 6045 6102 6160 762 6218 6276 6333 6391 6449 6507 6564 6622 6680 6737 763 6795 6853 6910 6968 7026 7083 7141 7199 7256 7314 764 7371 7429 7487 7644 7602 7659 7717 7774 7832 7889 756 7947 8004 8062 8119 8177 8234 8292 8349 8407 8464 766 8522 8579 8637 8694 8752 8809 886(5 8924 8931 9039 757 9096 9153 9211 9268 9325 9383 9440 9497 9555 9612 758 9669 9726 9784 9841 9898 9956 ..13 ..70 .127 .185 769 880242 0299 0356 0413 0471 0528 0580 0642 0699 0756 760 0814 0871 0928 0985 1042 1099 1156 1213 1271 1328 761 1385 14^42 1499 1556 1613 1670 1727 1784 1841 1898 762 1955 2012 2069 2126 2183 2240 2297 2354 2411 2468 763 2525 2581 2638 2695 2752 2809 2866 2923 2980 3037 764 3093 3150 3207 8264 3321 3377 3434 3491 3548 3606 765 3661 3718 3775 3832 3888 3945 4002 4059 4115 4172 766 4229 4285 4342 4399 4455 4512 4569 4625 4682 4739 767 4795 4852 4909 4965 5022 5078 6135 5192 5248 6305 768 5361 5418 5474 5531 5587 5644 5700 5757 5813 5870 769 6926 5983 6039 6096 6152 6209 6266 6321 6378 6434 770 6491 6547 6604 6660 6716 6773 6829 6886 6942 6998 771 7054 7111 7167 7233 7280 7336 7392 7449 7506 7561 772 7617 7674 7730 7786 7842 7898 7955 8011 8067 8123 773 8179 8236 8292 8348 8404 8460 8616 8573 8629 8655 774 8741 8797 8863 ,8909 8965 9021 9077 9134 9190 9246 776 9302 9358 9414 9470 9526 9582 9638 9694 9760 9806 776 9862 9918 0974 ..30 ..86 .141 .197 .253 .309 .866 777 890421 0477 0533 0589 0646 0700 0756 0812 0868 0924 778 0980 1035 1091 1147 1203 1259 1314 1370 1426 1482 779 1537 1593 1649 1705 1760 1816 1872 1928 1983 2039 780 2095 2150 2206 2262 2317 2373 2429 2484 2540 2695 781 2651 2707 2762 2818 2873 2929 2985 8040 3096 3151 783 3207 3262 3318 3373 3429 3484 3540 3595 3651 3706 783 3762 8817 3873 3928 3984 4039 4094 4150 4205 4261 784 4316 4371 4427 4482 4538 4593 4648 4704 4769 4814 785 4870 4925 4980 5036 6091 5146 5201 6257 6312 5367 788 6423 6478 5533 5588 5644 6699 5754 5809 6864 5920 787 5975 6030 6085 6140 6195 6251 6306 6361 6416 6471 788 6526 6581 6636 6692 6747 6802 6857 6912 6967 7022 789 7077 7132 7187 7242 7297 7362 7407 7462 7617 7672 790 7627 7683 7737 7792 7847 7902 7957 8012 8067 8122 791 8176 8231 8286 8341 8396 8451 8606 8561 8615 8670 792 8725 8780 8835 8890 8944 8999 9054 9109 9164 9218 793 9273 9328 9383 9437 9492 9647 9602 9656 9711 9766 794 9821 9875 9930 9985 .,39 ..94 .149 .203 .258 .312 795 900367 0422 0476 0531 0586 0640 0695 0749 0804 0859 796 0913 0968 1022 1077 1131 1186 1240 1295 1349 1404 797 1458 1513 1567 1622 1676 1736 1786 1840 I8y4 1948 798 2003 2057 2112 2166 2221 2275 2329 2384 2438 2492 799 2547 2601 2655 2710 2764 2818 2873 2927 2981 3036 OF NUMBERS 17 N. 903090 1 2 3 4 6 6 7 8 9 800 3144 3199 3253 3307 3361 3416 3470 3524 3578 801 3633 3687 3741 3795 3849 3904 3958 4012 4066 4120 802 4174 4229 4283 4337 4391 4445 4499 4553 4607 4661 803 4716 4770 4824 4878 4932 4986 5040 5094 6148 5202 804 5358 5310 6364 5418 5472 5526 5580 6634 5688 5742 805 5796 5860 5904 5958 6012 6066 6119 6173 6227 6281 806 6335 6389 6443 6497 6551 6604 6658 6712 6766 6820 807 6874 6927 6981 7035 7089 7143 7196 7250 7304 7358 808 7411 7465 7519 7573 7626 7680 7734 7787 7841 7895 809 7949 8002 8056 8110 8163 8217 8270 8324 8378 8431 810 8485 8539 8592 8646 8699 8753 8807 8860 8914 8967 811 9021 9074 9128 9181 9235 9289 9342 9396 9449 9503 812 9656 9610 9663 9716 9770 9823 9877 9930 9984 ..37 813 910091 0144 0197 0251 0304 0358 0411 0464 0518 0671 814 0624 0378 0731 0784 0838 0891 0944 0998 1051 1104 815 1158 1211 1264 1317 1371 1424 1477 1530 1584 1637 816 1690 1743 1797 1850 1903 1956 2009 2063 2115 2169 817 2222 2275 2323 2381 2435 2488 2541 2594 2645 2700 818 2753 2808 2859 2913 2966 3019 3072 3125 3178 3231 819 3284 3337 3390 3443 3496 3549 3602 3655 3708 3761 820 3814 3867 3920 3973 4026 4079 4132 4184 4237 4290 821 4343 4398 4449 4502 4555 4608 4660 4713 4766 4819 822 4872 4925 4977 5030 5083 5136 5189 5241 5694 5347 823 5400 5453 5505 5658 5611 5664 5716 5769 5822 5876 824 5927 5980 6033 6085 6138 6191 6243 6296 6349 6401 825 6454 6507 6559 6612 6664 6717 6770 6822 6875 6927 826 6980 7033 7085 7138 7190 7243 7296 7348 7400 7463 827 7506 7568 7611 7663 7716 7768 7820 7873 7925 7978 828 8030 8083 8185 8188 8240 8293 8345 8397 8450 8502 829 8655 8607 8859 8712 8764 8816 8869 8921 8973 9026 830 9078 9130 9183 9235 9287 9340 9392 9444 9496 9549 831 9601 9663 9706 9768 9810 9862 9914 9967 ..19 ..71 832 920123 0176 0228 0280 0332 0384 0436 0489 0541 0593 833 0845 0697 0749 0801 0853 0908 0958 1010 1062 1114 834 1166 1218 1270 1322 1374 1426 1478 1530 1582 1634 835 1686 1738 1790 1842 1894 1946 1998 2050 2102 2164 836 2206 2258 2310 2362 2414 2466 2518 2570 2622 2674 837 2725 2777 2829 2881 2933 2985 3037 3089 3140 3192 838 3244 3296 3348 3399 3451 3503 3656 3607 3658 3710 889 3762 3814 3865 3917 3969 4021 4072 4124 4147 4228 840 4279 4331 4383 4434 4486 4538 4589 4641 4693 4744 841 4796 4848 4899 4951 5003 5054 5108 5157 5209 5261 842 5312 5364 5415 5467 5518 5570 5621 5673 5725 5776 843 5828 5874 5931 5982 6034 6085 6137 6188 6240 6291 844 6342 6394 6445 6497 6548 6600 6651 6702 6754 6805 845 6857 6908 6959 7011 7062 7114 7165 7216 7268 7319 846 7370 7422 7473 7524 7576 7627 7678 7730 7783 7832 847 7883 7935 7986 8037 8088 8140 8191 8242 8293 8345 848 8396 8447 8498 8549 8601 8652 8703 8754 8805 8857 849 8908 8959 9010 9081 9112 9163 9216 9266 9317 9368 J 18 I^OGARITHMS N. 850 923419 I 1 2 3 4 5 6 7 8 9 . 9473 9521 9572 9623 9674 9725 9776 9827 9879 851 9930 9981 ..32 ..83 .W4 .185 .236 .287 .338 .389 86fi 98<>i40 0491 0542 0592 0643 0694 0746 0796 0847 0898 853 • 0949 1000 1051 1102 1153 1204 1254 1305 1366 1407 864 1458 1509 1660 1610 1661 1712 1763 1814 1865 1915 855 1966 2017 2068 2118 2169 2220 2271 2322 2372 2423 856 2474 2624 2676 2626 2677 2727 2778 2829 2879 2930 857 2981 3061 3082 3133 3183 3234 3285 3335 3386 3437 858 •3487 3538 3589 3639 3690 3740 3791 3841 3892 3943 869 3993 4044 4094 4145 4195 4246 4269 4347 4397 4448 880 4498* 4549 4699 4650 4700 4751 4801 4852 4902 4950 861 5003 5054 5104 6154 6205 6265 5306 6856 6406 6457 862 6507 5558 6608 6658 6709 5759 6809 5860 6910 5960 863 6011 6061 6111 6162 6212 6262 6313 6363 6413 6463 864 6514 6564 6614 6665 6715 6766 6815 6865 6916 6966 866 7016 7066 7117 71-67 7217 7267 7317 7367 7418 74G8 866 7518 7568 76lg 7668 7718 7769 7819 7869 7919 7969 867 8019 8069 8119 8169 8:^19 8269 8320 8370 8420 8470 868 8620 8570 8620 8670 8720 8770 8820 8870 8919 8970 869 9020 9070 9120 9170 9220 9270 9320 9369 9419 9469 870 9519 9569 9616 9669 9719 9769 9819 9869 9918 9968 871 940018 0068 0118 0168 0218 0267 0317 0367 0417 0467 872 0516 0566 i 0616 0866 0/16 0765 0815 0865 0916 0964 873 1014 1064 1114 1163 1213 1263 1313 1362 1412 1462 874 1511 1561 1611 1660 1710 1760 1809 1859 1909 1968 875 2008 2068 2107 2157 2207 2256 2306 ,2365 2405 2465 876 2504 2554 2603 2653 2702 2762 2801 2851 2901 2950 i 877 3000 3049 3099 3148 3198 3247 3297 3346 3396 3445 1 878 3495 3644 3693 3643 3692 3742 3791 3841 3890 3939 i 879 3989 4038 4088 4137 4186 4236 4286 4336 4384 4433 i 1 880 4483 4532 4581 4631 4680 4729 4779 4828 4877 4927 881 4976 6026 5074 5124 5173 6222 5272 6321 5370 5419 882 5469 6518 5667 6616 5666 5715 6764 6813 6862 6912 883 5961 6010 6059 6108 6157 6207 6256 6305 6364 6403 884 6452 6501 6551 6600 6649 6698 6747 6796 6845 6894 885 6943 6992 7041 7090 7140 7189 7238 7287 7336 7385 886 7434 7488 7532 7681 7630 7679 7728 7777 7826 7876 887 7924 7973 8022 8070 8119 8168 8217 8266 8315 8365 888 8413 8462 8511 8560 8609 8657 8706 8765 8804 8863 889 8902 8951 8999 9048 9097 9146 9195 9244 9292 9341 899 9390 9439 9488 9636 9586 9634 9683 9731 W80 9829 891 9878 9926 9975 ..24 ..73 .121 .170 .219 .267 .316 892 950865 0414 0462 0611 0560 0608 0667 0706 0754 0803 893 0851 0900 0949 0997 1046 1095 1143 1192 1240 1289 894 1338 1386 1436 1483 1632 1580 1629 1677 1726 1776 895 1823 1872 1920 1969 2017 2066 2114 2163 2211 2260 896 2308 2366 2405 2463 2502 2550 2699 2647 6696 2744 897 2792 2841 2889 2938 2986 3034 3083 3131 3180 3228 898 3276 3325 3373 3421 3470 3518 3566 3616 3663 3711 i 899 3760 3808 8856 3905 3963 . 4001 4019 40JS 4146 4194 OF NUMBERS. ]9 N. 1 3 3 4 5 6 7 8 ' 9 900 954243 4291 4339 4387 4435 4484 4532 4580 4628 4677 901 4726 4773 4821 4869 4918 4966 6014 5062 MTO 5158 902 6207 5256 5303 5351 6399 5447 5496 5643 5592 5640 903 6688 5736 5784, 5832 5880 5928 6976 6024 6072 6120 904 6168 6216 6265 6313 6361 6409 6457 6565 6553 6601 905 6649 6697 6745 6793 6840 6888 6936 6984 7032 7080 , 908 7128 7176 7224 7272 7320 7368 7416 7464 7512 7559 907 7607 7655 7703 7751 7799 7847 7894 7942 7990 8088 908 8086 8134 8181 8229 8277 8326 8373 8421 .8468 8516 909 8564 8612^ 8659 8707 8755 8803 8860 8898 8946 8994 910 9041 9089 9137 9185 9232 9280 9328 9375 9423 9471 911 9518 9566 9814 9661 9709 9757 9804 9852 9900 9947 . . 912 9995 ..42 ..90 .138 .185 .233 .280 .328 .376 .423 913 960471 0518 0566 0613 0661 0709 0766 0804 0861 0899 914 0946 0994 1041 1089 1136 1184 1231 • 1279 1326 1374 915 1421 1469 1616 1563 1611 1658 1706 1753 1801 1848 916 1895 1943 1990 2038 2085 2132 2180 2227 2275 2322 917 2369 2417 2464 2511 2559 2608 2653 2701 2748 2795 918 2848 2890 2937 2985 3032 3079 3126 3174 3221 3268 919 3316 3363 3410 3457 3604 3552 3699 3646 3693 3741 : 920 3788 3835 3882' 3929 3977 4024 4071 4118 4165 4212 ■ 921 4260 4307 4354 4401 4448 4495 4642 4590 4637 4684 922 4731 4778 4825 4872 4919 4966 5013 5061 5108 5155 923 5202 5249 5296 5343 6390 5437 5484 5531 5578 5625 924 5672 5719 5766 6813 5860 5907 5954 6001 6048 6095 925 6142 6189 6236 6283 6329 6376 6423 6470 6617 6564 926 6611 6658 6705 6762 6799 6845 6892 6939 6986 7033 927 7080 7127 7173 7220 7267 7314 7361 7408 7454 7501 928 7548 7595 7642 7688 7735 7782 7829 7875 7922 7969 929 8016 8632 8109 8156 8263 8249 8296 8343 8390 8436 930 8483 8530 8576 8623 8670 8716 8763 8810 8856 8903 931 8950 8996 9043 9090 .9136 9183 9229 9276 9323 9369 932 9416 9463 9509 9556 9602 9649 9695 9742 9789 9835 933 9882 9928 997^ ..21 ..68 .114 .161 .207 .254 .300 934 970347 0393 0440 0486 6533 0579 6626 0672 0719 0765 935 0812 0858 0904 0951 0997 1044 1090 1137 1183 1229 936 1276 1322 1369 1415 1461 1508 1564 1601 1647 1693 937 1740 1786 1832 1879 1925 1971 2018 2064 2110 2157 938 2203 2249 229S 2342 2388 2434 2481 2627 2573 2619 939 2666 2712 2758 2804 2851 2897 2943 2989 3035 3082 940 3128 3174 3220 3266 3313 3359 3405 3451 3497 8543 941 8590 3636 3682 3728 3774 3820 8866 3913 3959 4005 942 4051 4097 4143 4189 4235 4281 4327 4374 4420 4466 943 4512 4558 4604 4650 4696 4742 4788 4834 4880 4926 944 4972 5018 5064 5110 5156 5202 5248 5294 5340 5386 945 5432 5478 5524 ,5570 5616 5662 5707 5753 5799 5845 946 5891 5937 5983 6029 6075 6121 6167 6212 6258 6304 947 6350 6396 6442 6488 6533 6679 6925 6671 6717 6763 948 6803 6854 6900 6946 6992 7037 7083 7129 7175 7220 949 7266 7312 7358 7403 7449 7496 7541 7586 7632 7678 ' 2a 1 LOGARITHMS N. 1 S 3 4 5 6 7 8 9 950 977724 7769 7815 7861 7906 7952 7998 8043 8089 8135 951 8181 8226 8272 8317 8363 8409 8454 8500 854G 8591 96-2 8637 8683 8728 8774 8819 8866 8911 8956 9002 9047 953 9093 9138 9184 9230 9275 9321 9366 9412 9457 9503 964 9548 9594 9639 9685 9730 9776 9821 9867 9912 9968 955 980003 0049 0094 0140 0185 0231 0276 0322 0367 0412 956 0458 0503 0549 0594 0640 0686 0730 0776 0821 0867 957 0912 0957 1003 1048 1093 1139 1184 1229 1275 1320 958 1366 1411 1456 1501 1547 1592 1637 1683 1728 1773 959 1819 1864 1909 1954 2000 2045 2090 2135 2181 2226 9G0 2271 2316 2362 2407 2462 2497 2543 2588 2633 2678 961 2723 2769 2814 2859 2904 2949 2994 3(H0 3085 3130 962 3176 3220 3265 3310 3356 3401 3446 3491 3536 3581 963 3626 3671 3716 3762 3807 3852 3897 3942 3987 4032 964 4077 4122' 4167 4212 4257 4302 4347 4392 4437 4482 965 4527 4572 4617 4662 4707 4752 4797 4842 4887 4932 I 966 4977 5GG2 5067 6112 6157 5202 5247 6292 6337 5382 967 B426 5471 5516 6561 5606 5651 6699 5741 6786 5830 968 5876 5920 6965 6010 6056 6100 6144 6189 6234 6279 969 6324 6369 6413 6458 6503 6548 6593 6637 6682 6727 970 6773 6817 6861 6906 6951 6996 7040 7085 7130 7175 971 7219 7264 7300 7353 7398 7443 7488 7532 7577 7622 972 7666 7711 7756 7800 7845 7890 7934 7979 8024 8068 973 8U3 8157 820Q 8247 8291 8336 8381 8425 8470 8514 974 8559 8604 8648 8693 8737 8782 8826 8871 8916 8960 975 90a5 9049 9093 9138 9183 9227 9272 9316 9361 9405 &76 9450 9494 9539 9583 9628 9672 9717 9761 9806 9850 977 9895 9939 9983 ..28 ..72 .117 .161 .206 .250 .294 1 978 990339 0383 0428 0472 0516 0561 0605 0650 0694 0738 1 979 0783 0827 0871 0916 0960 1004 1049 1093 1137 1182 I 980 1226 1270 1315 1359 1403 1448 1492 1536 1580 1625 j 981 1669 1713 1758 1802 1846 1890 1935 1979 2023 3067 1 982 2111 2156 2200 2244 2288 2333 2377 2421 2465 2509 1 983 2554 2698 2642 : 2686 2730 2774 2819 2863 2907 2951 i 984 2995 3039 3083 3127 3172 3216 3260 3304 3348 3392 1 j 985 3436 3480 3524 3568 3613 3657 3701 3745 3789 3833 j 986 3877 3921 3965 4009 4053 4097 4141 4185 4229 4273 i 987 4317 4361 4405 4449 4493 4537 4581 4625 4669 4713 988 4757 4801 4845 4886 4933 ,4977 5021 6065 5108 5152 989 6196 6240 6284 5328 5372 ;6416 5460 6504 6547 6691 990 6635 6679 5723 5767 6811 5854 6898 5942 6986 6030 991 6074 6117 6161 6205 6249 ;6293 6337 6380 6424 6468 992 6612 6655 6599 6643 6687 6731 6774 6818 6862 6906 993 6949 6993 7037 7080 7124 :7168 7212 7255 7299 7343 994 7386 7430 7474 7517 7661 7605 7648 7692 7736 7779 1 1 995 7823 7867 7910 7954 7998 8041 8085 8129 8172 8216 996 8259 8303 8347 8390 8434 !8477 8521 8564 8608 8652 997 8695 8739 8792 8826 8869 8913 8956 9000 9043 9087 998 9131 9174 9218 9261 9305 9348 9392 9435 9479 9622 999 9665 9609 9662 9696 9739 9,783 9826 9870 9913 i 9957 ♦if TABLE II. Log. Sines Hud Tangents, (0°) Natural Sines. *1 T" Sine. D.IO" (Jos'iiie. in(F T«ng. ■ D.IO" Coiang. Infinite. N.8ine. N. COS. 0.000000 10.000000 0.000000 00000 lOOOOO 60 1 6.463726 000000 6.463726 13.536274 00029 100000 59 2 764756 000000 764756 235244 00058 lOOOOU 58 3 940847 000000 940847 059153 00037 100000 57 4 7.085786 000000 7.065786 12.934214 00116 100000 56 5 162696 000000 162696 837804 00145 100000 56 6 241877 9.999999 241878 758122 00175 100000 54 7 308824 999999 308825 691175 00204 100000 63 8 366816 999999 366817 633183 00233 I 00000 62 9 417968 999999 417970 582030 00262 100000 51 10 463725 999998 463727 536273 00291 100000 50 11 7.505118 9.999998 7.505120 12.494880 00320 99999 49 12 54290S 999997 542909 457091 00349 99999 48 13 677668 999997 577672 422328 00378 99999 47 14 609853 999996 609867 390143 00407 99999 46 15 639816 999996 639820 360180 00436 99999 45 16 667845 999996 667849 332151 00465 99999 44 17 694173 999995 694179 305821 00496 99999 43 18 718997 999994 719003 280997 00524 99999 42 19 742477 999993 742484 257516 00553 99998 41 20 764754 999993 764761 236239 00582 99998 40 21 7.785943" 9.999992 7.785951 12.214049 00611 99998 39 22 806146 999991 806155 193845 00640 99998 38 23 825451 999990 825460 174540 00669 99998 37 24 843934 999989 843944 156056 00698 99998 36 25 861663 999988 861674 138326 00727 99997 35 26 878695 999988 878708 121292 00756 99997 34 27 895085 999987 896099 104901 00785 99997 33 28 910879 999986 910894 039106 00814 99997 32 29 926119 999985 926134 073866 00844 99996 31 30 940842 999983 940858 059142 00873 99996 30 31 7.955082 9.999982 7.956100 2298 2227 2161 2098 2039 1983 1930 1880 1833 1787 1744 1703 1664 1627 1591 1557 1524 1493 1463 1434 1406 1379 1353 1328 1304 1281 1259 1238 1217 12.044900 00902 99996 29 32 968870 2298 S99981 0.2 968889 031111 00931 99996 28 33 982233 2227 999980 0.2 982253 017747 00900 99995 27 34 995198 2161 999979 0.2 996219 004781 00989 99995 26 35 8.007787 2098 999977 0-2 8.007809 11.992191 01018 99995 25 36 020021 2039 999976 0-2 0-2 020045 979955 01047 99995 24 37 031919 1983 999975 031945 968055 01076 99994 23 38 043501 1930 1880 1832 1787 1744 1703 1664 1626 1691 1567 1524 1492 1462 1433 1405 1379 1353 1328 1304 1281 1269 1237 1216 999973 0-2 0-2 043527 956473 ' 01105 99994 22 39 054781 999972 054809 945191 1 934194 1 01134 99994 21 40 065776 999971 0*2 065808 01164 99993 20 41 8.076500 9.999969 0'2 0-2 8.076531 11.923469 01193 99993 19 42 086965 999968 086997 913003 i 01222 99993 18 43 097183 999966 0'2 097217 902783 1 01251 99992 17 44 107167 999964 0'2 o;3 107202 892797 1 01280 99992 16 45 116926 999963 116963 883037 01309 99991 15 46 126471 999961 3 0.3 0.3 126610 873490 01338 99991 14 47 136810 999959 135851 864149 01367 1)9991 13 48 144953 999958 144996 855004 01396 99990 12 49 153907 999956 0.3 0.3 163952 846048 01425 99990 11 50 162681 999954 162727 837273 01454 99989 10 51 8.171280 9,999952 0.3 0.3 0.3 0.3 0.3 0.3 8.171328 11.828672 01483 99989 9 52 179713 999950 179763 820237 01513 99989 8 53 187985 999948 188036 811964 01542 99988 7 54 196102 999946 196166 803844 01571 99988 6 56 204070 999944 204126 795874' 01600 99987 5 56 211895 999942 211953 7&8047 01629 99987 4 57 219581 999940 0.4 0.4 219641 780359 j 01658 99986 3 58 227134 999938 227195 772805 ' 01687 9998() 2 59 234557 999936 0.4 0.4 234621 765379 01716 99985 1 60 241855 999934 241921 758079:, 01746 99985 Cosine. Sine. Coihmg:. Tnnar. ' N. cos. N. sine- 89 Degrees. | 22 hog. Sines aud TangentB. (l") Natural Sines. TABLE H. D.IO' Cosine. D.IO' Tang. DIG" Coiang. N. sine. N. cos. 10 11 12 13 14 16 16 17 18 19 20 21 22 23 24 26 26 27 28 29 30 31 8 32 33 34 36 36 37 38 39 40 41 42 43 44 45 46 47 48 49 60 61 62 53 54 66 56 57 68- 69 60 .241856 jj g 249033 „,, 256094 I;'' 26304-2 };°° 2€9881 {^^ 276(il4 {{r.'t 283243 |iX« 289773 V^l 290207 \ali 302546 \lf, 308794 \Yl^ .314954 }^, ' 321027 aoi 327016 QOR 332924 g^j 338763 q-q 344604 all 350181 qo^ 355783 Qoo 361316 Qin 366777 gg" .372171 S^o 377499 S°? 382762 ^^ 387962 ^L 393101 o2^ 398179 o^ 403199 007 408161 ofo 413068 or^ 417919 oXX .422717 ?^J 427462 7^2 432156 7^f 436800 7^R 441394 7^0 445941 III 450440 740 454893 7^ 459301 ^tj 463665 «on .467985 ':" 472263 ;l-\f 476498 gXX 480693 ^^l 484848 ^«fi 488963 l^l 493040 ^', 497078! ^^5 501080 i l^i 506045 i °^^ .608974 I °5q 612867 ! ^11 516726 ^^^ 620551 ; lii 524343 2^^ 528102 ^^^ 531828 : fi,^ 535623 ^}° 539186 «i.