$B E7fi fib? Vf IN MEMORIAM FLORIAN CAJORl V I I ^-'^^ vs. ^J F C/^t^a<.^^ CLgi^l.^-^ ELEMENTS OF ARITHMETIC, THEORETICAL AND PRACTICAL; ADAPTED TO THE USE OF SCHOOLS, AND TO PEIVATE STUDY. BY F. n. MA88LER. F. A. P. S. xrEW-YORs: l*ftlNTED AND PUBLISHED BY JAMES BLOOMFIELB. 1826. SOUTHERLY DISTRICT OF KEW-YORK, SS. BE IT REMEMBERED, That on the 6th day of Oc- L.S. tober, A. D. 1826, in the 51st year of the ludependence of the United Slates of America, F. R. HASSLER, of the said District, hath deposited in this oflBce the title of a Book, the right whereof he claims as Author, in the words following;, to wit : Elements of AriihmttiCf Theoretical and Practical ; adapted to tht use of Schools, and to Private Studi/. By F. R. HASSLER, F. A. P. S, la conformity to the Act of Congress of the United States, en- titled " An Act for the encouragement of Learning, by securing the copies of Maps, Charts, and Books, to the authors and pro- prietors of such copies, during the time therein mentioned." And also to an Act, entitled " An Act, supplementary to an Act, enti- tled an Act for the encouragement of Learning, by securing the copies of Maps, Charts, and Books, to the authors and proprietors of such copies, during the times therein mentioned, and extending the benefits thereof to thfi arts of designing, engraving, and etch- ing historical and o(,her prints.*' JAMES DILL, Clerk of the Southern District of M^v-York. CAJCRI INTRODUCTION Arithmetic contains the first elements of rea- soning upon quantity ; its principles take their rise in ideas so simple as to he adapted to the most un- tutored mind, and to the lowest capacity. It is at the same time so indispensahle for every human heing, not only in common life, but in the pursuits of the highest sciences, that it forms the most pro- per, and has always formed one of the principal branches of the earlier education of youth. By its very nature it furnishes the means of de- veloping the reasoning faculties, from the time of their first beginning to expand themselves, and of habituating them to correctness and precision. It therefore gives the human mind the power and dis- position to reason upon sound and correct principles. It is therefore the duty of the faithful teacher of youth, (not the mere teacher for his own private emolument,) to take advantage of this property of arithmetic, and apply it to cultivate the mind, and enligiiten the understanding of his scholars, by a proper reasoning in this elementary science ; he should not make it the object of the memory alone ^ a method that leaves no impression upon the mind^ whose results are therefore lost again as soon as the school is dismissed. To neglect to take this advantage of the study of arithmetic, is either a proof of ignorance, or an ac- tual dereliction of duty. This may appear strong to many people, but strength is the essential pro- perty of truth* I can safely appeal to those who 9181' *-i IV INTRODUCTIOBT. have in early youth been taught by the negligent method of mere rules, and have at a later period attained scientific eminence, to decide between this and any contrary assertion. The difficulties that the young experienc6 on en- tering upon any scientific studies, in colleges, or otherwise, are well known ; the path to be followed there, must be that of reasoning, and no prepara- tions are made for this by their previous education, for the cultivation of the memory alone, is, from the very constitution of the human mind, always detri- mental to the reasoning faculty. However the opportunity, as has been stated, exists, of cultivating the reasoning faculty at an earlier period, by familiarizing the scholar with the simple reasonings of elementary arithmetic. The step from that to higher or general arithmetic, usually called Mgebra, becomes by this mode, both short and simple, as in its nature it really is ; and the scholar who does not wish to go farther than common arithmetic, can alone obtain the knowledge of the propriety or principles of its application to any occurrence in common life, by a knowledge of it, founded upon correct reasoning. It is entirely wrong to say and act. upon the ground, *' Iwant to know how to do this or that," the principle must be, *< I wish to understand this or that," if ever any lasting good result shall be obtained. My object in undertaking this work was not to swell the number of elementary treatises on arith- metic, but may be stated as follows. 1st. I wish to smooth the path of the teacher and the scholar, by explaining and proving, the pro- priety and correctness of any step that is taken, by previous reasonings, leading to the discovery of the principle that ought to direct it, and therefore point- ing out the rule for the appropriate operation ; and I have, therefore, not been content to give the final INTRODUCTIOlf. V result alone, and the example for its proof, which is an individual, and consequently a defective me- thod, while reasoning always leads to general pro- positions and proofs. In this way we attain, step by step, to the real scientific structure of this ele- mentary science, and thus all the operations become satisfactory to the mind, and therefore agreeable to the growing intellect of the scholar. In carrying such a system through the whole extent, to that point where more general and exten- sive considerations, of a higher analytic nature, arc to guide us, 1 have even thought it possible to make a treatise which a man of science might look at with some satisfaction, and by which the young scholar would arrive at the entrance of his higher scientific studies, properly prepared by a correct habit of reasoning. 2d. The young and untutored mind, in truth, reasons analytically ; a boy, and in fact a man, asks always WHY; and as he enters more and more deep- ly into the investigation, continues to ask the rea- son of every thing that is said to him in the way of explanation. The reason of this lies in the nature of his situation ; he cannot proceed synthetically, because synJhesis needs some previous data, averr- ed, given, or adopted, on which to build the reason- ing to arrive at a conclusion. This does not y&t exist at this early stage of instruction. In following this mode, and grounding every conclusion upon inquiry, of whicli the ground lies, cither in the human mind itself, even untutored, or in the result of preceding investigations, I intend to make a book which a lad remote from cities, although he might not have had the benefit of a good early education, can take in hand usefully, and which a simple knowledge of reading, coupled with his own desire for improvement and instruc- tion, would induce him to take up, and undertake 1 # VI INTRODUCTION. to study, as both useful and agreeable ; useful, be- cause it would show him the means of accounting to himself for the result of his own labours ; and agreeable, because it would afford him a pleasing object of speculation for his winter evenings. I should be delighted to see several such lads, pass- ing an evening together, with this book between them, each his slate and pencil before him, discuss- ing, mutually giving and solving, the questions which they learn from it to form out of the