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About Google Book Search Google's mission is to organize the world's information and to make it universally accessible and useful. Google Book Search helps readers discover the world's books while helping authors and publishers reach new audiences. You can search through the full text of this book on the web at |http: //books .google .com/I I « I I COURSE OF MATHEMATICS. IN THREE VOLUMES. COMP08BD FOR THE USE OF THE ROYAL MILITARY ACADEMY, BT ORDBR or HIS LORDCHIP THE MASTER GENERAL OF THE OM)NANC£. BY CHARLES HUTTON, LL.D. F.R.S. J^TB PROFESSOR OF MATHEMATICS IN THE BOTAK. MILITARY ACADEMY. THE SIXTH EDITIOtr, XHLAROID AilD CORRBCTBI^ toL. I. LONDON: PRINTEB FOR F, C. AND J, RIVINGTON; O. WILKXB AN9 f, ROBINSON ; J. walker; G. ROBINSON; LACKINOTON, ALLBNf' AND CO.; VERNOR, HOOD, AND SHARPE ; 6. KSARSLBY ; LONGMANy HURSTy REES, ORME, AND BROWN ; CADELI AN1I DA VIES ; J. CUTHELL ; B. CROSBY AND CO. ; J. RICHARDSON ; J. M. RICHARDSON; BLACK^ PARRY^ AND I^^NOS^URY ; QAIA AND CURTIS ; AND J. JOHNSON AND CO. 1811. ' * »• J I , • « \ V •■ I * ' t * ' • • • F w >'^ *<r* T. Dttviwp, Lombard 8tice<» X PHEFACE. A SHORT and Easy Course of t&e Mathematical Sciences h^s* long been ^considered as a desideratum for the use of Students in the different schools of education: one that should hold a middle rank between the more voluminous ^nd bulky collections of this kind, and the mi^e abstract % and brief common-place forms, of principles and. memo* fandums. «^ For Ung experience, in all Seminaries of Learningji ha$ j> shown, that such a work was very much wanted, and would >l prove a gre^t and general benefit ; as, for want of it, re- 4iourse has always been oblig^ to be bad to a number of other books, by different authors; selecting a part from one and a part from another, as seemed most suitable to thef purpose in hand, and rejecting the other parts— a practice which occasioned much expence and trouble, in procuring and using such a number of odd volumes, of various forms and modes of composition; besides wanting the benefit of uniformity and referencei which are found in a regular series of composition. To remove these inconveniences, the Author of the pre- sent work has been induced, from time to time, to compose various parts of this Course of Mathematics; which the experience of many years' use in the Academy has enabled him to adapt and improve to the most useful form' and quantity, for the benefit of instruction there. And, to render ti^t benefit more eminent and lasting, the Master General of the Ordnance has been pleased to give it its present form, Ky ordering it to be enlarged and printed, for the use of the . Roayl Military Academy. A 2 As IT PREFACE. As this work has been composed expressly with the inten- tion of adapting it to the purposes of academical education, it is not designed to hold out the expectation of an entire new ipass of inventions and discoveries: but rather to collect ^d arrange the most useful known principles of mathe- matics, disposed in a convenient practical form, demonstrated in a plain and concis'e way, and illustrated with suitable ex- amples; rejecting whatever seemed to be matters of mere curiosity, and retaining only such parts and branches, as have a direct tendency and application to some useful purpose in life or profession. It is however expected that much that is new will be found in many parts of these volumes; as well in the matter, as in the arrangement and manner of demonstration! throughout the whole work, especially in the geometry, which is" ren- dered much more easy and simple than heretofore ; and in the conic-sections, which are here treated in a manner at once new, easy, and natural ; so much so indeed, that all the propositions and their demonstrations, in the ellipsis, are the very same, word for word, as those in the hyperbola, using only, in a very few places, the word sum, for the word differ^ ence: also in many of the mechanical and philosophical parts which follow, in the second volume. In the conic sections^ too, it may be observed, that the first theorem of each sec- tion only is proved from the cone itself, and all the rest of the theorems are deduced from the fint, or from each other, in a very plain and simple manner. Besides renewing most of the rules, and introducing every- where new examples, this edition is much enlarged in several places; particularly by extending the tables of squares and cubes, square roots and cube roots, to 1000 numbers, which will be found of great use in many calculations ; also by the table of logarithms at the end of the first volume, and of lo- garithms, sines, a(nd tangents, at the end of the second Yolume ; by the add;ition of Cardan's rules for resolving cubic equations I PREFACE. V equations; with tables and rules for annuities; and many other improvements in different parts of the work* Though the several parts of this course of mathecnatics are ranged in the order naturally required by such elements^ yet students may omit any of the particulars that may be thought the least necessary to their several purposes; or they may study and learn various parts in a di&rent order from their present arrangement in the book, at the discretion of the tutor. So, for instance, all the notes at the foot of the pages may be omitted, as well as many of the rules ; particularly the 1st or Common Rule for the Cube Root, p. 85, may well be omitted, being more tedious than useful. Also the chaf^ ters on Surds and Infinite Series, in the Algebra : or these might be learned after Simple Equations. Also Compound Interest and Annuities at the end of the Algebra. Also any part of the Geometry, in vol. 1 ; any ojf the branches in vol. 2, at the discretion of the preceptor. And, in any of the parts, he may omit some of the examples, or he may give more than are printed in the book ; or he may Tery pro- fitably vary or change them, by altering^the Aumbera oc» casionally.— -As to the quantity of writing ; the author would recommend, that the student copy out into his fair book no more than the chief rules which he is directed to learn off by rote, with the. work of pne example only to' each rule, set down at full length : omitting to set down the work of all thff other examples, how many soever he may be directed to work out upon his slate or waste paper. — In short, a great deal of the business, as to the quantity and order and maimeri must depend on the jtt4goient of the discreet and {Hmdeiit #utor or director* CONTENTS OF VOLUME I. GENERAL Pr^mmaty Prtnci^ P«f» ARITHMETIC. / Ihtaiiim and Nnmeratwn • • . • . 4 Rsman Notation . • • • * • 7 AMUim 8 jfuitraetion ....... 11 J/ukiptfcation • . .13 18 ^Bedmtion . . ^ 2S ' Gmpaund JMMon • . 92 ■ Subtracit&H .... S< M ^o -— Dioision . . . . « 41 Getien Bule^ or Stde of Three • . • . 44 Compound Pfopofpfion . . . . , 49 Vtdgar Fractions 51 Beduetion of Vulgar Fractions . . . , 54 Addition of Vulgar Fractions . . . , 62 Subtraction of Vulgar Fractions 62 MtdttpKcaiim of Vulgar Fractions . . 63 DMsiou of Vulgar Fractions- . . . 64 i?Ei& g/" 7%retf in Vulgar Fractions 65 Decimal Fractions 66 Addition of Decimak . . . ^ . , 67 Subtraction of Decinuils 68 Multiplication of Djscimals ib. C0N1!ENI3. «n r»€» Dwision of Decimals . • « . 70 JMuction of Decimals • • . . 75 Mute of Three in Decimals • T« Dmsdecimals ... . . TT Iwfolution •» Evolution ...... «• To extract the Square Soot «1 To extract the Cube Root . . , «5 To -extract any Root whatever «8 Table qfJPawers and Roots «0 Ratios f Propor turns, and Progressions , . , . 110 Arithmetical Proportion • • • 111 Geometrical Proportion lie Musical Proportion . • • . i» fiUasosh^fj or Partnership , ft. Single Fellowship . • . . ISO Btmble Fellowship . . . . m Smple Interest . . • . . IS* Vmpound Inttrest . • « . . . . U7 dU^gatim Medial U9 ^Hkgatwn Alternate . » - . . . • Jbh^/^ Position . . . . . . . iS5 Double Position 1S7 Practical Suestions • . . < • 140 LOGARITHMS. • i^nitim and Propertks vfJJDgari^nm r . . 14S To 'Compute Logarithms . ■ , ■ : . . 149 Description and Use cf Lagarkhim . « . i» Mnltiplieation-by LogariU^ms . 157 iiivisum-by Logarithms . . . . .158 involution by Logarithms , 159 iSvoluiion h/ Logarithm* . . . kW «m CCKNTENTS. ALGEBRA. Definitions and Notation Addition . . • . Subtraction . . Multiplication . . Division . . . . Fractions . , . Involution * Evolution • . . Surds Infinite Series ArUhmetical Proportion . Arithmetical Progression PUes of S/iot or Shells Geometrical Proportion . Simple Equations . . Suadratic Equations . Cubic and Higher Powers Simple Interest Compound Interest - . Annuities • . . • GEOMETRY. Definitions • • • . . Axioms , Remarks and Theorems , Of Ratios and ProportionS'^Definitions Theorems Of Planes and Solids-^Definitions Theorems . • . . ; Problems . Application of Algebra to Geometry , , Problems . • . . . Table qf Logarithms . . . • Paftf 161 165 170 171 174 178 189 198 196 203 208 210 213 218 28(1 23d 247 256 257 260 265 271 ib. S09 318 326 328 S43 359 360 S66 COURSE OP MATHEMATICS, 4-0. GENERAL PRINCIPLES. UANTITT, or Magnitude, is any thing that mil admit of increase or decrease ; or that is capable of any sort of calculation or mensuration : such as numbers, lines, spacey time, motion, weight. 2, Mathematics is the science which treats of all kincls of quantity whatever, that can be numbered or measured.-— Th^ part which treats of numbering is called Arithmetic; and that which concerns measuring, or figured extension, is called Geametfy^-^These two, which are conversant about multitude and magnitude, being the foundation of all the other parts, are called Pure or Abstract Mathematics; be- cause they investigate and demonstrate the properties of ab- stract numbers ^d magnitudes of all sorts. And when th^se two parts are applied to particular or practical subjects, they .eonsdtute the branches or parts called Mixed Mathematics.--^ Mathematics is also distinguished into Speculative and Prvnv Jica/c viz. Speculative, when it is concerned in discovering properties and relations ; and Practical, when applied to practice and real use concerning physical objects* \, Vo^L B 3.to 2 GEigERAL PRINCIPLES. 3. In Mathematics are several general terms orprinci^es; such as. Definitions, Axioms, Propositions, Theorems, rro- blems. Lemmas, Corollaries, Scholiums, &c. 4. A Definition is the explication of any term or word in a science ; showing the sense and meaning in which the term is employed. — Every Definition ought to be clear, and ex- pressed in words that are common and perfectly well under- stood. 5. A Proposition is something proposed to be proved, or something required to be done ; and is accordingly either si Theorem or a Problem* 6. A Theorem is a demonstrative proposition; in which some property is asserted, and the truth of it required to be proved. Thus, when it is said that. The sum of the three angles of any triangle is equal to two right angles, this is a Theorem, the truth of which is demonstrated by Geometry. —A set or collection of such Theorems constitutes a Theory. 7. A Problem is a proposition or a question requiring something to be done ; either to investigate some truth or property, or to perform some operation. As, to find out the quantity or sum of all the three angles of any^triangle, or to di'aw one line perpendicular to another. A Limited Pro» hlem is that which has but one answer or solution.- An Un^ limited Problem is that which has innumerable answers. And a Determinate Problem is that which has a certain num- ber of answers. S. Solution of a Problem, is the resolution or answer given to it. A Numerical or Numeral Solutiony is the answer given in numbers. A Geometrical Solution^ is the answer given by the principles of Geometry. And a Mechanical Solution^ is one which is gained by trials. 9. A Lemma is a preparatory proposition, laid down in order to shorten the demonstration of the main proposition which follows it. 10. A Corollary y or Consectary^ is a consequence draW|x immediately from some proposition or other premises. 11. A Scholium is a remark or observation made by some foregoing proposition or premises. • 12. An Axiom, or Maxim, is a self-evident proposition ; requiring no formal demonstration to prove the truth of it ; but is received and assented to as soon as mentioned. Such as, The whole of any thing is greater than a part of it ; or. The whole is equal to all its parts taken together: or. Two quantities that are each of them equal to a third quantity, are equal to each other. / 13. A GENERAL PRINCIPLES. S Id* A Pastulatey or Petition, is sometliing required to be donC) which is so easy and evident that no person will hesi- tate to allow it. 14. An Hypothesis is a -supposition assumed to be truei in order to argue from^ or to found upon it the reasoning and demonstration of some proposition. ^ 15. Demonstration is the collecting the several arguments and proofs, and laying them together in proper order, to «}iow the truth of the proposition under consideration* 16. A Direct, Positive, or Ajffirmative Demonstration, h that which concludes with the direct and certain proof of the proposition in hand. — This kind of Demonstration is most satisfactory to the mind ; for which reason it is called some- times an Ostensive Demonstration. 17. An Indirect, or Negative Demonstration, is that which shows a proposition to be true, by proving that some absur- dity would necessarily follow if the proposition advanced were false. Thrs is also sometimes called Reductio ad Absurdum; becausie it shows the absurdity and falsehood of all supposi- tions contrary to that contained in the proposition. 18. Method is the art of disposing a train of arguments in a proper order, to investigate either the truth or falsity of a proposition, or to demonstrate it to others when it has been found out. — This is either Analytical or Synthetical. 19. Analysis, or the Analytic Method, is the 2rt or mode of finding out the truth of a proposition, by first supposing the thing to be done, and then reasoning back, step by step, till we arrive at some known truth. — ^Thi$ is also called the Method of InventioHy or Resolution; and is that which is com- monly used in Algebra. 20. Synthesis, or the Synthetic Method, is the searching out truth, by first laying down some simple and easy princi- jdes, and pursuing the consequences flowing from them till we arrive at the condusion. — This is also called the Method rfCon^sjfion; and is^ the reverse of the Analytic method, as £bis proceeds from known principles to an unknown conclu- sion ; while the other goes in a retrograde order, from the thing sought, considered as if it were true^ to some knpwn principle or fact. And therefore, when any truth has been foupd out by the Analytic method, it may be demonstrated by a process in the contrary order, by Synthesis. B 2 ARITH- t * 3 ARITHMETIC. jljLRITHMETIC is the art or science of numbering ; be- ing that branch of Mathematics which treats of the nature and properties of numbers, — ^When it treats of whole num- bers, it is called Vulgar^ or Common Arithmetic; but when of broken numbers, or parts of numbers, it is called Frmctions. Unity J or an Unitj is that by which every thing is called one ; being the beginning of number ; as, one man, one baU^ one gun. Number is either simply one, or a compound of several units % as, one man, three men, ten men. An Integer^ or Whoh Number^ is some certain precise quantity ofunits ; as, one, three, ten.— These are so called as distinguished from Fractions^ which are broken numbers, or parts of numbers ; as, one-half, two-thirds, or three-fourths. NOTATION AND NUMERATION. Notation, or Numeration, teaches to denote or ex- press any proposed number, either by word$ or characters ; or to read and write down any sum or number. The numbers in Arithmetic are expressed by the following ten digits, or Arabic numeral figures, which were intitxlucea into Europe by the Moors, about eight or nine hundred ^ years since ; viz. 1 one, 2 two, 3 three, 4 four, 5 five, 6 six, 7 seven, 8 eight, 9 nine, cipher, or nothing. These cha* racters or figures were formerly all called by the general name of C^hers ; whence it came to pass that the art of Arithmetic was then often called Cohering. Abo the first nine are called Significant Figures^ as distinguished from the cipher, which is of itself quite insignificant. Besides this value of those figures, they have also another, which depends on the place they stand m when joined toge- ther; as in the following table : Units NOTATION AND NU»ffiRATION. S 9 Si ^ ' o CO « »3 r^* a J rS S*3 ,2 JS « &€. 98765432 1 98 7 65 4 3 2 9 8 7 6 5 4 3 9 B 7 6 .5 4 9 8 7 6 5 9 8 7 6 9 8 7 9 - 8 9 fl Here, any figure in the first place, reckoning from right to left, denotes pnly its own simple ralue ; but that in the second place, denotes ten times its simple value ; and that in the third place, a hundred times its simple value ; and so on : the value of any figure, in each successive place, being always ten times its former value. Thus, in the number 1796, the 6 in the first place denotes only six units, or simply six ; 9 in the second place signifies nine tens, or ninety ; 7 in the third place, seven hundred ; and the 1 in the fourth place, one thousand : so that the whole number is read thus, one thousand seven hundred and ninety^six. As to the cipher, 0, though it signify nothing of itself, yet being joined on the right-hand side to other figures, it in- creases their value in the same ten>fbld proportion : thus, 5 signifies only five ; but 50 denotes 5 tens, or fifty ; and 500 is five hundred ; and so on. For the more easily reading of large numbers, they are divided into periods and half-periods, each half-period con« sisting of three figures ; the name of the first period being units; of the second, millions; of the third, millions of millions, or bi*millipns, contracted to billions : of the fourth, millions of millions of millions, or tri-millions, contracted to trillions, and so on. Also the first part of any period is so many units of it, and the latter part so many thousand?. The 6 ARITHMETIC. The following Table contains a summarj ef the whole docUine. Periods* QuadrilL; Trillions; Billions; Millions; Units. th. un. th« un, th. un. th. un. th. un. 123,456; '78»,098; 765,432; 101,234; 567,890. Numeration is the reading of any number in words that is proposed or set down in figures ; which will be easily done by help of the following rule, deduced from the fore- going tablets and observations — ^viz. Divide the figures in the proposed number, as in the sum- mary above, into periods and half-periods ; then begin at the left-hand side» and r^ad the figures with the names set to them in the two foregoing tables. EXAMPLES. Express in WQr4s the fbllo\ving numbers ; viz^ 34 96 180 304 6134 5028 15080 72003 109026 483500 2500639 7523000 13405670 47050023 309025600 4723507689 274856390000 6578600307024 Notation is the setting dgwn ii| figures any number pro- posed in words ; which is done by setting down the figures instead of the words or names belonging to them in the sum- mary above ; supplying the vacant places with ciphers where any words do not occur. EXAMPLES. Set down iii figures the following numbers ; Fifty-seven. Two hundred eighty- six. Nine thousand two hundred and ten. Twenty-seven thousand five hundred and ninety-four. Six hundred and forty thousand, four hundred and eighty-one. Three millions, two hi^ndred sixty thousand, one hundred and six. Four NOTATION AND NUMERATrON. Four hundred and eight millions, two hundred and fifty-five thousand) one hundred and ninety-two. Twenty-seven thousand and eight millions, ninety-six thou- sand two hundred and four. Two hundred thousand and five hundred and fifty millions) one hundred and ten thousand, and sixteen. Twenty-one billions, eight hundred and ten millions^ sixty- four thousand) one hundred and fifty. Of the Roman Notation. The Romans, like several other nations, expressed their numbers by certain letters of the alphabet. The Romans used only seven numeral letters, being the seven following capitals: viz. I for one; Y for Jive; X for ten; lu for fifty; C for an hundred; D for five hundred; ^ for a thousand* The other numbers they expressed by various repetitions and combinations of these, aft:er the following manner : 1 = 2 = S = 4 = 5 = 6 7 8 9 10 50 100 500 1000 2000 5000 6000 10000 50000 60000 100000 1000000 2000000 &c. I II III mi or IV V VI VII VIII IX X L C Dor ID M or CIO MM As often as any character is re- peated, so many times is its value repeated. A less character before a greater diminishes its value. A less character after a greater increases its value. V or IDD VI X or CCIOO Lj>r IDOD LX C^or CCCI0D3 Mor CCCCIODDD MM &c. For every 3 annexed, this be- comes 10 times as many. For every C and O, placed one at each end, it becomes 10 times as much. A bar over any number in- creases it 1000 fold. ExPLA- $ ARITHMETia • • • Explanation of certain Characters* There are various characters or marks used in Arithmetic, amd Algebra, to denote several of the operations and proposiF- tions ; the chief of which are as follow : + signifies/Zi/x, or addition. — - •«■ minus, or subtraction. X or . - multiplication. -5- - - division. : :: : - proportion! = - - equality. \/ - - square ropt. -J/ • - cube root, &c. ^ - - diffl between two numbers when it is not known which k the greater. Thus, 5 + Sj denotes that 3 is to be added to 5. .6 <— 2, denotes that 2 is to be taken from 6. '7 X 8, or 7 . S, denotes that 7 is to be multiplied by S. 5 -r 4, denotes that 8 is to be divided by 4. 2 : 3 : : 4 : 6, shows that 2 is to 3 as 4 is to 6. 6 + 4 = 10, shows that the sum of 6 and 4 is equal to 10. V^S, or 3i, denotes the square root of the number 3. ^5, or 5% denotes the cube root of the number 5. 7^, denotes that the number 1 is to be squared. 8^, denotes that the number 8 is to be cubed. &c» OF ADDITION. Addition is the collecting or putting of several numbers together, in order to find their sum, or the total amount of the whole. This is done as follows : ^ . ^ Set or place the numbers under each other, so that each? figure may stand exactly under the figures of the same value, iha^ ADDITION. ft . thatisi units under units^ tens under tens, hundreds under hundreds, &c. and draw a line under the lowest number, to separate the given numbers from their sum, when it is found. -~Then add up the figures in the column or row of units» and find how many tens are contained in that sum.-— Set down exactly below, what remains more than those tens, or if nothing remains, a cipher, and carry- as many ones to the next row as there are tens. — Next add up the second row, together with the number carried, in the same manner as the first. And thus proceed till the whole Is finished, setting down the total amount of the last row. To PROVE Addition. • First Method. — ^Begin at the top, and add together all the rows of numbers downwards; in the same manner. as they were before added upwards ; then if the two sums agree, it may be presumed the work is right.— -This method of proof is only doing the same work twice over, a little varied. Second Method. — ^Draw a line below the uppermost number, and suppose it cut off. — ^Then add all the rest of the numbers together in the usual way, and set their sum under the num- ber to be proved.— r-Lastly, add this last found number aad the, uppermost line together ; then if their sum be the same as' that found by the first addition, it may be presumed the work is right.— This method of probf is founded on the plain axiom", that " The whole is equal to all its parts taken together.*' » Third Method.'-^Add the figures in the uppermost line together, and find example i. how many nines ^e contained in their sum. — Reject those nines, and 3497 g 5 set down the remainder towards the 6512 .S 5 • right hand directly even with the 8295 ^ 6 figures in the line, as in the annexed © — example. -r-Do the same with each 18304 g 7 of the proposed lines of numbers, set- -^ • ^ — ting all these excesses of nines in a co- W himn on the right-hand, as here 5, 5, 6. Then, if the excess of 9^s in this sum, found as before, be equal to the excess of 9's in the total sum 1 8804, the work is probably right.-* Thus, the sum of the right-hand column, 5, 5, 6, is 1^, the excess of which above 9 is 7. ' A^o the sum of the figures in the 10 ARITHMETIC the sum toul 18304, is 16, the excess of which above 9 is also 7 J the same as the former^. 2. 12345 OTHER EXAMPLES. 3. 12345 4. 12345 67890 98765 43210 12345 67890 67890 9876 543 21 9 876 9087 ^ \ 56 234 1012 302445 90684 •I 23610 290100 78339 11265 302445 90684 23610 * This method of proof depends on a property of* the number g, which, except the number 3, belongs to no other digit whatever ; namely^ that '^ any number divided by g, will leave the same re- mainder as .the sum of its figures or digits divided by 9 :** which may be demonstrated jn this manner. Demonstration. Let there be any number proposed, as 4658. This, separated into its several parts, becomes 4000 + 600 + 50 + 8. But 4000 = 4 X 1000 = 4 X (9«9 + 1) = 4 X 999 -J- 4. In like manner 600 = 6 xgg + 6', and 50 = 5 X 9 + 5. There- fore the given number 4658 = 4 X 999 +44-6x99 + ^ + 5 X9-P5 + 8=:4X999 + <5X99 + 5X9 + 4 + 6 + 5 + 35 and 4058 -r-p = (4 X 999 -f 6 X 99 + 5 X 9 + 4 + 6 + 5 + 8) -r- 9. But 4 X 999 + 6x99 + 5 X9i8 evidently divisible by 9, without a remainder 5 therefore if the given num- ber 4658 be divided by 9ji it will leave the same remainder as 4 + 6 + 5 + 8 divided by p. And the same, it is evident, will hold for any other number whatever. In like manner, the same property may be shown to belong to the number 3 ; but the preference is usually given to the number 9, on account of its being more convenient in practice. Now, from the demonstration above given, the reason of the rule itself is evident ; for the excess of 9*s in two or more numbers being taken separately, and the excess of 9's taken also out of the sum of the former excesses, it is plain that this last excess must be equal to the excess of 9's contained in the total sum of all these numbers 5 all the parts taken together being equal to the whole. T his rule was first given by Dr. Wallis in his Arithmetic, published in the year l657* Ex. SUBTRACTION. 1 1 'Ex.5. Add 3426 J 9024; 5106; 8890; 1204, together. Ans. 27650* 6- Add 509267; 235809; 72920; 8392; 420; 21; and 9, together. Ans. 826838. 7. Add 2; 19; 8l7; 4298; 50916; 730205; 9180634^ together. Ans. 9966891. 8. How many days are In the twelve calendar months? Ans. 365. 9. How many days are there from the 15th day of April to thie 24th day of November, both days included ? Ans. 224. 40. An army consisting of 52714 infantry*, or foot, 5110 horse, 6250 dragoons, 3927 light-horse, 928 artillery, or gunners, 1410 pioneers, 250 sappers, and 406 miners : what is the whole number of men? Ans. 70995. OF SUBTRACTION. • > SuBTKACtiON teaches to find how much one number exceeds another, called their drfferencty or the remainder^ by taking the less from the greater. The method of doing which is as follows : Place the less number under the greater, in the same man« ner as in Addition, that is, units under units, tens under tens, and so on ; and draw a line below them. — Begin at the right hand, and take each figure in the lower line, or number, from the figure above it, setting down the remainder below it,— But ifthe figure in the lower line be greater than that above it, first borrow, or add, 10 to the upper one, and then take the lower figure from that sum, setting down the remainder, and carrying 1 , for what was borrowed, to the next lower figure, with which proceed as before; and so on till the whole is fiqished. * The whole body of foot soldiers is denoted by the word Iri" fantry; and all those that charge on horseback by the word Cccvo/r^. — Some authors conjecture that the terra infantry is derived fi-om a certain Infanta of Spain, who, finding that the army commanded by the king her father had been defeated by the Moors, assembled a body, of the people together on foot, with which she engaged ^and totally routed the enemy. In honour of this events and to distinguish the foot soldiers, who were not before held in much estimation, they received the name of Infantry, from her own title of Infanta. To 12 ARITHMETIC. To psovs Subtraction. Add the remsunder to the less number^ or dut which b just above it; and if the sum be equal to the greater or upper- most number^ the work is right*. 1. From 5386427 Take 2164815 £XAMFL£S. 2. From 5386427 Take 4258792 S. From 1234567 Take 70297S Rem. 3222112 Rem. 1127635 Rem. 531594 Proof.5386427 Proof. 5336427 Proof. 1234567 4. From 5331806 take 5073918. 5. From 7020974 take 2766809. 6. From 8503602 take ^74271. Ans^. 257888. Ans. 4254165. Ans. 79291 3 U 7. Sir Isaac Newton was- bom in the year 1642, and he died in 1 727 : how old was he at the time of his decease ? Ans. 85 years. 8. Homer was bom 2543 years ago, and Christ 1810 years ago: then how long before Christ was the birth of Homer ^ Ans. 733 years. 9. Noah's flood happened about the yeai^ of the world 1656, and the birth of Christ about the year 4000: then how long was the flood before Christ? . Ans. 2344 years. 10. The Arabian or Indian method of notation was first known in England about the year 1150: then how long is it since tp this present year 1810 ? Ans. 660 years. 1 1 . Gunpowder was invented in the year 1 330 : then how long was this before the invention or printing, which was in 1441 f Ans. 1 1 1 years. 1 2. The mariner's compass was invented in Europe in the year 1302: then how long was that before the discovery of America by Columbus, which happened in 1492.^ Ans. 190 years. * The reason of this mediod of proof is evident; for if the diflerenoe of two numbers be added to the less^ it must manifestly siake up a sum e^ual to ibe greater. OF ikrLTlPLICATION. IS OF MULTIPLICATION. » Multiplication is a compendious method of Addition^ teaching how to find the amount of any given number when repeated a certain number of times; as, 4 times 6, which is 24<. The number to be multiplied, or repeated, is called the Mub^licand^'-^The number you multiply by, or the number of repetitions, is the Mubipiier.^^Aj^ the number found, being the. total ambunt, is called the Productr^Aho, both the multiplier and multiplicand are, in general, named the Termi or Factors. Before proceeding to any operatiops in this rule, it is ne« cessary to learn off very perfectly the following Table, of all the products of the first 12 numbers, commoidy called the Multiplication Table, or sometimes Fythagoras's Table^ from its inventor. Multiplication Table. 1 2 2 3 4 8 5 6 7 14 8 16 9 10 11 12 4 6 10 15 20 25 30 35 12 18 18 20 22 24 S 6 9 12 16 21 24 27 30 33 36 4 .8 12 24 30 28 32 36 40 44 48 60 5 10 15 20 24 28 35 40 45 50 55 6 7 12 18 36 43 48 54 60 66 72 14 21 4^ 49 56 63 70 77 84 96 108 8 16 24 32 40 48 56 64 72 72 80 B8 99 9 18 27 SO 86 40 44 45 50 55 60 54 63 81 90 10 U 12 20 60 66 70 77 80 90 100 110 120 22 24 33 88 99 110 121 132 36 48 72 1 84 96 108 120 132 144 T0 14 ARITHMETIC. To multiply any Given Number by a Single Figure, or by am ^^ ' Number not nJf than I'i. * ^^ ■ « * Set the multipliei^ under the units figure, or right-hand pl;(ce, of the multiplicand, and draw a line below it.*— Then, beginning at the right*hand, multiply every figxire in this by the multiplier.^ — Count how many tens there are in the pro- duct of every single figure, and set down the remainder di- rectly under the figure that is multiplied ; and if nothing remains, set down a cipher. — Carry as many units or ones as there are tens counted^ to the product of the next figures | and proceed in the same manner till the whole is finished. EXAMPLE. ' Multiply 9876543210 the Multiplicand. By - - - - 2 the Multiplier. 19753086420 the Product. To multiply by a Number consisting of Several Figures. \ Set the multiplier below the multiplicand, placing thenv as in Addition, namely, units under units, tens under tens, &c. drawing a line below it. — ^Multiply the whole of the multi- plicand by each figure of the multiplier, as in the last article; setting ^ The reason of this rule is the same as for the process in Addition^ in which 1 is car- ried for every 10, to the next place, gra- dually as the several products are produced^ one after another, instead of setting them all down one below each other, as in the an- nexed example. 5§78 4 ■ 32 280 2400 2000O 8X4 70 X 4 600 X4 5000 X 4 22712 =5678 X 4 f After having found the produce of the multiplicand by the first figure of the multiplier, as in the former case, the multiplier is supposed to be divided into parts, and the product is found for the secQud figure in the same manner : but as this figure stands in the place of tens, the product must be ten times its simple value ; and therefore the first figure of this product must be set in the place of tensi MULTDPLICATIOI?. IS setting down a line of products for each figure m the multi- ptier, so as that the first figure of each line ma^c stand straight under the figure multiplying by. — Add all the lines of pro- ducts together, in the order as they stand, and their sum will be the answer or whole product required. To PROVE Multiplication. There are three different ways of proving Multiplication, which are as below : . . First Method. — Make the multiplicand and multiplier change places, and' multiply the latter by the former in the same manner as before. Then if the product found in this way be the same as the former, the number is right. Second Method. — *Cast all the 9's out of the sum of the figures in each of the two factors, as in Addition, and set down the remainders. Multiply these two remainders together, and cast the 9's out of the product, as also out of tens ; or, which is the same things directly under the figure multi- plied by. And proceeding in this manner separately with all the ^ures of the multiplier, 1234567 the multiplicand, it is evident that we shall mul- 4567 tiply all the parts of the mul- — tipdicand by all the parts of 8641969= 7 times the mult, the multiplier, or the whole of 7407402 r= 60 times ditto, the multiplicand by the whole 6172835 r= 500 times ditto, of the multiplier : therefore 493826S =4000 times ditto. these several products being _ added together, will he equal 5638267489=4567 times ditto. tothewhole required product; ■■ ■■ as in the example annexed. * This method of proof is derived from the peculiar property of the number 9, mentioned in the proof of Addition, and tha reason for the one may serve for that of the other. Another more ampla demonstration <^ this rule may be as follows : — Let P and Q denote the number of 9^s in the factors to be multiplied, and a and 6 what remain ; then 9 P+a and 9 Gl+6 will be the numbers themselves, and their product is (9 P X 9 Gl) + (9 P X A) + (9 Gl X a) -f (a X h) i but the first three of these -products are each a preiCise number of 9's> because their factors are so, either one or both : these therefore being cast away, there remains only a x 6; and if the 9's also be cast out of this, the excess is the excess of 9*s in the total product : but a and h are the excesses in the factors them- Bcjves^ and a x 6 is their product ; therefore the rule is true., the 16 ARITHMETIC the whole product or answer bf the questioni reserving the renuinders of these last two, which remainders must be eauai when the work is right. — Ifote^ It is ccMnmon to set the lour remainders within the four angular spaces of a crossj at in the example below. Third 3/^^.-- Multiplication b also very naturally proved by Division ; for the product divided by either of the factors, will evidently give the other. But this cannot be practised till the rule of Division is learned. Muk. 3542 by 6196 21252 34878 3542 21252 21946232 Product. EXAMPLES. Proof. or Mult. 6196 by 3542 12392 247S4 30980 18588 21946232 Proof. OTHER EXAMPLES. Multiply Multiply Multiply Multiply Multiply Multiply -Multiply Multiply Multiply •Multiply Mukiply Multiply Multiply Multiply Multiply Multiply 123456789 123456789 123456789 123456789 123456789 123456789 123456789 123456789 123456789 302914603 273580961 402097316 82164973 7564900 8496427 2760325 by S. Ans. by 4. Ans. by 5. Ans. by 6. Ans. by 7. Ans. by 8. Ans. by 9. Ans. by 11. Ans. by 12. Ans. by 16. Ans. by 23. Ans. by 195. Ans. by 3027. Ans. by 579. Ans. by 874359. Ans. by 37072. Ans. 370370367. 493827156* 617283945. 7407407S4. 864197523. 987654S12. 1111111101. 1358024679. 1481481468. 4846633648. 6292362103. 78408976620. 248713373271. 4380077100. 7428927415293* 102330768400. COHTIUG- MULTIPUCATION. 11 Contractions in MoLTtrucATioM. I. Jf^im there an Ciphers in the Factors. If the ciphers be at the right-hand of the numbers ; mul- tiply the other figures only, and annex as many ciphers to the right-hand otthe whole prodocti as are in both the fac* tors.— When the ciphers are in the middle parts of the mul* tiplier } neglect them as before^ only taking care to place the first figure of every line of products exactly ttnd«r th* £gure multiplying with. EXAUPLES. Mult. 9001635 Mult. 3901S0400 by • 70100 by - 406000 ■■III- ■■■■■II .^ ■ ■ » — — — .— ^ 9001635 2844S284 63011445 15628816 631014613500 Products 158632482400000 S. Mukiply 81503600 by 7080. Ans. 572970308000. 4. Multiply 9030100 by 2100. Am. 18963210000. 5. Multiply 8057069 by 70050. Ans. 56439768345Q. n. When the Multiplier is the Proiiict of two or more Numbers in the Tabk; then ^ Multiply by each of those 'parts separately, instead pf the whole number at once. EXAMPLES. 1. Multiply 51307298 by 5Q^ or 7 times t. 61307298 7 359151086 ' 8 2873208688 f The reason of this rule is obvious enough $ for any number mttltif^lied by the component parts of another^ must give the same product as if it were multiplied by that number at once Thus, in tbe 1st example, 7 times the product of b by the giv^n number^ ifnakes 5Q times the same number, as plainly as 7 times 8 makes a6. Vol. I. C 2.Mul- It: ARITHMETIC 2. Multrpl7 3!7045d2 -by 86. Aus, .1141365512. S. Multiply. 2S753804 by 72. Ans 2142273888* 4. Multiply 71283^8 by 96. Ans. 684i52:3326. ^. A|i*hiply 160430800 by 10$, Aiis. 11^26526 \QO. ,«, Multiply 61835720 by 1320. Ans. 81623150400. . 7* There was aa army composed of 104 "^ battalions, each consisting q£ ^00 men j what was the number of men con* tained in the whole? Ans. ^2000* / •. 9f A conToy of ammnnkion f bread, consisting of 250^ waggonst and each waggon containing 320^ loaves, hzvinfr^ been intercepted and taken by the enemy ^ what is the num- ber of loaves lostA Ans. 80000. tsam OF DIVISION. ' Division is ,a kind of compendious ^lethod of Subtrac* tion, teaching to find how often one number is contained in another, or may be taken out of it : wiiidx is the same thing. The number to be divide is called the Dividend. — The number to divide by, is the Z>;wflr.4r— And the number of times the c^vi^nd contain^, the divisor, is called the Quo* tient. — iSometiiqes thfire b g Rnnaitider left^ after the division i^^nished. The usual manner of placing the terms> is, the dividend in the. middle^ haying.the cU visor en the left hand, and the<}uo* tient on the right, each separated by a curve line j as, ta divide 12 Iq^ 4, the quotient is 3, • Dividend ' ' ' - l2 Divisor 4) 12 f 3 Quotient; 4 subtr, showing that the number 4 \& 3 times — contained i9 12, t)E may be S times 8 subtracted out of it, as m the ipargin. 4 subtr* X RuU. — Having placec^ the divisor — before the dividend, as above .direct- 4 ed, find how often the 'divisor is con- 4 subtr* tained in as many figures of the divi- -— dend as are just necessary, and place the number on the right in the quotient. — Mul- *yi^^— W^ I ■ ■■ I H ■ II ,1 ^ ■ ■ I I III I I ■■ fcl^M^I—— *^^ll ■ * A battalion is a body of foot, censisting of 500, or 600, or 700- men, more or less. f The amitoufiitioii bread, is that which b provWedfor, ami ifi^ 'triboted to, the soldiers j t!^ u^u^ allowanee being- a leaf of ^ pounds to evety soldier, once in 4 days. ' ' ■> - Xlta this way the dividend b resolved into parts, end by trial i* *- - - ■ '• - ibdnd Multiply the divisor by this number, and set the product tinder the figures of the dividend before^mentioned.-*-*Sub« tract this product froro that part of the dividend undi^ which it stands, and bring down the n^xt figure of the dividend, ox* more if necessary, to join on the right of the temainder.—Di* vide this ntunber, scr increased, in the same n^aiinera;^ before | and so on till all the figures are brought down and used. • If. B, If it be necessary to bring down more figures than one to any remainder, in order' to make it as large as the divisor, or larger, a cipher must be set in the quotient for every Bf^ate so brou^t down more than one. To PRQvt Division* * Multiply the quotient by the divisor; to this product add the jpemainder, if there be any ; then the sum will be equal to the dividend when the work is right* found how often the divisor is contsllned in each of those patt8> one idler another^ arranging the several figureS'Of the qaotieutone after another^ into one numben When there is no remainder to a division, the^ quotient is the whole and perfect answer to the question. But when there is a re- mainder^ it goiea so much towards another time> as it approaches to the divisor : so» if the remainder be half the divisor^ it will go the half of ^ time iQore) if the 4th part of the divisor^ it will go. one fourth of a dq^e more ; and so on. Therefore, to complete the quotient, set the remainder at the end of it, above a small line, and toe divisor below it> thus forming a fractional part of the whole quotient. * This method of proof is plain, eoough : for since the quotient is the nimiber of times the dividend contains the divisor, the quo- tieni multiplied by the divisor must evidendy be equal to the dividend. There are also several other methods sometimes used for proving IHvision, some of the most useful of which are as follow : Stccftd Me/W.-*Snbtract the remainder from the dividend^ end 4ivide what is left by the quotient ; so shall the nei^ quotient £ro9i, this last divisi9n be equal to the former diviaor, when the wofk is tight. T^r4 Method, — Add together the remainder and all die pro- ducts of the several quotient figures by the divisor^ according to the order is which they stand ia the work $ and the sum will be ^ual to the dividend wl^en the work is right. C 2 EXAM- 2« ARrrHMETIC. EXAMPLES. I. Quot. 3} 1234567 (411522 12 mult. S 2. Quot. 37 ) 12345678 (333666 111 37 3 3 1234566 add 1 4 S 1 1234567 9 •■ 15 15 Proof. 6 7 6 Rem. ' 1 \ 2335662 1000998 rem. 36 124 135 . 1 1 1 1234567$ ^ 246 Proof. 222 247 222 258 222 Rem. 36 3. Divide 73146085 by 4. 4. Divide 5317986027 by 7. 5. Divide 5701^6382 by 12. 6. Divide 74638105 7* Divide 137896254 8. Divide 35821649 9. Divide 72091365 Ans. 1 828652 Ij^. Ans. 7597122894*. Ans. 47516365^. Ans.20l7246yy. Ans. 1421610^. Ans. 46886fJ^. Ans. I386I3WT. Ans. 80496|U?-^ by 37. by 97. by 764. by 5201. 10. Divide 4631064283 ^ 57606. 11. Suppose 471 men are formed into ranks of 3 deep, what is the number in each rank? Ans. 157. 12. A party, at the dbtance of 378 miles from the head quartersi receive orders to join their corps in 18 days : what number of miles must they march eadi dajr to obey their orders? Ans. 21. 13. The annual revenue of a gentleman being 88330/; kow ^uch per day is that equivalent to, there being 365 days in the year ? Ans. 104A CoNTHACTlONS IN DIVISION. . There are certain contractions in Division,, by which the operation in particular cases may be performed in a shorter jmanner : as follows : . . I. Divp DIVISION. « !• jPivision if any Small Number^ not greater tlian 12, vaxj be expeditioualy performedi bj multiplying and subtracting mentallyy omitting to set down the work, except only the quotient inunediately below the dividend. 3) 56103961 Quot. 18701320J. EXAMPLES. 4) 52619675 5) 1379192^ 6) 38^72940 7 > 81396627 3 ) 23718920 9 ) 439819^2 1 1 ) S76 14230 J2 ) 27980373 -«■ »■ ■ ' U n. * When ethers are annexed to the Divisor i cut off those ciphers from tt, and cut off&e same number of figures from the right*hand of the dividend; then divide with the remain- ing figures, as usual. And if there be any^ thing remaining after this division, place the figures cut off from the dividend to the right of it, and .the whole will be the true remainder; otherwise, the figures cut off only will be the remainder. EXlMPi.ES. 1. Pivide 3704196 by 20. 2. Divide 31086901 by 7100. 2,0) 370419,6 71,00)310869,01 (4378ff 284 Ol- Quot. 185209ii 268 213 556 599 568 SI ■y^ 3. Divide * This method is only to avoid a needless repetition of ciphers^ which wQuld happen in the coQimoa V9y. And ihe truth of the principle B2 AlUTHMETICt 3. Divide 7380964 by 23000^ Am. ilpii^^^ 4. Divide fli04109 fay 580a Am. 397if^. nif W7>en tie Dhuistnr is the ex^ct Product -rf two or moh jf the small Numbers not grettter than 12 : * Divide by eadl of those numbers Separately^ iiutead of the whole divisor sit pnce. N. B» There are commonly several remainderi in work*- ing by this rule, one to each division', and to find the true w whole remainder, the same as if the division had been per-r formed ^U at opce, proceed as follows; Multifdy the. last remainder by thf preceding divisor, or last but one| and to the product add the preceding remainder; multiply this sum by the n^xt .preceding divisor, and to the product add the nex^ preceding remainder i and so pn, till you have gone backward through all the divisors ^4 remainders to the firsts As in the example following : EXAMPILES. 1. I^ivide 31046895 l^y 56 or 7 times d. 7 ) 810^835 6 the iastremf tsxdttf 7 precpd^ iHvisof^ p ) 4435262--1 iirst rem. ; 48 554401 — fi second ren^. ^lid> 1 the I'st rem* Ans. 55440744 43 'whole rem. TS principle on which it is foande^, is evident ; for, pi^ti^Dg off the fame number of ciphers^ pr figures, from each> is the same at dividing each of them by 10, or lOO, or 1000, &c. according to the number of ciphers cut off; and it is e'^rdent, that as ofien as f be whole divisor is cpn^ined in the whole dividend, so often must ^ny part of the former be containcji in a like part of the latter. *' This fdlows from the second contraction in Multiplication^ being only the cpnverse of it; for the half of the third part of any thing, is evidently the same as the ^ixth part of the whole ; and 00 of ^ny other numbers.— The reaspn of the method of finding the whole remainder from the .several particular ones, will best appear from the- nature of Vulgar Fraptipps. Thus^ in the firs^ example abpvfs, the- first remainder being 1, when the divisor i^ 7| makes j- ; this must be added to the second remainder, 6^^ making 6f to the flivisor 8^ or to be divided by 8. But €f =: r— ^ = yi and thi^ divided by 8 gives ^^=:|^.. 2. P^vide r «. Divide 7014596 by 72. Ans. 97424f^ 3. Qisrid^ .5 130652 by .132. . . Ans^ SSiea^ZyV 4. Divide 83016572 hj 2ii9. Ans. 945502^. . IV. Common Divisign may be pftfirmed more condse/jj by omitting the several products, and setting down only the remainders; namely, x multiply the divisor by the quotient figures as before, and*^ without setting down the product^ subtract each figure of it from the dividend, as it is produc^ds mlvrays remembering to carry as many to tbft next figure as wbre borrowed before. [ EXAICPIES* U Divide 3104679 liy $33. «33 ) 3 104679 ( 3727^V» 6056 2257 5919 88 «. Divide 79165288 by 238* . Ans^ 332627^^^. 8. Divide 291S7062 by 5317. . . Ans. 5479mf. 4. Divide 62015735 by 7803- - Ansu 7?47f||f OF REDUCTION. Reduction is the changing of numbers firofcn oot name or denomination to another, Hithout altering their valqe.-— This is chiefly concerned in reducing' money^ weights^ and measures. . When tilt numfiers are to be reduced from a higher name to a lovrer, it is called Reduetkn DescenSngi mst wheii^ contrarywi^, from a lower name to a higher, it is Redudkn AscenHngm Before proceeding to the rules and qtresttc^fs of Rednctim, it will be proper to set down the usual Tables of mdns^^ vireights, and measures^ whick ar^ as follow : \ bf f4 ARITHMETIC (y MONET, WEIGHTS, amd MEAStTRES. Tables of Moi^et*, . 3 Farthings 4 Farthings )2 Penc9 20 ShilUngs =s I Halfpenny f ^* 1 Penny d «; I ShilUng / = 1 Pound £ 4 48 S60 S5r 1 / == 12 =^ 1 =; 240 = 20 £ = I PENCE TABLE. fHltLINGS TABLE, 1 d i d S l/ 1 20 is 1 8 1 is 12 • 30 — 2 6 2 — 24 40 -^3 4 3 -«* 36 30 - 4 2 4 «.» 48 60 — 5 5 — 60 1 70 — 5 10 6 -.^ 72 ' 80 — 6 8 7 «^ 84 i 90 —-7 6 8 --. 96 ' 100 — 8 4 9 ^^ lOS HO — 9 a 10 -~ 120 1 120 -, JO 11 — 132 1 Trot Gold. * £ denotes ponndi , « shillings, and d denotes pence. ^ denotes i farthing, or one quarter of any thing. \ denotes a halibenny^ or the half of any thing. ^ denotes 3 farthings, or tliree <|uarter8 of aoy thing. The full weight and value of the English gold and silver coin, i^ as here below : Weigh. dwt gr a 16| 1 8^ I The usual value of gold is nearly 4/ an ounce^ or 2<i a grain | an4 that cf ^silver is oearlv ^s an ounce. Alsoj the value pf any quantity of gold^ is to the value of the s^me weight of standard iiiver^ nearly as t. 5 to I « or more nearly as 15 and 1-1 4th to I. Pure gold, free from mixture with other metals, usually called \ $Dt gol4» U of so puie 9 Datur^> that it wiU endure the fire •' vitfeQut Vahe. £ » d 1 1 10 d A Guinea Half-guinea Seven Shillings 7 Quarter-guineaQ ^ 3 SlLVEH. Value. Weight. 9 d dwt gr A Crown 5 19 84 Half-crowa % 6 9 l^i ShiUing . 1 O 3 21 Sixpence Q 6 . 1 ^^i 24= 1 oz 480=r 20= 1 A 5760=240 = 12= r TA5US5 OF W^QHTS. 15 Trov Weight*. Grains - - marked ^r \ gr dnvt 24 Graim make i Pennyweight dwt 20 Pennyweights 1 Ounce «z 1 2 Ounces 1 Pound lb By this weight are weighed Gold, Silyer, and Jewels* Apothecahies' Wjeight. Grains - - marked gr ^0 Grains make 1 Scruple sc or 9 3 Scruples 1 Dram dr or 3 .8 Dr^mi. 1 Ounce w; or t 12 Ounces 1 Pbund lb or jl, " gr sc • 20 = 1 ir 60 ass 3 = 1 «5 480 = 24 = 8 = 1 lb . 5760 = 288 = 96 = 12. =3 1 This is the same as Troy weight, only having some dif- ferent divisions. Apothecaries make use of this weight in compounding th^ir M^icines i but they buy and sell their Drugs by Avoirdupois weight. AvoiR- widiout wasting, though it be kept contigually melted. But silver, not having the purity of gold^ will not endure the fire like it : yet £ne silver will waste but a very little by being in the fire any moderate timej whereas copper, tin, lead, &c. will not only waste, but may be calcined, or burnt to a powder. Both gold and silver, in their purity^ are so very soft and flexible (like new lead, &c.), that they are not so useful, either in coin or otherwise (except to beat into leaf gold or silver), as wnen they are allayed, or mixed and hardened with copper or brass. And though most nations dlifer, more or less, in the quantity of such allay, as well asHn the samis place at different times, yet in £ng- land the standard for gold and silver coin has been for a long time as follows — viz. That 2*JL parts, of fine gold, and 2 parts of copper, being melted together, shall be esteemed the true standard for gold coin : And that 1 1 ounces and 2 pennyweights of fine silver, and . 1 8 pennyweights of cop|)er^ being melted together, is esteemed the true standard for silver coin, called Sterling silver. ♦ The original of all weights used in England, was a grain or com of wheat, gathered o^t of the middte of the ear, and, being W^Jl dri^d^ 32 of them were to m^e one pennyweight, 20 penny- weights / t6 AR|THM£71C. Atoiildupois Weight. DraiQS • - naarked (ir 16 Drains make 1 Ounce f • - • OZ 16 Ounces -^ • • 1 Pound • • • a t9 Pounds - . • 1 Quarter 1 Hundred • - . 9^ 4 Quarters - 2(X Hundred 1 . . 1 Weight ^ nvt RTeight iTon p» - - Jr ' 'OZ ^ 16 = 1 a 256 = 16 = = 1 i^ 1 7l6« = 448 = = 28 =s 1 cwt 28^72 = 1792 = s H2 = 4 s 1 tm 573440 3e 35840 = = 2240 8= 80 = 20 s= 1 By this weight are weighed all things of a coarse or drossy nature^ as Com, Bread, Butter, Cheese, Flesh, Grocery Wares, and somt liquids ( also all Metak, except Silver and Cold.. OZ dtift gr N§if^ that la AToirdupcns at 14 U 154 Troy, las - - =s 18 54 Ur - - «= 1 34 LoN^ Measure. * • 3 Barley*coms make 1 Inch *> Li 12 Inches « 1 Foot » ' Ft 3 Feet • 1 Yard • rd 6 Feet ■■ 1 Fathom -/ •■• Fth S Yards and a half 1 Pole or Rod «» PI 4b Poles * 1 Furlong - J- Fur S Furlongs - «• 1 Mile • Mile 3 Miles m 1 League - • Lea 694 ^1^ nearly - 1 Degree - . Dfg at ". • wdghts one ounqe> and 1^ ounces one pound. But in later times, it was thought sufficient to divide Hhe same pennyweight into 24 equal parts, still called grains, being the least weight now in common use ; and from tb^ce the vest are computed, as in the Tables above. In TABUS «fM&A9C7K£8. ft In 2% 12 SSI 1 rd - 36' sa 3 s= 1 fl ■ 198 =r 1<54 ' zs 5i =: 1 Fur ^7920 =: 660 =: 220 ss 40 = I 13960 ss 6230 ^^ 1760 ss 320 s: 8 ClOTH MB4SyKB. Mik 2 Inches and a ijuart^r make 1 Nail <- * Nl *4 Nails r - sr 1. Quarter of ^ Tard Or 3 'Quarters .r - » 1 EU Flemish - £-F 4 Quarters - ^ ^ 1 Yard - * Td , 5 Qu;qpters r «. * 1 Ell English - EE 4 Quarters 14 Inch - 1 £11 Scotch ^ ^ £S » 144 Squarefochetmake 1 Sq Foot - R 9 Square Feet r 1 SqTard - rd 90i Square Tafds - 1 Sq Pol^ - Pok 40 Sq^are Poles - 1 Rood * Rd 4 Roods -pi Acw f jfcr ' SqLic ^Ft ; 144 « i SqTd 1296 = 9=1 SqPi 39204 .s 272^ s 30f r? 1 Jtd 1568160 s: 1O890 b 1210 s 4Q « i >/rr 6272640 = 4^^6Q b 4840 == 160 » 4 ^ i , By diis measure^ Land, and Husl|aadme& and Oardenen? irork are measured; also Artificers' work, ancl^ ^s Board, Glass, Pavements, PlaMering, Wainscoting, Til|j9g, Floorilig, and every dimension pf laigth smd breadth only; . When three dimensions arjs concemed| namely, length, breadthy aapd depth or thkkness, it is called cubic or solid measure, which is used to measure fllmbery Stone» &c. The cubic or s^lid Foot, which is 12- inches ia length and )>readth and thickness> contains 1728 cubic or sblici inches, u^d {27 solid feet make one solid yard. 29 ARnHMETia DrY) or Cork Measchb. 1 Quart < 1 Gallon - 1 Peck f Pifits make 2 Quart* - 1 Potde 2 Pottles 2 Gallons 4 Pecks 5 Bashels S Quarters 2 Weys ■■ « M Bushel Quarter 1 1 1 Wey» Load> or Ton 1 Last - Gai Pec Bu Wej JLaU Pts ' % 16 64 512 2560 5120 Gal 1 2 =x 8 = 64 = S20 = 640 =: PiC 1 4 32 160 S20 1 40 80 Or J 5 10 : 1 ZdRlf : 2 =: I By this are measured all dry wares> aS) Corn> Seeds, Rooft% Fruits, Salt, Coals, Sand, Oysters, &c. The standi Gsdlon dry-measur^ cpntauis268| cubic ot solid inches, and the Cdm or Winchester husl^el 21 50|- cubic inches j for the dimensions of the Winchester bushel, by the Statute, are 8 inches deep, and 1 8-^ iiv;hei wide or in diameter. But the Coal bushel must be 19^ inches in dlanneter; and 36 bushels, heaped up, make a London chaldron of coals> the weight of which is ^ 1 5 6 lb Avoirdupois* AhZ and Beeil Measure ^ 2 Pints make 4 Quarts - •• 36 Gallons ^. -w 1 Barrel aud a half f Barrels m 2 Hogsheads ^ 2 Butts; <* -r Pts Of 2 =; I Gat 8 =; 4 =?, 1 28^ = 144 =: 36 482 23 2li5 = 54 864 2= 432 = 108 1 Quart 1 Gallon 1 Barrel 1 Hogshead \ Puncheon 1 Butt \ Tw - Gal Bar Hhd ^ Pim fiuu Tut\ Bar : 1 mi 1 Sm 2 i= 1 NitCji The Ale Gallon contains ^82 cubic or soSd Lichee. Win* TABLES ot MEASURES amo TIME. 2» Wine Measure. •• 2 Pints make 1 Quart Qt 4 Quarts - *• 1 Gallon Gal 42 Gallons - . >- 1 Tierce Tter 63 Galkms or it Tierces 1 Hogshead - Hhd 2 Tierces - 1 Puncheon - Pun. 2 Hotheads 1 Pipe or Butt Pi 2 Pipes or 4Hhds I Tun - Tun Pti Qt V 2 = i Go/ « = 4= 1 : tter 9J6 = 168 r: 42 = I Hid 504 = 252;= 63 = lt= I P«/» 672 = 336 = ^4 = 2 = lt= 1 Pi 1 1008 :;= 504 = 126 = 3=2= lt= I Tim 2016 =: 1008 = 252 = 6=4=3 =a = 1 / .Notit By this are measured all Wines, Spirits, Strong- Waters, Cyder, Mead, Perry, Vinegar, Oil, Honey, &c. The Wine Gallon contains 231 cubic or solid inches. And it is remarkable, that the Wine and Ale Gallons have the same propprtipn to each other, as the Troy and Avoirdupois Pounds hare^ that is, as one Pound Troy is to one P4!>und Avoirdupois^ so is one Wine Gallon to one Ale Gallon* Of TIMiE. 60 Seconds or 60'' make • 1 Minute - Mot' ^ Minutes - 1 Hour Hr 24 Hours - - i Day Da* 7 Days * - - - 1 Week . ni 4 Weeks - - - • 1 Month - M, 13 Months I Day 6 Hours,) or 365 Days 6 Hours ) 1 Julian: Year Tr 4 ■ Sec Min , 60 =: 1 Hr , . 3600 = 60 = I ■ -^"^ „. • S6400 = 1440 = :■ 24. =s -4 IVk *• 604800 =: . lOOSO =s. 168 = 7 = 1 M» ■ . 1 2419200 =: 4.0320 = 672 = 28 = 4 .: = 1 ! 31557600 == 5250QQ ^ 8766 = 365t;=i llrfar. V Or ( y m ARITHBIEnC* Wk Da Hr Mo Da Hr Or 52 I 6 = 13 1 6 = 1 Julianrear Da Hr M Sec But 365 6 48 48 =r 1 Sflcir rear! ItULES FOR REDUCTION. ^ trim tie Numben are to he reduced from a Higher Denond^ ' nation to a Lower : Multiply the number m the highest denomination by as many as of the next lower make an integer, or I, in that higher ; to this jH'oduct add the number, if any, which was in this lower denominatioir before, a^ set down the aihonnt. Reduce this amount in like manner, by multiplying^ it by as many as of the next lower make an integer of this, taking in the odd parts of this lower, as before. And so pnroceed through all the denominations to the lowest ; so shall die number last found be the value of all the numbers which, were in the higher denominations, taken together*. EXAMPLE. 1. In 1234/ Idx 7i^ how many farthings I lid 1234 15 7- 20 24695 Shillings 12 296347 Pence 4 ^ Answer 1185388 Farthings. * The reason of this ru)e is very evident; for potmda are brought i^to shillings by maltiplying them by 20 ; ^hilltngs into pence« by multipljrtig them -bj 12; and pence into ^rthings^ by multiplying by 4 5 and tKe reverse of this nile by Division. — And the same^ it is evident^ will be true in the reduction of numbers consisting of any denominations whatever. ll.When RULES foil REDtJCnCMJ. St IL When tie Number $ are to be reduced from a Lower .Deftomt" nation to a Higher : m Divide the given number by as many as of that denomt'* nation make 1 of the next higher, and sec down what nemainsy as well as the quotient. Divide the quotient hj as many as 6f this denominatbn make 1 of the nett. higher } setting down the new quotient, and remainder, as before. / Proceed in the same manner through all tike denomina*- tions, to the highest ; and the quotient last fimnd, ti^ether with the several remainders, if any, will be of the same vahif s the first number proposed. V EXAMPLES. 2. Reduce 1185388 farthings into pounds, shillings, and pence. 4) 11853*8 1*1 I I 12) 296347^ ■ t ■' I !■ * 2,0 ) 9469,5 /-^7rf 1S^4/ 14^ *!d < ■ »»■»» 3. Reduce 24/ to farthinjgs. Ans. 23(Hai 4> Reduce 3375^7 farthings to pounds, &c. Ans. S6U I3i 6^. 5. Kow many futhings^nre in 3€ guineas? Ans. 36S89« €. In 36288 farthings how many guineas I Ans. 36. 7. In 59 lb 13 dwts 5^ how many gnttis ? Ans. 340157* 8. In 8012131 grains how many pounds, &c.? Ans. I390ib 11 oz 18dwt t9gr 9. In 35'ton I7cwt I qr 23 lb 7oz i3dr how many drams? Ans. 20571005. 10. How many barley-cornSx will rqadb round the earth, supposing it^ according to the best calculations, to be 25000 miles ? * Ans. 4752000000. 11. How inany seconds are in a solar year, or 365 days 5 hrs 49 min 48^ sec ? Ans. 3 1 556928. 12. In a lunar month, or 29 ds 12 hrs 44 min 3 sec, how many second 2 Ans. 2551443. COM. St ARTTHMETia COMPOUND ADDITION, CoMPOiTND ApDiTiON shows how to add or collect seTeral numbers of different denominations into one sum. RijL£.*-Place the numbers so» that those of the same de^* nomination may stand directly under eauch other, and draw z line below them» Add up the figures in the lowest denomi- nadon, and find, by Reduction^ how many units, or ones, of the next higher denomination are contained in their $um.-~-» Set down the remainder below its proper column, and cany those units or ones to the next denomination, which add up in the same manner as be£bre«r-*Proceed thus through all the denominations, to the highest, whose sum, together with tht several remainders, will give the answer sought. The method of proof is the same as in Simple Addition. EXAMPLES or MONET» I. 2. s. 4. / J d / X d / / d / / rf 1 13 3 14 1 5 15 11 10 J3 14 8 a 5 lot 8 19 2t 3 14 6 5 10 H 6 18 7 7 8 It 23 6 2| 93 11 6 2 H 21 2 9 14 d 4t 7 5 4^ 3 7 16 8f 15 6 4 IS 2 S n 15 4i 4 3 V 6 1^ 9^ 18 7 %9 15 H ' • • 32 2 . • 39 15 H / . 5. 6. . 7. 8- / / d . / s d / s i / J- d H n 37 15 s 613 2t 472 15 3 8 15 3 14 12 H 7 16 8 9 2 2t 62 4 7 ■ 17 14 9 29 13 10 J 27 12 6t 4 IT & 23 10 9t 12 16 2 370 16 2t 23 H 8 6 1 b^ 13 7 4 6 6 7 14 H 24 13 6 10 ■5t SI lOj 54 £ ii 5 10|. 30 111 » 1 \ • EXAH<^ / COMPOUND AjDOmON. ' aa Exam. 9. A nobleman, going cmt of tovn, is informed by his steward, that his butcher's bill comes to 197/ ISj '\\d: his baker's to 59/ .'J/ 2|^; his brewer's to 85/; his wine-mer- chant's to 103/ 13x; to his corn-chandler is due 75/ %d; \o his tallow-chandler and cheesemonger, 27 i IBs 11|</; and to his tailor 55i Ss B^d; also for rent, servants' wages,, aqd other charges, 127/ 3/; Now, supposing he would take 100/ with him, to defray his charges on the road, for what sum must he send to his banker ? Ans! 830/14/ 6^. 10. The strength of a regiment of foot, of 10 compani^, and the amount of their subsistence*, for a month bf 30 days, :flu:cording to l)he anaeKed Table, are r^uired i Numb. Rank. Sul^istence for - a Montih. * / # d 1 Colonel 27 1 1 7 11 Lieutenant Colond IVfajor Captains Lieutenants 19 10 17 5 78 15 57 13 * 9 1 \ 1 JEnsigns Chaplain Adjutant Quarter-Master 40 10 7 10 4 10 , 5 5 1 1 30 30 2P 2 Surgeon Surgeon's ]\{ate Seijeants Corporals Pruipmers Kfes 4 10 4 10 45 30 20 2 390 Private Men 292 10 5Qir Total 656 iO ■*■**• * Subsistence Money, is the money paid to the soldiers weekly which is short of their full pay, because their clothes^ accoutre-* ments, ^c. are to be accounted for. It is likewise, the money ad- vanced to officers till their account are made op*J||^<^^ is com- *monly once a year, when they are paid their arrearir^lhe follo)w- ing Table shows the full pay and subsistence of each rank on the ^English establishment . VoLiL D DAILT 34 ARITHMETIC. •- • o o CO ^ O o c > 00 '• n »-* p^ "2 £ O) QD -* (o t^ lo -i* »o »o "O k4 C3 g •H • -H ^^ »«^ O -It »- ^ o o o o t oooo o ^ O CO o o o <oo •£ '3S 9 o -• CO ^ ^ -^ M , 121.^ 1 c/> »H »-« o o o o o .000 b:. O GO CO 00 Cfl rt -E o. 1 1 1 12 2 1^ 00 1 c» »o O O O » 00 • o ^ CO co<o o ^ 1 00 1 en 'J 9 ^4 •H ^2 :2 1 ^ ' ^ 1 0)«0 • <^ - -H ^ ^ O O c > .0 00 ft b3 • o o oooo O y • 00 £. CO -H O "t O -H 1 -H 11*1 Lb •TS —1 r-l •-I ^i* '1 ^ U. S g c 3 o (s« ^ -H p- ph o o c q • O CO CO O '- <N ^ c ^ a 4>< in 1 t^ CO O QD C^ X 11^ 1 l'°l CO y5 w i »H f— oooo o c •H lO (X) 00 00 p ) 00 10 1 »o Ci 1 »o u - o O O O O CO 1 00 C ' > it o rz ooooo oooo < ^ •* O t^ "o 1 o t^'o *^ 1 1 1 12 l-^ >- O w c<i-«^oo oooo 'o < u .* o o CO iSiO O O C 1 -* 9 e 1— 1 2 I« 1 4 ^^ 1 Q^ "^ fH O OOOO o 1 00 00, o o O OOOO 00. 2^ ?: ^ 12^ 1 r-i »f5 O C^C» I • 1 CS 1 CO GO ""-^^^S mn •^ ^ OOOO 00 - • "'-N . . ' >•••,••• --§ : ; k 4 > • • • t^D • * • > . , , u • • • < 1 > < » 1 > 4 1 * '-< , . / -3 . . . . . S3 gco . . . fc S S • h^^ • • 3 c 13 "oJ <i> r > O o ^ "r pj O • • ^ ^ »L C t <^ CO fe 5 o«Jc U U — cs 1 « T3 ,^ ,«^ S ^o .9 a 3 ► 2 (3 eg pq vj I ° o •> 01 e o '5 a s o ;> o "2 o so c o a ■i '^ a ^ S •»••■ Cfa o S w Of " 1*^ 3f^ .J .tXAMfLSS COMPOUND ADPITION. S3 EXAMPLES OF WEIGHTS, MEASURES, t(e. ' TROY WEIGfiT. APOTHECARIES* WEIGHT. f K 2. 3. 4. lb oz dwt oz dwt gr lb OZ dr sc oz dr sc gr 17 3 15 37 9 3 3 5 7 2 3 5 1 17 , 7 9 4 9 5 3 13 ,? 3 7 3 2 5 O 10 7 8 12 12 19 6 2 16 7 12 ^ 9 5 17 7 8 9 1 2 7 3 2 9 i76 2 17 5 9 36 3 5 4 1 2 18 i23 11 12 3 19 5 8 6 1 36 4 1 14 AVOI«l>UPOIS WEIGHT. T.ONO MEASURE. 5. , 6. 7. . 8. lb oz dr ' ♦ cwt qr lb 1 mis fiir pis yds feetinc • 17 10 13 15 2 15 29 3 14 127 1 5 • 5 14 8 6 3 24 19 6 ^9 12 2 9 12 9 18 9 1 14 7 24 10 10 27 1 6 9 1 17 9 1^37 54 1 11 4 10 2 6 7 3 5 2 7 6 14 10 3 3 4 5 9 23 5. - CLOTH MEASURE. LAND 1 MEASURE. 9. 10. 11.. 12. yds qr nls el en qrs nls ac ro p ac ro p 26 3 1 270 1 225 3 37 19 16 13 1 2 57 4 3 16 1 25 270 3 29 9 1, 2 18 1 2 7- 2 18 6 3 13 217 3 3 2 4 2 9 53 34 9 1 10 1 42 1 19 7 2 16 55 3 1 4 4 1 7 6 75 23 WINK E^EASURB. — - ALE and BEER MEASURE. 13. 14 • 15. 16. t hdsgal hds gal pts hds gal pts hds gal pts 13 3 -15 15 61 5 17 37 3 29 43 5 8 1 37 * 17 14 13 9 10 \5 12 19 7 14 1 20 29 23 7 3 6 2 14 16 6 25 12 3 15^ 1 5 14 6 8 1 . 3 1 9 16 8 12 9 6. . .57 13 4 72 3 21 4 3.6 ' 6 8 42 4 5 6 D2 COM- ■"S6 ARfTHMETlC. COMPOUND SUBTRACTION. « Compound Subtraction shows how to find the difl&r- 0nce between any two numbers of different deaomin^tio^Sr To perform which^ ob^'erve the following Rule : * Place the less number below the greater, so that the parts of the same denomination may stand directly tinder each other ; and draw -a Kne below them. — Begin at the right-hand, and subtract each number ,or^part m the lower line, from the one just above it, and set the remainder .straight below it. — But if any number in the lower line be greater than that above it, add as many to the upper number is make 1 of the next higher denominatibn v then take the lower number from the upper one thus increased, and set down the remainder. Carry the unit borrowed to the next number in the lower line ; after which subtract this number from the one above ity as befofe; and so proceed till* the whole is finished. ' Then the several remainders, taken to^ jcther, will be the whole dffference sought. The method of proof is the same as in Simple Subtraction. iEXAMPLES OF MONET. I. - 2. 3. . 4. ^ I s i I s d I s d t s d From 79 17 8|. 103 3 2J- SI 10 J I 254, 12 Take 35 12 4f 71 12 5^ 29 13 S^ 37 9 4^ Rem. 44 5 4Jr 31 10 8^ ■ ' ' ■ * Proof 79 17 8| 103 -^ 2^ 5. Wh^t ia the difFerence between 73/ 5|^/ and 19/13/ lOrff Ans. 53/ 6* 7^. * The red-son of this Rule wiM easily appear from what* has hcenc said in Simple Subtraction; for the borrowing depends on the same principle, and -is only different as Hie;numberstoi>e^b<' Uacted: are of difiete&t deApmination^. COMPOUND SUBTRACTION. 93 Ex 6. A lends to B 100/, how much, is B. in debt afier^ has taken goods of him to the amount of 73/ 12/ 4Jr//* Ans. 26/ Is 7i(L , 7, Suppose' that my rent for half a year is 20/ I2/9.an<l ihat I haye laid out for the land-tax 14« 6di and for. severs^ repJlirs 1/3* 3 J^, what have I tppay of my half-year's rent ? Ans. 'l8/ 14j 2id. 8* A trader^ failing, owes to A 35/ Is 6d, to B 91/ 13/ id^ to C 53/ 7id, to D 87/ 5s, and to E 11 1/ 3/ 5^ When this happened, he had by him* in c^sh 23/ 7/ Sd^ in wares 53/ 1 1/ lOj^, in household furniture 63/ 171 7^, and in re- coverable book^-debts 25/ 7/ 5d. What will his creditors lose by him, suppose these things delivered to them ? . Ans. 2121. 5s Sid. EICAMPLES OF WEIGHTS, MEASURES, iS^C. TROV WfilGUT. 1. lb ozdwtgr Prom 9 2 12 10 Take 5 4 6 17 2. lb oz dwt gr 7 10 4 r7 3 7 16 12 Al'OTHECARIES WfilOHT- 3. lb oz dr scr gr 73 4 7 14 29 5 3 4 19 Rem. Proof AVOIRDUPOIS WEIOHT. LONG MEASURE. 4. 5. . ;6- 7. C qrs lb lb oz dr m fu pi yd ft in From 5 0.l7 71 5 9 14 3 17 96 4 Take 2 3 10 17 9 18 7 6 U 72 2,9 Rem. Prbof ^LOTH MEASURE. 8. . 9. yd qr nl yd qr nl From 17 2 1 9 2 T^e 9 2 7 2 1 LAND MEA817R£* 10. 11. ac ro p 17 1 14 16 2 8 ax: ro p 57 1 16 22 3 2d Rem. Proof WINE 38 ARITHMETIC. WINE MEASURE. ALE and BE 12. t hdgal From 17 2 23 Take 9 1 36 13. hdgal pt 5 4 2 12^ 6 14. hd gal pt 14 29 3 9 35 7 Rem. Proof IS. hd gal 71 16 19 7 pt 5 1 - ^ DRY MEASURE. From Take la 9 6 16. 17. qr bu bu gal pt 4 7 13 7 1 3 5 9 2 7 TIME. Hem. Proof -r*~ LP ™ ' * 18. 19. mo we da ds hrsmin 71 2 5 114 17 26 17 1 6 72 10 37 20. The line of defence in a certain polygon being 236 yards, and that part of it which is terminated by the curtain and shoulder being 146 yards 1 foot 4 inches; what then was the length of the f^ce of the bastion? Ans. 89 yds 1 ft S iq. "TT" COMPOUND MU^LTIPLICATION, Compound Multiplication shows how to find the amount of any given number of different denominations re^ peated a certain proposed number of times \ which is per** formed by the following rule. Set the multiplier under the lowest denomination of th^ multiplicand, and drjiw a line below it. — Multiply the num- ber in the lowest denomination by the multiplier, and find how many units of the next higher denomination are con- tained in the product, setting down what remains. — In like manner, multiply the number in the next denomination, and to the product carry or add the units, before found, and find how many units of the n^xt higher denomination are in thi^ , , f^mowit, COMPOUND MULTIPLICATION. 39 amount, which carry in Hke manner to the next product, settiiig down the overplus.*^Proceed thus to the highest de- nomination proposed : so shall the kst product, with the se« Verat remainders, taken as one compound number, be the whole amount required. — ^The method of Proof, and the reason of the Rule, are the same as in Simple Multiplication. EXAMPLES OF MONEY. i. To find the amount of 8 lb of Tea, at 5s 8 Id per lb. i- d • 8 jf 2 5 8 Answer. / / d 2. 4 lb of Tea, at Is Sd per lb. Ans. 110 8 3. 6 lb of Butter, at 9^4 per lb. Ans. 4 9 4. YlbofTobacco, at li 84rfperlb, Ans. 11 11^ 5. 9 stone of Beef, at 2^7 trf per St. Ans. 1 10 6. 10 cwt of Cheese, at 2/ i7« KWper cwt. Ans. 28 1 8 4 7. 12 cwt of Sugar, at 3/7/ 4rf per cwt. Ans. 40 8 CONTRACTIONS. I. If the multiplier exceed 12, multiply successively by its component parts, instead of the whole number at once. EXAMPLES. 1 . 15 cwt of Cheese, at 17^ 6d per cwt» / / V 17 6 S 2 12 6 5 13 2 6 Answer. / X d 2. 20 cwt of Hops, at 4/ Is 2d per cwt. Ans. ^7 3 4 3. 24 tons of Hay, at 3/ 7/ 6d per ton. Ans. 810 O 4. 45 ells of cloth, at is 6d per ell. . Ans. 3 7 6 Ex. 5. 4a ARITHMETRi t Ex. 5. 63 gallon J of Oil, at 2/ %d per galh Aas* 7 $. 70 iKirrels of Ale, at l/^j per t^urel. Ans. 84 *7. 84 quarters of Oats, at 1/ 12/ 8//per qr. Ans* 187 8. 96 quarters of Barley,at l/3x4rf)per qr. Am. 112 9. 120 days' Wages, at bs 9^ per day. . Ans; 34 10. 144 reamsof Paper, at 13/ 4^^ per ream. Ans. 96 II. If the multiplier cannot be exactly produced by the multiplication of simj^^ numbers, take the nearest number to it, either greater or less, which can b^ so produced,' aAmuI-^ tipiy by its parts, as before. — ^Then multiply the giveii mul- tiplicand by the difference between this assumed number and the multiplier, and add the product tq that before found, >rhen the assumed number is less than the multiplier^ bul^ subtract the same wh^n it \& greater. / 4 1 9 41 10 G EXAMPLES. f.' ■?^ y?irds of Cloth, at / • 3/ Ot^ per yard. / d 3 f 15 H , 5 f . 3 16 H # § 3 oi . 19 li ■Answer. / s d 2. 29 quarters of Corfi, sit 2/ 5/ S^d pet qr. Arts. 65 12 lO^ si 53 loads of Hay, at 3/ 15/ 2^ per ibad. Ans. 199 3 10 . 4. 79bushelsofWh^at,atll/5|.rf*^er bush. Ans. 45 6 lOf 5. 97 casks of Bepr, at 12/ 2^?peif cask. Ans. 59 O 2 6. 114 stone of Meat, at 1 5/ 3|tf per stone. Ans. 87 5 74. EXAMPLES OF WEIGHTS AND MEASURES. 1. • lb oz dwt gr 28 7 14 10 4 2. lb oz dr sc 2 6 3 2 • 8 cwt 29 3. qr lb oz 2 16 14 12 <^- • ■ """V ■ > •♦►•'■ COMPOUND DIVISION. 41 4. 5, 6. ttds fu pis yds yds qfs na ro po 22 5 . 29 6 4 126 3 1 7 28 3 27 9 T»" V. tuns hhd gal pts ^0 2 26 2 3 8, 9. we qr bu pe mo we da ho min 24 2 5 3 172 3 5 16 49 6 10 «•■ "W" *?iP COMPOUND DIVISION, CoMi^oUND Division teaches how to divide a number of sj^veral denominations by any given number^ or into any number of equal pjarts ; as follows : Fla^e the divisor on the left of the dividend^ as in Simple Division. — ^Begin at the left-hand^ and divide the number of the highest denomination by the divisor^ setting dgwn th^ quotient in its proper place.-«-If there be any remainder after this division, reduce it to the next lower denominattony which add to the number, if any, belonging to that denomi- nation, and divide the sum by the divisor. — Set down a^a tlfis 4dot}ent| reduce its remainder to the next lower deho-* mination again^ and so on through all th^ denominations tq Isfae last. r EXAMPLES OF MONBY. I. Divide 237/ 8/ 6d by 2. lid 2 ) 231 8 6 ;^118 14 3 the Quotient. 2. Divids . 4a ARTTHMEnC- I s d t s d ^i Divide 4S2 12 \\ by 3. Ans. 144. 4 a|. 3. Divide 507 3 5 by 4. Ans. 126 15 lOf 4. Divide 632 7 61- by 5. Ans. 126 9 6 5. Divide 690 14 S^J: by 6. Ans. 115 2 44- 6. Divide' 705 10 2 by 7. Ans. 100 15 8^ 7. Divide 760 5 6 by 8. Ans. 95 O 8:^ 8. Divide 761 5 7| by 9. Ans. 84 11 %\ 9. Divide 829 17 10 by 10. Ans. $2 19 ^\: 10. Divide 937 '8* 8|byll. Ans. 85 4 5 11. Divide 1145 11 4^ by 12. Ans. ^^ 9 3^. CONTRACTIONS. \p If the divisor exceed 12, find what simple numbers, multiplied together, will produce ity^and divide by them se-> parately, as in Simple Division, as below. ' . £XAMPJ^£5. I. What is Cheese per cwt, if 16 cwt cost 25/ 14j ^i? Isd 4) 25 14 8 4) 6 8 8 j^ 1 12 2 the Answer. / 2. If 20 cwt of Tobacco come to : 20 cWt ot lobacco come to 7 An « a in a 150/ 6/ 8i, what is that per cwt ?i ^^* ^ * 3. Divide 98/ 8j by 36. Ans. 2 14 8 4. Divide 71/ 13/ \Odhj 56. Ans. 1 5 7-5- 5. Divide 44/ 4 j by 96. Ans. 9 2t 6. At 3J / lOx per cwt, how much per lb } Ans. 5 7^ IL If the divisor cannot be produce4 by the multiplication of small numbei-s, divide by the whole divjisor at once, after the manner^of Long Division, as follows. EXAU- COMPOUND DIVISION. 43 EXAMPLES. 1. Divide 59/6/ Sid by 19. I s d I s d 19 ) 59 6 3| (3 2 51 An^. 57 2 20 46 (2 38 8 12 99 (5 95 • 4 4 . 19 (1 ■ t 2, 3. 4. 5. Divide 39 14 5| by 57. Divide 125 4 9 by 43- Divide 542 7 10 by 97. Divide 123 11 2t by J 27. / s Ans. 13 Ans* 2 18 Ans. 5 11 Ans. 19 Hi 3 10 EXAMPLES OF WEIGHTS AND MEASURES. 1. Divide 17 lb 9 oz dwts 2 gr by 7. Ans. 2 lb 6 oz 8 dwts 14gn 2. Divide 17 lb 5 oz 2 dr 1 scr 4 gr by 1 2. Ans. lib 5 oz 3 dr 1 scr 12 gr. 3. Divide 178 cwt 3 qrs 14 lb by 53. Ans. 3cwt Iqr 14lb. 4. Dividie 144 mi 4 fur 2 po 1 yd 2 ft in by 39. Ans. 3 mi 5 fur 26 po yds 2 ft 8 in. 5. Divide 534 yds 2 qrs 2 na by 47. Ans. 1 1 yds 1 qr 2iia. 6.- Divide 7 1 ac 1 ro 33 po by 5 1 . Ans. 1 ac 2 ro 3 po. 1. Divide 7 tu hhds 47 gal 7 pi by 65. Ans. 27 gal. 7 pi. 8. Divide 387 la 9 qr by 72. Ans. 3 la 3 qrs 7 bu. 9. Dividij 206 ino 4 da by 26. Ans. 7 mo 3 vire 5 ds. 44 ARITHMETIC. The golden RULE,' or RULE OF THREE. - The Rule of Three teaches how to find a fourth pro- portional to three numbers given : for which reason it is sometimes called the Rule of Proportion. It is called the Rule of ,Three, because three terms or numbers are given, to find a fourth. And because of its great and extensive use- fuhiess, it is often called the Golden Rule. This Rule is usually considered as of two kinds, namely. Direct, and Inverse. The Rule of Three Direct is that in which more requires more, or less requires less. As in this; if 3 men dig 21 yards . of trench in a certain time, how much will 6 men dig in the same time ? Here more requires more, that is, 6 men, which are more than 3 men, will also perform, more work, in the same tihie. Or when it is thus: if 6 men .dig 42 yards, how much will 3 men dig in the same time ? Here then, less re- quires less, or 3 men will perform proportionably less work than 6 men, in the same time. In both these cases then, the Rule, OK the Proportion, is Direct j and the stating must be thus. As 3 : 21 :: 6 : 42, or thus,' As 6 : 42 1 : 3 : 21, But the Rule of Three Inverse, is when nlore requires less, or less reouires more. As in this i if S m^o dig a celtain quantity or trench in 14 hours, in how many hour^ Will 6 men dig the Kke quantity ? Here it ii evident that 6 men, being more than 3, will perform an equal quantity of work in less time, or fewer hours. Or thus : if 6 men perform a certain quantity of work in 7 hours, in how many hours will '3 nien perform the same ? Here less requires more, for 3 men will take more hours than 6 to perform the same work. In both those cases then the Rule, or the Proportion, is Inv^i^ ; and the stating must be thus. As 6 ? 14 :: 3 : 7^ or thus. As 3 : 7 :: 6 : 14. And in all these statings, the £:nirth term is found, hj inukiplying the 2d and 3d terms together, and dividing the product by the 1 st term. ^ Of the three given numbers ; two of them contain the supposition, and the third a demand. And for stating and working questions of these kinds, observe th^ following ge- lieral Rule :.• Stat^ RULE OF THREE. *» State the qt^stion, by setting down.in astraigbtline the three given numbers, in the following mannery'viz. ao that the 2d term be that ntunbea* of supposition which is of the same kind that the answer or 4th term is to be ; 'making the other number of supposition the 1st term, and the demanding number the dd term, when the question is in direct propor- tion ; but contrariwise, the other number of supposition the Sd term, and the demanding number the 1st term, when th^ question has inverse proportion. Then, in both cases, multiply the 2d and Sd terms to* gether, and divide the product by the Ist, which will give the answer^ or 4th term sought, viz. of the same denomma- tion as the second term. Nate^ If the first and third t^ms consist of diffinrent deno^ minations, reduce them both to the same : and if the second term be a compound number, it is mostly convenient to re- duce it to the lowest denomination mentioned.^-^If, after di« vision, there be any remainder,, reduce, it to the next lower denomination, and divide by the same divisor as before, and the quotieilt will be of this last denominatioh. Proceed in the same manner with all the remainders, till they be re* duced to the lowest denomination which the second admits of, and the several quotients taken together will be the an- swer required. N^te also. The reason for thtf foregoing Rules will appear, when we come to treat of the nature of Proportions.— Some- times two or more statings are necessary, which may always be known from the nature of the question. * EXAMPLES. i. If 8 yards of Cloth cost 1/ 4^, what will 96 yards cost i yds 1 s yds 1 s As 8 : 1 4 *: : 9€ : 14 S the Answer. 20 24 96 144 216 8) 2304 2^0) 28,8* jf 14 8 Answer* £x. 2s 4^ ARITHMETrC% Ex. 2. An engineer having rabed 100 yards of a certain work in 24 days with 5 men ; how many men must he enw jploy to finish a like quantity of work in 15 days ? ds men ds men As 1 5 : 5 ': : 24 : 8 Ans. 15) ^20 ( 8 Answer. 120 3» What will 72 yards of cloth cost> at the rate of 9 yards for 51 12j ? Ans. 44/ 16/. 4. A person^s annual income being 146/; how much is that per day ? Ans. 8/. 5. If 3 paces or common steps of a certain person be equal to 2 yardS) how many yards wiU 160 of his paces make ? Ans. 106 yds 2 ft. 6. What length must be cut off a board, that is 9 inches broad, to make a square foot, or as much as 12 inches in length and 12 in breadth contains ? Ans. 16 inches. 7. If V50 men require 22500 rations. of bread for a month; how' many rations will a garrison of 1 200 men require ? • Ans. 36000. 