^ 542819 . ^"^ 9.999934 999932 999929 999927 999925 999922 999920 999918 999916 999913 999910 9.999907 999905 999902 999897 999894 999891 999888 9.999879 999876 999873 999870 999867 999864 999861 999858 999854 999851 9.999848 999844 999841 999838 999834 999831 999827 999823 999820 999816 9.999812 999809 999805 999801 999797 999793 999790 999786 999782 999778 9.999774 999769 999765 999761 999757 999753 999748 999744 999740 999735 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.5 0.5 0.6 0.5 0.6 0.5 0.5 0.5 0-6 0-6 0.6 0.6 0-6 0.6 0.6 0.6 0.6 0.6 0.6 0-6 0-6 0.6 0.6 0-7 0-7 0.7 0.7 0.7 0.7 .241921 249102 266166 263115 269956 276691 283323 289866 296292 302634 .3160-46 321122 327114 333026 344610 350289 356895 361430 .372292 377622 8ine. 388092 393234 398316 403338 408304 413213 418068 .422869 427618 432315 436962 441560 446110 450613 455070 469481 463849 .468172 472454 476693 480892 485050 489170 493250 497298 501298 505267 .509200 513098 516961 620790 624586 528349 532080 536779 539447 6 43084 Cotarifr- 1197 1177 1168 1140 1122 1105 1089 1073 1057 1042 1027 1013 999 985 972 959 946 934 922 911 879 867 857 847 837 828 818 809 800 791 783 774 766 758 750 743 736 728 720 713 707 700 693 686 680 674 668 661 655 650 644 638 633 627 622 616 611 606 11.758079 743835 736885 730044 723309 716677 710144 703708 697366 691116 11.684954 678878 672886 666975 661144 665390 649711 644106 638670 633105 11-627708 622378 617111 611908 606766 601685 596662 591696 686787 581932 11.577131 567685 563038 568440 553890 549387 544930 540519 636151 11.531828 527546 523307 519108 614950 610830 506750 502707 498702 494733 11.490800 486902 483039 479210 476414 471651 467920 464221 460563 456916 01742 01774 01803 01832 01862 01891 01920 01949 01978 02007 02036 02065 02094 02123 02152 02181 02211 02240 02269 02298 02327 02356 02385 02414 02443 02472 02501 02630 02660 02589 02618 02647 02676 02705 02734 02763 02792 02821 02850 99985 99984 99984 99983 99983 99982 99982 99981 99980 62 99980 99979 99979 99978 99977 99977 99976 99976 99975 99974 99974 99973 99972 99972 99971 99970 99969 99969 99968 99967 99966 99.9661 30 99965 99964 99963 99963 99962 99961 99960 99959 02879 99959 99958 99957 99956 99955 99954 99953 99952 99952 99951 99950 99949 03228 99948 03257 99947 03286 99946 03316 99945 03346 99944 03374 99943 03403 99942 02908 02938 02967 02996 03025 03054 03083 03112 99952 03141 03170 03199 03432 03461 03490 99941 99940 99939 Tang. N. COS. N.8ine 88 Degrees. TABLE II. Log. Sines and Tangents. (2°) Natural Sines. 23 8.542819 54G4'22 549995 553539 557054 560540 563999 567431 570836 574214 577666 8.580892 584193 587469 590721 593948 597152 600332 603489 606623 609734 8.612823 615891 618937 621962 624965 627948 630911 633854 636776 639680 8.642563 645428 648274 651102 653911 656702 659475 662230 664968 667689 8.670393 673080 675751 678405 681043 683665 686272 691438 693998 8.696543 699073 701589 704090 706577 709049 711507 713952 716383 _718800 Cosme. D. 10" 600 595 691 586 581 576 572 567 563 559 554 550 546 542 538 634 530 526 522 519 515 511 508 504 501 497 494 490 487 484 481 477 474 471 468 465 462 469 456 453 451 448 445 442 440 437 434 432 429 427 424 422 419 417 414 412 410 407 405 403 Cosine. 1.999735 999731 999726 999722 999717 999713 999708 999704 999694 999685 999680 999675 999670 999665 999660 999655 999550 999645 999640 999635 999629 999324 999619 999614 999608 999603 999597 999592 999586 999581 999575 999570 999564 999558 999553 999547 999541 999535 999529 ,999524 999518 999512 999506 999500 999493 999487 999481 999475 999469 ,999463 999456 999450 999443 999437 999431 999424 999418 999411 999404 D. 10" Tang. 0.7 0.7 0.7 0-8 0-8 0-8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.9 0.9 0.9 0.9 0.9 0.9 0-9 0.9 0-9 0.9 0-9 0.9 0.9 0.9 0.9 1.0 1-0 1.0 1.0 1-0 1.0 1.0 1.0 1-0 1-0 1.0 1.0 1.0 1.0 1.0 1.0 1.543084 546691 550268 553817 557335 560828 564291 567727 571137 574520 577877 .581208 584514 587795 591051 594283 597492 600677 603839 606978 610094 .613189 616262 619313 622343 625352 628340 631308 634256 637184 640093 ,642982 645853 648704 651537 664352 657149 659928 662689 665433 668160 .670870 673563 676239 678900 681544 684172 6-6784 689381 691963 694529 .697081 699617 702139 704246 707140 709618 702083 714534 716972 719396 Coians:. D. 10"i Coiang. |{N. sine. N. cos. 602 593 591 587 582 577 573 568 564 559 555 551 547 543 539 535 631 527 523 519 516 512 508 505 501 498 495 491 488 485 482 478 475 472 469 466 463 460 457 454 453 449 446 443 442 438 485 433 430 428 425 423 420 418 415 413 411 408 406 404 11.456916 453309 449732 446183 442664 439172 435709 432273 428863 425480 422123 11.418792 415486 412205 408949 405717 402508 399323 396161 393022 389906 11.386811 383738 380687 377657 374648 371660 368692 365744 362816 359907 11.357018 354147 351296 348463 345648 342851 340072 337311 334567 331840 11.329130 326437 323761 321100 318456 315828 313216 310619 308037 305471 11.302919 300383 297861 296354 292860 290382 287917 285465 283028 280604 03490 03519 99938 03548 03577 99936 03606 1 03635 1 03664 ■03693 103723 03752 1 03781 03810 03839 03868 03897 03926 03955 03984 04013 04042 104071 104100 103129 {04159 04188 04217 04246 04275 04304 04333 04362 04391 04420 04449 04478 04507 04536 04565 04594 04623 04653 99892 04682 99890 0471199889 04740 99888 04769 99886 04798 99885 Tans 04827 04856 9988 04885 04914 04943 04972 05001 05030 05059 05088 05117 05146 05176 06205 05234 99939 99937 99935 99934 99933 99932 99931 99930 99929 99927 99926 99926 99924 99923 99922 99921 99919 99918 99917 99916 99915 99913 99912 99911 99910 99909 99907 99906 99906 99904 99902 99901 99900 99897 99896 99894 99883 99881 99879 99878 99876 99876 99873 99872 99870 99869 9y867 99864 99863 N. COS. N.8ine 87 Degrees. 24 iOg. Sines and Tangciiis. (3°; Natural Sines, TABLE II. 2 3 4 5 6 7 8 9 10 11 •12 13 14 15 16 17 18 19 20 21 22 23 24 25 28 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 60 51 8 52 53 54 55 56 57 58 59 60 Cosine. 1 1 1. 1 1 1 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.2 Sine. |U. W Cosine. D71o .71880a 721204 723595 725972 728337 730{>88 733027 735354 737667 739969 742259 .744536 746802 749055 751297 753528 755747 757955 760151 762337 764511 .766675 768828 770970 773101 776223 777333 779434 781524 783605 785675 .787736 789787 791828 793859 795881 797894 799897 801892 803876 805852 ,807819 809777 811726 813667 815599 817522 819436 821343 823240 825130 ,827011 828884 830749 832607 834456 836297 838130 839956 841774 843586 401 398 396 394 392 390 388 386 384 382 380 378 376 374 372 370 368 366 364 362 361 359 357 355 353 352 350 348 347 345 343 342 340 339 337 336 334 332 331 329 325 323 322 320 319 318 316 316 313 312 311 309 308 307 306 304 303 302 .999404 999398 999391 999384 999378 999371 999364 999357 999350 999343 999336 .999329 999322 999315 999308 999301 999294 999286 999279 999272 999265 .999257 999250 999242 999235 999227 999220 999212 999205 999197 999189 .999181 999174 999166 999158 999150 999142 999134 999126 999118 999110 .999102 999094 999086 999077 999061 999053 999044 999036 999027 .999019 999010 999002 998993 998984 998976 998967 998950 998941 1.2 1.3 1.3 1.3 1.3 1.3 1.3 1.3 1.3 1.3 13 1.3 1.3 1.3 1.3 1.3 1.3 1.3 1.3 1.3 1.4 1.4 1.4 1.4 1.4 1.4 1.4 1.4 1.4 1.4 1.5 1.5 1.5 lanj; .719396 721806 724204 726588 728959 731317 733663 735996 738317 740326 742922 ,745207 747479 749740 751989 754227 766453 758668 760872 763065 765246 767417 769578 771727 773866 775995 778114 780222 782320 784408 786486 8.788554 790613 79-2662 794701 793731 798752 80'J763 802765 804858 80 )742 808717 810683 81-2641 814589 816529 818461 820384 822298 824205 826103 8.827992 829874 831748 833613 835471 837321 839163 840998 842826 844644 Cotanp. l». !(/ 402 399 397 395 393 391 389 387 385 383 381 379 377 375 373 371 3o9 367 365 364 362 360 358 356 355 353 351 350 348 346 345 343 341 340 338 337 335 334 332 331 329 328 326 325 323 322 320 319 318 316 316 314 312 311 310 308 307 306 304 303 Cotang. |(N. sine. 11.2806041 278194] 276796 273412 271041 I 268683 i 266337 I 264004 I 261683 I 259374; 1 05496 257078 I 05524 05234 05263 05292 06321 05350 05379 05408 06437 05466 11.254793 252521 250260 248011 245773 243547 241332 239128 236935 234754 11.232583 230422 228273 226134 224005 221886 219778 217680 215592 213514 11.211446 209387 207338 205299 203269 201248 199237 197235 195242 193258 11.191283 189317 187359 185411 183471 181639 179616 177702 175796 173897 11.172008 170126 168252 166387 164529 162679 160837 159002 167175 156366 05553 05582 05611 05640 05669 05698 0572 05766 05785 05814 05844 05873 05902 05931 05960 Tang. 06018 06047 06076 06105 06134 06163 06192 06221 06250 06279 06308 06337 06366 06395 08424 06463 06482 06511 06540 06569 06598 06627 06656 06685 08714 06743 06773 06802 06831 06860 06889 06918 06947 06976 'iIn. cos N.cos. 99863 99861 99860 99858 99857 99855 99854 99852 99851 99849 99847 99846 99844 99842 99841 99839 99838 99836 99834 99833 99831 99829 9982' 99826 99824 99822 99821 99819 99817 99815 99813 99812 99810 99808 99806 99804 99803 99801 99799 99797 99795 99793 99792 99790 99788 99786 99784 99782 99780 99778 99776 99774 99772 99770 99768 99766 99764 99762 99760 99758 99766 .V.Bine 86 Degrees. TABLE 11. .og. Sines and Tangents. (4°) Natural Sines. 25 D. W 300 299 298 297 295 294 293 292 291 2;)0 288 287 286 285 284 283 282 281 279 279 277 276 275 274 273 272 271 270 269 268 267 266 265 264 263 262 261 260 259 258 257 257 256 255 254 253 252 251 250 249 249 248 247 246 245 2'14 243 243 242 241 Cosine. 9.998941 998932 998923 998914 998905 998896 998887 998878 998869 998860 998851 9.998841 998832 998823 998813 998804 998795 998785 998776 998766 998757 9.998747 998738 998728 998718 998708 8.843586 845387 847183 848971 850751 852525 854291 856049 857801 859546 861283 8.863014 864738 866455 868165 869868 871565 873255 874938 876615 878285 8.879949 881607 883258 884S03 886542 888174 889801 891421 893035 894643 .896246 897842 899432 901017 902596 904169 905736 907297 908853 910404 8.911949 913488 915022 916550 918073 919591 921103 922610 924112 926609 8.927100 928587 930068' 931544 933015 934481 935942 937398 938850 940296 Cosine. 998689 998679 998669 998659 9.998649 998639 998629 998619 998609 998589 998578 998568 998558 9.998648 998537 998627 998616 998506 998495 998485 998474 998464 998453 998442 998431 998421 998410 998399 998377 998366 998355 998344 Sine. D. 10" Tang, D. 1 0" Cota ng. l|N. sine. N. cos 884530 886185 887833 889476 891112 892742 894366 895984 .897596 899203 900803 902398 903987 906570 907147 908719 910285 911846 .913401 914951 916495 918034 919668 921096 922619 924136 925649 927156 .928658 930155 931647 933134 934616 936093 937565 940494 941952 Cotang. 302 801 299 298 297 293 295 293 292 291 290 289 288 287 285 284 283 282 281 280 279 278 277 276 275 274 278 272 271 270 269 268 267 266 265 264 263 262 261 260 259 258 267 256 256 255 264 263 252 251 250 249 249 248 247 246 245 244 244 243 11 11.166366 153646 151740 149943 148164 146372 144697 142829 141068 139314 137667 , 135827 134094 132368 130649 128936 127230 125531 123838 122151 120471 118798 117131 116470 113815 112167 110524 108888 107258 106634 10-1016 102404 100797 099197 097602 096013 094430 092863 091281 089715 088154 11.086599 085049 083606 081966 080432 078904 077381 075864 074361 072844 11.071342 06976 07005 07034 07063 07092 07121 07150 07179 07208 07237 07266 07295 07324 07363 07382 07411 07440 07469 07498 07527 07556 07586 99756 99754 99752 99750 99748 99746 99744 99742 99740 99738 99736 99734 99731 99729 99727 99725 99723 99721 99719 99716 99714 99712 07614 99710 1 07643 07672 07701 07730 07759 07788 07817 07846 07875 07904 07933 07962 07991 08020 08049 08078 08107 08136 08165 08223 08252 08281 08310 108368 08397 08426 08455 1 08484 108513 1 08542 i 08671 108600 068863 066866 065384 063907 06243511 08629 |9y62 060968 059606 068048 Tang. 08658 08687 08716 99619 N. co,«. -N.eine 99708 99705 99703 99701 99699 99696 99694 99692 99689 99687 99685 99683 99680 99678 99676 99673 d9671 99668 99666 08194 99664 99661 99659 99657 99654 99652 99649 99647 99644 99642 99639 99637 99635 99632 99630 99625 99622 85 Degrees. 26 hog. Sines and Tangents. (5°) Natural Sines. TABLE II. 1 2 3 4 6 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 56 5] 62 63 54 65 56 57 58 69 60 Sine. 8.940296 941738 943174 944608 946034 947456 948874 950287 95169S 953100 954499 955894 957284 958670 960052 901429 962801 984170 965534 966893 968249 8.969600 970947 972289 973628 974962 976293 977619 978941 980259 981573 8.982883 984189 985491 986789 D. 10" 989374 990660 991943 993222 994497 B. 995768 997036 998299 999560 9.000816 002069 003318 004563 005805 007044 9.008278 009510 010737 011962 013182 014400 015613 016824 018031 019235 Cosine. 240 239 239 238 237 236 235 235 234 233 232 232 231 230 229 229 228 227 227 226 225 224 224 223 222 222 221 220 220 219 218 218 217 216 216 215 214 214 213 212 212 211 211 210 209 209 208 208 207 206 206 203 205 204 203 203 202 202 201 201 Cosine. (9.998344 998333 998322 998311 998300 998289 998277 998266 998255 998243 998232 9.998220 998209 998197 998186 998174 998163 998151 998139 998128 998116 9.998104 998092 998080 998068 998056 998044 998032 998020 998008 997996 9.997984 997972 997959 997947 997935 997922 997910 997897 997885 997872 ,997860 997847 997835 997822 997809 997797 997784 997771 997758 997745 997732 997719 997706 997693 997680 997667 997654 997641 997628 997614 D. 10"j Tang. 1.9 1.9 1.9 1.9 1.9 1.9 1.9 1.9 1.9 1.9 1.9 1.9 1.9 1.9 1.9 1.9 1.9 1.9 2.0 2.0 2,0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2,1 2.1 2,1 2.1 2,1 2.1 2.1 2.1 2.1 2.1 2.1 2.1 2.1 2.1 2,1 2.1 2.1 2.1 2.1 2.2 2.2 2.2 2.2 2.2 2.2 Sine. 941952 943404 944852 946295 947734 949168 950597 952021 953441 954856 956267 8.957674 959075 960473 961866 963255 964639 906019 967394 968766 970133 .971496 972855 974209 975560 976906 978248 979586, 980921 982251 983577 8.984899 986217 987532 988842 990149 991451 992750 994045 995337 996624 .997908 999188 ,000465 001738 003007 004272 005534 006792 008047 009298 010546 011790 013031 014268 015502 016732 017959 019183 020403 021620 D. 10"! Cotang. Co tang. 242 241 240 240 239 238 237 237 236 235 234 234 233 232 231 231 230 229 229 228 227 226 226 225 224 224 223 222 222 221 220 220 219 218 218 217 210 216 215 215 214 213 213 212 211 211 210 210 209 208 208 207 207 206 206 205 204 204 203 203 11.058048 056596 055148 053705 i 052266 I 050832 I 049403 i 047979 046559 I 0451441 043733 I 11.042326! 040925 i 039527 038134 036745 035861 033981 032606 031234 029867 11.028504 027145 025791 024440 023094 021752 020414 019079 017749 016423 11.015101 013783 012468 011158 009851 008549 007250 005955 004663 003376 11.002092 000812 10.999535 998262 I 996993 I 996728 I 994466 I 993208 I N. sine 08716 08745 08774 08803 08831 08860 08918 08947 08976 09005 09034 09063 09092 09121 09150 09179 09208 09237 09266 09295 09324 09353 99619 60 99617 99614 99612 99609 99607 99604 99602 99599 99596 99594 99591 99588 99586 99583 99580 99578 99575 99572 99570 99567 99564 99562 09382|99559 0941199556 09440J99553 09469199551 09498199548 09527199545 09556199542 09585i99540 99537 99534 99531 99528 99526 99523 99520 99517 09614 09642 09671 09700 09729 06758 09787 09816 09845|99514 09874199511 09903199508 09932 j9950i 09961 J99503 09990199500 10019J9949 10048J99494 10077199491 10106199488 991953 1110135 99485 990702 1 110164199482 10.989454 988210 686969 986732 984498 983268 983041 980817 979597 j 978380 I i 10192 99479 11022199470 110250,99473 110279,99470 10308 99467 Tang. 10337 10366 10395 10424 10463 99464 99461 99458 99455 99452 N. COS. Njsine. 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 ^6 26 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 6 4 8\ Degrees. TABLE IT. Log. Sines and Tangents. (6"') Natural Sines. 27 60 Sine. 9.019235 020436 021632 022825 024016 025203 026386 027567 028744 029918 031089 9.032257 033421 034582 035741 036896 038048 039197 040342 041485 042625 .043762 044895 046026 047154 048279 049400 050619 051635 052749 053859 9.054966 056071 057172 058271 059367 060460 061561 062639 063724 064806 9.066886 066962 068036 069107 070176 071242 072306 073366 074424 075480 076633 077583 078631 079670 080719 081759 082797 083832 084864 086894 I>. 10"[ Cosine. ID. 10' Cosine. 200 199 199 198 198 197 197 196 196 195 196 194 194 193 192 192 191 191 19a 190 189 189 180 188 187 187 186 186 185 185 184 184 184 183 183 182 182 181 J81 180 180 179 179 179 178 178 177 177 176 176 175 176 176 174 174 173 173 172 172 172 .997614 997601 997588 997574 997661 997647 997534 997520 997507 997493 997480 .997466 997452 997439 997425 997411 997397 997388 997369 997355 997341 .997327 997313 997299 997285 997271 997267 997242 997228 997214 997199 .997185 997170 997156 997141 997127 997112 997098 997083 997068 997053 .997039 997024 997009 996994 996979 996964 996949 996934 996919 996904 .996889 996874 996858 996843 996828 996812 996797 996782 996766 996761 Sine I 2.2 I 2.2 2.2 2.2 2.2 I 2.2 2.3 I 2.3 ! 2.3 2-3 2-3 2.3 2-3 2.3 2.3 2.3 2-3 2-3 2-3 2-3 2.3 2.4 2-4 2.4 2.4 2-4 2.4 2.4 2.4 2-4 2 2 2 2 2 2 2 2 2.5 2.5 25 25 2-5 2.6 2.6 2-5 2-6 2.5 2.6 2.6 2.5 2.6 2.6 2.6 2.5 2.5 2.6 2 6 2.6 2.6 Tang. ,J>. W' 9.021620 022834 024044 025251 026455 027655 028852 030046 031237 032425 033609 9,034791 035969 037144 038316 039485 040651 041813 042973 044130 045284 9.046434 047682 04872/ 049869 051008 052144 053277 054407 056635 056659 9.057781 058900 060016 061130 062240 063348 064453 066656 066655 067762 9.068846 069038 071027 072113 073197 074278 075356 076432 077505 078576 9.079644 080710 081773 082833 083891 084947 086000 087050 202 202 201 201 200 199 199 198 198 197 197 196 196 195 195 194 194 193 193 192 192 191 191 190 190 189 189 188 188 187 187 186 186 185 186 185 184 184 183 183 182 182 181 181 181 180 180 179 179 178 178 178 177 177 176 176 175 175 176 174 99354 99361 99347 99344 99341 99337 99334 99331 99327 99324 99320 99317 99314 99310 99307 99303 99300 99297 99293 99290 99286 99283 99279 99276 99272 99269 992G5 99262 99258 y9256 I Tang. Il N. cos. N.sine. 0453 0482 0511 0540 0569 0597 0626 0655 0684 0713 0742 0771 0800 0829 0858 0887 0916 0945 0973 1002 1031 1060 1089 1118 1147 1176 1205 1234 1263 1291 99452 99449 99446 99443 99440 99437 99434 99431 99428 99424 99421 99418 99415 99412 99409 99406 99402 99399 99396 99393 99390 Cotang. t N. sine. N. ccs. 10.978380! 