occur- rences around them. I can promise them more sa- tisfaction from it, than in their passing that time in the bar-room of a public house, or a grocery ; and more beneficial, economical results, from the ex- penditure in book, slate, and pencil, to assist their studies, (for they must write every thing,) than were they to lay out the cost in the vile liquor, that emptiness of mind leads them to call for ; they will soon be able to calculate : that they even make a saving, if they write their full studies, ideas, and questions, on paper, with pen and ink, in comparison with the expences of the deleterious pleasures of a bar-room. If I should succeed only in this part of my aim, I would consider my labour as sufficiently rewarded; and I would have the greatest enjoy- ment, to meet witli such a company, afford them assistance, and partake of their rational amusement. For the use of this book, I should like to advise, the teacher, as well as the student, first to peruse attentively the theoretical principles of any rule or subject, and then exercise his scholars, or himself, in the application, which will give him an opportu- nity to generalise, and clear up their, or his, ideas properly ; and after having gone through any of the principal subdivisions, to take a general view of the whole ; taking care to comprehend the leading prin- ciples, and the mode of considering the subject, that has been treated of; in this way he will be enabled UTTRODrCTIOK. Vll to make a proper use of it in the parts to be treated next. It is an unavoidable condition in every systematic work, that the subsequent parts shall be grounded upon the preceding ones, and therefore these must be supposed known in the progress of the work, as it proceeds. Therefore also the study of no systematic and good work, can be begun in any other part than at the beginning, by any scholar ; that is, a per- son not fully acquainted with the wliole subject of the book, but seeking instruction from it. If any person thinks he knows already some of the ele- mentary parts, and wishes to study only tbe sub- sequent part, it is necessary for him to read over, attentively, the parts with which he is acquainted ; to make himself acquainted with the manner in which the author expresses himself upon those subjects, which he has his own ideas upon. By comparing these together, he will be able to understand pro- perly, afterwards, those parts with which he is not acquainted^ and therefore read and study with success ; which otherwise will certainly not be the case. This is nothing else but what is necessary between all men, in any intercourse, that is, the necessity of being acquainted with each others' language. J\'<:tc-York, October, 1826. F. R. HASSLER. PAR T r FIEST ELEMENTS AND DEDUCTION OV THE FOUR BULES or ARITHMETIC. CHAPTER J. Fundamental Ideas of Q^iiantity, — System of JVumeration, § 1. QUANTITY, which is the object of Aritli* metic, is the idea that has reference to any thing whatever, arising from the consideration of its be- ing susceptible of being more or less ; without re- gard to the nature or kind of the thing itself. It is not therefore an absolute existence ; but a relative idea, that can be referred to any object whatever. § 2. No quantity therefore can be called great or small, much or little, in itself; it can be so only in relation to another quantity of the same kind, which would be smaller or greater. § 3. Objects of diiferent kinds cannot be com- pared with each other directly by their quantity only. When therefore Objects of diiferent kinds are to be considered in Arithmetic, it becomes necessary : that a certain relation be given between them, which is completely arbitrary as to quantity itself, and must be determined before any comparison can take place. § 4. The mutual relation of quantities to each other, under certain given conditions, is the object of arithmetic. In this general acception then it ad- mits any number of systems of combination, that the imagination can devise. 10 FUNDAMENTAL IDEAS OF IJUANTITT. § 5. To forn< a cSesir and distinct idea of arith- metic it is necessary, to impress the mind fully with these fiindamer/tal ideas, and the general principles that r&liii>v from thsm. By comparing every ope- ration of arithmetic with them, they will become always more and more clear and useful ; the whole system of arithmetic will become the more simple, the more its principles are generalized. $ 6. Common ai'ithmetic, which might also be called with propriety, determinate arithmetic, limits itself to the most simple combinations of quantities, and these are all grounded successively upon the first elementary idea of increase or decrease, or more or less, either simple or repeated successively, or according to certain determined laws. § 7. To express quantities we make use, in our system of common arithmetic, of ten figures only, by the means of which, and by their relative places, according to a certain law, we can express any quantity whatsoever. This law is called the sys- tem of numeration 5 and in particular the decimal system, from the individual circumstance, of its using ten different figures, nine of which are signifi- cant, and the tenth indicates the absence of the quantity (or thing, or object.) 5 8. These figures are in regular succession 1, 2, 3, 4, 5, 6, 7, 8, 9, and 0; this last is used to de- note the absence of a quantity ; the 1, denotes the unit of any object, of whatever kind or nature it may be ; the subsequent denote in regular succes- sion each one object more than the one before it. § 9. To denote quantities which exceed the num- ber of significant figures, (or above 9,) recourse is had to a law that assigns superior values to these figures, according to the order in which they arc placed, assigning to them a value as many times greater, in every successive change of pi ace from the right to the left, as the number of figures indicates, NUMEKATIOX . H and therefore in our usual system of ten figures, a tenfold value. This must itecesvsaiily be the law if the system be able to express all numbers, because any other law giving another relation of value to the places, tliaii the number of figures, would either leave a space of quarstity unexpressed, or occasion double expressions, if it were to increase in a greater or less ratio than that number. (The cir- cumstance of this increase taking place from the right towards the left originates in the fact, that this system is borr'owed from the Arabic or rather Asiatic nations, who have the habit of writing from the right towards the left, instead of our writing from the left towards the right.) § 1 0. Thence we have for the successive values of the numbers, in their successive places from th« right to the left, the denominations shown in the fol- lowing table : C (t> a 3 Ci- O 2 3 EJ -i ^3 H ,-* x "1 '"-' et o2a.s.D Q.D f3'-i=ffi'i2iiil?P" • • CO • C Ci. D- CO 03 Cft ai • 1 1 1, 1 1 1,1 1 1, 1 1 1,1 1 Such would be the value or denomination of any figure, placed irs any one of the places, and if no quantity of one or the other of these denominations is to be expressed, the place of it must be supplied with a 0, in order to give to the next figure its pro- per rank._ 12 NtlMERATIOIf. § 11. In reading the numbers we follow our usual way of reading, and therefore express the great- est quantities first; to render this reading more easy it is also customary in large numbers : to di^ vide off by a (,) every three figures, which divides them by hundreds ; so for example : 689,347 would read thus: Six hundred and eighty-nine thousands^ three hun- dred and forty-seven, (understood units.) 