8. If 7 cwt 1 qr of sugar cost 26/ 10/ 4^; what will be the price of 43 cwt 2 qrs ? Ans. 159/ 2s. 9. The clothing of a regiment of foot of 750 men amount- ing to 2831/ 5s; what will the clothing of a body of 3500 men amount to? Ans. 13212/ 10/; 10. How many yards of matting, that is 3 ft broad, will cover a floor that is 27 feet long and 20 feet broad ? Ans. 60 yards. 1 1 . What is the value of 6 bushels of coals, at the rate of \l I4fs 6d the chaldron ? Ans. 5s 9d. 12. If 6352 stones of 3 feet long complete a certain quan- tity of walling ; how many stones of 2 reet long will raise a like quantity ? Ans. 9528. 13. What must be given for a piece of silver weighing 73 lb 6 oz 15 dwts, at the rate of 55 9d per ounce ? Ans. 253/10/0|rf. 14. A garrison of 536 nien having provision for 12 months; how long will those provisions last, if the garrison be increased to 1 124 men ? Ans. 174 days and -tttt' 15. What will be the tax upon 763/ 15x, at the rate of 35 6d per pound sterling ? Ans^ 1 33/ 1 3/ If ^. 16. A RULE OF THREE. 47 16* A certain work being raised in 12 days, by working 4 liours each day ; how long would it hjive been In ra^ising bf %Krorking 6 hours per day ? . Ans. 8 days, IT. What quantity of com can I buy for 90 guineas, at the^ raje of 6s the bushel ? Ans. 39 qi^s 3 bu. 18. A person, failing in trade, owes in all 977/; at which time he has, in money, gobds, and recoverable debts, 420/ 6s S^d\ now supposing these things delivered to his creditors, lidw much will they get per pound ? Ans. 8x 7 j€f.> « 1 9. A plain of a certain extent having supplied a body of SOOO horse with forage for 1 8 days ; then how many days would the same plain have supplied a body of 2000 horse? Ans. 27 days. 20. Suppose a gentleman^s income is 600 guineas a year, and that he spends 25/ 6d per day, one day with another ; ' ho'w much will he have saved at the year's end ? Ans. 16^/ I2s 6d. 2i . What cost 30 pieces of lead, each weighing 1 cwt 121b, at the rate of I6j W the cwt ? Ans. 27/ 2s 6d. 22. The governor of a besieged place having provision for 54 days, at the rate of 1-^ lb ofbread ; but being d^irous to prolong the siege to 80 days, in expectation of succour, in that case what must the ration ofbread be ? Ans. l-^lb. 23. At half a guinea per week, how long can I be boarded for 20 pounds ? Ans. 38^?^ wks* 24. How much vrill 75 chaldrons 7 bushels of coals come to, at the 'rate of 1/ 13i 6^ per chaldron ? Ans, 125/ 19x0|J. 25. If the pormy loaf weigh 8 ounces when the bushel of wheat costs 7s 3rf, what ought the penny loaf to weighi|lhen the wheat is at 8/ ^dP Ans. 6 oz 15t^ dr. 26» How much a year will 173 acres 2 roods 14 poles of land give, at the rate oi Ills 8d per acre? - > Ans. 240/ 2s 1^^. • 27. To how much amounts 73 pieces of lead, each weigh- ing 1 cwt 3 qrs 7 lb, at 10/ 4j per fother of 1 9^ cwt? Ans. 69/ 4j 2i/ 111 q. /^ 28. How many yards of stuff, of 3 qrs wide, will line a ' cloak that is ^ yards in length and 3t yards wide ? Ans. 8 yds qrs 2| nl. '29. If 5 yards of cloth cost 14/ 2d, what must be. given for 9 pieces, containing each 21 yards 1 quarter? ^ Ans. 27/ \s \Q\d, '; 30. If a gentleman's estate be wortt 2i07/ 12/ a year j what may he spend per day, to save 5C0/ in the year ? Ans. 4/ Ss l-^Sd. 31. Wanting M ARTTHMETia 31. Wanting just an acre of land cut off from a piece ^which is 13^ pdes in hreadthj what length must the piece be? / Ans 11 po 4yds 2ft O^f in. 32. At 7s 9f «/ per yard, what is- the value of a piece of doth containing 53 ells English 1 qu. Ans. 25/ lb/ 1^. 33. If the carriage of 5 cwt 14 lb for 96 miles be 1/ 12/ 6ds Jhow far may I have 3 cwt 1 qr carried for the same money i Ans. 151 m 3 fur ^rrt^* 34. Bought a silver tankard, weighing 1 lb 7 oz 14 dwts i what did it cost me at 6/ 4^ the ounce ? Ans. 6/ 4/ 9 V« 35. What is the half year's rent of 547 acres of land, at 1 5s 6d the acre ? Aas. 21 1/ 19j SJ. 36. A wall that is to be built to the height of 36 feet, was^ raised 9 £set high by 16 men in 6 days; then how many men must be employed to finish the wall in 4 days, at the same rate of working ? Ans. 72 men* 37. What will be the chargfe of keeping 20 horses for a year, at the rate of 14^^ per day for each horse ? Ans. 441/Oj lOrf. 38. If 18 elk of stuff that is i yard wide, cost Z9s6d'} what will 50 ells, of the same goodness, cost, being yard wide-^ Ans. 7/ 6s :i||</. 39. How many yards of paper that is 30 inthes wide, will hang a room that is 20 yards in circuit an^ 9 feet high? Ans. 72 yards. 40. If a gentleman's estate be worth 384/ 16/ a year, and the lapd-tax be assessed at 2/ 9id per pound, what is his net aftnuar income? Ans. 331/ 1/ 94^. 41 . The circumference of the earth is about 25000 miles i at what rate per hour is a person at the middle of its sur£a€e carried round, one whole rotation being made in 23 hours 56 minutes ? Ans. i044-ry^ miles. 42. If a person drink 20 bottles of wine per month, when It costs 8/ a gallon j how many bottles per month may he drink, without increasing the expense, when wine costs 10/ the gallon? Ans. 16 bottles. 43. What cost 43 qrs 5 bushels of corn, at 1/ 8/ 6d the . quarter ? Ans. 62/ 3/ S^d. 44. How many yards of canvas that is ell wide will line 50 yards of say that is 3 quarters wide? Ans. 30 yds. 45. If an ounce of gold cost 4 guineas, whaC is the valfae of -a grain? Ans. 2-yT,it 46. if 3 cwt of tea cost 40/ 12/; at how much a pound must it be retailed, to gain loi^ by the whole ? Ans.3y|^^j. CX)MPOUN» C « 3 COMPOUND PROPORTKHf- Compound Proportion ^ows how to resolve such ques- tions as require two or more statings by Simple Proportion ; and these may be either Direct or Inverse. • • In these questions, there is always given an odd mimber of terms, either five, or seven, or nine, Sec. These are distm-. guished into terms of supposition, and terms of demand, there being always one term more of the former *than of the y^ latter, which is of the same kind with the answer sought. The method is thus : Set down in the middle place that term of supposition which is of the same kind with the answer sought.— -Take one of the other terms of supposition, and one of the demand- ing terms which is of the same kind with it ; then place one of them for a first term, and the other for a third, according to the directions given in the Rule of Three. — Do the same with another term of supposition, and its corresponding de- manding term ;, and so on if there be more terms of each kind ; setting the numbers under each other which fall all on the left-hand side of the middle term, and the same for the others on the right-^hand side. — ^Then^ to work By several Operations, — ^Take the two upper terms and the middle term, in the same order as they stand, for the first Rule-of-Three question to be worked, whence will be found a fourth term. Then take this fourth number, so found, for the middle term of a second Rule-of-Three question, and the next two under terms in the general stating, in the same order as they stand, finding a fourth term for them. And so on, as far as there are any i|iumber$ in the general Stating, making always the fourth number, resulting from each simple stating, to be the second term in the next following one. So shall the last resulting number be the answer to the question. By one Operathfi, -^Multiply together all the terms stand- ^ ^t^o^^^^Ur ing under eajch other, on the left-hand side of the middle I ^^eSL^ terni} and, in like manner, multiply together all those on the > ^^^^JgSul, right-hand side of it. Then multiply the middle term by \ ^f^iim^ jfthe latter product, and divide the result by the former pro- «^ ^^^^^-^^ 'duct } so shall the quotient be the answer sought. ^t^ij?^'*^ Vq|.. I, % l&^^AMPtES 50 ARITHMETIC. EXAMPLES. 1. How ms^y men can complete ^ trench of 135 yards long in 8 da^y when 16 men can dig 54 yards in 6 days ? General Stating. yds 54 : 16 :: 185 yds dfiys 8 6 days 432 810 16 4860 81 men 432 ) 1 2960 ( 30 Ans. by one opcratioq. 1296 ^' ' The same hy two Operations. 1st. As 54 : 16 :: 135 : 40 16 / 8IQ 135 54) 2160 (40 216 2d. As 8 : 40 : : 6 : 30 6 8 ) 240 ( 30 Ans. 24 O 2. If 100/ in one year gain 5/ interest, what will be the interest of 750/ for 7 years ? Ans. 2621 "iq^, 3. If a family of 8 persons expend 200/ in 9 months ; Jxow much will serve a family of 1 8 people 1? months ? . ^ Ans. 300/: 4. If 2*7s be the wages of 4 men for 7 days ; what will be the wages of 14 men tor ip days ? Ans. 6/ 15/. 5. If a footman travel 130 miles in 3 days, when the days are 12 hours long; in how many days, of 10 hours each, , may be travel 360 miles ? Ans. 9|| days. "i VULGAR FRACripNS, 54, # Ex. 6. If 120 bushels o£ com am s^rve 14 horyet SS daysi how inany days will 94 bushels serye, 6 horses ?, . • • ^ Ans. 102^ days.., 7. If 3000 lb of Ijeef serve ^0 mta^ IS Jays ^ how jiiany lbs win serve 120 men for 25 days ? >t)s.\ l764lb 1114.02^ ' 8. If a barrel of beer be sufficient to last a family of 8 per-. sons 12 days ; how many barrels will be .drank by i6 persons, in the space o^ year? - . ..AftSf.6P|. barrels*^ 9. If 180 men, infe d^ys^ of 10 hours ttcj^i, can dig a. trench 200 yards long, 3 wide* and 2 deep Zip hpw many, days, of 8 hours long, will 100 mth dig % treyh of 36 yards . ^Jong, 4 wide, and- 3 deep ? Ans. '^""*"* 9*" •"; -c. /•■.■.■; ■••■I , OF VULGAR FRACTIONS, part, with a line bet\^een them : _,_ 3 numerator i , . , . - « - , *• Thus, -- , > , which IS named 3»fourth§. 4 denominator J > The Denominator, or number placed below the line, shhws how m;^y equal ps^ts the whpl6(rqj|(i^f^it^ i^divid^qiflo ; and it represents the Divisor in D,iVjision«-t^An4 tH^/^^IMfifr tor, or numbeir set ^oye the line« shows, how m^y.^JFjtti^^ - parts are expressed by the Fraction; l^^ing the r^|n^^^^;|4ff ' after division. — Also, both these i^i^Jser^ ^c, in gen^r^ named the Tjerms of the Fracjt;iort. '> » ^ r! ; »t Fractions' are either. Proper* ImjMroper, Siajpjte>'jQf)aw 'pounds or Mixed. v . ; '^^ . f A Proper Fraction, is when the numerator is less than the denominator ; as, 4/ or f , or j-* ^c* y An Improper Fraction, is when the numerator is equal to, or exceeds, the denominatoj* ; as, j.,' Qrr^,.or ^s &c. V / ' A Simple FractiQjSy is a single exprei^ion, deiaodng any nuini>er of parts of the integer \ asi^,.er i^ A Compound Fraction, i$ the fir^Kririon of a fraction^ (r several fractions connected with the word ^ bet W^0n theii^; as, j^ of I, or -f of ^ of 3, &c* A Mixed Number, is composed of a whyle nipnb^ miA^ fraction together; at^.3^, or I24, &c. - E2 Sr A whole Hk A Krliote* nt hit^br htunlier may bef expressed like t frftc** tion, by writisg 1 btelo^'^it, ia a denominator ; so 8 is fs <^ 4is4i&c, A fraction denotes division ; and Its valtie is equal toliie qaotient obtained by dividing the nuinerator by the dtno^ ininator : so' '-^ is ecjud to'B, and V is eqnal to 4. Hence tlien, if the nomerator be lesfi than the denominator, the talne of the fraction is less tiian 1 . But if the numerator be the samfe iH the deno^ninator;,' the fiction is just eqUa! to 1 / And If the numerator be greater than the denraU^ natorj the ffattioji is greater than J^ tssst REDUCTION OF VULGAR FRACTIONS. REDi7CTX0N of Vujglt FractionS| is the bringing them out of.one form or denomination into another } commonly to {)repagre.theip for the o|>erations of Additionj Subtractioni ^c^ of vl^ch there ^^ure several cases. 1 I 4^ PROBLEM, 1 . ^ . . i • i m ToApd the Great f St Ccmmcn Measure ofTtuo or more Numbers* T|rtE Cotirtnoti Measure of two or more numbers, is that trulxiber which will divide them both without remainder ; so> f is a-^omtnon measure of 18 and 24 ; the quotient of the Ibrm^ being 6, and of the latter S. And the greatest num« ber that will do thisy is the grejttest common measure : so 6 is the greatest common measure of IS and 24 ; the quotient t>f tWe former being S, and ci the latter 4, whi^rh will not iHDtli diyidp furtJier. t^ there be two numbers only; divide the greater by thft tesfe \ l4ven divide the divisor by the remainder \ $md so on,^ dividing always the last divisor by the last remainder} t3l no- thing remains ; so shall the last diyisol^ of all be th^ greatest fimimon measure sought* When there are more than two numb^rs^ find the greatest Wttkfofsti miKisure of two of thom^ as bisfore ; then do the If^n^ for that cpmmpxi'^ineasure and another of the numbers} an4 REDUCTION w VUWJAH FRACTIONS. It wd io oo» through «U the uMibers; so uriU the great^t^ com- mon mtwmt last ibund be the answfr/ . . If it happen that the conunooi measure thus found' is If then the numbers are said to be incommensurabIp> Or uofi liif ing any comqion measure^ BXAM^L^. 1. To find the griNitest common measure of 1908> 936, and 630. 936) 1908 (2 So that 36 is the greatest common 1872 measure of 1808 and 9S6* 36 ) 936 ( 26 Hence 36 ) 630 ( 17 72 36 216 270 216 252 18) 36 (2 36 ^p Hence then 18 is the answer required. 2. What is the greatest common measure of 246 ind 372 ? Ans. 6* 3. What is the greatest common measure of 324, 61 9, and 1032/ ' A»3. 12. CASE I. T$ Abbrevii^ or Reduce Fractions U their Lowest Terms* * Divide the terms of the given fraction by any numher chat will divide .them irithput z remainder; tl^n divide these guotients * Tluit dtviding both the terms of the fraction by th^sam^ namt>er^ whatever it be, will give another fraction equal to the former, is evident A^d when these divisions' are performed as if^n as can be dobe^ or when the common divisor is the greatest possible^ tbf», terms of the resuhing fraction must be ^he least possible. Note, 1. Aiqr number endlag widi aoevea ijiuQiher, or a cipher^ n divisible^ or can be divided, by 2. 2. Any number ending with ^> or 0^ is divisible by 5, 3, If si ' Ai^ITHMETIC. c^ucei^tits again in the same manner ; and so on, till it appears that Uiere is no number greater than 1 which will divide thetti} then the fraction will be in its lowest terms. • • * . » Or^ divide both the terms of the fraction by their greatest common measure at once, and the quotients will be the terms of the fraction required, of the same value as at first. EXAMPLES. 1 . Rediuc^ a^ to its least terms.^ m-U- il = Tf =1 == ii the answer.^ Or thus : 216 ) 288 * { 1 Therefore 72 is the greatest common 216 taeasure; and 72; |4f =s ^ the An- swer, the same as before. 72) 216 {3 216 2. Reduce 3 . If the right-hand place of any numfier be O, the whole is di- visible by 10 3 if there be two ciphers, it is divisible by 100 j if three ciphers, by 1000 : and so on; which b only cutting offdiose ciphers. 4. If the two right-hand figures of any number be divisible by 4, the whole is divisible by 4. And if the thrqe right-hand figures be dtvisible by 8, the whde is divisible by 8. And so on. If 5. If the sum of the digits in any number be divisible by a, or by 9, the whole is divisible by 3, or by g. e. If the right-hand digit be even, and the sum of all the digits be divisible by 0, then the vlhole is divisible by 6. 7. A number is divisible by 1 1 , when the sum of the l st, 3d, 5th, . 3cc, or all the odd places, is equal to the sum of the 2d, 4tt, 6th, &c, or of all the even places of digits. 8. If a number cannot be divided by some quantity less than the sqliarfe root of the same, that number is a prime, o.^ cannot be di- vided by any number whatever, g. All priine numbers, except 2 and 5, have either 1, 3> 7, org, m the place of units ; and all other numbers are composite, or can be divided. 10. When REDUCTION OF VULGAR FRACTIONS. 55 t ^. Reduce f|^ to its lowest terms. Ans. ^* 3. Reduce 441 1® ^^^ lowest terms. Ans. \. 4. Reduce fyv ^^ ^^^ lowest terms. Ans. -I* CASE II. To Reduce a Mixed if umber to its Squivalent Improper Fraction. * MuLTiPLt the integer or whole number by the deno^ minator of the fraction, and to the product add the numera- tor ; then set that sum above the denominator for the frao^ tion required* EXAMPLES; 1 . Reduce 23| to a fraction* 23 5 115 Or, 2 (23x5)+2 in , . rs — , the Answer. 117 ^ ^ ' . 5 2. Reduce 12|- to a fraction. Ans. '^^ 3. Reduce 14t^ to a fraction. Ans. Vtf . 4. Reduce 183^ to a fraction. Ans. ^.t**. 10. When numbers, with the sign of addition or subtraction be^ tween them^ are to be divided by any number, then each of those , 4i ,, , . . rr.i 10-1-8—4 numbers must be divided by it. Thus — i =5 + 4'-2~7« 11. But if the numbers have the sign of multiplication between % them, only one of them muSt be divided. Thus, 10X8X3 _ 10X4X3 _ 10 X4X 1 _^ 10X2X1 _20^^ 6X2 "~ 6X1 ■" 2X1 " 1X1 ""T*"" \ . * This is* no more than first multiplyitig atjuaritity by some number, and then dividing the result l»ck again by the samd : which it is evident does not alter the value 5 for any fraction re- presents i. divisiori of the numerator by the detiominatot". . ' CASB g$ ARITHMETIC CASE III. T9 Reduce an Improper Fraction to its Equivalent Wb^ or Mixed Number, * DiVfDE the numerator, by the denominariior, and the quotient will be the whole or mixed number sought. EXAMPLES* 1. Reduce y to its equivalent number* Here y or 1 2 -r- 3 = 4, the Answer. 2. Reduce y to its equivalent number. Here V or 15 -r- 7 = 2|, the Answer. 3. Reduce V^^ to Its equivalent number. Thus, 17) 749 (44^ 68 69 So that ^ 3= 447V> the Answer. 68 1 4. Ileduce l^ to its equivalent number. Alls. 8. 5. Reduce ' ij* to its equivalent number. Ans. 54|4» 6. Reduce *^* to its equivalent number. Ans. 17147* i CASE IV. To Reduce a Whole Nundfer to an Equivalent Fraction^ having a Given Denmmnator^ f MULTIPLT tlie whole number by the given denominators then set the product over the said denominator, and it will form the fraction required* / * Tills Rule is evidently the reverse of the former ; and the reason of it is manifest from the nature of Common Division* f Multiplication and Division being here equally used> the re- sult must be the same as the quantity first proposed. EXAMPLES. REDUCTION op VULGAR FRACTIONS. S7 EXAMPLES. I 1. Reduce 9 to a fraction whose denominator shall he 7. • Here 9 X 7 s 63: then y is the Answer; For V = 63 -^ 7 = 9, the Proof. ^ Reduce 12 to a fraction whose denominator shall be IS. Aas. Vt • 3^ Reduce 27 la a fraction whose denominator shall be 1 U Ans. Vi^. CASE Y. To Reduce a Compound Fraction to an Equivalent Simple One* * Multiply ail the numerators together for a numerator, and aU the denominators together for a denominator, and they will form the simple fraction sought.' When part of the compound fraction is a whole or mixed number, it must first be reduced to a fraction bv one of the former cases. ; And, when it can be done, any two terms of the fraction may be divided by the same number, and the quotients used instead of them. Or, when there are terms that are conunon, they may be omitted, or cancelled. £XAMPL£S. 1. Reduce |^ of |- of |- to a simple fraction. ,- 1x2x3 6 1 , ^ **€^^:;: — :;; = :r: = rr> ^e Answer. 2x3x4 24 4' Or, ^ — - — 2 ~ T* ^y cancelling the 2's and i*s. ^ X Q X 4? 4 * The truth of this Rule may be shown as follows : Let the .compound fraction be -| of j . Now j- of -f is | -r 3» which is ^\-j consequently f of -^ will be ^V X 2 or ij--, that is, the numerators are muluplied togetlier, and ulso the denominators, as in the Rule. When the compound fraction consists of more than two single ones i having tirit reduced two of them as above, then the resulting Action and a third will be the same as a compound fraction of t«vo jparts^ and so on to the last of all. 2, Reduce 58 ARITHMETIC. 2. Reduce ^ of -^ of -{4 to a simple fraction. _, 2x3x10 60 13 4 , . ""'" sTI^TTT = 165 = 33 = IT' ^^^'^ ^"'^"'••. 2 X 3^ X jd 4 ^^» ^ — f — TT ~ 7T> ^^ ^^°^^ ^^ before, by cancelling the 3's, and dividing by 5's. S. .Reduce ^ of | to a simple fraction. Ans. 54* 4. Reduce f of f of 4 to a simple fraction. Ans. -J. 5. Reduce |. of -f- of Si to a simple fraction. Ans. {• 6. Reduce 7^ of ^ of -J of 4 to a simple fraction. Ans. i. 7. Reduce 2 and | of |^ to a fraction. Ans. |^. CASE VI, To Reduce Fractions of Different Denominators^ to Mquivatent Fractions having a CommoH Denominator. * Multiply each numerator by all the denominators ex- cept its own, for the new numerators : and. multiply all the denominators together for a common denominator. NotCy It is evident, that in this and several other (fperation^ when any of the proposed quantities are integers, or mixed numbers, or compound fractions, they must first be reduced, by their proper Rules, to the form of simple fractions* EXAMPLES. 1 . Reduce 4-> t> and |., to a common denominator. 1 X 3 X 4 = 12 the new numerator for 4. 2x2x4 = 16 ditto f. 3 X 2 X 3= 18 ditto f. 2 X 3 X 4 = 24 the common denominator. Therefore the equivalent fractions are ^, -J^, and i^. Or the whole operation of multiplying may be best per^ formed mentally, only setting down the results and given fractions thus: 4, 4, J- = H> ih H = A» tV> tV, by abbl'eviation. 2. Reduce y and ^ to fractions of a common denominator. Ans. II, 14. * This is evidently bo more than multiplying each numerator And its denominator by the same quantity, and consequently th6 value of the fraction is not altered. 3. keduce REDUCTION OF VULGAR FRACTIONS. v<9 5. Reduce •§-, ^i and ^ to a conunon denominator. Ans. ^, IJ, 1^. 4. Reduce i, 2^1 and 4 to a common denominator. AnQ 2 5 7 8 »ao Note 1 . When the denominators of two given fractions have a common measure, let them be divided by it ; then muhiply the terms of each given fraction by the quotient arising from the other's denominator, £/:• 2T and j^ =: -rrr ^^^ tttj ^7 niuhiplying the former 5 7 by 7 and the latter by 5. 2. When the less denominator of two fractions exactly divides the greater, multiply the terms of that which has the less denominator by the quotient. J?ar* f and -^ =: ^ and -^^ by mult, the former by 2. 2 3. When more than two fractions are proposed, it is some- times convenient^ first to reduce two o£them to a common denominator ; then these and a third ; and so on till they be all reduced to their least conunon denominator. £:e. I and J and I =;: I and J and | = ^^ and if and J^. CASE vir. To find the value of a Fraction in Parts of the Integer, Multiply the integer by the numerator, and divide the product by the denominator^ by Compound Multiplication and Division, if the integer be a compound quantity. Or, if it be a single integer, multiply the numerator by the parts in the next inferior denomination, and divide the pro- duct by the denominator. Then, if any thing remains, mul- tiply it by the parts in the next inferior denomination, and divide by the denominator as before \ and so on as far as necessary ; so shall the quotients, placed in order, be the value of the fraction required *i * The numerator of a fraction being considered as a remainder, in Division^ and the deriomitiator ^s the divisor, this rule is of the same nature as Compound Dtvisioni or the valuation of remainders in th« Rule of Three, before e&plained. EXAMPLES. tQ ARTTHMEnC. EXAMPLES. I. What is the f of 2/ 6s f" B J the former part of the Rule, 2/ 6s 5)9 4 Ans» 1/ 16/, 9d 2\q. 2. What Is the value of ^of 1/r By the 2d part of the Rule^ 2 20 3 ) 40 ( 13x 4rf Am. 1 12 3) 12 (4</ $• Find the value of -f of a pound sterling* Ans. Is 6d» 4. What is the value of f of a guinea ? Ans. 4i 8i^ Sm What is the value of ^ of a half crown i Ans^ 1/ lO^^. Ans* 1/ 1 l{d. Ans. 9oz 12dwts» Ans* Iqr 71b* Ans. 3 ro. 20 po* Ans* 7brs 12mi&» 6. What is the value of f of 4/ \0d? 7. What is the value of * lb troy ? 8* What is the value of i ^of a cwt ? 9. What is the value of {-of an acre? iO.. What is the value of -^ of a day ? CISE VJII. I Tq Reduce a Fraction from one Denomination to another. ^ Consider how many of the less denomination make one of the greater ; then multiply the numerator by that number, if the reduction be to a less name^ but multiply the denomir nator, if to a greater. EXAMPLES. 2 . Reduce |^ of a pound to the fraction of a penny, f X V X V == *r =^ 't""* ^^e Answer. * This is tiie same as the Rule of Keduclion in whole numbers from ooe denomiiiatign to another. 2. Reduce ADDITION OF VULGAR FRACTIONS. 6fr $• Reduce f of a penny to the fraction of a pound* I X A X ^ = ^, the Answer. 9. Reduce i%/ to the fraction of a penny. Ans. V^« 4* Reduce |q to the fribCtion of a pound. Ans. t^V?- 5. Reduce y cwt to the fraction df a lb. Ans. V^ 6. Re<htce f dwt to the fraction of a lb troy. Ans. -j^^* 7. Reduce |- crown to the fraction of a guinea.. Ans. ^^. 6« .Reduce ^ half-<:rown to th« fract. of a shilling. Ans. 44* 9. Reduce 2s 6d to the fraction of a £• Ans. 4« 10. Reduce 17/ 7rf S\j to the fraction of a £. *lmmmmiitmmmm^ttmmatagmmmmmmm*im ADDmON OP VULGAR FRACTIONS. tp the fractions have a common denominator; add all the liumerators together, then place the sum over the common denominator, and that will be the sum of the fractions required. * If the proposed fractional have not a common denomina- tor, they must be reduced to .one. Also compound fractions must be reduced to simple ones, and fractions of diderent denominations to those of the same denomination. Then add the numerators as before. As to mixed numbers, they may either be reduced to improper fractions, and so added with the others ^ or else the n-actional parts only added, and the integers united afterwards. * Before fractions are reduced to a common denominator, they are quite dissimilar, as much as shillings and pence are, and there- fore cannot he incorporated with one another, any more than these can. But when they are reduced to a common denominator, and made parts of the same thing, their sum, or difference, may then.be as properly expressed by the sum or difference of the numerators, as the sum or difference of any two quantities whatever, by the sum or difference of their iedividuals. Whence the reason of thQ B-ule is manifest, both for Additipn and Subtraction. When several fractions are to be collected, it is commonly best ^rst to add two of them together tuat mrst easily reduce to a com- fifwn dptiomn^^^f then ftdd their ^um and a thirds and so on. EXAMPLES. » \ 63 ARITHMETIC. EXAMPLES. 1. To add ^ and f together. Here ^ + ^ = ^ = 1 4, the Answer. • 2. To add 4 and | together. H-i = 44 + l^=^=lT4,the Answer. 3. To add 4 and 7i and 7 of |- together, 4. To add* f and 4 together. Ans. If. 5. To add ^^ and f together. ; Ans. IfJ^, 6. Add y and -/^ together. Aiis. ^2^. 7. What is the sum of -f and ^ and ^? Ans. 1tS4* 8. What is the sum of | and ^ and 2^? Ans. S|^. 9. What is the sum of 4 and ^ off, and 9,^ ? Ans. 10^^. 10. What is the sum of |- of a pound and 4- of a shilling ? Ans. '|5,or 13i 10^24^. 1 1 • What is the sum of 4 of fi shilling and -j^ of a penny ? Ans. Vt ^ or Id \^g. 12. What is the sum of 4 of a pound, and |. of a shilling, and TT'of a penny ? Ans. 4^^j or Ss Id 145?* SUBTRACTiqjf^ OF VULGAR FRACTIONS. ' Prepare the fractions the same as for Addition, when necessary 5 then subtract the one numerator from the other, and set the remainder over the common denominator, for the difference of thp fractions sought.* EXAMPLES, 1 . To find the difference between 4 and 4« Here 4 "^ 1^ ^ 4 == t> ^^^ Answer. 2. To find the difference between 4 and 4. T - V = T^ •- If = Ty» the Answer. 3. What MULTIPLICATION of VULGAR FRACTIONS. 63 S. what IS the difference between -^ and -f-^ ? Ans. ^. 4. What IS the difference between ^ and -f^i Ans, y^. 5. What Is the difference bel;ween ^ and ^Zj ? Ans. -rh- 6. What is the diffl between 5| and ^ of 4^ ? Ans. 4-,^^. 7. What is the difference between -^ of a pound, and J- of i of a shilling ? Ans. Vt -^ ^r lOx Id l^. 8. What is the difference between f of 5^ of a pound, and I of a shilling ? - Ans. lU-il or V Ss \ 1 -.y. MULTIPLICATION of VULGAR FRACTIONS. * Reduce ^ixed numbers, if there be any, to equivalent fractions ; then multiply all the numerators together for a numerator, and all the denominators together for a denomi« nater, which will give the product required. EXAMPLES* N 1 . Required the product of ^ and |.. Here t ^ i == t% ^ t> ^*^c Answer. Or^x J = 4x^ = i. 2. Required t}ie continued product of ^ 3--, 5, and | of -fi ,, ?f 13 ^ a 3 ISx 3 39 Here y x -x ^ X "^ X y = -;^;3^ =- = 4^, Ans. 3. Required the product of f and ■^. Ans. j^y. 4. Required the product of -^ and -j^. Ans. -rV. 5. Required the product of f , ^, and 44» Ans. ^*j. * Multiplication of any thing by a fraction, implies the taking some part or parts of the thing j it may therefore be truly expressed by a compound fraction; which is resolved by multiplying toge- ther the numerators and the denominators. Note, A Fraction is best inultiplied by an integer, by dividing the denominator by it -, but if it will not exactly divide^ then multiply the numerator by it 6. Required 64 ARITHMETia 6. Kequired the product ofi, jf^nd S* An$«> 1; 7. Required the product of ^, ^, and 4^ Ans. 2^. 8. Required the product of ^ and •§- of y. Ans. H^ 9. Required the product of ^, and f of 5, Ans, 20» 10. Required the product of | of j, and -J. of 3f . Ans. f J^ 1 1 . Required the product of 3| and 4^. . Ans, i 4444* 1 2. Required the product ef 5, |> f of f, and 4|. Ans. 2^. ■««» DIVISION or VULGAR FRACTIONS. ^ Prepare the fractions as before in MuhipKcation ; then divide the numerator bv the numeratorj and the denominator hj the denominator, it they will exactly divide : but if not» then invert the terms of the divisor, and multiply the dividenc) by h, as in Multiplication* ZXAUPtES. 1. Divide V by f . Here V "^ 4^ = t = Ht ^7 the first method* 2, Divide^ by ,5^. Here^^T^=:^x V=-f xi = V =-H- 5, It is required to divide 44 ''T 4* Ans. f* 4. It is required to divide -^ by ^. Ans. ^V 5. It is required to divide \^ by -^. Ans. l-f. 6. It is required to divide ^ by V • Ans. -^^ 7. It is required to divide y^ by ^. Ans. -f . .8. It is required to divide 7 by ^ Ans. -Jt « ^ Divison being the reverse of Multiplication^ the reason of the Rule is evident. Note, A fraction is best divided by an integer, by dividing the numerator by it ; but if it vyill not exactly divide^ then multiply the denominator by it* 9. ft RULE OF THREE m VULGAR FR A CTIONS. fi^ •9. It is required to divide V^ t)y 3. • Ans. -^ 10. It is required to divide 4 by 2. Ans. ,25^' 11. It is required to divide 7 j. by 9^. Ans. ff . 1 2. It is required to divide f ttf j- by f of 7f . Ans. Tfr- ' * ' " RULE OF THREE in VULGAR FRACTIONS. Makb the necessary preparations as before directed ; then multiply continually together, the second and third terms^ Mtkd the first with its parts inverted as in Divi^ion^ for the answer** Examples. 1 • If f of a yard of velvet cost | of a pound sterling ; vfhM Will 1^ of a yard cost ? S 2 5 S i S , , ^ y •• "5" •• li = T ^ 7^ ig =i'*^^«^^ Answer. ^i What trill S^ot bf silver cost^ at 6s 4i an ounce f Ans. 1/ li 44rf. 3. If T^ of a ship be worth 273/2/ 6d; what are ^r of her worth? Ans. 227/ 12/ Id. 4i What is the purchase of 1230/ b^k-stbck, at 108| per cent.? Ans. 1336/ UW. 5. What is the interest of 273/ 15/ for a year, at 3^ per cent.? Ans. 8/17/ ll|i/. .6. If I of a ship be worth 73/ 1/ 3/// what part of her is worth 250/ iOsP Ans. j-. 7. What length thust be cut off a bo^d thit is 7^ inches broadi to tontain a square foot, or as much as another piece of 12 inches long and 12 broad ? ' An^ I844 inched. S, What quantity of shalloon that is ^ of a ydrd wide^ will line 9i yards of cloth/ th^at is 24 yards widei Ans. 31|^ yds. » . ^ . . ■ . , "* This is only multiplying the 2d and 3d terms Oogether^ and diidding the product by the firsts as in the Rule o^^ree in whote ^umbera. >. Vol. h T ' 9. If 66 ' ' AWtllMlETIC: 9. If'tlie penny loaf weigli 6^<5K| when the prite bl w'heat is -5/ the bushel ; what CH^t it to weigh whe^ the wheat is 8s 6d the bushel? Ans. 4-^7- oz. 10. How much in length, of a piece of land that is ll^^* poles broad, will make an acre of land, or as much as 40 poles in length and 4 in breadth ? Ans. 13-^^ poles. 11. If a courier perform a. certain journey in %5i days^ travelling 13^- hours a day ; how long would he be in per- forming the same, 'travelling only I'l-ru liours a day ? Ansi 40|44 days. 12. A regiment 6f soldiers, tonsisttng of 976 mw, are to be iiew cloathed ; each coat to contain ^i yards of cloth that is i{ yzrft wide, and lined ifith shalloon ^ yard wide : how maily yaid$* of shalloon will line them ? • Ans. 4531 yds 1 ^ 2| nails. * .d' i /tiii; DECIMAL FRACTIONS. A Decimal Fraction, is that which has foi** its deno- minsttonaatmit (1), with as many ciphers annexed as the humet-atoi: ha^ places ; and it is usually expressed by setting down the numerator only, with a point before it, on the left- hand.- Thus, ^^ is -4, and -,^\ is •24,>and ^i-ttr is "074, arid ^^>5*y^^ is '00124 ; where ciphers are prefixed to make up as many places as are ciphers in the denominator, when there is a deficiency* of figures. • A mixed nuipber is made up of a whole number with some decimal iiraction, the one being separated from the .other by a point.; Thus, 3 '25 is the same as S-^y or 4^. ' Ciphers on the right-hand of decimals make no alteration in their value; for*4, or '40, or '400 are decimals having all the same value, each being == -/„, or |^. But when tliey are placed on the left-hand, they decrease the value in a ten-fold proportion: Thus, '4 is T^y* or 4 tenths ; but '04 is oiily -r^t or 4 hunciredfhs, and '004 is only -ri^j or 4 thousandths* The 1st place of deciinals, counted from the left-hand to- wards the^rightj j.s called the place of primes, or 1 Oths ; the 2d is the place of seconds, or lOOths ; the 3d is the place of thirdsj or lOOOths ; and so on. For, in decimals, as well ag in whole numbers,- the values of the places increase towards the left-hand, afid decrease towards the right, both in the same « same tenfold proportion ; as in the following Scale or Table «f Notation. CO § -9 .•i'.-!-.g,"S<'3-g;| iltM to ill I ^ •;■■*,■ .^-.i I ^ ^ § 3 3 3-3 3 3 3 •3.3 • 3 3 3 3 ADDtTlbN'oF DEfclMALS. S£t 't&e) ttic&bcfrs ttxubr each*oth«9 «ccdniiiig >to die faltte .of jAac >plac€js, like as ip^wljLple nipnb^K j. jp jwrb^^ta^tp tl^ tieqijiipgl .5^para^ing poinds will stand all exacriy ynder each other. Then, beginning at the ri;^t-hand, add u'p aH the columns of numbers as in integers ; and point off as many places, for decimals, as are in the greatest number of decimal places in any of the litres that are addled ; or place the point directly below all the other points. EXAMPLES. 1. Ta add together 29-0146, and"3146*5, and 2109, and ; . 29-0 146 3146*5 : » J > • . 2109- .. ,: . ' • , 14-16 5299*29877 ' the Sum. Ex.2. What is the sum of ^^6, 3 9 '2 13: 72014*9, 417, and 50S2? . ' • ' . 3. What is th^ sutti qf .7530y 16^201, «-0142,;957*X3, ^•72119 and 'OSOU*: 4. What is the syrii pf 312-09, 3-57llj 7195-6, 7l*4»8, 9739.215, 179, and -0027? F2 - SUBTRACTIOJI €i ARITHMETIC . SUBTRACTION or DECIMALS. Place the numbers under each other according to the value of their places^ as in the last Rule. Then^ beginning at the right-hand) subtract as in whole numbers, and point off the decimals as in Addition. EXAMPLES. 1. To find the difference betwfeen Sl-IS and 2^13«. 9r73 2-138 Aixs. 89*592 the Difference. q: Find the diff. between 1-9185 and 2*73. Ans. 0'81 15. '3. To subtract 4.-90U2 from 214*81. Ans* 209-9085&. 4. Find the diff. between 2714 and 'QIS. Ans. 2713-084. MULTIPLICATION of DECIMALS. / . • ^ Place the factors^ and multiply them together the same as if they were whole numbers.— -Then point off in the pro- duct just as many places of decimals as there are decimals in both the factors. Bufif there be not so many figures in the product) then supply the defect by prefixing ciphers. * The Rule will be evident from this example :-«Let it be re- quired to multiply *12 by '36l ; these tumbers are equivalent to ;^ and ^t}„ 5 the product of which is rUlhs = 04332, by the nature' (^ Notation^ which consists of as many places as there are ciphers, that is, of as many places as there are in both numbers. And in like manner for any other numbers. ' ' • EXAMPLES. MULTIPLICATION of DECIMALS. 69 EXAMPLES. 1. Multiply •321096 by '2465 1605480 1926576 1284384 642192 " 1 W ' ■ I P Ans. -0791501640 the Product. 2. Multiply 79-347 by 23-15. Ans. 1836'88305. 3. Multiply -63478 by -8204. Ans. .520773512- A. Multiply -385746 by -00464. Ans. -00178986144. CONTRACTION I. To multiply Decimal/ by 1 with any number of Ciphers^ as by lO, or 100, or 1000, isTc. ■ This is done by only removing the decimal point so many places farther to the right-hand,* as there are ciphers in the Qiultiplier ; and subjoining ciphers if need be. EXAMPLES* 1. The product of 51-3 and 1000 is 51300. •2. The product of 2-714 and 100 is 3. The product of -916 and 1000 is 4. The product of 21-31 and lOOOO is CONTRACTION 11. To Contract tie Operation^ sq as to retain only as many Decimals in the Prodsjct,f^ may be thought Necessary^ tuhen the Product would naturally contain several more f^lfu:es^ Set the units' place of the multiplier under that figure of the multiplicand whose place is %\^ same as is to be retained for the last in the product ; aind dispose of the rest of the figures in the inverted or contrary order to what they are usually placed in. — ^Then, in multiplying, reject all the figures that are more to the right-hand than each multiplying figure, and set down the products, so tliat their right-hand figures may iof ARmiRtETlC. may fall in a column straight below each other; but observing to increase the first figure of every line with what would arise from the figures omittedy in this manner, namely 1 from 5 to 14, 2 £om 15 to 24, 3 from 25 to 34, &c ; and the sum of all the lines will be the product as required^ com<« monly to the nearest unit.in the last figure. EXAMPLESi 1. To multiply 27^14986 by 92.41035, so as to retain only four j^ace^ of decimals in the product. Contracted Way* Common Way. 27-14986 » 27-14986 53014-^9 92-41095. 24434874 542997 108599 2715 81 14 2508-9280 13 574930 81 4.4958 2714 986. 108599 44 542997 2 24434874 250S-9280 650510 t • 2. Multiply 480*14936 by 9-72416, retaining only four decimals in the product. 3. Multiply 2490-3048 by -573286, retaining only five decimals in the product. 4. Multiply 325-701428 by -7218393, retaining oidythre^ decimals in the product* DIVISION OF DECIMALS. Divide as in whole numbers; and point off in the quo- . tfent as many places for decimals, as the dedi^i^^l places in the dividend exceed those in the divisor*. l^ II 1 1 1 1 1 1 ♦ The reasort' of t^its. Rule is evident 5 for, since the divlsoi: multiplied by the quotient gives the dividend, therefore the num- ber of decimar places in the dividend, is equal to those in the"di- vi^orand quotfettt, taken together, by the nature of Mnltiplfca- <fe& 5 and consequently thequoti«at itBelf must cohtdfe ds many as i(he dividend exceeds the divisor* Another DIVISION ct DXCiMAJ^. %■ Another way to know the place {gr the decimal point, is this : The first figure of the quotient must be made to oc- cupy the same place, of integers or decimals, as doth that figure of the dividend which stands over the unit's figure of the first product. When the places of the quotient are not so many as the Rule requires, the defect is tp be supplied by prefixing ciphers. When there happens to be a remainder after the division'; or when the decimal places in the divisor are more than those in the dividend ; then ciphers may be aivi?xed ('p the divi* dend, and the quotient carried on as far as required. EXAMPLES. . 1. nS) -48520998 (-00272589 1292 460 1049 1599 1758 156\ 2. •2639) 27-00000 (102-3114 6100 8220 3030 3910 12710 2154 J. Divide 123;70536 by 54*25, 4. Divide 12 by -7854. 5. Divide 4195*68 by 100. 6. Divide -8297592 by -153. Ans, 2-2802, Ans. 15-278. ' Ans. 41*9568, Ans. 5*4232. CONTRACTION I. When the divisor is an integer, with any number of ci-^ phers annexed : cut off those ciphers, and remove the deci--. mal point in the dividend as many places farther to the left as there are ciphers cut off, prefixing ciphers if need be; then proceed as before*. * This is no hjopc than dividing both divisor and dividend by* the same number, either 10^ or lOO, or 1000, &c, ^c^otrdibg to< the number of ciphers cut off, which^ leaving them in the same proportion, does not affect the quotienf . And, in the same way, the decimal point may be moved the same number of p^ces in both the divisor and dividend, either to the right or left, whetheif they have ciphers or not. tXAMPX-ES^ if ARITHMETIC. EXAMPLES, 1. Dividp 45*5 by 2100. 21-00) -455 (-0216, &C. V '^2 35 140 14 $. Divide 41020 by 32006. 5. Divide 953 by 2 1600, 4. Divide 61 hj 19000. CONTRACTION IJ. HbncEi if the divisor be 1 vith ciphers, as 10, 100, O]^ 1000, &c : then the quotient will be found by merely mov- ing the decimal point in the dividend so many places farther to the left, as the divisor has ciphers j prefixing ciphers if need be. 3EXAM?LES. So, 217^3 -r 100 ==2-173 And 419 -t- 10 = And 5-ll5 -r 100 = And -21 — lOpO = CONTRACTION III. When the^e are many figures in the djvisor; or when only a certain number of decimals arie necessary to be re- tained in the quotient ; then take only as many figures of the divisor as ^mW be equal to the number of figures, both ^l- tegers and decimals, to be in the quotient, and find how many times they may be contained in the first figures of the divia^nid, as usual. l(jpX each rjjn^aind^ b^ a ne^ dividend i^ and for ^very such <livi4fnd, Ipve oyt one figure mpr^ on the rightrhand side of tide divisoi: j' r^m^jmbering tQ parry for the increase ox the figures ciit p^, as in the ^ contractipn in Multiplication. Note. Wb^n there are not so many figures in the divisor, as are required to be in th^ quotient, begin the operation with all the figures, and continue it as usual till the number of figures ill the divisor be equal to those remaining to be found in the quoUent ; after which begin the contraction. EXAMPLES. 1. Divide 2508-92806 by 92'41035, so as to have only four decimals in the quotient, in which case tJ^e quotient will contain six figures. Contracted. REDUCTION OF DECIMALS. 73 Contracted, Common. PJ'4103,5)25O8'928,06(a7- 1498 660721 13849 4608 80 6 92-4 103,5)2508-928,06(27. 1 498 66072106 13848610 46075750 91116100 79467850 5539570 2. Divide 4109-2351 by 230*409, so that the quotient may contain only four decimals. Ans. 17*8345. 3. Divide 37' J 0438 by 5713-96, that the quotient may contain only five decimals. Ans. •00649. 4.Divide 913*08 by 2137*3, that the quotient may contain fmly threie decimals. flEDUCTION OF DECIMALS. CASE I. To reduce a Vulgar Fraction to ks equivalent Decimal. « Divide the numerator by the denominator as in Division pf Decimals, annexing ciphers to the numerator as far as necessary ; so shall the quotient be the decimal required. EXAMPLES. 1. Reduce ^ to a decimal. 24=4 X 6. Thei£4) 7* W) 1*750000. •291666 &c. 2. Reduce ^y and ^ and \<, \o decimals. Ans. '25, and '5, and *75. ' 3. Reduce -^ to a decimal, Ans, '625. 4. Reduce ^ to a decimal. Ans. -12, 5. Reduce ttt *o * decimal. Ans. '03^59. 6. Reduce ^:^ to a decimal. Ans.. '143155 &c. CASE 74 . ARITHMETia CISB II. To find the Value of a Decimal in terms of the Inferior Den^ minationSm Multiply the decimal by the number of parts in the next lower denomination ; and cut off as many places for a remainder to the right-hand, as there are places in the given decimal. Multiply that remainder by the parts in the next lower denomination again, cutting off for another remainder as before. Proceed in the same manner through all the parts of the integer ; then the several denominations separated on the left- hand, will make up the answer. Notey Tliis Operation is the same as Reduction Descending; in whole numbers. ' • exa'mples. 1. Required to. fin,d the value of ^775 pounds sterling* •775 90 / 15-500 \2 d 6-000 Ans. 15/ 6i: 2. What is the value of '625 shil ? Ans, 7^/. 3. What is the value of -8635/.^ Ans. 17/S-24A 4. What is the value of '0125 lb troy ? Ans. 3 dwts,.^ 5. What is the value of '4694 lb troy ? Ans. 5 oz J2 dwts 15*744 gr. 6. What is the value of '625 cwt ? ^Ans. 2 qr H lb» 7. What is the value of -009^43 miles ? ^ Ans. 17 yd 1 ft 5-93S48 inc. 8. What is the valu^f -6875 yd? Ans. 2 qr 3 nls. .9. What is the value of '3375 acr ? Ans. 1 rd 14 poles* 10. What is the value of '2083 hhdof wine? Ans. 13-1229 gal. iCASE REDUCTION OF DECIMALS. 75 CASK III. To reduce Integers or Decimals to Equivalent Decimals (f Higher De/iominations, Divide by the number of parts in the next higher deno- mination ; continuing the operation to as many higher deno^ minations as may be necessary, the s^me as in Reduction Ascending of whole numbers. EXAMPLES. 1. Reduce 1 dwt to the decimal of a pound troy* 20 12 1 dwt 0*05 oz 0004-166 &c.lb^ Ans, 2. Reduce 9fl? to the decimal of a pound: Ans. .0375/. 3. Reduce 7 drams to the decimal of a pound avoird. Ans. -02734.375^.- ' 4. Reduce '26^ to the decimal of a /. Ans. '0010333 &c. /. 5. Reduce 2*15 lb to the decimal of a cwt. Ans. -019196 + cwt. 6. Reduce 24 yards to the decimal of a. mile. ^ Ans. -013636 &c. mile. 7. Reduce *056 pole to the decimal of an acre. Ans. -00035 ac. 8. Reduce 1'2 pint of wine to the decimal of a hhd. Ans. '00238 + hhd. 9. Reduce 14 minutes to the decimal of a day* Ans. -009722 &c. da. 10. Reduce *21 pint to thedecimalof a peck. Ans. -013125 pec. 1 1 , Reduce 28" 1 2"' to the decimal of a minute. Note, When there are several numbers^ to be reduced all to the decimal of the highest : Set the given nuihbers directly under each otherj for divi- dends, proceeding orderly from the lowest denominatiori to. the highest. % Opposite to each dividend, on the left-hand, set such a number for a divisor as will bring it to the next higher name; drawing a perpendicular line between all the divisors and dividends. Begin at the uppermost, and perform all the divisions: qnly observing to set the quotient of each division^ as decimal parts. 16 ARITHMETIC.^ parts, on the right-hand of the dividend next below It } so shall the last quotient be the decimal required. EXAMPLES. 1. Reduce 17/ 9^rf to the decimal of a pomid* . 4 12 20 3 9-75 17*8125 £ 0*890625 Ans. 2. Reduce 19/ 17/ S^d to /. Ans. 19-86354fl66 &c. /. 3. Reduce 15/ 6d to the decimal of a /. Ans. '775/. 4. Reduce 74^ to t^e decimal of a shilling. Ans. '6S5s. 5. Reduce 5 oz 1 2 dwts 1 6 gr to lb. Ans. *46944 &c. lb; iSKTrnffTTT RULE OF THREE in DECIMALS. PiEPARfe the terms, by reducing the vulgar fractions to decimals, and any compound numbers either to decimals of the higher denominations, or to integers of the lower, also the iirst and third terms to the same name : Then multiply and tlivide as in whole numbers. Note^ Any of the convenient Examples in the Rule of Three or Rule of Five in Integers, or Vulgar Fractions, may be taken as proper examples to the same rules in Decimals. — ^The following Examplei which is the first in Vulgar Frac- tions, is wrought out here, to show the method. Jf I of a yard of velvet cost |/, what will ^-^ yd cost ? yd / yd / s d 1 = -375 . •375 : -4 :: -3125 : -333 &c. or 6 « •4 T — * •375 ) -12500 ( •333333 &c. 1250 20 rf 1^1? . _ / 6-66666 Sec: 12 « Ans. 6s Sd. d 7*99999 &c.=8rf. ]>UOD£* DUODECIMALS. 77 DUODECIMALS. 1 • f DuobsciMALs, or Cross Multiplication, is a rule used by workmen and artificers^ in computing the contents of tneir works. Dimensions are usually taken in feet, inches, and quarters ^ any parts smaller than these being neglected as of no conse- quence. And the same in multiplying them together, or casting up the contents. The method is as follows. Set down the two dimensions to be multiplied together, one under the other, so ^t feet may stand under feety inches under inches, &c. ' Mukiptyeach term in the multiplicand, beginning at the lowest, by the feet in the multiplier, and 6et the result of each straight under its corresponding term, observing to carry 1 for every 12, from the inches to the feet. In like mai^ner, multiply all the multiplicand by the inches and parts of the multiplier, and set the result of each term one place removed to the right-hand of those in the multipli- cand; omitting, however, what is below parts of inches, only carrying to these the proper number of units from the lowest denomination. Or, instead of multiplying by the Inches, take such parts <£ the multiplicand as there are of a foot. Then add the two lines together, after the manner of Compound Addition, carrying 1 to the feet for 12 inches^ when these come to so many# EXAMPLES. J- Multiply 4 f 7 inc by 6 4 2. Multiply Uf 9 inc. by 4 « 27 6 59 7 4i Ans. 29 0\ Ans. ^6 41- t 3. Multiply 4 feet 7 inches by 9 f 6 inc. Ans. 43 f. 6^ inc. y 4, Multiply 12 f 5 inc by 6 f 8 inc. Ans. 82 9J- 5. Multiply 35 f 4| mc by 12 f 3 inc. An?. 433 4 J. 6. Multiply 64 f 6 inc by 8 f 9J inc. Ans. b^o 8| INVOLUTION. 1$ AIUTHMETIC: INVOLUTION. Involution is the raising of Powers from tny given number, as a root. ' ' ' ' A Power is a quantity produced bytnultiplying any given number, called the Root, a certain number of times conti- nually by itself. Thiis, •' * 2 = g is the*0Qt, qr iBt ppwer of 2* 2x2= 4 is the 2d power, or square of'il^ , 2x2x2= 8 is, the 3d powerj or cubeiof 2. . 2 X '2 X 2 X 2 == 16 is the 4th. power of 2, &€• / jAnd in this tnanner may be caktflated the followitig Table of the first^nine powers of the fkaj 9 numbers. TABLE of th.^ jfirst Ninb Powers of Numbers. 1 1st 4 2c if 3d 9. 5 6 7 s 9 16 I 8 27 04 25 36 J25 40 2J6 343 64 512 _! |8l|/2,9 4th 16 81 256 625 I2g6 240] •5th j ^t1i'|'>ih ■t ' 1 32 64 128 243 J024 729 4096 2187 16384 3125 15625 7776 16807 409632768 0561 59019 46656 78125 . 2799^6 II 7649 823543 I 2621442097162 531441 8th 256 6561 65536 39062^ J 67961 6 ^•^'-••••^va 5764591 16777216 4 782969 4304672 J 4 * 512 .196.88 262144 19531 2 j: 10077696 40353607 ■S.t. 1«4217 »> 387420489 The r^ INVOLUTION. ^9 The Index or Exponent of a Powerj is the number de- noting the height or degree of that power; and it is 1 more than the number of muUiplications used in producing the same. So 1 is the index or exponent of the 1st power or rooty 2 of the 2d power or square^ 3 of the third power or cube, 4 of the 4th power, and so on. Powers) that are to be raised, are usually denoted bj placing the index above the root er first power. So 2* S3 4 IS the 2d power of 2. .' ■ 2'= 8is the. 3d power of 2. ' 2;* ac 16 is tb« 4th power of 2. 540* is the 4th power of 540, &c. When two or more powers are multiplied together, their product is that power whose index is the sum of the expo* nents of the faaors or powers multiplied. Or the multipH-* cation of the ppmrs, answers to the addition of >tfaje indices. Thus, in the following powers of 2, 1st 2d 3d 4th 5th 6th 7th 8th 9th 10th £ 4 8 16 32 64 123 256 512 1024 fflp ^' «• 2^ £♦ 25 2* 2' 2« go gto Here^ 4 X 4 ±=: 16, and 2 + 2 5= 4 its index; and 8 X 16 = 128, and 3 + 4 = 7 its index; ako 16 X 64 = 1024, and 4 + 6 = 10 its index. eTHfiR EXAMPLES. ' 1. What IS the 2d power of 45 ? . Ans. -2025. 2. What is the square of 4' 16 ? Ans*^ 17*3056. S. What is th« 3d power of 3*5 ? Ans. 42'875. 4, What is the 5th power of '029? Ans. '00000002051 1 149. 5, What is the square of-f- ? Ans. ^. 6. What is the 3d power of 4? Ans. fj^. 7. What is tjtie 4th ppw^r of 4: ? Ans. ^Vy* EVOLUTION. 80 ARITHMETIC. EVOLUTIO^f. EvoLUTiONi or the reverse of Involution^ is the extracting or finding the roots of any given powers, ■ , The root of any number, or power, is such a number^ 29 being multiplied into itself a certain number of times, wul produce that power. . Thus, 2 is the square root or 2d root of 4, because 2' = 2 x 2=4; and 3 is the cube root of 3d root of 27, because 3' = 3 x 3 x 3 = 27. Any powei: of a gfiven number or root toay be found ex- actly, namely, by multiplying the number continually inta itself. But there are many numbers of which a proposed root can never be exactly found. Yet, by means of decimals, we may approximate or approach towards the root, to any de- gree of exactness. Those roots which only approximate, are called Surd roots; but those which can be found quite exact, are called Rational Roots. Thus, the square root of 3 is a surd root; but the square root of 4 is a rational root, being cfqual to 2 : also the cube root of 8 is rational^ being eqvial to 2 ; but the cube root of 9 is surd or irrational. Roots are sometimes denoted by writing the charstcter 4/ before the power, with the index of the root against it. Thus, the 3d root of 20 is expressed by ^0 5 and the square root or 2d root of it is v'20, the index 2 being always omit- ted, when oiily the square root is designed. When the power is expressed by several numbers, with the sign + or ~ between them, a line is drawn from the top of the sign over all the parts of it: thus the third root of 45 -12 is ^45 - 12, or thus ^(45-12), inclosing the numbers in p^entheses. But all roots are now often designed like poweti, with fractional indices : thus, the square root of 8 is 8^, th6 cube root of 25 is 25"^, and the 4th root q{ 45 -^ 1« is 45- 1,8 J*, •r (45 -.18)^. TO SQUARE ROOT. M TO EXTRACT THE SQJIARE EOOT. * DfViDB the given number into periods of two figuret each, by setting k point over the place of units^ mother over the place of hundredsi and so on, over every second figure^ both to the left hand in integers, and to the right in de^ cimals. Find the greatest square in the first period on the left-hand, and set its root on the right-hand of the given number, after the manner of a quotient figure in Division. . ■»•»• * The reason for separating the figures of the dividend into periods or portions of two places each^ is^ thdt the square of any single figure never consists of more thui two places $ the square of . a number of two figures, of not more than four places, and so i^ So that there will be as many figures in the root as the given nUin- ber contains periods so divided or parted off. And the reason of the several steps in the operation appears from the algebraic form of the square of any number of terms, whether two or three or more. Tlius, (a + b)^ = a» + 2ab + 6* =: a* + (2a + b) b, the squareof two tffirms ; where it appears that a is the first term of the root, and b the second term ) also a the first divisor, and the new divisor is 2a 4- ^> or denize the first term increased by the s^copd. And hence the manner of extractipp is thvs : 1st divisor a) a^ ^ 2ab -f 6* {a + b the root* a* 2d divisor 2a -f & | 2a6 + b* b\2ab + bf Again, for a root of three parts, a, b, c, thus : (a + 6 + c)* =; a* + 2a6 + 6® + 2tfc + 2bc + e« = a^ + (2a + 6) 6 -f (2a + 26 + c) c, the squaie of three terms, where a is the first term of the root, b the second, and c the third term ; also a the first divisor, 2a -f ^ the second, and 2a -^ 2b + c the third, each consisting of the double fif the root increased by the next term of the same. And the mode of extraction is thus : 1st divisor a) a* + 2a6 ^ 6^ + 2ac + 2bc + c'^ {a + b +c the root. a?- 2d divisor 2a + b b 2ab + i* 2a6 + A» N 3d divisor 2a + 26 + c I 2flc + 26c + c* c I 2flc + 26c + c* Vol. I. G Subtract sr -Aum^/^tic. »% • ^ Subtract the square dius found from the said period, and to the remaindet annex tNi^>^two fTg^ures of the ntext following pQriodffbrfidivi4^. ' * Doiri>le the rootabove aaeadoned fisr a divisor ; and.ftid^ how oiften k is contained in the said dividend^ exclusive of its right-hand figure ; and set that quotient figure botl)^ in the quotient and divisor. Multiply the whole augmented divisor by this last quotient figure, and subtract the p]t)duct from the said dividend, bnnging down to it the next period of the given number, foraaew^Uvidend. . Rppeat tb^ same procesi o^r agaln^ viz. find another new dsmei^. by douUipg all the figures new hund in the root ( fixmi wludi, and the last dividend, find the next figure of the root as before ; and so on through all die periods, to the last. Note^ The best way of doubling the root, to form the new divisor's, is by adding the last figure always to the last divisor,^ as appears in the following examples. — Also, aftejr the figures belo^ng to the given number are all exhausted, the operas, tipn may be continued into decimals at pleasure, by adding any number of periods of ciphers, two in each period, EXAMPLES. i. To find the square root of 29506624. • • • • 29506624 ( 5433 the root. 25 104 4 450 416 1083 8 3466 3249 2 [21724 21724 Note, When the rooi is to he extracted to many places of figures ^ the work may be considerably shortenedy thus'/ * • . ' * Having proceeded in the extraction after the common me- thod> till there be found half the required number of figures •'in S(^tTAR£ ROCyr. » in the robt^ or one figure more; tb^> Ibr ib(t fM, dii^e the last remainder by its eori^e^pohdihg divl^rrafter the ittali-' aer of the thu*d contraction in Division of Deciihdisi tluis, 2. To find the root of 2 to nine placid of fi^M^ 2 ( 1*4142195^ the tooti 1 • • ^ 100 4j 96 2ii 1 400 3^1 2824 4 11900 11296 28282 2 60400 56564 v.* 28284 } 3. What 4. What 5. What 6. What 7. What i. WlKit 9. What 10. What 11. What 12. What id the is the is (he is the is the is the is the is the is the is the square square square square square sqtiarje square square square square S8S6 ( 1856 1008 160 19 root of 2025 ? root of 17-3056 ? root of •000729 f root of S ? root of 5 ? . root of 6 ? root of 7 ? ^ root of 10 ? root of II ? root of 12? Am. 45. Ans. 4*16. Ans. *027. Ans. 1-732050. Ans. 2*236068. Ans. 2*449489. Ans. 2*645751. Ans. 3162277. Ans. 3*316624. Ans. 3*464201. RULES tOK TB£ s'qpARE ROaTS OF VUXOAR FRACTIONS AND MIXED NUMBERS. FitLsT prepare all vulgar fractions, by reducing theni to their least terins^ both for this and all other roots. Then 1. Take tbe root of the numeratoi' and of the denominator for the respective terms of the root required. . Aiid fKb is the best way if the den6i£iiilator be a complete power: but if it be not, then 2. Multiply the numeraitbr and denominator together; take tlie root of th^ product : this root being zidade the nume^ ^ G 2 rater M ARITHMETia xatorto the denominator of the riven fraction, or mado tl^e denominator to the numerator of it> will form the fractio^al root required. That is, v^y f^a \/ah s^b b s/ab' And this rule will serve, whether the root be finite or infinite. 3^ Or reduc(e the vulgar fraction to a decimal, and extracf its root, 4. Mixed numbers may be either reduced to improper Actions, and extracted by the first or second rule» or the vulgar fraction may be reduced to a decimal, then joined tp the integer, ami the root of the whole extracted. SAMPLES. Ans. 4» Ans. f . !• What is the root of |4 ? 2. What i$ the root of^l S. What is the root of A ? Ans. 0-866025. 4. What is the root of T^j. ? Ans. 0-645497^ 5. What is the root of n| ? ^ns. 4-168333; By means of the square root also may readily be found the 4th root, or the 8th root, or the l^th root, &C| that is, the root of any power whose index is some power of the numoer 2 ; namely, by extracting so often the square root as is de* .noted by that power of 2; that is, two extractions fbr the 4th root, three for the 8th root, and so on. So, to find the 4th root of the number 21035*8, extract the square root two times as follows : 21035-8000 I ( 145-037237 (12-0431407 the 4th root, 1 24 4 -110 96 22 2 45 44 285 5 1435 1425 2404 I 10372 4 9616 ■▼ .- 29003 3 108000 24083 87009 3 75637 72249 20591(7237 3388 ( 140t 687 980 107 17 Jlx, 2. Wh^t is th? 4th root of 97--^ ^ I n ROOT; §5 TO EXTRACT THE CUBE ROOT. 4 I. By the Common Ruk^i . i. Having divided the given number into periods of three £gures each, (by setting a point over the place of units, and also over every third figure, from thence, to t&e left hand iri y^hole numbers, and to the right iirxtedmals), find the nearest less cube to the first period ^ set. its root in the quotient, and subtract the said cube ftoffl^the fifsrperiod f to the remainder bring down the second period, and call this the resolvend. 2. To three times the square of the root, jiist found, add three times the root itself, setting this one place more to the right than the former, and calltliis siiiri the divisor/ Thctn divide the resolvend, wanting the last figure, by the divisort for the next figure of the root, which annex to the former ; calling this last figure #, and the part of the root before found let be called a. 3. Add alltogether these three products, ns^ely, thrice ^uare multiplied by ii^ thrice a multiplied by e s()uare, and ^ cube, setting each of them one place more to the right than the former, and call the sum the subtrahend ; which must hot exceed the resolvend ^ bdt if it does, then lilake the last £gure e lessj and repeat the operation for finding the subtra- hend> till it be less than the resolvend. 4. From the resolvend take the subtrahend, and to there- tnainder join the next period of the given number tdr s hew resolvend ; to which form a new divisor from the wlfole rOot 'now found ; and froith ihence another figure of the root, as directed in Article 2, and so oii. ^ ♦ The reason, for pointing the givcaa number into periods of three figures eacb> is because the cube of one' figure never amounts io more than three pl^es. And, for a similar, reason, a given number is pointed into periods of four figures for the 4th root> tif £ve figures for the 5th root, and so on. And the reason for the-other parts of the rule depends on the algebraic formatioad of a ci^be : for, if the root consist of the two parts a -{- h, then its cube is as follows ; (a + 6)' = a' + 3a*6-f 3aA* + b^ j where ei is the foot of the first part a' y the resolvend is 3a*6 + 3«i* -f J?, which is aUo.the same as the three parts of {he subtrahend ; also the divisor Is 3a* + 3tf, by which dividing the first two terms ©f the re^lvend 3o*^ + ao*, gives b for the le^kond part of the root 3 and so on. ixAMPLK. tft AWTHMET^, J^XAMPLC To extsaet tke cube root of 48328-544. S X S* sr 27 S X ? 5B. .09 Divis^ 279 48228*544 ( 3$ '4 rpot. 27 ■WP*<PV«^ 21228 resoWend. ^ly^'fnr f " ) ' '^' S X S* X 6 =c 162 3x3 X 6* = 324 Vadd 6» = 216 3 X 36*. == 3888 3 X 36 sc 108 38988 19656 subtrahend 1572544 resolvend. 5' X 36^ X 4 = 3 X 36 X 'V = 4^ = 15552 nSSSadd 64 1572544 subtrahcn*. 0000000 remainder. Ex. 2/ Extraa the cube root erf" 57 1482' 19» Ex. 3. Extnct the cube root of 1628' 1582. £x« 4. Extract the cube root of 1332. * II. T* 49ctract tie Cube Root by a short ITay*. t. Bj trials, c»r by the table of roots at p. 90, &c, t^ke. the nearest rational cube to the given nunaber, whether it be greater or less ; and call it the assumed cube. 2. Then f •. * The method usually giveb for extracting the cube root, is so exeedingly t^edioos> and difficult to be remembered, that various pth^r appro'xiiuiiting rules have heeh invented, viz. by Newton, llaphson, H^lley^ De Lagny, Simpson, Emerson, and several other mathematioiaDs ; but; ho one that I have yet seen, is so simple in ^ts form, or seems to wiell adapted for general use, as ^hat above given. This rule is the same in effect as Dr. Bailey's, rational formula. hmtbm nai doaUe the assumsdrcube^ is to iht stm of the assumed cube and ctouble the given number, so isthe lOOt of the assumed cube, to the root required^ nearly^ Otf As the first sum Is to the difierence of the given and assumed cube^ so is the assumed root to the difference of the root» nearly. 3. Again, by using, in like manner, the cube of the root last found as a new asstiiTietrcube, another root wiQ be ob- tained still nearer. - And $o on as £u: as we please ; uring d- ways the cube^ of the last found root, for the assumed cube. EXAMPLE. To £nd<the ctdse root of 31035*8. Here we soon find that the root lies between 20 and 30, and then between 27 and 98. Taking th^erefore 27, its cube is 19683, which is the assumed cube. Then . 2 2 39366 4207 1-6' 21035*8 19683 »■ < ■■ As 60401-8 : .617S4-6 :: 27 : 27 '60471 27 f . k 4322822 1235092 60401-8) 1667374-2 (27-6047 th« root neadf. 4is;9338 284 42 formoki) but m or e co m med t o usl y eapiessed ' , tnd-^ie first-intesti^ gation of it was given in my Traets, p. 49. ^ Tl^e algsbraic fom of it is this : As P H^ 2a : A «f 29 ; : r ; a. Or^ . . \ As B + aA : p fio; a : : r ; a^ v> r j ] ^bere r is the given nittqber, .i^ Ite as^otattd^ieafrilt cobc r ^Ire- ^be root of a^ and a the root of ? sought. Again, 88 ARTTHMEnC. . Agam, for a secoad opdratlony the cube of tfaif rooit le 91035*318645155623^ and the process bj the ktter method will be thus : 21035-318(545, &c. ' • . 42070-637!29a 21035-8 21035-8 . 21085-318645, &c. As 63106-43729 : diff. -481355 ;: 27*6047 : thediff. -0002 10560 codseq. the r<>ot req. is 27-604910560. Ex. 2. To extract the cube root of •67. Ex. 3. To extract the cube root of *01. TO EXTRACT ANY ROOT WHATEVER*. Let p be the given power or number, n the index of t&e' power, A the assumed power, r its root, r the required root of p. Then say. As the sum of if + 1 times a and /i — 1 times p, is to the sum of » + 1 times ? and » — 1 times a^ so is the assumed root r, to the required root R. Or, as half the said sum of ii -f 1 times A, and »*- 1 times r, is to the difference between the given and assumed powers,- so is the assutned root r, to the difference bet'^een the true and assumed roots ; which difference, added or subtraaed^ as the case requires, gives the true root nearly, ' ■ ■ ■ ■■ ' ■ ■ ■■ ■ ■ ■ ii Thatis, «+l. A + «— 1. P :/f+l'. P. +«— 1. a :: r :R. Or, «+1.4a+«— I. -y* : PcQ A :: r : Rco r. And the operation may be repeated as often as we please, by using always the last found root for the assumed root, and its nth power for the assumed power A. mmmimim'mmm^aKt^ttm * This is a very general approximating rule« of which that fbc the cube root is a particuk.r case^ and is the best adapted for practice^ andfor meinorjr, of any that I have yet seen. It was first disooveied in this form by myself^ and the investigation and use of it.wefe^ven at large ia my Tacts, p. 45, Dec. EXAMPLE. d£N£RAL tOOrtS. i& fXAMPLE. "to extract the 5th root of 21035*8. ( Here it appears that the 5th root is between 7*3 and 7'4. ^Taking 7*3, its 5th power is 20130'7 1593. Hence we have p = 21035-8, « = 5, r =7*3 and A = 20730-71593; then «+l. iA + «— U 4^ • ^ CO A :: r : rco r, that is, 3x30730-11593+11x21035-8 : 305 084 :: 73 : 3 2 7-3 62192-14719 42071-6 1P4265-74779 42071-6 915252 2135588 2227- 11 32 (-02 13605 = 11 cor 7'3 = r, add. 1. What 2; What 3. Wliat 4. Wifiat 5. What 6. What 7. What 8. What 9* What lOi What 11. What 12. What 13. 'What OTHfiH iXAMPLES. s the 3d root of 2 ? sthe 3d root of 321 4? s the 4th root of 2? s the 4th root of 91 '4ft ? s the 5th root of 2 ? s the 6th root of 21035-8 ? s the 6 th root of 2? s the 7th root of 21035-8 ? s the 7th root of 2 ^ s the 8th root of 21035-8 ? s the 8th root of 2 ? s the 9th root of 2 1 0S5'*S ? sth^' 9th root of 2? 7-321360 = R, true to the last figure^ Ans. 1-259921. Ans. 14-75758. Axis. 1-189207. Ans« 3-1415999. A^s. I-148699. Ans. 5-254037. Ans. 1-122462. Ans. 4-145392- Ans. 1-104089. Ans. 3-470323. Ans. 1-090508. Ans. 3-022239. Ans. l-080059.f •riT llie following is a Table of squares and cubes, as also the Square, roots aild cube roots, of ah numbers from 1 to 1000, JKrhich will be found very useful on many occasions, in nu- meral calculations, when xoots or powers are concerned. A TAtiS fO A TABLE OF SQUAWJSb CUMS^ and ROOTS. ^. Number. Square. Cube. Square Roo t. « Cube Root. 1 I 1 1*0000000 1000000 3 4 8 1-4142136 1-259921 3 9 . 27 1-7320508 1-442250 4 l6 64 2'C iJi • 1 ?i 1*587401 5 25 125 ^^ ^^ ^^ -^y -^r- ^^ ^^ -^^ 2*2360680 ' 1*709976 6 36 %\6 2-4494897 1-^17121 7 49 343 2-6457513 1-912933- - s 64 . 512 2*82842^1 2K)octoob 9. . 81 729 . 3-ooooonf] 2*Qfi0nB4 10 100 1000 3*1622777 2*i8.4?l35 11 121 1331 3*3166248 2*223986 ji 144 1728 3*444 r 016 2*289428 13 169 2i97 3-6055513 2-35i335 14 196 2744 37416574 2*410142 g^5 2f5 3375 .3-8729833 J , _ ^_. .^ _ 2*466212 250 4096 4913 4-C i:i:i:i:t:i:« 2 510842 17 289 ^ -^^ ■mm' -^^ ■^^ -m^ ■m^ '^^ 4*1231056 - '2*571282 18 324 5833 4-2426407 2*620741 19 361 6859 4*3588989 2-668402 20 4CX) 8000 4'4721360 2-7144I8 21 441 9261 4-5825757 2-758923 22 484 10648 4-6904158 2*802039 23 529 12167 4*7958815 2-843867 24 ;?76 . 13624 4-8989795 2-884499 25' 625 156!15 5-0000000 2-924OJ 8 26 . ^7^ 1757I5 5-0990195 2-962496 27 7'^9 19683 5-1961524 3-000000* 28 784 21952 ' 5*2915026 3*036589 29 841 24389 5-3851648 3-072317 30 900 27000 5-477^^6 I 3-107232 31 961 29791 5*5077644 3^141381 32 1024 . 32768 5*6568542 > 3*174802 33 ' IO89 35937 6*7445626 3*207534 ' 34 1156 39304 5-8909519 " 3-239612 35 1225 42875 5-9160798 3*271066 36 1296 , 46656 6'OOOOOOCi 1 . 3*3019i27 ^37 1369 50653 6-082/625 3^332222 ^8 1444 54872 6-1644140 ► 3-361975 ^ 1521 59319 ^*:M49980 3*391211 40 1600 64000 6-3245553 3*419952 41 I68I 68921 6-4031242 3*448217 42 1764 74O88 6-4807407 3*476027 w 4a 1849 79507 6*5574a85 3*503398 44 19^6 85184 66332496 3*530348 45 2^ .911^ 6*7082039 3-556893 46 2116 97336 6*7823300 3*563048 47 220^ 103823 6*8556546 3*608826 48 2304 110592 6*9282032 3*634241 49 2401 117649 7-0000000 \ 3*659306 50 2500 125000 7*0710678 3*684031 • r. M ri ■ SQlTARtS, CUBES, AND ROOTS. 91 Tf umber. Square. 51 52 53 *54 *55 56 57 58 ,m 6k Gl 63 64 65 66 07 68 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 S7 88 89 90 91 '93 94 95 9^ 97 98 99 100 2601- 2704. 2809 2916 3025 3rl36 3249 3364 3481 36Cp 372i 3844 3969 4096 4225. ^4356 4489 4624 4791 4960 5041 5184 ^329 5476 5625 5776 5929 6084 6241 6400 6561 6724 68^ 7056 7225 7396 7569 77^ 7921 8100 8281 .8464 8649 8836 9025 9216 9409 9604 9801 10000 Cube. 132651 140608 148877 157464 166375 175616 195193 195112 205379 216000 22^81 238328 250047 2S2144 . 274625 287496 300763 314432 3285Q9 343000 357911 373248 389017 405224 421875 438976 456533 474552 493039 512000. 531441 551368 571787 '592704 614125 636056 658503 68I472 704969 729000 753571 778688 804357 830584 857375 884736 912673 941192 ^970299 1000000 Square Root. * Cube Bool. . •• \ 7'1414284 7'21 11026 . 7-2801099 7-3484692 7'4l6l985 7-4833148 7^40^44 7'6l5773i " 7*6811457 77459667 7\8iQ2497 7-8740079 ' 7*9372539 8-0000000 8-0622577 8*1240384 8-1853529 8-2462113 8-3066239 8-3666003 8*4261498 8*4852814 8-5440037 '^•6023253 8*6602540 S*7177979 ' 8*7749644 8-8317609 8-888 1944 8-9442719 9-0000000 9-0553851 9*1104336 9-1651514 9-2195445 9*2736185 93273791 9*3808315 9-43398 11 9*4868330 9*5398920 -9*5916630 9*6436508 916953597 9-7467943 9-7979590 9-8488578 9-8994949 9-9498744 10*0000000 I 3-708430 3*732511 3*7562^96 3779763 3-8O2953 3-825862 3*843501 3*870877 3*892996 3*914867 3-936497 3-957892 3*979057 4-000000 4-020726 4*041240 4-061548 4-081656 4-101566 4- 12 1285 4*140818 4*l60l68 4-179339 4-198336 4*217163 4*235824 4-254321 4-272659 4*290841 4*308870 4-326749 4*344481 4-362071 4'379519 4*396830 4*414005 4-431047 4-447060 4*464745 4*481405 4-497942 4-514357 4-530655 4-546836 4*562903 4-578857 4*594701 4-610436 4*626065 4-641589 ^ -• I s. »2 \ ARITHMETIC. N amber. Square. 10201 101 102 10404 103 10609 104 10816 105 11025 106 11236 107 11449 108 11664 109 11881 110 12100 111 12321 112 12544 113 12769 114 12996' 115 13225 116 13456 117 13699 118 13924 • 119 14J61 120 14400 121 14641 122 14884 123 15129 . 124 15376 125 15625 126 15876 127 16129 128 16384 129 16641 130 l6S0p 131 17161 132 17424 133 176S9 134- 17956 135 18225 136 18496 137 I8769 138 . 1 19044 139 19321 140 196QO 141 I988I 142 20164 143 20449 144 ' 20736 145 21025 146 21316 i47 2I6O9 148 21904 ug 22201 . 150 22500 Cube. 1030301 .IO612O8 1092727 1 1 24864 1157625 1191016 1225043 1259712 1295029 1331000 1367631 1404928 1442897 1481544 1520875 I56O896 1601613 1643032 1685159 1 728000 1771561 1815848 I86O867 1 906624 1953125 2000376 2048383 2097152 2146689 2197000 2248091 2299968 5K3 52637 2406104 2460375 2515456 2571353 2628072 2685619 2744000 2803221 2863288 2924207 2985984 3048625 3112136 3176523 3241792 3307949 3375000 Square Root. 00498756 0-p99i049 0-1488916 O-198O39O 0-2469508 0-2959301 0-3440S04 0-3923048 0-4403065 0*4880885 0-5356538 0-5830052 0*6301458 0-6770783 0*7238050 07703296 0*8166538 OS6278O5 0-9087121 0*9544512 rooooooo 10453610 1*0905365 1'1355287 1*1803399 1 224972a 1 2694277 1-3137085 1*3578167 1-4017543 1-4455231 1*4891253 1*5325626 1-57583^ 1-6189500 1-6619038 1*7046999 1 -7473444 1-789^261 1-8321596 1-8743421 1-9163753 1-95 82607 2-0000000 2*0415946 2*0830460 2-1243557 21655251 2-2065556 2-2474487 Cube Root. 4-657010 4-672330 4-687548 4702669 4717694 4*732624 4747459 4762203 4'77^56 4'791420 4*805896 4*820284 i 4-834588 4*848808 4-862944 4-876999 4*890973 4*904668 4*918685 4*932424 4*946088 4-959675 4-973190 4*986631 5*000000 5-013298 5-026526 5-039684 5-052774 5*065797 5*073753 5-091643 5-104469 5*117230 5-129928 5*142563 5-155137 5^167649 5*160101 5*192494 5-204828 5*217103 5-229321 5*241482 5-253588 5*265637 5-277632 5-289572 5-301459 5-313293 SQUARES, CUBES, akd ROOTS. 9$ 1 Sfjoare. Cube. « Square Root. CubeKoot. 151 22801 3442951 12-2882057 5325074 152 23104 3511808 12*3288260 ||-336803 ^•348481 153 23409 zb%\^^^ 1 2*3693 169 154 23716 3652264 12-4096736 5360108 155 24025- 3723875 12-4498996 5-371685 156 24336 3796416 12*4899960 5*383213 157 24649 386989a 12-5:^99641 5-394690 158 24964 3944312 12*5698051 5-406120 159 25281 4019679 126095202 '5-417501 160 25600 4096OOQ 12-6491106 5*428835 l6i 25921 4173281 12*6885775 5*440122 l62 26244 4251528 , 12 7?79221 $-451362 163 26569 4330747 127671453 5-462556 164 26896 4410944 12*8062485 5-473703 165 27225 4492125 1 2-8452326 5*484806 166 1155^ . 4574296 12-8840987 5-^5865 167 27889 4657463 12-9228480 $-506879 168 28224 A'J^lQ^'i }2'9614814 5-517848 169 28561 4826809 13-0000000 5'52S775 170 28900 4913000 13-Q384048 5*539658 171 29241 5000211 13-0766968 5*550499 T73 29584 . 5088448 13-1148770 5-561298 29929 5177717 13-1529464 5'572Q54 174 30276 52^5024 , 13-19(^060 1 5-582770 175 30625 5359375 13-2287566 1 j5 593445 17? 30976 M5\77Q 13*26(54992 5-604079 177 31329 5545233 13-3041347 5 6146/3 175 31684 5^3971^'^ 13-3416641 5»625226 179 3204! 5735339 ^13-3790882 5-635741 180 32400 5S320O0 13-4164079 5*646216 181 32761 5929741 13-4536240 5-656652 182 33124 6028568 13-4907376 5-66706I 183 33489 6128487 135277493 5-677411 184 i 33856 6229504 13-5646600 5-687734 185 34225 6331625 13-6014705 5-698OI9 186 > 34596 6434856 13*6381617 5*708267 187 3491^ 6539203 13*6747943 5-/18479 188 35344 6644672 137113092 5-728654 189 35721 67512^ 137477271 5-738794 190 36l(jp 6859000 13-7840488 5*748897 191 36481 6967871 13-8202750 5-75S965 192 36864 7077888 13-8564065 5768998 193 37249 71 89057 13-8924440 5-77S996 194 37636 7301384 13-9283883 5*788960 195 38025 7414875" . 13-9642400 5*798890 J 90 38416 7520536 ^Kvi:i:«:i:i iii^ 5-808786 5-818648 1 *^ 197 38809 7645373 14-0356688 198 39204 7762392 14-0712473 5-828476 190 39601 7880599 141067360 5*838272 J ^^ 4obqo 8O()0Qqo 14-1421356 5-848035 •« ARITHMETIC. Numb. Square. Ciibfe. Sqaare Root. Cube Root. 201 4O401 8120601 1417744^ 5-857705 202 40804 8242406 14-2126704 5-867464 203 41^ 8365427 14-2478068 6-87? 130 204 41616 S4Sg66^ 14-2828560 5-886765 205 42025 8615125 14-3178211 5-890308 206 42436 8*741816 14-3527001 5^905941 2G7 42849 8869743, 14-3874946 5-915481 208 43264 8998912 14^222051 -^•9^^4991 209 436S1 9123329 14-4568323 5-934473 210 44100 9261000 14-4913767 5-943911 211 44521 9393931 14-5258390 5-953341 212 44944 9528128 14*5602198 5962731 213 45369 ' 9^^3597 14-5945195 5-972091 214 45796 9800344 14-6287388 5-98U26 215 46225 9938375 , 14-6628783 5-990727 216 46656 10077696 14-6969385 6-000000 9 2J7 47O89 10218313 14-7309199 6-0Q9244 2ia 47524 10360232 14*7648231 6-018463 219^ 47961 10503459 14-7986486 6O27650 220 48400 10648000 14-8323970 6-036811 221 48841 10/93861 14-8660687 6-045943 . 222 49284 10941048 14-89960*4 6055048 223 49729 1 10895^ 14*9331845 6-064126 ' ' 224 50176 11239424 14-9666295' 6-073177 225 50625 II39O625 15-0000000 6-082201 226 51076 11543176 150332964 .6-09Hd9 : 227 51529 11 697083 15-0665192 6-100170 228 51984 11852352 15-0996689 6-109115 . 229 52441 I2OO8989 15-1327460 6-118032 230 5290a 12167000 15- 1657509 6-126925 231 53361 12326391 15-1986842 6135792 232 53824 12487168 15-2315462 6-144634 233 ^4289 12649337 15-2643375 6' 1 53449 , 234 54756 1281290* 15-2970585 6162239 235 55225 12977875 15-3297097 6-171005 236 65696 13144256 15-3622915 6-1797^7 237 56169 13312053 15-3948043 6-188463 238 56644, 13481272 15-4272486 6-197154 239 57121 ^ 13651919 15-4596248 6-205821 . 240 576OO 13824000 15-4919334 6-214464 241 58081 13997521 ^5-3241747 6*223083 242 5S564 14172488 15-556349^ 6-231678 243 59049 14348907 15*5884578 6-240251 244 5g536 14526784 15-6204994 6-248800. ' 245 60025 14706125 1^-6524758 6-257324 246 60516 14886936 15-6843871 6-205820 247 61009 15069223 15-7162336 6-274304 248 61504 15252992 15-7480157 6-282760 249 62001 15438249 15-7797338 6-291194 ; 250 62500 15625000 15*8113883 6-299604 SQUARES, CUtJES, and ROOTS: 95 '^imlb. Square. Cube. Square Koot. Cube Root 251 63001 15813251 15-8429795 6-307992 232 ' 63504 16003008 15-8745079 6-316359 253 64009 16194277 15-9059737 6-32470* 254 64516 16367064 15-9373773 6-333025 255 65025 16561375 15*9687194 6-341325 256 65536 16777^16 16-0000000 6349602 257 66049 16974593 16*0312195 6-357859 258 66564 17173512 16-0623784 6-366095 259 67O8I 17373979 16-0934769 6-3743 ID S<50 67600 17576000 16-1245155 6-382564 261 6812i 17779581 16-1554944 6-390676 262 68644 17984728 16*1864141 6-39S827 263 69169 18191447 16-2172747 6-406958 264 . 69696 18399744 1 6-2480768 6-415068 265 70225 186096^5 16-2788206 6-423157 266 70756 18821006 16-3095064 6-431226 267 71289 190341 )3 16-3401346 6-439275 266 71824 192468^2 16-3707055 6-447305 26g 72361 19465 1O9; 16-4012195 6-455314 270 72900 19683060 16-4316767 6*463304 271 73441 19902511 16-4620776 6-471274 272 73984 20123648 16-4924225 6-479224 273 74529 20346417 16-5227116 6-487153 274. 75076 20570824 16-5529454 6-495064 275 75625 20796875 16-5831240 6-502956 276 76X76 21024576 16-6132477 . 6-510829 277 76729 21253933 16-6433170 6-51 8694 ' 278 77284 21484952 16-6733320 6-526519 279 77841 21717639 16-7032931 6-534335 280 78400 21952000 16-7332005 6-542132 28i 78961 22188041 16-7630546 6-549911 282 79^24 22425768 16-7928556 6-557672 263 8OO89 22665187 l6-8226038 6-565415 284 80^56 22906304 16-8522995 & 573 139 285 81225 23149125 16-S8I943O 6-580844 286 8 1796 23393656 16-9115345 , 6-588531 287 82369 23639P03 16-9410743 6-596202 288 62944 23887872 16-9705627 6-603854 . 289 83521 24137569 17-0000000 6-61 1488 290 84100 24389000 17-0293864 6-619106 291 • 84681 24642171 . 17-0587221 6-626705 292 85264 24897088 17-0880075 6-634287 293 85849 25153752 17-1172428 6-64 F851 294 '86436 25412184 17-1464282 6-649399 295 87025 25672375 17-1755640 6-656930 296 87616 25934336 17-2046505 • 6-664*43 297 88209 26196O73 17-2336879 6-671940 298 88804 26463592 17-2626765 6-679419 299 89401 2673O899 17-2916165 6-686882 300 90000 27000000 17-3205081 6-694328 / p« ARITHMliTlC. 1 1 Nuiob. Square. Cube. Square Root. Cube Root^ ] 301 90601 27270901 17-3493516 6*701758 , 302 91204 27543606 17-378 1472 6*709172 3W 9I8O9 27818127 17-4068952 6-716569 304 92416 26O94464 17-4355958 6*723950 « 305 93025 28372625 17-4642492 6-731316 soej 93636 28652616 17-4928557 6'7SS66& . 