977166 I 975956 I 974749 I 973645 I 972345 j 971148 i 969954 968763 I 967676 i 966391 I 10.965209 964031 962856 961684 960516 959349 958187 957027 955870 954716 10.963566 952418 951273 950131 948992 947856 946723 945693 944465 943341 10.942219 941100 939984 938870 937760 936652 935647 934444 933345 I 932248 10.9311541 930062 928973 927887 926803 926722 924644 923668 922496 921424 10.920356 91929a »18227 917167 916109 915053 914000 912950 911902 jl 910856 99383 99380 99377 99374 99370 99367 99364 99360 1320'99357 1349 1378 1407 1436 1465 1494 1523 1552 1580 1609 1638 1667 1696 1726 1754 1783 1812 1840 1869 1898 1927 1956 1985 2014 2043 2071 2100 2129 2168 2187 28 Log. Sines and Tangents. (7°) Natural Sines. TABLE II. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 60 61 62 53 54 55 56 57 58 59 60 D. Ml Cosine 9.085894 086922 087947 088970 089990 091008 092024 093037 094047 09505G 096062 9.097065 098036 099065 100052 101056 102048 103037 104025 105010 105992 9.106973 107951 108927 109901 110873 111842 112809 113774 114737 115698 .116656 117613 118567 119519 120469 121417 122362 123306 124248 125187 9.126126 127060 127993 128925 129864 130781 131706 132630 133551 134470 9.135387 136303 137216 138128 139037 139944 140860 141754 142655 143565 Cosine. 171 171 170 170 170 169 1&9 168 168 168 167 167 166 166 166 165 165 164 164 164 163 163 163 162 162 162 161 161 160 160 160 159 159 159 158 158 158 157 157 167 166 156 156 155 156 154 154 154 153 163 163 152 152 152 152 151 151 151 150 996751 996735 996720 996704 996688 996673 996657 996641 996625 996610 996594 9.996578 9965G2 996546 996530 996514 996498 996482 996465 996449 996433 9.996417 996400 996384 996368 996361 996335 996318 996302 996286 996269 9.996252 996236 996219 996202 996186 996168 996151 996134 996117 996100 .996083 996066 996049 996032 996015 995998 996980 995963 995946 995928 9.995911 995894 995876 995869 996841 995823 996806 995788 995771 996753 2.6 2.6 2.6 2.6 2.6 6 6 6 6 6 7 7 7 2.7 2.7 2.7 2.7 2.7 2.7 2.7 2.7 2.7 2.7 2.7 2.7 2.7 2.7 2.8 2.8 2.8 Sine. 2.8 2.8 2.8 2.8 2.8 2.8 2.8 2.8 2.8 2.8 2.9 2.9 2.9 2.9 2.9 2.9 2.9 2.9 2.9 2.9 2.9 2.9 2.9 2.9 2.9 2.9 2.9 2.9 2.9 Taiig: mrw 9.089144 090187 091228 092266 093302 094336 095367 096395 097422 098446 099468 9.100487 101504 102519 103532 104542 106550 106556 107559 108560 109559 9.110556 111551 112643 113533 114521 115507 116491 117472 118462 119429 9.120404 121377 122348 123317 124284 126249 126211 127172 128130 129087 1.130041 130994 131944 132893 133839 134784 136726 136667 137605 138642 .139476 140409 141340 142269 143196 144121 145044 146966 146885 147803 Cotang. 174 173 173 173 172 172 171 171 171 170 170 169 169 169 168 168 168 167 167 166 166 166 165 165 165 164 164 164 163 163 162 162 162 161 161 161 160 160 160 159 159 159 168 158 168 167 157 157 156 155 156 156 165 156 154 154 154 153 153 163 Ootang. I'N. sine. N. con. 10.910856 909813 908772 907734 ! 906698 905664 i 9046331 903605 j 902578 j 901554 I 900532 I 10.899513 i 898496 ! 897481 I 896468 895458 I 8944501 893444 j 892441 12187 12216 12245 12274 12302 12331 12360 12389 12418 12447 12476 12504 99256 99251 99248 99244 99240 99237 99233 99230 99226 99222 99219 99215 12591 12620 12678 12706 99189 8914401 1 12735 890441 1 112764 10.889444 !j 12793 99178 88844911 12822 99175 887457 I i 12851 886467 885479 884493 II 12937199160 883509 ij 12966 12633 99211 12562 99208 99204 99200 12649 99197 99193 99186 99182 99171 12880 99167 12905 99163 882528 881548 880571 10.879596 878623 877652 876683 99156 ! 12995 99152 13024 13053 13081 99148 99144 99141 13110 99137 13139 99133 13168 99129 875716 j 13197 99125 874751 '113226 99122 873789 i! 13264)99118 872828 1 1 13283199114 871870 :ll3G12 99110 870913 |! 13341 99106 10.86995911 13370 99102 99098 99094 867107 I i 13456199091 866161 1113485 869.006 jilSG 99 868056 113427 865216! 864274 8633c!3 13514 13543 99079 13572 99075 862396 861458 10.860524 859591 868()60 857731 856804 855879 864956 864034 j 13860 8531 15 ''.13889 862197 ii 13917 13773 13802 Tang. ?9087 99083 13600 99071 13629199067 13658 99063 13687 99059 1371C 99055 13744,99051 99047 )9043 13831 99039 99035 39031 M9027 N. cos. N.eine. 82 Degrees. Log. Sines and Tanj^ents. (8°) Natural Sinca. 29 Bino. 9.143555 144453 145349 146243 147136 148026 148915 149802 150686 151569 152451 9.153330 154208 155083 155957 156830 157700 158569 159435 160301 161164 9.162025 162885 163743 164600 165454 166307 167159 168008 168856 169702 9.170547 171389 172230 173070 173908 174744 175578 176411 177242 178072 9.178900 179726 180551 181374 182196 183016 183834 184651 185466 186280 9.187092 187903 188712 189519 190325 191130 191933 192734 193534 194332 I Cosine. D. 10' 150 149 149 149 148 148 148 147 147 147 147 146 146 146 145 145 145 144 144 144 144 143 143 143 142 142 142 142 141 141 141 140 140 140 140 139 139 139 139 138 138 138 137 137 137 137 136 136 136 136 135 135 135 135 134 134 134 134 133 133 Cosine. 9.996753 995735 995717 995699 995681 995664 995646 995628 996610 995591 995573 9.995555 995537 995519 995501 995482 995464 995446 995427 995409 995390 9.995372 995353 995334 995316 996297 995278 995260 995241 996222 995203 995184 995165 995146 995127 995108 995089 995070 995051 995032 995013 9.994993 994974 994955 994935 994916 994896 994877 994867 994838 994818 .994798 994779 994759 994739 994719 994700 994680 994660 994640 994620 Sine. D. 10" 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.1 3.1 3.1 3.1 3.1 3.1 3.1 3.1 3.1 3.1 3.1 3.1 3.1 3.1 3.2 3.2 3.2 3.2 3.2 8.2 3.2 3.2 3.2 3.2 3.2 3.2 3.2 3.2 3,2 3.2 3.2 3.3 3.3 3.3 3.3 3.3 3.3 3.3 3.3 3.3 3.3 3.3 3.3 3.3 3.3 3.3 Tang. 9.147803 148718 149632 150644 151464 152363 153269 154174 155077 155978 156877 9.167775 158671 169666 160467 161347 162236 163123 164008 164892 165774 9.166654 167532 168409 169284 170157 171029 171899 172767 173634 174499 . 175362 176224 177084 177942 178799 179655 18060S 181360 182211 183069 . 183907 184752 185597 186439 187280 188120 188968 189794 190629 191462 . 192294 193124 193953 194780 196606 196430 197253 198074 198894 199713 D. 10' Cotang. |N. sine. N. cos, 153 152 152 152 151 151 151 160 160 150 150 149 149 149 148 148 148 148 147 147 147 146 146 146 145 145 146 145 144 144 144 144 143 143 143 142 142 142 142 141 141 141 141 140 140 140 140 139 139 139 139 138 138 138 138 137 137 137 137 136 Co tang. 10.852197 851282 850368 849456 848546 847637 846731 845826 844923 844022 843123 10.842225 841329 840435 839543 838653 837764 836877 835992 836108 834226 13917 13946 13975 14004 14033 14061 14090 14119 14148 14177 14205 14234 14263 14292 14320 14349 14378 1440 14436 14464 14493 10.8333461 114522 832468 831691 830716 829843 828971 828101 827233 826366 825501 10.824638 823776 822916 822058 821201 820345 819492 818640 817789 816941 10.816093 815248 814403 813561 812720 811880 811042 810206 809371 i 808538 10.8077061 806876 806047 I 806220 I 804394 1 803570 i 802747 801926 I 8011061 800287 14551 14580 14608 14637 14666 14695 14723 14752 14781 14810 14838 14867 14896 14925 14954 14982 16011 16040 15069 15097 15126 15155 i 15184 16212 15241 16270 15299 15327 16356 15385 Tang. 15442 15471 15600 16629 15557 16586 16615 15643 99027 99023 99019 99015 99011 99006 99002 98994 98990 98982 98978 98973 98969 98965 98961 98957 98953 98944 98940 98931 98927 98919 98914 98910 98906 98902 98897 98893 98876 98871 98867 98863 98858 98854 98849 98845 98841 98827 98823 98818 98814 98809 15414 98806 98800 98796 98791 98787 98782 98778 98773 98769 N. cos. N.siDe 81 Degrees. 30 Log. Sines aud Tangents. (9°) Natural Sines. TABLE n. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 IP) 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 BO 51 52 53 54 55 56 57 58 59 60 Sine. 9.194332 195129 1959-25 196719 197511 198302 199091 199879 200666 201451 202234 9.203017 203797 204577 205354 206131 208906 207679 203452 209222 209992 9.210760 211626 212291 213055 213818 214679 215338 216097 216854 217609 9.218363 219116 219868 220618 221367 222116 222861 223606 224349 225092 9.225833 226573 227311 228048 228784 229518 230252 230984 231714 232444 9.233172 233899 234625 235349 236073 236795 237515 238235 238953 239670 D. ny Ck>8ine. 133 133 132 182 132 132 131 131 131 131 130 130 130 130 129 129 129 129 128 128 128 128 127 127 127 127 127 126 126 126 126 126 125 125 125 125 124 124 124 124 123 123 123 123 123 122 122 122 122 122 121 121 121 121 120 120 120 120 120 119 Cosine. 9.994620 994600 994580 994560 994540 994519 994499 994479 994459 994438 994418 9.994397 994377 994357 994336 994316 994295 994274 994254 994233 994212 9.994191 994171 994150 994129 994108 994087 9940(36 994045 994024 994003 9.993981 993960 993939 993918 993896 993875 993854 993832 993811 993789 .993768 993746 993725 993703 993681 993660 993638 993616 993594 993572 .993550 994528 993506 993484 993462 993440 993418 993396 993374 993351 D. It)' Sine. 3.3 3.3 3.3 3.4 3.4 3.4 3.4 3.4 3.4 3.4 3,4 3.4 3.4 3.4 3.4 3.4 3.4 3,5 3.5 3.5 3.5 3.5 3.5 3.5 3.5 3.5 3.5 3.6 3.5 3.5 3.5 3.5 3.5 3.5 3.5 3.6 3.6 3.6 3.6 3.6 3.6 3.6 3.6 3.6 3.6 3.6 3.6 3.6 3.6 3.7 3.7 3.7 3.7 3-7 3.7 Taai;. 3.199713 200529 201345 202169 202971 203782 204592 205400 206207 207013 207817 ). 208619 209420 210220 211018 211815 212611 213405 214198 214989 215780 ). 216568 217356 218142 218926 219710 220492 221272 222052 222830 223606 1,224382 225156 225929 226700 227471 228239 229007 229773 230589 231302 '.232066 232826 233586 234345 235103 235859 236614 237368 238120 238872 .239622 240371 241118 241865 242610 243354 2440{}7 244839 245579 246319 CoUmj;. 136 136 136 135 136 135 136 134 134 134 184 133 133 133 133 133 132 132 132 132 131 131 131 131 130 130 130 130 130 129 129 129 129 129 128 128 128 128 127 127 127 127 127 126 126 126 126 126 126 126 125 125 126 124 124 124 124 124 123 123 Cotang. N. mne.lN. cos 10.800287 799471 798666 797841 797029 796218 796408 794600 793793 792987 792183 10.791381 790580 789780 i 788982 788185 787389 786596 785802 735011 784220 10.783432 782644 781858 781074 780290 779608 778728 777948 777170 776394 10.775618 774844 774071 773300 772529 771761 770993 770227 1 1 769461 ! 16643 15672 15701 15730 16758 1578 15816 16846 15873 16902 16931 16959 98769 98764 98760 98766 98751 98746 98741 98737 98732 98728 98723 98718 15988 98714 16017 16046 16074 16103 16132 16160 16189 16218 16246 16275 16304 16333 16361 16390 16419 16447 16476 16605 16533 16562 13591 16620 16648 16677 16706 98709 98704 98700 98695 98690 98686 98681 98676 98671 98667 98662 98667 98652 98648 98643 98638 98633 98629 98624 98619 98614 98609 98604 98600 98595 10.767935! 767174! 766414 I 765655 764897 I 764141 j 763386 1 762632 761880 I 761128 10.760378 759629 758882 758136 757390 756646 755903 756161 764421 763681 16734 98590 16763 98585 16792 9S580 16820,98576 16849 16878 16906 16935 16964 16992 17021 17050 198570 98565 98661 98556 98551 98546 98541 98536 17078198531 17107 98526 Tang. 98621 98616 98611 98506 98501 98496 98491 98486 98481 N. COS. N.sine, 17136 17164 17193 17222 17250 17279 17308 17336 17365 80 T).vaveK. TAFLE II. Log. Sinc« and Tar.gcnt.s. (10«-') Kjjturjil Sines. .31 8 9 10 11 I'i 13 14 15 16 17 18 19 20 22 23 24 25 2o 2/ I SO j3i i 32 j 33 j34 I 35 I 3;» ! 3/ I 38 I 39 I 40 41 I 42 i "^^ 4i 4t) 47 48 4y 50 51 52 53 54 55 5b 67 58 69 60 Sine. K 239670 240386 241101 241814 242526 243237 - 243947 244655 245363 246069 246775 ). 247478 248181 248883 249583 250282 250980 251677 252373 253067 253761 ). 254463 255144 255834 256523 257211 257898 258683 259268 259951 260633 >. 26 1314 261994 262b73 263351 264027 264703 265377 266051 266723 267395 1.268065 2t>8 /34 269402 2/0069 270735 271400 272064 272726 273388 274049 1.274708 276367 276024 276681 2^77337 277991 278644 279297 279948 280599 C().«inf'. D. 10" Cosiu 119 119 119 119 118 118 118 118 118 117 117 117 117 117 116 116 116 116 116 116 115 115 115 115 115 114 114 114 114 114 113 113 113 113 113 113 112 112 112 112 112 111 111 1.11 HI 111 111 110 110 110 110 110 110 109 109 109 109 109 109 108 1.993351 993329 993307 993285 993262 993240 993217 993195 993172 993149 993127 •.993104 993031 993059 993036 993013 992990 992967 992944 992921 992898 .992875 992852 992829 992806 992783 992759 992736 992713 992690 992666 .992643 992619 992596 992672 992549 992525 992601 992478 992454 992430 .992406 992382 992359 992335 992311 992287 992263 992239 992214 992190 .992166 992142 992117 992093 992059 992044 992020 991996 991971 991947 D. 10" 3.7 3.7 3.7 3.7 3.7 3.7 3.8 8 8 8 8 8 8 8 8 8 3.8 3.8 3.8 3.8 3.8 3.8 3.8 3.9 3.9 3.9 3.9 3.9 3.9 3.9 3.9 3.9 3.9 3.9 3.9 3.9 ,9 ,9 ,0 ,0 ,0 ,0 3 3 4 4 4 4 4.0 4.0 4.0 4.0 4.1 Tanir. 9.246319 247057 247794 248530 249264 249998 260730 261461 252191 252920 263648 9.254374 265100 255824 256547 257269 257990 258710 269429 260146 260863 9.261578 262292 263005 263717 264428 265138 265847 266555 267261 267967 9.268671 269375 270077 270779 271479 272178 272876 273573 274269 274964 9.275668 276361 277043 277734 278424 279113 279801 280488 281174 281858 9.282642 283225 283907 284588 285268 285947 286<>24 287301 287977 288662 Co tang. D. 1(»"| CoUm/. |.N..«ine.|N. M*, 123 128 123 122 122 122 122 122 121 121 121 121 121 120 120 120 120 120 120 119 119 119 119 119 118 118 118 118 118 118 117 117 117 117 117 116 116 116 116 116 116 116 116 116 116 115 115 114 114 114 114 114 114 113 113 113 113 113 113 112 il0. 753681 752943 752206 751470 750736 750002 749270 748539 747809 747080 74(i352 10.745626 744900 744176 743453 742731 742010 1736598481 17393 98476 17422.98471 17451 984{)6 17479 98461 17508 98455 1753798450 17565 98445 17594 98440 17623 98435 17651 198430 17680 98425 741290 740571 739854 739137 10.738422 737708, 736995 i; 736283 , 735572 |! 7348621; 734153 ] 733445 I 73273911 732033'! 10.731329' 730625 729923 729221 728521 ii 727822 727124 726427 725731 725036 jj 10.7243421 723649 I 722957" 17708 17737 17766 17794 17823 17852 17880 17909 17937 17966 17995 18023 18052 18081 18109 18138 18166 18195 18224 18252 18281 18309 98420 98414 98409 98404 98399 9831*4 98389 98383 98378 98373 98368 9b 36 2 98367 98362 98347 98341 98336 98331 98326 98320 98316 98310 18338 98304 18367 198299 18395198294 18424 18452 18481 98288 96283 98277 18509 98272 18538 98267 79 Dojfreos. 25 :« Log. Sines and Tangents. (11°) Natural Sines. TABLE II. blue. 9.280599 281248 281897 282544 283190 283836 284480 285124 285766 286408 287048 9.287687 288326 288964 289600 290236 290870 291504 292187 292768 293399 9.294029 294658 296286 295913 296539 297164 297788 298412 299034 299655 9.800276 300895 301514 302132 302748 303364 303979 304593 305207 305819 9.306430 307041 307650 308259 308867 309474 310080 310686 311289 311893 (9.312495 313097 313698 314297 314897 315495 316092 316689 317284 317879 Cosine. D. 10' 108 103 108 108 108 107 107 107 107 107 107 106 106 106 106 106 106 105 105 105 105 105 105 104 104 104 104 104 104 104 103 103 103 103 103 103 102 102 102 102 102 102 102 101 101 101 101 101 101 100 100 100 100 100 100 100 100 99 99 99 D. 10 " 1.991947 991922 991897 991873 991848 991823 991799 991774 991749 991724 991699 1.991674 991649 991624 991599 991574 991549 991524 991498 991473 991448 1.991422 991397 991372 991346 991821 991295 991270 991244 991218 991193 1.991167 991141 991115 991090 991064 991038 991012 990988 990960 990934 ' 990908 990882 990855 990829 990803 990777 990750 990724 990697 990671 1.990644 990618 990591 990565 990638 990611 990485 990458 990431 990404 Sine. 4.1 4.1 4.1 4.1 4.1 4.1 4.1 4.2 4.2 4.2 4.2 4.2 4.2 4.2 4.2 4.2 4.2 4.2 4.2 4.2 4.2 4.2 4.2 4.3 4.3 4.3 4.3 4.3 4.3 4.3 3 3 3 3 3 3 3 3 4.3 4.3 4.4 4.4 4.4 4.4 4.4 4.4 4.4 4.4 4.4 4.4 4.4 4.4 4.4 4.4 4.4 4.4 4.5 4.5 4.5 4.5 .288662 289326 289999 290671 291342 292013 292682 293350 294017 294684 295349 .296013 296677 297339 298001 298662 299322 299980 300638 301295 301951 .302607 303261 303914 304567 305218 305869 308519 307168 307815 308463 .309109 309754 310398 311042 311685 312327 312967 313608 314247 314885 9.315523 316159 316795 317430 318064 318697 319329 319961 320592 321222 321851 322479 323106 323733 324368 324983 325607 326231 326853 327475 Co tan R. Degrt^s. 112 112 112 112 112 111 111 111 111 111 111 111 110 110 110 110 110 110 109 109 109 109 109 109 109 108 108 108 108 108 108 107 107 107 107 107 107 107 106 106 106 106 106 106 lOJ 105 105 105 105 105 106 105 104 104 104 104 104 104 104 104 ix)t.ang. (N. sine. N. cos 10.711348 710674 710001 709329 708658 707987 707318 706650 705983 705316 704651 10.703987 703323 702661 701999 701338 700678 700020 699362 698705 698049 10-697393 696739 696086 696433 694782 694131 693481 692832 692185 691537 10-690891 690246 689802 688958 688315 687673 687033 19081 19109 19138 19167 19195 19224 19252 19281 19309 19338 19366 19396 19423 19452 19481 19509 19638 19566 19595 19623 19662 19680 19709 19737 19766 197y4 19828 19861 19880 19908 19937 19966 19994 20022 20061 20079 20108 20136 6863921120166 20193 20222 20260 20279 20307 120336 20364 20393 20421 20460 685753 6861161 10-684477! 683841 683205 ' 682570 681936 681303 680671 i 680^)39 i 679408 'j 20478 678778 i 1 20507 10.678149^120535 6776211 20563 676894 I ! 20692 676267 i 1 20(i20 675642 1 1 20649 675017 j 20677 674393!! 20706 673769 1 1 20734 673147 j 20763 6725261120791 Tang. 98163 98157 98152 98146 98140 98135 98129 98124 98118 98112 98107 98101 98096 98090 98084 98079 )73 98067 98061 98056 98060 98044 98039 98033 98027 98021 98016 98010 98004 979981 31 97992 97987 97981 97975 97969 97963 1 25 97958 97952 97946 97940 97934 97928 97922 97916117 97910 10 7905 97899 97893 97887 97881 97876 97869 97863 97867 97851 97846 97839 97833 97827 97821 97815 N. COS. IV.