13,842,167 reads thus: Thirteen millions^ eight hundred and forty-tw© thousands^ one hundred and sixty-seven. (It will be proper in this way to exercise the beginner in reading numbers, or what is called numeration.) § 12. It may be easily conceived : that other sys- tems might be formed upon the same principles, and of course with the same properties, as to the ex- pression of the greater quantities by the successive rank or place of the figures, and with any other greater or smaller numbeV of significant figures | besides the (0,) which must, like the unit, make part of every such system of numeration. If no other figures were used but (I) and (0,) that is presence or absence of the quantity indicated by any rank or place of figure, the value in each place will always be successively double of that in the pre- ceding place, and the whole of the calculation would become a mechanical mutation of places ; so for in- stance in this system the following numbers 111101, transcribed into our usual decimal system, would be 32, 16, 8, 4, and 1: or (61.) (It will be a very good exercise for the reflection of the scholar to try some of this kind of expressions in different systems.) § IS. It may also assist in clearing up the prin- ciples of the decimal system > to contrast it with the old Roman system of numeration. This consists in the use of seven letters having each a particular signification^ as : I BTUMEKATION, &C. 1$ M, for one thousand D, — five hundred C, — one hundred L, — fifty X, — ten V, — five I, — unity. In this system therefore, the numeration consistt* merely in writing as many of these letters as will make out the quantity desired ; and the whole arith- metic consisted, in placing or taking away, upon a black board, as many marks under the denomination of each of these letters, as the calculation required. A bad habit of the Romans, m later ages, introduced in- to this system anomalies arising from their consid- ering one of the figures of an inferior number, when placed before a higher one, as subtracted or taken away, from it, as for example ; IV was written for four, XC for ninety, and so on. CHAPTER II. General Ideas, and JSTotation of the Four Rules oj •Arithmetic, § 14. The first and simplest combination of quan- tities, and therefore also the first and simplest ope- ration in arithemetic, from which all others pro- ceed, is called addition. Of this we have already an example of the simplest kind in the system of fig- ures, that presents the successive additions of unity in their regular order of succession, and therefore also presents the combination of the quantities by addition as far as the sum 9. Mdition consists therefore in finding a quantity, or number, equal to two or more other quantities taken together '^ or, a? 2 14 DOTATIONS OF THE FOUR it is usually called : to find the sum of two or more numbers. § 15. If on the contrary the difference of two quantities or numbers is to be founds the operation is called subtraction. In this operation the smaller of two numbers, which is called the subtrahend, is taken away from the larger one, and the result is called the remainder ; it is evidently the opposite of the foregoing. Only two quantities or numbers can be concerned in a subtraction, for one result ; if more mimbers are to be subtracted, it must be done by a new operation. § 16. All the subsequent operations in arithme- tic, are combinations of the two preceding ones ac- cording to certain laws. § 17. The addition of the same number or quantity a certain number of times, is called multiplication. When this is treated in detail, the manner in which the principles of multiplication arc deduced from those of addition, will be shown. The two num- bers multiplied into each other are called factors, and the result is called the product. § 18. The opposite of the operation of multiplica- tion^ is called division. It represents a successive .subtraction of the same number, a certain number of times, from another. The number from which this successive subtraction is made is called the divi- dend ; the number repeatedly subtracted, the divi- sor; and the result, the quotient, it indicates how many times the divisor is contained in, or can he taken away from, the dividend. § 19. These four operations of arithmetic, addi- tionf subtraction, multiplication, and division, are called : the four rules of arithmetic. It has been ob- served that the second is the opposite (»r the first, and the fourth the opposite of the thira ; and such must be the case in any system cf combination of quantity that can be devised. In aU arithmetic, it is I RUIES OF ARITHMETIC. 15 always necessary, that both the direct and inverse operation shall be devised ; and directions or rules deduced and given for their execution. § 20. To facilitate the expression of the idea of these four operations or rules of arithmetic, certain signs are made use of, to indicate them in an abridged manner, which it is proper and very use- ful to understand ; their use will conduce to clear- ness in the expression of the operations of arithmetic. To denote an addition the sign (-j-) is used, as for instance, if 7 and 2 ai-e to be added, this will be writ- ten, 7+2, and in the same way for more numbers. To denote a subtraction, the sign ( — ) is used, so for instance, to indicate that from the number 7, the number 2 is to be subtracted, this will be written, 7—2. To indicate a multiplication the numbers are se- parated by a full stop, (.) or by this sign. ( X?) thus to indicate the multiplication of 7 by 4, we write 7. 4 or 7 X 4. If two or more quantities already united by -f- or — are to be affected by the mul- tiplication with one number, these quantities are inclosed in ( ) and the multiplier written to them, in. the same manner as before to the single number, for instance {7 -{- 5) X 13 is the sum of 7 and 5 to be- multiplied by 13. To indicate a division, two different signs are also made use of ; either by placing two dots, or the i colon, (:) after the dividend, and writing the divi- sor after ; or by writing the divisor under the divi- dend separating them by a horizontal line, thus 8:2 or f denotes that 8 is to be divided by 2. Besides these four signs we are yet in need of a sign to express the equality of two quantities ; this is done (by two horizontal parallel lines) thus, =. These signs will suffice here, for other forms of calculation, or combination, other signs are made use of ^ but it will be much easier to understand their 16 NOTATIONS OF THE TOUR meaning when the subject itself is treated ; it is therefore more proper to postpone their explanation for the present. § 21. As it will be proper for the scholar to ex- ercise himself in the expression of these signs, in order that he may become familiar with their im- port, and acquire clear ideas of arithmetical opera- tions, I shall here join a few examples of the four rules of arithmetic, which the teacher may after- wards multiply. In Addition. 