307 9^249 28934443 17-5214155 ^'7^5997 • 3^ 94864 292 181 12 17-5499288 6-753313 ' 309 95481 29503629 17-5783958 6-760614 310 96100 29791000 17-6068169 6*767899 311 96721 . 3008023 1 17-6351921 6'775lt}S 312 97344 303/1328 17'66S5'/.17 6-782422 i 3r3 97969 30664297 17-6918060 6'7eg66i M 314 QS5g6 30959144 17-7200451 6-796884 > 315 99'^2S 31255875 l7-748239a 6*804091 3)6 99S6(i 31554496 17-7763888 6-811284 317 100489 3185^013 17-8044938 6-816461 318 J01124 32157432 ^*S325545 6-825624 319 ^01761 3^^461759 17-8605711 6-832771 320 102400 3^768000 17-8i88543(^ 6-639903 321 103041 33076161 17-91<>4729 6*847021 . 322 103684 , 33386248. 17-i;443584 6-854124 323 101329 33698267 l/-i;722C08 6-861211 324 ia\97<^ 34012224 » 18-OUOOQOO 6-868284 .325 105625 34328125 18-027/504 6-875343 326 IO6276 34645976 180554/01 6-882388 327 IO6929 34fj65783 l&-083i413 0-889419 328 107584 35287552 18- 1107703 6-896435 ■ 329 108241 35611289 lb- 13835/1 6-903436 330 lOSgOO 35937000 18-1659021 ^-910423 331 IQ956I 36264691 16- 1934054 6-917396 332 ) 10224 36594368 I8r2/08072. 6924355 • 333 IIOS89 36926037 1 8*2482876 6-931300 334. 111556 37259704 18-2756669 6*938232 335 112225 37595375 18-3030052 6-945 ug 336 • 112896 37933656 18-3303028 6-952053 337 113569 38272753 18-35/5598. 6-9^8943 338 114244 38614472 • 16-3847763 6*965819 ■ 339 114921 38958219 18-41 19-)26 6'97^6S2 340 U5600 39304000 18-4390889 6-979532 341 116281 3965 1821 18*4601853 6-986369 342 II6964 4CKX)l688 18-4932420 0^993191 • 343 117649 46353607 18-5?02592 7-000000 344 118336 40707584 18r5472370 7-006796 ■ 345 119025 41063625 I8-574I 756 7-013579 1 346 119715 41421736 18^10752 7-020349 347 120409 41 78 1923 18-6279360 7-027106 348 121104 42144192 18-6547581 7-033850 1 349 121801 42508549 18-6815417 7-040581 1 350 122500 42875000 18*7082869 7-047208 1 ,,v» SQUARES," CUBES, and ROOTS. 97 Cu^elt OQt. Numb. \ Square. | Culie. 351 352 353 354 355 356 357 358 359 360 361 362 303 364 365 366 367 368 370 371 372 373 374 375 376 377 378 379 380 381 382/ 383' 384 335 386 387 38S 389 390 391 393 393 394 395 39S ^97 . 398 399 400 ■ ■ ■ m tA 123201 123i,04 124609 125316 126025 126736 127449 128164 128881 129600 130321 131044 I3i;69 132496 133225 133P56 134689 135424 136161 136900 137641 138384 139I29 135876 140625 141376 1421^29 142884 1436^11 144400 145 161 145924 -.146^89 147456 148225 148996 149769 150544 151321 152100 152881 153664 154449 15523*6 156025 156816 157609 158404 1.59201 160000 Square Root. 43243551 43614208 43986977 44361 864 44738875 45118016 45499293 45832712' 46268279 46656000 47045881 47437928 47832147 46228544 . 4862712^ 4902789<5 49430^3 49836032 50243409 50(i5 >000 5106481 1 51478848 51899117 52313624 52734375 58 157376 53582633 540(0152 54439939 54872003 5530634 1 557429* 56181887' 56623 104 57066625 575 1 2456 57960633 58411072 58863869 593.1 9000 59776471 60236288 6O69S457 61162984 61629875 62099136 ^257077^ 630i4792 63521199 64000000 H 87349940 8761 6630 87882942 1^*8148877 5-8414437 8*8679623 8«S944436 8*9208879 8*(H729.53 8*9736660 49*0000'X)J 9^0262 j76 9-0525589 9-0787840 9*1040732 Q* 13 11265 9*15724^1, 9*1833261 9*2093727 9*2353841 9-26l?60? 9*2873015 9-3132079 9-339079^ 9*3649167 9-3907194 9*4164878 9*4422^^21 9-4679^«3 9-4935887 9-5192213 9-5448263 9*5703858 9-5959179 9*62141(^ 9*6468327 9*6723156 9'%77^^Q 9*7230329 9*74^4177 9-7737199 97989899 9*8242276 9*84943J2 9-3740069 9-89974S7 9-92<18588 9'9m^73 9-9749844 20*0000000 7*0540O» 7*06069« 7067376 7*074043 7*060698 7*087341 7-093970 7* 100588 7- 107198 7' 1 13786 7-120367 7*12693i 7*133492 7*140037 7'iA6!i6g 7*153090 7159599 7*l6609« 7']7268i0 M 79054 7*185516 71919W 7' I984O5 7*204832 7*211247 7*217632 7*224045 7-230»l27 7*236797 7*243156 7-249504 7*255b44 /• 262107 7*268482 7274786 7-281071 7*287362 7*293638 7-299893 7-306143 7*312363 7*318611 7-324829 7*33 I0i7 7-337234 7-343420 7*349596 7•a5^762 7-36ISI17 r'368(>6i V mm J 98 ARraiMETlC. Nuinb" 401 • Square, jocaoi Cube. . 64^81201. Square Hoot. Cube Root. 20-0249914 7*374198 . 4K^2 -l6l(x)4 64964808 20*G49t>377 7-380322 ; 403 ' 1 $24091 65450827 20-0/48599 7'3fa6l37 1 4M J6321(i 6.939264 20'0997512 7-392542 ; 405 16^025 66430125 20- 1240 116 7*398636 ' 40«i 164836 66923416 20-1494417 7-404/20 1 j 407 165649 67419143 201 7424 10 7-410/94 I 40iJ 166464 679U312 20'\ijgoi)Qg 7-410859 i 409 167281 i- 684 17(^29 20-2237484 7-422914 1 410 468100 - 68921000 20-2484567 7-4*28958 ! 4U 468921 6942653 1 20-2731349 7-404993 4t2 16&Z4,4 6gg3452S 20-297783 1 7-44K)18 * 413 4 . I70&l)g 70^144997 20-3224014 7-447033 ! 414 i;j3y6 7^)957944 yO-34(i98Ci9 7-4;53039 • 415. •a723i»5;. .71473375 20-3715488 7-459036 4l(j 17^^056 71991296 20:396078 1 7-465022 . 417 . 17^«*89 72511713 20-4205779 7'V(m9 i 418 1747/i4 73034632 20-4'W5OW3 7'4f709(>^ ' 410 . }7^^^<il 7356CO59 20-4694895 7*482924 420 . 176400 74O8SOOO 20-4939015 7-488872 : 421 177241 74618401 20-5V82845 7-494810 422. 178Q84 75151448 20-5426386 7-500710 ■ 423 .178929 75<)i()ijb7 . 20-5669608 7-506660 1* 424 . \m)770. 702'>5(yM 20'59J 2603 7-512571 425 . . •,li)OGi25 . 707(>5ty25 20*6155281 7-518473 426 18X476 . 77'^i>^77^ 20-6397674? . 7-524365 427 . l?2y^ 778544»3 20-6639783 7-330248 : 128. » U3184, 78402752 20-688 1609 7-536121 4?^ IS4©'4l 7895:)589 , 20-71 23 W2 7*541986 h 430 . J8I9G9 7950;(X)0 20-/304414 7-547841 43\ 1 85/61 .8U(J02991 aO-7605395 7-553688 482 . .186624, 6U62 1 5ib •iO-7846(:97. 7-559525 ^33 V -187489 61 1 Q'17^7 20*8086520 7-565353 434 * . IB 8-356 B\7465iH 20-8326667 7-5/1173 ' 435 ; -189225 82S12S75 :ilO'SM}6536 7-576984 tf^) I9O096 82tiil856' 20-6906130 7-582786 437s ■i9<^^69 8:i4l^3453 20-9045450 7'5S657y \ 43S ,101644 84027672 20-9:84495 7'5ij4'M>3 430, 192721 84(;04519 20*9523268 7-600138 4*40, .. 193600 85I8400O 20-9761770 7'605g05 441 if;4-J8i . 85766121 2100(XXK0 7-611662 44^ 196364 86350S88 21 0337960 7-617411 44'Ji * l(;62t9 t-6i3S307 21-0475652 7-623151 1 4^4 nj7r^6, 875283r84 21-0713075 7-628683 ' 4-15 1 i)S025 88121125 21-0950231 7-034006 I I 440 . i9«yi6' ^ 83716*536 2l-Jlb7121 7-0 1 (.321 \ 447 I$9«09 . 89314023 : 21-1423745 7-646027 i 448 200704 89915392 21-16*'i010S 7't)5i725 ^ 44g 20t601, <)()5 18849 21 1896201 7-657414 202500 91 1 25000 21-2132034 7-663094 4 I SQUARES, CUBES, and ROOTS. 90 Numb. Square, Cube. Square Root. Cube Root- 451 203401 91733851 21-2367606 7'6m76& • 452. 204304 . 9234540s 21-2602916 7-6744^0 ■ 453 205209 92i^5gt)77 21/2837967 7'680085 ^ 454 206116 93576664 & 1-307275 3 7*685/32 455 20/025 9^19^375 21»33072C;0 7'(^9^3}i • 45^ 2079:i6 . 04818816 21-3541565 7*697002 457 20bb4^ 95443993 21-3775583 7-/0L1624 458 209764 9607l9Ja 21 -4009346 77oe23S 459 210681 96702579 21-424^2853 7*713«44 4(>0 211600^ 97336000 21-4476105 77x91*2 4^1 212521 d7972J8l 21-4709103 7-/25032 . 402 213444 9861 1 1 28 i2 1*494 1853 7730014 463. 214369 99252847 21-5174348 . 7-7^61^7 464 215296 99897344 21-5406592 7741753 465 216225 103544625 21-5638587 7747310 \ 466 217156 101194696 4 21*5870331 7752860 467 2I8O89 10184/563 ♦?l-6l(n828 7-758402 46S 21902^ 102503232 21*6333077 7763936 m 2i9'^6l 103161709 21*6564078 776t)462 470 220900 103823000 21-6794834 7774980 471 221841 104497111 217025344 7 7^t90 472 222784 105154048 21-7255610 7 795992 473 223729 105323817 21 -7485632 7791487 474 224676 ;06496424 21-7715411 7*796974 .475 225023 IO7I71875 21-7944947 7*602453 476 2265/6 107850176 21-8174242 7-807925 1 ^77 227529 108531333 . 21-8403297 7*813380 47s 228484 10:^2153 52 21-8632111 7*818845 479 229441 109902239 21-88606^6 7-824294 480 230 ICO 110532000 21-9O89023 7'829/35 • 481 231361 . 111284641 21-9317422 7*835198 482 232324 . lilQ80l68 * 2 1 '9544984 7-840594 ! 483 2332^9 1126/8587 21-9772610. 7'846013 .4S4. . 234256 U 3379304 22-0000000 7' 85 1424 .4S5 . 235225 ;i 14084125 22*0227155 7-856828 ' 1 486 236196 114791256 22-0454077 7-862224 487 237169 115501303 22-0680763 7'S676i3 488 23S14A 116214272 220907220 7-872994 ,489 . 239121. 116930169 22* 1 1 33(444 7-878363 490 • 240100 1 1764.0000 22-1359436 7*883734 :4i)i : 241081 3 18370771 2.-I585198 7*889094 1 1^92- . u 242064: . 119095488 5?2- 1 3 10730 7-8J4446 . 493; 2430*9 11982315;^, : 22*2036033 7-8c;9791 ^ 49* • 244036 . J 205^3734 i?2-226M0S 7-905 1 29 495 24.'>025 12128/375 22-2485955 7*910460 496- . 246016 122023936 22-2710575 7-9^0734 497 247009 122763473 122*2934968 7921100 498 24^004 -123505992 22-3 1 5/) 136 7-9264O8 .499>, 249001 124251499 2^-33830/9 7?9'5]7io ■ soo 2.0OCO3 1 25C0 XXX) 22-3606708 7-937005 ^ '•m • -«w^p^». > ll 2 " - ._.. • J It * 100 ARITHMETIC. * Numb, i b<\\\sire. 50I 502 503 504 505 5Q7 506 srp 510 511 512 513 514 515 Si\6 517 5m 519 520 521 ^22 523 524 525: 526 ' 527 528 52p 5:^a 531 632 ' 535 ' 53^ 535 5?>7 5S8 539 54U 541 *5 43 544 54 > 5llt; 518 54 a 4 I 251001 25200^1 2.W009 254016 255025 256(30 ^57049 258064 2^908! 200100 261121 262144 263 1 6^ 264196 263225 26eji25Q 2()7289 208324 269361 2;04(X) 27)441 2724 S4 273529 274576 275625 278784 27984 1 2^:0,00 281(j6l 283024 2840S9 285150 2S6225 28/296 288369 289444 290521 291<XX) 29268 1 293764 *2(|4849 ■295936 2y7025 298II6 299209 30030f 301401 ;i02500 Cube.' 11 ittii • 25751501 26506008 27263527 26024064 28787625 29554216 30323843 31(^512 31b72229 3265 10(X) .334328 51, 342l77'i8 350O5f;97 35796744 3659O875 373te096 38488413^ 38(JS1832 39708359 ' 4O60S0CO 41420/61 42236643 438/7824 44703125 4.' ^^ \ 576 40363183 47 J 97952 48035889 46877000 4972129! 5i)5j6S70'8 51419437 52273304 53130375 53990^).'!^ 54854153 5572O872 574b*4000 58340421 5925W)088 6(>l63(X)7 6C9S9I84 618/8625 62771336 63667323 645(56502 654601^9 66375000 Square Root. 2i*3Wit)293 22-40J35J5 22-4276615 22*449^)443 22 4/22051 22*4^)44438 22-5l666a5 22-5388553 22-5610283 22^83 1796 22-pO53091 22-6274 170 226495033 22*6715681 22 -0936 11 4 22-7156334 227376340 22759Q134 227515715 '22-b035085 22-8254244 22-84 73 ISA 22-8691935 22-891046^^ 22-912878^ 22-93 ^6899^ 22-95^806 22-9782500 23 OCOOOQO 23O2I72S9 230434372 23-065 1252 23 0867928 23- 1 084400 23-13a':6yO 23- 15 16738 2M73i^05 23-1948270- 23 •2163735 23-2379001 23-2594067 ' 23-2808935 23-3023604 23-S23807(} 2:^-345 235 1 23-3666429 23'38:?0311 23-4093998 23-4307490 23-4520788 *. -ti^^»-*mi^t,-mftmim Cube Uooi 75942293 7*947373 7'9^2«47 7-958114 7-963374 7-9C8'r-7 .7-97:>873 7-97i^ii2 7*984344 7-9895f;9 7*9t;4788 8'OCCOOO 6-005205 8-010^03 8'015595 8*020779 8*025957 8-03U29 8-036293 8-041451 8*046603 8-05 1 748 8-056886 6-00^018 806714a 8-072262 8-077374 8-082480 8*087579 8O92O72' 8-0()7758 8-102838 8-M7912 8*11-2980 8-118041 8- 1 23096 8M 28144- 8*133 186 8-138223 8- 143253 8*M8276 8* 1 53293 8*158304 8- 163309 8* 1 68308 8- 1 73302 8-176289 8-|832<^ 8-188244 8-193212 ■mh SQUARES, CUBES, a^j> ROOTS. 101 Numbr. .551 552 553 554 555 556 557 558 5 9 500 5b i 5(52 503 564 565 566 567 508 56g 57P ^572 573 574 575 576 577 57s 5bO 561 582 5^3 584 585 580* 587 688 589 5(J0 5^2 5ij3 ^y4 5i)5 SyG • 5ys .00 ) , Square. 303601 304/04 3058:k> 3001)10 308b-i5 3 9 MO 3102-19 311364 312-181 3136CX) 311721 315844 3\6j69 3 1 80i)(i 319225 32035d 321489 322J24 323761 324poO 3200.1 1 3^^84 328329 329476 330 J25 33177O 332929 33-1084 335241 33(>400 337561 338724 339889 34 To 36 342225 34339O' S4456'9 34^744 316921 348*00 34'>28 1 35(MC>4 35V649 35283 C) 354025 355216 355409 357604 ?588pi ii'iiiobo Cube. 10728415 1 lobigo'd'os 10*91 123/7 1704X^1464 170,51875 17I879'J10* 172S08693 17:^741112 174676B79 I756iCxx;0 176558481 17750-i328 178453547 i794J6i44 180332125 18l32M9'> 18228^263 183250432 184220009 i85l930J0 186169411 1 87149248 18eri325l7 1 89 119224 1130109375 191102976 I92IOCOJ3 193103552 194104539 1951i20JO 190*122941 197J37368 193155297 199 -70701 2(i()20l625 201 23005(3 2{y22(j20J3 203^j7472 2043:i64oy 305379O:xi 206125071 2074746%Q io^527^7 20i)5&4584 3lb:)4487d 3 1 1 7O87 J6 ^12776173 213847193 214921799 216000000 Sqn«re Reot.^ Cube Uuoi. iZ., m^lmtm '23-4733892 23-4946802 i>3-5 159520 23-5372046 2D-558433a 23-571:6322 i!3 -0006 174 230220^36 '23-6431 b08 23-6643191 23-6S54386 23-70J5392 237276210 237436842 2376J7286 23-790754.5 23-8117618 23-8327506 23-8537209 23-8746/28 23-8956063 23-9165215 23-9374184 £3-9582971 23-9791576 24-0000'JOO 24-02)6243 24^16306 24-062>18:i 24-0831892 24-1039416 21- 1*^46762 24' 1453:^29 24-y>>0t)19 24-1807732 24-2074369 24-22S0829 24-2487113 24-i6932ii 24-2899156 24-3104916 24-^310301 24-35 i5Dl 3 24-37:»U52 24*39a6:^id 24-4)31112 24-4335834 24-4540385 24-47447fi5 a, t 8-198175 H8-2^3131 8-208082 S-2IJO27 8-2179U5 8-222898 8-22/825 8-232746 8-237001 8-242570 8-2-17474 8-252371 8-2572d3 8-262149 8-267029 8-271903 8-276772 8-281635 8-28649$ 8-291344 8-296190 8-301030 8-305865 8-3lO.>94 8-315517 8-320335 8-SI25I47 8-329:'54 : 8-334755 8-33955 1 8-34434 1 8-349 12» 8-353901 8-358678 6*363446 8-368209 8-372966 S"J777^S 8*382465 8*3d7206 8-391942 8-396673 8-4013 v)8 8-f406U8- 8-^10832 8-(^ 15541 8*420245 8-424944 8-429638 I II-434327 / 102 ARITHMETIC. ).\ Square. | •m.!^ Square 6oot. . Cube KcMDit. i\umb Cube. . (101 603 (3CH 605 606 607 60S 009 610 611 612 613 I 6u 615 6l6 617 618 '619, 620 mi ' 6^2 . 623 624 Cii7 (52^ 629 6:jo ;63f i^ 6?i 635 630 .Qiit 64! 642 •643 jB44'* 7545 646- 361201 362404 363609 364316 366015 36/236 368449 36^664 370831 372100 373321 374544 378325 379^56 3S0639 381924 383161 384400 385641 386884 3881 29 389376' 390625 391 876 ' 393129 ; 391384 393C"^U 306900'' 3()3|6l , 399424' 400^89 401956 403225 404406 405769 : 407044 408321 . 409600 410881 412164 : 41344Q . 414736^ 416025* . 417^16*- 4 J 9904 42i2ai[ 217081801 21 8 167203 219256227 220348864 221445125 222543016 223648543 224755712 225^66529 22698 1 COO 228'.)9913l 229220928 230346397 231475544 232608375 233744896 23458.5113 236029032 237 1 76(^59 iJ3832S0 10 ' 2394Q306J 240641848 241804367 242970624 244140625 .245314376 216191883 '247073 1 .02 2488581 89 250047000 251239591 2524359(iS '? 53636! 37 2546-10104 2560478/5 257559456 .25847485^-." 259694072 ' 26091!^ J 19 "262144000 263374721 '26460c528S 26/5S47rq7 ' 26708^gir4 2683361:25 269586136 '270fe-f0Qi3 '272og7'T9'r .273359449 24-5153013 24-5356883 21-5560583 2i-5764115 24-5967478 24-61706731 24.6373700 24-65/6560 24-6779254 24-6981781 24718^1142 24-7386338 247588368 24-7790234 247991935 24-3193473 24-8394847 24 8596053 24-8797106 24-8997992 24-9193716 24-9399^78 24-9599679 24"97P9!)20 25hob()boo 25019992b 25039968 1 25-05'99282 25*0798724 25-0998008 2.5-1197134 25- 1396102 25 15949 13 25-1793566 25-l<i9i063 25-2190404 25'23885b9 25-2586619 25-2784493 25-2982213 25-3179778 25-3377189 25-35/4447 25-3771551 25-3^K)8502 25-416530! "25*4361947 25-4558441 25-4754784 25:4950076 - 8-439009 8-443687 8-448360 '8-453027 8-457689 8-462347 8-466999 B'47Wi7 8-476289 8-4S0926 6-485557 8-490184 8-494806 8-499423 6-504034 8-508641 8-5 13243 8-517840 8-522432 8'5270!8 8-5J16CO 8-53f>177 8-540749 8-545317 8-549879 8-5*4437 8-558990 8-563*537 &-5660S0 8'5726l8 8-577152 8*58 168O 8-586204 8-590723 8-595238 8*599747 8-6()4252 8 608752 8-613248 8-617738 8-622224 8-626706 8-631183 8-635655 8-640122 S-644585 S-649Q43 8653497 6-657(^6 8-662301 SQUARES, CUB£6, AMD ROOTS. lOS : NuQib. Square. iCiibe, Square Ro^t,. Cub« ituol 051 ^ 423601 275894451 25-5147016 8*666831 ()52 425104 ^ 277167BOS •25-5342()OX 8*671266 65^ 426409 27S4i5077 25-a53il?47 8-675697 654 427716. 2797^626* 25^7:^4137 . 8-68W23 655 429025 281011375 / 25Jj9i9678 . 8-684545 656 430336 232300 H 6 25-6124969/' «;6t^63 657 431649 2835^33 ^ 25-6310112. 8^693976 65S 432:^64 28489C»3I2 »5-65i510;e 8^697784 659 43t<i8l 286191179 ^5'67(m^ 8-702183 660 435600 287^99000 . 25-6^)54652 v 8-7o6iS7 661 436921 2dd80478b 257099303 s-7iO$«2 662 438244 29J1 175211 25-72oa!eto/ 8-7Ui73 .663 439569 29143i247 25-7487864 87l9r5.7 664 44O896 292754444 2^763195^^" 8724111 66 J 442i25 294C79;yi5 25-7375939 8-726518 666 m 443556 2954O8296 25'80ai9753 8-734891 607 444889 ' 296740963. 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Cube-Rodt. 701 49}iai . 344472101 26-4704O46 8-8832116 702 492804 34594SOO8 26-4942826 8*887438 , 703 494209 347428927 26*5141472 ,8-8ftl706 7<H 49^16 848913664 26*5329981 8*895920 705 4970^ 350402625 2&5518361 8*9001 30 706 493436 351895816 26*5706605 8*904336 707 49.)849 353393243 26*5894716 8-908538 7O8 50l2(S4 354B94912 26-6082694 8-9l27|5* 8-9i<»fl 7<» M20SI S5640082^ 26-6270539 7iQ A04100 35791 lOOQ 26-6458252 8-921121 7U 50S52I S59425431 26-6645833 8-925307 7U 50r)944 360944128 26-6833281 8-929490 ' 7«3 5033(!^ ^ 862467097 267020598 8-Q33668 7W 3q979<> 3*63994^4 267207784 8-937843 .7*5 511225 365525675 26-73©4S39 8-942014 7i« 512^56 367O61696 26-7581763 8-9 t6l 80 717 514089 36bd0l8i3 'iQ'77^bli7 8-^50343 7iS 515524 370146232 267955210 8-954502 ' 7W 5IQ96I 37i^94»« 26-8141754 8-958658 •720 5 J 8400 373248000 26-83^ 157' 8-962809 T^l 5h9841 3748C536i 26*6514432 QV^957 722 532284 370367048 Z6-8700577 8-97110O 723 52^729 377933067 2d-«8S6593 *»975240 724 524i7t> 379^03424 20-9072*81 8-979^7^ 725 5^^(>25 5j27076 381078125 2C)-92.56240 8-933^08 726 382657 « 76 26-f4438/2 8-C;87<>37 1%7 5'2S5'H| 3S4240583 26'9629.75 8-9917^2 728 52(}9ci4 385828352 26'9a 14751 8'(;9588a 729 .531441 367420189 27-poooo :o 9*000000 730 532^00 3P9O 17000 ' 27-0185122 9»004113 731 534361. 3ii0'il789l 270370117 9-00S222 . 732 535324 . 3f;2223l63 270554935 9»0 12328 733. 53. 289 393H32837 27-0739727 9-016430 734 53875<J 3 J544d9D4 27-09^43+4 y020529 735 540225 397<^'6.0375 27' 1108834 9-021.62^ 7.'« 541 696 396688256 271293199 9-028714 7'67 543I()9 400315553 271477439 9-032802 738 544f>14 4019172/2 27' 166 1^)54 9*036385 739 546121 403563419 27-i 845544. 9-040(:65 740 547600 40522^1000 27-20.94IO 9*045041 741 5^19081 . 46o8()'9021 27-2213152 9-0491 U 742 6bo:^<i^ 4G85l84»r^ 272396769 9-053 1 83 743 552049 4^0172407 27-2580265 9-05;2i8 744 553536 4U&307S4 27(2763634 9*O0*J309 745 556025. . 41349^625 27-}J946881 > 9*065367 746 5A6516 • 415160936 . 27- ) 130006 9:069422 7'^7 ^^W^^ 416832723 27- 3313007 ^ $^073^72 748 5S9504 418508992 ^30189749 27&495887 9^7519 7-^0 501091" , 27-3678644 i 9'081503 75C> i t 562.W0 . 4U1875O0O i -27-386127^ 9-O856C0 / SQUARES, CUBES, and ROOTS. 105 Numb. 7^1 753 75* 7^7 758 769 700 701 7d-i 7tf3 7W 765 766 767 76S 769 770 771 771 77^ 77^ 775 776 777 778 7» 780 781 7Q3 784 785 786 787 788 789 790 793 .794 795 797 798 799 800 Square V' Ctibev 3t>400X ^03^04 567009 5685 \6 570025 57i539 573049 B745(k 57Cf081 V760O 579121 5S0044 58'ilfi^ 5SJ60i 5833Ji5 48^5(3 588289 589824 5iniaoi 59?900 594441 •95984 597529 ^99^^ ^100625' 6m\76 6o37'J9 lk)5284 600841 608400 09961 611524 613089 614656 616225 617796 619^69 620944 622521 $24100 6256S1 627264 628849 . 63Q436 632a25 633616 6352Q9 636804 ^ 638r*01 640000 . 423564/5 1 425259008 426957777 428661064 430868875 432O8I216 433798O93 4355l9idl2 437245479 43897CiOOO 44071108.1 44245P72S 444194947 445943744 447697 i 25 449455096 43121/663 452984632 454756609 456533000 45831401,1 460099648 4618699*7 463684824 465484375 467288576 469097433 470910952 472729*39 474552000 476*379541 478211768 480048i)b7 43l890:iO'i 48:17360255 485587656 487443403 489303872 49n69069 493039000 494913671 496793088 498677257 $00566184. $02459«^75 $0^3^336 506261573 50^16^592 5IJDOV2399 5 li>000000 .- • Squa^ Root. | Cube Roou ^ 9-069639 9*093672 9-097701 9-101726 9-105748 9-10^766 9'l.J37«l 9*117793 9*121801 9.125895 9-129806 9ia38^ 9-137/97 9' 14.1 788 9-145774 9149757 9-153737 9-157743 9-161686 9-105656 '9-169622 y 173585 9:177344 9-181500 9185452 9-189401 9- i 93347 9-197^^9 9*20i228 9'i05l64 9'20CK)96 9*213(/25 9-216950 9-2Z0S72 9-224791 9-J2fel706 9-432618 9*237527 9-240433 9-2443ii5 9-248234 9-252130 9-2o6J22 9"25<;91 1 9-2^3797 9-207679 9-27 155() 9-275435 9'279^0& 0-283177 27*404379'* 274226184 27*4408455 27*4590604^ 27*4772633 ' 27*4954542 27-5 1 36330 275317998 27-5499546 27-5680975 27-5862284 27*6043475 27'6224546 27-6i05499 27*6586334 27-6767050 27-6947648 27712i*129 2773O8492 27-7488739 27-7C68b68 >277B4B880 27-b028775 27-8203555. 27-S3i!i82i8 27'S567766 27^8747197 27-8926514 27-9105715 27-9284801 27-9463772 27-96426/9 27-9821372 28-0000000 280178515 28-0356915 28-053j'iO3 28-07133/7 28-0891438 28-lo:i9386 28-1247222 28-1424946 28- 160 J657 28'17800?li 28-1957444 28-2m72Q 28-2311 8 i4 23-:j4 83938 :48*:t66588i 28-28427 1 2 106 arithmetk:. Nnmb. i Square.*- Cube. | Square Root. l •eoi 602 803 eo4 eo5 800 807 60S 8C9 810 811 6i2 813 814 $15 BW 817 6)8 Big 8;30 821 822 823 $24 825 826 627 828 839 830 8^1 832 833 834 835 836 837 638 839 840 841 842 843 844 845 846 847 848 849 850 '-■ 641601 (;43204 64^809 ^464 16- 648025 6\()()36 651249 652864 654481 65&]0C> 657721 659344 66O969 662556 664225 665B56 m4BQ 669124 670761 672400 674041 675684 677329 P78976 68062^5 ^82276 6b3929 68£«5S4 687241 688CjO0 6go56i 692224 693689 ()i)5S56 697225 698896 7oa569 702244 703921 705(iOO 70/281 70896^4 71O649 712336 714025 7i74Cg 719\04 72OSO1 ■722500 513922401 515849^08 5(77Sl<^'i7 6J9718464 52\6G0125 523606616 5255579-<3 627514112 529475129 53I4410CO 533411731 535367328 537^0^797 539353144 541343375 543338496 545336513 547343432 549353259 551368000 553387661 555412248 557441707 559476224 561515625 56S55gg7Q 565609283 667tG3552 5097227^9 57 I787COO 573856191 ' 57593036S 576OO9537 '580093704 582182875 584277056 586376253 5Se4fe0472 590589719 592704000 594823321 5960,47688 599077107 6(XJ 21 1,584 60535 1 J 25 ^0549573^ 60f 04 54 23 609tOOI92 611960049 6I41250CO 28-3019*34 28-3100045 26-^^7^546 26-35489^8 28-3/25219 28*6i,01 39 1 28-4077454 28''4253408 28-k4i9253 •28-40O49B9 26*4; 8061 7 28-4956137 28*;5U>549 28*t$306852 28')54 83048 26-5657137 28-5832119 28-t0069:^3 ^8'6l8]760 28-6356421 28-P53Q976 26-6705424 «8-e879766 28-^054002 28-7228132 28-7402157 28-^5?6077 C8-774&891 28-7923001 28-6037106 28-82;o706 28-8444102 28-8617394 28-8/90582 28-8963666 28-9136646 28-<)i09523 28-fi48i2(,»7 28-|654(,67 28-0827535 Q9h0QCOO0 290172363 29-*344623 29-*5 16?61 29-4688837 •V9-06()C'79l 2.g- 1032644 20- 12043 96- '29J370C46 ♦ S?9l5475(;5 Cube Book! 9-2^7044 9'2gOL}07 9^'.(|4767 9-2i;b0\!3 9-30i477 9^306327 9-310175 9'3 14019 9-317859 9'32l697 9-325532 9*329363 9*333191 9-337016^ 9-340838 9-3446^7 9-348473 9'35!2285 9-356095 9*359901 9{363704 * 9*367505 9»371302 9*375096 g-378887 9-362675 9*366400 ,9-390241 9*3^«20 y-.97796 9-401 569 9-405338 9-409105 9-412S69 9-416630 9-420387 t;-424l4l 9-427893 (}'43l643 9*435388 9*439130 9-44!i870 9-4 4 6607 9-4.C0341 9^*540/ 1 9';i57799 9-461524 9-J4 65247 gH46896to^ 0-472682 SQUARES, CUBES, and ROOTS. lOT Namb. 851 852 853 854 855 856 8S7 858 859 860 86l SdZ SO'3 8C)4 8b'5 SQd I 867 868 8(X) 87b S7J 872 873 874 875 87a 877 87s «79 880 881 882 883 884, 385 835' 887 888 889 890 891 8ij2 893 894 895 896 8fl7 .8!«8 .mo Square. 7^^4201 727609 729316 731025 732736 734449 7361 61 7:^7881 739600 7'<I32l 7430 N 744769 746496 748225 749956 751089 753424 755I6I 756900 75864 1 760384 762 129 763376 765()25 767376 769129 770884 77264 1 774400 776161 777t;'24 779689 781456 783225 784996 786769 788544 790321 792100 793881 7956(:4 ?97449 79^6 801025 80^816 QO4606 806404 808201 :^10QPO Cul?e. 6l6i05O)l 6184/0208 620650-177 622835864 625026375 627222016 629422793 631628712 633839779 636056000 638277381 640503928 6*2735647 4)44972544 647214625 64946I896 651714363 653972032 656234909 658503000 6607763 1 1 663054848 665338617 66762/624 66992 I 875 672221376 674526133 676836152 679151439 08 1472000 Ci8i797S4l 686»289<>8 688465387 69O6O7104 693154125 6i:5506456' 69786410 J 700227072 702595369 704969006 707'M797l 709732288 712121957 7if^5r6984 7i6c)i7375 7103231^6 7217^4273 72*150792 72^572699 7290000 JO Square Root. 29-1719043 29-18903f)0 29*2061 637 29-2232784 2'^-2403830 29'7'^74777 292745623 29^291^370 29308701 8 29-325/566 29*342801 5 293598365 29*3768616 29-39:^8769 29' 4 108823 29*4278779 29*4148637 29-4618397 29-4788059 29-4957624 29-5127091 29*5296461 29-5465734 29*5634910 29*5803989 29*5972972 29-6141858 29*6310648 29-6479325 29*6647939 29-08,l6442 2^-6984848 29*7153 150 '29-7321375 29*7**8(?496 29*7^57521 29*7825452 29-7993289 29*8»U1030 29'8328678 29-8496231 29-8663690 29-8831056 29-899«328 29*9165506 299332591 29-949958:^ 29-9366481 29*933^287 '30-0000000 Cube Hoot. 9-476395 9*180106 9-483 3 13 9487518 9-4'9rii9 9*491918 9-498614 9-502307 9-505998 9*50C}()&5 9*513369 9*517051 9*520/30 9-524406 9-528079 9*53 1 749 9-535417 9-5.9081 9-542743 9-516402 9-550058 9*553712 9-557363 9-561010 956^655 9'56S297 9-^719^7 907^574 9-^79208 9-582839 9*585468 9-590093 9593716 9-597337 9-600954 9604569 9-6081 81 9-611791 9-615397 9-619001 9-622t>03 9-626201 9-^X9797 9-633390 9-636981 9-640569 9-644154 9-647736 9-651316 9-654893 109 ARITHMETIC. • * "Nucijir. Square. Cube. Square ilobt. 300 J 66620 Cube Root^ goi ' 811801 731432/01^ 9-658468 ()C2 813^04 733870»08 30-0333148 ^•662040 903 815409 736314327 ;}0 0499584 9-665C09 904 8I7216 738763264 3O'O605f>28 9'^\7yy 905 8 19025 741217625 300832179' 9-672740 906 620836 743677416 30-099^339 9-676301 907 822049 746142643 30 1164407 9-679860 DOS. 824464 748013312 301330383 9-683416 909 826281 751089429 30-1496269 9-686970 910 828100 753571000 30-1632063 9-690521 9ii S2992I 756058031 30-1827765 9'6940t)9 gn 631744 758550528 301993377 9-697615 913 S33569 7610.8497 30 2158899 9-701158 914 835396 763551944 ' 35-2324329 97^^8 915 83/225 7660(X)875 30-2489C69 ' 9-703236 9ie> 839056 7^5Tb'2^ 30-2654919 9711772 917 840889 771095213 30-2820079 9-7 1 5305 918 842724 77362O032 30-2985148 9-718835 y»9. 8445^1 ;^76151559 30-3150128 9-722363 920 846400 7786860C'0 30-3315018 9-'725888 921 84824 1 731229,61 30-34798 1 8 9729410 92i 860084 78377/448 303644529 9732930 923 85 1929 . 7S6330467 30-3809151 9-736448 924 . 853/76 786S89024 30-3973^:^83 973c9i3 925 855625 791453125 30-4138127 9743475 92t) 85/476 79402277s 30-4302481 9-746965 ' 927 859329 79^597983 30-44^6747 9-750493 923 861184 , 799173752 30-4630924 9753998 . ^29 863041 80I/650S9 30-4795013 9757500 930 864900 804357000 30'49590M 9-761000 931 866/61. 806954491 30-5122926 9704497 ; 932 - 868624 809^75 j3 30-5286750 97S7992 933 ^ 870^89 8 12166237' 30-5450487 9771484 934' 872:56 814780504 30-5614136 9-774974 935^" 8/4225 817400375 30-5777(>97 977846 J . 93^' - 876oi>(i 820025856 30-5941171 978i9-^6 i)37 ^779^9 . 822656953 30-6104557 9*7854 2.S 938 87y844 825293*672 30-6267857 9788903 . 939 . 8S1721 82793COI9 30-6431069 9792336 • 940 883 coo 8305^4000 1 30-6594194 9795361 941 8S5481 83323/621 ; , 30-6757233 979.^33 • 942 887304 83^896888 30-6920 1 85 9-802303 1 943 8S9249 838561807 30-7083051 ' 9-^0027 1 , 944 S9II36 841232384 307245830 -9T8097;J6 ! . :94^, . 8ij3025 8439O8625 307408523 •9-313198 ! ; ^^^^^ 894916 S4p5y0536 30-757,1130 ,9181^59 . ^ cn48 . 8968OO . 84|,)2;58123 85I971392 3b-7 733651 . 91820U7 898704 3O-7896O86 9-,823572 ' ,-9^9 90o60ii. 854670349 30*tJ058436 9-pi70'^ 1 , 9025JP 8573/500) 3O*822«7Q0 1. 9*830475 jj SQUARES, CUBES, and ROOTS. 109 - Nurob. ^ Square. . Cube. Square Root Cube KtM)t <r83392:i y5l 9044m 86C0S535 I 30-S3S82S79 . y«2 90Q304 ■ 802801408 30*8544972 9*837369 953 CK^8209 8^5523177-^ 30-970098 1 (^*H0812 (,54 910116 868250664 30-88689^ 9-844253 1 . yfi5 9120^5 8/0.083875 • 36-9030743 y6»7692 956 yisgst) ^ 8737228 16 30-0192497 9-851128 y57 9iotM9 876467493 30-93 J4 1 Q6 9-854561 958 917704 S792}7g\2 ^ 309515751 9'857992 959 9^9561, 8319740/9 . 30-96'7725 1 9*861431 960 x^2 1 boo 384736000 ^ ' 30*9838668 9*b6: 848 9'n 92352 1 887503681 31'0CO00O0 9*868272 \ 902 925444 890277123 310161248 9-871694 903 y27o(ig 893056347 31 0322^1 3 9875113 90* 92929(> 895841344 31-0^183494 9-878530 f ycis 931225 8^86h2125 31*0644491 9^881945 . 906 933156 90l428(i96 31-08^405 9-885357 , 967 935O69 904231003 3 1 -0966236 9-88^/67 * 968 937024 907039232 31-1126984 9'8/il74 1 909. gssgOl . 90(;8 33209 31-1287648 9*8y5580 ■ 970 ^940900 912673000 31-1418230 9-898983 p 971 942841 915498611 31-16J8729 9*902383 972 944784 918330048 311769145 9-905781 s. - 97i 946729 921167317 31-1929479 9-909177 974 948676* U2-1010424 3 1 '208973 1 9-912571 p 975 950625 926859375 31-2249900 9-915962 » . 97fi g5257§ 929714176 3l-24099$7 9*9^9^5* 1 • 977 954529 932574833 31-2569992 9922738 973 9564S4 93A41352 31-2729^^15 9*92612-4 f 979 958441 938313739 31-2889757 9*920504 9««> 960400 941192001 3I-30J9517 9*932883 981 962361 944076 L41 31-3269195 9-'9362jl 98». 964324 946966168 3l-33<?8792 9'.^3i)636 1 9>^3 966289 9*19862087 3 1 -3528 ^08 9943009 yS'i gOBisQ 952763904 31-3687743 9*940379 9S5 970225 • 95^07 i(yl5 31-3847097 9949747 > 996 97-^19^' 958585256 31*4006369 9'953 1 13 <;87 974169 961504803 31-4165501 9'9-36477 .■988 976144 964430272 31 '1324673 9-t^59839 1 9S9 978121 967361669 31-4483704. 9*963198 ; 1 I 990 98OIOO 970299000 31*4642054 9-0'j65A4 991 982081 973242271 31-4901525 9-9699'^9 99a 984064 976191488 31*4960315 9-973262 993. 98^9 979I466&7 . 3 1-5 119025 9-976612 99* 9880^ 98210778* 31*52/7055 9-979959 9JS 990025 985074875 3 1 -5436206 9-983304 ■ 1 ' 99O 992016 9S8OI7936 31*5^94677 9-986 J4S f 997 99^*009 991026973 31*5753068 9*959990 998 99ti0O4 994011992 31 '591 1380 g-993328 S»9 990001 997002£)99 31-6069613 9'yyd6Q5 9 110 ARITHMETIC. Of ratios, proportions, and PROGRESSIOKS- Numbers arc compared to each other in two difFerent >yay$: the one comparison considers the difference of the two numbers, and is named Arithmetical Relation ; and the dif-* ference sometimes the Arithmetical Ratio 2 the other consi- ders their quotient, which is called Geometrical Relation ^ and the quotient is the Geometricat Ratio. So, of these two numbers 6 and 3, the difference, or arithmetical ratio> is 6 — S or 3, but the geometrical ratio is y or 2. There must be two numbers to form a comparison : the number which is compared, being placed first, is called the Antecedent ; and that to which it is compared} the Conse^ quenl. So, in the two numbers above, 6 is the antecedentf and 3 the consequent. If two or more couplets of numbers have equal ratios, or equal differences, the equality is named Proportion, and the terms- of the ratios Proportionals. So, the two couplets, 4*, 2 and 8, 6,^ are arithmetical proportionals/ because 4 — 2 = 8 -^ 6 ;= 2} and the two couplets 4, 2 and 6, 8, are geometri- cal proportionals, because 4 ^ 4 ^ ^> ^^^ same ratio. To denote numbers as being geometrically proportional a colon is set between the terms of och couplet, to denote' their ratio; and a double colon, or else a'mark of equality, betweeri the couplets or ratios. So, the four proportionals, 4, 2, 6, 3 &re set thus, 4 : 2 : : 6 : 3, uhicli, means, that 4 is to 2 as 6 is to 3 ; or thns 4 : 2 3= 6 : 3, or thus, 4 = yy both which mean, that the ratio, of 4 to 2, is equal to the ratio of 6 to 3. . Proportion is tUstinguished into Continued and Disconti-* nued. When ' the difference or ratio of the consequent of one couplet, and the antecedent of (he ndxt couplet, is not the ' same as the common difference or ratio of the coufplets, the proportion is discontinued. 5o, 4, 2, ?, 6 are in discontinued arithmetical proportion^ because. 4 — 2=fc8 — 6 =2, where- as 8—2 = €; and'4, 2, 6, 3 are in. discontinued geometrical proportion, because 4 « |. a 2,«btit f =' 3, which is not the same. ' * But when tlie difference or ratio ,qf every two succeeding terms is the same quantity, the proportioh i§ &ai4 to be Conti- nued, and the num^jers thenwelyes make a seriets p/Cpntinued . '• \ . ;, /i?jc6pQrtionals, ARITHMETICAL PROieORTION. ill Proportionalsr oc a progressiox^t So 2, 4» 6, 3 form an arith* metical progression, because 4 — 2 = 6—4 = 8—6 = 2, all the same common difierencc ; and 2, 4, 8, i6 a geometrical progression, because $ = J = y^ r= 2, all the. same ratio. When the following terms of a pro^rression increase, oy exceed each other, it is called an Ascending Progression, ot Series ; but when the terms decrease, it is a descfjhding one. Ko, 0, 1, 2, 3, 4, &c. is ^ja ascending arithmetical progisession^ but 9, 7, 5, 3, 1, &c. is a descending arithmetical progression. Also 1 , 2, 4, 8, 1 6, &c. is an ascending geometrical prop-ession, and 16, 8, 4, 2j^ 1, $cc* is a de^^ding geometrical profession. ,. ■.. i^ I i gStt ARITHMETICAL PROPORTJONj^iPROGRESSlON. In Arithmetical P/ogressionj^ the niimbers or terms hate all the same cbmmon difference. . Also, the first and last terms of a Progression, are called tlieTIxtremes j ai;id the? otter terms, lying between them, the Means. The mose useful part of arithmetical proportions, is tonttiined in the follow* ing theorems : , » • . ,. . '- TilUoREM 1. When four quahtities are in arithmetical proportion, the suaa^of the two ^tremes is equal to the dum of the two means. Thus, of the four % 4, d, 8, here 2 ^ 8=4 + 6 = iOi * f Thkokesvt 2. In any continued arithmetical progression, the som of the two exti%mes is equal to the sum of any two / means that are equally distant from them, or tqual to double the middle term when there is an uneven number of terms* Thps, in the terms I, 3, 5, it is 1 + 5 = S + 3 ;= 6., And in the series 2, 4, 6, 8, 10, 12, 14, it is 2 + 14 = 4 + 12 = 6+10 = 8 + 8=16. TtoEoKiSM 3. Th^ difference between the extreme^ terms p£ an arithmetical progression, is equal to the common dif- ference of the series multiplied by one less than the nuipber of the terms. So, of the ten terms, 2, 4, 6, 8, 10, 12, 14, J 6, IS, 20, tbet <fom»dn differtoce is 2, and one less thaa tho number of terms 9 j then.tli^ifferenceof th^ extremes is 20-2 = 18, and 2 X 9 = IS also. , '. i. Consequently, 112 AftltHiMETlC Consequently tW greatest ttrm is <^at t6 the least term ^dded to the product of the eommon 4»ereiiee multiplied hf '1 less than the humber of terms. Theorem 4. The sum of all the terms, of any arithiue-' ♦ical progression, is equal to tht sum of the two eiCtreines mul- tiplied by the* number of terms, and divided by 2; or the sum of the two iBxtrcmes multiplied by the number of the tcrms^ gives double the sum of all the termt in the series. ' This is made evident by setting the termi of the series in tin inverted order, under the same series in a direct order, and lidding the corresponding termi together in tliat order. Thus^ in the series I, 3, 5, 7, 9, n, l^t 15 1 ditto inverted 15, 13, !!» 9> % 5, ^8, 1 ; the sums are 16+ 16 + 16 + 16+ 16 + 16 + J6 + 16, which must be double the sum of the single series, and is equal to the sum of the extremes repeated as often as are the number of the terms. * ■ From these theoreftis may reacfilf be fcund any one of these Qye parts ^ the two exlremos, the number of terms, th^ common difference, and the ium of all the ternis^ when any three of them are given ; is in the following problems : PROBLEM I. Xiivmihe ExtfrmeSf antlthe Number ofTemi$t to Jmdthe Suth of^dl the Terms ^ Add the extremes together, midtipty the sum by the num^ ber of terms, and divide by 2. . ' EXAJ^WS. 1. The extremes being .3 and Wt and the nuic^wr of teriQs 9 \ required the sum of the terms ? 19 ' ^ 22 . Or, *— -; — X 9 = ~ X 9 = 1 1 )c 9 = 99, 2 ) 198 ** 1 * ' , ^ the same answti\ Ans. 99 - 2. It Is reqiwred t© find the number of all the strokes a common clock strikes in ^ifie whole irtvohition of the ind^x, or in W hours? Ans. 78v 1) ARITHIMETICAL PROGRESSION. US ( lEx. S, How m^ny strokes do the clock* of Venice strike in the compass of the dzj, which go continually on from I to 24 o'clock ? . Ans. 300* 4. What debt can be discharged in a^ ye^, by weekly payments in soithmetical progression, the first payment being 1/^ and the last or 52d payment 51 Si/ AnSj 135/ 4/. '^ PROBLEM It. Given the Extremes y and the Number <kf Terms ; to jind the Common Difference, SuBTjiACT the less extreme from the greater, and divide the remainder by I less than the number of terms; for th< i^ommoh difference. £XAMPL£2(. • ( 1. The extremes bein^ 3 and 19, and the number of terms ; required the tomimon difference ? 19 3 ^ 19- 3 16 , ^ 9-1 8 8) 16 Ans. 2 • / 2. If the extremes be 10 and 10, and the number of terftii t\ \ what is the. common difierence, and the sum of the series ? Ans. the com. diff. is 3, and the sum is 840. 3, A* certain debt can be discharged in one year, by weekly payments in arithmetical progression, the first payment being 1 J, and the last 5/ 3/ i what is the cdmmcin difference of the terms ? Ans« 2i PROBLEM III. ' Xjiven one o£ the Extremes^ the Common Difference^ and the Number of Terms : to find the other Extreme .^ and the Sum of the Series. Multiply the common difference by 1 less thati the num- ber of teriiis, and the product will be the difference of the fextremes i Therefore add the produtt to the less extreme, to give the greater ; ot subtract it from the greater, to give the less extreme* Vol. I. I EXAMPLES. I 114' AfttTHMETlC EXAMPLES. 1. Giv^n the fcasf term 3, the connnoit difference ^, of an arithmetical series of 9 terms ^ to finxl the greatest term, and the sam of the series: 8 16 19 the greatest tertii 3 the least 22 sum 9 number of terms. 2 ) 198 99 the sum of the series. .2. If the greatest term be 70, the common difference if and the number of terms 21, what is the least term, ai\d the sum of ^he series ? Ans. The least term is 10, and the sum is 840. 3. A debt can be discharged in a year, by paying I shilling the first week, 3 shillings the second, and so on, always 2 . shillings more every week ; what is the debt, and what will the last payment be ? Ans. The last payment will be 5/ 3/, and the debt is 135/ 45. PROBLEM IV. * To find an Arithmetical Mean Proportional between Two Givetr Terms, Add the two given extremes or term* together, and take half their sum for the arithmetical mean required. EXAMPLE. To find an arithmetical mean between the two numbers 4 and 14. TT Here 14 4 IT) 1» Ans. 9 the' mean required. ;>v^ P160BLE5* ARITHMETICAL PROGRESSION. 115 PROBLik V. to find Two Arithmetical Means between two Given Extremes. Subtract the less extreme from the greater, and divide the dijflference by % so will the quotient be the common dif- ference ; which being continually added to the less extreme^ or taken from the greater, gives the means. EXAMPLE. To find two arithmetical means between 2 and 8. Here 8 . 2 . 3)6 Then 24-2 = 4 the one mean; and 4 + 2 = 6 th« other mean; com. dif. 2 PROBLEM Vl. t$ find any Number of Arithmetical Means between Tw Given, Terms or Extremes. Subtract the less extreme from the greater, and divide the difference by 1 more than the ntlmber of means required to be found, wblch will give- the common difference ; then this being added continually to the least term, or subtracted from the greatest, will give the mean terms required. EXAMPLE. To find five arithmeticarmeans between 2 and 14. Here 14 . 2 6)12 Then by adding this com. dif. continually, the means, are found 4, 6, 8, 10, 12. ^dn. dif. 2 See more of Arithmetical progfesslbn in. the Algebra. I 2 GFOMETRICAL 116- ARITHMETIC. I / GEOMETRICAL PROPORTION *^ PROGRESSION, In Geometrical Progression the numbers or terms have all the same multiplier or divisor. The mpst useful part of G^metrical Proportions is contained m the following^ theorems* Theorem 1. ,When four quantities are in geometrical proportion, the product of the two extremes is equal to the product of the two means. . Thus, in the four 2, 4, 3, 6, it is 2 x 6 = 3,x 4 = 12. And hence, if the product of the two means be divided by one of the extremes, the quotient will give the other extreme. So, of the above numbers, the prodirct of the. means 12 -f- 3 = 6 the one extreme, and 12 -r- 6 =2 the other extreme \ and this is the foundation and reason of the practice m the Rule of Three. Theorem 2. In any continued geometrical progression, the product of the two extremes is equal to the product of any two means that are equally distant from them, or equal to the square of the mididle term when there is an uneven number of terms. ■ Thus, in the terms 2, 4, 8, it is 2 x 8 =c 4 x 4 = 16. And in the series 2, 4, 8, 16, 32, €4, 128, it is 2 X 128 = 4 X 64 = 8 X 32 = 16 X 16 = 256* Theorem 3. The quotient of the extreme terms of si geometrical progression, is eqtial to the common ratio of the series raised to the power denoted by 1 less than thi number of the terms. Consequently the greatesit term is equal tof the least terra multiplied by the said quotient. So, of the ten terms 2, 4, 8, \^^ 32, 64, 128, 256, 512j 1024, the common ratio is 2, and one less than the numbef of ternis is 9 ; then the quotient of the extremes is 10^4 4- 2 = 512, and 2^ = 512 also. Theorem • x: V GEOMETRICAL PROGRESSION. 117 TThborem 4. The suih of all the terms, of any geome- trical progression, is found by adding the greatest term to th^ difference of the extremes divided by 1 less than the ratio. So, the sum of 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 1024 — 2 (whoseratiois2),isl024+-— — -— = 1024 + 1022 = 2046, jL — 1 The foregoing, and several other properties of geometrical proportion, are demonstrated more at large in the Algebraic part of this work. A few examples may here be added of the theorems, just delivered^ with some problems concerning mean proportionala. EXAMPLES. !• The least of ten terms, in geometrical progression^ being 1, and the ratio 2 ; what is the greatest term, and the sum of all the terms ? Ans. The greatest term is 512, and the simi 1023. 2. What debt may be discharged in a year, or 12 months, by paying 1/ the first month,' 2/ the second, 4/ the third, and V so on, each succeeding payment being double the last^ and what will the last payment be ? Ans. The debt 4095/, and the last payment 2048/. PROBLEM I. t V - '. To find One Geemetrical Mean Proportional between any Tnv$ Numbers, 9 MuLTiPXT the two numbers together, and extract the square root of the product, which will give the mean propor- tional sought. ^ EXAMPLE* To find a geometrical mean between the two numbers S and 12.' 12 3 36 (6 the mean. 36 PROBLEM lia ARITHMETIC. PROBLEM II. To findr Two Geometrical ]Hean Proportionals iet^uneen any Ttu§ Numbers, Divide the greater number b7 the less, and extract tho. cube root of the quotient, which will give the common ratio of the terms. Then multiply the least given term by th<^ ratio for the first mean, and this mean again by the ratio foi' the second mean : or,- divide the^ greater of th^ two given terms by the ratio for the greater mean, and divide this agaiA by the ratiq for the less mean. E3CAMPLE. ' To find two geometrical means between 3 and 24f. Here 3 ) 24« (8; its cub^ root 2 is the ratio. Then 3 x 2 =: 6, and 6 x 2 =s 12, the two means. Or 2* -7- 2 = 12, and 12 — 2 = 6, the same. That is, the two means between 3 apd 24, are 6 and 12. PROBLEM III. s To find any Number of Geornetrical Mea^s between Two Numbers^ Divide the greater number by the less, and exta'act such root of the quotient whose index is 1 more than the number of means required ; that is, the 2d root for one mean, the 3d iTpot for two means, the 4th root for three means, and so on ; and that root will be the common ratio of all the terms. Then, with the ratio, multiply continually from the first term!.* or divide continually from the last or greatest term^ Example. To find four geometrical means between 3 and 96. Here 3 ) 96 ( 32 ; the 5th root of which is 2, the ratio. Then 3x2=6,&6x2 = 12, &12x2 = 24, &24x2=48. pr96-^2=48,&48-r-2=24,&24-T-2=:12, &a2-7-2=6. ' That is, 6,12, 24, 4?8, are the four means between 3 and 96. Of MUSICAL PROPORTION. |l» Of MlTSICiVL PROPORTION. There is also a third kind of proportion, called Musical^ which being but of little or no common use^ a very short ac- count of it may here suffice. Musical Proportion is when, of three numbers, the first has the same proportion to the third, as the difference between the first and second, has to the diderence between the second and third. As in these three, 6, 8, 12 ; where 6 : 12 :: 8 — 6 : 12 - 8, that is 6 : 12 :: 2: 4. When four numbers are in musical proportion ; then the first has the same ratio to the fourth, as the difference be- tween the first and second has to the difference between the third and fourth* As in these, 6, 8, 12, 18; where 6 : 18 :: H -^^ i 18 - 12> that is 6 : 18 :: 2 : 6. - When numbers are in musical progression, their recipro- cals are in/ arithmetical progression ; and the converse, that is, when numbers are in arithmetical progression, their reci^ f rocals are in musical progression. So in these musicals 6, 8, 12, their reciprocals 4i h A» are in arithmetical progressioo \ for 4 + A = A = T > ^i T + T^T'-Ta ^^^^ i^> ^^^ sum of the extremes is equal to double the mean, which is the property of arithme«- ticals. The method of finding out numbers in musical propor- tion is best expressed by letters iji Algebra. FELLOWSHIP, OR PAJITNERSHIP. Fellowship is a rule, by which any sum or quantity may be divided into any number of parts, which shall be in any given proportion to one another. * By this rul^ ar^ adjusted the gains or loss or charges of partnei;5 t 12« ARrrHMETia psurtners in company ; or the effects of bankrupts* or legacies in case of a deficiency of assets or effects ; or the shares of prizes ; or the numbers of men to form certain de- tachments; or the division of waste lands among* a number of proprietors. Fellowship is either Single or Double. It is Single, when the shares or portions are to be proportional each to one sin- gle given number only 5 as when the stocks of partners are all employed for the same time : And Double; when each portion is to be propoirtional to two or more numbers ; as when the stocks of partners are employed for different times. SINGLE FELLOWSHIP. GENERAL RULE. Add together the numbers that denote the proportion of the shares. Then say, As the sum of the said proportional numbers. Is to the whole sum to be parted or divided, So is each several proportional number. To the corresponding share or part. Or, as the whole stock, is to the whole gain or loss. So is each man's particular stock, To his particular share of the gain or loss. To PROVE THE Work. Add all the shares or parts to^ gether, and the sum will be equal to the whole number to be shared, when the work is right. EXAMPLES. 1 . To divide the number 240 into three such parts, a$ shall be in proportion to each other as the three numbers l^ 2 and 3. Here 1 + 2 + 3 = 6, the sum of the numbers. Then, as 6 : 240 : : 1 : 40 the 1st part, and as 6 : 240 : : 2 : 80 the 2d part, ' also as 6 : 240 :: 3 : 120 the 3d part. Sum of all 240, the proof. Ex. 2« \. SINGLE FELLOWSHfP. 121 Ex.'2. Three persons, a,b, c, freighted a ship with 540 tuns of wine ; of which, a loaded 1 10 tuns, B 97, and c the rest : in a storm the seamen were obliged to throw overboard 85 tuns I how much must each person sustain of the loss i Here 110+ 97 = 207 tuns, loaded by A and b j^ theref." 340 — 207 = 133 tuns, loaded by c. 110 110 : 27i tuns = a's loss j 97 : 24^ tuns =: b's loss ; . 133 : 33^ tuns =: ds loss ; Hence, as 340 : 85 or as 4 : 1 and as 4 : 1 ajiso as 4 : 1 Sum 85 tuns, th^ proof. 3. Two merchants, c and d, made a stock of 120/; of which c contributed 75/, and d the rest : by trading they gained 30/ ; what must each have of it ?. Ans. c 18/ 15/, and d 11/5/. 4. Three merchants, e, f, g, make a stock of 700/, of which E contributed 123/, F 358/, and G the rest : by trading they gain lg5/ 10/ ; what must each have of it ? Ans. E must have 22/ Is Od 2^^q. F - - - 64 S 8 Off. G - - - 39 5 3 l-/y. 5. A General imposing a contribution * of 700/ on four villages, to be paid in proportion to the number of inhabitants contained in each j the 1st containing 250, the 2d 350, the 3d 400, and the 4th 500 persons ; what part must each vil- lage pay ? Ans. the 1st to pay 116/13/ 4rf. the 2d^ - - 163 6 8 the 3d - - 186 13 4 the 4th - - 233 6 8 6. A piece of ground, consisting of 37 ac 2ro i4<ps, is to be divided among three persons, L, M, and n, in propor- tion to their estates : now if l's estate be worth 500/ a year, lid's 320/, and n's 75/ ; what quantity of land must each one have ? ^Ans. l must have 20 ac 3 ro 39 {4^ ps.. M - - •- IS 1 30,*^. ,N . - - 3 2344I. 7. A person is indebted to o 57/15/, to p 108/ 3/ 8i, to Q 22/ 10 J, and to r 73/; but at his decease, his effects . * Contribution is a tax paid by provinces, towDs^ villages, ^c. to excuse tbem from being plundered. It is pffid an provir sio)^$ or in money^ and sometimes in both. arc 123 ARITHMETIC are found to. be^ worth no more that! 170/ 14s ; how must it be divided among his creditors ? Ans, o must have 37/ 15j- Bd2^^^^, P ... 70 15 2 ^T^^. Q^ - . - 14 8 4. OAVtV R - - - 47 U 11 2t^V Ex. 8. A ship, worth 900/, being entirely lost, ofwhich -g-be-r longed to s, \ to T, and the rest to y ; what loss will each sustain, supposing 540/ of her were Insured ? Ans. s will lose 457, r 90/, and v 225/, 9. Four persons, w, x, y, and z, spent among them 25/, and agree" that w shall pay i of it, x 4> y |-, and z 4- ; that is, their shares are to be in proportion as -j-, ^, \y and ^ : what are their shares ? Ans. w must pay 9/ 8</ S^f^, • X - '.6 5 3ff. T - - 4 10 If^. Z - - 3 10 S-yV- 10. A detachment, consisting of 5 companies, being sent into a garrison, in which the duty required 76 men a day \ what number of men must be furnished by each company, in proportion to their strength ; the 1st consisting of. 5 4 men, the 2d of 51 men, the 3d of 48 men, the 4th of 39, and the 5th of 36 men ? * Ans. The 1st must furnish 18, the 2d 17, the 3d 16, the 4th 13, and the 5th 12 men*. BOUBLE FELLOWSHIP.. Double Fellowship, as has been said, is concerned in cases in which th r stocks of partners are employed or conti- nued for different times. * Questions of this nature frequentiy occurring in military >9ervice« General Haviland^ an officer of great merit, contrived an ingenious instrument, for more expeditiously resolving them; which is distinguished by the name of the inventor, being called ^ ]^avilaod. ^ RVLE, DOUBLE FELLOWSHIP. 123 • I Rule*.-— Multiply each person s stock by the time of its continuance ; then divide the quantity, as in Single Fellow- ship, into shares, in proportion to these products^ by sayings As the total sum of all the said products, Is to the whole gain or loss, or quantity to be parted. So is each particular product, T^p the correspondent share of the gain or loss. EXAMPLES. I. A had in company 50/ for 4 months, and b had 60/ for 5 months ; at the end of which time they find 24/ gained ; bow must it be divided between them ? Here 50 60 4 5 200 + 300 =: 500 Then, as 5C0 : 24 : : 200 : 91 zz 9/ 12j = a's share, and as 500 : 24 :: 300 : 14^ = 14 8 = B's share. 2. c and D hold a piece of ground in common, for which they are to pay 54/. c put in 23 horses for 27 days, and d 21 horses for 39 days ; hpw much ought each man to pay of the rent f Ans. c must pay 23/ 5s 9rf. p must pay 30 14 3 3. Three persons, e, f, g, hold a pasture in common, for which they are to pay 30/ per annum ; into which e put 7 oxen for 3 months, f put 9 oxen for 5 months, and G put in 4 oxen for 12 months ; how much must each person pay of the rent ? Ans. e must pay 5/ lOi 6d l-r^g^. F . . 11 16*10 O-j^. G - - 12 12 7 2t%. 4. A ship's company take a prize of 1 000/, which they agree to divide among them according to their pay and the time they have been on board : now the officers and midship- men have been on board 6 months, and the sailors 3 months ; s * The proof of this rule is as follows : When the times arc equal/the shares of the gain or loss ar^ evidently as the stocky, as JD Single Fellowship ; and when the stocks are equal, the shares are as the times; therefore, when neither are equal, the shares must be as their products. the 124 ARITHMETIC. the officers have 40j a month, the midshipmen SOSf and the sailors 22/ a month ; moreover there are 4 officers, 12 mid* thipmen, and 110 sailors : what will each man's share be ? Ans. each officer must have 2S/ 2s 5d O-^^^. each midshipman - 17 6 9 3i7^« each sieaman - - 6 7 2 Oj^, Ek. 5. H, with a capital of 1000/, began trade the first of January, andj meeting with success in business, took in i as a partner, with a capital of 1500/, on the fii^t of March fol- lowing. I'hree months after that they admit K as a third partner, who brought into stock 2800/. After trading toge- ther till the end o^^ the year, they find there has been gained 1776/ 10/ ; how must this be divided among the partners ? Ans. H must have 457/ 9s A^d J - - - 571 16 sy K - - - 747 3 14* 6. X, Y, and z made a joint-stock for 12 months; x at first put in 20/, and 4 months after 20/ more ; t put in at first SO/, at the end of 3 months he put in 20/ more, and 2 months afiier he put in 40/ more ; z put in at first 60/, and 5 months after he put in 10/ more, 1 month after which he took out 30/; during the 12 months they gained 50/'^ how much of it must each have ? > Ans. X must have 10/18/ 6d S^q. Y - - 22 8 1 0^. z - - 16 13 4 0. SIMPLE INTEREST. Interest is the premium or sum allowed for the loan, or forbearance of money. The money lept, or forkorn, is called the Principal. And the sum of the principal and its interest,^ added together, is called the Amount. Interest is allowed at so much per cent, per annum ; which premium per cent, per annum, or interest of 100/ for a year, is called the rate of interest ; — So, Whe» SIMPLE INTEREST. IfS When interest is at 3 per cent, the rate is 3; - - - 4 per cent. - - 4; - - - 5 per cent. - - 5; - - - 6 per cent. - - 6; But, by lawj interest ought not to be taken higher than at " the rate of 5 per cent. Interest is of two sorts ; Simple and Compound. Simple Interest is that which is allowed for the principal lent or forborn only, for the whole time of forbearance. As the interest of any sum, for any time, is directly propor- tional to' the principal sum, and also to the time of continu- ance; hence arises the following general rule of calcula- tion. As 100/ is to the rate of interest, so is any given principal to its interest for one year. And again. As 1 year is to any given time, so is the interest for a year, just found, to the interest of the given sum for that time. Otherwise. Take the interest of 1 pound for a year, which multiply by the given principal, and this product again by the time of loan or forbearance, in years and parts, for the interest of the proposed sum for that time. NoUf When there are certain parts of years in the time, as quarters, or months, or days : they may be worked for, either by taking the aliquot or like parts of the interest of a year, or by the Rule of Three, in the usual way. Also, to divide by 100, is done by only pointing off two figures for decimals. EXAMPLES. 1. To find the interest of 230/ 10/, for 1 year, at the rate «f 4 per cent, per annum. Here, As 100 : 4 ; : 230/ lOi : 9/ 4/ 4|rf. 4 100) 9,22 20 Ans. 9/ 4/ 4| J. 3-20 Ex.2, 126 ARITHMETIC. Ex. 2. To find the interest of 54^/ 1 5sy for 3 years, at 5 per cent, per annum. As 100 : 5 :: 547-75 : Or 20 : 1 :: 547-75 : 27-3875 interest for 1 year. 3 / 32-1625 ditto for 3 years. 20 J 3-2500 12 \ d 3 00 Ans. 82/ Ss Zd. 3. To find the interest of 200 guineas, for 4 years 7 months »nd 25 days, at 4Y*pcr cent, per annum. ds / ds 210/ As 365 : 945 :: 25 : / 44 or 73 : 9*45 :: 5 : -6472 5 840 105 73 ) 47-25 ( •64721 345 9-45 intei-est for 1 yr. 530 4 19 37-80 ditto 4v years. 6 mo =5 4 4-725 ditto 6 month, 1 mo = 1^ -7875 ditto 1 month. -6472 ditto 25 days* / 43-9597 20 s 19-1940 12 d 2-3280 4* Ans. 45/ ids 2^; q 1-3120 4. To find the interest of 450/, for a year, at 5 per cent, per annum. Ans. 22/ lOj. 5. To find the interest of 715/ i2s.6df for a year, at 4j. per cent, per annum. Ans» 32/ 4/ O^d. 6. To find the interest of 720/, for 3 years, at 5 per cent, per annum. Ans. 108/. 7. To find the interest of 355/ 15j for 4 years, at 4 per cent, per annum. Ans. 561183 ^d. , - Ex. 8. . ' COMPOUND INTEREST. 121 Ex. 8. To find the interest of 32/ 5s Sd, for 7 years, at 4|. per cent, per annum. Ans. 94 \2s \d. 9. To find the interest of 170/, for I4 year, at 5 per cent. per annum. Ans. 12/1 5j-. -^ 10. To find the insurance on 205/ 15j, for -J- of a year, af 4 per cent, per annum. Ans. 2/ 1/ \\d, ^ 1 1 . To find the interest of 319/ 6//, for 5^- years, at 3| per cent, per annuin. . Ans. 68/ I5s 9\d. 12* To find the insurance on 1 07/, fof 117 days, at 4| per cent, per annum. Ans. 1/ 1 2s Id. -— IS* To find the interest of 17/ 5/, for 117 days, at ^ per cent, per annum. , Ans. bs Sd.' ' r4. To find the insurance on 712/ 61, for 8 months, at 74 per cent, per annum. Ans. 35/ t2s S^d. Ni^^m The Rules for Simple Interest, serve also to calcu- late Insurances, or the Purchase of Stocks, or any thing else that is rated at so much per cent. See abo more on the subject of Interest, with the algebraical expression and investigation of the rules, at the ^nd of the Algebra, next following. 1 COMPOUND INTEREST. Compound I^nterest, called also Interest uppn Interest, Is that which arises from the principal and interest, taken together, tis it becomes due, at the end of each, stated time of payment. Though it be not lawful to lend mOhey at Compound Interest, yet in purchasing annuities, pensions, or leases in reversion, it is usual to allow Compound Interest to the purchaser for his ready money. Rules. — 1. Find the amount of the given principal, for the time of the first payment, by Simple Interest. Then con- sider this amount as a new principal for the second payment, whose amount calculate as before. And so on through all the payments to the last, always accounting the last amount as a new principal for the next payment. The reason of which is evident from the definition of Compound Interest* Or elsCi 2. Find the amount of 1 ^ound for the time of the first payment, and raise or involve it to the power whose index is denoted by the number of paynients. Then that power multiplied by the gi^cen principal, will produce: the whole amount. \ / 128 ARltHMETIC. , amount. From which the said principal' being ^ubti^actdil/ leaves the Compound Interest of the same. As is evident from the first Rule. EXAMPLES. I. To find the amount of 720/, for 4 years, at 5 per cent, per annum. Here 5 is the 20th part bf^.lOO, aiid the inter^t of 1/fora year is ^V or '05, and its amount 1*05. Therefore^ 1 . By the ist Rule. 2. By the 2d Rule. I s d r05 amount of l/« 20)720 1st yr*s princip. 1-05 36 1st yr's interest. 1 • 1025 2d pow^rof it*' 2Q) 736 2d yr's pr^ncip. 1'1025 37 16 2d yr's interest. 1 -2 1 5506 25 4th pow. of it, 20) 793 16 3dyr*sprincip. 720 39 13 94: 3d yt's interest. / 875-1645 20) 833 9 9^ 4th yr's princip. 20 41 13 5|- 4th yr's interest , ^ ^ / 3 -2900 £ 875 3 3|. the whole amo^ 12 ■ or ans. required. ■ ^3-4800 2. To find the amount of 50/, in 5 years, at 5 per centj per annum, compound interest. Ans. 63/ 16/ 3^rf. 3. To find the amount of 50/ in 5 years, or 10 half-, years, at 5 per cent, per annum, compound interest, the in- terest payable half-yearly. Ans. 64/ Os Id* 4. To find the amount of 50/, in 5 years, or 20 quarters, at 5 per cent, per annum, compound interest, the interest ^ payable quarterly. ' Ans. 64/ 2/ 0-|^. 5. To find the compound interest of 370/ forbom for 6 years, at 4 per cent, per annum. Ans. 98/ 3/ 4^. 6. To find the compound interest of 410/ forborn for 24 years, at 4^ pei* cent, per annum, the interest payable half- yearly. Ans. 48/ 4j 1 \\d* 7. To find the amount, at compound interest, of 217/, forbom for 2^ years, at ^5 per oent. per annum, the interest payable quarterly. Ans. 242/ i 3j 44rf. Nate. See the Rules for Compound Interest algebraically investigated, at the end of the Algebra. ALLIGATION* ALLIGATION* ■ , 129 ALLIGATION. 1 \ Alligation teaches how to compound or mix together several simples of different qualities, so that the composition may be of" some intermediate quality, or rate. It ia com- monly distinguished into two cases, Alligation Medial, and Alligation Alternate*. ALLIGATION MEDIAL. Alligation Medial is the method of find ing the rate or quality of the composition, from havii^g. the quantities and rates of qualities of the several simples given* And it is thus performed : ' * Multiply the quantity of each ingredient by 'its rate or quality; then add. all the produclts together, and add also all r • ■('... • ■ ' -- -- — ■■ - -. f • j ■•■ ' I I ■ ■ ■ , ■ - ■ ,- J- - ;-| -jq 1 |^_iJi«lM^M i ' ' - * " ' * Demonstration* The Rhle is thus riroyed by Algebra. Let flj bg c be the quantities of the ingredients, arid wi, «, p their rates, or qualities> or prices 3 then amy bn, cp iare their severar values, atid am + bn -f cp the sum of their values, also a -f 6 -f- c is the sum of the quantities, and if r denote the rate of the whole Compositioh^ . then a + 6 + c X r will be the value of the whole, conseq.a -{■ b + c X r zz am -^ bn -^ cp, and fit am + 4« + cp^a -\' i H-c, which is the Rule. • > ■ Note, If an ounce or any other quantity of pure gpld be reduced into I?4 equal, parts, these parts are called Caracts; but gold is often mixed with sonie base metal^ which is called the Alloy , and the mixture. is said to be of so many caracts fine, according to the. proportion of pure gold contained in it ; thus, if 22^ caracts of pare golfd, and 2 of alloy be mixed together, it is said to be 22 caracts fine. If any,one of the simples be of litde or no value with respect to the rest, its rat6 is supposed to be nothing ; as water mixed with wine, and alloy with gold and silver. VolL K ^ the ru the quantities together into another sum ; then divide the former sum by the latter, that is, the sum of the product* by the sum of the qutotities, and: the quotient will be the rate or quality of the composition required*^ EXAMPLES* I. If three sorts of gunpowder be imxed togetbeis ^rz* 50lb 9t I2d z pound, 441b at 9d, and 261b at Sd a pound ^ how much a pound is the composition worth i Here 50, 44, 26 are the quantities, and 12, 9, 8 the rates or qualities ; then 50 x 12 = 600 44,X 9 = S96 26 X 8=208 120 ) 1204 ( lOrJv = 10^ Ans. The rate or price is 10^^ the poimd. ^•2, A comporitioa bein|^ made of 5lb of tea at 7s per Ib^ 9lb.at 8r 6^ per Ib^ and. 14ilbat 5/ lOi/per lb; whacis»^ ♦lb of it worth ? Ans. 6s lO^rfr 5» Mixe4-4-gsfioa»-of-wfiie-at-4j' ICtf-pcr gatf, with"7 gal- k)ns at 5s Sd per, gall, and 9|- gallons at 5s Si/pjer^gall^ what is a gallon of this composition worth ? * Ans. 5s 4^. 4. A meahnan would mix 3 bushels of- flour Jat ^i 5d^ per bushel, 4 bushels at 5$ 6d per bushel, and 5 bushels at 4j8^per bu⪙ what is the: worth of. a bushel of this mixture ? • > Ans.- 4r iid^ 5. A farmer nvixes 10 bushels of wkeat. at .5s the bushd, Vith 18 bushels of rye at 3x the bushel, and 20 bushels of barley at 2s per l>ushel:' how much is a bushtel of the mixture worth ? Ans; di^ 6. Having melted tog^b^r. 7 *oz of gold of 22caracts fine, 1240Z of 21 caracts fine, aild 17 oz of 19 caracts^fine : I Would know the fineness of tbfe composition ? Ansl 20ff caraets fine. 1, Of what fineness Is that con^osit ion, which is made- by tiiixing,3lb of silyer of 9 oz fine, ,with 5lh 8 oz pf -lOoaSr. fine, fUid> lib lOoz of aUay.. Ansi 7|j;0Z» fine.' ALUGATIOK [ 131 ] ALLIGATION ALTERNATE.. : AttiOATiaN Alternate is the method of finding what quantity of any number of simples, whose rates -are given, will compose a mixture of a given rate. So that it is the re- verse of Alligation Medial, and may be proved by it. RULE I*. 1. Set the rates of the simples in ^ column under each Other.-^2. Connect, or link with a continued liae, the rate of' each simple) which is less than that of the compound^ with one» or any number, of those that are greater than the com-' ppund ;* and. each greater rate with one or any number of the le»rftr-^* Write the difference between the mixture rate^and^ that of each of the simples^ opposite the rate with which they al-e Unked.-*-4. Then if only one difference atand ;^ainst any rate, it will be the quantity belonging to that rate ; but if there be seV.eral, their suni will be the. quantity. , - TUfe' examples may be, proved by the rule for AlHgation MediaL f Demonst, By connecting the less rate to the gteikter, atid placing the difference between them and the rate alternately, tho quantities resulting are such, that there is precisely as much gtified by 6ne quantity as is lost by the other, and therefore the gain' and loss upon the whole is equal, and is exactly the proposed rate : and the same will be true of any other two simples managed according to the Rale^ In like manner, whatever the number of siibples may be, and with how ttiany soever every one is linked, since it is always a less with a greater than the ^roean price, there will be an equal balance of loss and gain between every two, and consequently an equal balance on the whole. - a. e* o. « .It is obvious, from this Rale» that questions of this sort admit of a gneat variety of answers \ for, having found one answer, we may £nd as many more as we please,, by only multiplying or dividing each of the quaptities found, by 2, or 3, or 4, ^c: the reason of which is evident : for^ if two quantities, of two simples, toake a balance of loss and gain, with respect to the mean price, so must also the double or treble, the 4 or |- part, or any other ratio of these quantities, and so on, 0(2 in^mYt/m. These kinds of questions are called by algebraists indeterminate or unlimited problems 3 and by an analytical process, theorems may be raised that will give all the possible answers. ' K2 EXAMPLES. X 133 ARITHMETIC EXAMPLES. 1. A merchant would mix wines at 16/, at 18j, and at ' 22s per gallon, so as that the mixture may be worth 20/ the gallon : what quantity of each must be taken i /^16"^ 2 at 16/ Here20< 18x j2 at 18/ ^ i^22j/ 4 + 2 =: 6 at 22/. \Ans. 2 gallons at 16/, 2 gallons at 18/, and 5 at 22x. 2. How much wine at 6s per gallon, and at 4s per gallon,^ must be mixed together, that the composition may be worth Ss per gallon ? Ans. 1 qt, or 1 gall, &c. 3. .How much sugar at 4rf, at 6rf, and zt \ld per lb, mUst be mixed together, so that the composition formed by them" may be worth Id per lb ? Ans. 1 lb, or 1 stone, or 1 cwt, or any other equal quantity of each sort. 4. How much corn at 2x 6rf, 3/ 8rf, 4j, and 4/ 8 J per bushel, must be mixed together, that the compound may be worth ifif lOd per bushel ? Ans. 2 at 2s 6d. 2 at 3/ 8 J, 3 at 4/, and 3 at 4/ Sd. 'I 5. A goldsmith has gold of 16, of 18, of 23, and of 24 caracts fine : how much must he take of each, to make it ,21 caracts fine ? Ans. 3 of 1 6, 2 of 1 8, 3 of 23, and 5 of 24.' 6. It is required to mix brandy at 12/, wine at 10/, cyder at 1/, and water at per gallon together, so that the mixture may be worth 8/ per gallon ? Ans. 8 gals of brandy, 7 of wine, 2 of cyder, and 4 of water. ' ' * . . . « RULE II. ^ * •)• When the whole composition is limited to a certain iquantity : Find an answer as before by linking ; th^en say, as the sum of the quantities, or differences thus determined, is to the given quantity ; so is each ingre4ient, found by link- ing, to the required quantity of each* examples. ^ . How much gold of 1 5, 17, 18, and 22 caracts fine, must be mixed together, to form a composition of 40 oz of 20 ca^- racts fiuie ? . Here ALLIGATION ALTERNATE. 133 - 2 - - 2 - 2 5 + 3 + 2 = 10 16 Then, as 16 : 40 :: 2 : 5 and 16 : 40 :: 10 : 25 Ans. 5 oz of 15, of 17, and "of 18 caracts fine, and 25 oz of 22 caracts fine*. Ex. 2. A vintner has wine at 4/, at 5/, at 5s 6i, and at 6s a gallon; and he would make a mixture of 18 gallons, so that it might be afforded at 5s 4rf per gallon ; how much of each sort must he take ? Ans. 3 gal. at 4 J, 3 at 5j, 6 at 5s 6t/, and 6 at 6/. * A great number of qtiestions might be here given relating to the specific gravities of metals, &c. but one of the most curious may here suffice. Hiero, king of Syracuse, gave orders for a crown lobe made entirely of pure gold ; but suspecting the workman had debased it by mixing it with silver or copper, he recommended the dis- povery of the fraud to the famous Archimedes, and desired to know the exact quantity of alloy in the crown. Archimedes, in order to detect the imposition, procured two other masses, the one of pure gold, the other of silver or copper, and each of the same weight with the former j and by putting each Separately into a vessel full of water, the quantity of water expelled by them determined their specific gravities 5 from which, andtheit given weights, the exact quantities of gold and alloy in the crowa may be determined. Suppose the weight of each crown to be lOlb, and that the water expelled by the copper or silver was 92lb, by the gold 52lb, and by the compound crown 64lb ; what will be the quantities of gold and alloy in th€ crown r The rates of the simples are 92 and 5*2, and of the compound 04 5 therefore • fii I 9'^"^ ^^ of copper "■* I 52-^ 28 of gold And the sum of these is 12 + 28 =: 40, which should have been but 105 therefore by the Rule, ^ 40 : 10 :: 12 : 3 lb of copper 1 , : 7lbofgold /t^e answer. 40 : 10 : ; 28 RULE IS* ARITHMETia HOLE III*, When one of the ingredients is limited to a certain quaii« tity ; Take the difference between each price, and the mean rate as before ; then say, As the difference of that simple, whose quantity is given, is to the rest of the differences se-^ verally, so is the quantity given, to the several quantities required. EXAMPLES* 1. How much wine at 5/, at 5s 6df and 6s the gallon, must be mixed with 3 gallons at 4s per gallon, so tbat thp multure may be worth Ss 4d per gallon ? + 2 = 10 Here 64 + ^ = 10 + .4 = 20 + 4 = 20 3 3 : 6 3 : 6 Ans. 3 gallons at 5sy 6 at bs 6^, and 6 at 6x« ft. A grocer would mix teas at 12j, 10/, and 6x per lb, with $!01b at 4/ per lb • how much of each sort must he tako to make the composition worth %s per lb ? Ans. 201b at 4/, lOlb at ^x, lOlb at lOi, and 201b at 12/. 3. How much gold of 15, of l7, and of 22 caracts fine, must be mixed with 5 oz of 18 caracts fine, so that the com- position may be 20 caracts fine ? Ans. 5 oz. of 15 caracts fine, 5 oz of 17, and 25 of 22« •♦*" « In the very same manner questions may be wrought when sc* veral of the ingredients are limited to certain quant ities, by finding first for one limits and then for another. The two last Rules can peed no demonstration^ as they evideody result from the firstj the 1^904 of wl^icli h^ been already explained* fOSITl[0H« SINGLE POSITION. 1« POSITION- Position is a method of performing certain questibnSf which cannot be resolved by the common direct rules. It h sometimes called False Position, or False Supposition} because it makes a supposition of false numbers^ to work with the same as if they were the true ones, and by. tl^ir means dis- covers the true numbers sought. It is sometimes ako called Trial-and«£rror, because it proceeds by trials of false num- bers, and thence finds out the true ones by a comparisoa of the /rrvr/.'— Position is either Single or Double. SINGLE POSITION. Single Position is that by which a questmi k resotved by means of one supposition only. Questions which have their result proportional to their suppositionSj|^ belong to jingle Position : such as those which require the multiplica- tion or division of the number sought by any proposed num* ber ; or when it is to be increased or dimiotthed bj ifiself^ «r any parts of itself, a cevtain proposed nutnbe» ot times* The rule i& as follows : Takb or assume any nwnbeF fiur that whick is required, sad perform, the some operatttons with it, as aM described or performed in the question* Thensay, As the resiite cS the said operation, is to the position, or number assumed^ so is the result in the question, to a fourth term, which will be the number sought*. ^1^ * The reason of this Rule is evident^ becanse it is auppoaad tbajt the results are proportional to the sapposUions. Thus, na I a i I nz I 2g a z or — : a : : — : z. n n a Q z z or — ± — &c : fl ; : — ± — &c s z, n tn u m and so on. EXAMP LES« 1S« ARITHMETIC. EXAMPLES. 1. A person after spending y and ^ of his money, has yet remaining 60/; what had he at first ? Suppose he had at first 120/. Proof. Now ^ of 120 is 40 i of it is 30 •§.ofl4«4is 4S i of 144 is 36 their sum is 70 rhich taken from 1 20 their sum 84 taken from 144 leaves 50 Then, 50 : . J20 : ; 60 leaves 60 as 144, the Answer. per question. 2. What number is that, which being multipied by 7, and the product divide4 by 6^ the quotient may be 21 ? Ans. 18, S. What number is that, which being increased by i^ f, ^nd ^ of itself, the sum shall be 15 i Ans. 36. 4. A general, after sending X)ut a foraging i and -J- of his men, had yej; remaining 1000 : what number had he in com- mand ? Ans. 6000. 5. A gentleman distributed 52 p€n9e among a number of ppor people, consisting of men, women, and children ; to each man he gave 6^, to each woman 4^, • and to each child Sd : moreover there were twice as many women as men, and thiice as ipany children as women. How many were there of each i . Ans. 2 men, 4 women,' and 12 children. 6. One being asked his 'age, said, if ^ of the years 1 hav^ lived, be multiplied by 7, and -f- of them be added to the product, the sum will be 219. What was his age ? Ais. 45 years. |>OUBl.E I 137 ] DOUBLE POSITION, Double Position is the method of resolving certain questions by means of two suppositions of false numbers. To the Double R ule of Position belong such questions as have'their results not proportional to' their positions : such are those, in which the numbers souglit, or their parts, or* their multiples, are increased or diminished by some given absolute number, which is no known part of the number sought* RULE I*. Take or assume any two convenient numbers, and proceed with each of them separately, according to the conditions of the question, as in Single Position ; and find how much earJi result is different from* the result mentioned in the question, calling thes« differences the errors ^ noting also whether the results are too great or too little. * Demonstr,^ The RuKs is founded on tliis supposition, namely, thai the first errqr is to the second, as 'the difference between the true and first supposed number^ is to the diffbr^ce between the true and second supposed number 5 when that is no( the case, the exact answer to the question cannot be found by this Rule.— That thot Rule is true, according to that supposition^ may be thus proved. I^t a and b be the two suppositions, and a and b their results, produced by similar operation \ also r and s their errors, or the differences between the results a and b from the true result n ; and let x depote the number sought, answering to the true result If of the question. Then is n — a = r, and n — b = *. And, according to the supposition on which the Rule is founded, r : s :: x— a: x—bs% hence, by multiplying extremes and means, rx — r6 = ^«— *a j 4hen, by transposition, rx — «x = rb — sa j and, by div«iio|i^ flf g(i • -* X — r iz the number sought, which is the rule when the^ r — « . results ^re both too little;. If the results be both too great, so that a and b are both greater than N 5 then n — • a z: — r, and n — bzz — *, orr and s are both negative ; hence — r ; — $ : nx — a : x ^b, but — r : —4 : : + r : -f *, therefore c ; s : : ^p— a ; x -^"b; and the rest will be ex- actly as in the former case. ^ But if one result a only be too little, and the other b too great, or one error r positive, and the other s negative, then the theoreiu becomes x = — ^ , wbioh la the Rule in this case, or when the errors are unl&e, Theo 13S ARITHMETia Then multiply each of the said errors by the contrary sup- position, namely, tlie first positiQn by the second .error, jmd the second position by the first error. Then, If the errors are aJike, divide the difference pf th^ products by the difierence of the errors, and the quotient will be tl^e answer. But if the errors are unlike, divide the siun of tl^e produces by jhe sum of th^ errors, for the answer, Noicy The errors are said to be alike, when tb^y are either both too great or both too little^ ^md unlike, ^\^^ we is too great and the other too Uttle. EXAMPLES. 1. What number is th^t, which being multiplied by 6^ %!t^ product increased by 18, and the sum divided by 9, the quotient shall ^e 20 ? Suppose the two numbers 18 and 30* Then, First Position. 18 Suppose 6 mult* iSecopd Position. 30. 6 Rroof. 27 108 1^ « 1 add 180 18 162 18 9) 126 div. 9) 198 9} 1^ 14 20 results tri^e res. 22 20 20, • OA pos. SO errors unlike -^ 2 mujt. ' 18 1st pos. Igr- ( 2 180. rors ( 6 36 « surn. 8 ) 216 sum of products 27 < Answer * . # sought. RULE 11. Find, by trial, two numbers, as near the true number as convenient, and work with them as in the question ; mark- ing the errors which arise from each of them. Multiply the difierence of the two numbers assumed,, or found by trial, by one of the errors, and divide th^. product by the difference of the errors, wlien they are alik;e, but by their sum when they are unlike. AAi the quotient, last found, to the number belonging to the said error, when that number is too little, but; subtracjt It »• DOUBLE PpSmON. . 18? It when too greats and the result will give the true quantitf jBOUght *# examples; 4 1. So, the foregoing example, worked by this 2d rulet will be as follows : t: I. 30 positions 18; their dif. 12 ^ 2 errors + 6 ; least error 2 « . 9 sum of errors 8 ) 24 ( 3 subtr. from the position 30 leaves the answer ^7 Ex. 2. A son asking |iis father how old he was, received this answer: Your age is no^ one-third of mine; but 4 years ago, your age was only one-fourth of mine.. What then are their two ages ? , Aas. 15 and 4^,. 3, A workman was hii^ed for 20 days, at 3/ per day, for every day he worked ; but with this condition, that for every day he played, he should forfeit* 1^. Now it so hap- pened, that upon t^e whole he had 2/ 4x to receive. How piany of the days did he work f Ans. 16, 4« A and B began to play together with equal sums of money : a first won 20 guineas, but afterwards lost back ^ of what he then had; after which, B had 4 times as much as A. What sum did each begin with ? Ans. lOO guineas. 5. Two persons^ A and B^ have both the same income^ A saves 4^ of his ; but b, by spending 50/ per annum more than A, at the end of 4 years finds himself 100/ in debt* What does each receive and spend per annum ? Ans. They recei^ 125/ per annmn; also A spends 1004 and B spends 150/ per annum. ■••^ 1 * for since, by the supposition, rtnix — a : x^b^ there* fore by divisioni r— ^ : t ; ; *— « ; «— ft, which is the ad Rule, "" PRACTICAI, iLd liO » * AfHTHMETIC. PRACTICAL QUESTIONS in ARITHMETIC. Quest. 1. The swiftest velocity of a cannon-ball, is about 2000 feet in a second of time. Then in what time, at that rate, would such a ball be in moving from the earth to the sun, admitting the distance to be 100 millions of miles, and tlie year to contain S65 days 6 hours ? Ans. Sy^^s^ years. Quest. 2. What is the. ratio of the velocity of light to that of a cannon-ball, which issues from the gun with a ve- locity of 150O feet per second; light passing from the sun to the earth in 14 minutes ? Ans. the ratio of 782222|- to I • Quest. 3. The slow or parade-step being 70 paces per minute, at 28 inches each pace, it is required to determine at what rate per hour that movement is ? Ans. Ixfl- miles. Quest. 