> 97100 97093 97072 97066 24079 97 06;S 97051 24136 97044 24164 97037 24192 97030 N. COS. N sine. '• 76 Degrees. 36 Log^ Sines and Tangents. (15°) Natural Sines. 1 2 3 4 6 6 7 8 9 la 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 21 28 29 30 31 32 33 34 35 36 87 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 64 55 56 51 58 59 60 Sine. J. 412996 413467 413938 414408 414878 415347 415815 416283 416751 417217 417684 ). 418150 418615 419079 419544 420007 420470 420933 421395 421857 422318 ). 422778 423238 4236^7 424166 424616, 425073 425530 425987 426443 426899 ). 427354 427809 428263 428717 429 L70 429623 430a76 430527 430978 431429 >. 431879 432329 432778, 433226 433675 434122 434569 435016 435462 435908 11.436353 436798, 437242 437686 438129 438572 439014 439456 4C9897 440338 CoiJine, D. 10" I Ijosinc. k. 984944 984910 984876 984842 984808 984774 984740 984706 984672 984637 984603 .984569 984535 984500 984466 984432 984397 984363 984328 984294 984259 .984224 984190 984155 984120 984085 984050 984016 983981 983946 983911 .983875 983840 983805 9837'^0 983755 983700 983664 983629 983594 983558 .983523 983487 983452 983416 983381 983345 983309 983273 983238 983202 .983166 983130 983094 983058 983022 78.5 78.4 78.3 78.3 78.2 78.1 78.0 77.9 77.8 77.7 77.6 77.5 77.4 77.3 77.3 77.2 77.1 77.0 76.9 76.8 76.7 76.7 76.6 76.5 76.4 76.3 76.2 76.1: 76.0 76.0 75.9 75.8 75.7 75.0 75.5 75.4 75.3 75.2 75.2 75.1 75.. 0, 74.9 74.9 74.8 74.7 74 6 74.5 74.4 74.4 74.3 74.2 74.1 74. ft 74.0 73.9 73.8 73.7 73.6 73.6 73.51 982950 982914 982878 982842 Sine. JX W\ Tang. 5.7 6.7 5.7 6.7 5.7 5.7 6.7 5.7 5.7 5.7 5.7 6.7 6.7 5.7 6.7 5.8 6.8 5. .8 5.8 5v8 5.8 5.8 5.8 5.8 6.8 5.8 5.8 5.8 5.8 5.8 5.8 5.8 6.9 5.9 5.9 5.9 5.9 5.9 6.9 5.9 5.9 6.9 6.9 6.9 5.9 5.9 5.9 6.9 6.0 6.0 6.0 6.0 6.0 6. a 6.0 6.0 6.0 6.0 6.0 6.0 9.428062 428557 429062 429561 430070 430573 431075 431577 432079 432580 433080 9.433580 434080 434579 435078 435576 436073 436670 437067 437563 438059 9.438554 439048 439543 440036 440529 441022 441614 442006 442497 442988 9.443479 443968 444458 444947 445.435 44a923 446411 446898 447384 447870 1.448356 448841 449326 449810 46Q294 460777 45L260 451743 452226 452706 .453187 453668 454148 464628 465107 465586 456064 456542 457019 457496 Cotanjj. a iiy; 84.2 84.1 84.0 83.9 83.8 83.8 83.7 83.6 83.5 83.4 83.3 83.2 83.2 83.1 83.0 82.9 82.8 82.8 82.7 82.6 82.6 82.4 82.3 82.3 82.2 82.1 82.0 81.9 81.9 81.8 81.7 81.6 81.6 81.5 81.4 81.3 81.2 81.2 81.1 81.0 80.9 SO. 9 80.8 80.7 8ft. 6 80.6 80.5 80.. 4 80.3 80.2 80.2 80. 1 80.0 79.9 9.9 79.8 79.7 9.6 79.6 79.5 Lotuup;. N. sine. N. COS. 96593 96585 96578 96570 96562 96665 96547 96640 96532 96524 10.571948! 25882 571443 i 269 10 570938 ': 2593 670434' 2596 6r>9930;i25994 569427 1 126022 668925! 126050 5684231126079 567921 126107 667420 ,26135 5669201126163 96517 10.566420: '26191 5669201:26219 5654311126247 564922 1126275 5644241126303 663927 1 126331 5634301126359 562933 I 26387 562437 [26416 561941 ! 26443 10.561446!! 26471 560952 1 1 26500 560457 1 26528 26556 S6584 26612 26640 26668 26696 559964 559471 658978 558486 657994 557503 557012 1 1 26724 10-.666521H 26762 556032 1126780 5555421126808 555053!! 26836 654566 554077 553589 553102 552616 652130- 0.. 561644 551159 26864 26892 26920 26948 26976 27004 27032 37060 650674 127088 96261 650190 127116 549706 127144 549223 127172 5487401127200 648267 ; 27228 96222 547776 ij 27256 547294 1 1 27284 10.5468131; 27312 546332 545852 545372 644893 544414 643936 543458 542981 542504 Tang. 96509 96502 96494 96486 96479 96471 96463 96456 96448 96440 96433 96425 96417 96410 96402 96394 96386 96379 96371 96363 96355 96347 96340 96332 96324 96316 96308 96301 96293 96285 96277 96269 96253 96246 96238 96230 96214 96206 961.98 7340 96190 27368 96182 27396 96174 27424 96166 27452 96158 27480 96150 27508 96142 27536 96134 27564 96126 N. cos.lN.sine. 74 Degrees. TABLE II. Log. Sinefl and Taflgcnts. (16°) Natural Sines. 37 Hine. ).4t0338 440778 441218 441658 442096 442636 442973 443410 443847 444284 444720 ). 445165 445590 446026 446469 446893 447326 447759 448191 448623 449054 ). 449486 449915 450346 450776 461204 451632 452060 452488 452916 453342 ). 453768 454194 454619 455044 455469 456893 456316 456739 467162 457584 3 468006 458427 458848 459268 459688 460108 460527 460946 461364 461782 9.462199 462616 463032 463448 463864 464279 464694 465108 466522 465935 U. 10" Coiiiio. D. lu Cosine. 73,4 73.3 73.2 73-1 73.1 73.0 72.9 72.8 72.7 72.7 72.6 72.6 72.4 72.3 72.3 72.2 72.1 72.0 72.0 71.9 71.8 71.7 71.6 71.6 71.6 71.4 71.3 71.3 71.2 71.1 71.0 71.0 70.9 70.8 70.7 70.7 70.6 70.5 70.4 70.4 70.3 70.2 70.1 70.1 70.0 69.9 69.8 69.8 69.7 69.6 69.5 69.5 69.4 69.3 69.3 69.2 69.1 69.0 69.0 68.9 9.982842 982805 982769 982733 982696 982660 982624 982587 982551 982514 982477 982441 982404 982367 982331 982294 982257 982220 982183 982146 982109 9.982072 982036 981998 981961 981924 981886 981849 981812 981774 981737 9.981699 981662 981625 981587 981649 981512 981474 981436 981399 981361 9.981323 981285 981247 981209 981171 981133 981096 981057 981019 980^)81 ,980942 980904 980866 980827 980789 980750 980712 980S73 980635 980596 Sino. 6.0 6.0 6.1 6.1 6.1 6.1 6.1 6.1 6.1 6.1 6.1 6.2 6.2 6.2 6.2 6.2 6.2 6.2 6.2 6.2 6.2 6.2 6.2 6.2 6.3 6.3 6.3 6.3 6.3 6.3 6.3 6.3 6.3 6.3 6.3 6.3 6.3 6.3 6.4 6.4 6.4 6.4 6.4 6.4 6.4 6.4 6.4 6.4 6.4 6.4 6.4 6.4 Tanj;. 9.457496 467973 468449 458925 459400 469875 460349 460823 461297 461770 462242 462714 463186 463658 464129 464699 466069 466539 466008 466476 466946 467413 467880 468347 468814 469280 469746 470211 470676 471141 471605 472068 472632 472995 473457 473919 474381 474842 475303 476763 476223 9 476683 477142 477601 478059 478517 478976 479432 479889 480345 480801 481257 481712 482167 482621 483076 483529 483982 484436 484887 485339 1>. lo' Coian;:. ; N.ainc. N. coa 79.4 79.3 79.3 79 2 79.1 79.0 79.0 78.9 10.542504! 27564 5420271' 27692 641651 i 27620 Cotan05 644310 644715 645119 9.545524 545928 546331 646735 647138 547540 547943 548345 648747 649149 9.540650 549951 550352 560752 551162 661552 551962 552351 j 562750 553149 9.553548 553946 554344 554741 656139 665536 555933 556329 556725 557121 1.557517 667913 558308 658702 659097 659491 659885 660279 560673 561066 "CotangT 68.4 68.3 68.8 68.2 68.2 68.1 68.1 68.0 68.0 67.9 67.9 67.8 67.8 67.7 67.7 67.6 67.6 67.5 67.5 67.4 67.4 67.3 67.3 67.2 67,2 67.1 67.1 67.0 67.0 66. 66. 66. 66.8 66.7 66.7 66.6 66.6 66.6 66.5 66.5 66.4 66.4 66.3 66.3 66.2 66.2 66.1 66.1 66.0 66.0 65.9 66.9 66.9 65.8 65.8 65.7 65.7 66.6 65.6 65.6 Cotan);. ijiN'. Binc.|;>i. COS. I 32557 32584 32612 10.463028 462618 j 462208 j 4617981 1 32639 461389! 32667 460980' 32694 460571 I 32722 460163 I 459765 i 459347 I 458939 i 10.4585321 458125 j 467719 I 457312 I 456900 I 450501 ! 456095 ! 455690 1 455285 454881 I 10.454476! 464072 j 453669 I 453265 ! 452862 I 452460 ! 452057 451655 451253 450861 LO. 450450 460049 ! 449648 449248 448848 448448 448048 32749 32777 32804 32832 32859 32887 32914 32942 32969 32997 33024 33051 33079 33106 33134 .33161 33189 33216 33244 33271 33298 33326 33353 33381 33408 33436 33463 33490 33518 33645 33573 447649 1 1 33600 447250 1 1 33627 446851 1 1 33665 33682 10.446452 446054 445656 445259 444861 444464 83710 33737 33764 33792 33819 444067 1133846 443671 I [33874 4432761133901 33929 3395b 33983 34011 34038 34065 34093 84120 3414 4393271134175 438934 1134202 Tanjr. !lN. cos. N.sine, 94552 94542 94533 94523 94514 94504 94495 94485 94476 94466 94457 94447 94438 94428 94418 94409 94399 94390 94380 94370 94361 94351 94342 94332 94322 94313 94503 94293 94284 94274 94264 94254 94245 94235 94225 94215 94206 94196 94186 94176 94167 94167 94147 94137 94127 94118 94108 94098 94088 94078 94068 94058 94049 94039 94029 94019 94009 93999 93989 93979 93969 70 Degre'js. TABLE II. Log. i;iire« and TanReiitn. (20<=>) >aluraJ Siuoh. 41 Sine. p. W 534052 534399 534745 535092 535438 535783 536129 55G474 536818 537163 537507 537851 538194 538538 538880 539223 539565 539907 540249 540590 540931 541272 541618 541953 542293 542632 542971 543310 543649 543987 544325 9,544663 545000 545338 545674 546011 546347 546683 547019 547354 547689 1.548024 548359 548693 549027 549360 649693 550026 550359 550692 551024 1.551356 551687 552018 552349 552680 653010 553341 553670 554000 554329 Cosine. 57.8 57.7 67.7 67-7 67.6 57.6 57.5 57.4 67.4 57.3 57.3 57.2 57.2 67.1 57.1 67.0 57.0 56.9 56.9 66.8 56.8 56.7 66.7 56.6 56.6 56.5 56.6 58.4 66.4 56.3 56.3 56.2 66.2 66.1 56.1 66.0 56.0 55.9 55.9 56.8 55.8 55.7 65.7 65.6 65.6 55.6 65.5 55.4 55.4 55.3 55.3 56.2 55.2 55.2 55.1 55.1 55.0 55.0 54.9 54.9 Oosiui*. ). 972986 972940 972894 972848 972802 972755 97270:1 972663 972617 972570 972524 ). 972478 972431 972385 972338 972291 972245 972198 972161 972105 972058 >. 972011 971964 971917 971870 971823 971776 971729 971682 971635 971688 (.971540 971493 971446 971398 971351 971303 971256 971208 971161 971113 >. 971 066 971018 970970 970922 970874 970827 970779 970731 970683 970635 1.970586 970538 970490 970442 970394 970345 970297 970249 970200 970162 D. 10' Sine. 7.7 7.7 7.7 7.7 7.7 7.7 7.7 7.7 7.7 7.7 7.7 7.7 7.8 7.8 7.8 7.8 7.8 7.8 7.8 7.8 7.8 7.8 7.8 7.8 7.8 7.8 7.8 7.9 7.9 7.9 7.9 7.9 7.9 7.9 7,9 7,9 7.9 7.9 7.9 7.9 7.9 8.0 8.0 8.0 8.0 8.0 8.0 8.0 8,0 8.0 8.0 8.0 8,0 8,0 8,0 8.0 8,1 8.1 8.1 JTunj^ 9.561066 661459 561851 662244 662636 563028 563419 663811 664202 564592 564983 9,565373 565763 566163 666542 566932 667320 567709 668098 568486 568873 9.569261 569648 670035 570422 570809 571196 571581 571967 672362 572738 9.573123 573507 573892 574276 574660 576044 675427 676810 576193 576676 9.576968 577341 677723 678104 578486 578867 579248 679629 580009 580389 9.580769 681149 581528 581907 582286 582665 583043 583422 583800 584177 Cotanp. D. 10 65.5 65.4 65.4 65 3 66.3 65.8 65.2 65.2 65.1 65.1 65.0 65.0 64.9 64,9 64.9 64.8 64.8 64.7 64.7 64.6 64.6 64.6 64,5 64,5 64,4 64,4 64.3 64.3 64,2 64.2 64.2 64.1 64.1 64.0 64.0 63,9 68.9 63,9 63.8 63.8 63.7 63.7 63,6 63.6 63.6 63,5 63,6 63,4 63.4 63,4 63,3 63,3 63.2 63,2 63,2 63,1 63.1 63.0 63,0 62.9 t'otang. >. Kine. N, cos. 10.438934 438541 438149 437756 437364 436972 436581 436189 436798 435408 435017 10.434627 434237 433847 433458 433068 432680 482291 431902 431514 431127 1 a. 430739 430362 429965 429678 429191 428806 428419 428033 427648 427262 10.426877 426493 426108 425724 425340 424956 424573 424190 423807 423424 10.423041 422659 422277 421896 421614 421183 420762 420371 419991 419611 10.419231 418851 418472 418093 417714 417336 416957 416578 416200 415823 Tanp;. , 3420-: 1 34229 i 34257 134284 134311 34339 34366 34393 34421 34448 34475 34503 84530 34657 34584 34612 84639 34666 34694 34721 84748 84776 34803 34830 84857 84884 34912 349G9 34966 34993 35021 35048 85076 35102 85130 8515 35184 35211 35289 35266 35293 35320 3584 35375 35402 35429 35456 35484 35511 35538 35565 3559i; 35619 3564'i 85674 35701 3572b 35755 3578i. 85810 35837 93077 93667 93657 93647 93637 93626 93616 93606 93596 93585 i)3576 93565 93555 93644 93534 93524 93514 93503 93493 93483 98472 J3462 93452 93441 93431 93420 98410 93400 93f,fc9 93379 98368 93368 i N. COP. N.sine. 60 I 59 68 57 56 55 54 63 L2 ol 50 49 48 47 46 45 44 43 42 41 40 39 38 37 86 85 84 33 32 31 30 29 28 27 26 25 24 23 22 21 :.0 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 09 Degrcps. 42 Log. Sines and Tangents. (21") Natural Sines. TABLE IL Sine. 9.554329 554658 554987 555315 555(J43 555971 556299 556626 556953 557280 557()06 9.557932 558258 558583 558909 559234 55y558 559883 560207 560531 560855 9.561178 561501 561824- 662146 562468 562790 563112 563433 563755 664075 9.564396 564716 665036 565356 565676 565995 566314 666632 566951 567269 9.567587 567904 568222 568539 568856 569172 569488 569804 670120 570435 9.570751 571066 671380 571695 572009 572323 672636 672950 673263 673575 Cosine. D. 10"| Cosine. 64.8 64.8 64.7 54.7 54.6 64.6 54.5 54.5 54.4 54.4 54.3 54.3 54.3 54.2 54.2 54.1 54,1 54.0 54.0 53.9 53.9 53.8 53.8 63.7 53.7 53.6 53 53 53 53 53 63 53 53 63 53 53.1 53.1 53,1 63.0 53.0 52,9 52.9 52.8 52.8 62.8 52.7 52.7 52.6 52.6 52.5 52.6 52.4 62.4 52.3 52.3 52.3 52.2 52.2 52.1 9.970152 970103 970055 970006 969957 969909 969860 9698 U 969762 969714 969665 9.969616 969567 969518 969469 969420 969370 969321 969272 969223 969173 (9.969124 969075 969025 968976 968926 968877 968827 968777 968728 968678 968628 968578 968528 968479 968429 968379 968329 968278 968228 968178 968128 968078 968027 967977 967927 967876 957826 967775 967725 967674 9.967624 967573 967622 967471 967421 967370 967319 967268 967217 967166 D. 1U"| Tang. 8 8 8.2 8.2 8,2 8.2 8.2 8.2 8.2 8.2 2 2 2 2 2 2 3 8.3 8.3 8.3 8.3 8.3 8.3 8.3 8.3 8.3 8.3 8.3 8.3 8.3 8.4 8.4 8.4 8.4 8.4 8.4 8.4 8.4 8.4 8.4 8.4 8.4 8.4 8.4 8.6 8.6 8.6 8.6 8.6 8.6 8.6 9.584177 584555 684932 585309 686686 686062 686439 586815 687190 587566 587941 9.588316 688691 5890d6 689440 689814 590188 590562 690935 691308 591681 9.692054 59242*6 592798 593170 593542 593914 594285 594656 595027 595398 595768 596138 696508 596878 597247 697616 597985 698364 598722 699091 9.599459 599827 600194 600562 600929 601296 601662 602029 602395 602761 1.603127 603493 603858 604223 . 604588 604953 605317 605682 60)046 6 06410 Cotang. D. 10"| Cotang. I N .sine.lN. cos. 62.9 62.9 62.8 62 62.7 62.7 62 62 62 62 62 62 62 62 62 62 62.3 62.2 62.2 62.2 62.1 62,1 62.0 62.0 61.9 61.9 61.8 61.8 61.8 61.7 61.7 61.7 61.6 61.6 61.6 61.6 61.6 61.5 61.4 61.4 61.3 61.3 61.3 61.2 61.2 61.1 61.1 61.1 61,0 61.0 61.0 60.9 60.9 60.9 60,8 60.8 60.7 60.7 60.7 60.6 10.415823 415445 416068 414691 414314 . 413938 413561 413185 412810 412434 412059 10.411684 411309 410934 410560 410186 409812 409438 409065 408692 408319 10.407946 407574 407202 406829 406458 406086 405715 405344 404973 404602 10.404232 403862 403492 403122 402753 402384 402015 401646 401278 400909 10.400541 400173 399806 399438 399071 398704 398338 397971 397605 397239 '■' 10.396873, 396507 I 396142 : 395777 : 395412 395047 '■ 394683 ; 394318 : 393954 393590 ■ 36837 93358 36864193348 35891 193337 35918 35945 36973 36000 36027 36054 36081 36108 36135 36162 36190 93222 36217 93211 i 36244 93201 36271 93190 36298 93180 36325 93169 36461 36488 36515 36542 93084 ! 36569 93074 ij 36596 93063 1 136623 93052 36731 36758 36785 j 1 368 12 92978 1 136839 92967 ! 136867 36894 92946 ! 36921 93327 93316 93306 93295 93286 93274 93264 93253 93243 93232 36352 93159 36379^3148 36406 93137 36434 93127 93116 93106 93095 I 36650 93042 36677193031 36704 93020 93010 92999 92988 92966 92935 Tang. I N. i 136948 92926 ! 36975 92913 137002 92902 137029 92892 i 37056 92881 ' 37083 92870 137110 92859 : 37 137 92849 37164 92838 3719192827 : 37218 92816 37245 92805 37272 92794 i 37299 92784 137326 92773 37353 92762 37380 92751 137407 92740 j 37434 92729 13746192718 Log. Sines and Tangents. (22'') Natural Sines. 43 Sine. 9,573675 574200 674512 574824 576136 575447 575758 576069 576379 576689 ►.576999 677300 577618 577927 578236 578545 678853 579162 579470 679777 1.580085 580392 580699 581005 681312 681618 681924 682229 582636 582840 .583145 683449 583754 684058 584361 584665 584968 686272 685574 585877 .586179 686482 686783 587085 587386 687688 687989 588289 588690 688890 i. 6891 90 589489 589789 590088 E90387 590686 590984 691282 591680 591878 Cosine. D7W> 52.1 52.0 52.0 51.9 51.9 61.9 51.8 51.8 51.7 51.7 51.6 51.6 51.6 61.6 51.5 61.4 51.4 51.3 51.3 51.3 51.2 51.2 51.1 61.1 51.1 51.0 51.0 60.9 60.9 50.9 50.8 50.8 60.7 60,7 50.6 50.6 60.6 60.6 50.5 50.4 60.4 60.3 60.3 60.3 50.2 60.2 60.1 50.1 50.1 60.0 50.0 49.9 49.9 49,9 49.8 49.8 49.7 49.7 49.7 49.6 Cosine. .967166 967115 967064 967013 966961 966910 966859 966808 966756 966705 966653 .966602 966550 966499 966447 966395 966344 966292 966240 966188 966136 .966085 966033 965981 965928 966876 965824 965772 965720 965668 965815 .965563 966511 965458 966406 965353 965301 965248 965195 965143 965090 965037 964984 964931 964879 964826 964773 964719 964666 964613 964560 .964607 964464 964400 964347 964294 964240 964187 964133 964080 964026 Sine. D. 10" Tan" 8.5 8,5 8.5 a,5 8.5 &.5 8.6 8.5 8.6 8.6 8.6 8.6 8.6 8.6 8.6 8.6 8.6 8.6 8.6 8.6 8.6 8.7 8.7 8.7 8.7 8.7 8.7 8.7 8.7 8.7 8.7 8.7 8.7 8.7 8.7 8.8 8.8 8.8 8.8 8.8 8.8 8.8 8.8 8.8 8.8 8.8 8.8 8.8 8.9 8.9 8.9 8.9 8.9 8.9 8.9 8.9 8.9 8.9 8.9 8.9 ). 606410 606773 607137 607500 607863 608225 608588 608950 609312 609674 610036 ). 610397 610759 611120 611480 611841 612201 612561 612921 613281 613641 >. 614000 614369 614718 615077 615435 615793 616161 616509 616867 617224 1.617582 617939 618295 618652 619008 619364 619721 620076 620432 620787 1.621142 621497 621852 622207 622561 622915 623269 623623 623976 624330 1.624683 626036 625388 625741 626093 626445 626797 627149 627601 627852 Co tang. 67 Degrees. D. 10' 60.6 60.6 60.5 60.6 60.4 60.4 60.4 60.3 60.3 60.3 60.2 60.2 60.2 60.1 60.1 60.1 60.0 60.0 60.0 59.9 59.9 59.8 59.8 59.8 59.7 59.7 69.7 59.6 69.6 59.6 59.5 59.5 69.5 69.4 59.4 59.4 59.3 59.3 59.3 59.2 59.2 59.2 59.1 59.1 59.0 59.0 59.0 58.9 68.9 68.9 58.8 58.8 58.8 58.7 58.7 58.7 58.6 58.6 58.6 58.5 Cotang. ; N . Hine.l N. 10.393590 393227 392863 392500 392137 391775 391412 391050 390688 390326 389964 10.389603 389241 388880 388520 388169 387799 387439 387079 386719 386359 10.386000 385641 385282 384923 384565 384207 383849 383491 383133 382776 10-382418 382061 381706 381348 380992 380636 380279 379924 379568 379213 10-378858 378503 378148 377793 377439 371085 376731 376377 376024 375670 10.375817 374964 374612 374259 373907 373665 373203 372851 372499 372148 f^iig." 37461 37488 37515 37542 37569 37695 37622 37649 37676 37703 37730 37757 37784 92718 92707 92697 92686 92675 92664 92653 92642 92631 92620 92609 92598 92587 3781192676 37838 37865 37892 37919 37946 37973 37999 38026 38053 38080 38107 38134 38161 38188 38241 38268 38295 38822 38349 38376 38403 38430 92565 92554 92543 92532 92521 92510 92499 92488 92477 92466 92456 j 36 92444 92432 92421 38215 92410 92399 92388 92377 92366 92355 92343 92332 92321 38456 92310 38483192299 38510J92287 38537 92276 38564 92265 38591 38617 38644 38671 38698 38725 92254 92243 92231 92220 92209 92198 38778 92175 38805 92164 38832 92152 3885992141 38886i92130 38912 38939 38966 38993 39020 39046 39073 N. CO?. 02119 92107 92096 92085 92073 92062 [92050 >^Bine. 35 I 34 03 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 i 7 : 6 5 4 , 3 2 1 44 I>og. Sine« and Tangents. (23°) Natural Sines. TABLK II. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1() 17 18 19 20 21 22 23 24 25 26 27 28 29 30 I 31 i 32 33 34 85 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 S ine. 9.591878 592176 592473 592770 693067 593303 593659 593955 594251 594547 594S42 9.595137 595432 595727 598021 596315 696609 596903 597196 597490 597783 9.598075 598368 598660 598952 599244 599536 599827 600118 600409 600700 9.600990 601280 601570 601860 602150 602439 602728 603017 603395 603594 9.603882 604170 604457 604745 605032 605319 605606 605892 606179 60S465 9.606751 607030 607322 607607 607892 608177 608461 608745 609029 609313 D W- Cotinu 49.6 49.5 49.6 49.5 49.4 49.4 49.3 49.3 49.8 49.2 49.2 49.1 49.1 49.1 49.0 49.0 48.9 48.9 48.9 48.8 48.8 48.7 48.7 48.7 48.6 48.6 48.6 48.6 48.6 48.4 48.4 48.4 48.3 48.3 48.2 48.2 48.2 48.1 48.1 48.1 48.0 48.0 47.9 47.9 47.9 47 8 47.8 47.8 47.7 47.7 47-6 47.6 47.6 47-5 47-6 47.4 47.4 47.4 47.3 47.3 Oosinc. 1.964026 963972 963919 963865 963811 963757 963704 963650 963596 963542 963488 .963434 963379 963325 963271 963217 963163 963108 963054 962999 962945 .962890 962836 962781 962727 962672 962617 962562 962508 962453 962398 .962343 962288 962233 962178 962123 962037 962012 961957 961902 961846 .961791 961735 961680 961624 961569 961513 961458 961402 961346 961290 .961235 961179 961123 961037 961011 960955 960899 960843 960786 960730 Sine. Ta ng. 9.627852 628203 628564 628906 629255 629606 629956 630306 6306.56 631005 631355 9.631704 632053 632401 632750 633098 633447 633795 634143 634490 634838 ,635l8o 635532 635879 636226 636572 636919 637265 637611 637956 638302 638647 638992 639337 639682 640027 640371 640716 641060 641404 641747 9.642091 642434 642777 643120 643463 643806 644148 644490 644832 645174 .645516 645857 640199 646540 646881 647222 647662 <347903 648243 648583 Cotanii. ^6 De^nvF. 58.5 58.5 58.5 58.4 58.4 68 58 58 58 58 58 58 58.1 68.1 58.1 58.0 58.0 58.0 57.9 57.9 57.9 57.8 67.8 67.8 57.7 67.7 57.7 57.7 57.6 57.6 57.6 57.5 67.5 57.5 57.4 67.4 57.4 67.3 57.3 67.3 57.2 57.2 57.2 57.2 57.1 57.1 67.1 57.0 •57.0 57.0 66.9 56.9 66.9 m.o 56.8 66.8 56.8 66.7 56.7 56.7 Cotang. I N. sine. N. COS. 10.372148 139078 92050 371797 '139100 3714461 [39127 371095!' 39153 370746 1139180 i 39207 1 39234 ! 39260 370394 370044 369694 369344 368996 368646 1 39341 10.368296 '39367 367947 i 39394 367599! 139421 92039 92028 92016 92005 91994 91982 91971 39287 91959 39314 91948 91936 91925 91914 91902 367250 II 39448 91891 366902 1 1 39474 '91879 366553 r 39501 366205 ii 39528 366857 j 139555 365610! 1 39581 3651621139608 91822 10.364815 '139636 364468 1 139661 364121 ! 363774 ! 363428 ! 363081 I 362735 i 362389 ' 362044 1 361698 ! 39741 39795 39848 39875 91868 91856 91845 91833 91810 91799 39688 91787 39715 91775 91764 39768 91752 91741 39822 91729 91718 91706 10.361353 !i 39902 91694 361008 II 39928191683 360G63 'I 39955 19 1671 360318 l| 39982 91660 91648 91636 [40062 91625 ,40088 91613 359973! 1 40008 359629 II 40035 359284 358940 358596!|40115|91601 358253 ij 40141 bloiiO 1 0. 357909 I j 401 68 b 167 8 357o66 ' 867223 I 356880 1 356537 I 366194! 14030) 40195191566 40221 191555 40248 191543 40275 91531 91519 365862 1140328 9 16(j8 36.5610 j 40355 91496 355168 j 40381 354826 10.354484 364143 353801 363460 353119 362778 352438 352097 351757 351417 40408 40434 91484 91472 91461 40461 91449 40488 40541 40567 40594 40621 91378 91366 91355 Tang, ji N. cop. N.siTic 40647 40674 91437 40514 91425 91414 91402 91390 60 59 58 57 66 55 54 53 52 61 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 •■il fcO 29 •2b 27 26 25 'J 4 23 22 21 2U 19 18 17 16 15 14 Ki 12 n 10 9 8 7 6 5 4 2 1 Log. Sines and Tangents. (2-40) Natural Sines. 45 Sine. 9.609313 609597 2 609880 3 610164 4 610447 5 610729 6 611012 7 611294 8 611576 9 611858 10 612140 11 9.612421 12 612702 13 612983 14 613264 15 613545 16 613825 17 614105 18 614385 19 614666 20 614944 21 9.615223 22 615502 23 615781 24 616060 25 616338 26 616616 27 616894 28 617172 29 617450 30 617727 31 9.618004 32 618281 83 618558 34 618834 36 619110 36 619386 37 619662 38 619938 39 620213 40 620488 41 9.620763 42 621038 43 621313 44 621587 46 621861 46 622135 47 622409 48 622682 49 622956 50 623229 51 9.623512 52 623774 63 624047 54 624319 56 624591 56 624863 57 626136 68 625406 59 625677 60 625948 D. 10' 47.3 47.2 47.2 47.2 47.1 47.1 47.0 47.0 47.0 46.9 46.9 46.9 46.8 46.8 46.7 46.7 46.7 46.6 46.6 46.6 46.6 46.6 46.5 46.4 46.4 46.4 46.3 46.3 46.2 46.2 46.2 46.1 46.1 46.1 46.0 46.0 46.0 45.9 45.9 45.9 45.8 45.8 45.7 45.7 46.7 46.6 46.6 45.6 46,5 45.5 46.6 45.4 45.4 45.4 45.3 45.3 46,3 45.2 46.2 45.2 Cosine. 1.960730 960674 960618 960561 960505 960448 960392 960336 960279 960222 960165 1.960109 960052 959996 959938 959882 959825 959768 959711 959664 959596 i. 959539 969482 959425 959368 959310 959263 969195 969138 959081 969023 .958966 D. ic/ 958850 958792 958734 958577 958619 958661 958603 958445 958387 958329 958271 958213 958154 958096 968038 957979 957921 957863 967804 957746 967687 957628 957570 957511 957462 957393 957335 957276 9.4 9.4 9.4 9.4 9.4 4 4 4 4 4 4 6 6 5 5 9.6 9.6 9.5 9.5 9.5 9.5 9.5 9.5 9.6 9.6 9.6 9.6 9.6 9.6 9.6 9.6 9.6 9.6 9.6 9.6 9.6 9.6 9.6 9.7 9.7 9.7 9.7 9.7 9.7 9.7 9.7 9.7 9.7 9.7 9.7 9.7 9.8 9.8 9.8 9.8 9.8 9.8 9.8 9.8 Tanft. 9.648583 648923 649263 649602 649942 660281 650620 660959 651297 651636 651974 9.652312 652650 652988 653326 653663 654000 654337 654174 665011 666348 655684 656020 666356 656692 657028 657364 667699 658034 658369 658704 9.659039 659373 659708 660042 660376 660710 661043 661377 661710 662043 9.662376 662709 663042 663375 663707 664039 664371 664703 665035 666366 3,666697 666029 666360 666691 667021 667352 667682 6()8013 668343 668672 D. W 66.6 66.6 66.6 56 6 66.5 66.6 59.6 66.4 56.4 66.4 66.3 56.3 56.3 56.3 56.2 56.2 66.2 56.1 56.1 56.1 66.1 66.0 56.0 56.0 56.9 66.9 55.9 55.9 65.8 55.8 Uotang. Cotang. 65 55 65.6 55.6 55.6 65.6 55.5 55 55 55 56 55 55 56 56.3 55.3 55.2 65.2 55.2 65.1 55.1 56.1 56.1 66.0 55.0 56.0 10.351417 351077 350737 350398 350058 349719 349380 349041 348703 348364 348026 10.347688 347350 347012 346674 346337 346000 345663 345326 344989 344652 10.344316 343980 343644 343308 342972 342636 342301 341966 341631 341296 10.340961 340627 I 340292 i 339958 339624 339290 338967 338623 338290 337957 10.337624 337291 336958 336625 336293 336961 336629 335297 334965 334634 10.334303 333971 333620 333309 332979 332648 332318 331987 331657 331328 N. sine. N. cos. 40674 40700 40727 40753 40780 40806. 40833 40860 40886 40913 40966 40992 41019 41045 41072 41098 41161 41178 41204 41231 41257 91365 91343 91331 91319 91307 91295 91283 91272 91260 91248 91236 91224 91212 91200 91188 91176 91164 41125 91162 91140 91128 91116 91104 91092 41284 91080 4131091068 41337 "91056 Tang. 41363 41390 41416 41443 41469 41496 41522 90972 41549 41675 90960 90948 41602 90936 41628 90924 41655 41681 41707 41734 41760 41787 41813 41840 41866 41919 41946 41972 41998 42024 91044 91032 91020 91008 90996 90984 90911 90899 90887 90875 90863 90861 90839 90826 90814 41892 90802 90790 90778 90766 90753 90741 4205190729 42077 90717 4210490704 42130190692 42166J90680 42183 90668 42209 90656 42235 90643 42262 90631 N. COS. N. sine. 65 Degrees. 46 Log. Siucs and Tant^enu?. (ti6 ) N^axiiral Sines. TAULE II. Cotaug. I N .siuc. N 10.331327! 330998 330Jei8 :' 380339 330009 329(i80! 329351;: 32902311 328694:; 328037;; 10.327709 I 3273811 327053;; 32672611 326398 ]> 326071 ': 325743 1 ; 32541611 325090! I 3247631; 10.324436;! 3241 10 j I 323784;; 3234571; 323131 i 322806 ji 322480 it 32215411 321829 li 321504 10.321179 320854 320529 3202061! 319880 i 319556 I 319232]! 31890811 318684!! 31826011 10.317937 i 317613 31729011 316967;: 316644/ 316321 i! 31599911 3156761: 3153541! 31503211 10.314710;; 314388!; 814066 i; 313745!; 313423;; 3131021: 312781 31246a 312139 311818 .Sina. D. 10" 9.625948 626219 626490 6267i)0 627030 627300 627570 627840 628109 628378 628647 9.628916 629185 629453 629721 629989 630257 630524 630792 631059 631326 9.631593 631859 632125 632392 632658 632923 633189 633454 6337191 1: 633984 9.634249 634514 634778 635042 635308 635570 635834 636097 636360 636623 9.636886 637148 637411 637673 637935 638197 638458 638720 638981 639242 9.639503 639764 640024 640284 640644 640804 641064 641324 641684 641842 Cosine. I 45, 45, 45, 45. 45, 45, 44. 44, 44, 44, 44. 44. 44. 44. 44. 44. 44. 44. 44. 44. 44. 44. 44. 44. 44. 4A. 44. 44. 144. 9.8 9.8. 9.& 9,8 9.8 9.8 9,9 9.9 9.9 9.9 9 9 Cosine. |0. 10' 9,957276 9572 17 967158 957099 957040 956981 956921 956862 956803 956744 956684 9.956625 956566 9565 J6 956447 956387 956327 956268 956208 956148 956089 9.956029 955969 956909 955849 955789 956729 955669 955609 955548 955488 9.955428 955368 955807 955247 955186 955126 9550G6 9550U5 95494-4 964883 9.954823 954762 964701 954640 954579 964518 954457 954396 954335 954274 9.954213 954152 964090 954029 953968 953906 953845 953783 953722 953660 Sin-'. 9 9 9.9 9 9 9 9 9.9 9.9 9.9 10.0 10.0 10.0 10.0 10.0 10.0 10.0 10.0 10.0 10.0 10.0 10.0 10.0 10.1 10.1 10.1 10.1 10.1 10.1 10.1 10.1 10.1 10.1 10.1 10.1 10.1 10.1 10.1 10.2 10.2 10.2 10.2 10.2 10.2 10.2 10.2 10.2 10.2 10.2 10.2 10.2 10.3 Tan g. 9.668673 669002 669332 669661 669991 670320 670649 670977 671306 671634 671963 9.672291 672619 672947 673274 673602 073929 674257 674584 674910 676237 9.675664 675890 676216 676543 676859 677194 677520 677846 678171 678496 9.678821 679146 67947J „679795 680120 680444 680768 681092 681416 681740 9.682063 682387 682710 683083 683356 683679 684001 684324 684646 684968 9.686290 085612 685934 686256 686577 686898 687219 687640 687861 6 88182 Co tan J. 64 Dcgrwfl. D. 10" 42262 4228fc 42816 4-J841 4236, 4239-1 424:.i0 42446 42478 42499 42525 4255'..' 42578 42604 42631 42667 42683 42709 4273() 42762 42788 42815 42841 42867 42894 42920 42946 42972 42999 43025 43051 43077 43104 43130 43166 43182 43209 48235 43261 43287 43313 43340 43366 43392 43418 43445 43471 43497 43528 43549 43575 43602 43G28 43654 43680 43706 43738 43759 43785 43811 43837 Tane. 90631 ^iHJlo J0594 90669 90557 90545 90532 )0520 90507 90495 90483 90470 90458 90446 90433 90421 90408 90396 90383 90371 90358 90346 90334 90321 9030y 90296 90284 90271 90259 90246 90233 90221 90208 90196 90188 90171 90158 90146 90188 90120 901 08 90095 90082 90070 90057 90045 90082 90019 90007 89994 89981 89968 89956 89943 89930 89918 89905 89892 89879 Log. Sine* and Tangents. (26°) Natural Sines. 47 Sine. D. 10" Cosine. 641842 642101 642360 642618 642877 643135 643393 643660 643908 644166 644423 644680 644936 645193 645450 645706 645962 646218 646474 64G729 646984 647240 647494 647749 648004 648268 648512 648766 649020 649274 649527 649781 650034 650287 650639 650792 651044 651297 651549 651800 652052 9.652304 652665 652806 653057 653308 653558 653808 654059 654309 654558 9.654808 665058 655307 655556 655805 656054 656302 656651 656799 657047 43.1 43.1 43.1 43.0 43.0 43.0 43.0 42.9 42.9 42.9 42.8 42.8 42.8 42.7 42.7 42.7 42.6 42.6 42.6 42.5 42.5 42.5 42.4 42.4 42.4 42.4 4a. 3 42.3 42.3 42.2 42,2 42.2 42.2 42.1 42.1 42.1 42.0 42.0 42.0 41.9 41.9 41.9 41.8 41.8 41.8 41.8 41.7 41.7 41.7 41.6 41.6 41.6 41.6 41.5 41.5 41.5 41.4 41.4 41.4 41.3 Cosine, j 26 9.953660 953599 953537 963475 953413 953352 953290 953228 958166 953104 963042 9.952980 952918 952855 952793 962731 952669 952606 952544 952481 952419 9.952356 962294 952231 952168 962106 952043 951980 951917 951854 951791 9.951728 951665 951602 951539 961476 961412 961349 961286 951222 951169 951096 951032 960968 950905 950841 950778 950714 950650 950586 950522 950458 950394 950330 950366 950202 960188 960074 950010 949945 949881 "sine. D. 10" Tang. 10.3 10.3 10.3 10.3 10.3 10.3 10.3 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10.5 10.6 10.6 10.5 10.5 10.6 10.5 10.5 10.6 10.6 10.6 10.6 10.6 10.6 10.6 10.6 10.6 10.6 10.6 10.6 10.6 10.7 10.7 10.7 10.7 10.7 10.7 10.7 10.7 10.7 10.7 688182 688502 688823 689143 689463 689783 690103 690423 690742 691062 691381 691700 692019 692338 692656 692976 693293 693612 693930 694248 694566 694883 695201 695518 695836 696153 696470 696787 697103 697420 697736 698053 698685 699001 699316 699632 699947 700263 700678 700893 .701208 701623 701837 702162 702466 702780 703096 703409 703723 704036 . 704350 704663 704977 705290 705603 705916 706228 706541 706854 707166 _10^ 53.4 53.4 53.4 53.3 53.3 53.3 63 63 63 63 63 53 53 53.1 53.1 53.1 53.0 53.0 53.0 63.0 62.9 52.9 62.9 52.9 52.9 52.8 52.8 52.8 62.8 52.7 52.7 52.7 52.7 52.6 52.6 62.(5 52.6 52.6 52.6 62.5 62.6 62.4 62.4 52.4 52.4 52.4 52.3 52.3 52.3 52.3 52.2 52.2 52.2 52.2 52.2 52.1 52.1 62.1 52.1 52.1 Cclang Degrees. Cotong. j|N. sine 10.311818 311498 311177 310857 310537 310217 309897 309677 309268 308938 308619 10.308300 307981 307662 307344 307026 306707 306388 306070 305752 305434 10.305117 304799 ?J 04482 304164 303847 303580 303213 302897 302680 302264 10-301947 301631 301315 300999 300684 300368 300058 299737 299422 299107 10-398792 298477 298163 297848 297684 297220 296906 2S6691 296277 295964 10.2^5660 295337 295028 294710 294397 294084 293772 293469 29ol46 292884 Tang 43837 43863 43889 43916 43942 43968 43994 44020 89790 N^oos 89879 89867 89854 89841 89828 89816 89803 44046 44072 44098 44124 44151 44177 44208 44229 44265 44281 44307 44333 44359 44885 44411 44437 44464 44490 44516 44642 44668 44694 44620 44646 44672 44698 44724 44750 44776 44802 44828 44854 44880 449U6 44932 44958 44984 45010 45036 145062 45088 46114 45140 45166 45192 45218 45248 45269 45295 45321 45347 45373 45599 N . «-os. N.sine. 89777 89764 89762 89739 89726 89718 89700 89687 89674 89662 89649 89636 89623 89610 89597 89534 89571 89558 89545 89532 89519 89506 89493 89480 89467 89454 89441 89428 89415 89402 89889 89376 89363 89860 89337 89324 89811 89298 89285 89272 89259 89246 89232 89219 89206 b9193 89180 89167 89153 89140 89127 89114 89101 48 Log. Sines and Tangent*. (27^) Natural Sines. TABLE n. 19.657047 667295 657642 667790 658037 668284 668531 668778 659025 659271 659517 9.659763 660009 660256 660601 660746 660991 661236 661481 661726 661970 9.662214 662469 662703 662946 663190 663433 663677 663920 664163 664406 9.664648 664891 665133 665375 665617 666859 666100 666342 666583 666824 9.667065 667305 667546 667786 668027 668267 668506 668746 668986 669225 669464 669703 669942 670181 670419 670668 670896 671134 671372 671609 Cosine. D. 10 41.3 41.3 41.2 41.2 41.2 41.2 41.1 41.1 41.1 41.0 41.0 41.0 40.9 40.9 40.9 40.9 40.8 40.8 40.8 40.7 40.7 40.7 40.7 40.6 40.6 40.6 40.5 40.6 40.5 40.5 40.4 40.4 40.4 40.3 40.3 40.3 40.2 40.2 40.2 40.2 40.1 40.1 40.1 40.1 40.0 40 40.0 39.9 39.9 39.9 39.9 39.8 39.8 39.8 39.7 39.7 39.7 39.7 39.6 39.6 .949881 949816 949762 949688 949623 949558 949494 949429 949364 949300 949235 .949170 949105 949040 948975 948910 948846 948780 948716 948650 948584 .948619 948454 948388 948323 948267 948192 948126 948060 947995 947929 .947863 947797 947731 947666 947600 947633 947467 947401 947336 947269 .947203 947136 947070 947004 946937 946871 946804 946738 946671 946604 .946538 946471 946404 946337 946270 946203 946136 946069 946002 945935 Sine. ). 707166 _» 707478 IJ.' 707790 IJ.' 708102 It' 708414 °f 708726 °j 709037 ?{• 709349 °J 709660 l\ 709971 l\- 710282 °J" >. 710593 l\- 710904 l\' 711216 °;- 711525 °;- 711836 l\- 712146 °J- 712456 l\- 712766 l\- 713076 ^}- 713386 l\- >. 713696 l\- 714006 l\- 714314 l\- 714624 l\- 714933 °f- .715242 l\- 715651 °J- 716860 l\- 716168 l\- 716477 °} • 1.716786 °)- 717093 I j; • 717401 1 1\ ■ 717709 l\- 718017 l\- 718325 ^f- 718633 l\- 718940 l\- 719248 ?;• 719555 ^|- .719862 ^;- 720169 °}- 720476 °}- 720783 l\- 721089 ?}• 721396 °\- 721702 v.- 722009 2|- 722316 °}- 722621 ?J- 1.722927 °\- 723232 °/;- 723638 ;°"- 723844^^. 7241491^0- 724454 I °" • 724759 I °" • 725066;°"- 726369^2- 726674 p"- Co tang. Degreea. Cotang. I N. sine. N. cos, 10 10 10 10 .292834 292622 292210 291898 291686 291274 290963 290651 290340 290029 289718 .289407 289096 288785 288475 288164 287864 287644 287234 286924 286614 .286304 285996 286686 285376 286067 284758 284449 284140 283832 283523 ,283215 282907 282599 282291 281983 281675 281367 281060 1 280752 1 280445 i 280138 279831 1 279524 , 279217 278911 278604 278298 277991 277686 277379 277073 276768 276462 276156 275861 275546 276241 274935 274631 274326 Tiing. 45399 46426 45451 45477 45503 45529 46664 45680 45606 45632 45668 45684 45710 46736 45762 45787 45813 45839 45865 45891 45917 45942 45968 45994 46020 46046 46072 46097 46123 46149 46175 46201 46226 46262 46278 46304 46330 46356 46381 46407 4<)433 46458 46484 46510 46536 46561 46587 46613 46639 46664 46690 46716 46742 46767 46793 46819 46844 46870 46896 46921 46947 N. COF. .^.^iln 89101 89087 89074 89061 89048 89036 89021 89008 88995 88981 88968 88955 88942 88928 88916 88902 88888 88875 88862 88848 88835 88822 88808 88795 88782 88768 88766 88741 88728 88715 88701 88688 88674 88661 88647 88634 88620 88607 88593 88580 88566 88653 88639 88526 88512 88499 88485 88472 88458 88445 88431 88417 88404 88390 88377 88363 88849 88336 88322 88308 88295 TABLE II. Log. Sines and Tangents. (28°) Natural Smes. 37 60 S ine. 9.671609 671847 672034 672321 672558 672795 673032 673268 673505 673741 673977 9.674213 674448 674684 674919 675155 675390 675624 675859 676094 676328 9.676562 676796 677030 677264 677498 677731 677964 678197 678430 6786S3 9.678895 679128 679360 679592 679824 680056 680288 680519 680750 680982 9.681213 681443 681674 681905 682135 682365 682595 682825 683055 683284 .683514 683743 683972 684201 684430 684658 684887 685115 685343 685571 D. 10" Cosino. Cosine. 