7-f9=16, means the addition of seven and nine is equal to 16, or the sum of 7, and 9, is 16. 7-f3 + 8 = 18;or the addition of seven and three and eight is equal to 18, or the sum of 7, and 3, and 8, is 18. In Suh traction. 13 — 7 = 6, means the dif- ference between 13, and 7, is equal to 65 or 7 taken from 13, leaves 6; for example: we shall have, joining both the pre- ceding notations, 1 3 -|- 9 — 8 = 22 — 8 = 14 5 which as is shown by the above example^ it will be easy for any scholar to express in words, but the idea conceived as it is writ- ten by signs is the best mode of expressing it. In Multiplication. As a repeated addition of the same number first, then as a mul- tiplication of factors, equal to a certain product, it will be expressed as in tlie following examples : 74.74-7-1-7 = 4. 7 = 4 X7 = 28. 54-54-54-5-1-5 = 5. 6 = 5x 5 = 25. and thus in any other case. RULES OF ARITHMETIC. If In Division, If we express division by a suc- cessive subtraction of one number a certain number of times from, another, we shall, in the case of this subtraction exhausting the number, reduce it to 0, and there- by show that tlie divisor is con- tained in the dividend, exactly as many times as it has been possible to subtract it, so we would have for instance, 3a — 6 -- 6 — 6 — 6 — 6 = 0. Which showing that six subtracted five times from 30, and its successive remainders, leaves nothing | therefore, if we express this as a division, having the result, or quotient, on the other side of the sign of equality we obtain in this case the expression 30 : 6 = 3_o = 5 If the successive subtraction of the divisor, should at last leave a number smaller than this divisor, it will give what is called a remainder, that is still affected with the sign of division by the divisor ; as for instance in the following example : 36 — 8--8--8 — 8 = %6=:4^i. This last part of the expression indicates a division that can no longer be executed, on account of the divisor being greater than the dividend ; it no longer gives a whole quantity in the result; these expressions are called fractions, and we thus have already the fundamental idea of a fraction, from which we shall hereafter deduce the principles of calculation that are adapted to them. § 22. By means of these explanations of the prin- ciples, and the notations, of arithmetic, it is proper for the teacher to introduce his scholars to the sub- iect, and prepare them for its future practical appli- 2 ^ 18 ADDITION-. cation, if he would not make it a study toilsome to the boy, and an equally toilsome task for himself. No teacher ever had a scholar who did not ask him {why 'I) when he directed him to do something; and •this why, the reasonable and faithful teacher must answer in a satisfactory manner ; this will be ren- dered easy by the preceding process, elucidating the principles of arithmetic. The reasoning of the child must be cultivated, if he is ever actually to understand arithmetic, and not forget it when out of school, or out of practice, as will be the case, if he has only committed to memory dead rules for which he saw no reason. By such a process arithmetic will ever be agreeable to the scholar, as an exercise of his intellect within the limits of his capacity. The time spent in explaining and reasoning with the scholar upon these principles will be amply gained by his more successful and regular progress in arithmetic, when applying it to each individual rule and case. CHAPTER III. The four rules of Arithmetic, in whole numbers. • § 23. ADDITION, has been defined as the me- thod of finding a quantity, equal to two or more quantities, taken together. Its expression as a problem, is therefore : to find the sum of two or inore quantities. /From what has been said of the principles of the system of numeration, in common arithmetic, it follows : that in order to prepare the given numbers for addition, they must be written under each other so as to bring the units of the one under the units of the other ; and so all the numbers successively higher in the order of the system of uumeration^ will each come under its equal deno- ADDITION. 19 mination ; by which means they may be added the more easily. Then the numbers are added together, in this order, beginning always with the unit and proceed^ ing until we reach the last on the left hand side. Example.— To add 176873 + 34719. Write these numbers thus ; 176873 34719 and draw a line beneath them ; then add the column of units, 12 tens, 8 hundreds, 16 thousands, 10 ten thousands, 10 hundreds of thousands, 1 211692 placing each particular sum so that the figure on the right hand shall be under the numbers added; then draw a line and add the numbers as they are now pla- ced. The result thus obtained will be the sum of the numbers added. It is evident, here, that whenever the sum of any one of these individual additions exceeds what can, in our system of notation, be written with a single figure, we had to place the figure coming to the left of it, under the next high- er order ; and in the second addition, these numbers were then added to the result of the addition next following. This can therefore he done at once, by the following process. Having, as in the example, found the first sum, 9 and 3, which is 12, (or 9 -f- 3 = 12) the 2 is placed under the unit, and the 1 is kept in memory to be add- ed to the next operation, in this case to the sum of the tens, (which is called carrying ;) so that in this next addition you say ; 7 + 1 + 1 = 9 or 7 and 1 is 8, (as marked in the example,) and 1 carried gives 9, which is immediately written to the left of the former 20 ADDITION. result, or under the tens ; this number can be writ- ten entirely, and therefore gives nothing to carry. The next or hundreds would -give 8 -f 7 =- 15, or 8 and 7 is 15 ; write 5 and keep 1 ; then the next, 6 and 4 is 1 0, and 1 kept is II; (or 6 + 4 -f 1 = U ;) and so on to the last figure on the left hand. § 24. If there is a greater number of figures to be added the same mosle of operation is used, only repeated as often as the number of figures given will require ; as for instance in the following exam- ple: To find the sum of 674-21 -}- 389 -f 641827 -f 30 -f- 4 + 7259 = Write these numbers all under each other so that the units fall in the same column, and the other numbers successively under their respective places, thus : 67421 389 641827 30 4 7259 71(3930 Then, having drawn a line beneath, begin again by saying, in the column of the units, 9 + 4 -}- -h 7 -f 9 + 1 = 30, write 0, and keep 3; then for the second column, or that of the tens, say : 3 -f 5 + 3 -j- 2 + 8 + 2 = 23; write three and keep two, and proceding in this manner to the last figure on the left hand; which will produce the sum found in the example, under the line. It is necessary to practice such examples sufficiently, until the scholar can exe- cute them with facility and accuracy, so that it becomes to him an easy mechanical practice. It is proper to mix the numbers of different