4. The quick-time or stcp^ in marching, being 55 paces per second, or 120 Y^r minute, at 28 inches each ; then at what rate per hour does a troop march on a route, and how Jong will they be in arriving at a garrison 20 miles distant, allowing a halt of one hour by the way to refresh? A r the rate is 3-rV miles an hour. ^^^' 1 and the time 7| hr, or 7 h 17| min. Quest. 5. A wall was to be built 700 yards long in 29 days. Now, after 12 men had been employed on it for 11 days, it was found that they had completed only 220 yards of the wall. It is required then to determine how many men must be added to the former, that the whole number of them may just finish the wall in the time proposed, at the same rate of working. Ans. i men to be added. Quest. €. To determine how far 500 millions of gui- neas will reach, when laid down in. a straight line touching one another ; supposing each guinea to bean inch in diameter, as it is very nearly. Ans. 7891 miles, 728 yds,, 2 ft, 8 in. ^ Quest. 7. Two persons, a and B, being on opposite sides of a wood, which is 536 yards about, they begin to go round it, both the same way, at the same instant of time ; a goes at the rate of 1 1 yards per minute, and B 34- yards in S minutes j the question is, how many times will the wood be gone round before the quicker overtake the slower ? J^ Ans. 1 7 times. Quest, . PRACTICAL QUESXIONS. Ut Quest. 8. a can do a piece of work alone in 12 days^ and B alone in 14"^ 'in what time wiil they btrth together per-f form a like quantity of work ? Ans. 6^ days* , . QupsT. 9. A person who was possessed of a ^-.share of a copper mhie, sold -J- of his interest, in it for \ 800/ v what was the reputed value of the whole at the same rate? Ans. 4000/* QifEsT. 10. A person after spending 20/ more than \ or his yearly income, had then remaining' 30/ more than the . half of it ; what was his jncome ? Ans. 200/. Quest. 11. The hour and mjnute hand of a clock are exactly together at 12 o'clock; when^are they next together? Ans. at I7V hr> or 1 hr, 5^^ rain-r Quest. 1 2., If a gentleman whose annual income is 1300/y. spend 20 guineas a week ; whether will he save or run in debt, and how much in the year ? i 'I Ans. save 403/* ' Quest. 13, A person bought •! SlQ.or^nge^ at 2 a penny, and 180 more at 3 a penny; after ,>viiich, ^selling them out again at 5 for 2 pence, whether did he gain or lose by the bargain? '' Ans. he lost 6 pence. . Qubst. 14. If a qujintity of provislopis s^ves ,1500 men 12 weeks, at thje^.|*ate of ?0' ounces a;4ay for each man ;> how many men will the .same provisions maintain for 20 weeks, at the rate of 8 ounces a day for each'm^n ? Ans. 225Q ttitxu Quest. 15. In the latitude*of London, the distance round the earth,'measured on. the parallel q£ latitude^ is about 15550 miles; now as the earth turns round in.23|iours 56 nijfij^t;es, at what rate per hour is the city of J^p;i<J(xp carried by.jthis motion from west to east J A,ns< ^4p|r||' tpiles anl?i>u^» Quest. 16. A father left his son a fdrtune, J- of which he ran through in 8 months': f of the remamder lasted him 12 months longer ; after which he had bare 820/ left. What sundr did the father bequeath his son?' > Ans/ 1913/6/ 8//. Quest. 17. If 1000 men, besieged in a town, with pro- visions for 5 weeks, allowing each man 16 ounces a day, be reinforced with 500 men more ; and supposing that they can- not be relieved till th^ er|d of 8 weeks, how many ounces a day must each man have, that the provision may last that time ? \ Ans. 6|. ounces. • Quest. 18. A younger brother received 8400/, which was just |- of his elder brother's fortune : What was the father worth at his death ? , Ans. 19200/. ' - Quest. U2 ARlTHMfetiC . Qvi,sr. 19. . A person, looking on his j^stchj v^s. a^k^d Vlis^t >yas the time of^th^ day, who ansv^ere^i.It. U.bp(Wjeea 5 and 6 ;. but a more particular answer being required, he uad that the hoitr and minute hands were then exaqtly toge- ther : What was the time ? Ans. 27-1^ >^i°' P^^ ^* .^ Quest. 20. If 20 men tin perform a piece of work in lb days, how many men will accomplish another ti^rice as - large in one-fifth of the time? Ans. 300* ... . ..... Quest* 21, A father devised y^^ of his estate tp one of his sons, and -^ of the residue to another, and the surplus to his relict for life, 'the children's legacies were found to be 5i4/ 6s 8d different: Then what money did he leave the widow the use of? ' Ans. 1270/ 1/ 9|4^. Quest., 22* A person, making his will, gave tp^ ouj? child . i% of his estate, and the rest to another. When these legacies caoie to be paid the one turned out 1 200/ more than the other : What did the testator die worth ? Ans. ^OOO/* .Quest. 23» Two persons, a and B, travel between £ondon and Lincoln, distant 100 miles, a from London, and B from Lincoln, at the same instant. ' After 7 hoUrs they meet on the road, when it appeared that a had rode \i miles an hour more than b. At what rate per hour, then did each of the travellers ride ? Ans. a 7|4V^nd b 6^4 miles. Quest. 24. Two. petrsons, A and b, travel betwGeti Lon- don and Exeter, a leaves Exeter at 8 o'clock in the morrj* ing'^ and walks at t}ie r^te of S miles an hour,-wtthout inter* mission; and b sets out from London at 4 o'clock the same evening, and walks for Exeter at the ritte of 4 miles an hopr cpnsfantly. , Now,.. supposing the distance betwejen the two cities to be 130 miles, whereaSouts on the road will they meet ? ; Ans. 691 miles from Exeter. • QuesTi|,25. One hiindred eggs -being placed on the ground, in ^ straight line, at the distance of a yard from each other : How far will a .perspn travel who shall bring them ope by one to a basket, which, is placed at one, yard from the. first egg ? Ans. lOlOO yards, or 5 miles and 1300 yds. Quest. 26. The ' clocks, of • Italy gd on to 24 hours: Then how many strokes do they strike in one complete re- volution of the index? Ans. 300. Quest. 27. One Sessa, an Indiaiijij having ipyented the game of chess, shewed it to his prince, who was so delighted with wftK iti tfekf he promised^ litnf aby iWstr'd Ire shbuld' isle ; on' wHith Sessa- requested that he might be allofwed one griiri'of^ wheat for the first square on the cjiess boards 2 for the secoxid^ 4 for the third, and so on, d9ubiing contihualiy, to 64; the whold number of squares. Now, supposing a pint to contain 7(180 of theje grains, and one quarter or 8 busheU to be worth ^Is 6dy it is required to compute the value of* all the corji ? Ans. 6450468216285/ lis 3d 3j4J4^* . Quest. 26* A' person increased his estate annually by 100/ more than the j- part of it ; and zt the end of 4 year^ found'that'his estate amounted to 10342/ 3/ 9rf. What had h6' at' first ? Ans: 46oO/. Quest. 29.' Paid 1012/ lOx for a principal of 750/,'takeii' irf^7 years before : at what rate per ceiit. p^r aftnum did I' piy interest ? Ansf. 5 pet cent* ■ • '> . - . '• Quest. 30. Divide 1 000/ among A, b,' c j so as to give A 120 more, and b 95 less than o Ans. A 445^ b 2S0, c ^25- QuEST. 3U A person l)eing asked the hour of the day, said, the time past noon is equal tooths of the time till midnight. What was the time r Ans. 20 min. past ^- QuEsT. 82. Suppose that I have^ -/y of a ship worth 1^00/; what part of her hiave I left after selling | of ^ of ftiy share, and what is it worth? Ans. -^^j worth 185A Quest. 33. Fart ItOO acres of land among A, b, q ; so fliat B may have 100 morfc thkn A, and c 64 more than b. Ans. A 312, B 412, c 476. Q0EST. 34i What number is that, from which if there be takien j- of |, ■ and 'to the remainder be 'added -^j of -j^, thfc^^utn will be 1 ? Ans. 9||- QCEsT; S3; There is a number which, if multiplied by |^ of 4 of li, will produce ,1 ; what is the square of that numbejr? ' ' • , ^ Ans. l-,2^. Quest. 36. What leugth must be cut off a board, S^- inclies broad, to contain a square fobt, cr as much as 12 incBkes in length and 12 in breadth ? Ans. 1614 i-^ches. J Quest. 37. What sum of money will amount to 1 3S/ 2s IBd, in 15 months, at 5 per cent, per annum simple interest i Ans. 130/. Quest. 3'8. A father divided his fortune among his three sons, A, B, c, giving a 4 as often as b 3, and g 5 as often as B 6 J 144 ARITHMETIC. B 6 ; what neas the whole legacy, supposing ^*s share wsfc* 4000/. Ans. 9500/. Quest. 39. A young hare starts 40 yards before a grey- houndy and is not perceived by him till she has been up 40 seconds ; she scuds away at the rate of 10 miles an hour, and the dog, on view, makes after her at the rate of 1 8 : hovr long will the' course hold, and^what ground wiU bcTrun ov^, tounting from the outsetting of the dog? Ans. 60^5y sec. and 530 yards run. UE^T. 40. Two young gentlemen, without private forr tiffle, obtain commissions at the same time, and at the age of ISy One thoughtlessly spends 10/ a year more than his pay ; butjfi shocked at the idea of not paying his debts, gives his creditor a bond for the money, at the end of every year, and also »^sures his life for the amount; each bond costs him. 30 shillings, besides the lawful interest of 5 per cent, and to in- sure liis life costs hhn 6 per cent. The other, having a proper pride, is determined never to run in debt ; and, that he may assist a friend in need, perse- veres, in saving 10/ every year, for*» which he obtains an interest, of 5 per cent, which interest is every year added to his savings, and laid out, so as to ansfwerthe effect of com- pound mterest. Suppose these two officers to meet at the age 'of 50, when each recieives from Government 400/ per annum ; that the one, Seeding his past errors, is resolved in future to spend no more thto. he actually has, after paying the interest for what he owes,«^d the insurance oh his life. The oth^lk having now something before hand, means in future, to spen4 his* full income, without increasiog his stock. It is desirable to know how much each has to spend per annum, and wliat money the latter has by him to assist the distressed, or leaVM^ those who deserve it ? Ans. The reformed officer has to spend 66l 19/ lf*5389i per annum. The prudent officer has to spend 437/ 1 2^ 1 l^-4379i per annum. And the latter has saved, to dispose of,7 52/ 1 9/ 9' 1 896^. END OF THE ARITHMETIC. [ 14i' 1 OF LOGARITHMS^ L rOG ARITHMS are ihade to facilitate troublesonie caIcH^ lations in numbers. This they do^ because they perform multiplication by only addition, and division by only subtrac- tidn, and raising of powers by multiplying the logarithm by the Index of the pbwer j and extracting of roots by dividing the logarithm of the number by the index of the root. For, lagarithms are numbers sd contri?ed| and adapted to othei^ numbers, that the sums and difierences of the former shall correspond to, and show, tlie products and <|uoti^nts of th^ fatter, &c. Or, niore generally, logarithms are the numerical ^tpp^ nents of ratios ; Or they are a series of numbers in arith^ metical Mhata * The invention of Logdritbchs is due to Lord Napier, Baron of Merckistton, in Scotland, and is properly considered as one of the inost useful inventions of modern times. A tabl^ of tbe<ie numben was first published by the inventor at Edinburgh, iii the year 16 1 4, in a treatise entitled Canon Mirificum liOgnrtthmorum ; which was eagerly received by all the learned thfotighoat Europe. Mr. Henry Briggs, then professor of geometry at Qresham College^ soon after the discovery, went to visit the noble nventor ; aft»r which, they jointly undertook the arduous task of computing new, tables on this subject, and redacing them to a more cdnvenieht form than that which was at first thought of. But Lord Napier dying soon after, the whole burden fell upon Mr. Briggs, who, with prodigious labour and great skill, made an entire Canon, 9.0 cording to the new form, for all numbers from i to 'iOOncs and from 9 XX,0\o 10 100, to 14 places of figures, and published it at London in the year 1(J24, in a treatise euiitled Arithmetica Loga<> jtlthmica, with directions for supplying the intermediate parts. Vol.1. L ' This U6 LOGARITHMS. metical progression, answering to another series of numbers in geometrical progression. Th /^* ^» ^* ^* ** ^* ^' Indices, or logarithms* tl> 2, 4, 8, 16, 32, 64, Geometric progression* ^ rO, 1, 2, 3, 4, 5, 6, Indices, or logarithms* ll, 3, 9, 27, 81, 243, 729, Geometric progression. {0, 1, 2, 3, 4, 5, Indices, or logs. 1, 10, 100, 1000, 10000, 100000, Geom. progres. Or Where it is evident, that the same indices serve equally for any geometric series ; and consequently there may be an This Canon was again published in Holland by Adrian Vlacq,. in the year l6U8, together with the Logarithms of all the numbers which Mr. £riggs had omitted -, but he contracted them down to 10 places of decimals. Mr. Briggs also computed the Logarithms of the sines^ tangents, and secants, to every degree^ and centesm, or 100th part of a degree^ of the whole quadrant ; and annexed them to the natural sines, tangents, and secants, which he had before computed, to fifteen places of figures. These Tables, with their construction and use^ were first published in the year 1633, after Mr. Briggs's death, by Mr. Henry Gellibrand, under the title of Trigonometria Britannica. Benjamin Ursinus also gave a Table of Napier's Logs* and of tines, to every 10 seconds. And Chr. Wolf, in his Mathematical Lexicon, sajs that one Van Loser had computed them to every single second, but his untimely death prevented their publica- tion. Many other authors have treated on this subject ;' but as their numbers are frequently inaccurate and incommodiously dis« posed, they are now generally neglected. The Tables in most repute at present, are those of Gardiner in 4to, first published in the year 1742 3 and my own Tables in 8vo, first printed in the year i78^> where the Logarithms cf all numbers may be easily found from 1 to lOCXXXXX) -, and those of the sines^ tangents, and secants, to any degree of accuracy required; Also, Mr. Michael Taylor's Tables iu large 4to, containing the common logarithms, and the logarithmic sines and tangents to cVery second of the quadrant. And, in France, the new book of logarithms by Calletj the 2d edition of which, in 1795, has the tables still farther extended, and are printed with what are called stereotypes, the types in each page being soldered together into a solid ^ass or block. Dodson's Antilogaritbroic Canon is likewise a very elaborate work, and used for finding the numbers answer'uig to any givea logarithm. ' endless tOGARITHMS. 147 endless T^rkty of systems of Ic^arithms^ to the samecosi- mon numbersy by onlyt chaoaging the second term^ 2, 3/or IG, &c.^ of the geometrical series of whole numbers; and by interpolation the whoie system of numbers may be made to enter: the geometric series, and receive^thieir proportkmal logarithms, whether integers or decimals. It is alao apparent, from the nature of these ^ries,, that if any two indices be added together, their sum. will be the index of that number which is equal to the product of the two terms, in the geometric progression, to which those in- dices bdpng^ Thus, the indices 2 and S> beifig* zsMfii. to- gether, make 5*; and the numbers 4t and .8, on the termts corresponding to those indiees) being multiplied togetherj tnake 32, which is the number answering to. i^e itidex 5. " ■ ' • * In like manner, if any one index. b6 subtracted from another, the difierence willbe the index of that number which is equal to the quotient of the twa tenns to which those indices belong. Thus, the index 6> minus the index 4, is = 2' ; and the terms corresponding to those indices are 64 and 16, whose quotient is = 4, which is the number answering to the index 2* For the same reason, if the logarithm of any number be multiplied by the index of its power, the product will be equal to t^^e logarithm of that powef. Thus, the index or logarithm of 4, in the above series, i» 2; and . if this number be nmltiplied by 3j the product will be =s 6 ; which is the logarithm of 64^ or'Ae' third pDwer of 4.- •' . / -■ ' . .' - .>' And, if the Icgaritbm of any number bexliyided by the index of its root,.' the qtiOtient. yrill b&. eqiial to' the logarithm of that root. Thus, the iskdex . or logarithm of:.64 .is..6; and if this number be divided by 2, the quotient will b<e = 3; whiihris the logaritHm of 8, or the sqi^arerroot of 64. ' - * « . < The logarithms most convenient for jpriactice, are such as are adapted" to a geometric series iricreasmg in a tenfold pro- portion, as in the last of the above forms *; and are those which are to be found, at present, in most of the common tables on this subject. The distinguishing mark of this system of logarithms is, that the index or logarithm of 10 is 1 5 that of 100 is 2 5 that of 1000 is 3 5 &c* And, in L t, decimals. \ . 148 LOGARITHMB. ^•cimils, the logarithm of *1 b — 1 ; that of '01 i$^9i that of OOl is— 3} Slc* The log. of 1 being in every lystem. Whence it fellows, that the logarithm of any numbtr be- tween 1 and 10, must be and some fractioittl partem s^ that of a number between 10 and lOQ, wiU be 1 and sopie fractional parts \ and so on, fer any other number whatever. ^d since the integral part of a logarithm, usually called the Index, or Characteristic, b always thus readily feund, it is commonly omitted in the tables ; being left to be suppiied by the operator himself, as occasion requires. Another Definition of Logaritiuns is, that the logarithm of any number is the index of that power of some other nmn^ her, which is eaual to the nven number. So, if there be N zs #*, then i» is the log. of M ; where n may be either pcK sitive or negative, or nothing, and the root r any number whatever, according to the different systems of k^sirithms. When nis TzOf then N is ss 1, whatever the value of r is-^ which shows, that the k)^. of 1 b always 0, in every systepi of logarithms. When jiis ::= 1. thenN b =? r/ so thatthe radix r b always that number whose log. b 1, in every system* When the raduL ris s 2*7l8281828<b59 &<:, the indices n are the hyperbolic or Napier's log* of the i^umtMurs - N; so that n is always the hyp. log. of the number N or (2-718 &c.)^ « But when die ra£x r is s 10, thaa the index n becomes the common or Briggs's log. of the number N: so that Ihe common log. of any number 10" or N, b n the index of thsit power of lO which is equal to the said number. Thus 100, being the second power of 10, will have 2 for its loga^ rithm i and 1000, being the third power of 10, will have 9 for its logarithm: hence also, if 50 be s: iO'-^^'^^ then b 1*69897 Ae cofdmon log. of 50. And, in^genentl, the following docuple series of terms, viz. 10*, lO', 10*, 10% 10*, 10-^, 10-*, 10"», lOr-*, or 10000, 1000, 100, 10, 1, -1, -01, -OOl, -OOOl, have 4, 8, 2, 1, 0, —1, —2, — S, —4, for their logarithms, respectively. And from this ^alf.of numbers and logarithms, the same properties itosily follow; ^ as above mentioned. I'ROBLSM* LOGARITHMS. H9 PROBLEM. To compute the Logarithm to any of the Natural Numbers 1, 2, 3, 4, 5, isfc. RULE 1*. Take the geometric series, 1, 10, IGO, 1000, 10000, &c. And apiply to it the arithmetic series, 0^ 1, 2, 5, 4, &c. as logarithms.— Find a geometric mean between 1 and 10, or between 10 and 100, or any other two adjacent terms of the series, between which the number proposed lies.-^Ii^ like manner^ between the mean, thus found, and the nearest ex- treme, find another geometrical mean ; and so on, tilli you arrive within the proposed limit of the number whose loga^ rithm is sought. — ¥ind also as many arithmetical means, in the same order as you found the geometrical pnes, and these will be the logarithms answering to the said geometrical means. EXAMPLE. Let it be required to find the logarithm of 9. Here the proposed number lies between 1 and 10. First, then, the log , of 10 il 1, and the log. of 1 is ; theref. ^_+0 -f- 2 = -J- ae -5 is the arithmetical mean, and v/ 10 X 1 = V 10 — 3*1622777 the geom. mean j hence the log. of 3-1622777 is '5. Secondly, tl ie log, o f 10 is 1, and the log. of 8-1622777 is 'B\ theref. 1 -f '5 ^ 2 = '75 is the arithmetical mean, and -v/10 X 3-1622777 = 5*6234132 is the geom. mean J hence the log. of 5*6234132 is '75. Thirdly, the log, of 10 is 1, and the log. of 5-62341 32 is -75 -, theref. 1 + '75 -r- 2 = -875 is the arithmetical mean^ and v^ 1 X 5-6235 1 32 =» 7 -4989422 the geom. mean ; hence the log. of 7-4989422 is -875. Fourthly, tfa elog. of lO ils 1 , and the log. of 7-49*9422 is '875 i theref. 1 -f -875 4- 2 =s -9375 is the arithmetical mean, and V iO X 7*4989422 » 8*6596431 the geonr. mean| hence the log. of 8*6596431 is *9S75. » *^M*4M4 * The reader who wishes to inform himself more particularly coBceming the history, Bature, and constraction of Logarithms, may oonsoll the introdnction to my Math^malical Tables, lately published, where he will §nd his curiosity amply gratiled. Fifthly, 150 LOGARITHMS. Fifthly, the l og.oflOi sl, and the log. of S*659643 1 is -9375; theref. 1 +'9375 ^-g = '96875 is the arithmetical mean, and v'lOx 8-65964'31 = 9*3057204 the geom. mean 5 hence the log. of 9*3057204 is -96875. Sixthly, the log. of 86596431 is -9375, and the, log. of 9- 30 57204 is '96875 ; theref. '9375 + '96875 -^ 2 = '95 3125 isthearith.mean, and -/8'6596431 x 9-3057204 = 8-9768713 the geo- metric mean ; hence the log. of 8-9768713 is '953125. And proceeding in this manner, after 25 extractions, it tri?l Ije fbimd^hat the logarithm of 8-9999998 is -9542425; which may be taBen for the logarithm of 9, as it differs so little from it, that it is sufficiently exact for all practical pur- poses. And in this manner were the logarithms of almost all the prime numbers at first computed. RULE II*. ' Let h be the number whose logarithm is required to be found ; and a the number next less than ^, so that h'^azz.Xy the logarithm of a being known \ and let s denote the sum of the two numbers a ^ h. Then , 1. Divide the constant decimal -8685889638 &c, by /, and reserve the quotient : divide the reserved quotient by the square of /, and reserve this quotient : divide this last quotient also by the square of /, and again reserve the quotient : and thus proceed, continually dividing the last quotient by the square of /, as long as division can be made. 2. Then write these quotients orderly under one another, the first uppermost, and divide them re-pectively by the odd numbers, 1, 3, 5, 7, 9, 8cc, as long as division can be made; that is, divide the first reserved quotient by 1, the second by S, the third by 5, the fourth by 7, and so on. 3. Add all these last quotients ^together, and the sum will be the logarithm of h -=-. a; thereJFore to this logarithm add also the given logarithm of the said next less number a, so will the last sum be the logarithm of the number h proposed* f For the demonstration of thi^ rule, s^ npy Mathematics Tables, p. 109, &c. / ' ' That LOGARtTHMS, J51 . That IS, /; 111 Log. oi b IS log. a'\--jx {1+ "57+ ■57 + '77^ + ^^'^ where n denotes the constant given decimal '8685889638 &c. EXAMPLES. Ex. 1 . Let it be required to find the log. of the number 2, Here the given number b is 2, and the next less number a is 1, whose log. is 0; also the sum 2 -|- 1 = 3 zz /, and its square ,s* = 9. Then the operation will be as follows : 3) •868588964 1 ) •289529654 ( [ -28952965* 9) •289529G54 3 ) 32169962 ( ; 10723321 9 ) 32169962 5 ) 3574440 ( [ 714888 9 ) 3574440 7) 397160 ( : 56737 9) 397160 9) 44129 1 ; 4903 9 ) 44129 11 ). 4903 ( [ 446 9 ) 4903 13 ) 545 '( ; 42 9) 545 15 ) 61 ( 4 9) 61 1 » " log. of T - -301029995 add log. 1 log. of 2 - 000000000 - -301029995 Ex. 2. To compute the logarithm of the number 3. Here 3 = 3, the next less number ^ = 2, and the sum « + ^ sr 5 = J, whose square s^ is 25^ to divide by which, always multiply by '04. Then the operation is as follows: 5 25 25 25 25 '^5 •868588964 173717793 6948712 277948 11118 445 18 1 3 .5 "7 9 11 173717793 6948712 271948 11118 445 . 18 log. of 4 - log. of 2 add •173717793 2316237 55590 1588 50 2; '176091^60 3Q1029995 log. of 3 sought -477121255 Then, because the sum of the logarithms of numbers, gives the logarithm of their product ; and the difference of ^he logarithms, gives the logarithm of the quotient of the numbers ; 15? LOGARITHMS. pumbers ; from the above two logarithms, and the logarithm of 10, which is 1, we may raise a great many logarithms', as in the following examples ; EXAMPLE S. Because 2x2 = 4, therefore to log, 2 - •30102!:'995| add log. 2- '301029995 J t^mmrn^^ I. ■ ■ ■ sum is log. 4 -6020599911 FXAMPLE 4. Because 2 x 3 = 6, therefore to log. 2 - -301029995 add log. 3. -477121255 sum is log. 6, '778151250 EXAMPt^ 5. Because 2^ = 8, therefore tnulu by 3 301029995^ EXAMPLE 6. Becayse 3* = 9, therefore log. 3 - -4771212541^ muk. by 2 2 gives log. 9 954242 09 EXAMPLE 7. Because V* = 5, therefore from log. 1 1 OOOOr^OOOO take log. 2 -3010299951 leaves log. 5 -69897000411 EXAMPLE ^r Because 3x4 = 12, thereferer to log. 3 - -477121255 add log. 4 -602055(991 gives log. 8 -903089987 gives log. 12 1-0791 S124« ■Wi W TT And thus, computing, by this general mle, the logarithms tb the other prim^ numbers, 7, U, 13, 17| 19, 23, &c, and then using composition and division, we may easily find as many logarithms as we please, or may speedily examine any logarithm in the taUe *• f^^^i^^mr^ TT^ >^»» * Th^te are, besides these, many other ingenious methods, which later writers have discovered for ^ding the logarithms of numbers, in a much tasiet way than by the ori^nal inventor ; bat> as ^bey cpoodt ^ u|iderstood w|thpci| a knowledge of some of the higher branches of t|ie mathematics^ it is thought proper to omit thefn» ana to r^fer the reader to those works which are written expressly on the subject. It would likewise much exceed the limits of this cbfnp^pdium, to point out all the peculiar artifices that are made use of for constructing an entire table of these num- bers } but any information of this kind^ which the learner may "ivtsh to obtain^ may be found in my Tables, before mentioned. Descrtftiw, LOOARlTHliMS. US Vescripiim and Use rfthe Tablr j/^Logae^thms. HavinC explained the manner of forming a table of the logarithms of numbers, greater than unity; tne next thing to be done is^ to show how the logarithms of fractional quan- tities may be found. In order to this^ it may be observed, ' that as in the former case a geometric series is supposed to increase towards the lefty from unity, so in the latter case it is supposed to decrease towards the right hand, still be- ginning with unit ; as exhibited in the general description, page 148, where the indices being made negative, still show the logarithms to which they belong. Whence it appear^^ that as -h 1 is the log. of 10, so — I is the log. of Vrr or '1 > and as + 2 is the log. of 100, so -- 2 is the log. of ^w or ^01 : and so on* Hence it appears in general, th^t all numbers which con« sist of the same figures, whether they be integral, or frac- tional, or mixed, will have the decimal parts of their loga* rithms the same, but diSering only in the index^ which will be more or Jess, and positive or negative, according to the place of the first figure of the number. Thus, the logarithm of 2651 being 3-423410, the log. rf tVj or i^y or twyj ^c, part of it 5 wil^ be as'fbllows : Numbers. Logarithms. 2 6 5 1 3 -4 2 3 4 1 2 6 5-i 2-423410 2 6-51 1-423410 2*6 5 1 0-423410 •2651 -1-423410 •0 2 6 5 J -2-423410 2 6$! -3 -423410 Hence it abo appears, diat the ixKlex of any logarithm, i% ftlways less by 1 than the number of integer figures which the natural number consists of; or it is equal to ihe distance of the first figure firom the f^e of iinit% ox first place of in- tegers, whether on the kft', cnr on the right, of it : and this index is constantly to b^ placed on the left-hand side of the decimal part of the logarithm^ When there are integers in the given number, the index IS always afiirmative ; but when there are no integers, the index is negative,' and b to be marked by a short line drawn l^efore it, or else above it. Thus, A number having 1, 2, 3, 4, 5, &c, integer places, l;|ie index of it& log. is 0, 1, % 3, 4| &c. or 1 less than those * places. ^ And 15* LQGAmTHMS. And a decimal fraction having its first figure in the 1st, 2d, Sd, '4th, &c, place of the decimals, has always — 1, —2, —3, —4, &c, for the index of its logarithm, , It may also be observed, that though the indices of frac- tional quantities are negative, yet the decimal parts of their logarithms are always affirmative. And the negative mark( — ) may be set either before the index or over it. 1. TO nND, IN THE TABLE, THE LOGARITHM TO AKY NUMBER*. 1. If the given Number be less than 100, or consist of only two figures \ its log. is immediately found by inspection in the first page of the table, which contains all numbers from 1 to 100, with their logs, and the index immediately annexed in the next column. So the log. oi5 is 0*698970. The log. of 23 is V361728. The log. of 50 is 1 -698970, And so on. 2. y^ the Number be more than 100 but less than 10000; that is, consisting of either three or four figures; the decimal part, of the logarithm is found by inspection in the other pages of the table, standing against the given number, in this manner; viz. the first three figures of the given number in the first column of the page, and the fourth figure one of those along the top line of it ; then in the angle of meeting are the last four figures of the logarithm, and the first two figures of the same at the beginning of the same line in the second column of the page : to which is to be prefixed the proper index, which is always 1 less than the number of integer figures. So the logarithm of 251 is 2*399674, that is, the decimal '399674 found in the table, with the index 2 prefixed, be- cause the given number contains three integers. And the log. of 34*09 is 1-532627, that is^ the decimal -532627 found in the table, with the index 1 prefixed, because the given number contains two integers. . 3. But if the given Number contain more than four figures^; take out the logarithm of the first four figures by inspection in the table, as before, as also the next greater logarithm^ subtracting the one logarithm from the other, as also their corresponding numbers the one from the other. Then say» As the difference between the two numbers. Is to the difference of their logarithms, So is the remaining part of the given number. To the proportional part of the logarithm. I I .1 1 ' I ■■■■■'■ JL I * See ,tfce table pf Logarithms^ after the Geometry, at; th^ endi of this volume. * . Whigh LOGARITHMS. 155 'Which part being addejd to th« less logarichm^ before taken out, gives the whole logarithm sought very nearly. EXAMPLE. To find the logarithm of the number 34t*0926. The log. of 340900, as before, is 532627. And log. of 341000 * - is 532754. Thediffs.^are 100 and . 127 Then, as 100 : 127 : : 26 : 33, the proportional part. . This added to •- - - 332627, the first log. Gives, witii the index, 1 •532(360 for the log. of 34'0926. 4. If the number consist both of integers and fractions, or is entirely fractional ; find the decimal part of the logarithm the same as if all its figures were integral ; then this, having prefixed to it the proper index, will give the logarithm re- quired. 5. And if the given number be a proper vulgar fraction : subtract the logarithm of the denominator from the loga-, rithm of the numerator, and the remainder will be the loga- rithm sought ; which, being that of a decimal fraction, must always have a negative index. 6. But if it be a mixed number ; reduce it to an improper fraction, and find the difference of the logarithms of the numerator and denominator, in the same manner as before. EXAMPLES. l.Tofindthe.log. off J, Log. of 37 - 1-568202 Log. of 94 - 1*973 123 Dif. logl of f|. - 1-595074 Where the index 1 isnegative. 2. To find the log. of 17||. First, n4iJ=\V- Then, Log, of 405 Log. of 23 Dif. log. of 17i^ 2-607455 1-361728 1-245727 II, TO FIND THE NATURAL NUMBER TO ANT GIVEN LOGARITHM. This is to be found in the tables by the reverse method to the former, namely, by searching for the proposed loga- riit]im among those in the table, and taking out the corre- sponding number by inspection, in which the proper number of integers are to be pointed off^, viz. 1 more than the index. For, in finding the number answering to any given logarithm, the index always shows how far the first figure must 156 logarithms/ icnst be removed from the place of units, viz. to the Ifcfi; hand, or" integers, v^hcn tlxe index is affirmative; but to the-' right hand, or decimals, when it is negative. EXAMPLES* So, the number to the log. P532888 » 34*1K And the number of the log. 1*5S2882 is *34i 1. But if the logarithm cannot be exactly found in the table ^ take out the next greater and the next less, subtracting the one of these logarithms froiii the dther, as also their natural numbers the one from the other, and the less logarithm from the logarithm proposed. Then say> As the difference of the first or tabular logarithms^ Is to the difference of their natural numbers. So is the differ, of the given log. and the least tabular log. To their corresponding numeral difference. Which being annexed to the least natural number above taken, gives the natural number sought, corresponding to the proposed logarithm^ EXAMPLfi* So, to find the natural number answermg to the gives logarithm 1-532708. Here the next greater and next less tabular logarithms, with their corresponding numbers, are as below : Next greater 532754 its num. 34iOOO; given log. 532108 Next less ,532627 its num. 340900; next less 532627 Differences 127 — 100 — 81 Then, as 127 : 100 :: 81 : 64 nearly, the numeral difiTer. Therefore 34*0964 is the number sought, marking oflTtwo integers, because the index of the given logarithm is 1. Had the index been negative, thus 1 -532708, its ccmtc- sponding number would hxvt been *S4Q064, whoUy d«- cimaL MWLTIPLI- £ 157 } MULTIPLICATION, bt LOGARITHMS* Ta^e out the logarithms of tlie j&ctorfi from the table^ then add them together, and their sum will be the logarithm of the product required. Then, bj^ means of the table, take out the natural number, answering to the sum, for the product sought. Ob^i^ving to add what is to be carried from the ijecimal part of the logarithm to the affirmative index or indices, or else subtract it from the negative. * Also* adding the indices together yrhen they are of the samfs kind, bom affirmative or both negative ; but subtract^ ing the less, from the greater, when uie one is affirmative gnd the other negative, >nd prefixing the sign of the greater to the remainder. EXAMPLES* . 1. To multiply 23-14 by 5-062. . Numbers. Logs. 23.14 - 1'364.363 5-062 - 0-704322 Product 1 17*1347 2*068685 2. To multiply ^-531 92^ by 3-457291. Numbers. Logs* 2-581926 - 0-41 1^44 3-457291 - 0-538736 Prod. 8-92648 - 0-9506W X To mult. 3-902 and 597-16 ^4 '0314728 all together. Numbers. Logs. 3-902 - p-591287 - 597-16 - 2-776091 •03 14728- 2-497935 >ro4. 7S-3333 - 1-865318 9 Here the — 2 cancels the 2, and the 1 to carry from the di^cimals is set down. 4.To mult.3*586, and2-1046, and 0-8372, and 029.4-ali together. Numbers. Logs. 3-586 . 0'55461Q 2-1046 - 0-323170 0-8372 -1-922829 00294 — 2-468347 Prod. 0'1857618--l-268956 Here the 2 to carry cancels the— 2, and there remains thir I — 1 to set dawn . DITISIOK r 158 ] DIVISION BY LOGARrrHMS. RULE. • * « From the logarithm of the dividend subtract the loga- rithm of the divisor, and the number answering to the re- mainder will be the quotient required. Observing to change the sign of the index of the divisor, from affirmative to negative^ or from negative to affirmative^ then take the sum of the indices if they be of the same name, cr their difference when of different signs, with the sign of the greater, for the index to the logarithm of the quotient. And also, when 1 is borrowed, in the left-hand place of the decimal part of the logarithm, add it to the index of the divisor when that index is affirmative, but subtract it when negative ; then let the sign of the index arising from hence be changed, and worked with as before. EXAMPLES. I. Tt> divide 24163 by 4567. Numbers. Logs. Dividend 24163 - 4-383151 Divisor 4567 - 3-659631 Quot. 5*29075 0-723520 3. Divide -06314 by -007241. Numbers. . Logs. Divid. -06314 -2-800.'^05 Divisor -007241 —3-859799 2.Tpdivide37-149by523-7e. Numbers. Logs. Dividend 37- 1 49 - 1-569947 Divisor. 523-76 - 2-719132 Quot. 8-71979 0-940506 Herie 1 carried from the Quot. -0709275 -2-8508 15 4.Todivide-7438byl2-9476. ' Numbers. Logs. Divid. -7438 -1-871456 ' Divisor 12-9476 1-112189 Quot. -057447 - 2-759267 Here the 1 taken from •the decimals to th6 —3, makes it — 1, makes it become —2, to becopie— 2,whichtakenfroi^ set down, the other — 2, . leaves re- ' maming. ,' Note. As to the Kule-of-Three, or Rule of Proportion, it is performed by adding fhe logarithms of the 2d and Sd tcims, and subtracting that of the first term from their sum. INVOLUTION ( 159 ] mVOLUTION BT LOGARITHMS. RtTLE. ' • . • '" Ta«e out the logarithm of the given number from the table. Multiply the log. thus found, by the index of the power proposed. Find the number answering to the pro- duct, and it will be the power required. Ifdie* In multiplying a logarithm with a negative index, by an affirmative number, the product will be negative. But what is to be carried from the decimal part of the loga^ rithtn,.will always be affirmative. And therefore their dif- ference will be the index of the product, and is always to be made of the same kind with the greater. ^ EXAMPLES. 1. To square the number .2-5791. Numb. I'Og. Root 2'5791 - - 0-411468 The index - - 2 Power 6-65174 0-822936 2. To find the cube of 3-07146. Numb. Log. Root 3-07146 - - 0*487345 The index - - S Power 28-9758 1 -462035 8. To raise -09163 to the 4th power. Numb. Log. Root -09163 —2-962038 The index - - 4 Pow. -000070494 - 5 -8^8 152 Here 4 times the negative index being — 8,and 3 to carry, the difference — 5 is thp index of the product. 4. To raise 1 -0045 to the 365th power. Numb. Log. Root 1-0045 - - 0-001950 The index - - 365 9750 11700 5850 Power 5-14932 0*711750 EVOLUTIOK [ 160 i 'ZVOLXmON IT LOCillUtllMSl. Take the log: of the fiv^ pmmber out of the tabled. Divide the log. thus round by the index of the root. Thent dur number antveriag to the quotientj iTillbf the root. Hoii, When the index of the logsurithmj to be divided, W negative, and does not exactly contain the divisor, without some remainder, increase the index by such a number as will make it exactly divisible by the index, carrying the units bor-* rowed, as so many teni> to the left-hand place of the decimalf and then divide as in whole numbers* Ex. 1 .To find the square root of 365. Numb. Log. Power 365 2)2*562293 Root 19-10496 1-28114.64 "Ex. 3. To find the 10th root of 2. Numb. Log. Power 2 - 10 ) 0*301030 Root 1-071173 0-030103 Ex. 5. To find V' -093. Numb. Log. Power -093 t ) - 2-9684'»3 Root -304^959 ~ 1-4842414 Here the divisor 2 is con- tained exactly once in the ne gative index —2, and there- Fore the index of the quotient is —I. Exi 2. To find the 8d foot of 12345. . Numb. Log* Power 12345 3)4091491 Rpot 23-1116 1-363830J. Ex. 4. To find the 365th root of 1045. Numb. Log* Powerl'045 365)0^019116 Root 1-000121 0-0000524 Ex. 6. To find the ^-00048, Numb. Log. Power '00048 3)^4'681241 RcfQt -0782973 - 2-893747 Here thedirisor 3, not beipjBT exact- ly contained lA — ^, it is augmented by 2, to malw up 6, in which the di-* visor it contained just 2 times; then the 2, thus bon;owed, being ci^rriedt^ the decimal figure (>, makes 2^f which divided by d, ^ives 8, &c. Ex. 7. To find 3-1416 x 82 x fj. Ex. 8. To find -02916 x 751-3 X -^ Ex. 9. As 7241 : 3*58 :: 20-46 : ? Ex. 10. As v'724 : v^4| : : 6-927 : ? ALGEBRA. [ i«i J -■•;• • A L G E B R A; DEFINITIONS AND NOTATION. 1. xjLLCxEBRA i^ the scieoce of computing by sy^nhplju It Is sometimes also called Analysis ; and is a general kind of arithlnetic, or universal way of computation. 2. In this science, quantiti^^ of all kindaar^ re^es^ented by the letters of the alphabet. Ahd tbeo(}^rations' to be per* ^ibrmed with them, as addition or sjbibtractiDQ^ &C9 aiie ile* noted by certain simple character^ instead; of being ^i^e^s^d hy words at^ength. ._..... 3. In algebraical questions, s6mt quantities^ are kh^w^ oit' given, viz. those whose values are known: and others-un- known, or are to be found otit, vils. thos^ whos^ values Ve not known. The fonrter of these are represented by the leading letters of the alphabet, a, b^ r, dy &c \ and the lattef » or unknoi^ quantities, by the final lettei^, z> j^ at, u^ &<f.y 4>. The fiiaracters lis^sd to denote the opiorations^ av^ chiefly the JFoUowing : ' 1. + signifies addition, and is named //wi** " -^ signifies subtraction, and is named minus. , X or . signifies multiplication, and is named into^ . j -i- signifies division, and is named by* •/ signifies the square root ; ^ the cube root s ^ thi| 4th root, &c i and ^ the /7th root. : : : : signifies proportion. • . ss signifies equality, and is named 9^iud iff. And so on for other operations. Thus a^b denotes that . the number tepresented by.^ is to be added to that represented by tfk a — b denotes, that the number represented by j is to; be subtracted from that repiiresented by ii. aKn b denotes the difference pf a ahd ^, when it, id not known which is the greater. - Vot. I. M n^, oli* 162 ALGEBRA. aBf or a X if or a.i^ expresses the product^ by multipli- cation^ of the numbers represented by a and i. # -r i^, or-T-, denotes, that the number represented by # is to be divided by that which is expressed by i. a : b :: e : dy signifies that tf is in the same proportion to t, as r is to d» X =^ a -^ b -i^ CIS 7kn equation, expressing that x is equal to the diiSerence of a and by added to the quantity r. ^a, or M^, denotes the square root of « j ^a, or ^i^, the cube root of ^ ; and ^^a^ or a^ the cube root of the square of « ; zho ^y or sT^f is the Mth root oia\ and ^a* or tf« is the iith power of the mth root of a, or it is a to the ~ power. d'^ denote* the square of as n^ the cube of ^ ; o^ the fotuth p0wer oi a.: and if^ the Mth power of a, d + ^ >^ r, or (rt 4- i^) <•, denoted the product of the compound ^uc^tit]^ n^h multiplied^ by the ^i^le quantity r. U»ng the bar , or the parenthesis ( ) as a vi^culum^ to codtieGt •Several aimjple quantities into one compound. / «.+ i-ra — ^*or-7 — y, expressed like a fraction, means flftft quotient oi<jL\ it.divided by a^b. i/a^-^^cd^ rov {fib^ + 4ri)^9 is the square root of the com* I. ,1. I poixn4 quantity /l^4«r^. And^v^^ + ^^f or r («^ + c£ff denotes the product'of c into the square r<>ot of the coxnpocUid quantity ^3 + r^. .,. • fl H- ^ — r , or (^j + ^ -^ f)', denotes the cube, or third power, of the compound quantity a -f ^ — r. 3tf denotes that the quantity a IS' to be taken 3 times, and 4 (o 4- ^) i^ 4* timed ^i + ^ And these numbers, 3 or 4y showing how often the quantities are td be taken, or multi- plied, are called Co-efficients. • . Also \x denotes that x is multiplcd by ^ ; thus f x * or - 5. Like Quantities, are those which consist of the same letters, and powers. As a and 3a ; ch* 2db and ^ab ; or 3flf*Ar and -r bs^bc^ 6. Unlike Quantities, are those which consist of different letters, or dffierent powers. As a and ^^ or 2tf and d^\ or %§ff' and %abc. • - 1. Simple DEFINITIONS' ANiJ NOTATlOiJ. iss it* Simple Quantities, are those which consist of On^ term onjy. As Say or Saby or Saic^ 8. CompQund Quantities, are those which consist of two qj* more terms. A& « 4- ^, or 2<i — Sc, or a + 2b -^ Sf. 9. And when, the compound quantity consists of tw^ terms, it is called a Binomial, as a + ^} when of three term^ ^ k is a Trinomial, as tf + 2^ ~ 3^r ; when of foiir terms^ a Quadrinomial, as 2a '^ $b -^ /^ -^ 4d \ and so - on. Also, a Multinomial or Polynomial, consists of many tefmsi 10. A Residual Quantity, is a binomial having one of the terms negative. As a -^ ^^. 1 1 . Positive or Affirmative Qu^titiesj are those which are to be added, oi' have the sign +. As a or + ^j, or ^J : for when a quantity is found without a sign, it is understood tq be positive, or have the sign +- prefixed. 12. Negative Quantities, are those which ^ are to be sub** tracted. As — ii, or -'2tf^, or -3ai% 13. Like Signs, are either all positive ( + ), (Jr all nega- tive (-). 14. Unlike Signs, are when some are positive ( *f ), and ©thers negative ( — ). 1 5. The Co-efficient of any quantity, as shown above, is the number prefixed to it. As 3, in the quantity Sab. 16. The Power of a quantity (a]y is its square {a^)f or cube («^), or biquadrate (^'*), &c; callfed also, the 2d power, or 3d power, or ^th power, &c. 17. The Index or Exponent, is the number which denotes the power or root of a quantity. So 2 is the exponent of the square or second power /»* ; and 3 is the index of the cube or 3d power ; and ^ is the index of the square root, a^ or v^^ 5 and \ is the index of the cube root, d^y or ^a, 18. A Rational Quantity, is that which Has no radical sign (-y/) or index annexed to it. Ag tf, or Sab. 19. An Irrational Quantity, or Surd, is that which has not an exact root, of is expressed by means of the radical sign v/. As v^ 2, or v^j, or^a^y or ai^. 20. The Reciprocal of any quantity, is that quantity in- verted, or unity divided by it. So, the reciprocal of tf, or a , I f. a , i — ,is — y and the reciprocal of "T" is — :• M2 21. The i64 ALGEBRA. 21 • The letters by which any simple quantity is expressed^ may be ranged according to any order at pleasure. So the product. of a and b,, may be either expressed by ah^ or ta ; and the product of a, i, and r» by either abcf or acb^ or hacy or bcof or cab^ or cba ; as it matters not which quantities are placed or multiplied first. But it will be sometimes found convenient in long operations, to place the several letters According to* their order in the alphabet, as abc^ which order adso occurs most easily or naturally ta the mind. 22. Likewbe, the several members, or terms, of which a compound quantity is composed, may be disposed in any order at pleasure, without altering the value of the signifi- cation of the whole. Thus, Sa — 2ab + Aabc may also be \\Titten 3fl -f ^abc — 2aby or ^abc -f 3 j — 2ab^ or — 2ab + 3« + ^akcy &C5 for all these represent the same thing, namely, ' the quantity which remains, when the quantity or term 2ab is subtracted from the sum of the terms or quantities Stf and 4ahc. But it is most usual and natural, to begin with a po* sitive term, and with the first letters of the alphabet. a SOME EXAMPLES FOR PRACTICE, In finding the numeral values of various expressions^ or combinations, of quantities. Supposing /7 = 6, and 4=5, and c = 4,. and rf = 1, anct / = 0. Then 1. Will a' + Zab-c^ =^ 56 + 90 - 16 = UQ. 2. And 2^3 -3^*3 -^ c^ = 432 -540 + 64 =—44. 3. And a^ X a-\- b-2obc = 3« x 11-240 =156. a^ 216 4. And r- + ^ =z — r- + 16 = 12 + 16 = 2S.. a -i- 3c 18" 5. Arid v" 2^7+7^ or 2ac + r]^ = y/64i = 8. 6. And x/c + ^ • \ =5? .+ — = 7. ^ 2ac ' + e- 8 / „ ., 1 ^'--/^'-^^ 36-1 _35_- 8. And s^h"- -ac^ V2ac + ^ = 1 + 8 = ^. 9. Andx/b^-ac + ^?^cT^ = ^25-24^ + S = 3. 10. And fl*^ + r - ^/ = 1^5. n. And 9/7^ - 10^* + r = 24. \ 12. And ADDITION. fSS fl'= tt. And — X rf =3 45. c 13. And-^X -7= ISi- c a ^ . ,flr + ^ if— 5 14. And 7- = 11^ c a 15. And 1-^=46. 16. And — X ^ = 9: 4^ ::r 17. And*-^x J— ^ = i- . ^ 18. And^i+^— r— rf = 8, 19. Andtf+^ — ^ — </ = 6»' 20. Andfl*r x J' = 144. 21. Andacd- d:=:2S* 22. And tfV + y'e + rf ^^^ ^-^ 23. And -1 X — ^ = 18J-. . 24. Andv^/i^- +^^"-i/tf* - ^* = 4-4936249. 25. And 3^^ + ^a^ - ^' == 292-497942. 56. And 4fl* — 3^1 ^a'--^ab = 72. ADDITION. Addition,^ in Algebra, is the connecting the qnaptities together by their proper signs, and incorporating or uniting into one term or sum, such as are similar, and can b^ united. As Sa + 2b — 2a = a + 2b j the sum. • The rule of addition in algebra, may be divided into three cases : one, when the quantities are like, and their signs like also ; a second, when the quantities are lite, but their signs unlike; and the third, when the quantities are unlike. Which are performed as follows*. CASE * The reasons on which these operations are fou tided, will rea- dily app^ar^ by a little reflection on the nature of the quantities to 16« ALOEBRA. CASE I. TVben tie Quantities are Liiey and have Like Signs. . « Add the co- efficients together, and set down the sum j after which set the common letter or letters of the like quantities, and prefix the common sign + or — • be added, or collected together. For, with regard to the first ex- ample> where the quantities are 3a and 5a, whatever a represents in the one term, it will represent the same thing in the other^ so that 3 tiroes any thiqg and 5 times the same thing, collected together, must needs make 8 times that thing. As if a denote a shilling \ then \\a is 3 shiirmgj5> and ba is 6 shillings, and their sum 8 shillings. In like manner, — 2a6 and — 7«^> or —2 times any thing, and —7 times the same thing, make —9 times that thing. As to the second case, in which the quantities are like, but the signs unlike ; the reason of its operation will easily appear, by reflecting, that addition means only the uniting of quantities to- g€Jther by means of the arithmetical operations denoted by their signs -(- and — , or of addition and subtraction ; which being of contrary or opposite natures, the one co-efficient must be sub- tracted from the other, to obtain the incorporated or united mass. As to the third case> where the quantities are unlike, it is plain that such quantities cannot be united into one, or otherwise added, than by means of their signs : thus, for example, if t^ be supposed to represent a crown, and h z shilling ; then the sum of a and b can be neither 2a nor 26, that is neither 2 crowns nor 2 shillings, but only 1 crown plus 1 shilling, that is a -f 6* in this rule, the word addition is not very properly used ; being much too limited to express the operation here performed. The business of this operation is to incorporate into one mass, or alge- braic expression, different algebraic quantities, as far as an actual incorporation or union is possible; and to retain the algebraic marks for doing it, in cases where the former is not possible. When we have several quantities, some affirmative and some ne- gative j and the relation of these quantities can in the whole or in part be discovered ; such incoi'por^^tion of two or more quantities into one, is plainly effected by the foregoing rules. It may seem a paradox, that what is called addition in algebra^ should sometimes mean addition, and sometimes subtraction. But the paradox wholly arises from the scantiness of the name giyen to the algebraic process; from employing an old term in a new and more enlarged sense. Instead of addition, call it incorporation, or union, or striking a balance, or any name to which a more ex- tensive idea may be annexed, than that which is usuj^Hy implied by the word addition ; and the paradox vanishes. Thus, f Thus, 2a added to 5a, mak;ps 8a. And —2ab added to — 7a^, makes — 9fl*. And 5a + 7^ added to la + Si, makes I2a + 10* M7 OTHER EXAMPLES IfOR PRACTICE. 3tf Sa I2a a 2a Sla \5z - Six - lix hxy 2hcf 5hxy kxy Shxy &xy 11 bxy 2z Sjt* + Sxy 2ax — 4; 22 x" + xy 4ax — J? 4z 2x' 4- 4.Afy «Af — Sy z 5x^ + 2afj? Sax — 5^ 5^ 4;v* ^ Sxy 7a* — 2y 15if* + 15*y l^ax — 15; Sxy X^xy 22xy 11 xy lixy ixy 12/ 7/ 2/ 4/ / 3/ 4/1 •^ 4# Ai - 5* ^a — * 3a - 2* 2tf - 7* 30 - ISacf - Sxy 23 — lOxY - 44ry 14 - 14*1 - 7xy 10 - 16;rt - 5xy 16 - 20Ar^ - xy Sxy — 3jp + 4ai Sxy -* 4* + 3a* 3iA[y •«" 5ac + 5a* xy — 2x + ab ifxy -^ ;if + lot i m» % > " > " ■ -^U^ i|ii . i m CASS \$$, ALGEBRA. CASl^ II. r tf^ifu tie Qjuantitiei ari Lite, but have Unlike Signs : Add all the affirmative co-efficients into one sum, and all the negative ones into another, when there ^e several of i kind. 'Then subtract the less sum, or the less co-efficient, from the greater, and to the remainder prefix the sign of the greater, and subjoin the common quantity or letters. So + 5a and — Sa, united, make + 2a. And — 5a and + Sa, united, make ^ 2a. m OTHEH EXAMPLES FOR PRACTICE. r- 5a + Sax^ -\- 4a + irax* + 6fl — Sax^^ — 3a — eax": -fa + 5ax^ + 8*3 + 3^ — 5x^ + 4y — 16*' + 5y + 3x^ - 1y + 2x^ — 2y + 3a — 2fl** - 8*3 -f i(5y — 3i»* + sby + 4tf* + f — 5a^ + 9by - 4ab + 12 ' / lO/i* — loby + lab ~ 14. + lOfl* — 19by + ab + $ + Ua^ — 2by - Sab - 10 ■ < I — Zax'^ + 10^ ax + 3;^ + ^^x^ + ^M^ — 3x/ax -^ y — ,5tf*^ I r + 5ax^ + 4x/ax + 4)^ + 2ax'^ - 6fl*^ — 12 V ax — 2y + 6^*^ CASE — / ADDITION. 1G9 CASE III. When the Quantities are Unlike. Having collected together all the like quantitlesj,^ as ia the two foregoing cases, set down those that are unlike> one aft^r another, with their proper signs. %xy 2ax Sxy 6ax SXAMPLES. — 40;'*+ 3xy + 4;r*— 2xy -3^7+ 4x* 4.r^— 8a:' 4^x— 130+30:^ 5x* +3/i-r+9jr' i Ixy -4jr^ + 90 '-2xy + Sax lax + 8 j:* + 7^^ 9r*/ -7x*)» + Saxy -4jr*j? • 14tfjr— 2.r* 5ax-^3xy 8/ -4aj' 3jr* + 26 9+lQV'/jjr-5; 2jr+ 7v/jrj> + 5j^ 5y+ 3^i7X*-4; 10— 4-v/Ar+4; • 4x*y — 6;r/ '+3/jr -Ix'y 4-v/x — , Sy 2v^a:y+14r Sx + 2j^ -9 + 3^jrjr 3/1* + 9 + a-^— 4 2/j -8+2/1*— 3j: 4a:f-2iJ*+18 —7 -12+ tf-3.r*-2y Add /f + i and 3/i — 5A together. Add Stf— 8a: and 3/i— 4a: together. Add 6a:-5* + /i + 8to — 5/i -4a:+4i-3. Add /I +: 2*-3r- 10 to 3i— 4/i + 5c + lOand 5*-r. Add fl + i and a— h together. Add 3/1 + i- 10 to c^d-^a and —4^ + 2^-3^—7. Add 3/1* + b'^-c to^*-3tf' + fc-4. Add ^' + ^V-*' tP aA* -/i^r + A*. Add 9tf— 8* + 10Ar-6//-7r + 50 to 2ar-3/i— 5r + 4A + W-10. SVBIE ACTION/ 170 ALGEBRA. SUBTRACTION. Set down in one line the first quantities from which the subtraction is to be made ^ and underneath them place all the other quantities comi>osing the subtrahend : ranging the like quantities under each other, as in AdditicMi. Then change all the signs ( + and — ) of the lower line, or conceive them to be changed ; after which, collect all the terms together as in the cases of Addition*. \ EXAMPLES* From 7tf*-3* 9x* - 4y + 8 SJcyS + 6jr— y Take 3a*- 8* 6x^ + 5;^ - 4 My-1 ^ 6jr-4y I r m. ^ i ' -i..i. .Ill Rem. 4a* + 5^ Sjt*— 19^+12 44:j^ + 4+ 12^ + Sy From 5xy—\ 6 4y*— Sj;— 4 — 20— 6j:-;5xjr Take -2ji'j^4- 6 2/ + 2j?+4 3xy-9j; + S^'2ay Rem. 7jry— 12 2/— 5;^-8 - 28 + St "Sa^y-^^ay -*~«— From 8jr*3f-f6 5^xy'{-2a:^xy lx^+2^x—l8 + Si Take -2x*j;+2 7 4/ xy -{- 3 - 2xy 9^-12 +5^+j: Rem. I * This rule is founded on the consideration, that addition and subtraction are opposite to each other in their nature and operation^ ^s are thje signs -h and — , by which they are expressed and repre- sented. So that, since to unite a negative quantity with a positive one of the same iiind^ has the effect of diminishing it> or subduct- ing an equal positive one from it, therefore to subtract a positive (which is the opposite of uniting or adding) is to add the equal negative qu'antity. In like manner, to subtract a negative quan- tity, is the same in effect as to add or unite an equal positive one. So that, by changing the sign of a quantity from -f to — , of from — to +, changes its nature from a subduottve quantity to an additive one j and any quantity is ip effect subtracted^ by barely changing its sign, 5xy MULTIPLICATION. 171 Sxy - 30 7^3 _2 {a + h) Sjt/ + QOax/{xy +^ 10) Ixy - 5Q 2;r=-4 {a + h) 4xy -j- V2a*y\xy + 10) From /J 4- ^> take «— ^. From 4ii + 4^, take 3 + <»• ' From 4tf — 4^, take 3a + 55. From 8^1—1 2jr, take 4/ar — Sjt. From 2jr— 4a~2^ + 5, take 8-5* + a + 6t. From Sa-^b + c-d- 10, take c ^^a-d, ' From 3a+h+ c-d- 10, takei- 10 4--3zi. From 2^l5 + Ir^—^c + be -by take ^a^'-c + b\ From tf3 + 35V + ab^-ahc, take b" + (^b^-abc. From l2x + 6a'-U 4<40, take 45 - 3<i + 4ar + 6i- 10. From 2j:— 3fl + 45 4-6^-50, take 9tf+^ +65-6r-4a From 6«-45- 12t: + 12jr, take 2x-8^ + 45-5^, MULTIPLICATION. This consists of several xases, according as the fattors ar? simple or compound quantities. CASE i^ When both the Factors are Simple Quantities : First multiply the co-eiEcients of the two terms together, then to the product annex all the letters in those termsj which will give the whole product required. Ngte ^. Like signs, in the factors, produce + > -suid unlike jsigns — , in th^, products, I {:XAMPLES, * That this rule for the signs is true, may be thus shown. 1. When + a is to be multiplied by + c; the meaning is, that -{- <2 is to be taken as many times as ther<^.are units in c ; and since the sum of any number of positive terms is positive^ it follows that -j- a X -}- c ip^kes -f- ac, % When 2 lOa 2b ALGEBRA. • EXAMPLES. - Sa la + 2^ -^c -ex i 1 1 20ab -Cab -2Sac -i'2iax 4ac -Sab Od'x 4x -2x^y — ifXy - xy -\2abc 36tf*jr* -6xy +4xy 9 -Sax 4x 1 — ax -6c +Sxy -4 — Sxyz — Sax CASE II. ' f ♦ When one of the Factors is a Compound Quantity ; Multiply every term of the multiplicand, or compound quantity, separately, by the multiplier, as in the former case; placing the products one after another, with the proper signs; and the result will be the whole product required. 2. When two quantities are to be multiplied together, the re- sult will be exactly the same, in whatever order they are placed $ for a times c is the same as c times a, and therefore, wjiep — a if to be multiplied by + c, or + c by — a : this is the same thing as taking - a as many times as there are units in + c ; and as the sum of any number of negative terms is negative, it foljows that — a X -|- c, or + '< X — c make or produce — ac. 3. When — « is to be multiplied by — c; here — a is to b« subtracted as often as there are units in c: but subtracting nega- tives is the same thing ^s adding affirmatives, by the demonstration of the rule for subtraction 5 consequently the product is c times a, •r + ac Otherwise. S^nce a —- a zr 0, therefore {a — a) X — c is also =: 0, because multiplied by any quantity, is still but 5 and since the first term of the product, or a X — c is 3: — ac, by the second case ; therefore the last term of the product, or — a x — c, niust be -{- ac, to make the sum iz 0, or — ac -^ ac zzQ', that is, — a X — c =: + ac EXAMPLES. MULTIPLICATION. ng EXAMPLES* Sa-Zc Sac- 4b 2a^-Sc+5 2a 3a be 10a*- •6ac • 12j7- 4a 2ac i 4xy X 9fl V - 1 2ab 2a^bc - Uc" + 5bc 25c -lb 4x-b + 3ab — 2a iab CASE III. When both the Factors are Compound Quantities; Multiply every term of the multiplier by eVery term of the mulriplicand, separately ; setting down the products one after or under another, with their proper signs ; and add the several lines of products all together for the whole product required. « + * Sx-i-Qy 2x^'\-xy--2f « + ^ 4x — by Sjt— 3y a^-k-ab I2x^+Sxy 6x^ -h Sx^y- 6x/ +fl4 + ** -IBxy—lOy"- - 6^*y - 3x/ + 6/ m^-\-2ab-\-b^ ISj;'*— 7xy-10f 6x^-3x^y-'9xy''^ey^ m-^-b x^-j^y a^-k-ab-^b — b ^^-{-y ' a -^b ^ :-_ -ab-b"^ +yx'' +/ - a^b - ab"" - b^ ^ ♦ - ^» ^*'-f2;!^*+/ a^ * * -b^ Notf> ~i 174 ALGEBRA. Note. In the multiplication of compound quantities, it is the best way to set them down in order, according to the powers and the letters of the alphabet. And in multiplying them^ begin at the left-hand side» and multiply from the left hand towards the ri^ht, in the manner that we write, which is contrary to the way of multiplying numbers. But in setting down the several products, as they arisei in the second and following lines, range them under the like terms in the lines above^ when there are such like quantkiee ; which is the easiest way for adding them up together. In many cases, the multiplication of compound quantities is only to be performed by setting them down one after another, each within or under a vinculum, with a sign of multiplication between them. As {a ^ b) x {a — b) x Sabf era -{■ b . a — i . Zab. exa;mples for practice. 1. Muhiply lOac by 2a. Am. 20d'c. 2. Multiply Sa^-2b by Zb. . Ans. 9a^b~6b\ 3'. Muhiply Sa + 2i.by 3a- 2b. Ans. 9a^-4l^. 4. Multiply jr* - ary + / by jr 4- ;>. Ans. a;^ + f. 5. Mukiply a^ -^ i^b + ab"- -^ P by a-^b. Ans. a*-K 6. Multiply a^ + ab + b''hYa''-ab + b\ 7. Muhiply 3j:^-'2^y + 5 byx^ + 2a:y'^6. 8. Multiply Sa^-2ax + 5jr* by Sd'-Ux-l:^. ' 9. Multiply Sx^ + 2jry + 3/ by 2:c^-Sxy + 3/. 10. Multiply d' + ab +b^bYa-2b. DIVISION. 4 \ Division in Algebra, like that in numbers, is the con^rse of multiplication j and it is performed like that of numberp also, by beginning at the Teft-hand side, and dividing all the p^rts of the dividend by the divisor, when they can be so divided ; or else by setting them down like a fraction, the dividend over the divisor, arid then abbreviating the fraction as much as can be done. This will naturally divide into the following particular cases. CISE DIVISION. 175 CASE I. When th^ Diwsor and Dividmd are both Simple QuantitUs; Set the terms both down as in division of numbers, either the 4ivisor before the dividend, or below it, li're the deno- minator of a fraction. Then abbreviate these terms as much as can be done, by cancelling or striking out all the letters that are common. :to them, both, and also dividing the one co-efficient by the other, or abbreviating them after the manner of a fraction, by dividing them by their common measure. ^ . Note.' Like signs in the two 'factors make + in the quo- tient ; and unlike signs make — ; the same as in multipli- cation *^ EXAMPLES. 1. To divide 6^^ by 3 J. Here Sab -f- 3tf, or Za ) Gab ( ox 6ab Sa " 2b. t 2. Also c-^c ^ — "^ 1 ; and abj: c -T- b.vy : atx a ^ bxy - ji • 3. Divide I6x*by 8^. Ans. 2jr. 4. Divide 12j*j:* by - 'id'x. Ans. — 4*r. 5. Divide — \5ay^ by ^ay. Ans. — 5j. 6. Divide ■- ISax^y by — Sdfjrz. . Ans. --- • • * Because the divisor multiplied by the quotient, must produce the dividend. Therefore, 1 . When both the terms are + , the quotient must be + ; be- cause + in the divisor X -f in the quotient, produces + in the dividend. 2. When the terms are both — , the quotient is also -f ; be« cause ^ in the divisor X + in the quotient, produces — in the dividend. 3. When one term is + and the other — , the quotient mast be —J because + in the divisor x — in the quotient produces — ki the dividend, or — in the divisor X + in the q^uotient gives -* Ih the dividend. So that the rule is general; viz. that like signs give -{-, and unlike signs give — > in the quotient. C4SE /» 176 ALGEBRA, CASE II. When the Dividend is a Compound Quantiify and the Divisor Simple one : Divide every term of the dividend by the divisor^ a$ in the former case. EXAMPLES. ab + b'' a + b 1. {ab + *^) -^2*,or --^ = -|- ^ia + ib. lOab + I5aa: ^, . „ 2. (lOab + I5ar) -r 5a, or j^ = 2* + 3.r. 30flfz-48z • . _ 3. (30^2 -48«) -r- z, or * = 30^-48. 4. Divide 6^?^- Stfo: +ahj 2a. 5. Divide 3^*- 15 + 6.r + Qa by 3r. 6. Divide 6fl^r + 12fl^J7-9fl'3 by 3tf*. 7. Divide. 10^*x— 15-r*— 25jr by 5^. 8. Divide lot^bc- iBacr" + $ad^ by - Sac. 9. Divide 15^ + 3ay- 18/ by 2U. 10. Divide -ZOd-b"- + 60^^' by-6a*.^ CASE III. , When the Divisor and Dividend are both Compound Quantities: 1 . Set them down as in common division of hurabers^ the divisor before the dividend, ttrkh a small curvecJ line between them, and ranging the terms according to the powers of some one of the letters in both, the higher powers before th6 lower. ^ 2. Divide the first term of the dividend by the first term of the divisor, as in the first case, and set the result in the cjuotient. 3. Multiply the whole divisor by the term thus found, and ' subtract the result from the dividend. 4. To this remainder bring, down as tnany terms of the dividend as are requisite ' for the next operation, dividing as . before ; and so »n to the end, as ih common arithmetic. Not~e4 F DIVISION. m Noie. If the divisor be not exactly contained in the divi- 'dend^ the quantity which remains after the operation is finishddy may be placed over the divisor, like a vulgar frac- tion, and set down at the end of the quotient, as in common arithmetic* EXAMPLES. tf*- ab a^-4? ^—2) a'— 6fl* + 12a-8 (fl*-4tf + 4 r + -8 ■8 az + + - 4<i- 4«- il «) a^ + «» (, + »' - N «3 P - • '.•^' Tot. I. N «+*) lis ALGEfeRA. 2^* a + x) a^-'ix* ( tf'-a'x + ff4P'-jr» *+* »* +v* • - 3«* — «**- «'«» /!»«». 1 -*»>- -«.'- 8** — ■2** EXAMPLES FOR PRACTltE. !. Divide u* + 4j;r + 4;ir* by a + 2;r. Ans. a + 2x. 2. Divide tf^-3fl*« + 3iia*-z' hy a— z. Ans. II*— 2/jz + »*• 5. Divide 1 by 1 + ii. Ans. 1 -tf + tf*-a' + &c, 4. Divide 12;j«*-ld2 by 3*-6. Ans. 4w» + 84r* + 16jr + 32* A. Divide «5-5fl** + I'Oa^*^ - .lOay + 5/j**- ** by n*- 24f* + F: Ans.. «' - 3^** + 3^** - ^. 6. Divide 4823-96tf«*— 6*0^2 + ISOtf' by 2a;— 3^. 7. Divide ^*-3**P»* + SS^x^^x^ by ^3-3*'x + 3**'^-«^ 8« Divide fl^— o;^ by ^ — x 9. Divide a' + 5a^x + Sax" + *' by a + x. 10. Divide a* + 4fl*J*- 32** by « + 2*. 11. Divide '24fl*-** by 3a- 2b. tSS! i^» i^i i M S -< ALGEBRAIC FRACTIONS. Algebraic Fractions have the same names and rules of operation, as numeral fractions in common aritjj^netic } as ..appears in the following Rules and Cases. CASE JfRACIIONS. 17SI CfSB Z. To Reduce a Mixed Quantity to ah Improper Fraction. Multiply the integer by the denominator of the fraction^ and to the product add the huinerator> dr cozmect it wit^ its proper sign, .+ of — ) tli^n the denominator beipg set under this sum> will give the improper friction required* 1* Reduce S4>.atid d — - t6 improper fractions. 3x5+4 15 +4 : 19 , ^ First, 5f ^ j-^ = — ^ = -J, the Answeir^ . ■ * ax x^b a^—i . . And. tf — -= — — r^— 2= the Answer. ^ X X X 2* Reduce a + — and a -^ t6 improper fractions; . o a fl* axb+d* at-^d" First, 4+-7- = — *-7 == t the Amwer. And, a — ^ — -^— = '^^ = the Answer. a . a a . 3. Reduce 5^ Jo an improper fraction^ Ans. y * 4. Redu(^e 1 to an improper fi^ction^ Ans. * ' fiC X 6i Reduce 2i» — — to an improper frractioh^ 4^ " ■ * 6. ite4uce 12 4 ^t to an impfopiir fraction* 7. Reduce * H — '- ^ to sin improper fraction* c 8« Reduce 4 ^2x ^ to an improper fraction' ' tASE It. To Redwte an Improper Praction to a Jf%Ie wr Mixed Quantity! DitiiJi; the numerator! by thcf denpniinator, for the in* te^al part ; and set the remainder, if any, oter the deno^^ xmnator, for the fractional part j the two joined together will be the mi^ed gji^antity required. ' 1$0 ALGEBRA. EXAMPLES. . 1. To reduce — and — -r^ to mixed quantities. 3 p First, y 5= 16 -r 3 = 54, the Answer required. And, f*±^= ^M^ -5- 4 = « + J. Answer. "2. To reduce !flzif! and 5fl±if! to mixed quantities. First, ^^^^^ = 2ar - 3^ ^ ^ = 2« - — . Answer. c c And, ^^"^^"^ = i^HK*^ -T-fl+or = 3a: + — . Ans. 3. Reduce — and -r- — — to mixed quantities. 3jr* Ans. 64> and 2* . 4. Reduce — - and -I--— to whole or mixed quan- 2a a-b tities. 3jr*^- 3/ • , 2j^^ - 2y3 - , . . 5. Reduce ^, and to whole or mixed quantities. 6. Reduce ■ — to a mixed quantity. 7. Reduce ^ to a mixed quantity. 3a3 + 2a*- 2a— 4? CASE III. To Reduce Fractions to a Common Denominator* Multiply every numerator^ separately, by all the deno^ minators except its own, for the new numerators ; and all the ^ denominators together, for the common denominator. When the denominators have a common divisor, it will be ^better, instead of multiplying by the whole denominators, to multiply only by those parts which arise from dividing by the common divisor. And observing also the several rules and directions as in Fractions in the Arithmetic* EXAMPLES. / FRACTIONS. 181 EXAMPLES. a - i 1. Reduce — and — to a common denominator. X z Here — and — = £- and — , by multiplying the terms of X % xz x% the first fraction by z, and the terms of the 2d by x* /I V b 2. Reduce , — , and — to a common denominator. X b c Here — , ---, and — = ^ — , — > and , by multiplying the X h c hex hex hex terms of the 1st fraction by hc^ of the 2d by cx^ and of the 3d by hx. I* 3. Reduce — and — to a common denominator* X 2c Ans. and — • 2ex 2eK 4. Reduce — anJ^^^^^I — to a common denominator. h 2c ifOe - , SaH-2i* *Ans. -rr—, and — -r . 2bc' 2bc. ' 5. Reduce — and — , and 4«f. to a common denominator* 3* 2? Ans, and and %cx Qex Qex 6. Reduce — and — and 2b + -7-. to fractions having a 6 4 b * common denominator. Ans. — r— and ^ and , . 243 243 243 7. Reduce — and and — III- to ^ commori deno- 3 4 fl+3 minator., 8. Reduce — and — and — - to a common denominator. 4a* 3a 2a CA8B n2 ALGEBRA, CASE IV. To find the Qriofist Common Meastfti rf the Tkrms jf # practian* Divide the greater term by the less, and thje last divisor by the last remainder, and so on till nothing remains ; then the divisor last used will be the common measure required 9 just the same as in common numbers. But note, that it is proper to range the quantities according to the dimensions of some letters, as is shown in division. And note also, that all'the Itoers or figures which are com- mon to each term of the divisors, must be thrown out of them, or must divide theip, before they are used in the operation. £3(AMPLBS. J. To find the greatest common measure of -^ rr. «* + **) ac" + i^ 6r m+b) ac" + 6^{c^ Therefore the greater Cbmmon measure isa + i. 2. To^nd the greatest common mea^r^e of ^ or a+ i )d' + 2aL+f(a+L ai + ^ Therefore <f + ^ is the greatest common divisor. 9« To find the greatest cozjunon devisor of-; — rrr, ao+29 Ans* «— 2« 4. Ta FRACTIONS. 183 4. To find the greatest common divisor of — ^ ^ ,^ « Ans. fl* - f. 5, Fiadtne greatest com. measure of ^ .t\ ^ •;r^ — t ^ % z ' CASE T. i^ I To Reduce a FractUn to its Lowest Terms* Find the greatest common measure^ as in the last pro- blem. Then divide both the terms of the fraction by tjie common nxeasure thus founds and it will reduce it to its lowest terms at once^ as was required. OriKvide the terms by any quantity which it may appear will divide them both, as in arithmetical fractions. \ EXAMPLES. ai + b* . 1. Reduce ■ . , ^ *o i^s lowest terms. or + or ab + i^) ac" + he* dra + b ) 0c* + be'{c' 0^ + ^V* Here ab + i^U divided by the common factor b» Therefore a + i^s the greatest common measure, and ab + b^ b hence a + b ) , , . = t*** the. fraction required- 2. To reduce ; .■ ^, — r-rr to its least terms. c^+2bc + b tf* + 2fc + *" ) r' - b^c ( c - 2*^V2*V) c' + 2bc + b^ orc + b) i^ + 2bc+b^(c + i ji'+ be be + b' bc + b'' Thtjrefore lU ALGEBRA. Therefore ^ 4- ^ is the greatest common measure) and hence r + *) , ^ = , , is the fraction required* S. Reduce -r — rr^ to Its lowest terms. Ans. ^ , , ^ ■ fl* — J* 1 4. Reduce — — rr to its lowest terms. Ans. ^ . ^ ^ fl* — ** 5. Reduce --r — ^ ,, , ^ ^ — n to its lowest terms. ^ 7. Reduce , . ^ . . ... to its lowest terms. a* + 2ai + b^ CASE vi. T^ tf^i/ Fractional Quantities together. If the fractions have a common denominator^ add 73\ the numerators together ; then under their sum set the common denominator, and it is done. If they have not a common denominator, reduce them to one> and then add them as before. EXAMPLES. T ^ J ^ 1. Let ~- and — be given, to find their sum. 3 * -T a A 4tf 3tf la Here -5- + t-"^ i^ ^ T^ ^ 75 '* ^'^^ ^^°^ required 3 4 12 12 12 .> 2i.nrl — 4'r\ 4\r\A 4-ViA«« ».«««« a b , c % Given -7-, — ^,anQ~7, to find their sum. PC a the sum required* 3. Lit ,-*\i\ FRACTIONS. IBS Sx* 2ax * 3- Let a r-and i H — ^ be added together. b c , ° ^x^ , Sax * 3rjr* , , 2a5j: ^ Here^-— +* + — =^-^^+*+->^j- 2abx~^3cx* - * = a + i H r , the sum required. ^. *., 4!X ^ 2x , . 20*j:+6flx 4. Add -r- and -rr together. Ans. — ■ ^ , ^. 5. Add ---J — and — together. Ans. ^a. ^ Ajj 2^—3 - 5tf , . . 9a — 6 6. Add and -e together, Ans. "^ 4 8^ , 8 7. Add 2a -J •— to 4a H ^ — ^. Ans, 6a + ,**na 5 '4 20 So" a+b 8. Add 6a, and -— r- and —-7- together. 5a 6a 3a + 2 9. Add — , and --- and — - — together. 3a a 10. Add 2a, and — and 5 + — together. , , . , , ^ , Sa - 5a 11. Add 8a + — and 2a— — together. 4 o CASE VH. * To Subtract one Fractional Quantity from another . Reduce the fractions to a common denominator, as in addition, if they have not, a* common denohiinator. Subtract the numerators from each othler, and under their difference set the common denominator, and the work is done. * In the addition of mixed quantities^ it is best to bring the fractional parts only^o a common denominator, and to annex theiv sum to the sum of the integers, with the proper sign. And the same lule n^ay be observed for mixed quantities in subtraction also. (XAMPUS. ^ 1S6 ALGEBRA. EXAMPLES. 1 . To find the difference of *f and ~. - 4 7 Hcrc^-l?=ii?^l^=,^is the dMTerence remiircd- . 4 7 28 28 28 £• To find the difference of and Here •4r 3^ 4c Sb I2bc \2bc 6tf?— 3ii— 124awr+ lefc . , ■»./*• • » — i ' IS the difference required. \2bc ^ 3. Required the difference, of — f and -^. 9 7 4. Required the difference of 6« and — . 4 5. Required the difference of — a^d — • 4 3 6. Subtract?* from ?l±f. c b 7.Take!l±.^fromll±i. 9 5 U. Take 2a^t^l^ from 4^i + £f. CASE Till. . Jb Multiply Fractional Qjianiities together* Multiply the numerators together for a new numerator, and the denominators for a new denominator*. * 1. When the numerator of one fraction, and the denominator of thf»-other^ can be divided byisome qqaptttyj which is common to both, the quotients may be used inAtfsad of them. a. When a fraction is to be n|ulti(^ie4 by an integer^ the pro- duct is found either by i^ultiplying the uum^i^tort or dividing th& denominator by it; and if the integer be the s^e with the deno- minator^ the numerator may be taken for the product. EXAMPLES. FRACnONB; i*r EXAMPLES. I. Required to find the product of --^ and — . Here -.— — r- = -r- 8x5 40 — the product required. 2d' 2. Required the'product of — , ^, and — . f^iliiiiif = i!f! = !i the i*oduct reqal^ ■ 3x4x1 «4 , J4r *^ 3. Required the product o£ — and • ^ ^ b 2a -^c Here i — - — i = ! the product requum. 4. Required the product of — and — . 5. Required the product of— and — - 4 3flf 6. To multiply -^, and — ^i and ---- together. 3r 7. Required the product of 2fl+ — and — , 2c b 8. Required the product of — «-- — »- and -^ — , Sbc a -{rb 9. Required the product of 8^, and ^ T , and ^ *" * 10. Multiply «+4 --^byar-iL + _?l. *^' 'Za 4<i* 24? 4j?* CASE IX. 2it Divide one Fractional Quantity by another. Divide the numerators by each other, and the denomina- tors by each other, if they will exactly divide. But^ if not^ then invert the terms of the divispr, and multiply by it exaetly is in muhi{£catiQ«i^. *'^^ — — ^^ — ^^^ — ^^^^ — ,^ — .._ ^ _ _, ^ ^^^^^ — ^_.,.^^^,^ ^^^.^ — . — ^__ — ^^.^ — .^ ^ 1. if tin £raedoot'to be divided have a csramon deomninator^ take Ibe mmefto r t£ tbe dividend for a new nuineratDO and the numeratQjT of the Avisos ibr the new denomMiAtor. 2. Wheii Ito ALGEBRA. EXAMPLES. 1- Required to divide — by-^. Sa 5c 2. Required to divide rr by t-v „ 3# 5r Sa 4d \2ad Sad , • 3a* 2a , 4. To divide ^ . .» by - — r-^. 3tf^ tf+i _ 8a*x(a+ ^) _ 3tf is the quotient r.equiredl 5. To divide --r by t*. 4 ^ 12 6. To divide — by 3x. 7. To divide — - — by — . 9 ^ o ' 8. To divide *- — - — : by -^. 2^:— 1' ' 3 , 4x Sa 9. To divide -r by -t- 5 ^ 5h 2a "b Sac 10. To divide —-J- by .-TT. 11 n- -^ 5a^^5b^ , 6a*+5a* 11. Divide ^ . ^ I 77 by r. 2a* - 4ai -h 24* ^ 4a - 4* i2. When a fraction is to i>e divided by any quantity, it is tho Wne thing whether the numerator be divided by it, or the deno- minator multiplied by it. 3. When the two numeratprs, or the two denominators, can be divided by some common quantity, Jet that be done, and thequo- tients used instead of the fractions first proposed. INVOLUTION. [ 18i> ] INVOLUTION. Involution is the raising of powers from any proposed root ; such as finding the square, cube, biquadrate, &c, of any given quantity. The method is as follows : * Multiply the root or given quantity by itself, as many tipes as there are units in the index less one, and the last pro- duct will be the power required.— Or, in literals, multiply the index of the root by the index of the power, and the result will be the power, the same as before. Note. When the sign of the root is +> all the powersof it will be 4- ; but when the sign is — , all the even powers will be .+f and all the odd powers — ; as is evident from multiplication. exampCes. «, the root tf* = square tf' z= cube tf * =r 4th power 41* = 5th power &c. — 2a, the root -f- 4^* = square '— 8/j^ = cube + 16^1* == 4th power J- 32a* = 5th power 2ax^ , the root + + 3A 4a^j?* =: cube 27*^ 1 6a*x* ■ I I «. 81** = 4th power. a*, the root a^ = square a^ = cube if = 4th power **°= 5th power &c. - Zab\ the root + 9a*i* = square - 27a'** = cube + Sia'^b* = 4th power. - 243a**'° = 5th power. , the root 2i ^ = square « W =: cube 16* - = biquadrate * Any powerof the product of two or more quantities, is equal to the same power of each of the factors, multiplied together. And an^ power of a fraction, is equal to the same power of the^p numerator, divided by the like [ibwer of the denominator. Also, powers or roots of the same quantity, are multiplied by one another, by adding their exponents ; or divided, by sobtract- ing their exponents. Hius, «^ X a* =««+*=; ff^. And a^^a« or --=: a = a. a* L 190 AIX;^£BRA. j:-'a = root iji+tfnrool ;r* + tfjr X 4- tf + aj^ +2fl*x + fl' «' + 3tfT* + 3i^x + a' jf*— 2ar + a* square s -^ a jc^— 2aj;* +<!** jr5-3Ar*+ Mx-a^ Ae cubes^ or third powers, olx^a and x + ^. EXAMPLES. FOR PRACTICE. 1. Required. the cube or 3d power of So'.' 2. Reqjuired the 4th powi^.of 2i7'j. 3. Re<|uired the 3d powej- of — 4iiV. 4. To find the biquadrat^ of — --^. 5. Retfuired the 5th power of a — 2x. 6. To find.the 6th power of 2d^. Sir Isaac ICiTewtom^s Rdle for raising a Binomial to any Power whatever *. 1. To find the Terms without the Co^efficiertts. The index of the first, or leading quantity, begins with the index of the given power, and in the succeeding terms decreases conti- nually by I, in every term to the last; and in the 2d or following quantity, the indices of the terms are O, 1, 2, 3, 4, &c, increasing always by 1. That is, the first teno will con-» tain only the 1st part of the root with the same index, or of — 1 -— — ' ' — ■■ I , ji ■ — ■ * Tbb ruW expressed in general tennsi is as follows t M 2 3 ^ ' 2 2 3 Nate* The sum of the co-e£ident»^ in every power, is equd to the number 2, wl^n raised to that power. Thus 1-4- 1 =: 2 in the first power ; I + 2 + 1 = 4 == 2^ in the square ; i 4. 3 4- 3 j+ 1 =: 8 =: 2^ in the cube, or third power ; and so on* ' the ItJVOLtfTIiDN. , -l$l I fllel^ame height as the inteticled 'power : and the kst teHn of the^ series will contain only the 2d part of the given teot, when raised aho to the same height'of the intended- p6wer: but an the other or intermediate terms will contain the ]pro^ ducts of some powers of both the members of the -root, in such sort, that the powers or indices of the 1st or leading •lembet will always decrease by 1» while those of the 2d member always increase by 1. 