1.945935 945868 945800 945733 945666 945598 945531 945464 945396 945328 945261 .945193 945125 945058 944990 944922 944854 944786 944718 944650 944582 .944514 944446 944377 944309 944241 944172 944104 944036 943967 943899 .943830 943761 943693 943624 943556 943486 943417 943348 943279 943210 .943141 9-13072 943003 942934 942864 942795 942726 942656 942587 942517 .942448 942378 942308 942239 942169 942099 942029 941959 941889 941819 D. 10' Sine. T ang. .725674 725979 726284 726588 726892 727197 727501 727805 728109 728412 728716 .729020 729323 729626 729929 730233 730535 730838 731141 731444 731746 .732048 732351 782653 732955 733257 733558 733860 734162 734463 734764 .735066 735367 735668 785969 736269 736570 736871 737171 737471 737771 738071 738371 738671 738971 789271 739570 739870 740169 740468 740767 741066 741365 741664 741962 742261 742559 742858 743156 743454 743752 D, 10' Cotang. I N. sine.lN. cos Cotang. 47690 47716 47741 1 47767 87854 47793 87840 4781837826 47844 47869 47920 47946 47971 47997 87896 87882 87868 87812 87798 47895 87784 87770 87756 87743 87729 48022(87715 48048 87701 i 48073 87687 48099 87673 48124 87659 4815087645 48175 87631 48201 48303 48328 48354 48430 48456 48481 Tang. I N. coo. N.sine 87617 48226 87603 48252|87589 48277 87575 87561 87546 87532 48379 87518 48405 87504 87490 87476 87462 37 61 Degrees. 50 Log. Sines and Tangents. (29°) Natural Sines. TABLE n. 1 2 3 4 6 6 7 8 9 10 11 12 13 14 16 16 17 18 19 20 21 22 23 24 25 26 27 28 i29 30 31 32 33 34 36 36 37 38 39 40 41 42 43 44 45 48 47 48 49 60 51 52 63 54 56 66 57 58 69 60 D. W 685571 686799 686027 686254 686482 686709' 686936 687163 687389 687616 687843 9.688069 688295 688521 688747 688972 689198 689423 689648 689873 690098 9.690323 690548 690772 690996 691220 691444 691668 691892 692116 692339 9.692562 692785 693008 693231. 693463 693676 69389& 694120 694342 694564 9.694786 695007 696229 695460 696671 695892 696113 696334 696664 696775 9.6969y5 697215 697435 697654 697874 698094 698313 698532 698751 698970 Coaine. 38.0 37.9- 37.9 37.9 37.9 37.8 37.8 37.8 37.8 37.7 37.7 37.7 37.7 37.6 37.6 37.6 37.6 37.6 37.5 37,5 37.5 37.4 37.4 37.4 37.4 37.3 37.3 37,3 37,3 37.2 37.2 37.2 37.1 37.1 37.1 37.1 37.0 37,0 37.0 37.0 36.9 36.9 36.9 36,9 36.8 36.8 36.8 36.8 36.7 36.7 36.7 36.7 36.6 36.6 36.6 36.6 36.6 36.6 36.6 36,6 Cosine. \D. IC/' 9.941819 941749 941679 941609 941639 941469 941398 941328 941258 941187 941117 941046 940976 940905 940834 940763 940693 940622 940651 940480 940409 9.940338 940267 940196 940126 940054 939982 939911 939840* 939768 939697 939625 939654 939482 939410 939339 939267 939195 939123 939052 938980 9.938908 938836 938763 938691 938619 938647 938476 938402 938330 938268 9G8185 938113 938040 937967 937895 937822 937749 937676 937604 937531 '11. 7 11.7 11.7 11.7 11.7 Sine. 11 11 11 11 11 11 11.8 11.8 11.8 11.8 11.8 11.8 11.8 11.8 11.8 11.8 11.3 11.8 11.9 11.9 11.9 11.9 11.9 11.9 11.9 11.9 11.9 11.9 11.9 11.-9 12.0 12.0 12.0 12.0 12.0- 12.0 12.0- 12.0 12.0 12.0 12.0 12.0 12.1 12.1 12,1 12.1 12.1 12.1 12.1 12.1 12.1 12.1 12.1 12.1 Tang. D. 10" 9,743752 744050 744348 744646 744943 746240 745638 745836' 746132 746429 746726 9.747023 747319 747616 747913 748209 748505 748801 749097 749393 749689 9.749985 75(0281 760676 760872 751167 761462 751767 752062 762347 752642 9.752937 753231 753626 753820 754116 764409 754703 754997 756291 765685 9,765878 756172 766465 766759 757062 767346 767638 757931 768224 768617 9', 768810 769102 759395 759687 759979 760272 760564 760866 761148 7 61431 Cotang. 49.6 49.6 49.6 49.6 49.6 49.5 49.5 49,5 49,5 49.5 49.4 49.4 49.4 49.4 49^4 49.3 49.3 49.3 49,3 49.3 49.3 4Q,2 49.2 49.2 49.2 49.2 49.2 49.1 49.1 49,1 49,1 49.1 49.1 49.0 49.0 49.0 49.0 49.0 49.0 48.9 48,9 48,9 48.9 48.9 48.9 48.8 48 ..8 48.8 48.8 48,8 48.8 48.7 48.7 48.7 48,7 48,7 48.7 48.6 48.6 Cotang. 10.256248 266950 255652 256356 266067 254760 264462 254166 253868 263571 263274 10.252977 252681 252384 252087 251791 261496 251199 250903 260607 250311 10.260015 249719 249424 249128 248833 248638 248243 247948 247653 247358 10.247063 246769 246474 246180 245886 246691 245297 246003 N,8ine, 48481 48606 48532 48667 48583 48608 48634 48659 48684 87462 87448 87434 87420 87406 87391 87377 87363 87349 4871087335 48736 48761 48786 48811 48837 48862 48888 48913 48964 48989 49014 49066 49090 49116 49141 49166 49192 49217 49242 49268 49293 49318 49344 49369 87321 87306 87292 87278 87264 87260 87235 87221 87207 87193 87178 87164 49040187150 49419 49446 87136 87121 87107 87093 87079 87064 87050 87036 87021 87007 86993 86978 86964 86949 86935 86921 244709 4947086906 244416 11 49495 10.2441221! 49621 243828 149646 24363&i|49571 243241 49596 342948 242655 242362 242069 241776 241483 10.241190 240898 240606 240313 240021 239728 239436 239144 238852 238661 I Tang. 1 49622 49647 49672 49697 49723 49748 49773 49798 86892 86878 86863 86849 86834 86820 86805 86791 86777 86762 86748 86733 86719 4982486704 49849j86690 4987486675 49899 49924 86661 86646 4995086632 49976 50000 N. cofi. N.Hiiic 86617 86603 60 69 58 57 66 55 54 63 52 61 60 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 60 Degreeg. TABLE n. Log. Sines and Taugeats. (30°) Natural Sines. 51 Sine. D. 10" 9.698970 699189 699407 699626 699844 700062 700280 700498 700716 700933 701151 j 9.701368 701585 701802 702019 702236 702462 702669 702886 703101 703317 703533 703749 703964 704179 704395 T04610 704825 705040 705254 705469 9.706683 705898 706112 706326 706539 706753 706967 707180 707393 707606 .707819 708032 708245 708458 70S670 708882 709094 709308 709518 709730 .709941 710163 710J64 710576 710786 710967 711208 711419 711629 711839 Cosine. 36.4 36.4 36.4 36.4 36.3 36.3 36.3 36.3 36.3 36.2 36.2 36.2 36.2 ,1 36 36.1 36.1 36.1 36.0 36.0 36.0 36.0 36.9 35.9 36.9 36.9 36.9 36.8 35.8 36.8 35.8 35.7 35.7 35.7 35.7 36.6 35.6 35.6 35.6 35.5 36.6 35.6 35.6 35.4 35.4 35.4 35.4 35.3 36.3 35.3 35.3 36.3 36.2 35.2 35.2 36.2 35.1 35.1 35.1 35.1 35.0 Cosine. D. 10'' 12 12 12 12 12 12 12 12 12 12 12 12 12.3 12.3 12.3 12.3 12.3 12.3 12,3 12.3 12.3 12.3 12.3 12.3 12.4 12.4 12.4 12.4 12.4 12.4 12.4 12.4 12.4 12.4 12.4 12.4 12.4 12.5 12.6 12,6 12,6 12.5 12.6 12.5 12 12 12 12 12 12 12 12.6 12.6 12.6 12.6 12.6 12.6 12.6 12.6 12.6 Tanp" .761439 761731 762023 762314 762G06 762897 763188 763479 763770 764061 764352 .764643 764933 766224 765614 766806 766095 766385 766676 766965 767265 .767546 767834 768124 768413 768703 768992 769281 769570 769860 770148 .770437 770726 771015 771303 771692 771880 772168 772457 772745 773033 .773321 773608 773896 774184 774471 774759 775046 775333 775621 776908 .776195 776482 776769 777065 777342 777628 777915 778201 778487 778774 10' 48.6 48.6 Cotang. Cotang. j N. sine. N. cos 10.238561 238269 237977 237686 237394 237103 236812 236521 236230 235939 235648 10.235357 235037 234776 234486 234195 233905 233615 233325 233035 232746 10.232455 232166 231876 231587 231297 231008 230719 230430 230140 229852 10.229563 229274 228985 228697 228408 228120 227832 227543 227256 22696.7 10.226679 226392 226104 225816 226529 225241 224954 224667 224379 224092 10.223806 223518 223231 222945 222658 222372 222085 221799 221612 221226 150000 50025 60050 50076 50101 60126 60151 50176 60201 50227 50252 50277 50302 50327 50352 50377 50403 1 50428 '60453 60478 50603 60528 86588 86573 86559 86544 86630 86616 86601 86486 86471 86457 86442 86427 86413 86398 86384 Tang. 86354 86340 86325 86310 86295 60653 86281 60678 50603 50628 50654 50679 50704 60729 50754 60779 60804 60829 .50864 .5087: 50904 50929 50954 50979 51004 51029 51054 51079 61104 51129 61154 51179 51204 51229 5126l 713617 t: 713726 ^l 713936 tl .714144 tj 714352 ll 714561 tl 714769 t; 714978 ^2 715186 '^2 715394 ^^ 7166Q2 ^2 715809 "il 716017 ^* .716224 fl 716432 q^ 716639 ^2 716846 r: 717053 ^^ 717259 tl 717466 ^t 717673 ^;: 717879 t: 718086 ^^ .718291 "ij 718497 ^J 718703 ^^ 718909 i '11 719114 ^^ 719320 7.1 719525 ^;: 719730 f: 719936 ^^ 720140 ^? .720346 ll 720549 !^1' 720754;^^' 720958 I tl 721162' 5; 721366 1 tl 721570 ^X 721774 ■ it 721978:^ 722181'^^ . 722385 ^^t 722588 ^^ 722791!^^ 722994^^ 723197:^^ 723400 „ 723603 to 723805^^ 724007^^ 724210! ^ Cosine. I 9.933036 932990 932914 932838 932762 932685 932609 932533 932457 932380 932304 9.932228 932151 932076 931998 931921 931845 931768 931691 931614 931537 931460 931383 931306 931229 931152 931075 930998 930921 930843 930766 9.930688 930611 930533 930466 930378 930300 930223 930145 930067 929989 929911 929833 929766 929677 929599 929521 929442 929364 929286 929207 929129 929050 928972 928816 928736 928657 928678 928499 928420 Sine. D. 10" 12.6 12.7 12.7 12 12 12 12 12 12 12.7 12.7 12.7 12.7 12.8 12.8 12.8 12.8 12.8 12.8 12.8 12.8 12.8 12.8 12.8 12.9 12.9 12.9 12.9 12.9 12.9 12.9 12.9 12.9 12.9 12.9 12.9 13.0 13.0 13.0 13.0 13.0 13.0 13.0 13.0 13.0 13.0 13.0 13.0 13.1 13.1 13.1 13.1 13.1 13.1 13.1 13.1 13.1 13.1 13.1 13.1 Tang. 1.778774 779060 779346 779632 779918 780203 780489 780775 781060 781346 781631 1.781916 782201 782486 782771 783053 783341 783626 783910 784195 784479 (.784764 785048 785332 785616 785900 786184 786468 786752 787036 787319 >. 787603 787886 788170- 788453 788736 789019 789302 789685 789868 790151 1.790433 790716 790999 791281 791663 791846 792128 792410 792692 792974 ). 793256 793638 793819 794101 794383 794664 794946 795227 795508 7 96789 Cotang. D. ll>" Cotang. N.sine. N. cos 47.7 47.7 47.6 47.6 47.6 47.6 47.6 47.6 47.6 47.6 47.6 47.6 47.6 47.6 47.6 47.5 47.5 47.4 47 47 47 47 47 47 47.3 47.3 47.3 47.3 47.3 47.3 47.2 47.2 47.2 47.2 47.2 47.2 47.2 47.1 47.1 47.1 47.1 47.1 47.1 47.1 47.1 47.0 47.0 47.0 47.0 47.0 47.0 47.0 46.9 46.9 46.9 46.9 46.9 46.9 46.9 46.8 10.221226 220940 220664 220368 220082 219797 2195U 219226 218940 218654 218369 10.218084 217799 217514 217229 216944 216669 216374 216090 215805 215521 10.215236 214952 214668 214384 214100 213816 213532 213248 212964 212681 10.212397 212114 211830 211647 211264 210981 210698 210416- 210132 209849 10.209567 209284 209001 208719 208437 208164 207872 207690 207308 207026 10.206744 206462 206181 205899 205617 205336 205055 204773 204492 204211 51504 86717 61529 86702 61554 85687 51579 85672 61604 51628 61653 51678 ! 61703 161728 1 61753 'I 61778 61803 51828 61852 61877 51902 51927 61952 85667 85642 85627 86612 86697 85682 85567 85551 86536 85521 85606 86491 86476 86461 86446 61977185431 62002185416 52026J85401 52061 62076 86386 85370 52101^85365 6Q126 62161 152175 ! 52200 ! 62226 i 52260 i 52276 85340 j 36 85325 I 34 86310 33 85294 85279 85264 86249 52299 85234 28 62324J85218 27 i 52349 85203 26 i 52374 62399 ; 62423 152448 i 52473 I 62498 85188 1 26 86173 86167 85142 85127 86112 I 52522 85096 i 52547 '^ ■"' 1 62572 ! 62597 1 62621 i 52646 ■62671 ! 52696 i 62720 i 62746 I 52770 i 52794 162819 62844 52869 8488; 52893 84866 52918 84861 85U81 85066 85U51 86036 85020 85005 84989 84974 84959 84943 849:^8 84913 84897 ! 62943 j! 62967 1162992 Tang. 84836 84820 84805 N. cos.JN.sine. 58 Degrees. TABLE n. Log. Sines and Tangents. (32°) Natural SincB. 53 Sine. 9.724210 724412 724614 724816 725017 725219 725420 725622 725823 726024 726225 726426 726626 726827 727027 727228 727428 727628 727828 728027 728227 9.728427 728626 728825 729024 729223 729422 729621 729820 730018 730216 9.730415 730613 730811 731009 731206 I 731404 731602 731799 731996 732193 9.732390 732587 732784 732980 733177 733373 733569 733765 733961 734157 9.734353 734549 734744 734939 735135 735330 735525 735719 735914 736109 D. 10" Ck)sine. Cosine. |D. ly^ 9.928420 928342 928263 928183 928104 928025 927946 927867 927787 927708 927629 927549 927470 927390 927310 927231 927151 927071 926991 92691 1 926831. 9.926751 926671 926591 926511 926431 926351 926270 926190 926110 926029 926949 925868 926788 925707 925626 925545 925465 925384 926303 925222 925141 925060 924979 924897 924816 924735 924654 924572 924491 924409 9.924328 924246 924164 924083 924001 923919 923837 923755 923673 923591 Sine. 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13.3 13.3 13 13 13 13 13 13 13 13 13 13 13 13.4 13 13 13 13 13 13 13 13 13 13 13 13 13.6 13.6 13.6 13.6 13.6 13.6 13.6 13.6 13.6 13.6 13.7 13.7 Tang. ). 795789 796070 796351 796632 796913 797194 797475 797765 798036 798316 798596 ). 798877 799157 799437 799717 799997 800277 800557 800836 801116 801396 ). 801676 801956 802234 802613 802792 803072 803361 803630 803908 804187 ). 804466 804745 805023 806302 805680 806859 806137 806415 806693 806971 ). 807249 807527 807805 80S083 808361 808638 808916 809193 809471 809748 ). 810025 810302 810580 810857 811134 811410 811687 811964 812241 812517 Cotanc. D. 10" 46.8 46.8 46.8 46.8 46.8 46.8 46.8 46.8 46.7 46.7 46.7 46.7 46.7 46.7 46.7 46.6 46.6 46.6 46.6 46.6 46.6 46.6 46.6 46.5 46.5 46.5 46.5 46.5 46.5 46.5 46.5 46.4 46.4 46.4 46.4 46.4 46.4 46.4 46.3 46.3 46.3 46.3 46.3 46.3 46.3 46.3 46.2 46.2 46.2 46.2 46.2 46.2 46.2 46.2 46.2 46.1 46.1 46.1 46.1 46.1 Cotang. I N. sine. N. cos 10.204211 203930 203649 203368 203087 202806 202525 202245 201964 201684 201404 10.201123 200843 200563 200283 200003 199723 199443 199164 198884 198604 10.198326 198046 197766 197487 197208 196928 196649 196370 196092 195813 10.195534 195255 194977 194698 194420 194141 193863 193686 193307 193029 10.192751 1S2473 192195 191917 191639 191362 191084 190807 190529 190252 189975 189698 189420 189143 188806 188590 188313 188036 187759 187483 "Tani?. 52992 53017 53041 53066 53091 53115 53140 153164 53189 53214 53238 53263 53312 53337 53361 63386 53411 53435 1 53460 1 53484 63509 1 63634 1 53568 i 63683 1 53607 53632 53656 53681 53706 63730 63754 53779 10. 84805 84789 84774 84759 84743 84728 84712 84697 84681 84666 84650 84635 84619 84604 84588 84573 84657 84542 84626 84511 84495 84480 84464 84448 84433 84417 84402 84386 84370 84355 84339 84324 84308 53804 84292 1 53828J84277 i 63853184261 1 53877[84245 1 53902 84230 ! 63926 84214 ! 63951 84198 I53975s84l82 54000 84167 I 64024|84151 154049 84135 1 54073,84120 54097184104 1 54122J84088 i 54146184072 1 54171 84057 i 64195 84041 1 54220,84025 I 64244:84009 154269183994 54293 83978 1 54317 83962 54342,8S946 ! 54366 83930 1 54G91 183915 ! 64415183899 ; 64440;83883 15446483867 60 59 58 57 66 55 54 53 62 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 S3 32 31 30 29 28 27 26 26 24 23 ^2 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 S 2 1 N. COP. N.fiir.e. 57 Degrees. 54 Log. Sines and Tangents. (33°) Natural Sines. TABLE IL J_ S ine. D. V^' Cosine. D. W Tang. D. 10" Cotang 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 64 55 56 57 58 59 60 9.736109 736303 736498 736692 736886 737080 737274 737467 737661 737855 738048 9.738241 738434 738627 738820 739013 739206 739398 739590 739783 739975 740167 740369 740550 740742 740934 741125 741316 741508 741699 741889 9.742080 742271 742462 742662 742842 743033 743223 743413 743602 743792 19.743982 744171 744361 744560 744739 744928 745117 745306 745494 745683 9.745871 746059 746248 746436 746624 746812 746999 747187 747374 747562 Cosine. 32.4 32.4 32.4 32.3 32.3 32.3 32.3 32.3 32.2 32.2 82.2 32.2 32.2 32.1 32.1 32.1 32.1 32.1 32.0 32.0 32.0 32.0 32.0 31.9 31.9 31.9 31.9 31.9 31.8 31.8 31.8 31.8 31 31.7 31.7 31.7 31.7 31.7 31.6 31.6 31.6 31.6 31.6 31.5 31.5 31.5 31.5 31.5 31.4 31.4 31.4 31.4 31.4 31.3 31.3 31.3 31.3 31.3 31.2 31.2 9.923591 923509 923427 923345 923263 923181 923098 923016 922933 922851 922768 9.922686 922603 922520 922438 922355 922272 922189 922106 922023 921940 9.921857 921774 921691 921607 921624 921441 921357 921274 921190 921107 9.921023 920939 920856 920772 920688 920604 920520 920436 920362 920268 9.920184 920099 920015 919931 919846 919762 919677 919593 919508 919424 9.919339 919254 919169 919086 919000 918915 918830 918745 918659 918574 I Sine. 13.7 13.7 13.7 13.7 13.7 13.7 13.7 13.7 13.7 13.7 13.8 13.8 13.8 13.8 13.8 13.8 13.8 13.8 13.8 13.8 13.8 13.9 13.9 13.9 13.9 13.9 13.9 13.9 13.9 13.9 13.9 13.9 14.0 14.0 14.0 14.0 14.0 14.0 14.0 14.0 14.0 14.0 14.0 14.0 14.1 14.1 14.1 14.1 14.1 14.1 14.1 14.1 14.1 14.1 14.1 14.1 14.2 14.2 14.2 14.2 9.812517 812794 813070 813347 813623 814175 814452 814728 815004 815279 9.816555 815831 816107 816382 816658 816933 817209 817484 817759 818036 .818310 818685 818860 819135 819410 819684 819959 820234 820608 820783 9.821067 821332 821606 821880 822154 822429 822703 822977 823250 823524 3.823798 824072 824345 824619 824893 825166 825439 826713 825986 826259 ). 826632 826805 827078 827351 827624 827897 828170 828442 828715 828987 Cotang. 46.1 46.1 46.1 46.0 46.0 46.0 46.0 46 46 46 46.0 45.9 45.9 45.9 45.9 45.9 45.9 45.9 45.9 45.9 45.8 45.8 45.8 45.8 45.8 45.8 45.8 45.8 45.8 45.7 45.7 45.7 45.7 46.7 45.7 45.7 45.7 45.7 45.6 46.6 45.6 45.6 46.6 46.6 45.6 45.6 45.6 45.5 45.5 45.5 45.5 45.5 45.5 45.5 45.5 45.5 45.4 45.4 45.4 45.4 10.187482 187206 186930 186653 186377 186101 185825 185548 185272 184996 184721 10.184445 184169 183893 183618 183342 183067 182791 182516 182241 181965 10.181690 181415 181140 180865 180590 180316 180041 179766 179492 179217 10.178943 178668 178394 178120 177846 177571 177297 177023 176760 176476 10.176202 175928 176656 175381 175107 174834 174561 174287 174014 173741 10.173468 173195 172922 172649 172376 172103 171830 171558 171285 171013^ Tai^. N. sine. N. cos. 54464 54488 54513 54537 54661 83867 83851 83835 83819 83804 54586 83788 54610 83772 54635 83756 54659 83740 54683 «3724 54708 64732 54766 54781 54805 83645 5482983629 54854 83613 54878 54902 54927 83697 83581 83566 54951 83549 54975 54999 55024 56097 55121 55145 55169 65194 55218 65242 83708 83692 83676 83660 83533 83517 83601 56048 83485 55072 83469 83463 83437 83421 83406 83389 83373 83356 5626683340 56291 65315 55339 56412 65436 56460 56484 55509 56533 55657 56581 83324 83292 65363 83276 83260 83244 83228 83212 83195 83179 83163 83147 83131 56605 83116 55630 ,'83098 55654 83082 56678|83066 65702J83060 55726 83034 5675083017 55871 55895 55919 83001 82S85 55775 66799 55823 55847 82953 82936 82920 82904 N. eos. N.sine, 60 59 68 67 56 65 64 53 52 61 60 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 16 14 13 12 11 10 9 8 7 6 5 4 3 2 56 Degrees. TABLE II. Log. Sines and Tangents. (34°) Natural Sinea- 55 I). 