orders, as above at once and not to distinguish separate cases, SUBTRACTION^. Q{ iu order that the scholar may seize the principles of the operation intellectually, and with reflection, and not by mere memory and habit. § 25. SUBTRACTION, as has been already said, is the opposite of addition ; its Problem is : to find the difference between two numbers. In common arithmetic it is always required, that the number to be subtracted be greater than the number from which it is to be subtracted ; otherwise the result would become, what in universal arithme- tic is called negative : that is to say in denying the possibility of the subtraction it would indicate the number from which it was intended to be subtracted to be so much too small to admit this subtraction, as the number found indicates. This operation is necessarily limited to two num- bers or quantities, if more should be concerned in a question, the result must be obtained by a repetition of the operation. § 26. Of this operation in simple numbers we have given the principle in the explanation of the signs, as in the case of addition ; when the numbers are larger the following is the preparation and the operation. Write the number from which the subtraction is to be made first, and the subtrahend under it, in such a manner that the unit comes under the unit, and the following numbers, to the left, each under its similar superior number, and draw a line undei- them thus ; 9643187 7532043 = Subtrahend, 2111144 = Remainder, 9643187 = Proof, then take the difference between each of the corres- ;ponding numbers, beginning by the unit, and write r22 STTBTRACTIOSr. the difference directly under these numbers, the number resulting therefrom will be the entire dif- ference between the two given numbers. As well from the principle that this operation is the opposite of the addition, as from the consideration of the preceding operation, it may easily be observed : that the proof of the correct execution of this opera- tion may be given, by adding the result, or remainder obtained, to the lower number above the line, or the subtrahend, wliich addition must give the first or npper number for its result. It is therefore proper to accustom beginners to make this proof, in order that they may have the satisfaction of verifying the correctness of their operation ; drawing therefore a line under the result, the two numbers immediately above are added, when the first number must again appear in the result. § £7. In this operation it may evidently occur : that, though the quantity from which another is to be subtracted may be greater, some of the in- dividual numbers, of the inferior order, in the sub- trahend ,• may be larger than those corresponding to them in the superior number. In this case it becomes necessary to supply the want by borrowing an unit from the next higher or- der of the upper number, which will of course then represent a ten in its corresponding order next infe- rior in place and value, and furnishing of course al- ways in addition to this number itself a larger number than that in the subtrahend, will admit the latter to betaken from it,* the remainder is then written in its proper place, and if even tlie preceding superior num- ber were an 0, the lending being considered as possi- ble from the preceding higher order, the operation would be the same, an unit would be borrowed from it, and the number afterwards called 9, again under the supposition before made of the lending being made from the next higher order, which, when reached. SUBTRACTIOBT. 2S is considered as diminished by an unit. It is evi- dent that if the superior number is larger than the inferior or subti^ahend, this lending will always be compensated before the end of the operation, what- ever be its extent, through the figures preceding the last on the left hand side. Let the following example be given. 600198056 — 336499278 Place the example as indicated, thus :\ 600198056 356499278 243698778 600198059 Here in the units the 8 cannot be taken from the 6, an unit is therefore borrowed from the 5 in the tens preceding the 6, which added to the 6, gives 16, from which the 8, being taken leaves 8, to be writ- ten in the place of the units. (For beginners it will be proper to mark every figure from which an unit has thus been borro\Ved, by a dot above it, which is done in order that it may not be forgotten to pay attention to it in proper time.) In the second place or the tens we have then only a 4, instead of a 5 ; we are therefore again un- der tlie necessity of borrowing from the next higher figure, though this be an 0, subtracting then 7, from 14, the remainder 7, is written in the proper plac In the place of the hundreds we have then, by the effect of the foregoing borrowing, which is trans- ferred to the place of the thousands a 9, from which the 2 subtracted gives the remainder 7. By the preceding borrowing, the 8, in the order of the thou- sands has now become a 7, and is again insufficient to admit of a 9 being subtracted from it ; the borrow- ing of an unit of the higher order gives here 17. S4 SUBTRACTION. from wliich 9 being taken gives 8, as remainder. The 9 in the next higher order has now, by the lending become an 8, in order to subtract the 9 below, from it, a unit of the next higher order is again borrowed, making it 18, subtracting 9 from it, gives 9, as the remainder to be written. The unit in the next higher order having been borrowed, the remain- ing, is made into a 10, by borrowing an unit from the next higher order, from which 4 being sub- tracted, leaves 6 ; the next higher number being a 9 by the supposed borrowing from the higher or- der, and the same being the case for the next fol- lowing 0, these two subtractions are made exactly like that in the hundreds, until ultimately the last left hand figure being higher than the number of the subtrahend under it, the subtraction is possible "which being done, the number 243698778 presents the full remainder required by the subtraction, or is the difference between the two given numbers. The proof of the correctness of this operation will again be found by the addition of the subtrahend and the remainder, which by carryings correspond- ing to the preceding borrowing, will again give the upper immber, as seen by the example. Proper attention to the example here explained will teach bow to act in every case tliat may occur in subtrac- tion, and it will be proper for the scholar to be exercised upon a sufficient number of examples, that he may acquire facility in this operation. § 28. There are two otlier ways to perform the operation to obtain the same result; but the above explained course of reasoning is the one most closely connected with tlie nature of the question, and the implied requisites of the operation ; it is therefore proper to keep the scholar to this con- sideration. When once he has gone through the whole course of arithmetic he will easily see the two other methods, which if taught at this stage MFI.TIPI.1CATI0N. 25 gf the study would contuse his ideas^ and are there- tore intentionally omitted here. $ 29. MULTIPLICATION, as has been stated, is the addition of a given number repeated as many times