2; To find the Co^iffictents. The first co-eifficient is always 1, and the second' is the same as the index of the intended ' jpew^er ; to find the 3d co-efficient, multiply that tA die ^d term by the index of the leadkig letter in the sameterm^ and divide the product by 2 ; and so on, that is, multiply the co- efficient of the term last found by the index of the leading quantity in that terhi, and divide ^he product by the number of terms to that place, and it will give the co-efficient of the term next following; which rule will find all the co-efficients^ one after another. , Nate. The whole number of terms Will be 1 more than the index of the given power i and when both tel^ms of the root are +, all the terms of the power will be + ; but if the '^e-r cond term be — , all the odd terms wilt /be' +,- afed all the even terms — , which causes the terms to be + and — alter-* nately. Also the sum of the two indices, in each term, is always the same number, viz. the index of the required power: and, counting from the middle of the series, both ways, or towards the right and left, the indices of the two terms are the same figures at equal distances, but mutually changed places. Moreover, the co-efiicients are the saihe numbers at equal distances from the middle of the series, towards the right and left ; so by whatever numbers the increase to the middle, by the same in the reverse order they decrease to the end. ' tiAMPLES. \\ Let d-^^x be involved to the 5th power. ' The terms without the co-efficients» by the ist rule, #iU be a\ it%, c^x^y c^x^y #jr*^ jt^ and the co-efficients, by the 2d irule, will be . , ^5x4 10X3...1P„x2 ^>^\ l.^> 2 ' "^^^ 4^5^ or, 1,5, 10, 10, 5', \s Therefore the Ath power altogether is #5 + Ba\v + 10#»^* + lO^^jr^ + Sax^ + x\ But y 192 ALGEBRA. But it is best to «et down both the cOhefficiehts and the powers of the letters at once, in ope line, without the inter- mediate lines in the above example^ as in the example here below. ^ 2, Let fl — X be involved to the 6th power. The terms with the co-efl5cients will be a^-^ea^x + 15tf*x*- 20^x3 + ISd'j^-Gax^ + jt*. • S. Required the 4th power of ^ — jr. Ans. fl* - ^x + e(^x^ - ^ap^ + «♦. ' And thus any other powers may be; set down ^t once^ in the same manner ; which is the be$t way. EVOLUTION. Evolution is the reverse of Involution^ being the method" of finding the square rootj cube root> &c>. of any given quantity, whether simple or compound. CASE I. To find the Roots of Simple QuMntities. Extract the root of the co-efficient, for the numeral part \ and divide the index of the letter or letters, by the mdex of the power, and it will give the root of the literal part ; then annex this to the former, for the whole root sought*. ■■iiiUi * Any even root of an affirmative quantity^ may be either 4- or — : thus the square root of + c^ is either -^ a,ov — a ; be« cause + a X + a = + a*, and —a X — a r= + a^ also. But an odd root of any quantity will have the same sign as the quarittty itself: thus the cube root of -f a* is -f a, and the cube root of— a^is — ajfbr + ax +ax+a=:+«^, and —a X — fl X -^ azu -- a^. Any even root of a negative quantity is impossible; for neither + a x-^a, nor —a X — a can produce —a*. Any root of a product, is equal to the like root of each of the fBictors multiplied together. And for the root of a fraction, take ^he root of the numerator, and the root of the denominator. EXAMPLES. EVOLUTION. 19.S EXAMPLES. 1. The square root of 4fl*, is 2/i, 3* 2. The cube root of 8«^ is 2a^ or 2a. 3. Ihe square root of "^^> ^'' V^"5^j js ^\/5. 4. ' The cube root of ' j , xs — r — ^2^, 5. To find the square root of 2^***. Ans. fli*v^- 6. To find thfe cube root pf — 64«^^ Ans. -4a**. 7. To find^the square root of-^-r-, Ans. 9,abA/^* 8. To find the 4th root of^U^b^. Ans. 3aby/b. 9. To find the 5th root of - 32a*^. Ans. —^ab^b. CASE II. To find the Square Root of a Compound Quantity* This is performed like as in numbersi, thus : i . Range the quantities according to the dimensions of one pf the letters^ and set the root of the first term in the quotient. ^ , 2. Subtract the square of the root, thus founds from the first term, and bring down the next two terms to tike re- mainder for a dividend) and take double the root for a divisor. 3. Divide the dividend by the divisor, and annex the re- sult both to the quotient and to the divisor. 4. Multiply the divisor, thus increased, by the term last set in the quotient, and subtract the pnx^uct from the dividend. And so on, always tlie same, as in common arithmetic. EXAMPLES. 1 • Extract the square root of j*- ^Q}h + e^j*i* -- 4o3* + *♦. a'^-^^a^ + ^a^b^-^d^ + ** ( fl*-2fli + i* the root. 2<l* - fitf* ) - 4tf ^^ + 6<i**** \ Vot. I. O 2. Find 1 194 ALGEBRA. < ^ 2. Find the root of a^ + 4^a^b + lOaH"^ + l2aP + *♦. 3. To find the square root of fl*'+ 4^' + 6«' + 4* + 1. Ans. fl* + 2« + l. 4. Extract the square root of a^ — 2a^ + 2£^—a + ^. " .Ans. a:*— jr +4. 5. It is required to find the square root of a* --at. CASE III. To find the Roots of any Powers in Genera/, This is also done like the same roots in numbers, thus t Find the root of the first term, and set it in the quotient. — Subtract its power from that term, and bring down the second term for a dividend. — Involve the root, last found, to the next lower power, and multiply it by the index of the given power, for a divisor. — Divide the dividend by the di- visor, and set the quotient as the next term of the root. — Involve now the whole root to the power to be extracted ; then subtract the power thus arising £rom the given power, and divide the first term of the remainder by the divisor first found; and so on till the whole is finished^. EXAMPLES. * As this method, in high powers^ ipay be thought too labo- rious, it will not be improper to observe, that the roots of com- pound quantities may sometimes be easily discovered, thus': Extract the roots of some of the most simple terms, and connect them together by the sign + or — , as may be judged most suit- able for the purpose. — Involve the componnd root, thus found, la- the proper power -, then, if this be the same with the given quan- tity, it is the root required. — But if it be found to differ oiily in some of the signs, change them fcom + to — , or from — to +, till its power agrees with the given one throughout Thus, EVOLUTION. 195 EXAMPLES. i. To find the square root of a*--2a^t+Sa'k'^2aP + i\ — — ■ I . I. ■■ • !■ I 2. Find the cube root of a^-^a^ + 21a^ - 44a' + 63^*- - 54fl + 27. f«-6^|i + 21a*-44tf'+ 63fl*— 54fl + 27 ( d'-2a + $. 3tf^)-6a5 ii* - 6^5> i2fl* - 8fl' = (^~2j)3 a**-6^* + 21fl*-44fl5+63a*-54a +27 = {a^^2a-3y. 3. To find the square root of «* - 2ab + 2ax + ^* - ^x + j^. . Ans. a—b + j:. 4. Find the cube root of a^ - Sn* + 9fl^-13/i' + ISa*- 12a +,8. Ans. fl*-tf + 2. 5. Find the 4th root of 81a* - 216a'* + 216a*** - 96a*» + 16*^ Ans. 3a -2*. 6. Find the 5th root of a^ - 10a* + 40a'- 80a* + 80/i — 32. . Ans. a— 2. !• Required the square root of 1 — :r*. 8. Required the cube root of l—x'. i«i«i Thusy in the 5th example^ the root 3a— 2*, Is the differenpe of the roots of the first and last terms ; aad in the 3d example, the root a—* + x> is the sum of the roots of the ist^ 4th^ aod 6th terms. The same may also be observed of the 6th example, where the root is found from the first and last terms. O 2 SURDS. 196 ALGEBRA. SURDS. SuKDs are such qutntkics m ha^« no exact rbot; 4nd are usually expre^ed by fractional indices, or by means of the radical sign a/. Thus, 3^, or ^/S, denotes the square root of S ; and 2^ or i/2S or V*, the cube root of the s^are of 2 ; where the numerator shows the power to trhich the qutotity is tb be raised, and the denoniinator its root,. pHoblem I. Ta Reduce a Rational Quantity to the Form of a Surd. Raise the given quartity to the power denoted by the. index of the surd ; then over or before this nevr qiljiatity set the radical sign, and it will be of the form required. EXAMPLES. J • To reduce 4 to the form of the sqyare root. First, 4* = 4 x 4:=16; then Vl6 is the answer. 2. To reduce Sa^ to the form of the cube root* First, 3^' X^d" X Za"- = {^a^f == 27a% then ^'Itf or (27/1^)"^ is the answer: 3. Reduce 6 to the form of the cube root. Ans. (216)7or^l0. 4. Reduce \ab to the form of the square root. Ans. /fiV. 5. Reduce 2 to the form of the 4th root. Aiw. (U)^. I 6. Reduce a'^ to the form of the 5th root. 7. Reduce a-^-x to the forhi of the square root. S. Reduce a—x to the form of the cube root, PROBLEM II. ' * I. " To Reduce^ Quantities to a Common Index. 1. Reduce the indices of the gi^eh quantities to a com-i- mon denominator, and involve each of mem to the power denoted by its numerator ; then 1 set ov^ the common ^^ npminator will form the common index. Or, 2. If I t SURDS. 197 2. If tht common index be given, divide the indices of the quantities by the given index, and the quotients will be the new indices for those quantities. Then over ' the $sud quantities, with tbeir new indices, set the given ind^x, and they will make the equivalent quantities SQughiji EXAMPLES. I 1 ' 1. Reduce 3^ and 5'^ to a common ind^x. Here 4 and I = ^^ and -1%. Therefore 3^ and 5tI=(3s)tV ^nd {5^)^=^'^S^ and '^S* = '°/243 and ^^25. 1 2. Reduce (^ and AT to the same common index -J* Here, 4.4-4 3=-|-x-^=3f the 1st indes^ and 1^4 = 1- X 4^ = f the 2d index. Therefore (tf®)^ and (H^)^, or x^cf and ^S^ are the quan- tities. 3. Reduce 4^ and 5^ to the common index ^. Ansl 2567)Tand2S^. II 4. Reduce d^ and x^ to the common index 4- Ans. (fl*)^and(.r^)"«'. - 5. ' Reduce a^ and .r^ to the same radical sign. Ans. iy/fl* and a/x^* 6. Reduce (a + x)'^ and (a— j?*)^ to a common index. I I 7, Reduce {a + ^)^ and (a— ip to a commoti mdex. PROBLEM III. , T& Reduce Surds to more Simpte Terms. Find put the gre^te^t power contained in, or to divide the given surd 5 take its root, and set it before the quotient pr the remaining quantities, with the proper radical sign b^ tween them. EXAMPLES* 1. To reduce -v/82 to simj^er terms. Here v^32=: ^16 x2=^\6 x V2 =^4xV2=:4^2. 2. To reduce ^320 to simpler terms. ^320 = 3/64 X 5 =4/^4 X 4/5 = 4 X 4/5 = 4^/5. , S. Reduc 19S ALGEBRA. 3. Reduce -v/ 75 to its simplest terms. ^ Ans. 5a/S. 4. Reduce %/f| to simpler terms. Ans. -rrV^^^* 5/ Reduce ^189 to its simplest terms. Ans. 3'/7. 6. Reduce V^^V to its simplest terms. Ans. :|-(/10, 7. Reduce ^ISe^b to its simplest terms. Ans. Sax^Sb. Nate, There are other cases of reducing algebraic siu>ds to simpler forms, that are practised on several occasions ; one instance of which, on account of its simplicity and usefulness, may be here noticed, viz. in fractional forms having com* pound surds in the denominator! multiply both numerator and denominator by the same terms of the denominator, but having one sign changed, fr9m + to — or from — to +> which will reduce the fraction to a rational denominator. xi r^ A ^2Q + ^/12. • ^/5rf^3 Ex. To reduce ^^^^3 , multiply it by z^s^%' *°^ . ^ . 16 + 2v^l5 , ' ^, .^3v/15-4^5 It becomes — ^ — -; =8+^^15. Also, if — ,, ^ . — 7--; , . , . , s/lS-x/S ... 65 - Yv'^S multiply It by ,, ^ ' — TT, and it becomes — = '^ ^ ^ -v/15 — ^5 15 — 5 65-35x/3 _ IS — 7^/3 10 ■* 2 ' PROBLEM IV. To add Surd Quantities together. » i. Bring all fractions to a common denominator, and reduce the quantities to their simplest terms, as in the last problem. — 2. Reduce also such quantities as have unlike indices to other equivalent ones, having a common index.— S. Then, if the surd part be the same in them all, annex it to the sum of the rational parts, with the sign of multiplica- tion, and it will give the total sum required. But if the surd part be not the same in all the quantities, they can only be added by the signs + and — . EXAMPLES. 1. Required toadd v'l^ and v'32 together. First, -1/^8= -/9x 2=3-v/2; and '•32=v^I6x 2=4-v/2: Then,.3-v/2 + 4-v/2 = (3 + 4) ^2 = 1^.2 =- sum required. 2. It is required to add ^375, and VI 92 together. First, V375=V125X3=:5V3; and ^102= V64 x 3 =4^3 : Then, 5^3 + 4V3 = (5 + 4) V3 = 9?/3 = sum required. 3. Kequired ' .; SURDS. isy S. Required the sum of V27 and V48. Ans". 7 V^3. 4. Required the sum. of V50 and V72. Ans. 11 \^ 2. 5. Required the sum of ./I and -/-^ Ans. 4 V-iV o** A^l^- 6. Required the sum of ^56 and ^189. Ans. 5^/7, 7. Required the sum of V|- and^^ ' Ans* ^l^. 8. Required the sum of 3 Va^b and 5Vl 6a^i» PROBLEM V. To find the Difference of &urd Quantities, Prepare the quantities the same way as in the last rule; then subtract the rational parts, and to the remainder annex the common surd, for the difference of the surds required. But if the quantities have no common surd, they can only be subtracted by means of the sign — . , EXAMPLES. I 1. To find the difference between V320 and V 80. First, v^320= v'64 x 5—8^/5', and-/80=: ViGx 5=:4^^5, • Then S^ 5 — 4 -• 5 = 4 ^Z 5 the difference sought. 2. To find the difierente between ^128 and ^54. First, 3/128=3/64 x 2=4^2 ; and 4/54= ^27 x 2=3J/2. Then 4^ — 3^ =v^2, the difference required. 3. Required the difference of ^15 and v^48. Ans. ^3. 4. Required the difference of 3/256 and ^32. Ans. 2?/4. 5. Required the difference of \/^ and ^^. Ans. ^V6. 6. Required the difference of ^/^ and 3/y . An*. -f^l$. 7. Find the difference of v'24tf^^* and ^/54ab\ ' * • Ans. {a-2by^{Sb''-2ab)^ 6a. ' PROBLEM VI. To Multiply Surd Quantities together • Reduce the surds to the same index, if necessary ; next multiply the rational quantities together, and the surds toge- ther ; tnen annex the one product to the other for the whole product required ; which may be reduced to more simple terms if pecessary. EXAMPLES. 200 ALGEBk A. EXAMPLES, !• Rei^ulred to find the product of 4^^/12 and Sv^2. Here, 4x3xv'12xV2 = A2v/12x2=12v'24=12^4x6 = 12 X 2 X \/6 = 24 v' 6. the product required, 2. Required to mukiply ^^i by |^|. Here ^x^>?/^x|/| = ^x^^=^\xl/ii=^^xix Vl8 = ^^18, the product required. 3. Required the product of 3^2 s^d 2V'8. Ans. 24. 4. Required the product of |^ and |V12. Ans. ^^6. 5. To find the product of j^s/i ^^^ Av^t« Ans. -^^ 15. 6. Required the product of 2^14 and S^4. Ans. 12^7. ft 4 7. Required the product of 2a^ »nd a^. Ans* 2a\ 8. Required the product of {a + 3)^ and {« + *) ♦. 9. Required the product of 01.^ + \/* and 2x— v^*. 10. Required t^ product of (a + 2v/i)^, and (« - 2-v/*)' 11 J, Required the product of 2x' and Sx"*. i J. 12. Required the product of 4;? and 25^** / PROBLEM mi. t y To Divide one Surd Qmntity by another. Reduce the surds to the same index, if necessary ; then take the quotient of the rational quantities, and annex it to the quotient of the surds, and it will give the whole quotient required 5 which may be reduced to more simple terms if requisite;. EXAMPLES. 1. Requh^d to divide 6^/96 by 3 v' 8. Here 6 -r 3 .V(96 -i- 8)ifc2v^l2 =: 2v/(4xa) :?? 2x2v/3 = 4v/3, the quotient required. a Required to divide 12^280 by 3^5. Here 12 -f- 3 = 4, and 280 -r 5 = 56 = 8 x 7 = 2^7 ; Therefore 4 x 2 x4/7 = 8^7, is the quotient i^equired. • 3. Let SURD8. rSW S. Let 4i/50)be divided by 2V'5. Ans. .2^ 10- 4. Let 6V100 be divided by 3^5. Ans. 2^20. 5. Let I-v^tW be divided by ^Vf. Ans. -^^y/B. 6. Let ^;^-j% be divided by |^|. Ans. -^l^SO. 7. Let 4 v'/i, or -Jo^, be divided by j^^. Ans. |^^. 8. Let ^^ be divided hja^. 9. To divide Sa^ by 4/i *. "* PROBLEM VIIX. To Involve or Raise Surd Quantities to any Power* ■ > Raise both the rationd part and the surd patrt. Or mul-» Uply the index of the quantity by the index of the power to which it is to be raised, and to the result annex the power of the rational parts^ which will give the power required* EXAMPLES. 1. Required to £nd the square of |ii\ Fin^t, (|-)*=ix 1=^2^, and {a^Y=za^^^^a^:^a. Therefore (-|^j^)* = iV^i ^s the square required. 2. Required to find the square of \a'^. Fkst, -i X 1 = ^, and {a^f = a^'-a\/a\ z Therefore {ia^Y ^ ^fi^l/a ia the square required. I * 3. Required to find the cube of |.v/6 or f x e''. First, {\y =z f X I X f z= ,V» and (fi^f =r 6* = 6^6 j Theref. (|--v/6)3 = ^s^ x 6^/6 = ^-^^s/^ytbQ cube required. 4. Required the square of ,2 V2. An«k 4^4. I • 5.. Required the cube of 3^, or -/S. Ans. Sy'S. 6« Required the 3d power of y V 3* . Ans, | v^ 3. 7* Required to find the 4th power of iy/2. Ans. 4» 8. Required 2M ALGEBRA. S. Required to find the mth power of tf". 9. Required to find the square of 2 + VS. PROBLEM IX. To Evolve or Extract the Roots of Surd Quantities*. Extract both the rational part and the surd part. Or divide the index of the given quantity by the index of the root to be extracted ; then to the result annex the root of the rational part ^ which will give the root required. EXAMPLES. / 1. Required to find the. square root of IGv'S. First, v'lS = 4, and (6^^= 6^"^^ = 6^ ; ' II ' ^ theref. (16 V^6)^ =: ^.6\zz 4jJ/6, is the sq. root required. 2. Required to find the cube root of -rr ^^* Fu-st, i/-,V = h and (^S)^ = 3^"="^ r: 3^ j theref. (^ V3)'''=j. . 3^ = ^5^3, is the cube root required. 3. Required the square root of 6^ Ans. 6\/6^ 4. Required the cube root of ^tf^i. Ans. ial/6, 5. Required the 4th root of I6a\ Ans. 2Va. X. 6. Required to find the mth root of x" • 7. Required the square root of a* ~ 6a Vi + 9^. * The square root of a binom ial or r esidual surd, a -^ b, m-^h, may be foand thus : Take \/a« — ^ =z cj or . 7 .flf + C fl—C then ^/a +6 = •— r— + v^-tt-; — ^ — r a -f-c a — c and v^o- 6 =z •/-— ^T"' Thus, the square root of 4 + 2v^3 = 1 -|- ^3 ^ and the square root of 6— 2\/5 := ^5 — 1 . . But for the cube, or any higher root, qo general rule is known. inhntte SURDS. 20S INFINITE SERIES. An Infinite Series is formed either from division, dividing by a compound divisor, or by extracting^ the root of a coih- pound surd quantity ; and is such as, being continued, would run on infinitely, in the manner of a continued decimal fraction. But, by obtaining a few of the first terms, the law of the progression will be manifest; so that the series may thence be continued, without actually performing the whole operation. PROBLEM I^ To Reduce Fr^actional Quantities into Infinite Series by Division. Divide the numerator by the denominator, as in commcm division; then the operation, continued as far as maybe , thought necessary, will give the infinite series required. EXAMPLES. 2ah 1 . To change r into an infinite series. a -^ a . . 2i* 2*^ 2** M+b)2iA..(2b h-T r + &<^- 2ab + 2b^ -2i* - 2**-— a 2*» a ■ 2i} 2b* 2i* 23' * ,3 .&c. 2. Let a04 ALGEBRA. t 2. Let- be changed into an infinite series. 1 — a I - tf ) I . (l+a + tf*+^ + a^ + &c. 1 - a a a* «*-a» a' S. Expand -^---^ into an infinite series* Ans. — X (1 - — h-i r + &c.) 4. Expand — ^. — 7 into an infinite series. a <r Or 1 - ^ . 5. Expand -r—r — into an infinite series. Ans. 1 — 2.r + 2j:*— 2;ir' + 2x*, &c j «* . . . €. Expand - — \ — rrr into an infinite series. .2* 34* 44' Ans. 1 ' 1 — r r> &c. fl ' a* a'* 1 7. Expand •; — ; — - = 4> into an infinite series. 1 4- 1 PROBLEM II* To Reduce a Compound Surd into an Infinite Series. Extract the root as in common arithmetic ; then the Operation, continued as far as may be thought necessary, will give the series required. But this method is chiefly of use in extracting the square root, the operation being too tedious for the higher powers. EXAMPLES. INFINITE SERIES. fiOi EXAMPLES. i. Extract theitoot of rf^— ;r* in an infinite $erte. 2a Sa^ I6a^ 128a7 a" '--Ta^-^ or* a:* . ;i"* 2^---:^)- 4a* 8a* 64a* a 4a3 ' 8«* 64a' 6 *• *» r*+i 8iJ* • 16a' &c. 640^ 2. Expand \/ 1 + 1 = \/^, into an ininite series, Abs. 1 + i - i + rt - x-It &c- 3. Expand V'l — i i^^^o ^^ infinite series. Ans. l-i-l-iV— i4t*^- 4. Expand ^/a^ + x into an infinite series. - ' - *---»- 5. Expand Va^--2bx — or* to an infinite series. PROBLEM III. 3i Epitract any Root of a Binmud : or to Riduce a ^ BimmicJ Surd into an Infinite Series* Thi jj trill bp done by substituting the particular letters of the binomial, with their proper signs, in the following general theorem or formula, viz. (p + PQ^) n =p n + - ^^^—— B(^+ —^ C(^+ &C. and 206 ALGEBRA. and it will give the root required : observing that p dehotes m the first term, (^the second term divided by the first, T the index of the power or root ; and A> b, c» d, 8cc, denote the several foregoing terms with theii: proper sighs. EXAMPLES. 1.. To extract the sq. root of a^+ ^> in an infinite series. Here p = tf*> Q = -«f and — = — : therefore m m r F n = (a*) Q = (a^)^ zz a ss At the 1st term of the series. — AQ = i X a X — = — 5= B, the 2d term. m^n 1-2 4* 4» 4* , , — - — B(i X — r- X :r" X "1 = — rr-, = ^j the 3d term 2« ^ 4 2fl a* 2.4a^ m-2« 1—4 4* 4* 34* - ,' -3^c^ ;- — X - — 3 X ^= 5;^:^ = D the.4th. > 4* 3.4* Hencea+g-^-53j^3 + 5;j;^ - ^^^^^' ' 4* 4* 4^ 54» 1 2. To find the value of ;^ r-, oritsequal (a— a;)"*kin (a— x)* an infinite series*. * Note. To facilitate the application 'of the rule to fiactional examples, it is prgper to observe, that any surd may be taken fi*om tb^ denominator of a fraction and placed in the numerator^ and rice versa, by only changing the sign of its index. Thus, -^ =1 X jr^ or only jr-« 5 and—— = 1 X (a +4)"^ of (« + 4)-« y and • ^ .^ = tt^a + jp)-« j and f! == x^ X «^ ; alse — 1 (^ _ ' ^.y i = («Hx*)* X (o«-^) 'i &c. H«rt Here p INFINITE SERIES. »)7 r =fl, o== = — ^~ •*"> and — = -— = — 2 ; there*. ♦ «», 1 p » =: fa)"* = a"* r:— :; = a, the 1st term of the series* * ^ .a* — AQ=:— 2 X — X = , = 2a"'^ =x B, the 2d term. 2« ^ ^ a^ ^ fl* w — 2« ^ Sot* -X 4ir' . -t , Hence at* + 2a"'a: + 3a"*jr* + 4flr*jr'. + &c, or 1 . 2x Sx^ AiX^ . 5x^ «.,.'• J ~7 H i- + —7- + — r- H s- ^c, IS the series required, ;«* ' ^' ' a* ^ ^5 ^ ^^ . ^ 3. To find the value of-^ — » in an infinite series. a — x x^ x^ s^ Ans; a + x -{ +-5+--r&c 4. To expand V ^^^^^ or ^-^,^^^4 in a series. . 1 X* 3x* 5^** , Ans. ^r-T + "—7- — TTij aCC. a* 5. To expand tt: in an infinite series* *^ (a — Of 2J 3i* 4J^ 5J* , Ans. 1 +— + ^ + ^ + "^ *^- a a* a* a* e. To expand Va^-x^ ox (a* -jt*)"^ in a series * x^ x?" 5x^ X ^ w %^— « ■*"*•''" 2a ~ 8a' T6a»~"l28a^ 7. Find the value ofMa'-i') or (a'-A')^ in a series. Ans.a- — --^-g^&c. S. To find the value of V(a^ + -^0 or («*+-^0^^"^ series. x^ 2x^° 6jr** Ans.a+-^--^^,+ -j5^&c. 9. To 90% ALGEBRA. 9. To find the square root of ' n in an infinite serief* I . b M x^ ^^ AnS. 1 --;;;- +-;;-£ '"^ ^^' 10. Find the cuW root of a' e^ + b^ 2a' in a series. 63 3J6 Ans. I — -— r + -r-x- &c. ARITHMETICAL PROPORTION. Arithmetical Proportion is the relaticxi between two numbers with respect to their difference. Four quantities are in Arithmetical Proporti(Mi, when the difference between the first and second is equal to the dif- ference between the third and fourth. Thus, 4, 6, 7, 9, and Uy a -{- di b^ b -{- d, are in arithmetical proportion. Arithmetical Progression is when a series of quantities have all the same common difference, or when they either increase or decrease by the same common difference. Thus> 2, 4, 6, 8, 10, 12, &c, are in arithmetical progression, hav- ing the common difference 2; and a, a -^ d, a + 2dy fl + 3</, a + 4^/5 a + 5dy &c, are series in arithmetical progression^ the eommon difference being d. The mpst useful part of arithmetical proportion is con- tained in the following theorems : « 1. When four quantities are in Arithmetical Proportion, the sum of the two extreme^ is equal to the sum of the two means. Thus, in the arithmetical 4, 6, 7, 9, the sum 4 + 9 = 6 + 7=:l3: and in the arithmetical fl, a+rf, A, p-k-d^ the sum a-^-b-^-d-zia + b+d. 2. In any continued . arithmetical progression, the sum of the two extremes is equal to the sum of any two terms at an equal distance from them. tAus, ARITHMETICAL PllOPORTION. *p^ Thus, ifAe ^^ries bfe 1, 3,5, 7, 9, 1 r, &c. Th^a 1 + 11 = 3 + 9 = 5 + 7 = 12. . ■ S. Tlie fast t^rm of an^ increasing arithmetical series, is equal to th« first term increased by the product of the common 'difference multiplied by the ntimber of tenn^ lessr one ; but in.a.4^rea^g>;Serie9r t,^^ is^t tQrmhiii<e^ual to. the first term lessened by i^inesaid ^pduct^ / Thus, the 20th tewti of the^ s«fies 1 , »V 5is7, 9j &c, is sss' 1 + 2 (20-- 1 J «= r 4- 2 i^ 19 = 1 -^^W^zz'SQ. •- And the «th term of tf, a^d, a^^Qd, a-r^Sd, «-'4i/,'&c, 4. The sum of all th^ fvrffisimHty series in arithmetical progression, is equal to half the sum of the two extreines multiplied by the number ojTterm^. ^ - ^ ,.. t. Thus, the sum.of 1, 3, 5,%4>i &ci coi^iinved to the lOfli . (1 + 19) X 10 '20x10 : • term, is =5^ -^ =: — r- — a: lo x lO =s 100, Andthisvmo£ntermso{a,a+d^a + 2d, a+^d^toa^-md, " her- ...: \ 1^ The first^tern)^,^ jaAHnKreasing arithmetti:al seriei^is 1, the comman, difference 2^ and thts namber of teritos'421 } re« <)uired the si^^pf ^^ series ?. , . ~ r i ^^ :: I First, 1 +2 x: 20^ 1 +40'== 41, isltHe lifet^term. 14:41' ' '^^" '-' ■ I ^ ^ ->- ^ Thenf ^ - ^ fe'«0 ==-21 x 20 =i: 420, {heWm required. •. .' *' ' i -- « L -w: j: 4 . .if/j-* .'■t ! ^'^ff /^ 2k The first term- of a decreosing^arithnietical so^es i# 1 99^ * like common dffiTcnrence'l^j and tht dumbe'r of tettxik 37 ; re-- quired th* ^m 6f the serieaf ? • :i '^j ' -^ ; JFirst, 1 9d - 3 . e^ = 1 dp -- ISiS h J, iS'lhe last term,' . ,;, , ^ Then ^ ^ X 67 = lOQ ?^ ^7 ;=,,6700, .th,e sum, ric^r quired.'-' ^^ ' ■ ■'■"'' ' ^ -••'^- • '• '-' '- •^^;- ^3. Tp find the^sjim pf JOO^terins ©fjtjbe. ns^fllil^nUi^i^bm Y>VL- 1. , P . ^« He<]vured 4. * Required the ^um pf 9^ terma of Ike ^d omilbers 1, 3, 5, 7, 9, &c. ._ . . . -. • ; An* 9811. 5. T2^ ^st term of a decres^ng.oH^ui^tic^i^rit^ls lO, the oamojQn. diff$i!9nce f, and the Dtin^r of te^m^Sl ) i^eqaked^.supapf, the serves? . r. Ana. 140. 6. <)biejhitiidl«d' stoned beiR^ 'plai!e4«ii the ^6uhd, ins straight line, at the dktanc^ of ^' yitrdft frtittt each bther ; how far . viU zi^spri ^ tra?el» . whp \Ah9li bring tben\ ooe by one to a iyA9kst%_y^ck is- pi^ed 2 y^ds ftom^ihe first stone f ^ . Ans. 11 miles ^d 840 yards* *v .'!..' /I *•• ! o? 'r. APPXICATJON OF ARITHMETICAjL PROGRESSION ' ' TO- MBLfTA^T 'AfTAIl^S. ^' ; ■ .1 X "i • ■• ■■ - • • ■ OCTESTION I. A Triangular Battalion f, consisting of thirty ranks, in which tiie first rank is forifled of vone man biify, the second * The sum of aoy miAiber in) df IMUH: 4f the arithmetical aeries of odd number \,Z,5f 7, 9, &C| is equal to the square (n^) pf that number. That is, If 1 1^ '3, ^^ ^i 9» he, be (he nnoiiMn/than ^!U ^ . ^^i %\t^A\ 6\ Uc, be the suiiB 0£ Bpa, d^ dse, tormi; Thus, + 1 =: 1 or I2,the*tfm^i tfertn, • I .lb ^3 f?i 4or2^^the8u*o68t^ms . - 4 + 52= o or 3^, the sum of 3 term9, , 9 + vj s: igor4>the-fium.of4 t6rm4>.icfr * . For, by the 3d theorem, 1+2 («— 1) = l'+ 2«— 2 = 2ii— 1 i8thQli(5tterm^wfaenitfa9e''Ot|mberof'jlbrms ig.n; ti» tBia hist t^ui 2»— \/Mi t^ flrat tpfro. h give^2n ^ sum of thexearfarciBes, or n half the sum of the extremes ; then, < by the 4tb thi^oretpi nXfi =r 71/^ is the sum of , all. the term?. Hence it appears in general> that half the suioa df the extremes;, is dlways iht satne as the Hub-*. ber of the tei^s n ^ ,^Dd that the sum of all the tcmk,'k the ^ame as ihi^squai^ of the same ndiriber, n^. See more on Arithmetical Proportion in the ArithoqtetiCj^. p. 111. 'y.^'iti^ti^lM bittiiron, is tp be understood, a body of Iropps. Wg^ in the form of 9 triangle^ in which the Jl^ks exdedd ^ch ARITHMETICAL t^ROGRESSION. 21 1 rtcond of S, the third of 5 5 and so on : What is the strei^[di of such a triangular battalion ? '^ Ans#eri 900 iiten. * J <^J£ST20K II, A detachment having 12 snccesshedays til march, with orders to advance the first docf only i ieagae$j the second S^, and so on, increasing I4 league each day's inarch: What is the length of the whole marth^ and what is the last day's inarch? Ansjver, the last day's march is I8j l^^igues, and 12S leagues is the length of the whole march* qjJESTION III. A brigade of sappers*^, having carried on 1$ yards of 3ap the first night, the second only 13 yards, and so on, decreasing 2 yards every night, till at last Uiey carried on in ' one night only 3 yards : What is the number of nights thef Were employed ; and what is the whole length of tne sap ? Answer, they were employed 7 nights, and thq length of the whole sap was 63 yards. n . ' » i other by an equal number of men : if the first ranl^ consist of one man only^ and the difference between the ranks be also 1, tbea its form is that of an^equilateral triangle -, and when the dljfbrenc^ between tlw ranks is more than i, its form may then be an isosceles or scalene triangle. The practice of forming troops fa this order, which is now laid aside^ was formerly held in greatet esteem than forming them in a tfolid square^ a^ admitting of a greater fronts especially when the troops were to make simplj a stand on ail sides. ^ A brigade of sappers, consists generally of 8 men» divided equally into two parties. While one of these parties is advancing the sap, the other is furnishing the gabions* fascines* and other necessary implements : and wnen the first party is tired, t^e second takes its place, and so on, till each man in turn has heed at the head of the sap. A sap is a snudl ditch, between 3 and -^ feet in breadth and depth -, and is distinguished froin the trench by its breadth only, the trench having between 10 and 15 feet, breadth. As an encouragement to sappers, the pay for all the work carried on by the whole brigade,, is given to the survivors* F 2 QJJESTfQN 212 ALGEBRA. QUESTION IV, A number •£ gabions ^ being given to he placed in she ranks, one above the other, in such a manner as that each rank exceeding one another equally, the first may consist of 4 gabions, and the last of 9 : What is the number of gabions in the six ranks; and what is the difference between each rank f Answer, the difference between the ranks will be 1» and the number of gabions in the six ranks will be 39. ' * « • QUESTION V. * Two detachments, distant from each other 37 leagues, and both 'designing to eccupy an advantageous post equi-distant from each other^s camp^ set out at different times ; the first detachment increasing e^^ery day's march 1 league and a ^half, and the second detachment increasing each day's march 2 leagues : both the detachments arrive at the same time ; the first after 5 days' march, and the second after 4* diiys' faiarch ; What is ^he number of leagues marched by each detachment each day ? The progression -^, 2^xy, 3-,^, 5-^, 6^y answers the con* ditions of the first detachment : and the progression 14> 3|<» ^Ti ^h smswers the conditions of the second detachment. QUESTION VI. A^eseitery in>his flight, travelling at the rate of 8 leagues a day ; and a detachment of dragoons being sent after him^ with orders tamiarch the first day only 2 leagues, the second 5 leagues, the third 8 leagues, and so on : What is the number of days necessary for the detachment to overtake the deserter^ and what will be the number of leagues marched before He is overtaken ? Answer, 5 days are necessary to overtake him \ and coij- ^quently 40 leagues will be the extent of the march. -*-•■ * Gabions are baskets^ open at both ends, made of ozicr twigs, smd of a cylindrical form : those made use of at the trenches are 2 feet wide, and about 3 feet high $ whicb^ being filled with earthy serve as a shelter from the enemy's fire: and those made use of to construct batteries, are generally higher and broader. There is another sort of gabipn, made use of to raise a low parapet : its height is from 1 to 2 ftpt, and 1 foot wide at top, but somewhat les^ at bottom, to give room for placing the muzzle of a firelock between them : these gabions sen^e instead of sand bags. A sand bag is generally made to contain about a cubical foot of earth. • QUESTION PILING OP BALLS. (UTESnOH TII. A convoy • tSstant 95 leagues, having orders to join its camp, and to march at the rate of 5 leagues per day; its escort departing at the same timci with orders to march the first day only hidf a league, and the last day 94 leagues; and both the escort and convoy arriving at the same time : , At what distance is the escort m>m the convoy at the end of «ach march ? OF COMPUTING SHOT OR SHELLS IN A FIHISHED PILE. 'Stiot and Shells are generally piled in three different forms, called triangular, square, or oblong piles, according as their base is either a triangle, a square, or a rectangle. Fig. 1. C G Fig. 2, ABCD, fig. 1 , is a triangular pile, EFGH, £g. 2, is a square pile. E A Fig. ABCDEP, iig. 3, is an oblong pile. * fi^ convoy is generally meant a snp^y of ammunition or provisions, conveyed to a town or army. The body of men that {Hard this supply^ is cslled Mcort. A triangular 314 ALGEBRA. A triangulir pile is formed by the continual Ikying of triangular horizontal co.urse^ of shot one above another, i% such a manner, as that the sides of these courses, called rows, decrease by unity from the bottom row to the top row^* which ends always in 1 shot. A square pile is £Drm^d by the continual laying of square horizontal courses of shot one above another, in such a man- ner, as that f he sides of these courses decrease by unity from the bottom to the top row, which ends also in 1 shot. In the triangular and the square piles, th& sides or faces being equilateral triangles, the shot contained in those faces form an arithmetical progression^ having for first tem^unity, and for lait term and number of terms, the shot contained in the bottom row ; for the number of horizontal rows, or the number counted on one of the angles from the bottom to the top, is always equal to those counted on one side in the ^ttom : the sides or faces in either the triangular or square piles, are called arithmetical triangles ; and the nuoaibers contained in these^ are called triangular numbers: abc, fig. 1, £FG, fig. 2, are arithmetical triangles. The oblong pile may be -conceived as formed from the square pile abcd ; to one side or face of which, as ai), a number of arithmetical^triangles equal to the face have been added : and the number of arithmetical triangles added to the square pile, by means of which the oblong pile is formed, is always one less than the shot in the top row ; or, which is the same, equal to the difference between the bottoooTrow of the greater ^ide and that of the lesser. QIJESTIOH Till. To find the shot in the triangular pile abcd, fig. ], the b6ttdm row ab consisting of 8 shot. SOLUTION. The proposed pile consisting of S horizontal courses, eack of which forms an equilateral triangle ^ that is, the shot contained in these being in an arithmetical progression, of which the first and last term, as also the number of terms, •are known ; it follows, that the sum or these particular ^ourses, or of the 8 progressions, will be the shpt contained in the proposed pile i then . The PILING OS BALLS. tlS Hie sW of the fitst or lowet > ^ trianj^oiaf coarse vttlbe 5 8+ 1 X 4 ss'^e the second - ' ( 1 * < 7+a X3^=g|f the itivd \ ' m .' ^ * «. 6 + 1x3 &: 2t § the fourth - 5 ^- I X 2i s= 15 the fiita •- •* • - ♦ + I xS = 19 the sixth - - - 3 + 1 X li =s $ the seventh * - 2 + 1x1 = 3 the eighth - - 1 + ix i* I X Total - I20sl J in the pile propo$e(4 qt^ESTioN IX. To find the shot of the square pile xfgHj fig. 2, the bot^ torn row £F consisting of 8 shot. SOLUTION. The bottom row containing 8 shot> and the second only 7 ; that isj the rows forming the progression, S, 1, 6, 5, 4, 5,2, 1 p in which each of the terms being the square root of the shot contained in each separate square course employed ill formin^r the square pile j it follows, that the sum of the squares of these roots will be the shot required : and the sum of the squares, divided by 9» 7, €, 6, 4> S^ 2, 1 , being 204, expresses the shot ia the proposed pile. QJTESTION X. To find the shot of the oblong pile abc^def, fig. 3 ; in which >F «s 16, sind xc =b 7. SOLUTION. The ^oblong pile proposed, consisting of the square pile ABCD, whose bottom, row is 7 shot ; besides 9 arithmetical triangles or progressions, in which the first and last term^ ?s also the number of term$, are known ^ it follows^ that, if to the contents of the square pile - 14Q we add the sum of the dth progression - 252 , their total gives the contents required - S92 shot* REMARK I. The shot in the triangulin: and the square piles, as also the dbiot in each, horizontal course, may at once be ascer<^ tained v^ ALGEBRA. tauned by the followmg table : the Optical c<diimn a, coiu* taint the shot in the bottom row, from 1 tx> 30. izicliisiTe ; the column b contains the triangular numbers, cr number of eslch course ; the column c contains the sum of the triangular numbers, that is, the shot eontaihed in a trian^ gular pile, commonly called pyramidal numbers ; the column B contains the square of the numbers of the column a, that is, the shot contained in each square horizontal course ; and the column £ contains' the sum of these squares or shot in a square pile.* c B A D E Pyramidal Triangular Natural Square of thj* vtAtuml Sumofthese square numbers. numbers. numbers. numbers. LUC llV«*iil|flil Qumbers. 1 I 1 . 1 I- 4 3 2 4 5 10 6 S 9 14 20 ^ 10 4 16 30 35 15 5 25 55 56 21 6 36 91 34 28 7 49 140 120 36 8 64 204 165 45 9 81 285 220 55 10 100 385 28(J 66 11 121 506 ♦ 364 78 12 144 650 455 91 13 169 8I9 5d0 105 14 19^ 1015 680 120 15 225 1240 816 136 16 ^56 1496 969 153 17 289 I78d 1140 171 18 324 2109 1330 190 19 ,361 2470 1540 i 210 20 400 2870 t ^Thus, the bottom row in a triangular pile, consisting of 9 shot, the contents will be 165 ; and when of 9 in the square pile, 285.— ^In the same manner, the contents either of a square or triangular pile being given, the shot in the bottom roynfc may be easily ascertained. The contents of any oblong pile by the preceding table may be also with little trouble ascertained, the less side not exceeding 20 shot, nor the difference between the less and the greater side 20. Thus^ to find the shot ia an oblong pile, ... the PILING OF BALLS. <11 the less side bein^ 15, and the ^eaten95, we are first to imd the contents of the square pile, by means of which the oblong pile may be conceived to be formed $ that is, we tare to find the contents of a square pile, whose bottom ro# is 15 shot ; which being 1240, we are, secondly, to add these 1240 to the product 2400 of the triangular number 120» answering to 15, the number expressing the bottom row of the arithmetical triangle, multiplied by 20, the number of those triangles; and their sum, being 3640, expresses the number of shot in the proposed oblong pile. REMARK II. \ The following algebraical expressions, deduced from the investigations ot the sufns of the powers of numbers in arithmetical progression, which are seen upon many gunners* callipers *, serve to compute with ease and expedition the shot or sheUs in any pile. • That serving to compute any triangular ) ^^^ + 2x /i -f 1 x^ pile, is represented by I 6 \ That serving to compute any square ) ^ -f 1 x 2fg 4- 1 x ft pile, is represented by ) 6 In each of these, the letter n represents the number in the bottom row : hence, in a triangular pile, the number in the bottom row being 30 ; then this pile will be 30 + 2 x SO + I X V ^ 49$0 shot or shells. In a square pile, the number in the bottom row being also 30; then this pile will be 30 + 1 X 60 + I X ?/ = 9455 shot. or shells. That serving to compute any obiong pile, is represented by 2/1+1 +3«ix« + 1 X « . ,1-1. 1 J ■ ' ■ • — i in which the letter n denotes * Callipers are large compasses, with bowed shanks, serving to take the diameters of convex and concave bodies. The gunners' callipers concisx of two thin rules or plates, which are moveable quite round ^ joint, by the plates folding one over the other : the length of each rule or plate is t> inches, the breadth about 1 inch It is usual to represent, on the plates, a variety of scales, tables, proportionH, 6kC, such as are esteemed useful to be kuovq by persons employed about aitillery ; but; except the nfMsuring of the caliber of shot and cannon^ and the measuring of saliant and re-entering at^gles, none of the articles, with which the callipers are usually filled^ are essentia^ to that instrument. the fit ALGEBRA. the number of coimcSy aBcl the letter m the number of shat^ less ODe> in the top row : hence, in an oblong pile the num- ber of courses being 30» and the top row 31 ; thb pile wiU be 60 + 1+90 X 30+1 x V" = 23405 shot or shells. GEOMETRICAL PROPORTION, Geometrical Proportion contemplates the relation of quantities con^dered as to what part or what multiple one is of another, or how often one contains, or is contained in, another.— Of two quantities compared together, the first is called the Antecedent, and the second the Consequent. Their ratio is the quotient which arises from dividing the one by the other. Four Quantities are proporticmal, when the two couplets have equal ratios, or when the first is the same part or mul- tiple of the second, as the third is of the fourth. Thus, 3, 6, 4, 8, and a, ar^ b, br, are geometrical proportionals. or bt For ^ = |. = 2, and — = — = r. And they are stated thus, 3 : 6 :: 4 : 8, &c. Direct Proportion is when the same relation subsists be- tween the first term and the second, as between the third and the fourth : As in the terms above. But Reciprocal, or Inverse Proportion, is when one quantity increases in the same proportion as another diminishes : As in these, S, 6, 8, 4 ; and these, a, ar^ br^ b^ The Quantities are in geometrical progression, or con- tinuous proportion, when every two terms have always the same ratio, or when the first has the same ratio to the second as the second to the third, and the third to the fourth, §cc. Thus, 2, 4, 8, 16, 32, 64, &c, and a, ar^ at^, ar^, ar^y ar^, &c, are series in geometrical progression. , The most useful part of geometrical proportion is con- tained rtr the following theorems ; which are similar to those in Arithmetical Proportion, using multiplication for addi- tion, &c. . 1, When r GEOMETRICAL PROPORTION. 819 1. W^jen four qujuitities are in geometrical j>roportion, the product of the two extremes i$ equal to the product of the two means. As in these, 3, 6, 4, 8, where 3x8=6 X 4f=i24:i aad in tht^se, a» ar% b^ br^ where ax kr^ar x 2. When four quantities are in geometrical proportion, the product of the means divided by either of the extremes gives the other extreme. Thus, if 3 : 6 : : 4 : 8^ then 6x4 6x4 ■ ■ =: 8, and- ■ = 3} also if tf : ar :i b : br, then = ir, or — r— = a* And this is the foundation of the a or Rule of Three. 9* In any continued geometrical progression, the product of the two extremes, sekI that of any other two terms, ^quaUy ^stanl: fr(»n them, are equal to each other, or equal to the square of the middle term when there is an odd number of them. So, in the series 1, 2, 4, 8, 1(5, 32, 64, &c, it is 1 x 64 = 2 X 32 = 4" X 16 = 8 x 8 = 64. 4. In any continued geometrical series, fhe last term is f qual to the first multiplied by such a power bf the ratio as is denoted by 1 less than the number of terms. 'Thus, in the series, 3, 6, 12, 24, 48, 96, 8cc, it is 3 x 2^ = 96. 5« The sum of any series in geometrical progression, is {bund by multiplying the last term by the ratio, and dividin|r the difference of this product and the first term by the difc ference between I and the ratio. Thus, the sum of 3, 6, 192 X 2 — 3 12, 24, 48, 96, 192, is — -^ = 384- 3 == 381. And 2— i the suro of n terms of the series j, or, ar^, ar^^ ar^, &c, to , . ar^"^ X r— tf af—a r" — 1 I jg rz ~ = -a. ' r—l r- 1 r — 1 6. When four quantities, «, tfr, ft, Ar, or 2, 6, 4, 12, are proportional ; then any of the following forms of those quan- tities are also proportional, viz. 1. Directly a : ar :: b : Ar ; or 2 : 6 : : 4:12. 2. Inversely, ar :a : : ir : A j ar 6 : 2 : : 12 : 4. 3. Alternately, j : 6 : : «r : ir; or 2 : 4 : : 6:12. 4. Coxn^ 9ib AtCEBIlA. 4. Comp6\md€dlyfO:a+ar::b:b+kp\ or2:S::4:I6. 5. Dividediy, a i ur^a iihi br~-b\ or 2 : 4 :: 4 : 8. 6. Mixed, ar^g znr^a :: br^bibr^b\ or 8 : 4 :: 16 : S. 7. Multiplication, aciarc: : fc : brc ; or 2.3 : 6.3 : : 4 : 12* 8. Division, — :—:;>: ir; or 1 : 3 :: 4 : 12. c c 9. The numbers a, b, c, d^ are in harmonical proportion, when A I d II a^nh I czo d\ or when their reciprocals 1111 •^% "T* ""■> ^> are in arithmetical proportion* EXAMPLES. 1. Given the first term of a geometrical series 1, the ratio 2> and the number of terms 12 ; to find the sum of the series? First, 1 X 2" = 1 X 2048, is the last ternu i„ 2048x2 — 1 4096 — 1 , , . Then — ss ■ == 4095> the sum required. 2. Given the first term of a geometric series -f-, the rati© I, and the number of terms 8 ; to find the sum of the series? Krst, I X {\y = I X TTT = TTTf »s the last term. Then (f ^3^ X 4) -4- {\-i) = (i-yW) ^ 1 = fH X x =: f 44'> ^^ ^^^ required. 3. Requited the sum of 12 terms of the series 1, 3, 9, 27, SI,&c. Ans. 265720. 4. Required the sum of 12 terms of the series 1, -f, ^t Tjy Tr> &€• Ans. ttHtt* 5. Required the sum of 100 terms of tha series I, 2, 4, 8, 16, 32, &c. Ans. 1267650600236229401496703205375. 3ee more of Geometrical Proportion in the Arithmetic* I I SIMPLE EQUATIONS. An Equaticm is the expression of two equal quantities, with the sign of equality (=•) placed between them. Thus, 10^4 =x 6 is an equation, denoting the equality of the quan- tities J.0 — 4 and 6. Equations SIMPLE EQUATIONS. -221 Equations are either simple or compound. A Simple Equation, is that which contains only one power of the un- known quantity, without including different powers. Thus, x^a Si b + c, or ax^ = 4, is a simple equation, containing only one power of the unknown quantity iZ*. But jt*— Sat :;^&^ is a compound one. Reduction of Equations, is the finding liie value of the unknown quantity. And this consists in disengaging that quantity, from the known ones ; or in ordering the eqtia^ tion so, that the unknown letter or quantity may stan<i alone on one side of the equation, or of the mark of equality, without a co^efficient ; and all the rest, or the known quan- tities, on the other si€le.«-4n general, the unknown quantity i$ disengaged from the known ones,> by performing always the reverse operations. So, if the known quantities are con- nected with it by + or addition, they must be subtracted ; if by minus (—), or subtraction, they must be added'; if by multiplication, we must divide by them ; if by division, we ^ust multiply ; wheQ it is in any power, we must extract the root ; and when in any radical, we must raise it to the power. As in th^ foUdwiiiff particular rules ) which are tbundedon the general principle of performing equal operas tiQits on equal quantities; in which case i| is evident that the results must still be equal, whether by equal additions, or ^ubtraoliops, or multiplicatipns^ or divisions, or roots, or powers, FARTXCULAIt RULE I, « When known quantities are connected with the unknown by + or — } transpose them to the other side of the ecjua- tion, and change their signs. Which is only adding or sqb-' tracting the same quantities on both sides, in order to get all the unknown terms on one side of the equation, and ail the icnow|i pnes on (he other side *« - - Thusy ' » Wii I " ■ ■ . ■! . ■'■■ * '■' ^- " ^1 ■■■»■>! ♦ Here it is earnestly recommended that the pupil be aq-f customed^ at every line or stejj in the reduction of the equation $». to name tbo particular operation to be performed on the equation in the last line^ in order to produce the next form or state of the eqoation> in applying each of these rules, according as the particular ^rm of th^ equation may require 5 applying them according to the. order 2^2 - ALGEBRA. • Thus, i(x 4- 5=S ; then transposing 5 givte jr=r8 — 5==$. And, if a; — 3 -{-7=^9 then transposing the 3 and 7, gives j:=:9 + 3— 7 = 5. Alsb, if X ^ s -f b=s cd: then by transposing dr ^nd 5, itisx =: a — b + cd. In like manner, if 5jc — 6 = 4x + 1 0, then by transposing 6 and 4jr, it is 5r-.4?a: = 10 + 6, or .r = 16, RUtE II. "When the unknown tferm is multiplifed by aiiy quantity ; idiride all the terms of the equation by it. Thus, if /7jr=^5 — 4a; then dividing by a, gives x ^i-^if. And, if 3x + 5 xs 20 ; then first tron^osing 5<fi^^ SJt: sc 15 } and then by dividing by 3, it is x = 5. In like manner, i{ax+3ab=4€^^f then by dividing by <j, it 4r* 4:^^ is x+Sb = ; and then transposing 8^, gives x =- — 3A RULE III. When the unknown term is divided by any quantity ; we tnust then multiply all the terms of the equation by that di- visor ', which takes tt away. Thus, if— =i= 3 +2 : then mult, by 4, gives a: = 12 4 8 = 20*. 4 And, if — = 3i + 2^ - rf; a • ' ■ then by mult, a, it gives x = Sab + 2ac — ad. 5 Then by trani^posing 3, it is ^x = 10. ^d multiplying by 5, it is 3x = 50. Lastly dividing by 3 gives j; = 16y. order in which tbey are here placed -, and beginning every line with the words Then by, as in the following specimen^ (^ Bt- amjples > which two words will always bring to his recoUectioo^ that He is to pronounce what particular operation he is to perform dn the last \\{ie, in order to give the next -, allotting atways a single line for each operation^ and ranging the equations neatly just under each otlier^ in the several lines^ as they are successively produced. Rule SIMPLE l£QUATIONS. S&S RULE lY. Whek the unkn(^#n; ^tlamtiiy is incloded in any root or surd : transposes tiie rest; of the terms, if thtsre be any, bjr Rule 1 ; then raise e^ch side to such a power as is deao^ by the index of the surdf viz* square each side when it is the square root ; cube each side when it is the cube ro9t; &c. which clears that radical. Thus, if V.r— 3 = 4f ; then transposing 3j gives Vt=s7 j And squaring both sides giires -r = 49. And, if V2J+To K « : . Then by squaring, it becomes 2x + 10 = 64 j And by transposing 10, it is 2x =: 54/^ Lastly, dividing by 2, gives x zz 27. Also, if3/3.r+4 + 3 =6: . v Then by transposing 3, it is ^3^ + 4 ^ 5;.'" And by cubing, it is 3;i' + 4 = ^7 ; * ^ Also, by transposing 4, it is Sjt ^=: 23 ; Lastly, dividing by 3, gives x ss 7.J, R«L£ V. I I % r Whe j? that side of the equation which contains the un- known quantity is a complete power, or can easily be reduced to one, by rule 1, 2, or 3 : then extract the rbbt of the said power on both sides of the equation ; that is, extract the square root when it is a square power, or th^ cube root when it is a cube, &c. Thus, if JT* + Sj? + 16 = 36, or {x + 4)' = 36 : Then by extracting the roots, it is r + 4 = 6 ; And by transposing 4, it is x ;= 6 — 4 = 2. Andif 3^*-19 = 2l +3^. Then, by transposing 19, it is 3j:^,r= 75 ; And dividing by 3, gives .r* =s 25 ; And extracting the root, gives jr = 5. ' Also, if fr*— 6=s 24. Then transposing 6, gives f x* = 30 ; And multiplying by 4, gives 3:r* = 120; Then dividiag by 3, gives x* = 40 $ Lastly, extracting the root, gives x = V40 = 6*324555« RULE .f2iy ALGEBRA. EULE VU :. When there is any analogy or proportion, it. is to be changed into an equation, by multiplying the two extreaae tdrois together ) and the two means together, and xmaking the one. product equal to the other. Thus, if 2x : 9 : : 3 : 5. Then, mult, the extremes and means, gives lOx =: 27 v And (dividing by 10, gives ^r- =: 2-^, And i{ ^x : a :: 5i :2c. Then mult, extremes and means gives icx = S^t ; And multiplying by 2, gives Sex si \Oab ; \Oab Lastly, dividing by 5r, gives a: =-- — • Also, if 10— X : fjr : : 3 : 1. Then mult, extremes and means, gives 10-<-r r= 2r ; And transposing ir," gives 10 = 30*^ Lastly, dividing by 3, gives 3 j. = ;c, nvm VII. When the same quantity is found on both sides of ait equ?.tion, with the same sign, either plus or minus, it may be left out of both : and when every term in an equation is either multiplied or divided by the same quantity, it may b^ strucl? ou^ of them all. Thus, i[3x + 2az^2a + b: \ , Then, by taking away 2^, it is 3x =: k^ And, dividing by 3, it is :r = ^6. Also if there be 4ax + 6ah = lac. Then striking out or dividing by fl, gives 4j: + 6i =; 7r, Then, by transposing 6^, it becomes ^x r: 7^—6^; And then dividing by 4? giv^s ;r = ^r— J^. Again.if|x-J=V>-.J. . ; , . Then, taking away the ^^ it becomes -|.r =: */ 5 And taking away the 3's, it is 2jr =: 10 ; Lastly, dividing by 2 gives ^ n 5, MISCELLANEOUS KXAMpLESt i\ 1. Given 7;r- 18 = 4«r + 6 ; to find the value of \r.^ First, transposing 1 8 and 5j; gives 3;v 2: 24 ) Then dividing by 3, gives jr =r 8. 1. . ^' Qlvesx SIMPLE EQUATIONS. . 2t& I % GivenSO— 4a:-12 s=s^— Ipjr; to find i*. Firtt transposing 20 and 12 and IOjT, gives 6x » 84 ; Then dividing by 6, gives j: = 14. ^. Let 4tfjr- 5^ = 3<& + 2<? be given ; to find x. First, by trans. 5* and 3dXi.it is ^ax-^Sdx = 5* + 2r; 5ft4-2r Then dividing by 4fl— 3</, pves a: = ■ ^> • 4* Let 5;r*— 12x c= 9 J? + 2jr* be given; to find *. . Fii^, by dividing by x, it is 5j: — 12 =: 9 + 2jr ; Then transposing 12 and 2x, gives ^jt rt 21 ;- Xastly, dividing by 3, gives j; =» 7. 5. Giveii 9fljr'— ISabx^ ^ 6ax^ + I2ax^ \ to find x. First, dividing by SoJi^, gives Sx— 5^ = 2jr + 4; Then transposing Bb zad 2Xf gives x :=:, 5b -h 4. * 6. Let -r --+-j- = 2be given, to find X. I 3 4 5 First, multiplying by S, gives x^-^x + ^x =s 6 ; Then multiplying by 4, gives x -{- Y^ == 24. Also njultiplying by 5, gves J7r = 120 ; Lastly, dividing Dy 17, gites x = 7^. ^ ^. ^ .r— 5 jr ; i*— 10 ^ " 7. Given — ; — + — = 12 — : to find x. 3 ^ ^ .. S- '■ ^ . . .^^., .^ . First, mult, by S, gives x—S -f 4x =,*36 — jr +'lO ; Thep transposing 5 and x, gives 2jr -f* l-^ = 51 ; And multiplying by 2, gives 7jr =: 1 02 ; Lastly, dividing by 7, gives jrt=:14|, , Sx ' ' ^ S^ Let V'^T" + '^ ~ ^^> ^ givai ; to find x. First, transposing 7frgiyes ^^x = 3 ; Then squaring the equation, g|lves -j^ tsz>9\ 'Then dividing by 3, gives ^ as= 3 ; 1/astlyi multipiying by 4, gives a: = 1 2. . 9.'' Let* 2^ + 2 Vi?47r* =^J^=L, be^gitreti; tb firid x. Fiht, mult, by ^tf*+ jr*, gives 2xVa* + a:* + 2^* + 2^"- Then transp. 2a* and2jr%gives2;rv'a* + jr*=3«*— 2x'5 xoVoL-I. Q Th'^ \ i*r 226 ALGEBRA. i • I. -ii'V .H'ft. Then by squiring, it is 4a:*x^+ x*^3tf* - ar* 5 That is, ^a^x^ + 4x* = 9tf * - 1 2/i*a;* + 4jr* j By taking 4x* from both sides, it is 4tfV=9ii*— 12tfV j Laftly extracting the root, gives v ^^ CXIMPLBS FCfR Pfticmd^ ' 1. Given ^»— 5 + 16 = 21 ; to find n. Ato. at = 5* 2. Given 9* — 15 £rT + 6j to find x. Ans. x = 4|.. 3. Given 8-3i'+ 12=30— 5x+4; to find ;r. Ansjr=7; 4. Given x + -pr— i* =185 to find x. Ans, or r: 12. 5. Given S*'+4;» +2=5* - 4 j to find at. Ans. * = 4; 6. Givgn 4fljr + ^a— 2 cr ^jr— *x j to find x. Ans. X =: 9a+3^ 7. Giv€4i \x^\x + T^ = T> ^o fi»d r. Ans. ^ ^/if*J 4ji 8. Given v^4+x = 4 — v^jr; to find jr. Ans. xzz2\. X* 9. Given 4ta + x ^ -r — -— ; to deter, x* Ans. ^= — 2a. 4a+x V 10. Given V4ji* + jr*x=;J/4** + jr*j to find x. Ans. X tSL'V- 2^ ^MMM 4a 11. Given V;c + V^2a + x zr- ^ ; to find x. Ans. 4? irfiPir 12. Given r-~*" +t — ^ =2 2* ; to find x. l+2x 1 — ; 3* Ans. X fSr^V— -r—- 13, Given j + x = Va* + x VW + x*| to find k. Ans. j: ts a^ a w SIMPLE'EQUATIONS. «27 OV REDUCING DOUfiLB, TRIPLE, .ftc. £<^A'fiIONs/ CON- TAINING TWO, THREE, OR MORS UNX3K0WN jqUAN- TITIES. i»ROBL$M I. ; 7i Exterminate Two Unknown Sluantities; Or^ to Reduce the Two Simple ^puLtiom tontetining fkentfto a Single 4m. , RULE L Find the T?lue pf one of the pn^nown letters, in terms -W ^e other qiantjti^^ in eskfix 4af the (equations, by the lðods already explained. Then put those two values equal to each other tor a new equation, with only oi&e yn- known quantity in it, iiriiose value is to be found .as .before. N&te. It is ev ident that we must first begin to find the values of that letter which are easiest to be found in the two proposed equati^is* EXAMPLES* 1. Given [ll t%Zu]^ '^ ^^ * ^^^- 17 — Sv In the lstequat.traf»p.SyanddAv.by2,gives;rs%i-m--^; 14 4- 22/ Jbi the 2d transp. 2v and div. by 5, gives * =s r^J Putting these two values equal, gives "^^ ^» Then mult, by 5 and 2, gives 28 + 4y =z SS-- 15yj Transposing 28 and 15y, gives 19y = 57; And dividing by 19, gives ^ =ai S. And hence, ffp = 4. Or, to do the same by finding two valuc^ of ^, thus: In the 1st equat. tr. 2k and div. by ^, gives^ = s 5:^— 14 In the 2d tr. 2yand 14, anddiv. by 2, ja^vesj^ ;=■ art «,,.,, 1 1 . 5«— 14 17-2* Jrutting these two values equal, gives -t— -rr= . ■' ■ Mult, by 2^ and by 3, gives lBx^.^2. =s. ^^ -t'4^| " Q2 Transp. 2U ALGEBRA^ ^ Tramp. 42 and 4r, gives^ 19x cr J6 ; Dividing by 19, gives x ;= 4. Hence j^ = 3, as before. ' d. Given J|^j;|j;=;^> to find* and;. Ans. * = « + *, and j^ = ^—44- St Given 3* + ^ = 22, and Sy+«f = 18 ; to find x and >. Ans. X s 6} and j^ :^ 4. 4. Given {jj + ?{= ^, { ; to findirand,. Ans. jr = 6, and v = 3. '^. 2* 3v 22 3*p 2ir 67 - . ,5. Given — -l-'f = — , and— +-^«rr; to find /? and y. Ans. x = 3, and ; = 4. 6. Given * + 2j^ sz /, and i^*- 4/ = i/* ; to find x and jr. Ans. X = — ^, and jr =— jj— 7. Given ;» — 2v n rf, and xiyiiaib; to find x and y. ad , W Ans. X = r, and f = -r RULE U. Find the value of one of the unknown letters, in ooly one of the equations, as in the former rule ; and substitute this value instead of that unknown quantity in the other equation, and there will arise a new equation*^ with only one unknown quantity, whose value is to be found as before. " . ... * Nate. It is evident that it i5,l?est to begin first with that letter whose value is easiest found in the given equations. EXAMPLES* ' , 1. Given ^l^ 1 1 Zu]> to find ' and j. This will admit, of fom* ways of solution; thus : First, 17— 3v in the 1st eq. trans. 3y and div. by 2, gives x =s — ^~^* . 85/-^ I5y This val. subs, for x in the 2d, gives — ^ — 2y = 1 4 j Mttlt. by 2, this becomes 8S - liy-- 4y =: 28 j Transpv SIMPLl EQUATIONS. 22§ Transp. 15y and 4y and 28, gives 57 c= 19y ; , And dividing by 19> gives 3 szy. Then^=: — s~ = 4. 2dly, in the 2d trans! 2y and div. by S^ gives ^ = — ^-A 5 This subst. for :r in the 1st, gives 2! + 3j^;s 17 j Mult, by 5, gives 28 + 4y + 15y = 85 j Transpos. 28, gives 19y =: 57 ; And dividing by 19, gives y =: 3. Then x = ' ' = 4, as before. 3dly, in the 1st trans. 2r and div. by 3, gives j>= — ^^^^5 3 This subst* for ^ in the 2d, gives 5:c — = 14 . Multiplying by 3 gives 15*1;' -^ 34 + 4jr s= 42 $ Transposing 34, gives . 1 9x = 76 ; And dividing by 19, gives x = 4. „ 17-2;r ^ Men^e^ 1=: — - — <*ss 3, as before. 4thly,in the 2d tr. 2^ and 14 and div. by 2, gives y=: ^"^ \ This substituted in the 1st, fives 2x + = IT; Multiplying by 2, gives i 9x — 42 =; 34 j Transposing' 4<2, gives 19x == 76 ; And dividing by 1 9, gives x r: 4, 5x— 14 Hence y = — r — =: 3, as beforfc. 2. Given 2;r + 3y =: 29, and 3x - 2jy — 11 ; to find * andy. . Ans. jt = 7, and j^ == 5. , . 3. Given {J 1 J f ^2} ' '" ^'^ -^ '"'^> Aris. a? = 8, and v = 6. 4. Given 299 AIGBBRA* 4. GiTCi! {^'J^jl I '^1} ; to fftid T^iy. ' Am^ jr =: 6, and y = 4. 5. Given -r- + 3^ = 21, and ~- + S>r = 29 ; to find ;r and jf^ Ans. *r ss 9, and j^.= 6. «. Given 10 - ^ -i + 4, and ^ + ~ - 2 = — I 1 ; to find X and y. An$. x zz 8, and v s: 6, 7. Given x : y : : 4 : 5^ and ap'— / = 37 j to find x and jr. Ans. tT = 4j and y = 3. I.ULE III. I/ET the given equations be so multiplied, or divided^ ScCt^ , and by such numbers or quantities, as will make the terms which contain one of the unknown quantities the same ia both et[uations ; if they afe not the sarte when £rst pro- posed. Then by adding or siibtracting the equations, according as the signs may require, there will remain a new equa- tion, with only one unknown quantity, as before. That isj^ add the two equations when the signs are unlike, but sub- tract them when the signs are alike, to cancel that common term. Note. To make two unequal terms become equal, as abovcn multiply each term by the co-efficient of the other. EXAMPLES. ^^^^" {S + S^ = le} ' ^^ ^^ * ^^^ J'- Here we msly either make the two first terms, containing n^ equal, or the two £d terms, containing j^, equal. To make the two first terms equal, we must multiply tji€ 1st equation by 2, and the 2d by 5; but to mlike the two 2d terms equal, we must multiply the i st equation by 5, ancl the 2d by 3 5 as follows* 1. B7 SIMPLE EQUATIONS. • 231. i 1. Bj making the two first terms equal : Mult, the 1st equ. by 2, gives lOor— Gy ss 18 ^ And mult, the 2d by 5, gives 10.r + 25j> = 80 y Subtn the upper froi^ the under^ gives Sly a 62| ' And dividing by 51| gives ys£ 2* d + 3f Hfince> from the Ist given equ* .r s: ■ / ■ ■ f ' ==. 9i» • £• By making the twQ 24 terms equal: Mult, the 1st equat. by 5, gives 2Sx — 1 5jf =5 45 ;j And mi|lt. the 2d by 3, gives 6ur -f 15; =;; 4S ; Adding these two, gives Six s 93 ; And dividing by 31,^ gives x =s S. ^ 5x — 9 ^ Hence^ from the 1st equt^y = — ''^— s: 2* MISCELLANEOUS fiXAMPLSS* ;r-hS v-f6 1. Given — - + 6t =^ 21, and ^-~ + 5x ;a 2S 5 to » s find X and jiv Ans. x ^ 4, and ;^ =9 S* 2. Given — --^ + 10 = 13, an4 -^ J - + 5 = 12 ; to find X and y. Ans*> str 5, and jr s: 3. 3. Given— j-^+— =slO,and ^ + ^-^^9 to find X and y. Ans. ^ = 8, and y =: 4* 4. Given Sx + 4y = 38,and4x— Sy = 9; tofindxandy. Ans. x :;;: 6, and y sb 5« FROBLSM I|. To ExterffunaU Three or More Unknown Quantities i Or, to Reduce tie Single Equations, containing them, to a Singly one. nxsiA. This may be done by any of the thret methods i^ the Ia|t problem: viz. 1. After the manner of the first rule in the la$t problemi find the value oip one of the unknoven letters in each of th« given equations : next put Vrro of these values equal to each other, and then one of these and a third value equal, and so on for all the values of it ; which gives a ntwsct of equations, with i3« * ALGEBRA. i I t with which the same process is to be repeated, and so on till there is only one equation, to be reduced bj the rules for a single equation. 2. Or, as in the 2d rule of the same problem, find the value of one of the unknown quantifies in one of the equa-» tions only; then substitute this value instead of it in the other equations ; which gives a new set of equations to be resolved ^ before, by repeating the operation. 3. Ots as in the 3d rule, reduce the equations, by multi- plying or dividing themi so as to make som^ of the terms to agree: then,. by adding or subtracting them, as the signs may require, one of the letters may be exterminated, &c, as before. EXAMPLES. 1. Given -I ;ir + 2j/ + 32 = 16 >• ; to find 4f,^, and a» (Ar+3j/ + 4z = 2l) 1. By the 1st method : Transp. the terms containing^ and z in each ^ua.. gives • - » =s 9 — J/ — SB, 4?= 21 — 3y — . 42} Then putting the 1st and 2d values equal, and the 2d and 3d values equal, give 9 — y - as = 16 — 2j/ - 3«, 16 _ 2y - 32S = 21 - 8j^ - 4z ; In the 1 St trans. 9, 25, and 2yj gives j^ = 7 — 2z ; In the 2d trans. 16, 3z, and 3y, gives j/'= 5 — - a;; Putting these two equal, gives 5—2= 7 — 2zj Trans. 5 and 3z, gives 2 = 2.. Hence 3^ = 5-2 = 3, and x = 9— j^— 2 =; 4. Jdly. By the 2d method : From the 1st equa. x = 9—^ — 2; This value of «: substit. in the 2d and 3d, gives 9 -f y + 22 == 16, .U + 2y+.32 = 21; In the 1st trans. 9 and 2z, gives 3/ =; 7 — 23^; This substit. i|i the last, gjves 23 — 2 == 21 j. Trans, z and 21, gives 2 = 2. Pence again J/ =; 7 — 2z == 3, and x = 9— 3^-2 = 4. 3dly. By SIMPLE EQUATIONS. 3SS Sdly. By the 3d method : subtracting the 1st tqu. from the 2d, and the 2d from the 3d, gives J/ + 25f = 7, y + z=: 5i Subtr. the latter from the former, gives z = 2, Hence y = 5 — z = 3, and x =^ 9—7/—Z =i 4-. C x+ 1/+ z^lSl \. Given < x + 3j/ + 25r = 38 >.; L ^+j;y + i2 = ioy to find X, j^, and s, « Ans. a? = 4, ^ = 6, 5r = 8. = 271 ^. Given ^ x + yj/ + |z = 20 >• ; to find jr,3^, and s. z = 163 Ans. X = 1, j^ = 20, at = 60. 4, Given jt — j/ = 2, jr — ;a =: 3, and y — a = 1 j to find a;*, ^, and z, Ans. r = 7 ; j/= 5 ; 2; = 4. 2jr + 3y + 4z = 34T 5. Given ^Sx + iy + 5z = 46^ i to find x, y, ajid z. 4^ + 5y+ 6z =583 A COLLECTION OF QUESTIONS PRODUCING SIMPLE ^ EQUATIONS. Quest. 1. To find two numbers, such, that their sum shall be 10, and thair difference 6. Let JT denote the greater number, and 1/ the less *. Then, by the 1st condition x + j/ = 10, And by the 2d - - a; — 3/ = 6, Transp. y in each, gives jr = 10 — ^, and X = 6 + y J Put these two- values equal, gives 6+^ = 10-*j/; Transpjos. 6 and — t/^ gives - 2^ == 4 j Dividing by 2, gives - - ^ = 2. And hence - - - - x =:^ 6 +j/ = 8. * In all these solutions^ as many unknown letters are always used as there are unknown numbers to be founds purposely the better to exercise the modes of reducing the equations : avoiding the short ways of notation^ which> though g'^ving a shorter solu- tion, are for that reason less usefiil to the pupil, as affording less exercise io practising the several rules in reducing equations. Quest. 2. SS4 ALGJEBRA. Quest. 2. Divide lOQ/. simong A» u, c, se tlut a may have 20/. more than b, and b 10/. more thaa c. Let A' = a's share, j^ = b's, and z = c's. Then :t' + j^ + » = 100, X =cj/ +20, j^ =5 z 4- 10. In the 1st substit. y + 20 for x^ gives 2y + 2 + 20 = 100; In this substituting 2 + 10 for j<, ^ves Ss; + 40 :s 100 ; By transposing 40, gives • Sf; ^ 60 ; And dividing by S, gives - - z = 20. Htocej^ = 2 + 10 = 30, and x = j/ + 20 = 50. « Quest. 3. A prize of 500/. is to be divided between two persons, so^^as their shares may be in proportion as 7 to 8; required the share of each. Put X and y for the two shares ; then by the question, 7 : 8 : : jr :y, or muk. the extremes and the means, ly = Sx, and a:H-j/ = 500; Transposing J/, gives x = 500 — j^ ; This substituted in the 1st, gives 7j/ = 4000 — St/^ By transposing 8y, it is 15j^ = 4000 ; By dividing by 15, it gives y = 266^-1 And hence x = 500— j/ =' 233|-. Quest. 4. Whatnumber is that whose 4th part^exceeds its 5th part by 10 ? Let X denote the numba* sought. Then by the question ^x — |jr = 10 ; - By mult, by./4, it becomes x — ^:r = 40 j By mult, by 5, it gives x = 200, the number sought. Quest. 5. What fraction is that, to the numerator of which if 1 be added, the value will be 4 } but if 1 be add^ to the denominator, its value will be j- ? X Let --- denote the fraction. ^ 3/ Then by the quest. = », and —^ — = '. The 1st mult, by 2 andy, gives 2^ + 2 = v ; The 2d mult, by 3 and J^ + 1, is ,3^ =:^ + J ; The upper taken from the under leaves .r— 2 = 1 j By transpos. 2, it gives x z=z s. , And hence J/ = 2^ + 2 = 8 ; and the fraction is |. Quest. «. \ SIMPI^'EQUATIONS. 285 Quest. 6. A hbourer engaged to serve for SO days on these conditions : that for every day he worked^ he was to receive 20d, but for wu^ry day he played, or was absent, he was to forfeit \()d* Now at the end of the time he had to receive just 20 shillings, or 240 pence. It is required to find how many days he worked^ and how many lie was idle ? Let j: be the days worked, and^ the days idled«^ Then 20x is the pence earned^ and iOy th^ forfeits ; Hence, by the question - x +j/ ^ 80, and 20x — IC^ = 240; The 1st. mult, by 10, gives lOx + lOj^ = 300 j • These two added give - 30jr = 540 ; • This div. by 30, gives - ;r = 18, the days worked j Hence . - j^=30— a:=sl2, the days idled. Quest. 7. Out of a cask of wine, which had leaked :^way |, 80 gallons were drawn ; and then, being gaged, it jappeared to be half full; how much did it hold? Let it be supposed to have held x gallons. Then it would have leaked -^x gallons, Conseq. there had been taken away ^x + 30 gallons. Hence ix=zix + 30 by the question. Then mult, by 4, gives 2x = a: -+- 120; - And transposing x, gives x 2= 120 the contents. Quest. 8. To divide 20 into twd such parts, that 3 times the one part added to 5 times the other may make 76. Let X and^ denote the two part^. Then by the question - - :r + 3^ = 20, and Sx + 5t/ = 76. Mult, the 1st by 3, gives - 3-r + 8j^ =s 60; Subtr. the latter from the former, gives 2y = 16 ; And dividing by 2, gives - - ^ = 3. Hence^ from the 1st, - x s= 20 — j/ = 12. , QUBST. 9. A market woman bought in a certain number pf eggs at 2 a penny, and as many more at 3 a penny, and sold them all out again at the rate of 5 for two-pence, and by so doing, contrary to expectation, found she lost 3^.; what number of eggs had she ? Let X ss number of eggs of each sort. Then will ^x = cost of the first sorty And ^ = cost of the s^coi^sott ^ But 285 ALGEBRAr Bat 5 r 2 : : 2r (the whole number of eggs) : ^ ; Hence ^x = price of both sorts, at 5 for 2 pence ;• Then by the question ^r + jx— 4ir = 3 ; Mult, by 2, gives - *• + f^— 1-^ = 6 ; And mult, by 3, gives 5x — V-^ = 18; Also malt, by 5, gives x = 90, the number of eggs ol each sort. Quest. 10. . Two persons, a and b, engage at play. Before they begin, a has SO guineas, and b has GO. After a certain number of games won and lost between them, a rises with three times as. many guineas as B. Query, how many gttineas did A win of B ? Let X denote the number of guineas A won. ' Then a rises with 80 + x. And B rises with 60— x ; • Theref. by the quest. 80 + x = 1 80 - Zx\ Transp. 80 and 3 a;, gives 4x = 100 ; Anc^ dividing by 4, gives x =: 25, the guineas won. QUESTIONS FOR PRACTICE. 1. To determine two numbers such, that their difference may be 4, and the difference of their squares 64. Ans. 6 and 10. 2. To find two numbers with these conditions, viz. that half the first with a 3d part of the second may make 9, and that a 4th part of the first with a 5th part of the se- . cond may make 5. Ans. 8 and 15« . 3. To divide the number 20 into two such parts, that a 3d of the one part added to a fifth of the other, may make 6. Ans. 15 and 5. 4. To' find three numbers such, that the sum of the 1st and 2d shall be 7, the sum of the 1st and 3d 8, and the sum of the 2d and 3d 9. Ans, ^^ 4, 5. 5. A father, dying, bequeathed his fortune, which was 2800/. to his son and daughter, in this manner ; that for every half crown the son might have, the daughter was ta have a shilling. What then were their two shares ? Ans. The son 2000/. s^nd the daughter 800/. 6. Hiree persons. A, B, c, make a joint contribution, which in the whole amounte to 400A : of which sum b con- tributes SIMPLE EQUATIONS. ?37 tribiiites twice as much as a and 20/. more \ and c as much as A and b together. What sum did each contjribute ? Ans. A 60A B UO/. and c 200/1 7. A person paid a bill of lOOA with half guineas and crowns, vsing in all 202 pieces ; ..how many pieces were there of each sort ? Ans.. 180 half guineas, and 22 crowns. 8. Says A to B, if you give me 10 guineas of your money, I shall then have twice as much as you will have left : but says B to A, give me lO of your guineas, and then I shall liave 5 times as many as you. How many had each ? Ans.- A 22, B 26: 9. A person goes to a tavern with a certain quantity of money in his pocke't, where he spends 2 shillings; he then borrows as much money as he had left, and going to another tavern9 he there spends 2. shillings also; then borrowing < again as much money 'as was left, he went to a third tayerQ^ whi^e likewise h^ speixt 2 shillings; and thus repeating. the. « same at a &urth tavern, he then had nothing remaining. What sum had he at first ? Ans. 3/. 9 J. 10. A man with his wife and child dine together at an inn. The landlord charged 1 shilling for the child; and for the woman he charged as much as for the child and i^% much as for the man ; and for the man he charged as much as for the woman and child together.. How much was that for each ? Ans. The woman 20d. and the man 32rf. 4 11. A cask, which held 60 gallons, was filled with a mixture of brandy,. wine, and cyder, in this manner, viz* the cyder was 6 gallons more than the brandy, and the wine was as much as the cyder and \ of the brandy. How much was there of each ? '^S.' Ans. Brnndy 15, cyder 21, wine 24- 12. A general, disposing his army into a square form, finds that he has 284 nien more than a perfect square ; but increasing the side by 1 man, he then wants 25 men to be a complete square. Then how many m^n had he under his command ? Ans. 24000. 13. What number is that, to which if 3, 5, and 8, be severally added, tl^e three sums shall be in geometrical pro- gression? Ans. 1. 13. The stock of three traders amounted to 860/. the shares of the first and second exceeded that of the third SS8 ALGEBRA. hf 240 ; and the sum of tbe 24 and Sd extdeded the fint by 260, What was the share of each ? ^ Ans. The 1st 200, the 2d 300, the 3d 260. 15*. What two ntimbers are those, whicht being in* the ratio of S to 4, their product is equal to 12 times their swni Ans. 2 i and 29. 16.' A certain company at a tavern, when they came to settle their reckoning, found that had there been 4 more in company, they might have paid a shilling a-piece less than they did ; but that if there had beea 3 fewer in company. tliey most have paid a shilling a-piece more than they did. What then Was the number of persons in company, what each paid, and what was the whole reckoning ? Ans. 24 persons, each paid 7i. and the whole reckoning 6 guineas. 17* A jockey has two horses ; and also two saddles, the one valued at 18/. the other at 3/. Now when he sets the better saddle on the 1st horse, and the wor#e on theM, it makes the first horse worth double tbe 2d : but when he places the better saddle on the 2d horse, and the worse on the first, it makes the 2d horse worth three times the 1st,. What then were the values of the two horses t Ans. The Ist (i/., and the 2d 9/. IS. What two numbers are as 2 to 3, to each of which if 6 be added, the sums will be as 4 to 5 ? Ans. 6 and 9» 19fc What are those two numbers, of which the greater is to the less as their sum is to 20, and as their difference is to 10? Ans. 15 and 45. '20. What two number*^ are those , whose difference, sum, and product, are to each other, as the three numbers 2, 3, 5 ? Ans. 2 and lO. 21. To find three numbers in arithmetical progression, of which the first is to the third as 5 to 9, and the sum of all three is 63. Ans. 1 ^, 21, 27. 22. It is required to divide the number 24 into two such parts, that the quotient of the greater part divided by the less, may be to the quotient of the less part divided by the greater, as 4 to 1. Ans. 16 and 8« 23. A gentleman being asked the age of his two sons, answer<»d, that if to the sum of their ages 18 be added, ^ the result will be double the age of the elder ; but if 6 be ' taken ^o QUADRATIC EQUATIONS. Wl^ ttffeen fitfm the difference of ttueir age«, the remainder will be edual to the age of the yotmger. What then were their nfges? ^ Ans. SCand 12. 24. To find four numbers such, that the sum of the 1 st, 2d, and Sd shall be 13 ; the sum of the 1st, 2d, and 4th, 15 ; the sum. of the 1st, 3d, and 4th, 18 ; and lastly th^ sum of the 2d, 3d, and 4th, 20. Ans. 2, 4, 7, 9. 25. To divide 48 into 4 such parts, that the first increased liy 3, the second diminished by 3, the third multiplied by 3, and the 4th divided by 3, may be all equal to each other. Ans. 6, 12, 3, 27. QUADRATIC EQUATIONS. Quadratic Equations are either simple oh compound. A simple quadraticequatton, is that which involves' the jliqtare of the unknovOvi iquantityonly.. Asax^ = h. And the ^ktion of such quadratics has been aheady given 'in simple equattoms. A compound qtladr^atic equation, is that^which contains the square of the unknown quantity in one term, and the ^ ifilfst power in' another term. As ax^ -^^ hx ^c. All compound quadratic equations, after being properly r^dticed, fall under the thr^e following forms, to which they miliist ' always be reduced by preparing them for sola- tion. 1. x^ 'Y oxsz b 2. a^ -- ax 1=: b 3. jr* - oT = - A The general method of solving quadratic equations, is by what is caUed completing the square, which is ^s follows : 1. Reduce the proposed equation to a proper simple form, as usual, such as the forms above ; namely, by transposing all the terms which <C6ntain the unknown quantity to one side of the elation, and the known terms to the other; placing the square term first, and the single power second ; dividing the equation by the co*efficient of the square or first term, if it has one, and changing the signs of all the terms, when that terhi happens to do negative, as that term must always be made positive before the solution. Then the proper solution is by completing the square as folio W8| viz. 2. Complete xYr S40 ALGEBRA. 2. Complete the unknown side to a squarei in this man- ner»' viz. Take half the co-efficient of the second term, and square it } which square add to both sides of the equation, then that side which contains the unknown quantity will be a complete square. 3. Then extract the square root on both sides of the equation *, and the value of the unknown quantity will b^ determined. * ^l^l^he square root of any quantity may be either + or — , therefore all quadratic equations admit of two solutions. Thus, the square root of + n« is either + nor-.ii5 for+nx+it and — n X — n are each equal to + n*. But the square root of — n*, or \/— n% is imaginaiy or impossible, as neither + n nor — n, when squared, gives — n\ So, in the first form, x^+ax:nb, w here x - f ia Is found =: ^b + i^*, the root may be either + ^FTT^, or — i/hTj^S since either of them being multiplied by itself produces b -f ^\ And this ambiguity is expressed by writing the uncertain or double Vign ± before •/^ + |^' ; thus x =z ± •^ + ^a" — ^a. • In this form, where x r: ± \^b + ^a^ — itf» the first value of X, viz. X zr + \/b + Ja« —^a, is always affirmative; for since ^* 4- ft is greater than ^*, the greater square must neces- sarily have the greater root ; therefore \/ft + ^a* will always be greater than •Jo*, or its equal ia-, and consequendy + ^b + .^* — ^ will always be affinnative. The second value, viz. x iz — ^b -f- i«* — i^ will always be negative, because it; is composed of two negative terms. There- fore when x' + ax — b, we shall have x zz + ^/b + Ja' — ^a for the affirmative value of x, and x r: — ^ft -f -Ja^ — ^a for the negative value of x. % * . In the second form, where x = ± v^6 4. ^^ -f ^a the first value, viz. x = + v^ft i- |a^ + |fl is always affirmative, since it is composed of two affirmative terms. But the second value, viz. JP = -^ -v/^ + -Ja^ + io, will always be negative j for since ft + Jos is greater than Ja^ therefore ^/b + ^a* wi ll be gr eater than \/ia% or its equal ia 5 and consequently -^ v^ft + ia' + ia is always a negative quantity. » Therefore, QUADRATIC EQtJA^tONS- Sit determined, making the' root of the known side either + or — , which will giv6 two roots of the Equation, or two values of the unknown quantity* 'Notey 1. The root of the first side of tKe equation, is always equal to the rpdt of the first term, with half the co-efficient -of the second tei^m joined to it, with its sign, whether '+ or — • ' ' * 2. All equations, in, which there are two terms including the unknown qijiantjt jr, and which have the index of the one just double that^ of the other, are resolved like quadratics, by completing the square, as above. 1" Thus, x^ + ^-^ =* K or ^^ + ^^ ™ *> o** ^ + ox^ :2:^ are the same as quadraticsi and the value ,of the unki^qwo quantity may be determined accordingly* ./ t ■ ^ i < ■ M <*■ III I ' 1*1 I II I II ■ I III II ■ ' I I II I U ' 1 • Therefore, when a^ — ax.zzb, we shall have x zf - f . y^^ 4- ^ + la for the affirmative value of a:; and j? =* — y6 -f ia^+^a for the negative vake of a? j i^ that' in both the first and second forms> the unknown quanti^ has alwi^sbtro values^ one of which is positive^ and the otb^r negative. But, in the third form, where ar =z ± v'ifl* — ^ + 1«* both^tho values of X will be positiv e/ when Ja^ is greater tli^n 0, FoxthfS first vjflue, viz. ar' =: -f ^/^a^ - b + -Ja will then be affirmative, being composed of two affirmaiiveteirmd; !