10" 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 36 36 37 38 39 40 41 42 43 44 46 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 Sine. 9.747562 747749 747936 748123 748310 748497 748683 748870 749056 749243 749426 9.749615 749801 749987 750172 750358 760543 750729 750914 761099 751284 9.751469 761654 751839 762023 752208 752392 752576 752760 752944 753128 9.763312 753495 763679 753862 764046 764229 754412 754595 I 764778 '■ 754960 I 9.7661431 766326 I 755608 I 765690 : 755872 j 756054 i 756236 I 756418 ! 756600 i 756782 ! 9.756963! 757144 I 767326 757507 I 757688 I 767869 : 758050 : 758230 i 768411 ! _758591 ! Cosine. I 31.2 31.2 31.2 31.1 31.1 31.1 31.1 31.1 31.0 31.0 31.0 31.0 31.0 30.9 30.9 30.9 30.9 30.9 30.8 30.8 30.8 30.8 30.8 30.8 30.7 30.7 30.7 30.7 30.7 30.6 30.6 30.6 30.6 30.6 30.5 30.5 30.6 30.5 30.6 30.4 30.4 30 30 30 30 30 30 30.3 30.3 30.3 30.2 30.2 30.2 30.2 30.2 30-1 30.1 30.1 30.1 30.1 Cosine. 9.918674 918489 918404 918318 918233 918147 918082 917976 917891 917805 917719 9.917634 917648 917462 917376 917290 917204 917118 917032 916946 916859 .916773 916687 916600 916514 916427 916341 916254 916167 916081 915994 .916907 915820 916733 915646 915559 915472 915385 915297 915210 915123 .915036 914948 914860 914773 914686 914698 914610 914422 914334 914246 9.914158 91407.0 913982 913894 913806 913718 913630 913541 913453 913365 Sine. D. 10" Tang. 14.2 14.2 14.2 14.2 14.2 14.2 14.2 14.3 14.3 14.3 14.3 14.3 14.3 14.3 14.3 14.3 14 14 14 14 14 14 14 14 14 14 14.4 14.4 14 14 14 14 14 14 14 14 14 14.5 14.5 14.5 14.6 14.6 14.6 14.6 14.6 14.6 14.6 14.6 14.6 14.6 14.7 14.7 14. 14. 14. 14. 14. 14. 14. 7 7 7 7 7 7 7 14.7 ,828987 829260 829532 829806 830077 830349 830621 830893 831166 831437 831709 831981 832263 832626 832796 833068 833339 833611 833882 834154 834426 834696 834967 836238 835509 835780 836061 836322 836693 836864 837134 9.837406 837675 837946 838216 838487 838757 839027 839297 839568 839838 9.840108 840378 840647 840917 841187 841457 841726 841996 842266 842635 9.842805 843074 843343 843612 843882 844161 844420 844689 844958 846227 Cotang. D. 10 45.4 45.4 45.4 45.4 45.4 45.3 45.3 46.3 45.3 45.3 46.3 45.3 46.3 45.3 45.3 46.2 45.2 45.2 45.2 45.2 46.2 45.2 46.2 46.2 45.2 45.1 45.1 46,] 45.1 45.1 46.1 46.1 45.1 45.1 45.1 46.0 45.0 45.0 45.0 45.0 45.0 46.0 45.0 45.0 44.9 44.9 44.9 44.9 44.9 44.9 44.9 44.9 44.9 44.9 44.9 44.8 44.8 44.8 44.8 44.8 Cotang. IjN.sine 55919 56943 55908 56992 56016 56040 56064 56088 56112 56136 66160 10. 16-8019!! 56184 N. COS.) 167747 ! 167475 i 167204 166932 166661 166389 166118 165846 165576 10.165304 166033 164762 164491 164220 163949 163678 163407 163136 162866 10.162595 162325 162054 161784 161513 161243 160973 56208 56232 56256 66280 56305 56329 56353 56377 56401 56425 56449 56473 56497 56621 56545 56569 56593 56617 66641 56665 66689 66713 56736 56760 56784 66808 160703 !i568S2 16043211 56856 160162 1 156880 10.159892 169622 159353 159083 158813 168543 158274 168004 157734 157465 10.157196 166926 166657 156388 156118 155849 155580 155311 155042 154773 Tang. 56904 56928 56962 56976 57000 67024 57047 67071 57095 57119 57143 67167 67191 57215 57238 57262 67286 67310 57334 57358 82904 82887 82871 82855 82839 82822 82806 82790 82773 82767 82741 82724 82708 82692 82675 82659 82643 82626 82610 82593 82577 82661 82544 82528 82511 82495 82478 82462 82446 82429 82413 82396 82380 82363 82347 82330 82314 82297 82281 82264 82248 82231 82214 82198 82181 82165 82148 82132 82115 82098 82082 82065 82048 82032 82015 81999 81982 81965 81949 81932 81915 N. COS. N.sine, 56 Log. Sines and Tangents. (35°) Natural Sines. TABLE II, 9.758591 758772 758952 759132 759312 "V 759492 i:{" • 759672^"- 759852 ;;;!• 760031 :;^ • 760211 .;,^ • 760390 x;; • .760569 hfo 760748 i^^ • 760927 h;Q • 761106 2^ • 761285 ^;! • 761464 ^g • 7616421^^- 761821 j^^ • 761999 U;q' 7621771^^* .762356 ^^• 762534 f- 762712 f- 762889 ^^• 763067 :;'• 763245 ;^- 763422 :;^- 763600 Z,' 763777 ^^• 763954 ^^• .764131 f- 764308 ^q 764485 ^^• 764662 ^^• 764838 Z,' 765016 '^• 765191 f- 765367 -^• 765544 *^- ^^^-20 f^- 29. 29. 29. 29. 29. 29. 29. 29. 29. 29. 29. 29. 29 765. 887302 887198 887093 886989 88G885 886780 886676 886571 886466 886362 9.886257 886152 886047 885942 885837' 885732 885627 885622 885416 885311 9.885205 885100 884994 884889 884783 884677 884572 884466 884360 884264 D. 10" Sine. 17.0 17.1 17.1 17.1 17.1 17.1 17.1 17.1 17.1 17.1 17.1 17.2 17.2 17.2 17.2 17.2 17.2 17.2 17.2 17.2 17.3 17.3 17.3 17.3 17.3 17.3 17.3 17.3 17.3 17.3 17.4 17. .4 17.4 17.4 17.4 17.4 17.4 17.4 17.4 17.4 17.5 17.5 17.6 17.5 17.6 17.5 17.6 17.5 17.6 17.6 17.6 17.6 17.6 17.6 17.6 17.6 17.6 17.6 17.6 17.6 Tang. 9.908369 90S628 90S886 909144 909402 903660 909918 910177 910435 910693 910951 9.911209 911467 911724 911982 912240 912498 912766 913014 913271 913529 9.913787 914044 914302 914660 914817 916075 915332 916590 916847 916104 9.916362 916619 916877 917134 917391 917648 917905 918163 918420 918677 9.918934 919191 919448 919705 919962 920219 920476 920733 920990 921247 ). 921503 921760 922017 922274 922630 922787 923044 923300 923657 923813 Cotang. D. 10" 43.0 43.0 43.0 43.0 43.0 43.0 43.0 43.0 43.0 43.0 43.0 43.0 43.0 43.0 43.0 43.0 43.0 43.0 42.9 42.9 42.9 42.9 42.9 42.9 42,9- 42,9 42.9 42.9 42,9 42.9 42.9 42,9 42.9 42.9 42.9 42.9 42.9 42.9 42.8 42.8 42.8 42.8 42.8 42.8 42.8 42.8 42.8 42.8 42.8 42.8 42.8 42.8 42.8 42.8 42.8 42.8 42.8 42.8 42.8 42.7 Cotaug. 10.091631 091372 091114 090856 090598 090340 090082 089823 089505 089307 089049 10.088791 0S8533 088276 088018 087760] 087502 087244 ! a8S986 086729 086471 1 iO. 036213 036956 085698 085440 035183 034925 084668 084410 084163 033896 10.083638 083381 083123 082866 082609 082352 082096 081837 081680 081323 10-081066 080809 080552 030295 080038 079781 079524 07926? 079010 It 64033 078753 j 1 64056 10.0784971164078 07824011 64100 077983 |i6412G 077726 077470 !i 64167 077213;; 64190 0769561164212 076700 1 j 64234 07644311 64266 0761871164279 N. sine. N. cos. 77716 77696 77678 77660 77641 77623 77605 77586 77668 77550 77531 77513 77494 77476 77458 77439 77421 77402 77384 77366 77347 77329 77310 77292 77273 77255 77236 77218 77199 77181 77162 77144 77125 77107 77088 77070 77051 77033 77014 76996 76977 76959 62932 62955 62977 63000 63022 63045 63068 63090 63113 63135 63158 93180 63203 63225 63248 63271 63293 63316 63338 63361 63383 63406 63428 63451 63473 63496 63518 63540 63663 63585 63608 63630 63653 63675 63698 63720 63742 63765 63787 63810 63832 63854 63877 76940 63899 76921 6392: 60 59 58 57 66 65 64 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 ! 32 31 30 29 28 27 26 25 24 76903 63944 176884 j 63966 76866 1 6398:.' 76847 164011,76828 76810 76791 76772 76754 76736 64145 76717 76098 6679 76661 76642 76623 76604 Tang. U N. pou. N.Bine 50 Degrees. TABLE n. Log. SineB and Tangents. (40*) Natural Sines. 61 dTio^ Sine. D. IC 810017 810167 810316 810465 810614 810763 810912 811061 .811210 811358 811507 811655 811804 811952 812100 812248 812396 812544 .812692 812840 812988 813135 813283 813430 813578 813725 813872 814019 .814166 814313 814460 814607 814753 814900 815046 815193 815339 815485 1.816631 815778 815924 816069 816215 816361 816507 816652 816798 816943 Cosine. 25.1 25.1 25.1 25.0 25.0 25.0 25.0 25.0 26.0 24.9 24.9 24.9 24.9 24.9 24.9 24.8 24.8 24.8 24.8 24.8 24.8 24.8 24.7 24.7 24.7 24.7 24.7 24.7 24.7 24.6 24.6 24.6 24.6 24.6 24.6 24.6 24.5 24.6 24.5 24.5 24.5 24.5 24.5 24.4 24.4 24.4 Coaiae. jD. 10" 24 24 24 24 24 24 24,3 24,3 24.3 24.3 24.3 24.2 24.2 24.2 883723 883617 883510 883404 883297 883191 .883084 882977 882871 882764 882667 882550 882443 882336 882229 882121 .882014 881907 881799 881692 881584 881477 881369 881261 881153 881046 . 880938- 880830 880722 880613 880505 880397 880289 880180 880072 879963 .879855 879746 879637 879529 879420 879311 879202 879093 878984 878875 .878766 878656 878547 878438 878328 878219 878.09 877999 877890 877780 Sine. 8.0 8.0 8.0 8.0 8.0 8.0 8.0 8.0 8. 8. 8. 8. 8. 8. 8. 8. 8. 8. 8.2 8.2 8.2 8.2 8.2 8.2 8.2 8.2 8.2 8.3 8.3 8.3 8.3 Tang. 1.923813 924070 924327 924583 924840 925096 925352 925609 926865 926122 926378 I.926G34 926890 927147 927403 927669 927915 928171 928427 928683 928940 .929196 929452 929708 929964 930220 930475 930731 930987 931243 931499 '.931755 932010 932266 932522 932778 933033 933289 933545 933800 934056 .934311 934567 934823 935078 935333 9365S0 535844 936100 936356 936610 1.936866 937121 937376 937632 937887 938142 938398 938653 938908 939163 Cotang. Cotang. i N .«ine. N. cos. 10.076187 076930 076673 076417 075160 074904 074648 074391 074135 073878 073622 10.073366 073110 072863 072597 072341 072085 071829 071573 071317 071060 10.070804 070548 070292 070036 069780 069526 069269 069013 068767 068601 10.068245 0679901 067734 t 067478 I 067222 I 066967 066711 066455 066200 065944 10.065689 065433 065177 064922 i 064667 I 064411 I 064166 i 063900 ! 063646 i 063390 I 10.0631341 062879 I 062624 I 062368 062113 061868 061602 061347 061092 060837 64279 64301 64323 64346 64368 64390 64412. 64436 64457 64479 64501 64624 64646 64668 64690 64612 64636 64657 64679 64701 64723 64746 '64768 64790 64812 64834 64856 64878 64901 64923 64945 64967 64989 65011 65033 65065 65077 65100 65122 65144 65166 65188 65210 65232 65264 66276 66298 65320 65342 65364 65386 65408 65430 65452 66474 65496 65518 65540 65562 66584 66606 Tang. 76604 76686 76567 76648 76630 76611 76492 76473 76466 76436 76417 76398 6380 76361 76342 76323 76304 76286 76267 76248 76229 76210 76192 76173 76164 76135 76116 76097 76078 76059 76041 76022 76003 76984 75965 75946 76927 76908 76889 76870 76861 75832 75813 75794 76775 75766 75738 76719 75700 756S0 75661 76642 75623 75604 75585 75666 75547 75528 75609 76490 76471 N. COS. N.sine. 49 Degrees. Log. Sines and Tangents. (41°) Natural Sines. TABLE IL Sine. D. 10" C!osiue 9.816948 817088 817233 817379 817524 817668 817813 817958 818103 818247 818392 9.81S636 818681 818825 818969 819113 819257 819401 819545 819689 819832 9.819976 820120 820263 820405 820550 820693 820836 820979 821122 821265 9.821407 821550 821693 821835 821977 822120 822262 822404 822546 822688 9.822830 822972 823114 823256 823397 823639 823680 823821 823963 824104 9.824245 824386 824627 82-1668 82-1808 824949 826090 825230 825371 825511 Cosine. 24.2 24.2 24.2 24.2 24.1 24.1 24.1 24.1 24.1 24.1 24.1 24.0 24.0 24. 24.0 24.0 24.0 24.0 23.9 23.9 23.9 23.9 23. y 23.9 23.9 23.8 23.8 23.8 23.8 23.8 23.8 23 23 23 23 23 23 23 23 23 23 23 23 23.6 23.6 23.6 23.6 23.5 23.6 23.5 23.6 23.6 23.6 23.6 23.4 23.4 23.4 23.4 23.4 23.4 .877780 877670 877560 877450 877340 877230 877120 877010 876899 876789 87667 .876568 876457 876347 876236 876125 876014 876904 875793 876682 876571 ,876469 876348 876237 876126 876014 874903 874791 874680 874568 874456 ,874344 874232 874121 874009 873896 873784 873672 873660 873448 873336 873223 873110 872998 872886 872772 872659 872547 872434 872321 872208 872095 871981 871868 871755 871641 871528 871414 871301 871187 871073 "si]ie.~ 9.87 D. 10" 18.4 18 18 18 18 18 18 18 18 18 18 18 18,6 18.6 18.6 18.6 18.6 18.6 18. & 18.6 18.6 18.7 18.7 18.7 18,7 18.7 18.7 18.7 18.7 18.7 18.7 18.8 18.8 IS. 8 18.8 18.8 18.8 18.8 18.8 18.3 18.9 18.9 18.9 18.9 18.9 18.9 18.9 18.9 18.9 'I'ang. 9.939163 939418 939673 939928 940183 940438 940694 940949 941204 941458 941714 9.941968 942223 942478 942733 942988 943243 943498 943752 944007 944262 9.944517 944771 945026 945281 945535 945790 946045 946299 946554 946808 9.947053 947318 947572 947826 948081 948336 948590 948844 949099 949353 9.949607 949862 950116 950370 950625 950879 951133 951388 951642 951896 3.952150 952405 952659 952913 953167 953421 953675 963929 954183 964437 Co tang. D. 10" 42.5 42.6 42.5 42.6 42.5 42.6 42 42 42 42 42 42 42 42 42.5 42.6 42.6 42.6 42.6 42.6 42.6 42.6 42.4 42.4 42.4 42 42 42 42 42 42 42 42.4 42.4 42.4 42.4 42.4 42.4 42.4 42.4 42.4 42.4 42 Cotang. I iN. sine. 42 42 42 42 42 42.4 42.4 42.4 42.4 42.4 42.4 42.4 42.3 42.3 42.3 42.3 42.3 10.030837 i! 66606 060582 j 66628 060327 165650 060072 I j 65672 059817 !JG5694 05956-2 i I 65716 059306 jj 65738 059051 65759 058796 1 1 65781 0585421165803 0582S6!j65825 10.0580321166847 057777 1 1 65869 067622! 1 65891 057267 057012 056757 056502 056248 055993 055738 10.055483 055229 66913 65935 65966 65978 66000 66022 66044 66066 66088 054974 1 1 66109 0547191166131 054465 166153 054210 053965 053701 063446 053192 10.062937 062682 052428 i 66176 166197 [66218 166240 1 66262 66284 1 66306 166327 052174 j 66349 06191911 66371 051664 1 1 66393 0514101 66414 051166 66436 050901 66458 050647 I 66480 10 050393 66601 050138 I j 66623 049884 1166645 049630 049376 049121 048867 1 66632 0486121 1 66653 048358 166676 048104 j 66697 10.047850 1 66718 047595 1 1 66740 047341] 166762 047087 '66783 66666 C6688 66610 046833 046679 046325 046071 045817 045563 66805 66827 66848 66870 66891 66913 75471 76462 ?5433 75414 76396 75375 76366 76337 75318 76299 75280 75261 76241 75222 76203 75184 75166 76146 76126 75107 76088 76069 76050 75030 76011 74992 74973 74963 74934 74915 74896 74876 74857 74838 74818 74799 74780 74760 74741 74722 74703 74683 74663 74644 74625 74606 74586 74667 74548 74622 74509 74489 74470 4461 74431 74412 74392 74373 74353 4334 74314 N. COS. N.sine 48 Degrees. TABLE n. Log. Sines and Tangents. (42°)* Natural gines. 63 Sine. D. 10" Cosine. D. 10" Tang. D. 10' Cotang. ;N. Bine. IN. COB .825.^11 826651 825791 825931 826071 826211 826351 826491 826631 826770 826910 .827049 827189 827328 827467 827606 827745 827884 828023 828162 828301 .828439 828578 828716 828S55 828993 829131 829269 829407 829545 829683 .829821 829959 830097 830234 830372 830509 830646 830784 830921 831058 .831195 831332 831469 831606 831742 831879 ! 832015 832162 882288 832425 832561 832697 832833 832969 833105 833241 833377 833612 833648 833783 23.4 23.3 23.3 23.3 23.3 23.3 33.3 23.3 23.3 23.2 23.2 23.2 23.2 23.2 23.2 23.2 23.2 23.1 23.1 23.1 23.1 23.1 23.1 23.1 23.0 23.0 23.0 23.0 23.0 23.0 23.0 22.9 22.9 22.9 22.9 22.9 22.9 22.9 22.9 22.8 22.8 22.8 22.8 22.8 22.8 22.8 22.8 22.7 23.7 22.7 :22.7 122.7 122.7 1 22.7 '22.6 !22.6 122.6 122.6 ^22 1 22.6 Co.siiie. .871073 870960 870846 870732 870618 87050-i 870390 870276 870161 870047 869933 .869818 869704 869589 869474 869360 869246 869130 869015 868900 868785 .868670 868555 868440 868324 868209 868093 867978 867862 867747 867631 .867515 867399 867283 867167 867051 866935 866819 866703 866586 866470 .866853 866237 866120 866004 865887 865770 865653 865536 865419 865302 1.865185 865068 864950 864833 864716 864598 864481 864363 864245 864127 19.0 19.0 19.0 19.0 19.0 19.0 19.0 19.0 19.0 19.1 19.1 19.1 19.1 19.1 19.1 19.1 19.1 19.1 19.2 19.2 19.2 19.2 19.2 19.2 19.2 19.2 19.2 19.3 19.3 19.3 19.3 19.3 19.3 19.3 19.3 19.3 19.4 19.4 19.4 19.4 19.4 19.4 19.1 19.4 19.5 19.5 19.6 19.6 19.5 19.5 19.5 19.5 19.5 19.6 19.6 19.6 19.6 19.6 19.6 19.6 .964437 964691 954945 955200 955454 955707 955961 956215 956469 956723 956977 .957231 957485 957739 957993 958246 958500 958764 959008 959262 959516 .959769 960023 960277 960531 960784 961038 961291 961545 961799 962052 .962306 962560 962813 963067 963320 963574 963827 964081 964335 964588 .9t>4J542 9(i5095 965349 965602 965856 966109 966362 966616 967123 .9S7376 967629 967883 968136 968389 968643 968896 969149 969403 969656 42.3 42.3 42.3 42.3 42.3 42.3 43.3 42.3 43.3 42.3 43.8 43.3 43.3 43.3 43.3 43.3 43.3 43.3 42.8 42.3 43.3 43.3 43.3 43.3 42.3 42.3 42.3 42.3 42.3 42.3 42.3 42.3 42.3 42.3 42.3 42.3 42 42 42 42 42 42 42 42 42 43.2 42.2 42.2 42.3 42.2 43.3 42.2 42.2 42.3 43.2 43.3 43.2 42.2 42.2 42.2 10.045563 045309 045065 044800 044546 044293 044039 043785 043531 043277 043023 10.042769 042615 042261 042007 041764 041500 041246 040992 040738 040484 10.040231 039977 039723 039469 039216 038962 038709 038466 038201 037948 10.037694 037440 037187 036933 036680 036426 036173 035919 035665 035412 10.036168 034905 034651 034398 034145 033891 033638 033384 033131 032877 10.032624 032371 032117 031864 031611 031357 031104 030851 030597 030344 ||66913|74314 1 1 66935174295 j! 66956174276 ''66978174266 i! 66999 74237 ii 67021 167043 !' 67064 'i 67086 ii 67107 i! 67129 ;: 67151 !| 67172 i 1 67194 67215 1 1 67237 i! 67258 74217 74198 74178 74169 74139 74120 74100 74080 74061 74041 74022 74002 1 67280|73983 Cotang. Tang. 3963 3944 73924 73904 73885 73865 73846 73826 73806 73787 3767 73747 73728 73708 73688 73669 73649 73629 73610 73590 73570 73551 73531 73611 73491 73472 73452 73432 73413 73393 373 73353 73333 73314 73294 73274 73254 73234 73216 73195 73175 73155 73136 N. CO?. N.Hin«, 1167301 I {67323 167344 i 1 67366 167387 167409 67430 ! 67452 67473 i 167495 i 1 67516 167638 1 167559 i 1 67580 67602 ii 67623 ij 67646 i 167666 67688 67709 67730 67762 67773 67795 67816 I 67837 ! 67859 167880 167901 I 67923 67944 1:67966 1 1 67987 1 1 68008 i 68029 ii 68051 I 68072 68093 168115 1 68136 68157 iG8179 i 68200 47 Degrees^ 27 64 Log. Sines and Tangenta. (43°) Natural Sines. TABLE n. S ine. 9.833783 1 2 3 4 6 6 7 6 9 10 U 12 13 14 16 J 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 833919 834054 834189 834325 834460 834595 834730 834865 834999 835134 9,835269 835403 835538 835672 835807 835941 836075 836209 836343 836477 9,836611 836745 836878 837012 837146 837279 837412 837646 837679 837812 9,837945 838078 838211 838344 838477 838610 838742 838875 839007 839140 9,839272 839404 839536 839668 839800 839932 840064 840196 840328 840459 9,840591 840722 840854 840986 841116 841247 841378 841509 841640 8 41771 Cosine. D. ly^ l Cosine. i D. m Tang. D. 10" | Cotang 22.6 22.5 22.5 22.5 22.5 22.5 22.5 22.5 22.5 22.4 22.4 22.4 22.4 22.4 22,4 22.4 22.4 22.3 22.3 22.3 22.3 22.3 22.3 22.3 22.2 22.2 22.2 22.2 22.2 22.2 22.2 22.2 22.1 22,1 22.1 22.1 22,1 22,1 22.1 22.1 22,0 22.0 22,0 22.0 22.0 22,0 22,0 9 21,9 21.9 21.9 21.9 21.9 21.9 21.9 21.8 21.8 21.8 21.8 21.8 1.864127 864010 863892 863774 863656 863538 863419 863301 863183 863064 862946 .862827 862709 862590 862471 862353 862234 862115 861996 861877 861758 .861638 861519 861400 861280 861161 861041 860922 860802 860682 860662 .860442 860322 860202 860082 869962 859842 859721 859601 869480 859360 859239 859119 858998 858877 868756 868636 868514 868393 858272 858151 ,858029 857908 857786 857665 857543 857422 857300 857178 857056 856934 Sine. 19.6 19.6 19.7 19.7 19.7 19,7 19.7 19.7 19.7 19.7 19.8 19.8 19.8 19.8 19.8 19.8 19.8 19.8 19.8 19.8 19.9 19.9 19.9 19.9 19.9 19.9 19.9 19.9 19.9 20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.1 20.1 20,1 20.1 20,1 20,1 20.1 20.1 20,2 20.2 20.2 20,2 20,2 20.2 20.2 20.2 20,2 20.3 20.3 20.3 20.3 20.3 20.3 970162 970416 970669 970922 971175 971429 971682 971935 972188 9.972441 9.969656 .