as another number contains units, or indi- cates; thus every number is in itself the pro- duct of that number into the unit. It is indiiferent which of the two numbers be considered as acting the one or the other part in the operation ; therefore they are both equally called Factors ; the result of the operation is called the product. It is necessary, in order to perform this operation with ease, in more complicated calculations, to com- mit to memory the product of the nine numbers ex- pressed by our numerical symbols. It is needles3 for written operations to go any farther, because the higher multiplications overreach, in writing, oxsy system of numeration. We have already seen that our system of numera- tion is a successive addition of the unit below 9, which being the last symbol of quantity, the next quantity is expressed by a change of place. If now we treat every one of the nine symbols in the same way, by the successive addition of itself, we obtain^ successively, the product of each of these symbols in. a similar manner, forming what is commonly called the multiplication table. Writing therefore the re- gular series of numbers as far as 9, in a horizontal line, add each of them to itself, writing the result under it, then to this sum adding again the number^ and so in succession, until the whole 9 symbols are exhausted, we shall have the following system of re- sults : 9.6 MriTIPMCATlON'. '<><>0<>0'<><><><><>C<><>i.><><><><^^ 8 10 12 12 16 15 20 ><>C<>000<>0-0<><><>< 4^ 10 12 1^ 18 20 24 30 25 30 36 35 42 40 48 54 14 21 28 35 42 49 56 63 •C:.<»<>0<'-0-0-0-00< Considering the preceding table, we find that the first column to the left, which again contains the se- ries of natural numbers of our system of symbols, by the successive addition of the unit, keeps an account of all the other successive additions ; or that it indi- cates how many times this addition has been repeat- ed, and that the result of any number of such addi- tions, of any one of the successive numbers, is al- ways found in the meeting of the horizontal and vertical lines of the two numbers taken as factors ; thus, for instance, under 7, and where the horizontal line marked 6, in the first column, meets it, we find 42, that is, the addition of 7 six times repeated gives tlie result 42. In like manner under 6, opposite to the 7 in the first column, will again be found 42. So 6 times 7, and 7 times 6, (such is the usual ex- pression,) are equivalent ; as has been stated above t and such is the case with any other number. MULTIPLICATION. 27 The regular progression of the different results is easily observable, and some attention to it will assist in fixing them in the memory ; it is best not to load the beginner with a longer table, for which he has no use, until he may, in practical ap- plication, wish to calculate from memory, without writing,' when the circumstance of its possessing interest and usefulness will make that task easy, which at this stage of instruction is a dry and use- less labour. § 30. We must now suppose : that the scholar has acquired some facility in the use and application of the results of the preceding table ; and shall proceed to show the details of multiplication by examples. Be it given to multiply 357279 by 6 ; or to exe- cute what is expressed by the sign of multiplication, thus: 6 X 357279. Write the smaller factor, in this case the 6, under the other, so that the units stand under each other ; then execute the multiplication of each of the num- bers of the larger factor successively, and write the result under the horizontal line drawn below the factors, so that the right hand figure of the pro- duct shall always stand under the number multi- plied, thus : 367279 6 64 42 12 42 30 18 2143674 then adding up all these products, the sum result- ing will be the general product of the whole multi- |)lication. 28 MrLTIPLICATION. The inspection of this detailed execution of the preceding example, shows that we may again apply, in this case, the mode of abridgement that has been pointed out in addition. We would, therelbre, in the preceding example say, (analogous to what has been done in addition,) 6 times 9 is 54 ; write 4, and keep (or carry) 5 ; then keeping this 5 in mind., we would next say, 6 times 7, is 42, and 5, is 47 ; writing again the 7, and keeping the 4 to be added to the next product; then 6 times 2, is 12. and 4, is 16 ; when writing the 6, and keeping 1, and proceeding thus to the end of the number, we obtain at once the same numbers that appear above, in the final re- sult. This mode of proceeding is therefore the usual mode of operating, with each of the numbers of the factor that is chosen, for the purpose of taking the multiples of the other by it ; for which, as said before, it will be best to choose the smaller one, be- cause it gives the shorter example in writing. § 31. But when both factors are compound num- bers, it is evident that the multiplication of each of the numbers of the one, cannot be made at once with all the numbers of the other ; therefore we must pro- ceed with each number of the one factor, exactly as shown above with the single number ; and in order to give to each individual result its proper place, we must begin to write the first number of each pro- duct on the right hand side, exactly under the num- 'ber of the multiplier of which it is the product ; as its proper unit. The sum of all these partial products is then made, by the addition of all the numbers in the regular order in which they stand under each other, as this has been done in the preceding example, with the partial products of the simple number. This shall be shown in the following example, in which it is required to perform the multiplication 174392 X 6435; writing the factors properly under each other, so that the units stand under each other^ and the other numbers follow in their regular order. MULTIPIICATION-. 29 the successive results in their proper places, will be as follows : - 174392 6435 871960 523176 697568 1046352 1122212520 In this manner, every example, whatever quantit) of figures it may be composed of, will stand. If any of the figures in the number to be multi- plied, which is called the multiplicand, should be an 0, its product into any number whatsoever, is = 5 because times any number whatever, always indi- cates that the number is not there ; the place will, therefore, receive only that number which may be t^arried over from the preceding multiplication, and if none be carried, only an 0. If an 0, occurr among the numbers by which the multiplication is to be performed, or the multi- plier, the whole row of figures to be multiplied by it producing a result = 0, the place where the first number would stand will only be marked by an 0^ and the multiplication by the next following num- ber is begun in the same row, immediately after, thus placing each result in its proper place. The following example will explain both the above cases, where the effect of the two O's, in the multi- plier is shown by the removal towards the left of the two latter rows of figures. 