«> The secQnd value, viz* .^r 5= .;--t;. Vi^^ — 6 4- . J^ y a^jna* tive also 5 *for since ^o^ ,5 jrreatei; than Ja* — 6^, therefore, Vi** ot ^a is greater than \/Ja^ -r 6^ aqd consequently -r \/Ja*— 6.^ fa will always be an affirmative quantity. So that, when afi ^ ax = ^— 5, we shall have a: =: 4- \/|o* — ^ + ia, and,alsp x = — v^Jo* — 6 + f aj for the valuesDf X, botti positive^ * ' But in thifc third forpa, if b be greater than ^a?, the solution of th6 proposed -question wilt be impossible/ For ^ince the sqUdre of any quantity (whether jthat quantity be affirmative or negative) is always amrinative, the square 'robt^of a negdfcive quaotily ik im- possible, and cannot be assigned. But when b is greater than Ja^ thenirt*'— b is a negative quantity j add therefore its root V'Jo* — ^ is impossible, or imaginary; consequently, in that case, j: n ^a ± ^^a^ — b, or thfe two toots or values of Jf, are\>otli impossible, 01 imaginary qu^tities; .1 Voii. L R ' EXAMPLES. ^^ 34f ALGEBRA. EXAMPLES. 1. Given jt* + 4j: = 60 •, to find r. First, by completing the square, or^ '\'4fX + 4 ^ i4\ Then, by extracting the root, jr + 2 =z ± 8 j Then, transpose 2, gives x = 6 or — 10, the tw« roots. 2. Given x*— 6x + 10 = 65 ; to find 4% First, trans. 10, gives j;^— 6r = 55 ; Then by complet. the sq. it is x^—'Sx + 9 =: 64 j And by extr. the root, gives x-^S =: ±: S'f Then trans. 3, gives j* = II or — 5. 5. Given 2-r* + ar-30 as 60; to find x. ^ - First by transpos. 20, it is 2jr* + Bar rr 90;' Then div. by 2, gives .r* + ^'*'= 4-5*; And by compL the sq. it is jt* + 4jr -f- 4 = 49; "-Then extr. th e roo t , it is ar +-^ =^ ± 7}' " ' «. Ahd iransp. 2, gives Jr = 5 or — 9. 4.' Given So:'- - Ix V '^* = ^4-^ to find .r. \ . . ' . . First diy. by 3, giv^s ,ar*— jt + 9f= 21-; . Tbeja transpos. ^3, giye« jt*— itr ni'— I:} i ' : • -J . i And compl. the sq. gives ar^-^x + ^ =s ^; t Then extr.- the root gives or — 4^ = dt -Jr* ^ And transp. 4^, gives x = J or ^. J5. Given i^*— ^ +.30i =,52* ; tp find jt. First by transpos. 30 j., it is -Jjr*— ^jr as 22^; Then mult, by 2 gives jr*— l^r = 44^ ; And by compl. the sq! it is x*— |!rH- ^ = 44$; .' Then extr. the root, gives \r—^ = -t 6^; And transp. -J-, gives o^ = 7 or — 6-y. 6. Given ijr*— ^jr = r ; to find x. he First by div. by a, it is 1:* x » •^; ' Then compl. the sq. rives x* x + -:~x = 1 r-» And extrac. the root, gives X --=. ± ^-^j^-; r,^, i . ,4ac + i* A Then transp. —, gives x = ± ^-_^_+ — 7.. Given x* — 2<wr* =: h\ to find r. First by compl. the sq. gives x^ -* 2a4r* + a* i= a* +*s ^ QUADRATIC EQUATIONS. 24^ And extract, the root, gives x*- ^a'=i ±, ^/et + h\ Then transpos. j, gives a;*=±v'^* + * + *» iri tm ■*■ And extract, the root, gives ^ = ± v'* i V^.^ + ^* And thus, by always tising similar words at each lihe, the pupil will resolve the following examples. EXAMPLES. FOR PRACTICE. ) I.. 1. Given x*— 6x— 7 == 33 j to find ^r, i^wJf^ 10. 2. .Given x*— 5jr— 10 = 44; to find x. Ans. ar = 8. 3. Given 5jr* + 4x — 90 = 114 ; to find x, Ans. x = 6- 4. Given -J^x*— ^x + 2 » 9; fo find x. . A6s. > = 4. 5. Given 3x*— 2x* = 40 ; to find x. Ans. x == 2. 6. Given -fr— ^J-V^*^ = ^il to find x,. Ans. x = 9. 7. Given ix* + |x = | ; to find x. Ans. x = '727766. S. Given 05^ + 4x^ =» 12 i to find .r. Ans. .r =^ = 1-259921. 9. Given x* + 4x = tf* + ^5 to find x. ' . Ans. jr = y a*^^- 2. •<^lSTIpNS PRODUCING qUADRATiC EqUATIOMS. 1. To find two numbeps v^hose difference is 2, and product 80. Let X and y denote the two required numbers *. * Then the first condition gives x—yzz 2, f And the second gives xy = 80. Then trsnsp. y in the 1st gives x » y + 2 ; This value of x substitut. in the 2d, i8^*4-2y 5= 80; Then comp. the square gives j^ + 2y + l=8i; And extrac. the root gives ^ +1=9^ And transpos. 1 gives y = 8; « And therefore x ssj/ 4-2 = 10. * These questions^ like those in simple equations^ are also solved by using a^ many unknown letters, as are the numbers required, for the better exercise io reducing equations ; not aim« ing at the shortest modes of solution, which would not' afford so much useful practice. R 2 2. To 244f ALGEBRA. 2. Tq' divide the number 14- into two such parls> that' their product may be 48. ■ > • Xet X and 7/ denote the two numbers. . T^ien i^e Ist condition gives x + t/ « 14, . j,Apd the 2d gives xy ==48. Then transp. ^ m the 1st gives * = 14 -3/ ; This value subst. for x in the 2d, is 143/-^ = 48 ; Changirig all the signs, to make the square positivejj ' ^vesy-14y=-48; - Then compl. the square gives /— 14j; + 49 = 1 > And extrac. the root gives 5^— 7 = i 1; Then tnnspos. 7> gives > .= aor «> the two parts. » , . ■ ' ' S. Given the sum of two nunibers = 9, and the sum of their' sqiiareV = 45 ; to find those niimbers. Let jr and y denote the two numbers, * Then by* the 1st condition x + y = 9. . A"nd'bythe-2d:r*+/^45. ' ' Then transpos. y ixK tho. 1 st.gives xz^S-^y^ Tl^% value '.sjbst. in jJ>e^2d gives 81 - 18;; + 2/ = .45 ^ Then transpos. .8J, give? 2/- ISy = — 36 •, And dwiding by 2 gives / — 9y ±= — 1 8 -,' 8 Z — 9 . ,: The^ con>pL the sq. gives y^-9y -^ V And extrac. the root gives ;;— | = ± i ; Then transpos. ^ gives j? == 6 or 3, the two numjpers., 4. What two numbers arc those, whose sum^ pro4uct,' and difference of their squares, are all e^ual to each other ? ' Let .«• ^ndj/ denote the two numbers. Then the ist^md 2d expression give x + y sz xy^ And the 1 st and 3d give at + j? = *"*—/• Then the la^ eqoa. div. by x + y^ gives 1 ^ x^-y ^ 'Axi^ -transpos. y^ gives y -^ \ i^ x\ This vaL substit. in the 1st gives. 2;; + 1.3±:y +31^ And transpos. 2^, gives 1 3= /— y 5 Then complet. the sq. gives i| isc y* —)••{- ^f- ; And extracting the root giVea iV 5 = y — 4 j And transposing ^ gives 4>v/5 + t == y > And therefore \r = y -f- 1 = ^^ 5 + 4. And if. thc^e. expressions be turned intp numbers, by ex- tracting the root of 5, .&c, they give ar = 2*6I8a + , andy= 1-6180 +• .- ' * ' - 5. Theye are four numbers in arithmetical progression, of which QUADRATIC "E^^UATIONS. fe4S wMch the product of the two extremes is 22f,'and that of the means 40; what are the numbers? . ^ Let X = the less extreme, and J/ ^ the common diflferencc^' - Then x, .r+y, jr+2y, jr+Sy, will be thefourntJmbcrs. Hence by the Irt c<»dition i* + diry « Sa, And by the 2d x^ -{- 3xj/ + 2y =? 40,< Then subtracting the first from the 2d ^ives 2y* =5 18 j And dividing by 2 giv^y* — 9j And extracting the root gives y = 3. . Then substit. 3 for y in* tie 1st, gives 4:* .+ 9t = 22; And completing the square gives :r* + 9.r + y =s '|»j Then extracting the root gives > + 4 =*: V,> And ilransposing f give^ x =» 2 the least number. Hence the four numbers «re 2, 5, 8, 11. 6. To find S numbers in geometrical progression^ whose fjMnn shall be 7, and the sum of their squares 21, . Let XyT/j and z denote the three numbers sought. Then by the 1st condition 0% ±=j/% And by the 2d x + 1/ + ;ar =7, And by the 3d jr* +y + «* = 21. Transposing y in the 2d giy?es x + z =7 —3/$ Sq. this equa. gives x"^ + 2xz +;8^ = 49 — 14j^+j/^} Substi. 2y* for 2jr2f, gives x'^+ 2y*+ ;2^= 49 — 14y -|-y* ; Siibtr. j/* from each side, leaves ^* + y* + ^= 4?9 — 1 4y ; Putting the two values of x' + v* + 2* 7 ^ i ^ n ia.. .0 • 1 V •^ ' ' > 21=49-' 14V; equal to each other, gives 3 "^ ' Then transposing 21 and 14y, gives 14y = 28; And dividing by 14, gives y = 2. Then substit. 2 fory in the 1st equa. gives xz =; 4, And in the 4th, it gives ^4-^ = 5; Transposing z in the last, gives x ^ S^z%' This substit, in the next above, gives 5lf—z^ :^ 4; : Changing all the signs,.gives «*— 5a == --4^ . Then completing the square, gives «* — 5a; + V ^ ii And extracting the root gives 3 — 4- = i'4» Then transposing 4, gives z and x = 4 and 1} the twa , . other numbers; So that the three numbers are }, 2, 4» ' QUESTIONS FOR PRACTICE. - 1. Wha,t number is that which added to its square makes ♦2? Ans. 6. a- Ta S46 ALGEBRA. 2, To find two numbers such» that the less may be to the greater as the greater is to 12j and that the sum of their squares may be 45. Ans. S and €• S. What two numbers are those, whose difference i$ S^ and the difference of their cubes 98 ? Ans. S and 5* 4. What two numbers are those whose sum is 6, and the sum of their cubes 72 ? Ans. 2 and 4. 5. What two numbers are those, whose product is 20^ and the difference of their cubes 61 ? Ans. 4 and 5. 6. To divide the number 1 1 into two suqh parts, that the product of their squares may be 784. Ans. 4 and 7. 7. To divide the number 5 into two such parts, that the sum of their alternate quotients may be 44, that is of the two quotients of each part divided by the other. Ans. 1 and 4. 8. To divide 12 into two such parts, that their product may be equal to 8 times their difference. Ans. 4 and 8. 9. To divide the number 10 into two such parts, that the square of 4 times the less partf may be 112 more than the ^uare of 2 times the greater. Ans. 4 and 6. 10. To find two numbers such, that the sum of their squares may be 89, and their sum multiplied by the greater may prbduce 104. Ans. 5 and 8. 11. What number is that, which being divided by the product of its two digits, the quotient is 5^ ; biit when 9 is subtracted from it, there remains a numberil|Naving the same digits inverted ? Ans. 32. 12. To divide 20 into three parts such, that the continual product of all three may be 270, and that the difference of the first and second may be 2 less than the difference of the second and third. Ans. 5, 6, 9. 13. To find three numbers in arithmetical Sprogression, such that the sum of their squares may be 56^ and the sum arising by adding together 3 times the first and 2 times the second and 3 times me third, may amount to 28. ^^ Ans.^'2, 4, 6. 14. To divide the number IS into three such parts, that their squares may have equal differences, andj that the sum of those squares may be 75. * Ans. 1, 5, 7, 15. To find three numbers having equal differences, so that their sum may be 12, and the sum of their fourth powers 962. Ans. 3, 4, 5. 16. To CUBIC, &c. EQUATIONS. 2« 16. To find three numbers having equal differences, and such that the square of the least added to the product of the two greater may make 28, but the square of the greatest added to the product of the two less may make 44. ' Ans. 2, 4, 6. 17. Three merchants, A, b, c, on comparing their gains find, that among them all they have gained 1444/.; and that 9^s gain added to the square root of a's made 920/. ; but if added to the square root of c^s it made 912. What were their several gains ? Ans. A 400, B 900, c 144. "18. To find three numbers in arithmetical progression, so that the sum of their squares shall be 93 ; also if the first be multiplied by 3, the second by 4, and the third by 5, the sum of the products may be 66» Ans 2, 5, 8. 10. To find fpur numbers such, that the first may be to the second as the third to the fourth ; and that the first may be to the fourth as 1 to 5 ; also the Second to the third as 5 to 9 ; and the sum of the second and fourth may be 20. , Ans. 3, 5, 9, 15. 20. To find two numbers such, that their product added ^o their sum may make 47, and their sum taken firom the sum of their squares may leave 62. Ans. 5 and 7. RESOLUTION OF CUBIC AND HIGHER EQUATIONS. A Cubic Equation, or Equation of the 5d degree or power, is one that contains the third power of the unknown quantity. As a?^— (ur* -j^ bx =:c, A Biquadratic J or Double Quadratic, is an equation that contains the 4th power of the unknown quantity : As x^ — ax^ -f bx^—cx = d* An Equation of the 5th Power or Degree, is one that contains the 5th power of the unknown quantity : As pe^-^ax* + bx^-^cxP' + dx =^ e. And so on, for all other higher powers. Where it is to be noted, however, that all the powers, or terms, in the equation, are supposed to be freed from surds or fractional exponents* There are many particular and prolix rules usually given (pr the solution of some of the above-mentioned powers or (tiS ALGEBRA. . :or equations* • But they may be all readily solved by the following easy rule of Double Position, sometimes called Trial-and-Error. RULE. « 1. Find, by trial, two numbers, as near the trtie roofas you can, aiid substitute them separately in the given equation, instead of the imknown quantity 5 and find how much the terms collected together, according to their signs + or — , differ from the absolute known term of the equation, mark- ing whether these errors are in excess or defect. 2. Multiply the difference of the two numbers, found or taken by trial, by either of the errors, and divide the pro- duct by the difference of the errors,. when they are alike^ but by their sum when they are unlike. Or say. As the difference or sum of the errors, is to the difference of the two numbers, so is either error to the correction of its sup- posed number. 3. Add the quotient, last found, to the number belonging to that error, when its supposed number is too little, but subtract it when too great, and the result will give the true root nearly. 4. Take this root and the nearest of the two former, or any other that may be found nearer \ and, by proceeding in like manner as above, a root will be had still nearer than before. And so on to any degree of exactness required. Note ] . It is best to employ always two assumed numbers that shall differ from each other only by unity in the last jBgure on the right hand j because then the difference, or multiplier, is only I. It is also best to, use always the least error in the above operation. Note 2. It will be convenient also to begin with a single figure at first, trying several single figures till there be found the two nearest the -truth, the one too little, and the other too great ; and in working with them, find only one morQ figure. Then substitute this corrected result in the equation^ for the uftknown letter, and if the result prove too little, substitute also the number next greater for the second sup- position ; but contrarywise, if the former prove too great, then take the next less number for the" second supposition ; and in working with the second pair of errors, continue the quotient only so far as to have the corrected number to four places of figures. Then repeat the same process again with ^us last corrected number, and the next greater or less, *s th^ CUBIC, &c. EQUATIONS, 249 the case may require^ carrying the third corrected number to eight figures; because each new operation commonly doubles the number of true figures. And thus proceed to any extent that may be wanted. Examples. Ex. 1. To find the root of the^ubic equation x^ +**+ 9c = IDO, or the value of * in it. Here it is soon found that x lies between 4 and 5. As- sume therefore these two num- bers, and the operation will be 3IS fellows : 1st Sup. 2d Sup. 4 - X - 5 16 - ^* . 25 64 X' 84 - sums - 100 but should be 125 155 100 Again, suppose 4*2 and 4'3, and repeat the work as foU lows: 1st Sup. 2d Sup. 4'2 - jr - 4*3 17-64 ^ js* ^' 18-49 74-088 - x' - 79-507 95-928 100 sums 102-297 100 — 16 - errors - +55 the sum of which is 71. Then as 71 : 1,:: 16 : -2. Hence »r = 4'2 nearly. — 4-072 errors + 2-297 the sum of which is 6*369. As 6-369 fl :: 2-297 : 0*036 This taken fi-om - 4* 300 leaves x nearly = 4*264 Again> suppose 4-264, and 4-265, and work as follows: ' 4-264 ^ X - 4-265 18*181696 - ;^* - . 18*190225 77-526753 99-972448 100 X sums 77-5S1310 100-036535 100 -0-027552 - errors - +0O36533 the sum of which is -064P87. Then as -064087 : -001 : : -027552 : 0-0004299 To this adding - 4-264 gives X very nearly = 4-2644299 The 250 ALGEBRA. The work of the example above might have been much shortened, by the use of the Table of Powers in the" Arith- metici which would have given two or three figures by in- q>ection. But the example has been worked out so particu- larly as it is, the better to show the method. Ex. 2. To find the root of the equation x^ — 15jr* + 63x ss 50, or the value of x in it. Here it soon appears that x is very little above 1. Suppose therefore 1*0 and 1-1 > and work as follows : 1-0 - 1-1 63*0 - -15 1 *- 6^x - -15jr* x^ - sums - errors - sum •f the :1 ::1:'03 100 69-3 -18-15 1-331 49 - 50 52'4S1 50 -1 - 3-481 As 3*481 ; + 2-451 errors. correct. Hence x'= 1*03 nearly. Again, suppose the two num* bers 1-03 and 1-02, &c, ad follows : 103 - AT - I '02 64-89 - 63Jf 64*26 - 15-9135 — 15;v*- 15-6060 1-092727 x^ 1-061208 50-069227 sums 49*715208 50 50 + -069227 errors — •284793 •284792 As -35401 9 : -01 : : -069227 : -0019555 This taken from 1 '03 leaves 4: nearly = 1*02804 Note 3. Every equation has as many roots as it contains dimensions, or as there are units in the index of its highest power. That is, a simple equation has only one value of the' root ; but a quadratic equation has two values or roots, a cubic equation has three roots, a biquadratic equation ha$ four roots, and so on. And when one of the roots of an equation has been found by approximation, as above, the rest may be found as follows. Take, for a dividend, the given equation, with the known term transposed, with its sign changed, to the unknown side of the equation ; and, for a divisor, take x minus the root just found. Divide the said dividend by the divisor, and the quotient will be the equation depressed a degree lower th^n the given one. Find CUBIC, &c. EQUATIONS. 251 Find a root of this new equation by approximation, as before, or otherwisei and it will be a second root of tlye original equation. Then, hj means of this root, depress the second equation one degree lower, and' from thence £nd a third root, and so on, till the equation be reduced to a quadratic ; then the two roots of tjiis being found, by the method of completing the square, they will make up the remainder of the roots. Thus, in the foregoing equation, having found one root to be 1 "02804, connect it by minus with r for a divisor, and the equation for a dividend, &c, as follows : X - J -02804) x^ -• ISx" + eSx - 50 ( x* — 13-97l96r + 48-63627 = 0. Then the two roots of this quadratic equation, or ' X* - 13'97196jr se - 48*63627, by completing the square, are 6'57653 and 7-39543, which are also the other two roots of the given cubic equation. So that all the three roots of that equation, viz. x^— 15x^ + 6Sx = 50, d fi« 576*58 J^^d the sum of all the roots is found to b^ j»T«ofi'.iofl5, beine equal to the co-efficient of the and 7-39543 V ^ 1 ^r 1 • 1 • t_ 1 i- >2d term of tlje equation, which the sum of 15-00000 V^^ roots always ought to be, when thejr / are right. sum Note 4. It is also a particular advantage of the foregoing rule, that it is not necessary to prepare the equation, as for other rules, by reducing it to the usual final form and slate of equations. Because the rule may be applied at once to an unreduced equation, though it be ever so much embarrassed by surd and compound quantities. As in the following example : Ex. 3. Let it be required to find the root x of the equation Vl*4**-(*' + 20)'- + -v/i96A?"--(*' + 24)* =z 114, or the value of X in it. By a few trials, it is soon found that the value of x is but little above 7. Suppose therefore first that at is = 7, and then X = B. First, t$2 ALG£BRA. Firsts when .r s= 7j Second, when x ±z B. I 47-906 - ^144x*— (a:* + 20)» - 46'476 65-384 - v^ 196^* -(a:* +24)2 . 69*283 ■ > ■ ■ ■ ■ I 'I >' II 1 1 3-290 - the sums of these - 1 1 5*759 114*000 - the true number - IH'OOO —0-710 - the two errors - +1*759 + 1-759 ■ V As 2-469 : 1 :: 0*710 : 0*2 nearly 70 Therefore t = 7*2 nearly Suppose again jt = 7-2, and then, because it turns out toa great, suppose jc also = 7*1, &c, as follows : Supp. jr' = 7*2 Supp. ;r =: 7'I 47*990 - *Vl44:r*-~(^* + 20)* ^. 47-973 66*402 - ^/196jr*~(a:* + 24)* - . 65*904 114*392 - the sums of these - 113*877 1 1 4*000 - the true number - 1 1 4*000 +0*392 - the two errors - -0*123 0*123 ■ I ' As -515 : -1 :: '123 : -024 the correction, 7*100 add Therefore X = 7*124 nearly the root required. Note B, The same rule also, among other more difficult forms of equations, succeeds very well in what are called exponential ones, or those which have an unknown quan* tity in the. exponent of the power ; as in the following example : • Ex. 4. To find the value of x in the exponential equation x"" = 100. For more easily resolving such kinds of equations, it is convenient to take the logarithms of them, and then com-, pute the terms by means of a table of logarithms. Thus, the logarithins of the two sides of the present equation are ff X log, CUBIC, &c. EQUATIONS. 235 X X iog. of iT = 3 the log. of 100. Then, by a few trials, it is soon perceived that the value of x is somewhere be- tween the two numbers S and 4, and indeed nearly in the middle between them, but rather nearer the latter than the former. Taking therefore first x = S'5, and then n 2*6, and working with the logarithms, the operation will be as follows : First Supp. X =« 3*5. Log. of 3-5 =: 0-54.4068 then 3*5 x log. 2^'S^ 1*904238 the true number 2-OQOObO error, too little, — •095762 002689 Second Supp. x = 3'6. Log. of 3-6 = 0-556303 then 3-6xlog.3-6=2-00&689 the true number 2-000000 error, too great, +.002689 •098451 sum of the errors. Then, As '098451: '1 ; : '002689 : 0-00273 the correction taken from 3-60000 leaves - 3-59727 = x neatljr. On trial, this is found to be a very small matter too little. Take therefore again, x = 3-59727, and next = 3*59728, and repeat the operation as follows : * Firtt, Supp. i" 2= 3-59727. Log. of 3-^9727 is 0*555973 3-59727 X log. - of 3-59727 = 1-9999854 the true number.2-0000000 error^ too little, -0-0000146 - 0-0000047 Second, Supp. or = 3-59728. Log. of 3-59728 is 0*555974 3*59738 X log. of 3-59728 = 1-9999953 the true number 20000000 error, too little, - 0-0000047 0-0000099 diff. of the errors. Then, As -0000099 : -00001 : : -0TO0047 : 0*00000474747 the cor. added to - 3-59728000000 gives nearly the value oi.^ = .3-59728474747 Ex. 5. To find the value of x in the equation jt^ + lO^r* + 5x = 260. Ans. x = 4-1 179857. Ex. 6. To find the value of x in the equation ;r' — 2;r = 50. Ans. 3-8648«i. ^^^ Ex. 7. H -. U4 ALGEBRA. Ex. ?• To find the value of x in the equation jr' + 2x* — 2Sx = TO. Ans. x = S'lS^S?. Ex. 8. To find the value of x in the equation x^ - IIa:^ + 54* = 350. Ans. x = 14"95407. Ex. 9. To find the value of x in the equation x^— 3x* — 75x = 10000. Ans. x = 10-2609. Ex. 10. To find the value of x in the equation 2x*- 16x' + 40x*— SOx = - 1. Ans. x = 1-284724. Ex. 1 1 . To find the value of x in the equation x^ + 2Af* + 3*3 + 4x* + 5;t = 54S21. Ans. x = 8-414455. Ex. 12. To find the value of x in the equation x* =-- 1234567^9. Ans. x = 8*6400268. Ex. 13- Given 2x*-7x5 -f llx^-Sx = 11, to find x. Ex. 14. To find the value of x in the equation {3x''- 2v/ Jr + 1 )^ - (t* - 4Xv^x + 3s/x)^ = 56. Ans. X = 18-3i0877. 2a resolvf Cubic Equations by Cariarfs Rule. •Though the foregoing general method, by the applica^ tion of Double Position, b^ the readiest way, in real practice^ of finding the roots in numbers of cubic equations, as well as of all the higher equations universally, we may here add fhe particular method commonly called Cardan's Rule, for resolving cubic equations, in case any person should choose occasionally to employ that method. * The form that a cubic equation must necessarily have, to be resolved by this rule, is this, viz. z^ :|e js =» 5, that is, wanting the second term, or the term of the 2d power z\ There/ore, after any cubic equation has been reduced down to its final usual form, x^ + px^ 4- ^x = r, freed from the coefficient of its first term, it will then be necessary to take away the 2d term px^ ; which is to be done in this manner : Take \py or \ of the coefficient of the second term, and annex it, with the contrary sign, to another tmknown letter s, thus z—-jpf then substitute this for x, the unknown letter in the original equation x' +^* + ^x = r, and there will result this reduced equation z^ ^ a» zz i, of the form proper for applying the following, or Cardan's rule. Or take r = far, and d = 4^, by which the reduced equation takes this form, ;&' )|c ^cz^ 2d. Then CUBIC, &c. EQUATIONS. 255 Then substitute the values of ^r and d in this form, z =l^d+ V(^* + ^0 +{/^- V(^*+f^), or * = V^+V(^T^-|^^7^^, and the value of the root z, of the reduced equation s^ 4b az = i, wjll be obtained. Lastly, take x = «— -j^* which will give the value of x, the required root of the original equation x^ + px^ + qx czr^ first proposed. One root of this equation being thus obtained, then de- pressing the original equation one degree lower, after the manner described p. 250 and 251, the other two roots of that equation will be obtained by means of the resulting quadratic equation. N^e, When the coefficient a, or c, is negative, and c* is greater than d''^ this is called the irreducible case, because then the solution cannot be generally obtained by this rule. Ex. To find the roots of the equation x^^Sx^ + lOxsrS. First, to take away the 2d term, its coeflicient being ^ 6y its 3d part is — 2 ; put therefore ;r = « + 2 ; then 4r' = z3^6x*+ 12a -f 8 - 6x^ = - ez^ - 24.Z - 24 -J^lOx = + lOz + 20 theref. the sum z^ ♦ — 2z + 4=8 or z' * — 22 = 4 Here then «= — iS, ^ = 4, ^== — |, rf = 2. Tiieref. 4/£/4--v/K+^) = e/^i5+ v/(4-,V)= ^2 + -/ i^o=: i/2 + V^V^ = 1-57735 and 3 /i-^(^H-^ ^) = >^^2-^/(4-:^V)=^2 - ^/'^^ = 4/2 — V>/3 = 0-42265 then the sum of these two is the value of z = 2. Hence :r zz z + 2 = 4, one root of x in the eq. r'— 6x* + lOo: = 8. To find the two other roots, perform the division, &c, as in p.. 251, thus : x-4) J^— 6jr*+ 10jr-8(jr"-2;r + 2=B0 -2x^+ IOt -2ar*+ Sjt 2ar— 8 2ar— 8 Hence tS6 ALGEBRA. * Hence x^—2it = — 3, or a?f — 2ir +1 r: — 1, and r — 1 = ± \/-'l I r c=: 1 + v' —1 or = .1 - v' - *» the two other sought. Ex.2. Tofindtherootsof jt'— 94:*.+ 28Jr = 30. Ans. X :i= 3, or = 3 +v/ — 1, or =3 — -v/ — I. Ex. 3. To find the roots of x'^r-T-f* + 14r =t 20. Ans. x = 3, or = i + ^ — 3, or =* 1 — iy/' — 5. OF SIMPLE: INTEREST. As the interest of any sum*, ft)r any-time, is directly, pro- portional to the priocijxil sum^ and. to the time ; there^re • the interest of i pound) for 1 yoar> beine multiplied by any given principal sum', and by the time of its foH^earance^ in y^ars and part$> will give it^ interest for that time. That isj if there be put : # r = the rate of interest of 1 poimdper annum, p = any principal sum lent, / = the time it is lent for, and a = the amount or sum of principal and interest; then IS prt = the interest of the sum p; for th^time /, and conseq. p +prt or p X {I +rt) = a, the'amount for that time. From this expression, other theorems can easily be de- duced, fo}r finding arty -of the qtiaiui£jes abpVe mentioned: ^ which theorems, collected together,. will be as below i 1st, a "=: p + prfy the amount, ^^' ^ =" 1 4- rt *^^^ principal, ^ - a—p , 3d, r = — -y the rate, pt 4th, / = , the time. pr » For Example. Let it be required to find, in what time any principal sum will double itself, at* any rate of simple interest. In this case, we must use the first theorem, a = /) + prty in which the amount a must be made = 2p, or double the principal, that* i», /) + prt ;= 2/?, ov-prt = /;, or r/ = 1 ; and hence / «= — . r Here, COlaPOUND INTEREST, $Sl Here^ r being thfe iaterest of 1/^ for 1 yearj it follows, that the doubling at simple interest, is equal to the quotient t)f any sum divided by its interest for 1 yfear. So, if the Ihate o^ interest be 5 per cent, then 100 -^ 5 =i^2d, is^the time of doubling at that rate. Or the 4th theorem give^ at 6nc6 a—p ^p — p 2—11 ^ = -— ^=± -r~ = — 1—==— i the same as before* pr pr TV k! Compound iNXEREST? •■ . » Besides the quiantities concerned io Simple ]^tfrest| hamely^ p ^,the principal sum, * r =r the ratie or interest of 1/. for 1 year, a = the whole aittoiim of the principal and interest, i = the time, th^re is another qUatltity employed in Compound^ Interest, viz. the ratio of the x^te of interesti which is the amount pf 1/. for 1 time of payment, and which here let be denoted by R, viz. R. ax i 4- ^j the amount of lU for 1 time. Then the particular, amounts for the several tiities rpay he thus coiiipflted, viz; As I/« is to its amount for any time, isb is any proposed priiicipal sum> to its amount for die same time \ diat is, as 1/. : R : : p : pR, the Ist yearns amount, 12. : R : : pR : j^R*, the 2d year's amount, 1/. t R : ; pR* t j^R', the 3d year's amount, and so on. . , . Therefore, in general, pBl^ =i a is the amount for the t year,, or t time of piiyiiaent. Whence the following general theorems are deduced : , Ifit, a = /RV the amount^ a 2d, ^ = n^t ^^ principal, 3d, R = .J/—, the ratio, ^ _ log. of <7 — log. of^ , : Vol. I. S From 1>5« ALGEBRA. ,v. From which, anyone 6( th^'qu&mmes imify be foElndr Wh^n the rcsft are giteh^ A3 to the whole interest, it is found.by barely subtracting the principal p from the amount a. Example\ Suppose it be required to find, jn how many years any principal, sum will double itself, at any proposed rate of cote[^Qttn4 interest. In this case the 4th theorem must be employed, making 41 = 2/ ; and then it is log. fl— log./ log. 2^— log. p _ log. 2 "" log. Rk "" log. R. " log. R' So, if the rate of interest be 5 per cent, per annum; then lR BB 1- + *05 rr 1*05 j and hence . ^ . log. 2 -301030 , ^ ^^^^ , . ' / = .:; — ^-— : = ■ ^ ,^^ = 14-2067 nearly ; log.J-05 -021189 ^ ■ that is, any sum doubles' itself in 14^- years nearly^ at the rate of 5 per cent, per anmiixi compound interest* . Hence, and from the like question iti Simple Interest^ ' above given, are deduced the times in Which any sum doubles itiself, at several rates of interest, both simple and compound; Viz. ' • . ii^mt^^i At' 2 24 3 3i 4 5 6 7 8 9. 10 ''AtSimp.Int > per cent, per annum interest, 1/. 6r any other sum, will double itself in the following years. m 50 40 33| 28f 25 22| 20 16|. 14| 12^ IH 10 At Comprint. •*»—«• ill 35-0028 . 28-0701 23-4498 20-14S3 17-6730 . 15-7473 fr 14-2067 § 11-8957* !0'2448 '9-0065 8-04S^ 7-2725 .1 The f COMPOUND INTEREST- 2SB The following Table wili very mttdi facilitate caloilitipiis of compound interest on any sum, for any number of yeacSf at various rat«s of imef est- The Amounts of 1/. in any Number of Tears. Yrs. 3 34 .4 ■■4,i . 1 '^i > 1*0300 1*0350 10400 1^0450 1*0500 1-06DO 2 1-0509 10712 IO8I6 10920 11025 1-1236 3 10927 1-1087 ]-124g 1-J412 1*1576 11910 4 V1255 1-1475 1-1699 1-1925 1-2155 1*2625 5 1.1593 1-1877 1-2167 1-2462 l-!?763 1-3392 6 11941 1*2293 1-2653 1*3023 1*3401 1-4165 7 1*2299 1-3723 1*3159 1-3609 1*4071 1*5036 ^ 8 1*2668 1-3168 1-3686 1*4221 1*4775 15939 9 1-3048 1 -3629 1*4233 1*4661 1*5513 1 i*669il 10 1-3439 1*4106 1*4802 1-5530 1-6289 17909 n 1-3842 1-4600 j-5«9^ xm^ 17103 1-8933. 12 1-4258 1*5111 I'60lO 1-6959 17959 2-0l2» 13 1*4685 J -5649, 1-6651 l'77'^2 1*«856 2-13'^- 14 1-5126 l*6l87* 1*7317 1-8519 1-9799 2-2609 15 1-5580 1-Q753 1-8O09 1*9355 2-0789 2-3966 16 1-6047 1-7340 1*8730 2*0224 2*1829 2-5404 17 1-6528 17947 1-9^79 2-1134 22920 2-6928 18 1-7024 1-8575 20258 2*2035 2-4066 2-8543 19 1-7535 1-9225 2- 1068 2*30/0 2-5270 3-0256 20 1-8061 i'989B 21911 2-4U7 2-6533 3-2071 \ The use of this Tafeje, which contains all the poivers, :E^j to the 20th power, or the anjounts of 1/, is cniefly tp c;^T- culate the interest, or the aoiount of any principal sum, for any time, not more than 20 years. For example, let it be required to find, to how much 92$I* will ai^punt in 15 years, at the rate of 5 per cent per fttmuisi ^oropQund interest. In the table, on the line 15, and in the column 5 per cent. is the amount of 1/, viz. - - 2*0789 this multiplied by the principsil - 523 ^ gives the amioimt or and therefoi*e the interest is 10S7-2647 1087/. 5/. '$IJ. 564/. 5s. 3^, Noie 1. When the rate of interest i» to be determined tt^ any other time than a fear ; its suppose to 4- a ye^r, 4^^^a year, &c ; the rules are still th« Mfiie \ '• but theii^ s^mHH S 2 express ^260 ALGEBRA. « express, that time» and r must be taken the amount (or that .time also. Note 2. When the compound interest, or amount, of any sum, is required for the parts of a year ; it may be deter- mined in the following manner : . 1//, For any time which is some aliquot part of a year :— Find the amount of 1/. for 1 year, as , before } then that root of it which is denoted by the aliquot part, will be the amount of 1/. This amount being multiplied by the prin- cipal sum, will produce the amount of the given sum as required. 2</, When the time is not an aliquot part of a year : — Reduce the time into days, and take the 365th root of the amount of 1/. for 1 year, which will give the amount of the same for 1 day. Then raise this amount to that power whose index is equal to the number of days, and it will be the amount for that time. Which amount being multiplied by the principal sum, will produce the amount of that sum as before.— And in these calculations, the operation by loga- rithms will be very useful. ' OF ANNUITIES- ANNUITY IS a term used for ""any periodical income^ arising from money lent, or from houses, lands, salaries, pensions, &c. payable from time to time, but mostly by annual payments. "" Annuities are divided into those that are in Possession, and those in Reversion : 'the former meaning such as have _ commenced ; and the latter such as will not liegin till some particular, event has happened, or till after some certain time has elapsed. When an annuity is forborn for some years, or the pay*- moits not made for that time, the annuity is said to be in Arrears. An annuity may also be for a certain number of years ;. or it may be without any limit, and then it is called a Per- petuity. The Amount of an annuity, forborn for any number of •years, is the sum arising from the addition of all the annul- .f i^s for that number of years j together with the interest due Upon eAchjafter it becomes due. . ^i. : 4j The ANNUITIES.' 261 The Present Worth or Value of an annuity, is the price or sum which ought to be given for it, supposing it to be bought off^ or paid all at once. Let a = the annuity, pension, or yearly rent ; n = the number of years forborn, or lent for ; ft = the amount of 1/. for 1 year i m = the amount of the annuity ; V = its value, or its present worth. Now, 1 being the present value of the sum R, by propor- tion the present value of any other sum a, is thus found : a • as R : 1 : : tf ; — • the present value of a due 1 year hence. In like manner — is the present value of u due 2 years a a A u o hence ; for r : 1 : : — : ~y- So also — ^ -^, — j, &c, will be the present values of a, due at the end of 3, 4, 5, &:c, years respectively. Consequently the sum of all diese, or — +-I+ A+r: + &c = {— + -5+-i + ri&<^-)x«» R R* R' k* ^R R* R^ R* ' continued to n terms, will be the present value of all the n years' annuities. And the value of the perpetuity, is the sum of the series to infinity. But this series, it is evident, is a geometrical progression, .1 . ' . liaving — both for its first term and common rsrtio, and the R ~ .•» number of its terms n ; therefore the sum v of all the terms^ or the present value of all the annual payments, will be Jl \^ l^ R R R" R" — 1 tf « = — -y— x«,or=j^3^x- 1 R -^ . ■■ When the annuity is a perpetuity; n being infinite, R** . 1 is also infinite, and therefore the quantity — becomes = 0, therefore " x -^ also = ; consequently the exprei- R — 1 R sion becomes barely v = — jr ; that is; any annuity divided by the interest of 1/. for } year, gives the value of .the per- petuity. So, if the rate of interest be 5 per cent. Then lOO/i -r- 5 = 20a is the value of the perpetuity at S per cent : Also IQOa -r- 4 = 25a is the value of the per- petuity r- 26^ ALGEBHA. • • • • petuifj' at 4 f)6r teni : Atid 100^ -r 3 £= SS-^ h tlie valut of the perpetuity ^t 5 pcp cent t ^nd so on. Again, because the amount of 1/. in n years, fe H", its increase in that time will be R*^— I ; but its interest for one single ytdr, or the annuity answering to that increase, is R — 1 ; therefore as R — 1 is to r" — 1, so is a to iw ; that r"— 1 is, m =: r X a. Hencc^ the several cases relating to R — 1 Annuities in Arrear, will be resolved by the foUowipg equations : m = X a ■= vR" i R — 1 . • R" i~ 1 a m R - i . R - 1 ^ |t» - 1 R" - 1 WR — W -f tf ■ log. log. >yf — log. v _^ ^i "" log. R ^ log. R * log. ^ »- log. V . X.pg. R =2 — -^ ' ■•r " # n 1 1 fl ^R.P V^ R - 1 '.Ih this last theorem, r denotes the present value 6f an annuity in reversion, after p years, or not commencing till after the first j& years, being found by taking the difference between the two values 7 x — r and x --;;:, for R — 1 R" R — I RP n jeTLi^ and p years. But the amount and present value of any annuity for any number of years, up to 21, will be most readily found by the two following tables. I - TABLE AumimES, 2$$ TABLE I. The Amoiint of an Anhiiity of 1/. at Compound Interest. im atSperc. M «^-^ "^ 1 2 r 3: 4 5 6 7 8 9 10 11 12 J3 14 15 16 ■.17- a8 19 20 21 3f per cJ 4 per c. .10000 ^3'OaOO 1*0000 2T03i0 .^fOpQgrl 3-iQQ2. 4-1836 5-^091 d-4684 7*6625 d-8923 10-1591 1 1 -4^9 12-8078 14-1920 15-6178 1 70863 18-5989 20-1569 >2l'5?6l6 33-4144 26-8704 28-6765 4-2149 5-3625 6-5502 7.7794 9-0517 10-3685 11*7314 13-1420 14-6020 16-1130 17-6770 19-2957 20-9710 227050 24-^^7 ^-3572 28-2797 1-0000 2-0400 a-i2i6 4-2^465 5-4163 6-6330 7*89.83 9-2142 10-5828 12*0061 13*4864 15:0258 16-6268 18-2919 20-3236 21-8245 S3-^75 25*6454 «7*6712 297781 4^ per c. 30-2695131-9692 1-0000 2*0,450 3-1370 4-278:? 5-4707 6-7169 8*0192 9*3800 10*8021 12-2892 13-8412 15-4640 171599 1 8-9321 207841 22-7193 24-7417 26*8J51 290636 31'3714 '33-7831 5 per c. 6 per c. 1-0000 20500 3-1525 4-3101 5-5256 6*8019 8-1420 9*5491 11-0266 i«*5779 14-2068 15*9171 17-7130 19-5986 21-5786 23-6575 25-8404] 28-1324 30-5390 33*0660 135*7193 1*0000 20600 6-1836 4-3746 5-6371, 6-9753 8-3938 j 9*8975 11-4913 13-1808 U»97i6 16-6699 18*8821 21-0151 23-2760 25-6725 28-2129 30-9057 33-76001 S6'7856l i 39-9927 1 tab;l^ 11. Th^ Pr^s^t ^Vs^lue of an Annuity of 1/. Yrs. 1 2 3 4 5 6 7 8 9 10 11 12 \3 14 15 16 17 18 19 20 21 at9p^c< 0-9709 l-9i35 2-8286 3-7171 4-5797 5-4172 6-2303 70197 7*7861 8-5302 9-2520 9-9540 10-6350 11-2961 11 -9379 12-5611 131601 13-7535 14-3238 14-8775 15*4150 3t per^.[4 per c 09662 1*8997 2-8016 3*6731 4-5151 5-3286 6-1145 6-8740 7:0077 8-3166 9*01 16 9-6633 iG-3027 :i 0-9205 11-5174 12*0941 12-6513 13*1897 13-7098 14-2124 14*6980 0-9615 1-8861 2*775 1 3-6299 4-4518 5-2421 6-0020 6-7327 7-4353 8-110J9 8-7605 9-3851 9*9857 10-563 1 11-1184 11-6523 12- 1657 li-6593 131339 13-5903 140292 4t per c. 0-956& 1*8727 2-7490 3-5875 4-3900 5-1579 5-89i7 6-5959 7-268S 7-9127 8*5289 9- 1186 9*6829 10-2228 10-7396 1 1 -2340 11-7072 12-1600 12-5933 130079 13-4047 5 perc. ii< >i 0*9524 1-8594 2-7233 3-5460 4-3295 5-0757 5->864 6-4632 71078 7*7217 8*3054 8-B633 9*3936 9*8986 10*3797 10-8378 11-2741 11-6896 12*0853 12-4622 •12*8212 li ■ per e. 0-9434 1*8334 2-673^ 3-4651. 4-2124< 4*9173 5-5824 6-2098 6*8017 7*3601 7*8869 8-3838 8-8527 9-29501 9*7^23 10*1059 10-4773 10-8276 11*1581 11*4699 117641 Ta sH ALGSBRA. n find thf Ammnt cf any annuity forbore a eertain numbir tf years* TiiKE out the amount of 1/. from the £r$t table^ for the proposed rate and time ; then multiply it by the given annuity; and the prodyct will be the amount , for the same number of years, -and rate of int^r$st^-^And the converse to find the rate or time, £xaf^. To £nd how much an annuity of 50/. will amount to in 20 years, at Si per cent, compound interest. On the line of 20 years, and in the column of 3, f^r cent, $tands 28*2797^ which is the amount of an annuity of l/« for the 20 years. Then 28-2797 x 5Q, givw 1413-985/. a:; ■ 1413/. 19/. 8d. for the answer required. * To find the Present Value of anyannwtt^foranj number of j^^tfrx.— Proceed here by the 2d table, in the same manner as iabove for the Ist tabl^, and the present; vorth require will be found. Exam, \. To find the present value of an annuity of SOU which is to continue 20 years, at 3^ per cent.-f-^y the table, the present value of 1/. for the given >ate and time, is 14-2124 ; therefore 14-2124 x 50=710*62/. or 710/. 12/. 4rf. IS the present value required^ * . Exam. 2. To find the present value of an annuity of 20A to commence 10 ye^rs hence, and then to continue for l\ jezef longer, or to terminate 2 1 years hence, at 4 per cent. mterest. — In such cases as this, we hate to find the difierence between* the present values of two equal annuities, for the two given time? y which therefore will be done by $ubtract-i ing the tabular value of the one pieriod from thatdf the^ Other^ and then multiplying by the given annuity. Thus, tabular value for 21 years 14*0292 ditto for - 10 years 8 1105^ .■I . ■ 1 the difference 5'918i multiplied by 20 H. ' I' " ■ I gives - '1J8466/. or - - 1 1 8/. 7/. 34*/. the zns9ftr\ ENP OP THE ALGEBRA. •a sess i^MM*^ 55SS55 !te«9MBavs GEOMETRY. • - DEFINITIONS. 1. jhL point is that which has position, h%it no magnitude, nor dimensions ^ neither length, breadths nor thickness. 2. A Line is length, without breadth or thickness. 3. A Surface or Superficies, is an extension or a figure of two dimensions, length and bri^adth ', but without thickness. 4. A Body or 6oli4, is a figure of three di- mensions, namely, lepgth, breadth, and depth, or thickness. 5. Lines are either Sight, or Curved, or JVIixed of these two. 6. A Right Line, or Straight Line, lies all in the same direction, between its extremities ^ ^d is the shortest distance between two points. When a Line is mentioned simply, it means t Right Line* 7. A Curve continually changes its direction between its extreme points. 8. Lines are either Parallel, Oblique, Per- pendicular, or Tangential. 9. Parallel Lines are alw&ys at the same per- pendicular distance; and they nevermeet, though ever so far produced. 10. Oblique lines change their distance, and would, meet, if produced on the side of the least distance* IL One line is Peipendicular to another, when it inclines not more on the one side than t:v — * 266 GEOMETRY. •— / -.^«-. than the ©ther, or when the angles on both sides of it are equaL 12. A line or circfe is Tangehtial, or a Tangent to a circle, or other curve, when it touches it, without cutting, when both are pro- duced. 13. An Angle Js the inclination or opening of two lines, having difierent directions, and meeting in a point* 14. Angles are Right or Oblique, Acute, or Obtuse, 15. A Right Angle is that which js made by cuae iine. perpendicular to another. Or when the angles on each side are equal to One an* Qther^ they are right angles. ; 16. An Oblique Angle is that which is made by two oblique lines ; and is either le$s or greater than a nght angle. l7. An Acute Angle is less than a right angle. 1. IS. An Obtuse Angle is greater than a right angle. /■ v ' ' i9. Superficies are either Plane or Curved. 20. A Plane Superficies, or a Plane, is that with which a right line may, every way, coincide. Or, if the line touch the plane in two points, it will touch it in every point. But, if Bot, it is curved. 21. Plane Figures are bounded either by right lines or curves. 22. Plane figures that are bounded by right lines have names according to the number of their sides, or of their angles ; for they have as miny sides as angles ; the least number being three. 23. A figure of three sides and angles is called a Triangle. And it receives particular denominations from the relations of its sides and angles. 24. An Equilateral Triangle is that whos^ three sides are all .equal. 25.. An Isosceles Xnangle is that which has? two sides equal. 26. A DEFINITIONS. 26t ^ zx [ * 26. A Scalene Triangle is that whose three sides are all unequal: ^ . ' 27. A Right-angled Triangle is that which has one right-angle. > .28. Other triangles are Oblique- angled, and are either Obtuse or Acute. ^ 29. An Obtuse-angled Triangle has one ob- tuse an<rle. SO. Alt Acute-Angled Triangle has all its diree angles acute- 8K A figure of Four sides and angles is called a Quadrangle, or a Quadrilateral- '32. AParallelogram is a quadrilateral which &as both its pairs cf opposite sides parallel. And it takes the following particular name$» viz. Rectangle, Square, Rhombus, Rhomboid. 33. A Rectangle is a parallelogram having a right angle. 34*. A Square is an equilateral rectangle; having its length and breadth equal. 35. A Rhomboid is an oblique-angled pa- rallelogram. S6. A Rhombus is an equilateral rhomboid; having all its sides equal, but its angles ob- lique. 37. A Trapezium is a quadrilateral which hath not its opposite sides parallel. '* . . ' . 38. A Trapezoid has only.one pair of oppcH site sides paralleK 39, A Diagonal is a iitae joining any two opposite angles of a quadriiateraL 40^ Plane figures that have more than four sides are, in general, called Polygons : and they receive other particular names, according to the number of their sid,es or angles. Thus, 41. A Pentagon is a polygon of five sides; a Hexagon, of six sidfcs;*a Heptagon, seven; an Octagon, eight; a No- fiagon, nine ^ a Decagon, ten ; an Undecagon, eleven ; and a Dodecagon, twelve sides» 42. A C7 i6i GEOMETRY. 42. A Regular Poljgon has all its skies and all its angk§ equol. — If they are not both equal, the polygon is Irregular. 43. An Equilateral Triangle is also a Regular Figure of three sides, and the Square is one of four j the former being also called a Trigon, and the latter a Tetragon. 44. Any figure is equilateral, when all its sides are equal : and it is equiangular when all its angles are equal. When both these are equal, it is a regular figure. 45. A Circle is a plane figure bounded by a curve line, called the Circumference, which 55 every where equidistant from a certain point within, called its Centre. The circumference itself is often called a circle, and also the Periphery. 46. The Radius of a circle it a line drawn from the centre to the circumference. 47. The Diameter of a circle is a line drawn through the centre, and terminating at the circumference on both sides. 4S. An Arc of a circle is any part of the circumference. 5 I 49. A Chord is a right line joining the extremities of an arc. 50. A Segment is any part of a circle bounded by an arc and its chord. 51. A Semicircle is half the circle^ or a segmeilt cut off by a diameter. The half circumference is sometimes called the Semicircle. 52. A Sector is any part of a circle which IS bounded by an arc, and two radii drawn to i$s extremities. 53. A Quadrant, or Quarter of a circle, is a sector having a quarter of the circumference for its arp-, and its two radii are perpendicular to each other. A quarter of the circumference » sometimes called a Quadrant. •j^ 5*. Th« JJEPlMrtlONS. 26!l /k D K V-. 54. The Height or Altitude of a figure is ft perpendicular let fall from an angle, or its vertex, to the opposite side, called the base. 65, In a right-angled triangle, the side op- posite the right angle is* called tlie Hypothe- nuse 'y and the other two sides arecafted the X^egs, and sometimes the Base and Perpends- . cukr. 56. When an angle is denoted by three letters, of which one standi at the angular point, and' the other two on the two sides, that which stands at the angular point is read in the middle. 57. The cirCunrference of every circle is supposed to be divided into 360 equal parts, called Degrees : and each degree into QO Mi- nutes, each minute into 60 Seconds, and so on. Hence a semicircle contains 1 80 degrees, 2XidL a quadrant 90 degrees. 58. The Measure pf an angle, is an arc of any circle contained between the two lines which form that angle , the angi,ilar point being the centre; and it is estimated by the number of degrees contained in that arc. 59. Lines, or chords, are said to be Equi- distant from the centre of a circle, when per- pendiculars drawn to them from the centre are equal. . 60. And the right line on whichthe Greater ^ Perpendicular falls, is said to be farther from the centre. . 6 k An* Angle In a segment is that which is contained by two lines, drawn from any point in the arc of the segment, to the two extremities of that arc, > 62. An Angle On a segment, or an arc, is that which is contained by two lines, drawn from any point in the opposite or supplemental part of the circumference, to the extremi- ties of the arc, and containing the arc between them. 63. An angle at the circumference, is that whose angular point is any where in the cir- cumference. And an angle at the centre, is that whose angular point is at the centre, 1.^ \ . 6% A 270 GEOMETRY. 64. A right»Uned figure is Inscribed in a circle^ or the circle Circumscribes it, when all the angular points of the figure are in the circumference of the circle. 65. A right-lined figure Circumscribes a circle, or the circle is Inscribed in it, when all the sides of the figure touch the circumference of the circle. 66. One right-lined figure is Inscribed in another, or the latter Circumscribes the former, when all the angular points of the former are placed in the sjuies of the latter. 67. A Secant is a line that cuts a circle, lying partly within, and partly without it. 68. Two triangles, or other right-lined figures, are said tp be mutually equilateral, when all the sides of the one are equal to the cotresponding sides of the other, each to each r and they are said to be mutually equiangular, when the angles of the one are respectively equal to those of the other. 69. Identical figures, are such as are both mutually equi- lateral and equiangular ; or that have all the sides and all the angles of the one, respectively equal to all the sides and all the angles of the other, each to each ^ so that if the one figure were applied to,, or laid upon the other, all the sides of the one would exactly fall upon and cover all the sides of the other; the two becoming as it were but one and the same figure. • 70. Similar figures, are those that have all the angles of the one equal to all the angles of the other, each to each, ind the sides about the equal angles proportional; 74. The Perimeter of a figure, is the sum of all its sides taken together. 72. A Proposition, is something which is either proposed to be done, or to be demonstrated, and is either a problem or a theorem. 4 73- A Problem is something proposed to be done. 74. A Theorem , is something proposed to be demonstrated. 75. A Lemma, is something wliich is premised, or demon- strated, in order to render what follows more easy. 76. A CoroUary, is a consequent truth, gained immedi- ately fi-om some preceditig truth, or demonstration. * 77. A Scholium, is a remark or observation* made upon something going before it; ' * '^ *' ' # -^ AXIOMS. [ 211 ] , « AXIOMS. » • . ■ ' • i. Thincs which are equal to the sam^ thing are equal to each other. 5. When equals are added to equals^ the wholes are equal. , 3. When equals are taken from equals, the remains arc equal. " * , \ 4. When equals are added to unequals, the wholes are unequal. -5. When equals are taken from unequal% the remains are : unequal. ' -y 6. Things which are double of the same tbiag) or eqml things, are equal to each other. , 7. Things which are halves, of the samd thingi tsfe eqUaL • ' . , ^. Every whole is equal to all its parts taken together. ' . 9. Things which coincide, or fill the same space, are iden- tical, or inutually equal in all their parts. 10. All right angles are equal to one another.. 1 1» Angles that have equal measures, or arcs, are equaL THEOKEM I. • If two Triangles have Two Sides and the Included Angle in the one, equal to Two Sides and the Included Angle in the other, the Triangles will be Identical, or equal in all respects. In the two triangles aec, def, if the side ac be equal to the side df, and the side Bc equal to the side ef, and the angle c equal tq ^he angle f \ then will the two triangles be identi- cal, or equal in all wsp^ts, ■ ^ , ; .. for xioiiceive the triatijjfe. £BC it^ be -applied to, of ^e«d On^ the t]4angle VB^yitk iMin a ttiAnii^ that tho^poiiit fc may S72 GEOMETRY. - \ coincide with the point f^ and the side ac with th< $ide Jiff which is equal to it* Then, since the angle F k equal to the angle c (by hyp.)^ the side bc will fall on the side £f. Also, because AC is equal to df, and BC equal to £F (by hyp.)> the point A will coincide with the point d, and the point B with the point £ ; consequently the side ab will coincide witfi the side db. Therefore the two triangles are identical^ and have all their other corresponding parts equal (ax. 9), namely, the side ab equal to the side D£, the angle A to the angle t>, and th^ angle B to the angle e. <^ £. d, THEOREM tU When Twb Triangles have Two Angles and the included Side in the one, equ^ to Two Angles and the included Side in the other, the Triangles are Identical, or have their other sides and angle equal. ^ Let the two triangles aBc, t)EF| have the angle A equal to the angle D, the angle B equal to the angle £, and the side ab equal to the side D£; then these two triangles will be idetl^ tical. « For, cenceive the triangle abc to be placed on the triangle PEF, in such manner that the side ab may fell exactly on the equal side de. Then, since the angle a is equal to the angle D (by hyp,)> the side ac must fall on the side df ; and, in like manner, because the angle B is equal to the. angle £, the side BC must fall on the side ef. Thus the three sides of the triangle abc will be exactly placed on the thr^e sides of the triangle def: consequently the two triangles are identical (ax. 9), having the other two sides ac, Bfc, equal to. the two DF, EFy and the remaining angle c equal to the remaining angle f. <^ e, d. theorem III. In an Isosceles triangle, the Angles at the Base are equal. Or, if a Triangle have Two Sides equal, their Opposite Angles will also be equal. U the triangle abc have the side AC equal to the side bc; then will the angle d be equal to the angle a. For, conceive the angle c to be bisected, or divided into two «qua^>aru, by the line CD, making the angle A(ii> equidi.to the angfe*BCD. Then, THEOREMS. 273 Then, the two triangles acd, bcd, have two sides and the comained angle of the ,one, equal to two sides and the contained angle of the other, viz. the side ac equal to bCj the angle acd equal to bcd, and the side cd common; therefore these two triangles are identical, or equal in all respects (th, 1); and consequently the angle a equal to the angle b. q.. e. d. CotqL 1. Hence the line which bisects the verticle angte of an isosceles triangle, bisects the base, and is also perpendi- cular to it. Corol. 2. Hence too it appears, that every equilateral tri- angle, is also equiangular, or has all its angles equal. THEOREM IV. When a Triangle has Two of its Angles equal, the Sides Opposite to them are also equaL If the triangle abc, have the angle a equal to the angle B, it will also have the side AC equal to the sideBc. For, conceive the side a B to be bisected in the point d, making ad equal to db; and join dc, dividing the whole triangle into the two triangles acd, bcd. Also conceive the triangle acd to be tiu-ned over upon the triangle bcd, so that ad may fall on bd. Then, because the line ad is equal to the line db (by k0.), the point A coincides with the point B,.and the point d with the point d. Also, because the angle a is equal to the angle B (by hyp.), the line ac will fall on the line bc, and the ex- tremity c of the side ac will coincide with the extremity c of the side bc, because dc is common to both ; consequently the side ac is equal to BC. Q- E. D. CoroL Hence every equiangular triangle is also equila* ' teral* THEOREM v. ^ When Two Triangles have all the Three Sides in the one, equal to all the Three glides in the other, the Triangles are Identical, or have also their Three Angles equal, each to each. Let the two triangles abc, abd, haVe their three sides respectively . equal, viz. the side ab equal to ab, AC to AD, and BC to bd j then shall the two triangles be identical, or have ^ their angles equal, vijs. those afigles Vol. I. T thuc (ijrt>t4^l^t^ » * i74 GEOMETkY. that are opposite to the equal sides ; namely, the angle bac to the angle BAD, the angle abc to the angle abd, and the angle c to the angle d« For, conceive the two triangles to be joined ^ together by their bngest equal sides, and draw the line cd. Then, in the triangle acd, because the side AC is equal to AD (by hyp.), the angle acd i^ equal to the angle adc (th. 3). In like manner, in the triangle bcd, the angle BCD is equal to the angle bdc, because the side Be is equal to BD. Hence then, tne angle acd being equal to the angle ADC, and the angle bcd to the angle bdc, by equal addi« tions the sum of the two angles acd, bcd, is equal to the sum of the two adc, bdc, (ax. 2), that is, the whole angle ACB equal to the whole angle adb. Since then, the two sides Ac, CB, are equal to the two sides AD, DB, each to each, (by hyp.), and their contained angles acb, adb, also equal, the two triangles abc, abd, are identical (th. 1), and have the other angles equal, viz. the angle bac to the angle bad, and the angle ab& to the angle abd. q. £• d. THEOREM VI. When one Line meets another, the Angles which it makes on the Same Bide of the other, are together equal to Two Right Angles. Let the line ab meet the line cd : then will the two angles abc, abd, taken to- gether, be equal to two right angles. For, first, when the two angles abc, ABD, are equal to each other, they are both of them right angles (def. 15). But when the angles are unequal, suppose BE drawH per- pendicular to CD. Then, since the two angles ebc, ebd, are right angles (def. 15), and the angle ebd is equal to the two angles eba, abd, together (ax. 8), the three angles, EBC, EBA, and abd, are equal to two right angles. But the two angles ebc, eba, are together equal to the angle abc (ax. 8). Consequently the two angles abc, abd» are also equal to two right angles, q. £. D* CoroL 1. Hence also, conversely, if the two angles ABC, abd, on both sides of the line ab, make up together twa right angles^ then cb and bd form one contixraed right Ijne CD. Corol. THEOREMS. ?75 CoroL 2. Henccy all* the injgles which can be made» at any point B, by any number of lines, on the same side of the right line cd, are^ when taken all together^ equal to twQ right anglesi. \ Corol. S. And, as all the angles that can be made on th« other side of the line CD are also equal to two right angles ; therefore all the angles that c^m be made quite round a point s, by any number of lines, are equal to four right angles. CoroL 4. Hence also the whole circumfer- ence of a circle, being the sum of the mea- sures, of all the angles that can be made about the centre f (def. 57), is the measure of four right angles. Consequently, a semicircle, or « 180 degrees, is the measure of two right angles ; and a quadrant, or 90 degrees, the measure of ene right angle. THEOREM VII. When two Lines Intersect each other, the Opposite Angles are equal. Let the two lines ab, cd, intersect in the point E; then will the angle aec be equal to the angle bed, and the angle ABD. equal to the angle ceb. For, since the line CE meets the line AB, the two angles aec, beg, taken together, are equal to two right angles (th. ^). In like manner, the line be, meeting the Jdne CD, majces the two angles bec, bed, equal to two right angles. , \ Therefore the sua\ of the two angles aec, bec> is equal to the sum of the two bec, bed (ax. 1). < And if the angle bec, which is common, be taken away 'from both these, the remaining angle aec will be equal to the remaining angle bed (ax. 3). . And in like manner it may be shown, that the angle aed is equal to the opposite angle bec. THEOREM VIII. When One Side of a Triangle is produced, the Outward Angle is Greater than either of the two Inward Opposite . Angles. T2 Let 276 GEOMETRY. Let ABC be a triangle, having the side AB produced to d ; then will the outward angle cbd be greater than either of the inward opposite angles a •r c. For, conceive the side bc to be bi- sected in the point £, and draw the line ACT, producing it till bf be equal to A£i and join bf. Then, since the two triangles A EC, bef, have the side AE =s the side ef, and the side ce = the side be (by suppos.) and the included or opposite angles at s also equal (th. 7), therefore those two triangles are equal in all^ respects (th. I), and have the angle c = the corresponding angle EBF. But tlie angle cbd is greater than the angle ebfj consequently the said outward angle cbd is also greater than the angle c In like manner, if cb be produced to G, and AB be bi- sected, it may be shown that the outward angle ABG, or its equal CBD, is greater than the other angle a. J>B THEOREM IX. The Greater Side, of every Triangle, is opposite to the Greater Angle ; and the Greater Angle opposite to the Greater Side. Let ABC be a triangle, havbig the side AB greater than the side ac ; then will the angle acb, opposite the greater side AB, be greater than thp single B, opposite the less side ac. For, on the greater side ab, take the part AD equal to the less side ac, and join cD. Then, since BCD is a triangle, the outward angle adc is greater than, the inward opposite angle b (th. 8). But the angle acd is equal to the said outward angle adc, because ad is equal to AC (th. 3). Consequently the angle acd also is greater than the angle B. And since the angle acd is only a part of i\cB, much more must the whole angle ACB be greater than the angle b. q. e. d. Again, conversely, if the angle c be greater than the angle B, then will the side ab, opposite the former, be greater than the side, ac, opposite the latter. For, if ab be ^ not greater than AC, it must be either equal to it, or less than i^. But it cannot be equal, for then THEOREMS. 277 • thtn the angle c would be eqyal to the "angle B (th. 3), which it is not, by the supposition. Neither can it be less, for then the angle c would be less than the angle B, by the former part of this ; which is also Contrary to the supposi- tion. The side ab, then, being neither equal to AC, nor less than it, must necessarily be greater, q^. fi. D. « THEOREM X. v The Sum of any Two Sides of a Triangle is Greater than the Third Side. Let ABC be a triangle ; t^en will the » p ^um of any two of its sides be greater than ...••* i the third side, as for instance, AC + cb greater than ab. For, produce AC till CD be equal to cb, or AD equal to the sum of the two AC + cb; and join bd: — ^Then, because CD is equal to cB (by constr.), the angle D is equal to the angle cbd (th. 3). But the angle abd is greater than the angle cbd, consequently it must also be greater than the angle d. And, since the greater side of any triangle is op- posite to the greater angle (th. 9), the side ad (of the tri- angle abd) is greater than the side ab. But ad is equal to AC and cd, or ac and cb, taken together (by constr.) ; therefore ac + cb is also greater than ab. <^ e. d. Cors/. The shortest distance between two points, is a singly right line drawn from the one point to the other. THEOREM XI. The Difference of any Two Sides of a Triangle, is Less than the Third Side. Let ABC be a triangle ; then will the difference of any two sides, as ab— ac, be less than the. third side bc. For, produce the less side ac to d, till AD be equal to the greater side ab, so that CD may be the difference of the two sides AB — AC ; and join bd. Then, because ad is equal to ab (by constr.), the opposite angles d and abd are equal (th. 3). But the angle cbd is less than the angle abd, and consequently also less than the equal angle d. And since the greater side of any triangle is rts GEOMETRY. IS opposite to the greater angle (th. 9), the side co (of ihoj triangle bcd) is less than the side bc. q. £• D. THEOHEM XII. When a Line Intersects two Parallel Lines, it m^es the Alternate Angles Equal to each other. Let the line ef cut the two parallel lines AB, CD $ then will the angle aef be equal to the alternate angle efd. For if they are not equal, one of them must be greater than the oth^ ; kft it be £FD for instance which is the greater, if poissible ; and conceive the line fb to be drawn ; cutting off the part or angle efb equal to the angle JLEF ; and meeting the line AB in the point B. Then, since the outward angle aef, of the triangle bef, is greater than the inward opposite angle efb (th, 8)^^ and tunce these two angles also are equal (by the constr.) it follows, that those angles are both equal and unequal at the same time : which is impossible- Therefore the angle efd is not unequal to the alternate angle aef, that is, they are equal to each other. <^ £. d. CoroL Right lines which are perpendicular to one, of two parallel lines, are also perpendicular to the other. T.HEOB.EM XIII. When a Line, Cutting Two other Lines, makes the Al- ternate Angles Equal to each other, those two Lines are Pa^ rallel. Let the line ef, cutting the two lines as, CD, make the alternate angles aef, PFE, equal to each other; then will ab be parallel to cd.. For if they be not parallel, let some other line, as fg, be parallel to ab* Then, because of these parallels, the angle aef is equal to the alternate angle efg (th. J 2). But the angle aef is equal to the angle efd (by hyp.) There- fore the angle efd is equal to the angle efg (ax. 1) ;.that is, a part is equal to the whole, which is impossible. There-^ fore no line but cd can be parallel to AB. q. e. d, CoroL Those lines which are perpendicular to the same line, are parallel to each other. theorem ^■V i THEOREMS. m THEOREM XIV. When a Line cuts two Parallel Lines, the Outwar4 Angle is Equal to the Inward Opposite one, on the Same Side ; and the two Inward Angles, on the Same Side, equal to two Right Angles. Let the line £F cut the two parallel lines AB, CD; then will the outward angle EGB be- equal to the inward opposite angle ghd, oh the same side of the line £v; and the two inward angles bgh, GHp, taken together, will be equal to two right ^gles. For, since the two lines ab, cd, are parallel, the angle agh is e<^ual to the alternate angle ghd, (th. 12). But the angle agh is equal to the opposite an?l^ £GB (th. 7). Therefore the angle ZGB is also equal to the angle ghd (ax. l). (^ e. d. Again, because the two adjacent angles EGB, bgh, are together equal to two right angles (th. 6) ; of which the single EGB has been shown to be equal to the angle ghd; therefore the two angles bgh, ghd, taken together, are ^so e<|val to two right angles. v Coroi. 1. And, conyers^y, if one line meeting two other lines, make the angles on the same side of it equal, those two lines are parallels* CoroL 2. If a line, cutting two other lines, make the sum of the two inward angles, on the same side, less than two right angles, those two lines will not be parallel^ but will meet each other when produced. G H D THEOREM XV, Tho3|L LInfs which are Parallel to the Ssune Lines tfC Parallel to each oth^r* Let the Lines ab^ cd, be each of them parallel to the line ef ; then shall A. '* the lines ab, cp, be parallel to each q^ other, _ For, let the line gi be perpendicular ^^ to EF. Then will this line be also per- pendicular to both the lines ab, cp (corol. th. 12), andcon- i^(|uentl^ the two lines aB| cd, are parallels (cor<^- th. 13), Qi E. d; THEOREM —J feftO GEOMETRY. THEOREM XVI. When one Side of a Triangle is produced, the Outward Angle is equal to both the Inward Opposite Angles taken together. Let the side AB, of the triangle ABC, be produced to d; then will the outward angle CBDbe equal to the sum of the two inward opposite angles a and c. For, conceive be to be drawn p;v- rallel to the side ac of tlie triangle. Then bc, meeting the two parallels AC, ^E, makes the alternate angles c ^d cbe equal (th. 12). And ao, cutting the same two parallels ac, bp, makes the inward and outward angles on. the same side, A and ebd, equal to each other (th. 14). Therefore, by ecfual additions, the sum of the two angles a and c, is equal to the sum of the two cbe and ebd, that i$| to the wl^ol^ angle CBD (by ax- 2). <^£. D, THEOREM XVII, In any Triangle, the sum of all the Three Angles is tqual to Two Right Angles. Let ABC be any plane triangle ; then the sum of the three angles a + b + c i^ equal to two right angles. For, let the ride ab Be produced to d. / ^ ^^ Then the outward angle cbd is equal J> !> to the sum of the two inward opposite angles a + c (th. 16), To each of these equals add the inward angle b, thep will the sum of the three inward angles a + b + c be equal to the sum of the two adjacent angles abc+cbd (ax. 2). But the sum of these two last adjacent angles is equal to two right angles (th. 6). There- fore also the sum of the three angles ofthe triangle a+b + c is equal to twp right angles (ax, 1). q. E. d. • Corgi. 1. If two angles in one triangle, be equal to two angles in another triangle, the third angles will also be'equal (ax. 3), and the two triangles equiangular, Corol. 2. If one angle in one triangle, biB equal to on^ angle in another, the sums of the remaining angles will also be equal (ax. 3), Cord. THEOREMS. «si V \ Coroh 3. If one angle of a triangle be right, the sum of the other two will also be equal to a right angle, and each of them singly will be acute, or less than a right angle. CoroL 4. The two least angles of every triangle are acutet or each less than a right angle. THEOREM XVIII. In any Quadrangle, the sum of all the Four Inward Angles, is equal to Four- Right Angles. Let ABCD be a quadrangle; then the sum of the four inward angles, a + B + c + D is equal to four right angles. Let the diagonal ac be drawn, dividing tl>e quadrangle into two triangle, aec, adc. Then, because the sum of the three angles of each of these triangles is equal to two right angles (th. 17} ; it follows, that the sum of all the angles of both triangles, which make up the four angles of the quadrangle, must be equal io four right angles (ax. 2). q. E. D. Coroh L Hence, if three of the angles be right ones, the fourth will also be a right angle. Cor^l. 2. And, if the sum of two of the four angles be equal to two right angles, the sum of the remaining two will also be equal to two right angles. « THEOREM XIX. In any figure whatever, the Sum of all the Inward Angles, taken together, is equal to Twice as many Right Angles, wanting four, as the Figure has Sides, Let .ABCD^ be any figure ; then the sum of all its inward angles, a + b + c + D + E, is equal to twice as many right angles, wanting four, as the figure has sides. For, from any.point p, within it, draw lines PA, PB, PC, &c, to all the angles, dividing the polygon into as many tri- angles as it has sides. Now the sum of the three angles of each of these triangles, is equal to two right angled (th. 17) ; ^erefbre the sum of the angles of all the triangles is equal to twice as many right angles as the figure has sides. . But the sum pf all the angles about the point P, which are so ^ ' ' many 282 GEOMETRY. many of the angles of the triangles, but no part of the in- ward ai}«{k^s of the polygon, is equal to four right angles (corol. 3, :h. ^), arid niUbt be deducted out of the former Sum. II<'nc( ii follows that the sum of all the inward angles c ^' T.or^rori alone, a + b+c + d + e, is equal to twice as ..^y iight angles as the figiu*e has sides, wanting the sai«i iour right angles, q. £. D* THEOREM XX. "When every Side of any Figure is prq^uced out, the Sum of all the Outward Angles thereby made, is equal to Four Right Angles. Let A, B, c, &c, be the outward singles of any polygon, made by pro- ducing all the sides ; then will the sum A + B + c + D + Ej of all those outward angles, be equal to four right angles. For every one of thes^ outward angles, together with its adjacent inward angle, make up two right angles, as A+a equal to two. right angles, being the two angles made by one line meeting another (th* 6). And tbere being as many outward, or inward angles, as the figure has sides ; therefore the sum of all the inward and outward angles, is equal to twice as many right angles as the figure has sides. But the sum of all the inward angles, with four right angles, is equal to twice as many right angles as the figure has sides (th. 1 9). Therefore the sum of all the in<* ward and all the outward angles, is equal to the sum of all the inward angles and four right angles (by ax. 1). From each of these take away all the inward angles, and there remains all the outward angles equal to four right angles (by ax. 3). THEOREM XXI; A Perpendicular is the Shortest Line that can be drawn from a Given Point to an Indefinite Line. And, of any other Lines drawn from the same Point, those that are Nearest the Perpendicular, are Less than those More Remote. If AB, AC, AD^ &c, be line^ drawn from the given point a, to the indefinite line de^ of which AB is perpendicular. Then shall the perpendicular AB be less than ac, and AC less than ad, &c. ji c Jft II foFf ib^ ^gle B bemg a rigbt one^ the angle THEOREMS. sas angle c is acute (by cor. 3, th. 17), and therefore less than the angle B. But the less angle of a triangle is subtended by the less side (th. 9). Therefore the side AB is less than the side ac. Again, the angle ACB being acute, as before, the adja- cent angle aCd will be obtuse (by th. 6) ; consequently the angle d is acute (corol. 3, th, 17), and therefore is less than the angle c. And «ince the less side is opposite to the lesi angle, therefore the side ac is less than the side Ap. t q. E. !>• CoroL A perpendicular is the least distance of a givea point from a line. THEOREM XXII. The Opposite Sides and Angles of any Parallelogram ai^ equal to each other 5 and the Diagonal divides it into two Equal Triangles. Let ABCD be a parallelogram, of which the diagonal is bd ; then will its opposite sides and angles be equal to each other, and the diagonal bd will divide it into two equal parts, or triangles. For, since the sides ab and dc are pa- jrallel, as also the sides ad and Bc (defin. 32), and the line bd, meets them ; therefore the alternate angles are equal (th. 12), namely, the angle abd to the angle CDB, and the angle adb to the angle cbd. Hence the two triangles, having two angles in the one equal to two angles in the other, have also their third angles equal (eor. 2, th. 17), namely, the angle A equal to the angle c, which are twp of the opposite angles of the parallelogram* Abp, if to <tbe equal angles abd, cdb, be added the equal angles cbd, adb, the wholes will be equal (ax. 2)^ hamdy, the whole angle abc to the whole adc, which are the other two opponte angles of the parallelogram. , (^ E. D, Again, since the two triangles are mutually equiangular ,i ^nd have a side in each equal, viz. the common side bd 5 therefore the two triangles are identical (th. 2), or equal in all respects, namely, the side ab equal to the opposite sidef DC, and AD equal to the opposite side BC, and the wiiole triangle abd equal to the whol^ triangle scd. iq. s. b. Carets 284 GEOMETRY. ^ CoroL 1 . Hence, if one angle of a parallelogram be a right angle, all the other three will also be right angles, and the parallelogram a rectangle. CoroL 2. Hence also, the sum of any two adjacent angles of a parallelogram is equal to two right angles. THEOREM XXIII. Evert Quadrilateral, whose Opposite Sides are Equal, is a Parallelogram, or has its Opposite Sides F^llel. Let ABCD be a quadrangle, having the opposite sides equal, namely, the side ab equal to pc, and ad equal to bc ; then shall these equal sides be also parallel, and the figure a parallelogram. For, let the diagonal bd be drawn. Then, the triangles, abd, cbd, being mutually equilateral (by hyp.), they are also mutually equiangular (th. 5), or have their correspond- ing angles equal ; consequently the opposite side$ are paralUl (th. 13); viz. the side AB parallel to Dc, and AD parallel to BC, and the figure is a parallelogram, q. £. d. THEOREM XXIV. ' ^HOSE Lines which join the Corresponding Extremes of two Equal and Parallel Lines, are 'themselves Equal and Parallel. Let AB, DC, be two equal and parallel lines ; then will , the lines ad, bc, which join their extremes, be also equal and parallel. [See the fig. above.] For, draw the diagonal bd. Then^ because ab and dc are parallel (by hyp.), the angle abd is equal to the alternate angle bdc (th. 12). Hence then, the two triangles having two sides and the contained angles, equal, viz. the side ab equal to the side DC, and the side BD common, and the con- tained angle abd equal to the contained angle bdc, they have the remaining sides and angles also respectively equal (th. I); consequently AD is equal to Bc, and also parallel to it (th. 12). (^ E. D. THEOREM XXV. Parallelograms, as also Triangles, standing on the Same Base, and bettyeen the Same Parallels, are equal to ^ach other. ' . Let THEOREMS. UBS Let ABCD, ABEF, be two parallelo- jrams, and abc,- abf, two' triangles, staiiding on the same base ab, and be- tween the same parallels ab, de ; then will the parallelogram abcd be equal to the parallelogram abef, and the triangle ABC equal to the triangle abf. For, since the line de cuts the two ^ parallels af, be, and the two ad, bc, it makes the angle e equal to the angle aid, and the angle d equal to the angle BCE j(th. H) ; the two triangles, adf, bce, are therefore equiangular (cor. 1, th. 17); and haying the two correspond- ing sides, AD, BC, equal (th. 22), being opposite sides of a parallelogram, these two triangles are identical, or equal in all respects (th. ^). If each of these equal triangles then be taken from the whole space abed, there will remain the parallelogram abef ip the one case, equal to the parallel* ogram abcd in the other (by ax. 3). Also the triangles ABC, Abf, on the same base ab, and between the same parallels, are equal, being the halves of the said equal parallelograms (th. 22). (^ e. d. Corol. 1. Parallelogramsv or triangles, having the same base and altitude, ai*e equal. For the altitude is the same as the perpendicular or distance between the two parallels, which is every where equal, by the definition of parallels. Corol. 2., Parallelograms, or triangles, having equal bases ^nd altitudes, are equal. For, if the one figure be applied with its base on the other, the bases will couicide or be the same> because they are equal : and so the two figures, having^ the same base and altitude, are equal. THEOREM XXVI. If 2^ Parallelogram and a Triangle stand on the Same Base, and between the Same Parallels, the Parallelogram will be Double the Triangle, or the Triangle Half the Pa- rallelogram. Let abcd be a parallelogram, and abe a triangle, on the same base ab, and between the same parallels ab, dej then will the pa- rallelogram ABCD be double the triangle ABE, or the triangle half the parallelogram .