„ o 969909 ^Z-i 972948 973201 973454 973707 973960 974213 974466 974719 9.974973 975226 976479 976732 975985 976238 976491 976744 976997 977250 .977503 977766 978009 978262 978615 978768 979021 979274 979527 979780 9.980033 980286 980538 980791 981044 981297 981560 981803 982056 982309 9.982562 982814 983067 983320 983573 983826 984079 984331 984584 9 84837 Cotang. 1 42.2 42.2 42.2 42.2 42,2 42,2 42.2 42.2 42.2 42.2 42.2 42.2 42.2 42,2 42,2 42.2 42.2 42.2 42.2 42.2 42,2 42.2 42.2 42.2 42.2 42.2 42.2 42.2 42.2 42.2 42.2 42.2 42.2 42.2 42.2 42.2 42.2 42.2 42.2 42.2 42.2 42.2 42,2 42,1 42.1 42.1 42.1 42.1 42,1 42,1 42.1 42,1 42,1 42,1 42,1 42,1 42,1 42.1 42.1 I- 10.030344 030091 029838 029584 029331 029078 028825 028571 028318 028066 027812 10.027559 027306 027062 026799 026546 026293 026040 025787 025534 025281 10.025027 024774 024521 024268 024015 023762 023609 023266 023003 022750 10.022497 022244 021991 021738 021485 021232 020979 • 020726 020473 020220 10.019967 019714 019462 019209 018956 018703 018450 018197 017944 017691 I 10.017438 017186 016933 016680 016427 016174 015921 015669 015416 015163 "fang. |N .sine. N. cos 68200 68221 68242 68264 68285 68306 68327 68349 68370 68391 68412 68434 68455 68476 68497 68618 68639 68561 68582 68603 68624 68645 68666 68709 68730 68751 68772 68793 68814 68836 68857 68878 68899 68920 68941 68962 73135 73116 73096 73076 73056 73036 73016 72996 72976 72957 72937 72917 72897 72877 72857 72837 72817 72797 72777 72757 72737 72717 72697 72677 72657 72637 72617 72697 72677 72557 72537 72517 72497 72477 72457 72437 72417 68983 72397 6900472377 6902672367 69046 72337 69067 69088 69109 69130 69151 69172 72317 72297 72277 72257 72236 72216 69193 72196 169214172176 69235 72156 69256I721S6 69277 i721 16 69298172095 69319 72075 6934072055 6936172035 69382 72015 69403 69424 69445 69466 71995 71974 71954 71934 N. cos.lN.sine. 46 Degrees. TABLE IT. Log. Sines and Tangents. (44°) Natural Sines. 65 S ine. ,841771 841902 842033 842163 842294 842424 842555 842685 842815 842946 843076 .843206 843336 843466 843595 843725 843855 843984 844114 844243 844372 844502 844631 844760 844889 845018 845147 845276 845405 845533 846662 9.845790 845919 846047 846175 846304 846432 846560 846688 846816 846944 .847071 847199 847327 847454 847582 847709 847836 847964 848091 848218 .848346 848472 848599 848726 848852 848979 849106 849232 849486 Cosine. D. 10" 21.8 21.8 21.8 21.7 21,7 21.7 21.7 21,7 21.7 21.7 21.7 21.6 21,6 21,6 21.6 21.6 21.6 21.6 21.6 21.5 21.5 21.5 21.5 21.5 21.5 21.5 21.6 21.4 21.4 21.4 21.4 21,4 21.4 21.4 21.4 21.4 21.3 21.3 21.3 21.3 21.3 21.3 21.3 21,3 21.2 21,2 21.2 21,2 21,2 21.2 21.2 21.2 21.1 21.1 21.1 21.1 21.1 21.1 21.1 21.1 Cosine. 9.866934 866812 866690 856568 856446 866323 866201 866078 855956 866833 855711 855588 855465 855342 865219 855096 854973 854850 854727 854603 854480 9.854356 854233 854109 853986 853862 863738 853614 853490 853366 853242 853118 852994 852869 862745 862620 852496 852371 852247 852122 861997 9.851872 851747 851622 851497 851372 851246 851121 850996 850870 860745 850619 850493 850368 850242 850116 849990 849864 849738 849611 849485 Sine. D.10" 20.3 20.3 20.4 20.4 20.4 20.4 20.4 20.4 20.4 20.4 20.6 20.6 20 20 20 20 20 20 20.6 20.6 20,6 20.6 20.6 20,6 20.6 20.6 20.6 20.7 20.7 20,7 20.7 20.7 20,7 20.7 20.7 20.7 20.8 20,8 20,8 20.8 20.8 20.8 20.8 20.8 20.9 20.9 20,9 20.9 20,9 20,9 20,9 20.9 21,0 21.0 21.0 21.0 21.0 21.0 21.0 21.0 Tang. 984837 985090 985343 985696 985848 986101 980354 986607 986860 987112 987365 987618 987871 988123 988376 988629 988882 989134 989387 989640 989893 9.990146 990398 990661 990903 991156 991409 991662 991914 992167 992420 992672 992926 993178 993430 993683 993936 994189 994441 994G94 994947 9.995199 996452 995705 995967 996210 996463 996716 996968 997221 997473 ,997726 997979 998231 998484 998737 998989 999242 999495 999748 10.0 00000 Co tang. D. 10" Cotang. N. sine 10.015163 014910 014667 014404 014162 013899 013646 013393 013140 013888 012636 10.012382 012129 011877 011624 011371 011118 010866,1 0106131 010360,' 010107 i 10.009855 ii 009602 009349 009097 008844 608691 008338 008086 007833 007580 10-007328 007075 006822 006570 006317 006064 005811 i 005559 005306 005063 10.004801 004648 004296 004043 003790 003537 003285 003032 002779 i 002527 ! 10.002274 I 002021 001769 001516 001263 001011 000758 000505 000263 000000 69466 69487 69508 69529 69549 69570 69591 69612 69633 71772 69654 71752 Tang. 69675 69696 69717 69737 69758 69779 69800 69821 69842 698G2 69883 69904 69925 69946 69966 69987 70008 70029 70049 70070 70091 70112 70132 70153 70174 70196 70215 70236 70257 70277 70298 70319 70339 70360 70381 70401 70422 70443 70463 70484 70505 70525 70546 70567 70587 70608 70628 70649 70670 70691 N. cos 71934 71914 71894 71873 71853 71833 71813 71792 71732 71711 71691 71671 71650 71630 71610 71590 71569 71549 71529 71508 71488 71468 71447 71427 71407 71386 71366 71345 71325 71305 71284 71264 71243 71223 71203 71182 71162 71141 71121 71100 71080 71059 71039 71019 70998 70978 70957 0937 70916 70896 70876 70856 70834 70813 70793 70772 70752 70731 7071170711 N. COS. N. mm- 45 Degrees. 66 LOGARITHMS TABLE HI . LOGARITHMS OF NUMBERS, FROM 1 TO 110, INCLUDING TWELVE DECIMAL PLACES N. I Log. N. 36 Log. 0. 000 000 000 000 1. 656 302 500 767 3 0. 301 029 995 644 37 I. 568 201 724 067 3 0. 477 121 254 720 38 1. 579 783 596 617 4 0. 602 059 991 328 39 1. 591 064 607 264 6 0. 698 970 004 336 40 1. 602 059 991 328 6 0. 778 151 250 384 41 1. 612 783 846 720 7 0. 845 098 040 014 42 I, 623 249 290 398 8 0. 903 089 986 992 43 1. 633 468 465 679 9 0. 954 242 509 440 44 1. 643 452 676 486 10 1, 000 000 000 OOO 45 U 663 212 513 775 H 1. 041 392 685 158 46 1. 662 767 831 682 12 1. 079 181 246 048 47 1. 672 097 857 936 13 1, 113 943 352 309 48 1. 681 241 237 376 14 1. 146 128 035 678 49 1. 690 196 080 028 15 1. 176 091 259 059 60 1. 698 970 004 336 16 1. 204 119 982 656 51 1, 707 570 176 098 17 1, 230 448 921 378 52 1. 716 003 243 635 18 1. 255 272 505 103 53 1. 724 275 869 601 19 1. 278 753 600 953 54 I. 732 393 769 823 20 1, 301 029 995 664 55 1. 740 362 689 494 21 1, 322 219 294 734 56 1 748 188 027 006 22 1. 342 422 680 822 67 1. 756 874 855 672 23 1. 361 727 836 076 58 1. 763 427 993 663 24 1. 380 211 241 712 69 1. 770 852 Oil 642 25 1. 397 940 008 672 60 1. 778 161 250 384 26 1. 414 973 347 971 61 I. 785 329 835 Oil 27 1. 431 363 764 159 62 1. 792 391 689 492 28 1. 447 158 031 342 63 1.' 799 340 649 464 29 1. 462 397 997 899 64 1, 806 179 973 984 30 1. 477 121 254 720 65 1. 812 913 366 643 31 1. 491 361 693 834 66 1. 819 543 935 542 32 1. 505 149 978 320 67 1. 826 074 302 701 33 1. 618 513 939 878 68 1. 832 608 912 706 34 1. 531 478 917 042 69 1. 838 849 090 737 35 1. 544 068 044 350 70 1. 845 098 040 014 OF NUMBERS. 67 ■ N. Log. N. Log. 71 851 258 348 719 91 959 041 392 321 72 857 332 496 431 92 968 787 827 346 73 863 322 860 120 93 968 482 948 654 74 869 231 719 731 94 973 127 853 600 75 875 om 263 392 95 977 723 605 289 76 880 813 592 281 96 982 271 233 040 77 886 490 725 172 97 986 771 734 266 78 892 094 602 690 98 991 226 075 692 79 897 627 091 290 99 995 635 194 59S 80 903 089 986 992 100 2. 000 000 000 000 81 908 485 018 879 101 2, 004 321 373 783 82 9i3 813 852 384 102 2. 008 600 171 762 83 9i9 078 092 376 103 2. 012 837 224 705 84 924 279 286 062 104 2. 017 033 339 299 85 929 418 925 714 105 2. 021 189 299 070 8G 934 498 451 244 106 2. 025 305 865 265 87 939 519 252 619 107 2. 029 383 777 685 88 944 482 672 150 108 2. 033 423 755 487 89 949 390 006 645 109 •2. 037 426 497 941 90 954 242 509 439 110 2. 041 392 685 158 LO GARITHMS OF THE PRIME NUMBERS FROM IK ) TO ] 11^9. I NCLUDING TWELV] K DE CIMAL PLACES N. Log. N. Log. 113 2. 053 078 443 483 197 2. 294 466 266 162 127 2. 103 803 720 956 199 2. 298 853 076 410 131 2. 117 271 295 656 211 2. 324 282 455 298 137 2. 136 720 567 156 223 2. 348 304 863 222 139 2. 143 014 8G0 254 227 2. 356 025 857 189 149 3. 173 186 268 412 229 2. 359 835 482 343 151 2. 178 976 947 293 233 2. 367 355 922 471 157 2. 195 899 653 409 239 2. 378 397 902 352 163 2. 212 187 604 404 241 2. 382 017 042 576 167 2. 222 716 471 148 251 2. 399 673 721 509 173 2. 238 046 103 129 257 2. 409 933 123 332 179 2. 252 853 030 980 263 2. 419 955 748 490 181 2. 257 678 574 869 269 2. 429 752 261 993 191 2. 281 033 367 248 271 2. 432 969 290 877 193 2. 285 557 309 008 1 277 2. 442 479 768 999 68 LOGARITHMS N. Log. N. Log. 281 2. 448 706 319 906 601 2. 778 874 471 998 283 2. 451 786 435 523 607 2. 783 188 691 074 293 2. 466 867 523 562 613 2. 787 460 556 130 307 2. 487 138 375 477 617 2. 790 285 164 033 311 2. 492 760 389 026 619 2. 791 690 648 987 313 2. 495 544 337 650 631 2. 800 029 359 232 317 2. 601 069 267 324 641 2 806 868 879 634 331 2. 519 827 993 783 643 2! 808 210 973 921 337 2. 627 629 883 034 647 2. 810 904 280 666 347 349 2. 540 329 475 079 663 2. 814 912 981 274 2. 642 826 426 673 659 2. 818 885 490 409 353 2. 647 774 138 016 661 2. 820 201 459 485 359 2. 655 094 447 578 673 2- 828 015 064 225 367 2. 664 666 064 254 677 2. 830 588 667 946 373 2. 571 708 831 809 683 2. 834 420 703 630 379 2. 678 639 209 957 691 2. 839 477 902 551 383 2. 583 198 773 980 701 2. 845 718 017 237 389 2. 589 949 601 323 709 2. 850 646 235 112 397 2. 598 790 506 763 719 2. 856 728 890 383 401 2. 603 144 372 687 727 2- 861 634 410 855 409 2. 611 723 296 019 733 2. 865 103 970 639 419 2. 623 214 m2 971 739 2. 868 643 643 162 421 2. 624 282 085 835 743 9. 870 988 813 759 431 2. 634 477 268 999 761 2. 876 639 937 004 433 2^ 636 488 016 871 767 2. 879 095 879 497 439 2. 642 464 520 242 761 2. 881 384 656 769 443 2. 646 403 726 235 769 i'. 885 926 339 800 449 2. 652 246 388 777 773 2. 888 179 493 917 467 2. 659. 916 200 064 787 2. 896 974 732 358 461 2. 663 70& 925 389 7a7 2- 901 468 321 400 ■ 463 2.. 665 580 994 012 809 2. 9f77 948 459 773 467 2. 669 317 88S 008 ; 811 2. 909 OQO 864 210 479 2. 680 335 513 415 821 2I 914 343 157 120 487 2. 687 628 961 120 823 2. 915 39» 835 203 491 2. 691 081 487 026 827 2^ 9^17 505 509 487 499 2. 698 100 545 623 829 2. S.18. 654 530 558 503 2. 701 567 985 083 839* 2. 923 761 960 830 1 609 ; 2.. -zoe 717 782 345 853 2. 980 949 CGI 1G3 1 621 2. 716 837 623 304 857 2. 932 980 821 917 523 2. 718 602 688 873 859 2. 933 903 163 838 541 2v 733 197 26B 134 863 2. 936 010 794 546 547 2. 737 987 326 358 877 2. 942 999 593 360 557 2. 745 855 195 li92 881 2. 944 975 908 412 563 2. 750 508 395 940 : 883 2. 946 960 703 512 569 2, 755 112 178 598 887 2. 947 923 619 839 571 2. 756 636 108 333 907 2. 957 607 287 059 677 2. 761 175 813 171 911 2. 959 518 376 972 587 2. 768 638 004 465 919 2. 963 315 513 6C0 593 2. 773 054 693 364 929 ^. 968 015 713 997 599 2. 777 427 303 257 937 2. 971 739 590 780 1 OF NUMBERS. 69 941 ii. 947 2. 953 2. 967 2. 971 2. 977 2. 983 2. 991 2. 997 2. 1009 3. 1013 3. 1019 3. 1021 3. 1031 3. 1033 3. Log. 973 689 620 234 976 349 979 055 979 092 900 639 985 426 474 084 987 219 229 907 989 894 559 717 992 553 512 733 996 073 604 003 998 695 158 313 003 891 170 203 005 609 445 427 008 174 244 007 009 025 742 086 013 258 660 430 014 100 321 518 N. Ix>g. 1039 3. 016 615 647 658 1049 3. 020 775 488 195 1051 3. 021 602 716 026 1061 3. 025 715 383 898 1063 3. 026 533 264 623 1069 3. 028 977 705 205 1087 3. 036 229 513 712 1091 3. 037 824 749 671 1093 3. 038 620 157 372 1097 3. 040 206 627 671 1103 3. 042 575 612 437 1109 3. 044 931 546 149 1117 3. 048 053 173 103 1123 3. 050 379 756 239 1129 3. 052 693 942 370 It is not necessary to extend this table, as the loj^arithm of any one of the higher numbers can be readily computed by the fol- lowing formula, which may be found in any of the standard works on algebra, namely : Log. (2-|-i)=log. z-f 0.8685889638 I -i ) The result will be true to ten decimal places for all numbers over 1000, and true to twelve decimals for all numbers over 2000. The logarithms of composite numbers can be determined by the combination of logarithms already in the table, and the prime numbers from the formula. Thus, the number 3083 is a prime number, find its logarithm, true to ten places of decimals. We first find the logarithm of 3082. By factoring this num- ber, we find that it may be composed by the multiplication of 46 into 67. Log. 46 1. Log. 67 1. Log. 3082 3. Log. 3083=3.4888321343 662 757 8316 826 074 3027 488 832 1343 Now 0-8685889fi38 61 6d We give a few additional prime numbers : 1151 1153 1163 1171 1181 1187 1193 1201 1213 1217 1223 1229 1231 1237 1249 1259 1277 1279 1283 1289 1291 1297 1301 1303 1307 1319 1321 1327 1361 1367 1373 1381 1399 1409 1423 1427 1429 1433 1439 1447 1461 1453 1459 1471 1481 1483 1487 1489 1493 1499 1511 1523 1531 1543 1549 1553 1559 1667 1671 1579 70 AUXILIARY LOGARITHMS. AUXILIARY LOGARITHMS*. N. Log. N. Log. 1. 009 0. 003 891 170 2031 1. 0009 0. 000 390 576 304] 1. 008 0. 003 461 627 188 1. 0008 0. 000 347 233 698 1. 007 0. 003 030 465 635 1. 0007 0. 000 303 836 798 1. 006 0. 002 597 985 739 1. 0006 0. 000 260 435 661 1. 005 0. 002 166 071 750 [►A 1. 0005 0. 000 217 099 966 Ib 1. 004 0. 001 733 722 804 1. 0004 0. 000 173 690 053 1. 003 0. 001 300 943 017 1. 0003 0. 000 130 268 803 1. 002 0. 000 867 721 529 1. 0002 0. 000 086 850 213 1. 001 0. 000 434 077 479 J 1. 0001 0. 000 043 427 277^ — N. Log. 1. 00009 0. 000 039 084 741 1 1. 00008 0. 000 034 742 166 1. 00007 0. 000 030 399 546 1 . 00006 0. 000 026 066 884 1. 00005 0. 000 021 714 178 1. 00004 0. 000 017 371 430 1. 00003 0. 000 013 028 638 C 1. 00002 0. 000 008 685 802 1. 00001 0. 000 004 342 923 (a) 1. 000001 0. 000 000 434 294 (b) 1. 0000001 0. 000 000 043 429 (c) 1. 00000001 0. 000 000 004 343 (d) 1. 000000001 0. 000 000 000 434 (e) 1. 0000000001 0. 000 000 000 043 (f)J N» imber. Log. 0. 43. 12944819 —1. 637 784 298 This decimal number is the modulus of our system of logarithms. | Its loga rithm is very useful in correcting other logarithms, as may be seen in the Chapter on Logarithms. TAULfci V. TAULE VII. "! Dip of the Sea Horizon. Mean Refraction of Celestial Objects. '.^S b2 \^'» g2 Alt Rcfr. || Alt.|Uefr , Alt llffr.;| Alt. Refr. Alt. Refr »d5- o-o ?.| §0 / rf\ f f 1 "o / II 1 '^ / 1 II —FT 0^ 5'S, 55^ 5- a c 33 OlllO C 5 16 >( 2 35 32 C )1 30 67 24 P^ H ^2. P D* IC 31 32 IC 6 10 ' 1( 2 24 40 1 29 68 23 ~t If „ 2C 29 60 20 5 05 2( 2 22 33 C 1 28 69 22 1 59 38 6 4 6 18 6 32 6 45 6 58 7 10 7 12 3C 28 23 30 5 00 3( 2 21 2C 1 26 70 21 2 I 24 1 42 41 44 4C 27 OG 40 4 66 4( 2 29 4C 1 26 71 19 3 5C 25 42 60 4 61 5( 2 28 34 C 1 24 72 18 4 1 58 2 12 2 25 47 50 53 1 C 24 29 11 4 47 21 ( 2 27 2C 1 23 73 17 6 10 23 20 10 4 43 IC 2 26 4C 1 22 74 16 6 2C 22 15 20 4 39 2C 2 25 35 C 1 21 76 15 7 2 36 66 3C 21 15 30 4 34 3C 2 24 20 1 20 76 14 8 2 47 69 7 24 4C 20 18 40 4 31 4C 2 23 40 1 19 77 13 9 2 57 62 ' 745 50 19 25 60 4 27 5C 2 21 36 1 18 78 12 10 3 07 66 7 66 2 18 35 12 4 23 2^ C 2 20 30 1 17 79 11 11 3 16 68 8 07 10 17 48 10 4 20 IC 2 19 37 1 16 80 10 12 3 25 71 8 18 20 17 04 20 4 16 20 2 18 30 1 14 81 9 13 3 33 74 8 28 30 16 24 30 4 13 30 2 17 38 1 13 82 8 14 3 41 77 8 38 40 16 45 40 409 40 2 16 30 1 11 83 7 15 3 49 80 8 48 50 15 09 60 4 06 50 2 15 39 1 10 34 6 16 3 56 83 8 58 3 14 34 13 4 03 23 2 14 30 1 09 85 5 17 4 04 86 9 08 10 14 04 10 4 00 10 2 13 400 1 08 86 4 18 4 11 89 9 17 20 13 34 20 3 57 20 2 12 30 1 07 87 3 19 4 17 92 9 26 30 13 06 30 3 54 30 2 11 41 1 05 88 2. 20 4 24 95 9 36 40 12 40 49 3 61 40 2 10 30 1 04 89 1 21 4 31 98 9 45 60 12 15 60 3 48 50 2 09 42 1 03 90 22 4 37 JXl 9 54 4 11 61 14 3 45 24 02 08 30 1 02 23 4 43 104 10 02 10 11 29 10 3 43 102 07 43 1 01 24 4 49 107 10 11 20 11 08 20 3 40 202 06 30 1 00 26 4 55 110 10 19 30 10 48 30 3 38 30 2 06 44 69 26 5 01 III 10 28 40 10 29 40 3 35 40 2 04 80 58 27 5 07 116 10 36 50 10 11 60 3 33 60 2 03 45 57 28 29 5 13 5 18' 119 122 10 44 10 62 5 9 64 15 3 30 25 2 02 30 66 30 5 241 125 11 00 10 9 38 10 3 28 10 2 01 46 55 31 5 29 1 128 11 08 20 9 23 20 3 26 20 2 00 30 54 82 5 34 131 11 16 30 9 08: 30 3 24 30 1 59 47 53 33 5 39 134 11 24 40 8 54 40 3 21 40 1 58 30 62 34 5 44 137 11 31 50 8 41 50 3 19 50 1 67 48 61 35 5 49I 140 11 RQ 6 8 28 16 3 17 26 1 66 30 60 10 8 16 8 03 10 20 3 16 3 12 10 20 1 65 1 56 49 30 49 49 20 TABLE VI. 30 7 15 30 3 10 30 154 60 48 Dip of the Sea Horizon at 40 7 40! 40 l^ 40 1 63 30 47 different Distances from it. 50 7 7 30: 7 2O1 60 17 3 06 3 04 50 27 1 62 1 61 51 30 46 45 10 fT 1 li 10 20 30 3 03 3 01 15 30 1 50 1 49 52 30 53 44 ) 44 Dist. Hight of Eye in i't.l IV 1 xii 20 T noJ in Miles. 5 10 16 2 0J25 30 30 40 6 63| 2 69 46 28 1 48 43 T ~7~ ~T ~ - — ~ 6 45| 40 2 67 1 47 30 O 42 i 11 22 34 4 5 '56 68 50 6 37i 60 2 56 15 1 46 54 1 45 56 3 41 6 17 2 2 '28 34 8 6 29i 18 2 64l 30 } 40 4 8 12 1 5J19 23 10 6 22| 10 2 52 45 1 44 56 }38 I 4 6 9 li 2!l5 17 20 6 15 20 2 51i 29 1 42 57 ) 37 U 3 5 7 < ) 12 14 30 6 08 30 2 49 20 1 4ll58 3 35 H 3 4 6 I I 9 12 40 6 01 40 2 47 40 I 40|!59 ) 34 2 2 3 5 ( ) 8 10 50 5 56 50' 2 46 30 I 38|;60 ) 33 2i 2 3 6 ( 5 7 8 9 5 98 19 0; 2 44 20 I 37JI61 ) 32 3 2 3 4 I ) 6 7 10 5 42 10 i 243; 40 I 361^2 ) 80 3i 2 3 4 £ ) 6 6 20 6 46 20^ 2 41 31 I 35; 63 I 33' 64 I 32 165 0( ) 29 4 2 3 4 ^ I 5 6 30 6 41 30 5 J 40 20 )28 5 2 3 4 4 I 5 5 40 6 26 40 S \ 38 40 ) 26 6 2 3 4 4 I 5 6| 60 6 20! 60 2 37il32 0| I 3ll^6 0( ) 25 1 1 >w. 1 3f43 37 -.^ ^'» ^^