3603904 50203 10811712 72078080 180195200 3 * 180926792512 30 DIVISION-. § 32. It will be proper to ©xercise the scholar in a variety of examples, until he has become accus- tomed to the operation, and is able to make any multiplication without error : the younger the scholar may be, the easier the examples must be in the beginning, and must gradually increase in diffi- culty, by the combination of different cases, and larg- er numbers. Still, in this it is to be observed : that when the beginner has performed examples gradu- ally with the whole series of the nine simple num- bers, it will be proper to show him only, what is the effect of a compound multiplier, as a repetition of the similar operation of one number only, and the addition of the different partial products into one whole; and not to follow servilely the augmenta- tion by one number, (or place of figures,) that he may not, as often happens, consider that he has every time a new difficulty to overcome, but must himself come to the observation, that multiplication by a num ber of places of figures is a mere repetition of the operation he knows, requiring nothing but a little more attention, and more accuracy in the placing of the figures. § 33. DIVISON, is an operation the opposite of Multiplication, as has already been stated ; its pro- blem is therefore : to find how many times a givea number is contained in another given number, which is thus considered as a product of the first and the quantity sought. The table of products, or multiplication table, ^iven above, may therefore be here applied inverse- ly; a ready and habitual knowledge of its results is tlierefore also constantly applied in this rule, by the comparison of its results with the quantities pre- senting themselves in an example. While all the preceding operations have begun at the unit, this on the contrary must begin by the highest number, or order of symbols; for the greater BIVISIOX. number of times, which one quantitj^ may bfr con- tained in another is necessarily to be taken out, or considered, first, the inferior numbers will then fol- low in their regular order, and keeping account of the value of any remainder from the preceding ope- ration in its proper rank, as in the following example, which we shall express in the manner that has been shown in § 20, in order to accustom the learner to keep the systematic language of the operation itself, which is always the most preferable method ; with this view we shall draw a horizontal line under the dividend, under which we shall place the divisor, and the result, or quotient, will be written on the right hand side of the sign of equality which follows them, thus : 842316 = 280772 3 3 842316 24 24 23 21 21 6 6 Here we say S, in 8, is contained twice, and having written the 2, as the first number to the quotient, we must make the product of it by the divisor, write it under the corresponding num- ber of the dividend, and subtract it from it,* this pro- duct being 6, in this case the subtraction leaves 2, as a remainder. Now, for tho sake of easier distijic- 32 DIVISION. tion we place the next number by the side of this remainder, which being 4, gives for the next num- ber to be divided 24. Now 3, is in £4, contained 8 times ; placing the 8 in the quotient, multiplying the 3 by it, the product of 3 times 8, placed under the S4, being also £4, leaves no remainder ; placing the next number 2 down, we find, that 3 not being con- tained in it, we must indicate this by an 0, in the quotient, for the rank or order of the numeric sys- tem corresponding, which being Jone, tlie next num- ber, 3, is taken down to the right side of the 2, which making 23, we say 3 in 23 will be contained 7 times ; writing the 7 in the quotient, multiplying the 3 by it, and subtracting tlie product 21 from the 23, we obtain the remainder 2 ; taking down the 1 which gives 21, we say again, 3 in 21, is con- tained 7 times, and the product 3 times 7 being equal to 21, leaves iio remainder; lastly, bringing down the 6, we find 3 in 6 twice, and writing the 2 in the quotient, and subtracting its product by 3, from the 6, we obtain the exact quotient 280772. Division being the opposite of multiplication, we have tlie means of proving this result, by the multiplication of the quotient by the divisor : the product of which must be equal to the dividend, as is evident from the definitions given of this ope- ration. Writing then the divisor under the quotient, and performing the multiplication, the product resulting will be equal to the dividend, if the whole operation has been rightly performed. 5 34. If the divisor is not contained an exact whole number of times in the dividend there will remain at the end of the division, a number smaller than this divisor, which is called the remainder. In order to indicate fully the actual result of the division, this number is yet to be placed at the end of the quotient, with the divisor written under it, and it horizontal Divisioif. 35 line between them, to indicate that this division should yet be made. Such numbers as indicate a division which can- not be executed, are called proper fractions, while every division, indicated as above, of a number larger than the divisor, is, in comparison with these, oalled an improper fraction : and, when considered in this point of view, the number corresponding to the dividend, is called the numerator, and the num- ber corresponding to the divisor is called the de- nominator; while the quotient, whatever it maybe, will always represent the value of the fraction. This general idea of fractions, the origin of which it is proper to show here, will hereafter be the fun- damental idea from which the calculation of this kind of quantities is to be deduced. The following is an example that will show such a division, and the mode of operating in the case. Being given to divide 7835921 = 979490 l- 8 72 63 66 7835921 75 72 39 32 72 71 01 In this example : we see that thejfirst number pf the highest order being smaller than the divisor^ wo 54 DIVISIOK-. must take it jointly with the next following lower numher and say : 8, in 78, is contained 9 times ; and the 9 is written as the first numher in the quotient ; then making the product 8.9 = 72, and writing ii under the 78, from which it is subtracted, and leaves 6, which being written below the line, and the next lower number 3, written down to it, gives 63 for the next number, to be divided by 8, which being con- tained 7 times in it, 7 being written in the quotient^ the product 7 X 8 = 56 is written under the 63, the subtraction performed, and the 5 or next following number placed down to the 7 that remains from the subtraction ; the operation is thus continued, exactly as in the former example, until when the last num- ber, 1, is^set down at the side of the 0, we find that 8 is no longer contained in 1, and therefore write an in the quotient, and having no more numbers in the dividend, we find that 1 ought yet to be di- Tidedby 8, which we write in the quotient, as stated above, | like an unexecuted division, or a proper fraction. When we make the proof of this example, as has been done in the preceding one, we consider the 1 as a remainder, and in the multiplication of the quotient by the divisor, add it to the product ; so that we would here say 8 times 0, is 0, and the remainder 1 added, gives 1 for the first number of the product, exactly as in the dividend, and then continue the multiplication through the whole quotient obtained, as in the former example. § ^5, When the divisor is a number composed of more than one figure, the principles of the opera- tion remain the same ; but it becomes necessary to pay attention to the effect of the multiplication of the quotient into the whole number of the divisor : which may render it necessary to take this quotient smaller than might appear from a mere compari- son of the first numbers of the divisor and the divi- DIVISION. 