- For, draw the diagonal ac of the paral- lelogram, dividing it into two equal parts (th* 22). Then because the triangle?, abc, ABE, 286 GEOMETRY. ABE, on the same base, and between the same parallek, are equal (th. 25) ; and because the one triangle abc is half the parallelogram abcd (th. S2), the other equal triangle abb is also equal to half the same paralielogram abcd. a. £. d. Cor<^» 1 . A triangle is equal to half a parallelogram of the same base and altitude^ because the altitude is the perpendi- cular distance, between the parallels, which is eriery where equal, hj the definition of parallels. CoroL 2. If the base of a parallelogram be half that of a triangle, of the same altitude, or the basq of the triangle be double that of the parallelogram, the two figures will be equal to each other. THEOREM XXVII. Rectangles that are contained by Equal Lines, are Equal to each other. Let BD, FH, be two rectangles, having the sides ab, bc, equal to the sides ep, FG, each to each ; then will the rectangle BD be equal to the rectangle fh. For, draw the two diagonals ac, eg, dividing the two parallelograms each into two equal parts. Then the two triangles ABC, EFG, are equal to each other (th. 1), because they have the two ^ides ab, bc, and the contained angle B, equal to the two side's ef, fg, and the contained angle f (by hyp). But these equal triangles are the, halves of the respective rectangles. And because the halves, or the tri- angles, are equal, the wholes, or the rectangles db, if f» are also equal (by ax. 6). <^ E. D. . CoroL The squares on equal lines are also equal ; for every square is a species of rectangle. N ' THEOREM XXVJkl. The Complements of the Parallelograms, which are about the Diagonal of any Parallelogram, are equal to each other. ' Let AC be a parallelogram, at} a dia^ gonal, EiF parallel to ab or dc, and GiH parallel to ad or bc, making ai, ic complements . to the parallelograms EG, HF, which are about the diagonal DB : then will the complement Al be equal to the complement ic. For, I) Gr C eN '- /r >r A JI 15 THEOREMS. S81 For, since the diago&al )>B bisects the thtee parallelograms Ac; EG, HF (th. 22); thcffcfore, ^ the whdte triangle dab being equal to the ^^hole triangle dcb, and the parts DEly IHB, respectively equal to the parts DGi, ifb^ the remaining parts Ai, ic, must akio be equal (by ax. S). q. E. d. THEOKEM XXIX. A TRAPEZOID) or Trapezium having two Sides Parallel, is equal to Half a Parallelogram, whose Base is the Sum of those two Sides, and its Altitude the Perpendicular Distance between them. Let ABCD be the trapezoid, having its two sides ab, dc, parallel; and in ab produced take be equal to dc, so that AE may be the sum of the two parallel sides; produce dc also,^and let ef, gc, BH, be all three parallel to ad. Then is AF a parallelogram of the same altitude with the trapezoid ABCD, having its base A E equal to the sum of the parallel sides of the trapezoid; and it is to be proved that the trapezoid ABCD is equal to half the parallelogram af* Now, since triangles, or parallelograms, of equal bases and altitude, are equal (corol. 2, th. 25), the parallelogram dg is equal to the parallelogram he, and the triangle CGB equal to the triangle chb ; consequently the line BC bisects, or equally divides, the parallelogram af, and abcd is the half of it. Q. E. D. (^ , H C THEOREM XXX. • • The Sum of all the Rectangles contained undar one Wh<^ Line, and the several Parts of another Line, any way divided, is Equal to the Rectangle contained undei* the Two Whole Lines. • Let AD be the one line, and AB the other, divided into the parts kE, ef, fb ; then will the rectangle contained by AD ahd AB, be equal to the sum of the rectangles of ad and ae, and ad and ef, and AD and fb : thus expressed, ad . ab *s AD . AE + AD . EF + AD . FB. For^ make the rectangle Ac of the two whole lines ad, •AB ; and draw eg, fh, perpendicular to AB, or parallel to AD, to which they are equal (th. 22). Then the whole rectangle AC is made up of all the other rectangles ag, ^ BH, EjnB S8S GEOMETRY. S-ELC A — t i -j EH» Fcv But these rectangles afe contain- ed by ADand AB» eg and bf, fh and fb ; which are equal to the rectangles of ad and A^ AD and £Ff ad and FB9 because AD is equal to each of tlie two, bg^'fh. Therefore the rectangle ad • ab is equal to the sum of aU the ether rectangles aj> . ae, ad . bf, ad . FS. q^ e. d. CoroL If a right line be divided into any two parts ; the square on the whole line> is equal to both the rectangles of the whole line and each of the parts. I "n THEOREM XXXI. The Square of the Sum of two Lines, is greater than th6 Sum of their Square s> by Twice the Rectangle of the said lines* Or, the Square of a whole Line, is equal to the Squares of its two Parts, together with Twice the Rectangle of those Parts. * Let the line ab be the sum of any two £ H p lines AC, CB : then will the square of ab be equal to the squares of AC, cB, together ^ with twice the rectangle ®f Ac . cb. That is, ab* = AC* + CB* + 2AC . CB. For, let ABDE be the square on the sum or whole line ab, and acfg the square on the part AC. Produce cf and gf to the other sides at H and I. From the lines en, gi, which are equal, being each equal to the sides of the square ab or bd (th. 22), take the parts CF, OF, which are also equal, being the sides of the square af, and there remains fh equal to fi, which are also equal to dh, di, being the opposite sides of a parallelo- gram. Hence the £gure hi is equilateral : and it has all its angles right ones (corol. 1, th. 22); it is therefore a sqtiare on the line fi, or the square of its equal CB. Also the figures ef, fb, are equal to two rectangles under AC , and CB, because gf is equal to ac, and fh or fi equal to CB. But the wjkole square ad is made up of the four figures, viz. the two squares af, fd, and the two equal rect- angles EF, FB. That is, the square of ab is equal to the squares of AC, cb, togeth^ with twice the rectangle of AC, CB. q. E. D. CoroL Hence, if a line be divided into two equal parts ; the square of the whole line, will be equal to four times the square of half the line, * . « THEOREM 1?HEOR£M&N 289 THEOREM XXni. Thb Square of the Difference of tw^o lines, is less tLaii tke Sum of their Squares^ by Twice the Rectangle of the. said Lines. - Let Ac» Bc> be any two lines, and ab their diSerence : then will the square of ab be less than the squares of ac, bc, by twice the rectangle of ac and bc. Or, AB* s= AC* + BC* — 2AC . BC. For, let ABDE be the square on the di& ference ab, and acfg the square on the line AC. Produce £d to h ; also produce Db and Hc, and draw ki, making bi the sqiislr^ of the other line bc. Now it is visible that the square ad is less than the two squares af, bi, by the two f ectangles ef^ di; But gf is equal to the one line ac, and gE or fh is equal to the other line BC ; consequently the rectangle bf, contained under eg and GF, is equal to the rectangle of ac and bc. Again, fh being equal to ci or bc or dh, by adding the common part hc, the whole Ht will be equal to the whole FCy or equal to ac ^ and consequently the figure di is equal to the rectangle contained by Ac and bc Hence the two figures ef, di, are two rectangles of the two lines AC, bc ; and consequently the square of ab is less than the squares of ac, bc, by twice the rectangle AC . BC. Q^ E. D* THEOREM XX^llt. TttB Rectangle under the Siim and Difference of two Lines, is equal to the Difference of the Squares, of those Lines. Let ABj AC, be any two unequal lines ; E ly If then will tl>e difference of the squares of AB, AC, be equal to a rectangle undet their sum and difference. That is. & TJ b i> 3 A AB* — AC*?=AB + AC . AB -T AC. For, let ABDE be the square of ab, and ACFG the square of ac. Produce db till BH be equal to ac ; draw hi parallel to AB or ED, and produce Fc both ways to I and K. Then the difference of the ttliro squares AD, Af, is evi- VoL.L U dendy I 290 CEOMETRY. dently the t^o rectangles ef, rb. But the rectangles XiPy^- Biy are equal, bei!ng contained under eq[ual lines ; lor bk and BH are each equal to AC, and ge is equal to cb, being each^ equal to the difference between ab and AC, or their equals AE and AG. Therefore the two bf, kb, are equal to the two KB, Bi, or to the whole kh ; and consequendy kh is equal to the difference of the squares ad, af. But kh is a rect- angle contained by DH, or the sum of ab and AC, and by KS^ or the difference of ab and AC, Therefore the difference of the squares of AB, AC, is equal to the rectangle under tfaeiff sum smd difference* q,. £. D. THfiORKM XXXIV. In any Right-angled Triangle, the Square of the Hypo* fhenuse, is equal to the Sum of the Squares of the other tw« Sides. Let ABC be a right<<uigled triangle, having the right angle c ; then will th« square of the hypothenuse ab, be equal to the sum of the squares of the other two sides AC, CB, Or ab* s= ac* + BC*. For, on AB describe the square ae, and on ac, cb, the squares A6, bh; then draw CK parallel to ai> or be ; and join ai, bf, cd, cb. Now, because the line AC meets the two CG, cb, so as tq make two right angles, these two form one straight line gb (corol. 1, th. 6). And because the angle fac is equal to the angle dab, being each a right angle, or the angle of a square ; to each of these equals add the common angle bac, so will the whole angle or. sum fab, be equal to the whole angle or sum cad. But the line fa is equal to the line ac, and the line ab to the line ad, being sides of the same square ; so that the two sides fa, ab, and their included angle fab, are equal to the two sides ca, ad, and the contained angle cad, each to each ; therefore the whole triangle afb is equal to the whole triangle acd (th. 1). But the square ag is double the triangle afb, on the same base fa, and between the same parallels fa, gb (tt. 26); in like manner, the parallelogram *k is douNe the triangle ACD, on the same base ad, and between the same parallels ad, ck. And since the deubles of equal things, are equal (by ax. 6); therefore the square AG is equal t« the parallelogram *AK. In THEOREMa. 29i In like manner, the other square Bk is proved equal to the. other parallelogram bk. Consequently the two squared AG and BH together, are equal to the two parallelograms ak and BK together, or to the whole square ae. That is, the sum of the two squares on the two less sides, is equal to th» square on the greatest side. q. e. d. CoroL 1 . Hence, the square of either of the two less sides. Is equal to the diflFerence of the squares of the hypothenuse and the other side. (ax. 3);. or, equal to the rectangle con- tained by the sum and diflFerence of the said hypothenuser 9nd other side (th. 33). .CoroL 2. Hence also, if two right-aingled triangles have two sides of the one equal to two corresponding sides of the other; their third sidefs will alsd be equal, and th^ triangles identical; THEOREM XXXV. In any Triangle, the Difference of the Squares of the two slides, is Equal to the Diderence of the Squares of thc^ Begihems of the Base,- or of the two Lines, or Distances, include between th^ Extremes of the Base and the Perpen^*^ dicuiar. Let ABd b^ any triangle, having CD perpendicular to Ai ; then will the difference of the squares of ac,. BC, be equal to. the difference of the squares of AD, BD; that is^' Ac* — BC* = AD* — BD*. For^ since Ac* is equal to ad* + Of \ >t ; „.^^ and Bc* is equal to bD* + cD* f Theref. the difference between ac* and Bd*, is equd to the difference between ad* + cd* and BIT* + CD%' * or equal to the difference Between ad* and bd*, by taking away the common square cd* q. b. d«: Corol, The rectangle of the sum and difference of the two sides of any triangle, is equal to the rectangle of the ^um and difference of the distances between the perpendK» culai: and the two extremes of the base, or equal to the rectangle- of the base and the difference or sum of the segments, according as ,the perpendicular falls within or ' tirithout the triangle, i . F 2 - That r m GEOMETRY. That IS, AC + BC • AC -* BC =5 AD + BD . AD — BD ————*< ■ -■ ■ Or, AC 4* BC • AC — BC = AB. AD — BD in the 2d figure. And AC -{- BC . AC — BC = AB. AD + BD iQ the Ist figure. THBOREM XXICVI. In any Obtuse-angled Triangle, the Square of the Side subtending the Obtuse Anglei is Greater than the Sum of the Squares of the other two Sides, by 'Twice the Rectangle of the Base and the Distance of the Perpendicular from the Obtuse Angle* \ Let ABC be a triangle, obtuse angled at B, and cd perpen- dicular tp AB ; then will the square of ac be greater than the squares of ab, bc, by twice the rectangle of ab, bD. That is, AC* = AB* + BG* + 2ab . BD. See the 1st fig. above» or below. * For, since the square df the whole line ad is equal to the squares of the parts ab, bd, with twice the rectangle of the same parts ab, bd (th. SI); if to each of these equsds there be added the square of cD, then the squares of ad, cd, will be equal to the squares of ABy BD^ CD, with twice the rectangle of ab, bd (by ax. 2). . But the squares of ad, cd, are equal to the sqnare of ac; and the squares of bd^ cd, equal to the square of BC (th. 34) ; therefore the square of ac is equal to the squares of ab, bc,. together with twice the reaangle of ab^ bd. (^ k. d. THEOREM JCXXVII. In any Triangle, the Square of the Side subtending an Acute Angle, is Less than the Sqiuares of the Base and the other Side, by Twice the Rectangle of the Base and the Distance of the Perpendicular from the Acute Angle. Let ABC be a triangle, having Q 'C the angle A acute, and cd perpen- y dicttlartoAB; thenwill the square y^ j ©f BC, be less tjian the squares of jT I AB, AC, by twice the rectangle a W of AB, AD. That is, bc* = AB* + AC* — 2AB . ad. J. *. fi.V, THEOREMS. «93 For, in fig. 1, AC* is = bc» + ab* + 2ab . bd (th. 36). To each of these equals add the square of AB, ^ then is AB* + Ac* = bc* + 2ab* + 2ab . bd (ax. 2), or = bc* + 2ab . AD (th. 30). Q.£. D. Again, in fig. 2, AC* is = ad* + DC* (th. 34). And AB* = AD^ + db* + 2 ad . db (th. 31). Theret ab*+ ac* = bd* + dc* + 2ad* + 2 ad , db (ax. 2), ^or = bc* 4- 2ad* + 2ad . db (th. 34), or = BC* + 2ab .ad (th. 30). Q^ E. D. THEOREM XXXVIII. In any Triangle, the Double of the Square of a Line drawn from the Vertex to the Middle of the Base, together with Double the Square of the Half Base, is Equ<il to the Sum of the Squares of the other Two Sides. Let ABC be a triangle, and cd the line c drawn from the vertex to the middle of the base ab, bisecting it into the two equal parts AD, DB ; then will the sum of the squares of AC, cb, be equal to twice the £ DlirB $um of the squares of CD, bd •, or ac^ -f ' ^ CB* = 2cD* -j- 2db*. For, let CE be perpendicular to the' base ab. Then, since (by th. 36) AC* exceeds the sum of the two squares ad* and CD^ (or bd* and cd*) by the double rectangle 2aj> . D£ (or 2bd . D£) ; and since (by th. 37 )^ bc^ is less than the same .sum by the said double rectangle :; it is mani- fest that both AC* and bc* together, must be equal to that sjun twice taken ; the excess on the one part making up the ilefiect en the other, q. e. d. THEOREM XXXIX. In an Isosceles Triangle, the Square of a Line drawn from the Vertex to any Point in the Base, together with the Rectangle of the Segments of the Base, is equal tQ the Square of one of the Equal Sides of the Triangle. Let ABC be the isosceles triangle, and CD a line drawn from the vertex to any point D in the base : then will the square of AC^ be equ^ to the square of cb, together with the rectangle of ad and db. That iS| Ac* = cao* + AD • db. £§♦ GEOMETWt. For, let CE bisect the vertical ai^e ; then ^ill ft alsf bisect the base AB perpendicularly, making ae =r eb (cor. 1, th. 3). But, in the^triangle Acp, obtuse angled at d^ ^he squarp ac* is ^ CD* + AD* + 2ad . DE (th. 36), or 5= CD* + AD . AD 4- 2de (th. 30), or = CD* + AD . AE + DE, or = CD' + AD . BE + D^i or = CD* 4- AP . DB. Q. E. Df ITIEOllEM XL* In ahy Parallelogram, the two Diagonals Bisect each pther ; an^ the Sum of their Squares is equal to the Sum of the Squares of all the Four Sides of the Parallelogram* Let ABCD b^ a parallelogram, whos^ diagonals intersect each other in e : then will A^ be equal to ^c, and be to ed ; and the sum of the squares of ac, bd^ will be equal to the sum of the squares of ab, Bc, CD, da. That is, AE = jEc, and Bp == ED, and Aq* + BD* =;: AB* + BC* + cp* + DA*. For, the triangles a^^b, dec, are equiangular, because they have the opposite angles at e equal (th. T), and the two lines AC, bd, meeting 'the parallels ab, dc, make the angle bajb equal to the angle dce, and the angle ab]^ equal to the angle CD^, and th^ side ab equal to the side Dp (th. 22); therefore thes^ two triangles are identical, and have their corresponding sides equal (th. 2), viz. ae = EC, and BE = ED. :" ^ * ' Again, sinde ac is bisected in e, the sum of the squares AD^ + DC* = 2a^^ + 2de* (th. 38). - In like manner, ab* + bc* == 2ae* + 2be* or 2de*. • Theref. fB-'-f bc*+cd*+pa* = 4?ap*+4de* (ax. 2). But, because the square of a whole line is equal to ^ ^mes the square of half the. line (cor. tl^. 31), that 13, ac* =5 jl^AE*, and bd* ±= 4de*. ' ' ' ' Theref. ab* + bc? + CD* + da* s= ac* '+ Bf)« (ax. 1 ). ' ^ ■■'■ ■ '"' ^'" ' . ■"-'■■■ ^ • (J. E.D. THEOREM THEOREMS. 391 TpiBOUBM XU. If? a Line^ drawn through or from the Oetitre of a Circle, Bisect a Chord, it will be Perpendicular to it ; or, if it be Perpendicular to the Chord, it will Bisect both the Chord and the Arc of the Chor4. Let AB be any chord in a circle, and cd , a line drawn from the centre c to the chord. Then, if the chord be bisected in the point d, cd will be perpendicular to For, draw the two radii CA, cb. Then, the two triangles acd, bcd, having CA , equal to CB (def. 44), and cd common, also AD equal to db (by hyp.); they have ail the three sides of the oqe, equal to all the three sides of the other, and so* have their apglea also equal (th. 5). Hence then, the angle, ADC being equal to the angle bdc, these angles are right angles, and the line cd is perpendicular to ab (def. 1 1). Again, if CD be perpendicular to ab, then will the chord AB be bisected at the point d, or have ad equal to db ; and the arc A£b bisected in the point ^, or have ab equal eb. For, having drawn CA, cb, as before,. Then, in the tri- angle ABC, because the side cA is equal to the side cb, their opposite angles a and b are also equal (th. 3). Hence then, in the two triangles acd, bcd, the angle a is equal to the angle 9f and the angles at d are equal (def. 11); therefore their third angles are also equal (corol. I, th. 17). And having the side cd common, they have also the side ad equal to the side db (th. 2). Also, since the angle ace is equal to the angle bch, the arc AE, which measures the former (def. 57), is equal to the arc be, which measures the latter, since equal angles must have equal measures. CoroL Hence a line bisecting any chord at right angles, |Kisses through the centre of the circle. THEOREM XLIU If More than Twp Equal Lines can be drawn from any Point within a Ciycle to the 'Circumference, that Point wiH he the Centre. ms GEOMETRY. Let ABC be a circlei and d a point within it : then if any three lines^ ^9 JOB, DC, drawn from the point d to the circumferenqe, be equal to e^h other^ the point D will be the centre. For, draw the chords ab, BCj which let be bbected in the points £, c, and join DE, DF. I Then, the two triangles, dais, dbi, have the side da equal to the side db by supposition, and the side ae equal to the side eb by hypothesis, also the side DE common : therefore these two triangles are identical, an4 have the angles at e equal to each other (th. 5) ; conser quentiy D£ is perpendicular tp the middle of the chord ab (def. 11), and therefore passes through the centrie of the circle (corol. th. 41). In like manner, it may be showi) that df passes thrqugh the centre. Consequently the point d is the centre of the circle, and the three equal lines da, db, dc, are radii. ft; ?• ^? theorem xliii. > If two Circles touch one another Internally, the Centres of the Circles and the Point of Contact will be all in the Same Right Line. Let the two circles abc, ade, toucl^ one another internally in the point a ; then will the point A and the centres of those circles be all in the s^me ri&;ht line. For, let F be the centre of the circle ABC, through w^iich draw the diameter AFC. Then, if the centre of the other circle can be out of this line ac, let it be supposed in some other point as G ; through which draw the line FG cutting the two circles in b and d* Now, in the triangle afg, the surti of the two sides fg, GA, is greater than the third side af (th. 10), or greater than its equal radius fb. From each of these take away the common part fg, and the remainder ga will be greater than the remainder GB. But the point p being suppo^ the centre of the inner circle, its two radii, ga, gd, are equal to each other 5 consequently gd will also he greater than GB. But ade being the inner circle, cu is necessarily ' less V » THEOREMS- 3sn less tlian cb. So that cd is both greater and less than CB; whkh is absurd. Con^^uently the centre g cannot be ont of the line afc. a. £. d. THEOREM XLIV. J If two Circles Touch one another Externally, the Centres !of the Circles and the Point of Contact will be ^ in the Same Right Linew Let the two circles abc, ade, touch one another externally at the point a ; then will thje point of contact a and the centres of the two circles be all in the same right line. ' For, let F be the centre of the circle abc, jhrough which draw the diameter afc, and produce it to the other circle at E. Then, if the ccntreof the other circle ade can be out .of the line fe, let it, if possible^ be supposed in some other point as G ; and draw the lines AG, FBDG, cutting the two circles in b and d. Then, in the triangle afg, the sum of the two sides af, AG, is greater than the third side fg (th. 10). But, f and c being the centres of the two circles, the two radii GA, CD^ are equal, as are also the two radii af, fb« Hence the s;um of ga, af, is equal to the sum of gd, bf ; and therefore this latter sum also, gd. bf, is greater than gf, which is absurd. Consequently me centre G cannot be out of the )ine ef. q. e. p. THEOREM XLV. Any Chords in a Circle, which are Equally Distant from the Centre, are Equal to each other ; or if they be Equal . to each other, they will be Equally Distant from the Centre. Let ab, cd, be any two chords at equal distances from this centra g; then will these two chords ab, cd^ be equal to each other. For, draw the two radii ga, go, and the two perpendiculars ge, gf, which are the equal distances A-om the centre G. Then, the two right-aneled triangles, GAE, Gcf, having the side ga equal the side gc^ ana the side gs equal the side 49S GEOMETUT; sde GWf said the angle at x equal to ib0 abgle at f, therefore the two trtanfi^es GAB, GCF, are identical (cor. 2, th. S4)« and have the line ae equal the line cf. But AB is the douUe of ae, and cd is the double of CF (th. 41)^ therefore a8 b equal to cd (by ax. 6). q. e. d. Again> if the chord ab ^ equal to the chord co ; then will their distances from the centre, G^t gf, also be equal to each other. ToVf since ab is equal cd by soppoeition, the half abi^ equal the half of. Also the radii ga, gc, being equal, as well as the right angles £ and f,. therefore the third sides are equal (cor. 2, tlu 34), or the distance GE equal the diis- tance gf. q. e. d» - 2UEJI THEOREM XL VI. A Une Feq>endicular to the Extremity of a Radius, is a Tangent to the Circle. L'et the line adb be perpendicular to the radius CD of a circle ; then shall ab touch the circle in the point d only. For, from any other point £ in the line AB draw cfe to the centre, cutting the . circle in f. Then, because the angle D, of the triangle CDE, is a right angle, the angle at E is acute (th. 17, cor. 3), and consequently less than the angle d. But the greater side is always opposite to the greater angle (th. 9); there- fore the side ce is greater than the side cd, or greater than its equal cf. Hence the point e is without the circle ; aUd the same for every other point in the line ab. Consequently the whole line is without the circle, and 'meets it in the' point D only. .w- *'., . ~ THEORBiif THEOREMS. «d» THEOREM XLTn. - - I I When a Line is a Tangent to ia Circle, a Radius drawn to the Ppint of Contact is Perpendicular to the Taagent. Let the line ab touph the circumference of a circle at th^ point d; then will the radius cd be perpendicular to thi* tangent ab. [See the last figure.] ' For, the line , ab being wholly without the circumference except at the point d, every other line, as ce drawn froni the centre c to the line ab, must pass out of the circle to arrive at this line. The line cd is therefore the shortest that can be drawn from the point c to the line AB, and conse- quently (th. 21) it is perpendicular to that line. CoroL Hence, conversely, a line drawn perpendicular to a tangent, at the point of eontact> passes through die centra of the circle. THEOREM XLVITI. The Angle formed by a Tangent and Chord is Measured bjr Half the Arc of that. Chord. Let ab be a tangent to a circle, and cd a chord drawn from the point of contact c \ then is the angle' bcd measured by half the arc CFD, and the angle acd measured by half the arc cgd. For, draw the radius ec to the point of contact, and the radius £F perpendicular to the chbrd at h. . ' Then, the raius ef, being perpendicular to the chord CD, bisects the arc cfd (th. 41)« Therefore cf is half the arc CFD. in the triangle ceh, the angle h being a right one, the sum of the two remainiiig angles £ and c is equal to a right angle (cofoL 3, th. 17), which is equal to the angle bce, because. the radius c£ is perpendicular to the tangent. Froia each of these equals take awaf the common part or angle c^ and there remains the angle £ equal' to the angle bcd. But the angle e is measured by the arc of (def. 57;, which Is the half of^CFD \ therefore the equal angle bcd must also have the same measure^ namely, half the arc cfd of ^$he chord CD. Again, soo GEOMITRY. Again> the line gef, being perpendicular to tnc chord CD, bisects the «rc cg0 (th» 41). Therefore co is half Ae arc CGD. Now, since the line ce^ meeting SG, makes the sum of the two angles at E^ equal to two right angles (th. 6)» and the fine XJX makes with ab the sum of the two angles at c equal to two right angles } if from these^ twa equal sums there be taken away the parts or angles ceH and mcH, which have been proved equal, there remains the angle CEG equal to the angle ach. But the former of these, C£G» being an angle at the centre, is measured by the arc CG (def. 57) ; consequently the equal angle acd must also bave the same measure CG> which is half the arc qgd of the chord CD. Of e« d. CorcL 1. The sum of two right angles U n^asured bjr lialf the circumference. For the two angles qcd, Acb» which make up two right andes, are measured by the arcs CF> cG, which make up halt the circumference^ fg being a diameter* CdroL 2. Hence also one right angle must have for it^ measure a quarter of the circunuerencej or 90 degrees. THEOREM XLIX. An Apgle at the Circumference of a Qrcle, is measured by Half the Arc that subtends it. Let bac be an angle at the circumference; It has for its measure, half the arc Bc. which subtends it. For, suppose the tangent »e passing tlirough the point of contact A. Then, the angle dag being measured by half the arc ABC, and the angle dab by half the arc ab (th. 48); it follows, by equal subtraction, that the difference^ or angle bag, must be measured by half the arc bc, which it stands upon. q. £• p. x. THEOREM t. THEOREMS, 901 THEOREM t- AU Angles in the Same Segment of a Circle^ or Standing on ' the Same Arc, are Equal to each othen Let c and d be two angles in the same segment acdb, or, which is the same thing, standing on the supplemental arc aeb ; then win the angle c be equal to the angle d. For each of these angles is measured by lialf the arc aeb ; and thus, having equal measures^ they are equal to each other (ax. 11). * THEOREM U. An Angle at the Centre of a Orcle is Double the Angle at the Circumference, when both stand on the Same An:. Let c be an angle at the centre c, and D an angle at the circumference, both stand- ing on the same arc or same chord ab: then will the angle c be double of the angle D| or the angle d equal to half the angle c. For, the angle at the centre c is measured by the whole arc aeb (def. 57), and the angle atthecircum* ference d is measured by half the same arc aeb (th. 49) % therefore the angle D is only half the angle c, or the angle c double the angle o. THEOREM LU. f An Angle in a Semicircle, i;^ a Right Angle. If ABC or adc be a Semicircle ; then any angle D in that semicircle, is a right angle. For, the angle n, at the circumference, is measured by half the arc abc (th. 49), that is, by a quadrant of the circumference. But a quadrant is the measure of a right angle (corol. 4, th. 6 ; or corol. 2, th. 48). Therefinre the angle d is a right angle* THEOREM aos GEOMETRT. THEO&SM LIII. The Angle formed bjr a Tangent to a Circle, and a Chord drawn fsom the Point of Contact, is Equal to the Anglt , in the Alternate Segment* If AB be a tangent, and AC a chord, dnd ]> any angle in the alternate segment adc ; then will the angle D be equal to the angle BAG made by the tangent and chord of the lore AEC. For the angle d, at the circumference, is measured by half the arc aec (th. 49) ; and the angle bag, made by the tangent and chord, is also measured by the same half arc Asc (th. 48) ; therefore these two angles are equal (ax. 11)* THEOREM tlV. The Sum of any Two Opposite Angles of a Quadrangle Inscribed in a Circle, is Equal to .Two Right Angles. iiET ABCD be any quadtilaterat inscribed in a circle ; then shall the sum of the two opposite angles a and c, or b and i>, be equal to two right angles. For the angle A is meascrf ed by half the ;rrc dcb, which it stands on, and the angle t by half the arc dab (th^ 49) ; therefore Ae sum of the two angles a and c is measured by half the turn of these two arcs, that is, by half the circumference. But half the circumference is the measure of two right angles (corol. 4, th. 6) ; therefore the sum of the two oppo- site angles a and c is equal to two right angles. In like inanner it is shown, that the sum of the other two opposite angles, d tnd b, is equal to two right anglesw <^ £• b. THEOREM LV. If any Side of a Quadrangle, Inscribed in a Circle, be Produced out, the Outward Angle will be Equal to the Inward Opposite Angle; If the side Ab, of the quadrilateral AECD, inscribed in a circle, be produced to £ ; the outward angle dae will be equal . to the inward opposite angle c* For, \ / •THEOREMS. .463 r For, the sum of the two adjacent angles DAfi ancToAB is equal to two right angles (th, 6) ; and the sum of the two opposite angles c and "bkh is also equal to two right angles (th. 54) ; therefore the former sum, of the two angles dae and DAB, is equal to the latter sum, of the two c and dab (ax. l). From each of these equals taking away the com- mon angle dab, there remains the angle dae equal the angle c. q. £. p. TttEOREM LVI* Any Two Parallel Chords Intercept Equal ArcsSi LfiT the two chords ab, c&, be parallel : then will the arcs ac, bi>, be equal; or AC = bo. ■ ^ For, draw the line bC« Then, becaiise the lines ab, cd, are paraUel, the alternate angles b and c are equal (th. 12). But the angle at the circumference B, is measured by half the ar« AC (th. 49) ; and the other equal angle at the circumference c is measured by half tiie arc bd : therefore the halves of thef arcs AC, bd, and consequently the arcs themselves^ are alsa equal. Q. £. D« *rME0REM Ltir. "When % Tangent and Chord are PaJr^llel to each odieri thejr Intercept Equal Arcs. Let the tangent abc be parallel to the chord DF* ; then are the arcs bd, bf, equal y that is, bd = BF. For, draw the chprd bd. Then, be- cause the lines ab, df, are parallel, the al- ternate angles D and b are equal (th. 12). But the angle b, formed by a tangent and chord, is measured ■b^ half the arc BD (th. 48) ; and the other angle at the cir- cumference D is measured by half the arc bf (tn. 49); there- for^ the arcs bd, bf, are equali . (^ B. d. ' THEOHEM 5M GEOMETRY. THEOREM LVIII. The Ancle formed, Within a Circle, by the Intersection of two Chords, is Measured by Half the Sum of the Two Intercepted Arcs* Let the two chords ab, cd, intersect at the point e: then the angle aec, or deb, is measured by half the sum of two arcs ac. For,, draw the chord af parallel to cd. Then, because the lines af, cd, are parallel, and AB cuts them> the angles on the same aide a andi\D£B are equal (th. 14). But the angle at the circumference A is measured by half the arc bf, or of the sum of FD and db (th. 49) ; therefore the angle £ is also measured by half the sum of fd and db* Ag2un, because the chords af, cd, are parallel, the arcs ac^ FD, are equal (th. 56) i therefore the sum of the two arcs ac, DB» is equal to the sum of the two fd, db ; and consequently the angle e, which is measured by half the latter ^nm, is also measured by half the former, q. £. d. THEOREM LIS. The Angle formed. Without a Circle, by two Secants, h Measured by Half the Difference of the Intercepted Arcs* Let the angle X be formed by two se-> cants EAB and ecd; this angle is measured by half the difference of the two ztcs AC, DB, intercepted by the two secants. Draw the chord af parallel to CD. Then, beca^ise the lines af, cd, are parallel, and AB cuts them, the angles on the same side a and BED are equal (th. 14). But the ^igle A, at the circum* ference, is measured by half the arc bf (th* 49), or of the difference of Df and DB : thenefore the equal angle £ if also meas.ured by half the difference of df, DB. Again, because the chords af, cd, are parallel, thf arcs AC> FD, are equal (th* 56) ; therefore the difib^nce of the two TttEOREMS. soi ttro arcs At, DB, IS equal to the. difference of the twopp, DB. Consequently the angle E, which is measured by half the latter difference;, is also measured by half the former. q. £. 01 THEOUBM LX. Th^ Angle formed by Two Tangents, is Measured by Hilf the Difference of the two Intercepted Arcs. Let £B^ £b^ be two tangents to a circle at the points a, c; then the angle £ is measured by half the difference of the two arcs CFA, CQA* r For, draw the chord af parallel to ed; Then, because the linies af, ed, are pa- rallel, iand eb meets them, the angles on the same side a and £ are equal (th. 14'). But the angle A, formed by the chord af ahd tangent AB, is measured by half the arc AF (th. 48) j therefore the* equal angle E is also measured by half the same arc Af, or half the difference of the ards cfa and cf, or cga (th. 57). CoroL In like manner it is proved, that the angle E, formed by a tangent ec», and a secant eab, is measured by half the difference of the two intercepted arcis t A and CFB, ' J) F Theorem lxu When two Lines, meeting a Circle each in two Points, Cut one another, either Within it or Without it ; the Rect- angle of the Parts of the one, is Equal to the Rectangle of the Parts of the other ; th^ Parts of each being measured from the point of meeting to the two intersections with ^ the circumference. Vol. L X Let X)^ GEOMfeTRt. Let* the two lines ab, ^d, meet each other in E^ then the rectangle of ae, eb, will be equal to the rectangle of CE, EP. Qr^ A£ • EB = C£ . £0. FoTf through the point £ draw the dia- meter FG ; also, from the centre h draw ihe radius dh« and drai;^ hi perpendi- cuhr to CO. Then, since deh is a triangle, and the perp. HI bisects the chord cd (th. 41), the line CE is equal to the difference of the segments di, ei, the sum of them being DE. Also, because h is the centre of the circle,' and the radii dh, fh, gh, are all equal, the line Bc; is equal to the sum of the sides dh, he ; and ef is equal to their difference. But the rectangle of the sum and difference of the two tildes of a triangle, is equal to the rectangle of the sum and difference of the segments of the base (th. 35) ; therefore the rectangle of fe, eg, is equal to the rectangle of cE, ED. In like manner it is proved, that the same rectangle of fe, XG, is equal to the rectangle of ae, eb. Consequently the rectangle of ae, eb, is also equal to the rectangle of CE| £|> (ax. 1). (^ E. D. Corol. 1. When one of the lines in the second case, as de, by revolving about the point E, comes into the position of the tan- gent EC or ED, the two points c and D nmning into one^ then the. rectangle of cE, ED, becomes the square of ce, because cb and DE are then equal. Consequently the rectangle of the parts of the secant, ae . eb, is equal to the square of the tangent, €e\ Carol. 2. Hence both the tangents ec, ef, drawn from the saine point £, are equal ; since the square of each is equal to the same rectangle or quantity ae • eb* THEOREM LXU. ta Equiangular Triangles, the Rectangles of the Correspoiid- ing or Like Sides^ taken aljiernately, are Equal. Lfit THEOREMS: S07 , L^T ABCf DEF, be two equiangular triangles, having the angle a = the angle d, the angle b •= the angle e, afld the angle c =^ the anglp f ; also the like sides ab, de, and ac^ df^ being those opposite the equal angles: then will the rectangle of ab, df, be equal to the rectangle of Ac, de. In BA produced take AG equal to df ; and thrdugi the three points b, c, g, conceive a circle bcgh to be described^ ftieeting CA produced at H, and join Gif . Then the angle G is equal to the angle C on the same arc BH, and the angle h equal to the angle b on the^same arc cg (th. 50) ; also the opposite angles at a are equal (th. 7) : therefore the trlanglfe agiI is equiangular to the triangle acb, and consequently to the triangle dfe alsoi But the ' two like sides AG, df, are also equal by supposition; conse- quently the two triangles agh, dfE, are identical (th. 2), having the two sides AG, AH, equal to the two df, db, each to each. • But the rectangle ga . AB is equal to the rectangle HA . AC (th. 61): consequently the rectangle d^ * ab is equal the rectangle de . AC q. e. d. . THEOREM LXIII. iThe Rectangle of the^two Sides of any Triangle, is Ecjual to the ReCtangl^ of the Perpendicular on the third Side and the Diameter of the Circumscribing^ Circle. . Let CD be the perpendicular, and cU the diameter of the circle about the triangle ABC 5 then the, rectangle CA . cb is ±s the rectangle cd . cB. For, join BE : theii in the two triangles _ACD, ECB, the angled A and E are equal, standing on the same arclBC (th. 50) j also the right angle D is equal the angle b, which is also a right angle, being in a semicircle (th. 52) : therefore these two triangles have also their third angles equal, and are equiangular. Hence, ac, CE, and CD, cb, being like sides, subtending the equal angles,' the rectangle AC . cb, of the first and last of them, is equal to the rectangle ce . cd, of the other two (th. 62). X 2 THEOREM N io# GEOMETRY. tHEOKEM LXIV. The Scpare of a line bisecting any Angle of a Tnangfe^ together with the Rectangle of the two Segments of the opposite Side, is Equal to the Rectangle of dbe two other Sides including the bisected Angle. LsT CD bisect the angle, c of the triangle IBc J then the square cd* + the* rectangle AD . DB is 2=: the rectangle Ac . cB. For» let CD be produced to meet the cir- cumscribing circle at e, and join ae. Then the two triangles ace, bcd, are equiangular : for the angles at c are equal by suppositioDf and the angles B and E are equal, standing on the same arc AC (th. 50) v consequently the third angles at A and d are eqvtal (coro). I, th. 17): also ac, cd, and GE, CB, are like or corresponding sides,, being opposite to- equal angles : therefore the rectangle ac . Cb is = the rectangle cd • ge (th. 62). But the latter rectangle cef. ce is = cr^ + the rectangle cd . de (th. 30) ; therefore also the former rectangle AC • cb is also = cd* -|- cd. • db, or equal to CD*^ + AD . db,. since C2> . de is = ad . db (th. 61)« q. E« D. THBOREM LXV. The Rectangle of the two Diagonals of gny Quadrangle Inscribed in a Circle, is equal to the sum of the two Rect- angles of the Opposite Sicbss* Let abcp be any quadrilateral inscribed in a circle, and ag, bd, its two diagonals : then the rec^tangle AC . bd is = the rect- angle ab . DC + Jthe rectangle ad . bc- For, let CE be drawn, making the angle BCE equal to the angle dca. Then thetwa triangles acd, bce, are equiangular ; for the angles A and b are equal, standing on thesame.arc dc; and theangles* dca, bcb, are equal by supposition ; consequently the third angles adc, B£c,,are also equal t also, Ac, bc, and ad^ be> are like or corresponding sides, being opposite to the equal angles : therefore the rectangle AC . BE is =s^ the rectangle I A.D . bc (th.. 62)» Again, THEOREMS. 3<» Again, the two triangles abc, D£C| are eqniangidar : for tlie angles bag, bdc, -are equal, standingon the same arc bc; and the angle dce is equal to the angle fiCA, by adding the common tingle ace to the two «qua:l angles dca, bc£ ; there- fore the third angles £ and abc are also equal : but AC, Dc> and AB, DE, are the like sides : therefore the cectangle AC • "DE is = the rectar^le ab . dc (th. 62), -Hence, by equal additions, the sum of the rectangles AC • BE + AC . DE is = AD . SO -|- AB . DC. Bllt the •ibrmer «um of tb^ rectangles AC . be + ac . 0E is = the rectangle AC . bb (tb« 30): therefore the same rectangle AC . bd is equal tp the latter suro^ the rect. ad . Bc + the rect. AB . DC (ax. I). <^ e. n. OF RATIOS AND PROPORTIONS, DETlNiTIONS. ' Def. 76. Ratio is the proportion or relation which one magnitude bears to another magnitude of the same kind, with respect to quantity. JVi^. The measure> or quantity, of a ratio, is conceived, by considering what part or parts the leading quantity, called the Antecedent, is of the other, called the Consequent ; or what part or parts the number expressing the quantity of the •former, is of the number denoting in like manner the latter. So, the ratio of a quantity expressed by the number 2, to' a like quantity expressed by the number 6, is denoted by -6 divided by 2, or | qr 3 : the number 2 being 3 times con- tained in ^, or tlie third part of it. In like manner, the ratio of the quantity 3 to 6, is measured by ^ or 2 ; the ratio of .4 to 6 is ^ or 1|.; that of 6 to 4 is f or |; &c. 77. Proportion is an equality of ratios. Thus, 78. Three quantities are said to be Proportional, when the ratio of the first to the second is equal to the ratio of the second to the third. As of the three quantities A (2), b (4), c (8), where 4 = J. = 2, both the same ratio. 19. Four quantities are said to be Proportional, when the ratio of the first to the sec6nd,*is the same as the ratio of the third to the fourth. As of the four, a (2), b (4), c (5), D (10), •where i- = V* = ^> ^^^ ^^® ^^"^^ ratio. «ie GEOMETRY. ■ Noie. To denote that four quantities, a, b> c, d^ are pro* portional, thej are usually stated or placed thus, a : b : : c : Q; ^nd read thus, a is to B as c is to d. But vhen three €|uantities are proportional, the middle one is repeated, and they are written thus, a : B : : fi : c. 80. Of three proportional quantities, this middle one is said to he a Mean Proportional between the other two j ^d .the last, a Third Proportional to the first Jtnd second. 81. Of four proportional quantities, the last is said to be a Fourth Proportional to the other three, taken in orfier. ' ' ' ' . • . ' . 82. Quantities are said to be Continually Proportional, or in Continued Proportion, when the ratio is the same bietween every two adjacent terms, viz. when the first is to the second, a^ the second to the third, as the thij-d to the fourth, as the fourth to the fifth, and so on, all in the same common ratio* As in the quantities 1, 2, 4, 8, 16, &c; where the com« mon ratio is equal to 2. - ' -' - 83. Of any number of quantities, A, b, c, d, the ratio of the first A, to the last D, is said to be Compounded of the ratios of the firs^ to the second, of the second to the third, and so on to the last. . * , 8'k Inverse ratio is, when the antecedent is made the consequent, and the consequent the antecedent. — ^Thus, if 1 : 2 : : 3 : 6 5 then inversely, 2 : 1 : : (5:3. > ■ - * • SS, Alterrjate proportion is, when antecedent is compared with antecedent, and '^consequent with consequent. — Asy if 1 : 2 : : 3 : 65 then>.by alternation, or permutation, it will be 1 :3 ;;2 ;6. ' *. » S(), Compounded ratio is, when the sura of the antecedent and consequent is conipared, either with the consequent, or with the antecedent.— Thus, if 1 : 2 ! : 3 : 6, then by compo- sition, 1 + 2 : 1 : : 3 + 6 : 3, and J + 2 : 2 ; : 3 + 6 : 6. 87. Divided ratio, is when the'difference of the antecedent and consequent is compared, eikher with the antecedent or with the consequent. — Thus, if 1 : 2 : : 3 : 6, then, by division, 2-^1 : 1 :: 6-3 : 3, and 2— 1 : 2:: 6-3 : 6. ^ ■ I « I Note. . The term Divided, or Division, Here means sub- tracting, or parting; being used in the sense opposed to^com- pounding, or adding, in def. 86. TH£OR|SM THEORIJMS, a>I THEOREM LXVI. Equimultiples of any two Quantities have the same Ratio as the Quantities themselves,. Let a and b be any two quantities, and f»A, wb, any equimultiples of them*, m being any number whatever : then will mA and mB h^ve the same ratio as a and B;, or A : B : : mA : mB. ror — ^ = •— , the sa:me ratio. Corol. Hence, like parts of quantities have the same ratio as the wholes ; because the wholes are equimultiples of th« like p9rt5i or A and i ax^e like parts of i»a and tn^* THEORBM LXVJI. If Four Quantities, of the Same Kind; be Proportionals^ thf y will be in Proportion by Alternation or Permutation, or th^e Antecedents will have the Same Ratio as the Con^ ' sequents. Let a : b : : f»A : mB ; then will a '" mWi^ x mB, For 5;: |», an^ — • ^ niy both the same ratio, THEOREM LXVnx. If Four Quantities be Proportional ; they ^ill be in Pro* portion by Inversion, or Inversely, Let a : b : : mA : ms ^ then will B : a : : /»B : mA^ mtmm A A For — = - — , both the same ratio. ;;;b B theorem lxix. Jf Four Quantities be Proportional ; they will be , in Pro* portion by Composition and Division* Let a : b :: mA : mB'f Then will B ± A : A : : /wB ± iwA : mA, and B ± A : b : : /wb ± mA : i»b. _ mA A mB B ffor, _, , — = 2"-; — 9 and m^±mA, » ± a' mB±mA » ;t A 318 GEOMETRY. Coroi. It appears from hence, that the Sum of the Greatest and Least of four proportional quantities, of the same kind, exceeds the Sum of the Two Means. For, since — - - A : A + B : : ffifA : niA + nm% where A is tiie lea^t, and mK + w» the greatest ; then f« + 1 • A + mB, the sum of the greatest and least, exceeds « + i . A + P the sum o^ the two means. THEOREM I-XX. If, of Four Proportional Quantities, thiire be taken mj Equimultiples whatever of the two Antecedents, and any Eqoimnltiples. whatever of the two Consequents ; the 'quantities resulting will still be proportional. Let A : b : : ffiA : /«b ; also, let px antl pmi^ be any equimultiples of the two antecedents, and ^B and qm^ any equimultiples of the two - consequent* \ then will - r - - - ^A : ^B : : pmA. : qm'B, ' for - — = ^^, both the jame ratio. pmk pA THEOREM LXXI. If there be Four Proportional Quantities, and the iwfl Consequents be ,eifher Augmented, or Diminished by Quantities that have the Same Ratio as the respective Antecedents ; the Results and the Antecedents will still be Proportionals. Let A : b : : mA : /tie, and tiA and nmA any two quan-: pities having the sai; e ratio as the two antecedents j then will A : B ± «A ; : fwA : mB ± nmA. wB ± nmk B ± «A ^ , , ror = , both the same ratio. mA K ^ THEOREM LXXIJ. If any Number of Quantities be. Proportional, then any one of the Antecedents will be to its Consequent, as the Sum of all the Antecedents is to the Sum of all the Con- sequents. Let a \ b :i tnA ', ntfi \\ nA '. wB, &c ; then will * - - ^ A : B : : A + wA 4" ^A : : B + ^B + //B, &c. _ 5 + WiB + «B B _ J or -— =c •— , the same ratio, A + IWA + /7A 4 theorem THEOREMS. 4 It THEOKEU IXXIir. |f a Whole Magnitude be to ?i Whole, as a Part taken from tbe first, is to a Part taken from the other ; then th^ Re- mainder wijl be to the Remainder, as th^ whpl^ to the whole. JLet a : 3 : : — A : — B ; n . n tjpten will a:b::a -a:b B^ n n B-^B B For •■ ; ■ - ^ ■ ' ' = — , both the same ratia^ • • A-rr-^ A A ' » - THEOJIEM J.iXIV. If any Quantities be Proportional ; ^heir Squares, or Cubes^ or any Like Powers, or Roots, of them, will also be Pro- portional. Let a : b : : »a : ms ; then will a* : B° : : iw^a" : jw^b**. For --—-; = — -, both the same ratio. «j"A° a' ■ THEOREM LXXV. * ^ there be tvo Sets of Proportionals 5 then the Products 01? Rectangles of the Corresponding Terms will also be Pro- portional. JjET a 2 b : : ota : fwBj and p : D : : «c : «D J ^hen will AC : bd : : mnKC : mftBD^ rfmtBT> bd , - , ' or ' = — , both the same ratio. mtJAC AC THEaHEM LXXVI. Jtf Refer Quamtities be Proportional; the Rectangle or Product of the two Extremes, will be Equal to the Rectangle or' Product of the two Means. And the converse. X»^r A : 8 : : mA : mB ; then & A X nmszu X mAss foxB. as is evident. theor;^ «M GEOMETRY. THEOREM .LXXVIT. If Three Quantities be Continued Proportionals ; the Rect- angle or Product of the two Extremes, will be Equal to the Square of the Mean. And the conveise. Let a, mA, w*A be three proportionals, or A : mk : : wa : m^K ; ' then b A X /»*A == «i'A% as is evident. THROREM LXXVIII. If any Number of Quantities be Continued Proportionals \ the Ratio of the First to the Third, will be duphcate or the Square of the Ratio of the First and Second; and the Ratia of the First and Fourth will be triplicate or the cube of that of the First and Second ; and so on. . Let a, mA, f»^A, im'a, &c, be proportionals $ . OTA then IS = iM , nP'A m' = iM ; but =; «*; and =5 s &c TBEOREU LXXIX. Triangles, and also Parallelograms, having equal Altitude^ are to each other as their Bases. Let the two triangles aoc, def, have &e same altitude, or be between the same parallels ae, cf ; then is the surface of the triangle adc, to the surface of the triangle def, as the base ad is to the base DS. Or, ad : de : ; the triangle ADC, : the triangle def. For, let the base ad be to thp base de, as any one nupa- ber m (2), to any other number » (3) ; and divide the respec- tive bases into those parts, ab, « bd, dg, gh, he, all equal to one another ; and from the pcrints of division draw the lines bc, fg, f», to the vertices c and f. Then will these lines divide the triangles adc, def, into the. same number of parts as their bases, each- equal to the triangle ABC, because those triangular parts have equal bases and altitude (corol. 2, th. 25) ; namely, the triangle abc, equal to each of th^e triangles bdc, dfg, gfh, hfb« Sq that the triangle -adC| is 10 the triangliei qfE| aithe nuimber of THEOREMS. S15 parts « (2) of the former, to the number n (3) of the latter, chat is, as the base ad to the bas^ djk (cl6f. 79). ^ In like manner, the parallelogram adki is to the parallel- pgram defk, as the base ad is to the base de; each of these having the same r;itio as the number of their partSj m to «. Qi ;e. p. THEOREM LXXX, Triangles, and also Parallelograms, having Equal Bases, to each other as their Altitudes, > Let ABC, BEF, be two triangles having the equal bases ab, be, and whose altitudes are the perpendiculars CG, FPl \ then will the triangle ABC : the triangle bef : : cg : fk. ' -For, lei BK be perpendicular to AB, and equal to €G; in which let there be taken Bx. s? fh ;- drawing Ak and AL« » Then, triangles of equal bases and heights being equal (corol. 2, th. 25), the triangle abk is = ABC, ind the tri* atigle ABL == B^F. But, considering now abk, abl, as two triangles on the bases bk, bl, and having the same altitude AB, these will be as their bases (th, 19), namely, the triangle ABK: the triangle abl : : bk : bl. But the triangle abk = abc, and the trialigle abl = bef« also BK = CG, and bl = fh. Theref. the triangle abc : triangle bef :: cg : fh. « And since parallelograms are the doubles of these triangles, having the same bases and altitudes, they will likewise have to each other the same ratio as their altitudes, q. £. D. Coral. Since, by this theorem, triangles and parallelo^ams, ^hen their bases are equal, are to each other as thew alti- tudes 5 and by the foregoing one, when their altitudes are equal, they are to each other as their bases ; therefore uni- versally, when neither are equal, they are to each other in the compound ratio, or as the rectangle or product of their }>ases and altitudes. THX0S.EU 31$ GEOMETRY. THEOEEU LXXXI. If Four Lines be Proportional ; the Rectangle of the Ex- tremes will be Equal to the Rectangle of the Means. And) conversely, if the Rectangle of the Extremes, of four Lines, be equal to the Rectangle of the Means, the Four Lines, taken alternately^ will be FfoportionaL Let the four hues a, b, c, d, be ^oportionals, or A : b : : c : d; a,— then will the rectangle of a and d be i^ fequal to the rectangle of b and c j j^' or the rectangle A ^ d = 9 • c A 1 f U *^ i For, let the four lines be placed with their four extremities meeting in a common point, forming at that point four right angles ; and draw lines parallel tb them to complete the rectangles p, q, r, where p is the rectangle of A and D, Q^the rectangle of b and .c> and % the rect* angle of b and d. Then the rectangles p and R| being betFe^n the same p3trallels> are to each other as their bases a and B (th. 7:9/; and the rectangles q and r, being between the same pa- rallels, are to each other as their bases c and D* 3ut the ratio of a to b^ is the same as the ratia of c to p, by hypo* thesis 'j therefore the ratio of p to r, is the same as the ratio «of <^ to R 5 and consequently the rectangles P and q are tqual. Q. B. D. Again, if the rectangle of a and d, be equal to the rectangle of b and c; these Imes will be proportional^ qr A : b : : c : o. For, the rectangles being placed the sjime as before : then, because parallelograms between the same parallels, are to one another as their bases, the rectangle P : g : : a : Bi anil Q^ : R :: c : D. But as p and q^are equal, by supposition, they have the same ratio to r, that is, the ratio of a to MP equal to the ratio of c to d, or a : b :: c : d, q^ e. d. Corol, I. When the two means, namely, the second and third terms, are equal, their rectangle becomes a square of the second term, which supplies the place of both the second and third. And hence it follows, that when three lines arQ proportionals, the rectangle of the two extremes is equal ta a ' THEOREMS. z\1 the square of the mean ; and, conversely, if the rectangle of the extremes be equal to the sq^uare of the meaa, the three lines are proporti(»ials. CoroL 2. Since it appears, by the rules of proportion lit Arithmetic and Algebra, that when four quantities are pr«i- portionai, the prockict of the extremes is equal to the product of the two means ; and, by this theorem^the rectangle of the extremes is ^qual to the rectangle of the two mean^ » it fol- lows, that the area or ^pace of a rectangle is represented 03f expressed by the product of its length and breadth multiplied together. And, in general, a rectangle in geortietry is simi- lar to the product of the measures of its two dimensions of length and breadth, or base and height. Also, a square, it similar to, or represented by, the measure of its side multi- plied by itself. So that, what is shown of such products, is to be understood of the sqtiares and rectangles. CoroL 3. Since the same reasoning, as in this theorem, holds for any parallelograms whatever, as well as for the rectangles, the same property belongs to all kinds of paral- lelograms, having equal angles, and also to triangles, which are the halves of parallelograms ; namely, that if the sides about the equal, angles of parallelograms, or triangles, be reciprocally proportional, the parallelograms or triangles will be equal ; and, conversely, if the parallelograins or triangles be equal, their sides about the equal angles will be reciprocally proportional ' Corol. 4. Parallelograms, or triangles, having an angle in each equal, are in proportion to each other as the rectangles of the sides which are about these equal angles. THEOREM LXXXII. If a Line be drawn in a Triangle Parallel to one of its sides, it will cut the two other Sides Proportionally. Let de be parallel to the side bc of the triangle abc ; then will ad ;db : : ae : ec. For, draw be and CD. Then the tri- angles dbe, dce, are equal to each other, because they have the same base de, and are between the same parallels j^e, bc (th. 25). But the two triangles ade, bde, on the bases adi, i>h, ha?e the saotie alti- tudei ) J SIS GEOMETkt- tude; and the two triangtes ade, cde, on the bases ae, ec, have also the! same altitude; and because triangles of the same altitude are to each other as their bases> therefore the triangle Ade : bdk : : ad : db, and triangle adr : cde : : ae : £c. But BDE is ts CDE } and equals must ha^e to equals the same ratio; therefore ad : db ;: ae : £c* q. e. d* CS^ol. Hence, also, the whole lines ab, Xc, are propor- tional to their corresponding proportional segments (corol< VIZ* AB : AC and AB : ac • * ad bd AE, C£« THEOREM hXXXllU A Line which Bisects any Angle of a Triangle, divides the opposite Side into Two Segments, which are Propottional to the two other Adjacent Sides. LtT the angle acb, of the triangle abc, ^ be bisected by the line cd, making the angle r equal to the angle / : then will the segment ad be to the segment db, as the ,iide AC is to the side cb^ Or, - - - - AD : db : : AC : CB. For, let BE be parallel to cd, meeting AC produced at £• Then, because the line bc cuts the two |mrallels cd, be, it makes the angle cbe equal to the alter-*^ Hate angle / (th. 12), and thereifore also equal to the angle r, which is equal to / by the supposition. Again, because the line ae cuts the two parallels dc, be, it makes the angle E equal to the angle r on the same side of it (th. 14). Hence, in the triangle bce, the angles b and e, being each equal to the angle r, are equal to each other, and conse- quently their opposite sides cb, ce, are also equal (th, 3). But now, in the triangle abe, the line cd, being drawn par^lel to the side be, cuts the two other sides AB, A £, pro- portionally (th. 82), making ad to db, as is ac to cb or to Its equal CB. Q. £. D* THEORBM THEOREMS. 519 Theore:^ Lxxxir. . Equiangular Triangles are Similar^ or have their Like Sidct Proportional, Let ABC, DEF, be two equiangular tri- angles, having the angle a equal to che^ angle D, the angle B to the angle £, and consequently the angle c to the angle v, then will ab : ac : : D£ : d^. For, make dg = ab, and dh = ac, and join gh. Then the two triangles ABC, DGH, having the two sides ab, ac, equal to the two dg, dh, and the con- tained angles a and d also equal, are iden- tical, or equal in all respects (th. 1 ), namely, the angles b and c are equal to the angled G and if. Bat the angles b and c are equal to the angles £ and f by the hypo* thesis ; therefore also the angles g and h are equal to the' angles £ and f (ax. 1), and consequently the line gh is pa^ rallei to the side ef (cor. 1, th. 14). Hence then, in the triangle def, the line gh, being pa- rallel to the side ef, divides the two other sides propor- tionally, making dg : dh : : de : df (cor. th. 82). fiut DO and DH are equal to ab and ac ; therefore also - - - "^ JB : AC : : DC : df. q. e. p. THEOREM LXXXV. Triangles which have their Sides Proportional, are Equi* angular. In the two friangles abc, def, if AB : DE : : AC : df : : bc : ef ; the two triangles" will have their corresponding angles equal. For, if the triangle abc be not equian- gular with the triangle def, suppose some Other triangle, as deg, to be equiangular with ABC. But this is impossible : for if the two triangles abc, deg, were equi-* angular, their sides would be proportional (th. 84). So that, ab being to de as AC to dg, and ab to de as Bc to eg, it follows tliat OG and SG, being fourth proportionals to the same three quantitiest Cr F 880 Gfi(»fETRY* as well as the two pr» ef, the former D<f, £G, irdoici hi equal to the latter, df, ef. Thus then, the tiKro tmngk^ DBF, d£g, having their three sides equal, lifroilld be identkal (th« 5); which is absurd^ since their angles ate^ un^qual^ THEOREM LXXXVt- Triangles, which have an Angle in the on^ Equal to zh Angte in the other, and the Sides about these angles Proportionals Equiangular. Let ABC, DEF, be two triangles, having the angle a = the angle d, and the sides AB, AC, proportional to the sides D^, Dt: then will the triangle abc be equiangular with the triangle def» For, make dg = ab, and dh = AG| and join gh. Then, the two triangles abc, dgh, having two sides equal, and the contained angles a and d equal, are identical and equiangular (th. l), having the angles G and H equal to the angles B and c« But, since the side^^ DG, DH, are proportional to the .ides de, df» the line GH is parallel to ef (th. 82); hence the angles e and f are equal to the angles g and H (th. 14), and consequently to their equals m and c* q« s. d. THEOREM LXXXVII. In a Right- Angled Triangle, a Perpendicular from the Right Angle, is a Mean Proportional betweex^ the Segments of the Hypothenuse ; and each of the Sides^ about the Right Angle, is a Mean Proportional between the Hypothenuse and the adjacent segment. Let ABC be a right-angled triangle, and . CD a perpendicular from the right angle c to the hypothenuse ab; then will CD be a mean proportional between ad and db ;. AC a mean proportional between ab and ad ; Bc a mean proportional between ab and BD. .' Or^^AD : CD : : CD ; DB$ and ab : »c : : ac : bd | attd AB : AC : : AC : AD. For J THEOREMS* i^i For, the two triangles abc, adc, having the right angles at c and D equal, and the angle a common, have their third wangles equal, and are equiangular (cor. 1, th. 17). In like manner, the two triangle^ abc, bdc, having the right angles at C and d equal, and the angle b common, have their third angles equal, and are equiangular. Hence then, all the three triangles abc, apc, bdc, being equiangular, will have their like sides proportional (th. 84); VIZ. Ab : ci> * : CD : db; and A£ t AC :: AC : AD} and AB : BC : : BC : BD. ^ <^E. D* CoroL Because the angle in a semicircle is a right angle (th. 52) ; it follows, Ihat if, fi-om khjr point c in the peri- phery of the semicircle, a perpendicular be drawn to the diaxtieter ab ; and the two chords ca, dB^ be drawn td the extremities of the diameter : then are ac, bc, cd, the mean proportionals as in this theorem, or (by th. 17)^ *• * ^ tD* ac AD . DB; AC* ss AB • AD; and BC^ = AB . dDk THEOREM LXXXVIII* jEquiangular or Similar Triangles, are to each other as the 'Squareis of their Like Sides. Let abc, def, be two equi- angular triangles, ab and de being two lik6 sides : then will the triangle' abc Ije to the tri- angle DBF, as the square of AB is to the $quaxe of D£, or as AB* to de\ For, ht At and dn be the isquares on ab and dk; also draw their diaj^onals bk, eM, and the perpendicuhrs cg, fh, of the two triangles. Then, since equiangular triangles have their like sides proportional (th. 84), in the two equiangular triangles abc, def, the side ac : df : : ab : de ; and in the two acg, DFH, the side Ac : df : : cg : fh ; therefore, by equality CG :, FH : : ab : de, or cg : ab :: fh : DE. But because triangles on equal bases are to each . other as their altitudes, the triangles ABC, Abk, on the same base ab, are to eath other, as their altitudes cg, ak, or ab: Vol. L Y and Z22 GEOMETRY. and the triangles def^ dem, on the same base PE| are as their altitudes fh> dm^ or d£ ; that isy triangle abc : triangle abk ; : CG : AB9 and triangle def : triangle dem : : f h : ds. But it has been shown that cG : ab : : FH : i>£ } theref. of equality A abc : aabk : : abef : Adem,. or alternately, as a abc : adef : ; >^ abc t A dem. But the squares al, dn, being the double of the triangles ABKj D£M« have the same ratio with them \ therefore the A abc : adef : : square al : square dn* . , (i. E. i^ THEOREM LXXXIX. All Similar Figures are to each other, as the Squares of their Like Sides. Let abcd£| fghik, be any two similar figures, the like sides being ab, fg, and BC,'GH,and so on in the same order : then will the figure ABCDEbe tothefigurcFGHiK, as the square of.AB to the square of eg, or as ab* to fg*. For, draw £e, bd, gk, g^x dividing the ^^es into a» equal number of triangles, by Knes from two equal angles- b and G^ The tsffo figures being similar (by stippos.)i they are equi- angular, and have their like sides proportional (def. 67). Then, since the angle A is ^^ the angle f, and the side^ ab, A£, proportional to the sides fg, fk, the triangles ABE, fgk, are equiangular (th. 86). In like manner, the two triangles bcd, ghi^ having the angle c = the angle H^ , imd the sides «c, CD, proportional to the sides gh. Hi, are also ecfuiangdar. Also, if from the equal angles aed, fki> there be taken the equal angles aeb, fkg, there will remain the equals bed, GKI ; and if from the equal angles cde, HiK, be taken away the equals cde, hig, there will remain the equals bde, gik 5 so that the two triangles bde, gik, having two angles equal, are also equiangular. Hence each triangle of* the one figure, is equiangular with each corre* sponding triangle of the other. But equiangular triangles are similar, and are propottfofial to the squaj'es of their tike sides (tit. 88).* TWeforr THEO&tMS. t2% Therefore th^ A ab» : a fgk : t ab* :sg% and A BCD :' A ghi : : bc* : gh* and A bdb : A gik : : de^ : IK .> But ds the two polygons are similar^ their like sides are pro- portional, iand consequently their squares also proportional i so that all the ratios ab* to pg% ^d bc* to gh*, and de* to 1K% are equal among themselves, and consequently the cor- responding triangles also, abe to fgK) and bcd to ghi, and jBDE to GIK, have all the same ratio, viz. that of ab'^ to fg* : and hence all the antecedents, or the figure abode, have to mil the con^quents, or the figure fghik, still the «ame ratio^ Viz. that of AB* to FG* (th. 72). q. e. d. ^ THEOREM XC« Similar Figures Inscribed in Circles, have their Like Side«, and also their Whole Perimeters, in the Same Ratio- as this- Diameters of the Circles in which they are Inscribed. iiET ABCDE, fghik, be two similar -figures, inscribed in the circles whose diametof's are al * and FM ', then will each side A6, BC, &c, of the t)ne figure be to the like dde gf, gh^ &c> of the "other figure, or the whole perimeter ab + bc + &c, of tlie one figure, to the whole perimeter fg + GH 4" &c, of the Other figure, as the dis^meter al to the diameter f^. For, draw the two corresponding diagonals AC, fh, as also the lines bl, gm. Then, since the polygons are similar, they are equiangular, and their like*5ides have the same ratio (def. 67) ; therefore the two triangles abc, fgh, have the angle b =st the angle G, and the sides ab, bc, proportional to the two sides fg, Ch, consequently these two triangles are equiangular (th. 86), and have the angle acb = fhg. But the angle acb = alb, standing on the same arc ab; ^ and the angle fhg = fmg, standing on the same arc fg; .therefore the angle alb = fmg (ax. l). And sjince the wangle abl. = fgm, being both right angles, because in a .semicircle ; therefore the two triangles abl, fgm, having two anglCj* ecjoal, are equiangular ; and cpnsequentiy their .if 4 GEOMETRr. like sides afe profjortional (th. 8*) ; hence AB : ><J : ; the diameter al : the diameter fm. In like manner, each side bc, cd, &c, has to each side CH,m, &c, the same ratio of al to fm; and consequently the sums of them are still in the same ratio j viz, AB + bc -f c©, &c : FG + CH + HI, &c : i the diam. AV : the diami. FM (th. 72). £. B. D. . t THEOREM XCi; Similar Figures Inscribed in Circles, are to each other as the Squares of the Diameters of those Circles. Let abcdb, fghik, be two similar figures, in-' scribed in the circles whose diameters are al and FM ; then the surface of* the polygon abcde will be to the surface of the polygon fghik, as AL* to fm*. For, the figures being similar, are to each other as- the / squares of their like sides, ab* to FG* (th. 88). But, by the last theorem, this sides ab, fg, are as the diameters al, FM ; and therefore the squares of the sides ab^ to fg*, as the squares of the diameters al* to fm* (th. 74). Consequently the polygons abcde, fghik, are also to each other as the squares of the diameters al* to fm* (ax. l). <^ £• d. theorem xcii. The Circumferences of all Circles are to each other as their Diameters. Let d, di denote the diameters of two circles, and c, <:, their circumferences ; then will d :.i/ : : c : r, or D : c : : rf : r. ' For (by theor. 90), similar polygons inscribed in circles have their perimeters in the same ratio as the diameters of those circles. Now, as this property belongs to all polygons, whatever the number of the sides may be ; conceive the numbe^r of the sides to be indefinitely great, and the length of each inde- finitely smalU till they coincide with the chrcumference of the THEOREMS. 3-25 t the circle, and be equjd to it, indefinitely near. . Then the perimeter of the j)olygon of an infinite number of sides, is the same thing as the circumference of the circle. Hence h: appears that the circumferences of the circles, being the same as the perimeters of such polygons, are to each other in the i^ame ratio as the diameters of the circles, q^ s. d. THEOREM XCllI. t The Areas or Spaces of Circles, are to each other as the Squares of their Diameters, or of their Radii. Let a, ay denote the areas or spaces of two circles, aad JD, dy their diameters; then a :^ : : d^ : i/^. Fot (by theorem 91) similar polygons inscribed in circles are to each other as the ^squares of the diameters of th^ circles. Hence^ conceiving the numher of the sides of the poly- gons to be increased more and more, or the length of the aides to become less and less, the polygon apprpaches nearer and nearer to the circle, till at length, by an infinite ap<> proach, they coincide, and hecome in effect equal ; and then it follows, that the spaces of the circles, which are the same as of the polygons, will be to each other as the squares of the diametersof the circles. (^ £. J>. • CorQh The spaces of circles are also to each other as the squares of the circumferences ; since the circumlFerences are in the same ratio as the diameters (by theorem 92). THEOREM XCIV. The Area of any Circle, is Equal to the Rectangle of Half its Circumference and Half its Diameter. Conceive a regular polygon to he ^^f^ 'inscribed in the'ciTcle^; and radii drawn to //\ -all the angular points, dividing it into as j/ . xnaAy -equal triangles as the polygon has K . . ^idesji one of which is abc, of which the \x / jiltituid^ is the perpendicular CD from the -S^F^li centre to the base ab.' Then the triangle abc, being equal to -a rectan^e bif half the base and equal altitude (th. 26, cor. 2), is equal to the^ectangle of the half base ad aod the a^titudeoD ; £onse«> I S^6 GEOMETRY, consequently the whole polygon, or all the triangles added together which com- pose it, is equal to the rectangle of the conunop altitude cd, and the halves of all the sides, or the half perimeter of the po- lygon. Now, conceive the number of sides of the polygon to ht indefinitely increased ; then will its perimeter coincide with the circumference of the circle, and consequently the alt^ tude CD will become equal to the radius, and the whole polygon equal to the circle. Consequently the space of the circle, or of the polygon in that state, is equal to the rect^ angle of the radius and half the circumference/ ^ £• n. 9e OF PLANES AND SOLffiS, \ • DEFINITIONS. DjtF. S8. The Common Section of two Pl^es, is the line in which they meet, to cut each other* 89. A Line is Perpendicular to a Plane, when it is per-* . pendicular to every line in that plane which meets it. 90. One Plane. is Perpendicular to Another, when every Kne of the one, which is perpendicuJar to the line of their common section, is perpendicular to the other* 91. The Inclination of one Plane to another, or the ^gle they form between them, is the angle contained by two lines, drawn from any point in the common section, and at right angles to th« same, one of these lines in each plane^ 92. Parallel Planes, are such -as being produced ever so far both ways, will never meet, or which are evcf]^ where at an equal perpendicular distance* 9&. A. Solid Angle, is that which is made by three or more pltoe irfglcs, meeting tac-h pthcr in the sam^ point. 94f. Similar J DEFINITIONS- 327 .'94. SiirvilafS^Uds^ contained by plane figures, 3re such as liave all their so)id angles equal, each to each, and are bound- ed by the same number of similar planes, alike placed* ^ 95. A Prisrn, is a solid whose ends are parallel, equal, and like plane figures^ and its sides, connecting those ends, are parallelograms. - • > • 96. A Prism takes particular names according to the figure of its base or ends, whether triangular, square, rectangular, pentagonal, hexagonal, &c* ?7. A Right or Upright Prism, is that which has the planei, of the jides perpendicular to the plane* of the endU or bi^e. t^ hi 9a» A Porallelopiped, or Parallel opipedon, is .^ . '^m bounded by six parallelograms, every o. , lie two of which are equal, alike, and pa- ■•'9. A Rectangular Parallelopipedon, is that whose bound- ing i lanes are all \rectangles, which are perpendicular to each ©her. 100. A Cube, is a square prism, being bounded by six equal square sidei or faces, and are perpen* dicalar to each other. I (;1 . A Cylinder is a round prism, havij^g cir- cles for its ends ; and is conceived to be formed by I he rotation of a right line about the circum- ferences of two equal ^nd parallel circles, always parallel to the axis. 1 02. The Axis of a Cylinder, is the right line joinirrg the centres of the two parallel circles, about vhich the ligure is described. 10^. A Pyramid, is a solid, whose base is any right-lined jiane figure, and jts sides triangles, having all their vertices meeting together in a point above the base, called the Vertex of the pyramid. 104. A pyramid, like the prism, takes particular names '.from the figure of the base. ' ^ 105. A Cone, is a round pyramid, having a cir- .cula^ base, and is qonceived to be generated by the rotation of a right line about the circum- .ference of a circle, one ^nd of which is fixed at . a point above the piw^e pf tjiat circle. ' ' • -^ 106. The' 528 GEOMETRY. J Of, The Axis of a cone, is the right line, joining th« vertex, or fixed point, and the centre of the circle about H^hich the figure is described. 107. Similar Cones and Cylinders, are such as have their altitudes and the diameters of their bases proportional. 108. A Sphere, is a solid bounded by one curve surface^ which is every where equally distant from a certain point within, called the Centre. It is conceived to be generated by tlie rotation of a semicircle about its diameter, which re- mains fixed. 109. The Axis of a Sphete, is the right line about which th^ semicircle revolves; and the centre is the same as that of the revolving semicircle. 110. The Diameter of a Sphere, is any right line passing through the centre, and terminated both ways by the surface. 1 H. The Altitude of a Solid, is the perpendicul^ drawn from the vertex to the opposite side or base. THEOREM XCV, A Perpendicular is the Shortest Line which can be drawfi from any Point to a Plane. Let ab b^ perpendicular to the plane ju DE ; then any other line, as AC, drawn \ from the same point a to the plane, will \ be longer than the line ab. ^ ^"ul \ In the plancf draw the line Bc, j^oining <?^ the points B, o. ' ' - 'Then, because the line ab is perpendi- cular to the plane de, the apgle B is a right angle {def. 90), and coi^sequently greater than the angle c ; therefore the line AB> opposite to the less angle, is less than any other lint AC, opposifjB the greater angle (th. 21), q. E. D. THEOREM XCVI. A Ferpendic^lar Measures the Distance of any Point from 4 PJane, .The distance of one point from another is measured by a right line joining them^ because this is the shortest line which can be drawn from one point to another. So, also^ the ^stance from a point to a line, is measured by a perpendi- ipular^ l^ecause this line is the shortest which can be drawn fronat THEOREMS. S29 from tlie point to the line. In like manner, tbc distance from a point to a plane, must be measured by a perpendicular jdrawn from that point to the plane, because this is the shortest line which can be drawn from the point to the j>tane» THEOREM XCVII. . The Common Section of Two Planes, is a Right Line, Let acbda, akbfa, be two planes cutting each other, and A, b, two points £, in which the two planes meet ; drawing . \ F-* the line ab, this line will be the common intersection of the two planes. For, because the right line ab touches the two planes jn the points a and b, it — ^ touches them in all other points (def. 20): this line is therefore common .to the two planes; That is, * ^he cpmmon intersection of the two planes is a right line. ^ ^ u $ THEOREM XCVllh I \i a Line be Perpendicular to two other Lines, at their Common Point o!F Meeting ; it will be Perpendicular to the Plane of those Lines. Let the line ab make right angles with the lines AC, ad 5 then will it be per- pendicular to the plane cde which passes jthrough these lines. If the line ab were not perpendicular to the plane cde, another plane might pass through the point a, to which the line ab ^ would be perpendicular. But this is im- possible -, for, since the angles bag, bad, are right angles^ this other plane must pass through the points g, d.. Hence, this plane passing through the two points a, c, of the line AC, and through the two points a, d, of the line ad, it will "pass through both these two lines, and therefpre he the same jplane with the former, q. e. d. TH]£0R|(M sso GEOMETRY. THJEbUBM XCIX. If Two Lines be Perpendicular to the Same Plane, thejr wM be Parallel to each other. S > Let the two lines ab, cd, be both per- pendicuVar to the same plane ebdf ; then will Afi ha paralLed to cD. « For, join B, i>, by the line bd in the ' plane. Then, because the^ lines ab, cd, . are j^^erpendicular to the plane £F, they are both perpendicular to the line bd (def. 90) in that plane^ and cor sequent ly they are parallel to eacl^ other (corol. th. I'i). Q. E. D. CoroL It two lines be parallel, and if one of them be perpendicular to any plane, the other will also be perpendi- cular to the same plane. THEOREM C If Two Planes Cut each other at Right Angles, and a Line be drawn in one of the Planes Perpendicular to their . Common Intersection, it will be Perpendicular to the . other Plane. Let the two pknes acbd, abbf, cut each other at right angles; and the line^ CG be perpendicular to their common sec- tion ab ; then will cg be also perpendicular to the other plane aebf. For, draw eg perpiendicular to ab. Then, because the two lines gc, gEj are perpendicular to the common intersection ^ ABy the angle cge is the angle of inclination of the two planes (def. 92). But since the two planes cut each other . perpendicularly, the angle of mclination cge is a right smgle. And since the line cq is perpendicular to the. two lines GA, ge> in the plane aebf, it is therefore perp^ndi* cular to that plane (th. 9^)* q. e. v. TKBORSM THEOREMS- SSI THEOI^EM CI. If one Plane Meet another Plane, it will make Angles with that other Plane, which are together equal to two Right Angles. - Let the plane acbd meet the plane asbf; these planes xnake with each other two angles whose sum is equal to two right angles. For, through any point g, in the common section ab^ draw CD, ef, perpendicular to ab. Then, the line co ^akeswjth EFtwo angles together equal to two right angles. But thesie two angles are (by def. 92) the angles of inclina- tioii of the two planes. Therefore the two planes make jingles with each other, which are together equal to two right angles* CoroL In like manner, it may be demonstrated, that planes icfhich intersect, have their vertical or opposite angles equal; also, that parallel planes have their alternate angles equal ; (ind so on, as in parallel lines. THfcOUEM CII. y If Two Planes be Parallel to each other ; a Line which is Perpendicular to one of the Planes, will also be Perpendi-» Cular to the other. I tiET the' two planes CD, ef, be parallel, jind let the line ab be perpendicular to the plane cD ; then shall it also be perpendi- cular to the other plane ef. For, from arty point G, in the plane £F, draw GH perpendicufar to the plane cd, and 4raw^AH, BG. Then, because ba, gh, are both perpendicular to the plane cd, the angles k and h are both right an'gles. Attd because the planes cd, j^f, are parallel, the perpendiculars 3A, gh, are equal (def, 93). Hence it follovrs that the lines ;3G, AH, are parallel (drf. 9). And the line ab being perpendicular tp the line ah, is also perpendicular to the parallel line bg (cor. th. 12). In'likemanner it is ptoved, that the line ab is p«rpen- 4i£Dlar to aU otl^r 4ines whic^^a be drawn from^ the point b in 532 GEOMETRY, in the plane ef* Therefore the line ab is perpendicular tm the whole plane £f (def. 90). q. b. d. THEOREM cm. If Two Lines be Parallel to a Third Line, though not in the same Plane with it ; they will 6e Parallel to each other* Let the lines ab, cd, be each of them ^ . parallel to the third line £F, though not in the same plane with it ; then will ab be pa* lallel to cj>. For, frem any point G in the line ef, let GH, Gi, be each perpendicular to ef, in the planes eb, ed, of the proposed parallels. Then, since the line ef is perpendicular to the two lines GH, gi, it is perpendicular to the plane ghi of those lines (th. 9S). And because SF is perpendicular to the plane <yHi, its parallel ab is also per- pendicular to that plane (cor» th. 99), For the same reason, the line cd is perpendicular to the same plane ghi. Hence, because the two lines ab, cd, are perpendicular to the sam(^ plane, these two lines are paiydlel (th. 99). q. e. d* theorem cit. If Two Lines, that meet each other, be Parallel to Twq other Lines that meet each other, though not in the sanje ^Plane with them j the Angles contained by those Lin^s will be equal. Let the two lines ab, bc, be parallel to the two lines de, ef ; then will the ajigle ABC be equal to the angle def. For, make the lines ab, bc, de, ef, all lequal to each other, and join ac, df, ad« BE, cf. Then, the lines ad, be, joining the equal and parallel lines ab, de, are equal and parallel (th. 24). For the same reason, cf, be, are equal and parallel. Therefore adj cf, are equal and parallel (th. 15); and consequently also ac, df (th. 24). Hence, the two triangles abc, d35f, having aU^heir sides equ^l, THEOREMS- S3* ^ach to <^acli> have their angles also equal, and consequently the angle ABC = the angle def. (^ £. d. THEOREM cr. The Sections made by a Plane cutting two other Parallel Planes> are also Parallel to each other. 3LBT the two parallel planes ab, cd, be cut by the third plane efhg, in the lines fXSL=::^K £F, GH : these two sections ef, GH,.will be parallel. Suppose EG, FH, be drawn parallel to each other in the plane efhg ; also let EI, FK, be perpendicular to the plane cd ; and let IG,KH, be joined. Then eg, fh, being parallels, and ei, fk, being both perpendicular to the plane CD, are also parallel to each other (th. 99) ; conisequently the angle hfk is equal to the angle CEi (th. 104). But the angle fkh is also equal to the angle fcio, being both right angles; therefore the two triangles are equiangular (cor. 1 th. 17) ; and the sides fk, ei, being the equal distances between the parallel planes (def. 93), it follows that the sides fh, eg, are also equal (th. 2). But these two lines are parallel (by suppos.}, as well as equal; consequently the two lines e^, gh, joining those equal pa- Tallek, are also parallel (th. 24). <^ e. d. theorem cvr. If any Prism be cut by a Plane Parallel to its Base, the Sectioa will be Equal and Like to the Base. LsT AG be any prism, and il a plane ' parallel to the base AC ; then will the plane IL be equal and like to the base ac, or the two planes will have all their sides and all their angles equal. For, the two planes ac, il, being paral- lel, by hypothesis ; and two parallel planes, cut by a third plane, having parallel sections (th. 105); therefore IK is parallel to ab, and kl to BC, and I,M to CD, and im to ad. But ai and bk are parallels (by def. 95) ; consequently ak is a parallelogram ; and the opposite sides ab, ik, are equal (th. ^2). In like manner, it S34 GEOMETRY. It is shown that kl is =s bc» attd IM = ci>y uid IM ^ AD» 'or the two planes Ac, il, are mutually equilateral. But these two planes, havingtheir corresponding sides parallel, have tlie angles contained by them also equal (th. 104), namely, the angle A = the angle t, the angle B =: the angle K, the angle c = the angle. L, and the angle d = the angle m. So that the two planes AC, tL, have all their corresponding sides and angles equal, or they arc equal and like. Qi £. d» s THEOREM CVII. If a -Cylinder be cut by a Plane Parallel to its Base> the Section will be a Circle, Equal fo the Base. Let af be a cylinder, and ghj any section parallel to the base abc; then will CHI be a circle, equal to ABC. For, let the planes ke, kf, pass through the axis of the cylinder mk, and meet the section GHi in the three points il, I, 4- -, and join the goints as in the figure. Then, since kl, ci, are parallel (by def» l0'2) ; and the plane Ki, meeting the two parallel planes ABC, GHi, makes the two sections KC, Lr, parallel (th. 105) j the figure klic is therefore a paraI-» lelogram, and consequently has the opposite sides li, kg, equal, where KC is a radius of the circular base. In like manner, it is shown that lh is equal to the radius KB ; and that any other lines, dr;iwn from the point L to the circumference of the section ghi, are all equal to radii of the base 5 consequently Giii is a circle, and equal to ABC. (^ £. D. THEORElvr CVIII. All prisms and Cylinders, of Equal Bases and Altitudes^ ave Equal to each other. . Let AC, DF, be two prisms, and a cylinder, en equal bases ab, de, and having equal alti-» tudes £c, FF ; then will ^the solids AC, DP, be -cquaL FoT) let FQ, Rs, be any C J^---^ Q, IB i^C3s THEOREMS. ■ 355 any two sections parallel to the bases, and e^idlstant from them. Then, by the last two theoi^ms, the section pq^is equal to the base ab, and the section ks equal to the base ■ DK. But the bases ab. DE, are equal, by the hypothesis; therefore the sections pa, rs, are equal also. In like manDer, it m»y be shown, that any other corresponding sections are equal to one another. Since then every section in the prism ac, is equal to its _ corresponding secfion in the prism or cylinder DF, the prisms and cylinder themselves, which are composed of an equal sumheror all these equal sections, must also be eiual H..E.D. Carol. Every prism, or cylinder, is equal to a rectangular parallelopipedon, of an equal base and altitude. TUSORBM cix. Rectangular Parallelopipedons, of Equal Altitudes, are to each other as their Bases. \XT AC, EG, be two rectan- rularparallelopipedons, having the equal altitudes ad, eh ; thea will the solid ac be to the solid EG, as the base ab is to the base ef. For, let the proportion of the base AB to the base ef, be that of anyone number m (3) to any Other number n (2). And conceive AB to be didded into m equal parts, or rectangles, ai, i.k, mb (by dividing an into that number of equal parts, and drawing II, K^f, parallel to Bn). And let ef be divided, in like manner, into n equal parts, or rectangles, eo, pf : all of these pans of both bases being mutually equal among themselves.. And through the lines of division let the plane sections lr, ms_, pv, pass parallel to aq, et. "nien, the parallelopipeJons ar, ls, mc, ev, pg, are all equatj having equal base; and altitudes. . Therefore the solid AC is to the solid eg, as the number of parts in the former, to the number of equal parts in the latter ; or as the number of parts in ab to the number of equal parts in ef, that is, as the base ab tOjbe base Et. q. e. d. CsrU. From this theorem, ajid .the corollary to the last, it appears^ that all prisms and cylinders of equal alfitudest »» 336 GEOMETRY. to each other as their bases ; every prism at^d cylinder beiagf equal to a rectangular parallelopipedon of an equal base and altitude. THEOREM ex. Rectangular Parallelopipedons, of Equal Bases^ are to each other as their Altitudes. L B ^' Ct ^. 4 a C Let ab, cd, be two rectan- gular paraUelopipi^ons, stand- ing on the equal bases ae,cf; then will the solid ab be to the solid CD, as the altitude £ B is to the altitude fd. For> let AG be a rectangular parallelopipedon on the base ae^ and its altitude eg equal to the altitude fd of the solid CD. Then ag and en are equal, being prisms of equal ba^S and altitudes. But if hb, hg, be considered as bases, the solids ab, AGy of equal altitude ah, will be to each other as those bases hb, hg. But these bases hb, hg^ being parallelograms of equal altitude he, are to each other a^ their bases eb, eg ; therefore the two prisms ab, ag, are to each other as the lines eb, eg. But ag is equal to CD, and EG equal to fd; consequently the prisms aC|CD, are to e^ch other as their altitudes eb, fD^ that is^ - - ** 'ab : CD :: eb : fd, q^ e. d. CoroU 1. From this theorem, and the corollary to theorem 108, it appears, that all prisins and cylinders, of equal bases^ are to one another as their altitudes. CoroL 2. Because, by corollary 1, prisms and cylinders are as their altitudes, when their bases are equal. And, by the corollary to the last theorem, they are as their bases, 'when their altitudes are equal. Therefore, universally, when nei- ther tire equal, they are to one another as the product of their bases and altitudes. And hence also these products are the proper numeral meaisures of their quantities or magnitudes^ THEORBM CXI. Simijar Prisms and Cylinders are to each other, as the Cubes of their Altitudes^ or of any other Like Linear Di* mensions. Let abcd, efgh, be two similar prisms ; then will the prism CD be to the prism gh, as, ab^ to sif' or ad' to £h'. - ^ ' For o p Pot the solids M'e to e^ch other tis the product of their bases and alti- tudes (th. 110, cor. 2), that Ts, as AC . Alb to £G . £H. But the bases,' i>6ing similar planes, tire to each other as the squares* of their like -^^ [y'*' j; sides, that is, ac to eg as ab^ to £F^; therefore the solid cd is to the solid Gii, a^ ABf . ad to ^f' . EH. - But KD and fh, being . similar planes, have their like side^ proportional, that is, ab : ef : : ad : bh, - - - - - ^ qr AB* : £f* : : ad*: eh**: thereCore ab*. ad : ef*. sh : : ab^ :,ef% or : : ad' : eh' ; conseq. the solid cd : solid gu : ; ab' : 15f' ;: AD* t bh'* <^ e. d. 6i THEOREM exit In any Pyramid, a Section Parallel to the Base is similar to the Base; and these two planes are to each other as the Squares of their Distances from th<» Ydrtexi Let abcd be a pyramid, and e^o a sec'^ tion parallel to the base BCD, also Aiii a line perpendicular to the two planes at H and I : then will bd, ]sg, be two siniilar planes, and the plane BD will be to the plane bg, as AH* to AI*. For, join ch, ti. TheUj because a plane cutting two parallel planes, makes parallel sections (th. 105), therefore the plane ABC, meeting the two parallel planes bd, eg, mak^ the secticrns Bc, ef, parallel : In like manner^ the plane acd makes the sections CD, fG, paraUel^ Again, because two pair of parallel lines make equal angles (th. 104), the two ef, fg, which are parallel to BC, CD, make the angle bfg equal rthe angle bcd. And in like mannef it is shown^ that each angle in the plane eg is equal to each angle in the plane BD, and consequently those two planes are equian-^ gular. Again, the three lines ab, Ac, ADj making with the parallels bc, eF, and cb, fg, tqual angles (th. 14)j and the angles at a being common, the two triangles ABC, aef, &re equiangular, as also the two triangles ACD, afg, and have thcrerore their like sides proportional, namely, - - - VOL.L Z AC d58 GEOMETRY. AC : AF BC BP : : CD : VG. And in like manner it may be shown, that aU the lines in the plane fg, are proportional to all the corresponding lines in the base bd. Hence these two planes, having their angles equal, and their sides proportional, are similar, by def. 68. Ip-- — ^ But, similar planes being to each other as the squares of their like sides, the plane bd : eg : : bc* : ef% or : : AC* : AF*, by what is shown above. Also, the two triangles AHC, AiF, having the angles H and i right'ones (th. 98), . and the angle A common, are equiangular, and have there- fore their like sides proportional, namely, ac : af : : ah : ai, or AC* : AF* :: ah* : ai*. Consequently the two planes BD, EG, which are as the former squares ac*, af*, will be also as the latter squares ah*, ai'^ that is, - - - - - BD : EG :: ah* : Ai*. Q. E. D. THEOREM CXIII. X t In a Cone, any Section Parallel to the Base is a Circle ; and this Section is to the Base, as the Squares of their Distances from the Vertex. . Let abcd be a cone, and ghi a section p^vallel to the base BCd; then will ghi be a circle, and bcd, ghi, will be to each other, as the squares of their distances from the vertex. For, draw alf perpendicular to the two paralld planes; and let the planes ACE, ADE, pass through the axis of the cone AK.E, meeting the section in the three points H, I, K. Then, since the section ghi is parallel to the base BCD, and the planes CK, dk, meet them, hk is parallel to ce, and IK to DB (th. 105). And because the triangles formed by these lines are equiangular, kh : EC : : ak : ae : : Ki : ed. But EC is equal to ed, being radii of the same circle ; there- fore KI is also equal to kh. And the same may be shown of any other lines drawn from the point K to the perimeter of the section Gj^i, which is therefore a circle (def. 4?4). Again, by similar triangles, al : af : : ak : :: KI : ED, hence al* : af* :: Ki* : ed*; but Ki* circle ghi : circle bcd (th. 03); therefore A L* : circle ghi : circle bcd. q. e. d. AE or ED* : : af"^ ;: THEOHEM THEOR^MS^ S30 TkEOR^M ckiv; . ^ All Pjramidsy and Cones, of Equal Bases and Altitudes, art* Equal io one another^* Let abc^ def, ie any pyramids and cone, of ^qiial ba^es BC, £F, and equal , altitudes AG, dh: then will the pyra-, mids and cone abc and DEF, be equal* For, parallel to the bases and at equal distances AK, po, from the vertices, suppose the planes tK, lm, to be di^wn. ' ....... Then, by the two. preceding theorems, -------* DO* : DH* i: LM : sf, and AN* 2 AG* : : IK : BC. But since an*, ag*, are equal to d6% dh*, therefore iK : BC : : lm :.ef. But bc is equal to fcp, by hypothesis j therefore ik is also equal to lm. « In lik^ thaiinef it is shown, that any other sections, at equal distance from the vertex, are equal to each other. Since theii, every section in the cone, is 6qual to the cor- responding section iii the pyramids, and the heights are equal,' the solids abc^ d£f, /:omposed of all those sections, must be equal also; q. IS. jb. TilEOR^M CXV.' Every Pyramid is tlie l*hird Part of a Prism o^ the Same' Base and Altitude. Let abcd£f bef a prism, and ubsp a ^ pyramid, on the same triangular base d£^: then will the pyramid BDef be a third part tf the prism abcdef* For, in the plancfs of the thi-^e sidrfs df thcf ?rism, draw the diagonals bf, bd, cd. i'hen the two planes bdf, bcd, divide the whole prism into the three pyramids bdef, ^abc, 'iJBCF^^ ^hich are proved to be all equal to one another, as follows. Since the opposite end^ of the prism are equal to each other, ihe pyramid nfhbse base is abc and vertex D, is equal to the a 2 pyraxjrwd f Sit) CEOMETRT. pyramid whose base is def and vertex B (th. 114), being pyntmids of equal base . and altitude. But the latter pyramid, whose base is B£F and vertex b> is the same solid as the pyramid whose base is bef and vertex t>i and this is equal te the third pyramid whose base is bcf and vertex d, bei^ig py- .ramids of the same altitude and equal ba^es BEF, seF. Consequently all the three pyramids, which compose the prism, are equal to each other, and each pyramid is the third part of the prism, or the prism is triple of the pyra- mid, q. E. D. Hence also, merj pyramid, whatever its figure may be, is the third part of a prism of the same base and altitude ; since the base of the prism^, whatever be its figure, may^ divided into triangles, and the whole solid into trian^lar prisms and pyramids. CoroL Any cone is the third part oJF a cylinder, or of a prism, of equal base and altitude ; since it has been proved that a cylinder is equal to a prism, and a . cone equal to .a pyramid, 6f equal base and altitude. ScholU/m, Whatever has been demonstrated of the propor- tionality of prisms, or cylinders, holds equally true of pyra- mids, or cones ; the former being always triple the latter j* viz. that similar pyramids or cones are as the cubes of their like linear sides, or diameters, or altitudes, &c. And the -same for all similar solids whatever, viz. that they are in pro- portion to each other, as the cubes of their like linear dimen- sions, since they are composed of pyramids every way similar. THEOREM CXVI, If a Sphere T^e cut by a Plane, the Section will be a Grcle^ Let the sphere aerf be cut by the plane abb ; then will the Sisction adb be a circle. Draw the chord ab, or diameter of the seotion ; perpendicular to which, or to the section adjb, draw the axis of the sphere ecg.f, thrpugh the centre c, which will bisect the chord ab in the point G (th. 41). Also, join cA, CBj an^ THEOREWS. 341 9nd draw cd^ cd^ to any point d in tb^ pemmetm? of the section AOB. Thenj because CG is perpendicular to the plane adb, it is perpendicular both to ca and gd (def. 90). So that cga, COD are two right-angled triangles/ having the perpendicular CG common y and the two hypothenuses^cAy cd^ equal, being both radii of the sphere ; therefore the .third sides ga^ gd, are also equal (cor. 2» th. 34). |p like manner it is shown, that any other line, drawn from the centre G to the circum- ference of the secdcm adb, is equal to ga or gb ; conse- quently that section is a circle. CfTol* The section through the centre, is a circle having the same centre and diameter as the sphere, and is called a great circle of the sphere ; the oiher plane sections being Uttle Circles. THEOREM CXVII. Every Sphere is Two-Thirds of its Circumscribing Cylinder. . Let abcd be a cylinder, circum- scribing the sphere efghj then will the sphere efgh be two-thirds of the cylinder abcd. For, let the plane ac be a section of the sphere and cylinder through the centre i. Join ai, bi. Also, let fih be parallel to ad or bc, and eig and KL parallel to ab or DC, the base of the cylinder ; the latter line kl meeting Bi in M, and the circular section of the sphere in n. Then, if the whole plane hfbc be conceived to revolve about the line hp as an axis, the square fg will describe a cylinder ag, and the quadrant ifg will descrihe a hemi- sphere efg, and the triangle ifb will describe a cone iab. Also, in the rotation, the tliree lines or parts kl, kn, km, as radii, will describe corresponding circular sections of those solids, namely, kl a section of the cylinder, kn a section of the sphere, and km a section of the cone. Now, fb being equal to Fi or IG, and kl parallel to fb, then by similar triangles IK is equal to km (th. 82). Arid since, in the right-angled triangle ikn, in* is equal to ik* + KN* (th» 34; J and because kl is equal to the radius ig or 343 GEOMETRY. or IN, and KM rr IK, therefore kl* is jequal to km* + kn*, or the square of the longest, radiusi of the sud circular sections, is equal to the sum of the squares of the two others. And ber cause circles are to each other as the . squares of their diameters, or of their raidii, therefore the circle described by KL is equal to both the circles de- scribed by KM and kn ; or the section of the cylinder, U equal to both the corresponding sections of the sphere and cone. And as this is always the case in every parallel posir tion of KL, it follows, that the cylinder eb, which is com- posed of all the former sections, is equal to the hemisphere SFG and cone iab, .which are composed of all the latter sections. But the cone iab is a third part of the cylinder eb (cor. 2, th. 115) 5 consequently the hemisphere efg is equ4 to the remaining two-thirds ; or the whole sphere bfgh equal to two-thirds of the whole cylinder abcd. q^ e. d. CoroL 1. A cone) hemisphere, and cylinder of the same base and altitude, are to ieach other as the numbers 1, 2, 3. CoroL 2. All spheres are to each other as the cubes of their diameters ; all these being like parts of their circumscribing cylinders. Corol. 3. From the foregoing demonstration it also api^ pears, that the spherical zone or frustrum egnf, is equal to the difference between the cylinder eglo and the cone IMQ, all of the 3ame common height ik. And that the spherical segment pfn, is equal to tl^e difference between the cylinder ablo and the conic frustrum. A(^I9> all of tbo iwne common altitude FK, V * » ♦ ■ * fROBLEMS. ■-■:.x fy^V^'i—^' t s" ] PROBLEMS. PROBLEM !• Ar To Bisect a Line ab ; that Is, to divide it into two Equal Parts. From the two centres a and B, with any «qual radii, describe arcs of circles, in- tersecting each other in c and d; and draw the Une cd, which will bisect the given line ab in the point £. For, draw the radif ac,,. bc, ad, bd. Then, because all these four radii are equal, and the side cd common, the two triangles ACO, BCD, are mutually equilateral : consequently they are also mutually equiangular (th. 5), and have the angle ag^ equal to the angle bce. Hence, the two triangles ace, bce, having the two sides AC, cE, equal to the two sides Be, ce, and their contained angles equal, are identical (th. 1), and therefore have the ^de A£ equal to eb« q^ b. d. w problem II. To Bisect an Angle bac. From the centre a, with any radius, de- scribe an arc, cutting off the equal lines AD, AB ; and from the two ceni^res d, e, with the same radius, describe arcs intersect- ing in F ; then draw af, which will biisect the angle a as required. For, join dp, ef. Then the two tri- angles ADF, AEF, having the two sides AD, DF, equal to the two AE, EF (being equal radii), an4 the side af common, they are mutually equilateral ; conse* quently they are also mutually equiangular (th. 5), and have the angle baf equal to the angle caf. . Sciolium. In the same manner is an ^c of a circle b)^ PROBLEM 3H GEOmTBLY. PROBLEM III> t At a Given P<Mnt c, in a Line ab, to Erect a Perpendicular^ From the given point c, with any radius^ cut off any equal parts cp, C3j of the given line; and, from the ty^ centres d and e, with anyone radius, describe arcs intersecting is F ; then join cf, which will be perpendi- cular as required. For, draw the two equs^ radii df, £F. Then die tw^ triangles cdf, cef, having the two sides cd, of, equal tp die two CB, EF> and cf common, are mutually equilajteral; consequently they are also mutually eqmangular (th. 5), and have the two adjacent angles at c^qual to each other; there^ fore the line cf is perpendicular to ab (def. 1 1). Otherwise. Wh^n the Given Point c is near the End of the lin^. From any point d, assumed above the line, as a centre, through the given point c describe a circle, cutting the given line at E ; and through e and the centre D, draw the diameter edf ; then join cF, which will be the pci^ndicular required. For the angle at c, being an angle in a semicircle, is a right angle, and therefore the line cf is a perpendicular (bydef. 15), PROBLEM ly. From a Given Point a, to let fall a Perpendicular on a ' given Line Be. * From the given point A as a centre, with ^ ^ any convenient radius, describe an arc, cut- ting the giving line at the two points d and E ; and from the two centres n, E, with any radius^ describe two arcs, intersecting at F 3 then draw agf, which will be per- pendicylar to ec as required. For, draw the equal radii An, AE, and DP, £.F. Then the two triangles adf, aef, having; the two sides AD> df, equal to the two ae, ef, and af common^ are mutually %: PHOBIXMS. SU inBtually equilateral; consequently they are also mutuallf equiangular (th. 5), and have the angle dag eqnal the angle BAG. Hence then, the two triangles adg, aeg, haying the two sides AD, ACry equal to the two ae, AGt and their included angles equals are therefore equiangular (th. 1), and have the angles at G equal; consequently a g is perpendicular to BC (def. 11). " Otherwise. When the Given Point is nearly Opposite the end of the Line. From any point d, in the given line B€, as a centre, describe the arc of a circle through the given point A, cutting 3—. /''' J -JgC 90 in B ; ai3 from the centre b, with the *• .. ^\ radius ea, describe another arc, cutting the former in f ; then draw agf, which will be perpendicular to bc as required. For, draw the equafl radii da, df, and ea, ef. Then the two triangles dae, dfe, will be mutually equilateral ; conse- quently they are also mutually equiangular (th. 5), and,have the angles at d equal. Hence, the two triangles dag, dfg, having the two sides da, dg, equal to the two df, dg, and the included angles at d equal, have also the angles at G equal (th. l); consequently those angles at G are right angles, and the line AG is perpendicubr to dg* PROBLEM v« At a Given Point a, in a Line ab, to make an Angle Eqpial to a Given Angle c. From the centres a and c, with any one radius, describe the arcs de, fg. Then, with radius de, and centre f, describe an arc, cutting fg in o. Through G draw the line AG, and it will form the angle re- quired. For, conceive the equal lines or i:adii, DE, FG, to be drawn. Then the two triangles cde, afg, being mutually equilateral) are mutually equiangular (th. S), and have the angle at a equal to the angle c« FROBLBM S46 GEOMETRT. PROBLEM VI. Through a Given Point a, to draw a Line ParaQel ta a Given Line Bc. From the given point a draw a line ad to any point in the given line bc. Then draw the line eaf making the angle at A equal to the angle at d (by prob. 5); so shall £P be parallel to bc as required. For, the angle d being equal to the alternate angle A) the lines BC> bf, are parallel, by th. 1 9. PROBLEM VII. To Divide a Line ab into any proposed Number of Equal Parts. Draw any otlier line ac, forming any angle with the given line ab ; on which set off as many of any equal parts, ad, de, £F, PC, a^ the line ab is to be divided into. • Join BC ; parallel to which draw the other lines FG, £H, Di : then these will divide AB in the manner as required. — ^For those parallel lines di-» vide both the sides ab, ac, proportionally^ by th. 82. PROBLEM VIII. To find a Third Proportional to Two given Lines ab/ac. Place the two given lines ab, ac, forming any angle at a j and in ab take a a also AD equal to ac. Join bc, and A C draw DE parallel to it ; so will AE be r the third proportional sought. ^^^^C\ For, because of the parallels bc, dk, ^'^^ — Tiri the two lines ab, ac, are cut propor- tionally (th. 82) ; so that ab : ac : : ad or Ac : AE ; there- fore A£ is the third proportional to ab, ac. m PROBLEM ly. . ' To find a Fourth Proportional to three Lines ab, Ac, ad. Place two of the given lines ab, ac, making any angle at A; also place ad on ab. Join bc; and paralld to PROBLEMS. 3« to It draw ds : $o shall ae be the fourth proportional as required. For, because of the parallels Bc, de, the two sides ab, ac, are cut propor- . tionally (th. 82) ; so that - - » - - AB : AC : : AD : AE. PROBLEM X. To find a Mean Proportional between Two Lines ab, bc Place ab, ^c, joined in one straight a rB line AC : on which, as a diameter, describe 15 — c the semicircle adc ; to meet which erect the perpendicular £p \ and it will be the mean proportional sought, between AB and bc (by cor. th. 87). ^ A^ oTlfe , .1) PROBLEM XI. To find the Ceptre of a Circle. Draw any chord ab ; and bisect it per- pendicularly with the line ep, which will be a diameter {th. 41, cor.). Therefore cd bisected in o^ will give the centre, as re- quired.. PROBLEM XII. To describe the Circumference of a Circle through Three Given Points a, b, c. From the middle point b draw chords BA, Bc, to the two other points, and bi- sect these chords perpendicularly by lines meeting in o, which will be the centre. Then from the centre o, at the distance of any one of the point3, as OA, describe ^ circle, and it will pass through the two Other points b, c, as required. For, the two right-angled triangles oad, obd, having the sides ad, pB, equal (by constr.), and od common with the included right angles a^ d equal, have their third sido« OA, OB^ also equal (th. I ). And in like manner it is shown, that oc is equal to ob or OA. So that all the three oA, be, pc, being equil^ will be radii of the same circle. ' PROBLEM MS GEOMETRY. 1 PROBLEM XIII. To draw a Tangent to a Circle, through a Given Point a» When the given point a is in the cir- cnmference of the ciicle : Join a and the centre o ; perpendicular to which draw BAc» and it will be the tangent, by th, 46. But when the given pdint a is out of the circle: Draw ao to the centre oj on which as a diameter describe a semi- circlei cutting the given circumference in D ; through which draw badc, which will be the tangent as required. For, join do> Then the angle ado» in a semicircle, is a right angle^ and con- sequently AD is perpendicular to the ra- dius ix>> or is a tangent to the circle (th. 46). PROBLEM XIV. On a Given Line b to describe a Segment of a Circle> to^ Contain a Given Angle a At the ends of the given line make angles dab, dba, each equal to the given angle c. Then draw ae, B£> perpendicular to ad, bd ; and with the centre £, and radius ea or eb, describe a circle ; so shall afb be the segment required, as any angle f made in it will be equal to the given angle c. For, the two lines ad, bd, being perpendicular to the radii ea, eb (by c<Mistr.), are tangents to the circle (th. 46) ; and the angle A or b, which is equal to the given angle c by construction, is equal to the angle f in the alternate segment apb (th. 53). problem XV. To Cut off a Segment 'from a Circle, that sihall Contain a Given Angle G. Draw any tangent ab to the given circle ; and a chord ad to make the angle Cab equal to the given angle c ; then dea will be the segment required, any angle E made in it being equal to the given angle c. For PROBLEMS. S4f For 4he angle A, made by the tangent and chord, which is equal to the given angle c by construction, is also equal to any angle E in the alternate segment (th. 53). ' P|LOBL£M 2;VI« To make an Equilateral Triangle on a Given Line A3. « • From the centres a and b, with the distance iLB| describe arcsy intersecting in c. JDraw AC, bc, and abc will be the equi- lateral triangle. For the equal radii ac, bc, are, each of them, equal to ab. PROBLEM XVII.' To make a Triarigle with Three Given Lines ab, Ac, bc With the centre a, and-distance ao, describe an arc. With the centre B» and distance B€, describe another su'c, cutting the former in c. Draw ac, bc,. and ABC will be the triangle required. For the radii, or sides of the triangle, Ac, BC, are equal to the given lines AC| ^c, by construction. PROBLEM XV m. To make a Square on a Given Line ab« Raise ad, bc, each perpendicular and ^qual to AB ; and join dc ; so shall abcd ibe the square sought. For all the three sides ab, ad, bc, are «qual) by the construction, and dc is equal ^ and parallel to ab (by th. 24); so that all the four sides are equa]^ and the opposite ones are parallel. Again, the angle A or B, of the parallelogram, being a right angle, the angles are all right ones (cor. 1, th. ^2), Hence, then, the ii^re, having all its sides equals and all its angles xightf is a square (def. 34). 4 J \ PROBLEM S50 GEOMETRY. PROBLBM XIX. • To make a Rectangle, or a Parallelogram} of a Given Lengtfi and Breadth, ab, bc. Erect ad, bc» perpendicular to ab, and each equal to bc ; then join Dc, and it is done» The demonstration is the same as the last problem. And in the same manner is described any oblique paral- lelogram, only drawing ad and BC to make the given oh" lique angle with ab, instead of perpendicular to it* PROBLEM XX. To Inscribe a Circle in a Given Triangle ABC. Bisect any two angles a and b, with the two linesvA d, bd. From the inter- section D, which will be the centre of die circle, draw the perpendiculars de, DF, DG» and they will be the radii of the circle required. For, since the angle DAE is equal to the angle dag, and the angles at £, g, right, angles (by constr.), the two triangles ade, adG, are equiangular ; and, having also the side ad common^ they are identical, and have the sides de, dg, equal (th. 2). In like manner it is shown> that dp is equal to de or dg. Therefore, if with the centre D, and distance DE, a drcle be described, it will pass through all the three point* E, F, G, in which points also it will touch the three sides of the triangle (th. 46), because the radii de^ df, dg, are per- pendicular to them. ^ PROBLEM XXI. To Describe a Circle about a Given Tjiangle abc^ Bisect any two sides ieith two of the perpendiculars db, df, dg, and d will ho^ the c^itre. For, join da, db, dc. Theft the two right-angled triangles DA£,DBE,have the two sides DE, ea, equal to the two de, ^. ^ . ^ EB, and the included angles at e equal : those two triangles are therefore identical PROBLEMS. 351 (th. 1), ^nd have the side da equal to DB. In like manner it !s shown, that dc is also equal to da or db. So that all the thre^ da, db, dc, being equal, they are radii of a circle passing through A, B, and c. * . , PROBLEM XXII. To Inscribe an Equilateral Triangle in a Given Circle. • Through the^ centre c draw any dia- meter AB. From the point b as a centre, with the radius bc of the given circle, cLescribe an arc dce. Join ad, ae, de, and ad£ is the equilateral triangle sought. For, join db, do, eb, ec. Then dcb is an equilateral triangle, having each side equal to the radius of the given cir- cle. In like manner, bc£ is an equilateral triangle. But the angle ade is equal to the angle ABB pr cbe, standing on the same arc ae % also the angle aed is equal to the angle cbd, on the same arc ad ; hence the triangle dae has two of its angles, ade, aed, equal to the angles of an equilateral triangle, and therefore the third angle at a is also equal to the same ; so that triangle is equiangular, and therefore equilateral. problem XXIII. To Inscribe a Square in a Given Circle. Draw two diameters Ac, bd, crossing at right angles in the centre e. Then join the four extremities a, B, c, d, Vith right lines, and these will form the in- scribed square abcd* For the four right-angled triangles aeb, bec, ced, dea, are identical, be- cause they have the- sides ea, eb, ec, ed, all equal, being radii of the circle, and the four included angles at e all equal, be- ing right angles, by the construction. Therefore all their third sides ab, bc, cd, da, are equal to one another, and the figure ABCD is equilateral. Also, all its four angles, a, b, c, d, are right ones, being angles in a semicircle. Consequently, the figure is a square. problem nsi GEOMETRY. prob;.£M XXIV. Tp OescrU)e a Square about a GIvoi Circle; Draw ^Dro cliameters ac, BD|Cro$smg at right angles in the centre e. Then through their four extremities draw.FG, IH, parallel to AC^ and ft, gh, parallel to BDj and they will form the square FGHI. Forj the opposite sides of parallelo- grams being equals fg and IH are each equal to the diameter Ac^ and fi and gh each equal to the diameter bd ; so that the figure is equilateral. Again, be- cause the opposite angles of parallelograms are equal, all the four angles f^ g> h, i, are right angles, being equal to the opposite angles at £. So that the figure fghi, having its sides equal, and its angles right ones, is a squarej and its sides touch the circle at the four points a, b, c, d, being perpen- dicular to the radii drawn to those points. PROBLEM XXV. To Inscribe a Circle in a Given Square* Bisect the two sides ^o, fi, in the points a and s (last fig.)» Then through these two points draw ac parallel to FG or iH, and bd parallel to fi or CH. Then the point of intersection e will be the centrcj and the four lines^EA, £B, EC, £D, radii of the inscribed circle. For, because the four parallelograms ef, eg, eh, ei, have their opposite sides and angles equal, therefore all the four lines EA, £B, EC, ED, are equal, being each equal to half a side of the square. So that a circle described from the centfe X., with the distance £A, will pass through all the points A, 9, c, D, and will be inscribed in the square, or will touch its four sides in those points, because the angles there are right ones. PROBLEM XXVI. To Describe a Circle about a Given Square, (see fig. Prob. xxiii). Draw the diagonals /c, bd, and their intersection t will be the centre. For the diagonals of a square bisect each other (th. 40), making E4, eb, eg, ed, all equal, and consequently thes^ are radii of a circle passing through the four points a, b, c, d. PROBLKM * ■i /- .. .» 1^1 1»RDBLEM XXYU; .!■ '} Td Cut a CiVen L»n« ja Ejttreme and Me^an Ratb» XjsT. AB be the eiven line to be divK in e^ictreine and mean ratio, that is, so divided so as tKat the whole line maybe to the greater . p^t,»a$ the greater part is to the less part. I>ra1?:BC perpendicular to. AB, and equal to .half AB* Join AC Vrapd with tentre c . and disftance Qit, describct the circle. sd| then with centre A arid distance ad, 4e* scribe the arc DB ; so shall ab be divided in £ in extreme and mean ratioj or so that AB : A£ :: AE : eb'. . For, produce Ac to thd circumferencef at ^. Then, ABfF being a secant, and ab a tangent, because b U a right angle : ^ therefore the rectangie^AF.AD is^qual to ab* (cor. 1 th. Gl); consequently the means and extremes of these are proportional (th. 17), viz. AB : af or ad -f- rip : : ad : Ab. But a« is equal to ad by construction^ and ab = 2bc r: d^j therefore, ab : Ate' + ab :: a^ : ab^ mid by division^ ab : ▲£ : : ab ; sb; « t PROBLEM XXYllU To Inscribe an Isosceles Triangle in a Givefn Circle, that shall have each of the Angles at the Base Double the' Angle at the Vertex. ^ iDKAW any diameter ab of th^ givea - circle ; and divide thji radius CB, in thef point d, in extreme and mean ratio, by the last problem. From, the pomt b apply the chords bB, bf, each equal to the greatef part CD. Then join Afe, af, ef j and Aef will be the triangle requif ed* For, the chords Bg,' b1*, befing equals their arcs are equal j therefore the supplemental arcs and chords AE, AF^ are also equal 5 consequently the triangle ajsp is isosceles, arid haa the angle B equ^l to the angle F$ aiscX the angles at G are right angles. "V Draw cf and Dt. Then, »c : «d : : cd : Bd, or Be : Bf : : bp : fiiD'by constr. And ba : bf : : bf : bg ^by th. S7)i But bC = ^^ba j therefore bg = ^bd = gd j therefore the two triangles cbf, cpFi are identical (th. 1),! Vox.. L A a and SI4 GEOMETRY. and each equiangular to ibf and act (th. 87). ^ Tberefori their doubles, BFb, af£| are ^isosceles and eqtiiangfiitar> as well as the triangle bcf ; having the two sides bc» cf, equals and the angle b common w^h the triangle bfiX Buf cet is ^ DF or BF ; th^efore the angle c == the angle dfc (th. 4) ; consequently the angle BDi-, which is equal to the sum of these two equal angles (th. 16), is double of one of them c; or the equal angle b or cfb double the angle C. So that cbf is an isosceles triangle, having each of its two equal angles double of the third angle c. Consequently the triangle abf (which k has been shown is equiangular to the triangle cbf) has sdso each of its mg\e» at the base double the angle a at the vertex^ PROBLEM XXIX. To Inscribe a Regular Pentagon in a Given Crcle. Inscribe the isosceles triangle abc having each of the angles abc, acb, double the angle ]6ac (prob. 28). Then bisect the two arcs adb, aec, in the points D, E ; and draw the chords ad, d^, a£, £€, so ^ shall ADBCE be the inscribed equilateral pentagon required. For, because equal angles stand on equal arcs, and, double angles on double arcs, aho the angles arc, acb, being each double the angle bag, therefore the arcs adb, aec, subtending the two former angles, one each double the arcs Be subtending the lat;ter. And since the two former arcs are bisected in D and £, it follows that all the five arcs ad, db, bc, ce, ea, are equal to each other, and consequently the chords also which subtend them, or the five sides of the pentagon, are all equal. Note* In the construction, the points d and e are most easily found, by applying bd and C£ each equsri to bc^ problem' XXX. To Inscribe a Regular Hexagon in a Circle. Apply the radius Ao of the given circle 3s a chord, ab, bo, cd, &c, quite round the circumference, and it will complete the re- gular hexagon abcdep. For, draw the r.uiii ao, bo, co, i5o, Eo, Fo, completing six equal triangles } of* which any one, as Abo, being equilateral (bf PROBLEMS^ iU (jby constr.) its i£ree angles ^ all equal (cor. 2, th. 3), and any one <»f them, as-^bs, is one-third of the whole, or of tw^ right iandes (th. IT), or bne-sixtii of four right angles. But the whole circumfeirence is the measure of fchir right angles (ipor. 4, th. 6). Therefore the arc ab is one-sixth of the tircumference of the circKj and consequently 4ts chord Ap one side of an isquilateral hexagon inscribed in the circle* And the sanie bf the other thor<k* Cpr'of. The side pf a regular hexagoh is equal to th6 radius bf the circumscribing circle, or to ihi chord of oii^si^h {tet df thie circumference: ipROBtfiM xtxi: - To descnbe a Regular Pentagon or Hexs^n about a Circle. In the given circle inscAbe i^ regular polygoh of the szm6 name br hiuhber of sid^s, ' as abcde, by one of the foregoing problems. Then to all its angular points draw tangents (by' prob. 13), and these will form the cir* cumscribing polygon required. For, all the chords, or sides 6t / . the insc^ribing figure, ab. Be, &c, beiiig equal, and all thl* radii Oa, ob, &c, being equal, all the vertical angles about the point 6 are equal. But the angles OBF, oaf^ oXd, obc,- made by the tangents and radii, are right singles; therefore b£F 4- OAF = tWd Hght iUigles, and oAg + 6qg = two right angles j consequently, also, Ab& + Af5 =£ two right angles, and Aob + agb :£: twd right angles (cor. 2, th. 18): Hence, then, the angles Aofi + Afs being =at aob + acb» of which aob is = aoe } consequently the remaining angles F and G are also equal; In the same manhclr it is shown^ that all the angles f, G| Hy I, k, are equal. Again, the tangents from the same point #e, f a^ ari equal> as alsb thie tstogents AG, gb (con 2) th. 61,) ; and the angles F and G of the isosceli^ triangles afe, ag^, are 6qual^ as weir as their opposite sides ae, ab j coiisequently those two triangles are identical (th. l)i and have thWr oth^r sidei £F, fa, Ag, gb, dll equal, and fg equal to the double of iny one of them: In like manner it is shown, that all the bther sides gh, hi^ ik, kF, are equal tb FcT, or double of th^ tangents gb, bh, &c. Hence, then, the circumscribed figure is bcith equilaterat and equiangular, which was tb be showa* A a 2 C,ro/. ceotfisfrsuH 'Cor$l Tl\ei icirdc touches^theiBftddlnQfdie:^j€t To Iivscribe a Circle In a Regular Polygon* BisjEOT any two sides of the polygon by the perpendiculars go^ fp» and tmir intersection o. will be the centre of the inscribed circle^ and OG or of will b<r the radius. For the perpendicul^s tq the t^ngei^ts AF, AG, pass through the centre (cor. th« 47); and the insciibed circle touches . the middle points f, G| by the last corollary, .AlsOf the twm sides AG, AO, of the right-angled. triangle aoO« being equa) to the two sides AFf Ao,. of the right-angled triangle aof, the third sides of, og, will also be equal (cor. th. 45). Therefore the circle described witH the centre o and radius oa, will pa^ through f, and will touch the sides in the points g ami F. And the same for all the other sides of the figure^ PROBLEM XXXIII. To Describe a Circle about a Regular Polygon. BisBCT any two pf the angles, c and Dj \vith the lines co, do 5 then their inter- section o will be the centre of the cir- cumscribing circle } and QC, or OD, will be the radius. For, draw ob, oa, oe, &c, to the angular points of the giten polygon. Then the triangle ocb is isosceles, having the angles at c and D equal, being the halves of the equal angles of the polygon BCD, ODE ; therefore their opposite sides Co, do, are equal (th. 4)« But the two triangles ocd, ocb, having the two sides oc, cd, equal to the two oc, cb, and the in- cluded angles ocdj ocb, also ^qual, will be identical (th. l), and have their third sides bo, od, equal. In like manner it is shown, that all, the lines. o a, ob, oc, od, oe, ai'c equal. Consequently a circle described with the centre o and radius oa, will pass through all the other artgular points, B, c, D, &c, and will circumscribe the polygon. pkoblsm ■FROSLEMS. 857 5 IJ » PROBLEM ZXXIV. To make a Square Equdi to the Sum ol two ^w I^ore Given ^Squares^ Let ab and ac be the sides of two given squares. Draw two iadefihite lines AP, AQ9 at right singles 'to each other I in which place the sides ab^ ac, of the given square^ ; jom bc ; then a square described on bc, will be equal to the sum of the two squares described oh AB and AC (th. S4). in the ^ame-lnanaer, a sqiiare may beiisnadie jequai , to tbi^ sutli of the tfaoree ($rfiii>re giVen sqi^ares. -Foiyif 4^9 AQgAHh be taken as the sides of the given square^^i then,^ makio^ ' A£ = Bc^ AD =: AD, and drawing de^ it is evident that the square on de will be equal to the sum of the three squares on AB, AC5 AD. Andjjo on for more squares. PkOBLlE^ XXXV. • To make a Square Equal to the Difference x^yf two Given Squares. Let ab and A'c, tikett in the sa'me straight line> be equ^l to the sides of th^ two given squares.-^Fh5iil th^ cfehtte A, with the distance ab, describe a circle ; and make cd perpendicular to ab, meeting the circumference in d : so shall a square described on cD be equal to ad^— ac% or ab*— ac% as requiri^ ^cpr, th* 34); pRbBLfini xxxf I. To make a Triangle Eqtod to ^ Given Qtiadn^de ABcb. Draw the diagonal ac, ana parallel to it i^, meting ba produced at £, and. join C£ ; then will the triangle C£B be equal to th^ given quadrilateral abcd. For, the two triangles ace, acd, her ing on the same base ac, and between the same parallels ac, D£, are equal (th^ 25) ^ therefore, if ABC be added to each^ it wjU znake bce equal to abc^ (ax. 2). PiU)BL£M S5» GEOMETRt. PROBLEM ZXXVir. To make a Triangle Eqoal l;o a Given Pentagon abcde* DiCAw DA and db, and also ef. eC) parallel to them, meeting ab pro- duced at F and g ; then drai^ df and T>G ; so shall the triangle dfg be equal to the given pentagon abcd'e. • For th^ triangle d^a = deAi and the triangle dgb ss dcb (th. 25)} therefore! by adding DAB'to the equals^ the tinms are equd (ax. 3), that is, dab «f DAF + DBG SB' DAB 4- p^E + DBC> or the triangle dfg as to the {>entagon ABODE* ^ t-^ PHOBUEM XXXVIlt. f t To make a Rectangle Equal to a Given Triangle Ape. ' Bisect the base ab In d; then raisf D£ aiid BF perpendicular to ab, and meeting cf parallel to ^b^ at B and f : so shall DF be the rectangle equal to the given trismgle abc (by cor. 2, th. 26). PROBLEM XXXIX. To make a Square Equal to a Given Rectangle abjqO* FB.0DUCE one side ab, till ^e be equal to the other side bc. On as as a diametei* describe a circki meeting BC produced at f : then will bf be the side bf the square bfgh, equal to the given rectangle bd, as required; as ap^ pears by cor. th. 87, and th. 77. vT ^.«'* .* ATTT *.; » APPLICATION i 350 ] •t * APPLICATION o» ALGEBRA TO - ' GEOMETRY. W H E N it is proposed to resolve a gtometrical problem algebraically, or by algebra, it is proper, in the first place^ to draw a figure that shatltepresent the several parts or con- 'ditions of the problem^ and to suppose that figure to be the true one* Then, hairing considered attentively the nature of the problem, the figure is next to be prepared for a solution, if necessary, by producing or drawing such lines in it as ap« j)ear mcfitx^onducive to tbit end. This done, the usual sym« bols or letters, for known and.unknown quantities, are em* ployed to denote the several parts of the figure, both the knbwn and unknown parts, or as many of them as necessary; as also such unknown line or lines as may be easiest found, whether required ot not. Then proceed to the operation, by observing the relations that the several parts of the £gure have to each other \ from which^ and the proper theorems in the foregcMug elements of geometry, make out as many equations independent of each other, as thel« are unknown quantities employed in them : the resolution of which equa- tions, in the same manner as in arithmetical problems, will, ^determine the unknown quantities, and resolve the problem proposed. » . As no general rule can be given for drawing the lines, and selecting the fittest . quantities to, substitute for, so as always to bring out the most simple conclusions, because different problems requure different mode^ of solution^ the best way to gain experience, is to try the solution of the same problem in different ways, and then apply that which succeeds best, to other cases of the sanaeicind, when they afterwards occur. The following particular directions, however, may be of some use. \4t% In preparing the figure, by drawing lines, let them bo either parallel or perpendicular to other lines in the Bgurey Or so as to form similar triangles. And if an angle be given; it will be proper to let the perpendicular be opposite to that anglei ftnd to fall from one end of a given line^ if possible. 2rf, SCO APPLICATION or AtGEBRA 2di In selecting the qu^mities proper to substitute fopj^ those are to be phosen, whether required or not, which li^ nearest the known or given parts of the figure, and by means of which this 9j^jt gdj^ceqt p^t$Aiy3^Iexp/jpss!Bd by addi- tion and subtraction only, without using surds. 3^9 When two lines or quantities are alike related to other parts of the figure or problem, the best way i§, not to make Dse of either* pf them separately, but to substitute for their sum, or diflfer^nce, or rectangle, or the sum of their alternate quotients, or for some line or lines, in the figure, to ^hidi tKey have botji thp ^aipe t^hx\on*^. • \ t , . . 4/A, When thei ayea^ or the p^metpr, of a. figure, b p^^n^ pat such parts of it as have <^y .^ temotfi .rdation /to the parts required: it is sometime! .of pse to assume asuathies figure similar to the proponed om^f having. one side iiqualito linity, or some other known qnastity. Fpr, hepce th» qthtc parts of the figure may be &und, by tbe/kifow2\ proportions pj'the like sides, Or parts, ind h> an equ;iti6a hei«btainfid|? For {examples, takf tJb^ following .pcobleins, : j - '« Jn a RighUangkd Triangle^ having f^ven thf JB^sf (&)^ and iik .^ Sum 7f the Hyfoihenusc and Berpfndict^Iar (9) ; to, find ba^ these two Sides. ' . > , , Li^T. ABC represent the proposed triangle, right-angled at b. Put the base ab = 3 =:^ ^ and the sum AC + pc of tlie hypothehus.^ ' ^ ihd perpendicular = 9 == j; also, let 'jr iae- pote the hypothenuse AC, and jj; the perpen- dicular BC. Then by the question r-'^-f+j'—'Cj and by theorem 34-, - - " - - or* =i / -1- ^J By transpos. y in the 1st jcqu. gives ^ = / — j;, "JThis valup of X substi. in the 2d, -gives - - - - - ^* - '2/y -J* / =: f + (% Takingaway/onbothsideslcayes r — 2sy = ^% Bv tr^spo?, ^sy and i% gives - /* — i^ == 2/v, ■ ' ^ J* - ^* And dividing hy2s, gives -r •. ■ •.'. ■ =^ jp ?s 4?=?:]9C» Hfnce x=^j — y==5;= Ap. ' N. B. In thi§ solQt.}pns and the^fQlIowipg onfs^ ^^ iiota^; ^opi§ miidc? b$ jising ^ m^l tmb^9V© l^Umi ^ ?ftd jr^ a% ther^ there are tHikpearo ;»cleff^f the triangle^ a^par^e letter far each; ippceft^igiKfi |,o ais^ng .only -pije ^\nknoMrn letter iicir ox\f «de, Jttid i^pfcee^ng ^e othfff mknown side in ticvms of that letter ai\d the^iyen sum or difference of tli^e sides; though this'l^at4:er, way Y'^ould render the sqlation shorter and SQDlier J because the fbrn^er way, gives occasion for more and better practice in reducing equations, which is the very end aQd reason for which these problems are given at all« t * JP.ROBLEM II- /« a Right-angled Triangle J having given the Hypothenuse (5) ; and the Sum of the Base and P&rpendicular (7) ; tofiftd both these two Sides. ^ Let A3C represebtt the prp.posed triartgle, right-angled at p. Put the given hypQthen.usie Ac = 6 zr ^?, and the sum AB + BC of the base and perpendicular = 7 = / ; also let -r denote the base ab, and j> the perpendicular fiC. . Then by the question <- •- • - ;r rf- y-^ s < ^A {yy tl^^rem 34 . - .-^ t *• ^* +/ == «* ,JJy Irajnspos. yin.tb^ 1st, gives • :c v?: s -r- y Byaabstitu. th^v^lu. for -r, gives, x* -r 2jy + 2j?* = tf* By transposing /% gives - ^ 2/ — ^J)? = a* — j* By dividipg.t)y"2, gives - - - f — sy zn ^a^ — ^ JBy .cpmpl^^iiig tjbte s<jLiare; gives / — j;y + ^^^ = 4^* — i^, By e|Etracjfeij3g the root, gives - y ^ is ^s/^d^ — ^s*- * J^^^ transposing -JsT, gives - - y = -Jj" ± V'i^^ — :J/ = 4 and 3, the values of x and j». PROjBLEM III. In a Rectangle i having giveft the Diagonal {10\ and the Penmen terj or Sum of all the pour Sides (28) 5 toJin4 (^'^h of the Sides severally. Let abcd be the proposed rectangle; and put the diagonal AC = 10 = dy and half the perimeter AB + BC or Ap + PC = i 4 = a ; also put one side ab = .r, and the other side BC =? y. Hence, by -^ right-angled triangles, - - - - .r* + / = //* And by the question -- - - ^+^ = a Thk^n by transposing)} in the 2d, givers x :ss a — y Thisvaluesubstitutedinthe lst,£iye* a^ -r 2^ + 2/ =» ^ . Transposing 362 APPUCATION OP ALGEBRA Transposing tf*, gives - - - 2)>* — ^ay nt' -^ if And dividing by 2, gives - - j^ — jrjr =: 4rf* — 4** . By completing the square, it is / — «^ + ^ =s -^ * * ^a* And extracting the root, gives y — ■J^srv'i^— J^* And transposing ia, «vcs - y = i^ ± -/i^ "• i** = •^ or 65 the values oix and y. PROBLEM IV. Having given the Base and Perpen£eular of any Triangles ta find the Side of a Square Inscribed in the same* Let ABC represent the given ,triangle» and EFGH its^ inscribed square. Put the - base AB = ^/the perpendicular CD = ^ and the side of the square gf or gh =* Di == X i then will ci = CD — di =x « — or. ' Then, because the like lines in the Similar triangles abc, gfc, are propor* tional (by theor. 84, Geom.)> ab : cD : : GB : ci, that hf h I a ', \ X \ a ^ X* Hence» by multiplying extremes and means, ai^ix sz ax, and transposing ix, gives ai ^ax + hx^ then dividing by a + i, gives x = — -j--j = or or GH the side of the inscribed square : which therefore is of the same magnitude, whatever the species or the angles of the triangles may be. PROBLEM V. In an Equilateral Triangle, hatting given the lengths of the three Perpendiculars, drawn from a certain Point within, on the three Sides ; to determine the Sides. Let ABC represent the equilateral tri- angle, and D£, DF, DG, the given per- pendiculars from the point d. Draw the lines DA, DB, DC, to the three angular points; and let fall the perpendicular cu on the base ab. Put thje ttu*ee given per- pendiculars DE = tf, DF = h, DG = c, and put jr =z AH or bh, half !th« side of the equilateral triangle. Then is ac or bc = 2 x, and by right -angled t riangles the perpendicular CH = v^AC*— ah* = V4.r* — X* SB v'S^r* = x^ a. Now, % TO GEOMETRY. M Now, since the area or space of a rectangle, is expressed by the product of the base and height (cor. 2, th. ^1 GeQlD.)!^ and that a triangle is equal to halfa rectangle of equal base and height (con 1, th. 26), Jt follows that, the whote triangle abc is =tAb x ch =ir x x-/S=n4r*^St the tfiangle^BD 5= ^ab x t)G = j: x r = «r, the triangle bcd =^ ^bc x de = o: x « = «x, the triangle acd = ^ac x dp = a:* x * = ^x. But the three last triangles make up, or are equal tm^ llie whole former, or great triangle; that is, ^v^3 :=: ax -^-hx ^ cx'^ hence, dividing byx^i^ X ^3 :=: a +i +^> and dividing by -/f, " X = -jjr — ^, half the side of the triangle sought. .. Also, 3ince the whole perpendicular ch is = x^% it it therefore =: ^ + J + ^. That is, the whole perpencBcofar CH, is just equal to the sum of all the three smaller perpen- diculars DE + DF + DO taken together, wherever the point D is situated. PROBLEM VI. In a Right-angled Triangle, having given the Base (S)» and the Difference between th^ Hypothenuse and Perpendi- cular (1) 3 to £nd both these two Sides. PROBLEM VII, In a Right-angled Triangle, having given the Hypotbenme (5), and &e Difference between the Base and Perpendknhr (J } $ to determine both these two Sides. PROBLEM VXII. Having given the Area, or Measure of the Space, <if a Rectangle, inscribed in a given Triangle \ to iletermiiie the Sides of the Reetangle. ... . . ^ - - , . ^ PROBLEM IX. In a Triangle, having ^ven the Ratio of the two Sidfls^ together with both the Segments of the Base, made by a t^erpendicular from the Vertical Angle ; tp determine the Sides of the Triangle. PILOBLEM X. In a Triangle, having given the Base, the Sum of the other two SidtS| and the Length of a Lint drawn from the Vertical • I SM APPLICATIOH 69 ALGEBRA Vertical Angle to the Middle tkftheiBate; to find the odes of the Triangle* PR09JUEM XI. In -tt Triangle, having given the two Sides abotit At Vertical Angle, with the Line bisecting that Angle^ and tei^ ininating in iht Base i to find the Base* PROBIJSM XIZ. To determine a Right-angled Triangle; Imving given the Lengths of two Lines drawn from the acute angles^ to the Middle of the opposite Sides. PROBLEM Xllt. To determine a Right-angled Triangle; hiiving giretl the Perimeter^ »ad the Radius of its Inscribed Circle. PROBLEM XIV. T^o determine a Triangle ; having given the Base> the Perpendicular, and the Ratio of the two Sides. PROBLEM XV. To determine a Right-angled Triangle ; having given the H jpothenuse> and the Side of the biscribed Square. * • PROBLEM XVI. To determine the Radii of three Equal Circles, described in a giv^n Circle, to touch each other and also the Circum- ference of the given Circle. ^ PROBt-EM XVlI. In a Right-angled Triangle, having given the Perimeter, or Sum of all the Sides, and the Perpendicular let fall from the Right Angle on the Hypothenuse ; to deterihine the Tri- angle, that is, its Sides. PROBLEM XVin.. To. determine a Right-^angled Triangle; having given the Hypothenuse, and the Difference of two Lines drawn from tlie two acute* angles to the Centre of the Inscribed Cii-cle. PROBLEli /" ' / TO GEOMETRY. 365 PROBLEM XIX. To determine a Triangle; having given the Base, the Per- pendicular, aikd the Difi^nce of tHe tgvvt) other Sides. PROBLEM XX. • To determine a Triangle; having given the Base, the Per- pendicular, and the Rectangle or Product of the two Sides. N PROBLEM :^xi. To determine a Triangle ; having given the Lengths of diree Lines drawn from the three Angles, to the Midcfie o£ the opposite Sides. PROBLEM XXII. In. a Triangle, having given all the.thsee Sides; to find the Radius or the Inscribed Circle. PROBLEM XXIII. To determine a Right-angled Triangle; having given the Side of the-Inscribed Square, and the Radius of the Inscribed Circle. PROBLEM XXIV. To determine a Triangle, and the Radius of the Inscribed Circle ; having given the Lengths of three Lines drawn from the three Angles, to tlie Centre of that Circle. PROBLEM XXV. To detefirSnc a Right-angled Triangle ; having given the Hypothenuse, and the Radius of the Inscribed Circle. PROBLEM XXVI- To determine a Triangle; having given the Base, the line bisecting the Vertical Angle, and the Diameter of the 7 Circumscribing Circle. LOGARITHMS LOGARITHMS OF THE NUMBERS tROtt 1 tQ 1000. aBBSB N. Log. N. Log. N. Log. N. 76 Log. 1 26 1-414973 51 1-707570 1-880814 riT. ♦ »:rt:fi « 2 0-3O103O 27 1-431364 52 1-716003 77 1-886491 S 0*477121 28 1-447158 53 1-72*276 78 1-8^2095 4 0*602060 29 1-46239S 54 1-732394 79 1-897627 5 0-698970 'so 1 1-477121 56 1-740363 80 1-903090 € 0-778151 Isi 1-491362 56 1-748188 81 1-908485 7 0-845098 32 1-505150 57 1-755875 82 1-913814 8- 0-903090 33 1-518514 58 1-763428. 83 1-919078 9 0^54243 34 1-531479 59 1-770852 84 1-924279 10 1-000000 35 1-5440P8 60 1-778151 85 1-929419 11 1041393 36 1-556303 «1 1-785330 86 1-934498 12 1'079181 37 1-568202 62 1-792392 87 1-939519 13 Ml 3943 i38 1-579784 63^ 1-799341 68 1-944483 14 1-146128 139 1-591065 64 1 -806 1 80 89 1-949390 15 1-176091 40 1-602060 65 1-812913 90 1-954243 16 1-204120 41 1-612784 66 1-819544 9iL 1-959041 17 1 ^30449 42 1-623249 67 1-826075 92 1-963788 IS 1-255273 43 1-633468 68 1-832509 93 1-968483 19 1 -278754 44 1-643453 69 1*838849 94 1-973128 20 1-301030 45 1-653213 70 1-845098 95 1-977724 21 1-822219 46 1-662758 71 1-851258 96 1-982271 22 1-342423 47 1 -672098 72 1-857833 97 1-986772^ 25 1-361728 48 1-681241 73 1-863323 98 1-991226 24 1-380211 49 1-690196 74 1-869232 99 1-995635 25 1*397940 50 1 -698970 75 1-875061 '100 2-000000 N« B. In the following table, in the last nine colunlns of eech page, where the first or leading figures change from 9's to O's, large dots are now in- troduced instead 9f the 0*s through the rest of the line, to catch the eye^ and to indicate that from thence the corresponding natural number ia the^rst column stand's in the next 'lower line, and its annexed first tiiv'C* figures of the Logarithm ia the second ^column; LOftARrrHMS. >67 N.' 1 2 3 4 5 6 7 8 9 100 DOOOOO 0434 0868 1301 1734 2166 2598 3029 3461 31891 101 4^4321 4751 5181 5609 6038 6466 6894 7321 7748 8174 102 8600 9026 9451 9876 •300 •724 1147 1570 1993 2415 103- 012831^ 3259 3680 4100 4521 4940 5360 5779 6197 6616 104 7033 7451 7868 8284 8700 9116 9532 9947 •361 •775 105 021189 1603 2016 2428 2841 3252 3664 4075 4486 4896 106 5306 5715 6125 6533 6942 7350 7757 8164 8571 8d78 107 9384 9789 • 195 •600 1004 1408 1812 2216 2619 3021 lOS 033424 3826 4227 4628 5029 5430 5830 6230 6;£29 7028 109 7426 7825 8223 8620 9017 9414 9811 •207 •602 •998 no 041393 1787 2182 2576 2969 3362 3755 4148 4540 4932 111 5323 5714 6105 6495 6885 7275 7664 8053 8442 8830 112 9218 9606 9993 •380 •766 1153 1538 1924 2309 2694 113 053078 3463 3846 4230 4613 4996 5378 5760 6142 6524 114 6905 7286 7666 8046 8426 8?<05 9185 9563 9942 •320 115 060698 1075 1452 1829 2206 2582 2958 3333 3709 4083 116 4458 4832 5206 5580 5953 6326 6699 7071 7443 7815 117 8186 8557 8928 9298 9668 ••38 •407 •776 1145 1514 118 071882 2250 2617 2985 3352 3718 4085 4451 4816 5182 119 .5547 5912 6276 6640 7004 7368 7731 8094 8457 8819 120 9181 9543 9904 •266 •626 •987* 1347 1707 2p67 2426 121 082785 3144 3503 3861 4219 4576 4934 5291 5647 6004 122 6360 6716 7071 7426 7781 8136 8490 8845 9198 9552 123 9905 •258 •611 •963 1315 1667 2018 2370 2721 3071 124 093422 3772 4122 4471 4820 5169 5518 5866 6215 6562 ,125 6910 7257 7604 ,7951 8298 8644 8990 9335 9681 •026 126 100371 0715 1059 1403 1747 2091 2434 2777 3119 3462 127 3804 4146 4487 4828 5169 5510 5851 6191 6531 6871 128 7210 7549 7888 8227 8565 8903 9241 9579 9916 •253 129 110590 0926 1263 1599 1934 2270 2605- 2940 3275 3609 130 3943 4277 4611 4944 5^78 5611 5943 6276 6608 6940 131 7271 7603 7934 8265 8595 8926 9256 9586 9915 •245 132 120574 0903 1231 1560 1888 2216 2544 2871 3198 3525 133 3852 4178 4504 4830 5156 5481 5806 6131 6456 6781 134 7M)5 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 270^ 139 143015 3327 3639 3951 4263 4574 4885 5196 5507 5818 140' ^ -6128 6438 6748 7058 7367 7676 7985 8294 8603 8911 141 9219 9527 9835 • 142 •449 •7^6 1063 1370 1676 1982 142 152288 2594 2900 3205 3510 3815 4120 4424 4728 5032 143 5336 5640 5943 6246 6549 6852 7154 7457 7759 8061 144 8362 8664 8^65 9266 9567 9868 • 168 %^^^ •769 1068 145 1^1368 1667 1-967 2266 2564 2863 3161 3460 3758 4055 14^ ,*4353 4650 4947 5244 5541 5838 6134 6430 6726 7022 147 7317 7613 7908 8203 8497 8792 9086 9380 9674 9968 148 170262 0555 0848 1141 1434 1726 2019 2311 2603 2895 149 3186 3478 3769 4060 4351 4641 4932 5222 5512 580i> > 9 36S LOGARITHBte. N. 1 2 8 6959 4 I^J^l € . 7 ' 8 f ^ 150 176091 6381 6670 7248 7536 7S25 8113 8401 86S9 151 8977 9264 9552 9839 • 126 •413 •699 •986 1272 1558 152 181844 2129 2415 2700 2985 3270 3555 3839 4123 4407 .153 4691 4975 5259 5542 5825 6108 6391 6674 6956 7239 154 7521 7803 80H4 8366 8647 8928' 9209 9490 9771 ••51 155 190332 0612 0892 1171 1451 1730 2010 2289 2567 2846 156 3125 3403 3681 3959 4237 4514 4792 5069 5346 5623 151 5899 6l7d 6453 6729 7005 :728fl 7556 7832 8107 8382 158 8657 8932 9206 9481 9755 ••29 •303 •577 •850 1124 159 201397 1670 1943 2216 2488 2761 3033 3305 3577 3848 160 4120 4391 4663 4934 5204 5475 5746 6016 6286 6556 1 161 6826 7096 7365 7634 7904 817'3 8441 8710 8979 9247 162 9515 9783 ««51 •319 •586 •853 1121 1388 1654 1921 163 212188 2454 2720 2986 3252 3518 37»3 4049 4314 4579 .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 1153 1414 1675 1936 2196 2456 1G7 2716 2976 3236 3496 3755 4015 4274 4533 4792 5051 168 5309 5568 5826 6084 6342 6600 6858 7115 7372 7630 169 7887 8144 8400 8657 8913 9170 9426 9682 9938 • 1*93 no 230449 0704 0960 1215 1470 1724 1979 2234 2488 2743 171 2996 3250 3504 3757 4011 4264 4517 4770 5023 5276 172 5528 5781 6033 6285 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 3286 3534 3782 4030 4277 4525 4772 5019 '5266 176 5513 5759 6006 6252 6499 6745 6991 7237 7482 7728 177 7973 8219 8464 8709 8954 9198 .9443 9687 9932 • 116 178 250420 0664 0908 1151 1395 1638 1881 2125 2368 2610 179 2653 3096 3338 3580 3822 4064 4306 4548 4790 5031 180 5273 5514 5755 5996 6237 647'7 6718 6958 7198 7439 181 7679 7918 8158 8398 8637 8877 9116 9355 9594 9333 182 260071 0310 0548 0787 1025 1263 1501 1739 1976 2214 183 2451 2.688, 2925 3162 3399 3636 3873 4109 4346 4582 184 4818 5054 5290 5525 5761 5996 6232 6467 6702 6931 185 7172 7406 7641 7875 8110 8344 8580 8812 9046 9279 186 9513 9746 9980 •213 •446 •679 •912 1144 1377 1609 187 271842 2074 2306 2538 2770 3001 3233 3464 3696 3927 188 4158 4389 4620 4S50 5031 5311 5542 5772 6002 6232 189 6462 6692 6921 7151 7380 7609 7838 8067 8296 8525 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 3753 3979 4205 4431 4656 4882 5107 5332 193 5557 5782 6007 6232 6456 6681 6905 7130 7354 7578 194 7802 8026 8249 8473 8696 8920 9143 9366 19589 9812 195 290035 0257 0480 0702 0925 1147 1369 1591 1813 2034 196 2256 2478 2699 2920 3141 3363 3584 3804 4025 4246 197 4466 4687 4907 5127 5347 5567 5787 6007 6226 6446 198 66()5 6884 7104 7323 7542 7761 7979 8198 841.6 8635 199 8863 9071 9289 9507 9725 9943 • 161 • 878 •595^*813 0^ NUMBERS- S09 N. 1 2 3 4 5 6 7 8 9 200 301030 1247 1464 TesT 1898 2114 2831 2547 2764 2980 201 S196 3412 3628 3844 4059 4275 4491 4706 4921 5136 202 5351 5566 5781 5996 6211 6425 6639 6854 7068 7282 203 7496 7710 7924 8137 8351 8564 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 5551 5760 207 5970 6180 6390 6599 6809 7018 7227 7436 7646 7854 208 8063 8272 8481 8689 8898 9106 1 9314 9522 9730 9938 209 320146 0354 0562 0769 0977 1184 1391 1598 1805 2012 210 2219 2426 2633 2839 3046 3252 3458 3665 3871 4077 211 4282 4488 4694 4899 5105 5310 5516 5721 5926 6131 212 6336 6541 6745 6950 7155 7359 ^563* 7767 7972 8176 213 8380 8583 8787 8991 9194 9398 9601 9805' • • • o .211 214 330414 0617 0819 1022 1225 1427 1630 1832 2034 2236 215 2438 2640 2842 3044 3246 3447 3649 3850 4b51 4253 216 4454 4655 4856 5057 5257 5458 5658 5859 6059 6g60 217 6460 6660 6860 7060 7260 7459 7659 7858 8058 8257 218 8456 3656 8855 9054 9253 9451 9650 9849 ..47 .246 219 3404.44 0642 0841 1039 1237 1435 1632 1830 2028 2225 220 2423 2620 ^817 3014 3212 3409 3606 3802 3999 4196 221 4392 4589 4785 4981 5178 5374 5570 5766 5964^ 6157 922 6353 6549 6744 6939 7135 7330 7525 7720 7915 8110 223 8305 8500 86<i4 8889 9088 9278 9472 9666 9860 ..54 224 350248 0442 0636 0829 1023 1216 1410 1603 179t5 1989 225 2183 .2375 2568 276J 2954 3147 3339 3532 3724 3916 226 4108 4301 4493 4685 4876 5068 ,5260 545^ 5643 5834 227 6026 6217 6408 6599 6790 6981 7172 7363 7554 7744 228 7935 8125 8316 8506 8696 8886 9076 9266 9456 9646 229 99S5 ..25 .215 • 404 .593 .783 .972 1161 1350 1539 230 361728 1917 2105 2294 2482 2671 2859 3048 3236 3424 2*1 3612 3800 3988 4176 4363 4551 4789 4926 5113 5301 232 5488 5675 5862 6049 6236 6423 6610 6796 6988 7169 23S 7356 754i2 7729 7915 8101 8287 8473 8659 8845 9030 234 9216 9401 9587 9772 9958 . 143 .328 .513 .698 .883 235 371068 1253 1437 1622 1806 1991 2175 2360 2544 2728 23$ 2912 3096- 3280 3464 S647 3831 4015 4198 4382 4565 237 4748 4932 $115 5298 5481 5664 •5846 ^m^ 6212 6394 23% } 6577 6759 6942 7124 7306 7488 7670 7852 8034 8216 ?39^ 8398 9580 876118943 9124 9306 9487 9668 9849 ..30 240 3802 U 0392 0573 0754 0934 1115 1296 1476 1656 1837 241 2017 2197 2377 2557 2737 2917 3097 3277 3456 3636 248 8815 3995 ♦174 4353 4533 4712 4891 5070 5249 5423 243 5606 5785 b^^^ 6142 6321 6499 6677 6856 7034 721-2 244 •390 7568. 7746 7923 8101 S279 8456 8634 8811 8989 245 .9166 9343 9520 9698 9875 ..b\ .228 .405 .5&2 .759 246 S90935 U12 1288 1464 1641 1817 1993 2169 234^ 2521 847 2697 2873 3048 3224 3400 3575 3751 39.2(5 41Q1 4277 248 ^U52 4627 4802 4977 5152 5326 5501 5676 5850 6025 249 6199 6374 6548 6722 6896 7071 7245 7419 7592 7766 . VJ QUU.Ii \ JBl } I »' :rtor L0GAR1THM& K 1 2 3 4 ^ 5 6 7 8 9 t50 397940 8114 8287 8461 8634 8808 8961 9154 9328 9501 ^51 9674 9847 ..20 .192 .365 .538 .711 .883 1056 1228 252 401401 1573 1745 1917 2089 2261 2433 2605 2777 2949 bf 3121 3292 3464 3635 3807 3978 4149 4320 4492 4663 254 4834 5005 5176 5346 5517 5688 , 1 5858 6029 6199 6370 255 V 6540 6710 6881 7051 7221 7391 7561 7731 7901 8070 £56 8240 8410 8579 8749 8918 9087 9257 9426 9595 1 9764 \ 257 9933 . 102 .271 .440 .609 .777 .946 1114 1283 1451 258 4ll6i20 1788 1956 2124 2293 2461 2629 2796 2964 3132 259 ' 3300 3467 3635 8803 3970 4137 4805 4472 4639 4806 260 4973 5140 5307 5474 5641 5808 5974 6141 6308 6474 261 6641 6807 6973 7139 7306 7472 7638 7804 7970 8135 262 8301 8467 8633 8798 8964 9129 9295 9460 9625 9791 263 .^ 9956 ; 121 .286 .451 .616 .781 .945 1110 1275 1439 264 421604 1768 1933 2097 2261 2426 2590 2754 2918 3082 265 3246 3410 3574 3737 3901 4065 4228 4392 4555 4718 266 4882 5045 5208 5871 5534 5697 5860 6023 6186 6349 267 6511 6674 6836 6999 7161 7324 7486 7648 7811 7973 268 8135 8297 845? 8621 8783 8944 9106 926g 9429 9591 269 9752 9914 ..7^5 .236 .398 .559 .720 .881 1042 1203 270 431364 1525 1685 1846 1?007 2167 2328 2488 2649 2S09 27} . 2969 3130 5290 3450 3610 3770 3930 4090 4249 4409 272 4569 4729 4888 5048 5207 5367 ^526 5685 5844 6004 27*3 6163 6322 6481 6640 6800 61957 7116 7275 7433 7592 274 7751 7909 8067 8226 8384 8542 8701 8859 ?017 9175 275 9333 9491 9648 9806 9964 • .122 .279 .437 .594 .752 276 440909 1066 1224 1381 1538 1695 1852 2009 2166 2323 277 2480 2637 2793 2950 3106 3263 3419 3576 3732 3889 278 4045 4201 4357 4513 4669 4825 4981 5187 5293 5449 279 '5604 5760 5915 6071 6226 6382 6537 6692 6848 7003 280 7158 7313 7468 7623 7778 7933 80S8 8242. 8397 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 4235 •4387 4540 4692 285 4845 4997 5150 5302 5454 5606 5758 5910 6062 6214 286 6366 6518 6670 6821 6973 7125 7276 7428 7579 7781 287 ■7882 8033 8184 8356 8487 8638 8789 8940 9091 9242 288 9392 9543 9694 984.5 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 3*893 4042 4191 4340 4490 4639 4788 4936 5085 5234 292 5383 5532 5680 5829 5977 6126 6274 6423 6571 6719 293 6868 7016 7164 7312 .7460 7608 7756 7904 8052 8200 294 8347 8495 8643 8790 8988 9085 9233 9380 9527 9675 ' 295 9822 9969 . 116 .263 .410 .557 .704 .851 .998 ^1145 296 471292 1438 1585 1732^ 187S 2025 2171 2318 2464 2610 297 2756 2903 3049 3195 3341 3487 3633 8779 3925 4071 298 4216 4362 4508 4653 4799 4944 5090 5235 5381 5526 299 sell 5816 5962 6107 6252 6397 6542 6687 6832 6976 OF NUMBERS. 371 x^ • ■■■■"' 0' 1 2 3 4 5 6 7 8 8278 9 : 300 477121 7266 7411 7555 7700 7844 7989 8133 8422 ; ;^01 %8566 8711 8855 8999 9143 9287 9431 9575 9719 9863 302 480007 0151 0294 04^8 0582 0^25 0869 1012 M56 1299 303 . 1443 1586 1729 1872 2016 2159 2302 2445 2586 2731 304 2874 3016 3159 3302 344i> 3587 3730 3872 4015 4157 305 4300 4442 4585 4727 4865 5011 5153 5295 5437 5579 306 5721 5863 6005 6147 62S9 6430 6572 6714 6855 6997 ' 307 7138 7280 7421 7563 7704 7845 7986 8127 8269 8410 308 8551 8692 8833 8974 91U 9255 9396 9537 9677 9818 30i^ 9958 ..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 3597 3737 387^ 4015 312 4155 4294 4433 4^572 4711 4850 4989 5128 5267 5406 313 5544 5683 5822 5960 6099 6238 6376 6515 6653 6791 314 6950 7068 7206 7344 7483 7621 7759 7897 8035 8173 315 8311 8448 8586 8724 8862 8999 9137 9275 9412 9550 316 9687 9824 9962 ..99 .236 .374 .511 • 648 .785 . 922 317 501059 1196 1333 1470 1607 1744 1880 2017 2154 2291 318 2427 2564 2700 2837 2973 3109 3246 3382 3518 3655 319 3791 3927 4063 4199 4335 4471 4607 4743 4878 5014 320 5150 5286 5421 5557 5693 5828 5964 6099 6234 6370 321 6505 6640 6776 6911 7046 7181 7316 7451 7586 7721 «22 7856 7991 8126 8260 8395 8530 8664 8799 8934 9068 323 9203 9337 9471 9606 9740 9874 . • . ^ . 143 .277 .411 324 510545 0679 08J3 0947 1081 1215 1349 1482 1616 1750 •325 1883 2017 2151 2284 2418 2551 2684 2818 2951 3084 •326 3218 3351 3484 3617 3750 3883 4016 4149 4282 4415 327 4548 4681 4813 4946 5079 5211 5344 5476 5609 5741 328 5874 6006 6139 6271 6403 6535 6668 6800 ^932 7064 329 .7196 7328 7460 7592 7724 7855 7987 8119 3251 8382 a30 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 1530 1661 1792 1922 2053 2183 2314 :I33 2444 25175 2705 2835 2966 3096 3226 3356 3486 3616 334 3746 3876 4006 4136 4266 4396 4526. 4^56 4785 4915 335 5045 5174 5304 5434 5563 5693 5822 5951 6081 6210 S36 6339 6469 6598 6727 6856 6985 7114 7'243 7372 7501 337 7630 7759 7888 8016 8145 8274 8402 853 L 8660 8788 338 8917 9045 9174 9302 9430 9559 9687 a815 9943 ..72 339 530200 0328 0456 0584 0712 0840 0968 1096 1223 1351 340 1479 1607 1734 1862 1990 2117 2245 2372 2500 2627 341 2754 2882 3009 SI 36 3^64 3391 3518 3645 3772 3899 342 4026 4153 4280 4407 4534 4661 4787 4914 5041 5167 343 5294 5421 5547 5674 ^800 5927 6053 6180 6306 6432 344 6558 6685 6811 6937 7063 7189 7315 '7441 7567 7693 345 7819 7945 8071 8197 8322 8448 8574 8699 8825 8951 346 9076 9202 9327 9452 9578 9703 9829 9954 ..79 .204 347 540329 0455 0580 0705 0830 0955 1080 1205 1330 1454 348 1579 1704 1829 1953 2078 2203 2327 2452 2576 2701 349 2825 2950 3074 3199 3323 3447 3571 3696 3820 3944 372 350 544068 351 5307 352 6543 353 7775 354 9003 355 550228 356 1450 357 2668 358 3883 359 5094 360 6303 361 7507 362 8709 363 9907 364 561101 365 2293 366 3481 367 4666 368 5848 369 7026 370 8202 371 9374 372 570543 373 1709 374 .2872 375 4031 376 5188 377 6341 378. 7492 379 8639 380 9784 381 580925 382 2063 383 3199 384 4331 385 5461 386 6587 387 7711 388 8832 389 9950 390 591065 391 2177 ?92 '3286 393 4393 294 5496 395 6597 396 7695 397 8191 398 9883 999 600973 LOGARTTHMS 4192 5431 6666 7898 9126 0351 1572 2790 4004 5^15 6423 7627 8829 ..26 1221 2412 3600 4784 5966 7144 8319 9491 0660 1825 2988 4147 5303 6457 7607 8754 9898 1039 2177 3312 4444 5574 6700 7823 8944 ..61 U76 2288 3397 4503 5606 6707 7805 8900 9992 1082 4316 5555 6789 8021 9249 0473 1694 2911 4126 5336 6544 7748 8948 • 146 1340 2531 3718 4903 6084 7262 8436 9608 0776 1942 3104 4263 5419 6572 7722 8868 .. 12 1153 2291 3426 4557 5686 6812 7935 9056 .173 1287 2399 3508 4614 5717 6817 7914 9009 . 101 1191 4440 4564 5678 6913 8144 9371 0595 0717 1816 3033 4247 5457 6664 7868 9068 .265 1459 2650 3837 5021 6202 7379 8554 9725 0893 2058 3220 4379 5534 6687 7836 8983 • 126 1267 2404 3539 4670 5799 6925 8047 9167 .284 1399 2510 3618 8024 9119 .210 1299 5802 7036 8267 9494 1938 3155 4368 5578 6785 7988 9188 .385 1578 2769 3955 5139 6320 7497 8671 9842 1010 2174 3336 4494 5650 6802 7951 9097 .241 1381 2518 3652 4783 5912 7037 8160 9279 .396 1510 2621 3729 4724 4834 5827 5937 6927 7037 8134 9228 .319 1408 4688 5925 7159 8389 9616 0840 2060 3276 4489 5699 6905 8108 9308 .504 1698 2887 4074 5257 6437 7614 8788 9959 1126 2291 3452 4610 5765 6917 8066 9212 .355 1495 2631 3765 4896 6024 7149 8272 9391 .507 1621 2732 3840 4945 6047 7146 8243 9337 .428 1517 4812 6049 7282 8512 9739 0962 2181 3398 4610 5820 7026 8228 9428 .624 1817 3006 4192 5376 6555 7732 8905 .076 1243 2407 3368 4726 5880 7032 8181 9326 .469 1608 2745 3879 5009 6137 7262 8384 9503 .619 1T32 2843 3950 5055 6157 7256 8353 9446 .537 1625 r 4936 6172 7405 8635 9861 1084 2303 3519 4731 5940 7146 8349 9548 .743 1936 3125 4311 5494 f6673 7849 9023 .193 1S59 2523 3684 4841 5996 7147 8295 9441 .583 1722 2858 3992 5122 6250 7374 8496 9615 .780 1843 2954 4061 5165 6267 7366 8462 9556 .646 1734 8 5060 6296 7529 8758 9984 1206 2425 3640 4852 6061 7267 8469 9667 • 863 2055 3244 4429 5612 6791 7967 9140 .309 1476 2639 9800 4957 6111 7262 8410 9555 .697 1836 2072 4105 5235 6362 7486 8608 9726 .842 1955 3064 4171 5276 6377 7476 8572 9665 .755 1843 5183 6419 7652 88S1 . 106 1328 2547 3762 4973 6182 7387 8589 9787 .982 2174 3362 4548 5730 6909 8084 9257 .426 1592 2755 3915 5072 6226 7377 8525 9669 .811 1950 30S5 4218 5348 6475 7599 8720 9838 .953 2066 3175 4282 5386 6487 7586 8681 9774 .864 1951 J OF NUMBERS. ^ 373 'N. . (y 1 2 3 4 5 6 'j « 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 51:97, 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 5845 413 5950 6055 6160 6265 6370 6476 6581 6686 6790 6895 414 7000 7105 7210 7315 7420 7525 7629 7734 7889 T943 415 8048 8153 8257 8362 8466 8571 8676 8780 8884 8989 416 9093 9198 9302 9406 9511 9615 9719 9824 9928 ..32 417 620136 0240 0344 0448 0552 0656 0760 0864 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