3^5 dend ', all the rest of the operation is only an exten- sion of the operations explained in the preceding ex- amples, which have heen described in detail, with the express view of giving a full explanation of the first elementary principles. Reasoning with the same details upon the following example, the opera- tion of a division, with a divisor composed of more than one figure, will also be clear. The following division being given 64U59213 = 84510 ^^ 758 758 G064 34 1 9 422550 3032 591570 676080 .50 633 3872 3790 64059213 821 758 633 Here in considering only the first number of the divisor, and comparing it with the two first of the dividend, we would find 7 in 64 contained 9 times; but we must take into consideration the multiples of the numbers which follow the hundreds. The 5 tens, or 50, multiplied by 9 would give 45 tens, or 450, and 7 X 9 = 63 would leave only 1, which, considered as hundreds, as must be done in this case, would not allow us to take the 4 hundreds; from it. We find, therefore, that the quotient 9, is too large. Taking 8, we find that 7 X 8 = 56, leaves 8 as remainder; and if we consider now^the 58, as multiplied by 8, we find that the 4, which roracs here again as hundreds to be snhtnacted fr^na ^6 BIVISIOS-. the 8, eau be taken away with a considerable remainder. Writing then 8, as the first num- ber in the quotient, we make tlie product 8 X 758, and place it under the respective numbers of the di- vidend, so that the product of the first number of the divisor, that is to say, 7X8 may stand under 64, and the otliers follow in their regular order ; we now make the subtraction, in the same manner as has been often before shown, which leaves 34 1 as remainder,* as this is less than the divisor it also proves that no greater number could have been taken for the quotient ; to this we join, as in the preceding examples, the number of the dividend next after those used in the last subtraction, which is here 9 ; and now proceed as before to compare the products of 7 with the 34, as the number presenting itself here for division, in the same rank as the 7 of the divi- sor ; this shows 4 as the nearest factor producing with 7 a multiple, (28,) inferior to 34, and leaving 6 as remainder, while 4X58 giving only a 2 to carry to the place of the hundreds, leaves sufficient room for the whole product ; we thus obtain the remain- der 387, that is again smaller than the divisor, and placing after it, the next following number, 2, we say first 7 into 38 is contained 5 times, and the product, 5 X 7 == 35, taken from 38 leaving 3, the product 58 X 5, giving only 2 to carry to the place of the hundreds, will leave a sufficient quantity for the subtraction ; this being performed, and the 5 placed in the quotient, we have the remainder 82 ; then placing the 1 down after it, the resulting 821 contains the divisor, evidently, only once. Placing 1 in the quotient, the subtraction leaves 73 ; when the last figure, or 3, is written after tliis, the num- ber 733, that results, being less than 758, the lat- ter will not be contained in it; this gives an in the quotient, for the last whole number ; and the unexe- eutable division ^^ as a fraction or remainder, as in the second or foregoing example. - DIVISION. ST The proof of this example is again made in the same manner as in the last ; multiplying 8450 x 758 and adding 733 to it, the dividend will again be obtained, as seen in the example. The remark which has been already made, upon the propriety of practising any of the elementary opera- tions until a competent dexterity is acquired, of course, also applies here. The detailed manner shown hi re, is what is usu- ally called long division ; and even experienced cal- culators may often find it proper to apply it, when the number of places of figures in the divisor is great. § 36. For common calculation it is often desired to spare writing out the numbers for the subtrac- tion, and writing only the remainders. This is carried on as in the follovving example. Given 9460753 = 10763/^V 879 879 6707 6545 96867 2713 75341 76 8610476 9460753 Here the divisor is contained once in the three first numbers of the dividend ; the 1 being placed in the quotient, the subtraction is immediately made from them, and only the remainder placed below ; which being 67, and the next number, the 0, being put down to it, the divisor, 879, being larger than 670, the next number in the quotient becomes a 0. After w riting it, the next number, 7, is taken down from the dividend, and in the resulting 6707 the divisor is contained 7 times. Now the divisor is multiplied by this, and the subtraction of the result made in the memory immediately, and again only 4 38 Divisioir. the remainder wintteii down, thus : say 7 times 9 is 63, subtracted from 67^ which the number above must be supposed to reprrsent, in order to allow the subtraction of the product of the unit or first number, leaves 4, which is written down as a re- mainder under the 7, and the 6, which the num- ber in the next higher rank has been supposed, is kept in memory, ajid added to the next product of the tens, or next higher order of numbers, with which it is then again subtracted ; therefore, con- tinuing the multiplication, we say : 7 times 7 is 49, the 6 kept being a(hled makes 55, that subtracted from 60, which we suppose to bi- the number above, having the in the first place to the right, the re- mainder, 5, is written under the 0, and 6 is kept to add to the next following product; for which we say 7 times 8 is 56, and 6 carried is 62, taken trom 67 leaves 5. Bringing now the 5 from the dividend down to the remainder 554, we have for our next di- vidend 5545, in which we say : 8 in 55 is contained 6 times ; and as 6 X 8 = 48, leaves 7 in the place of the hundreds, for the carrying of the lower numbers following it, is evidently small enough to allow the subtraction of the whole product; so w^e say again, 6 X 6 = 54, from 55, leaves 1 ; write it, and carry 5 ; then 6 X 7 = 42, and 5, is 47, from 54, leaves 7; write 7 down, and carry 5 ; lastly, 6 X 8 = 48, and 5 is 53, from 55, leaves 2 ; tlie remainder, presents therefore, 271 ; to which the 3, as next lower num- ber in the dividend, being written, we find 7 in 27, is contained 3 times, or 3X7 = 21, leaves 6, a sufiicient remainder in the hundreds, for the carry- ing of the product 3 X 79 ; so we say again, 3 X 9= 27, from 33, leaves 6, and 3 to carry ; then 3X7= 21, and three added gives 24, from 31, leaves 7, and 3 to carry ; then 3 X 8=24, and 3 is 27, which g subtracts without a remainder, from the 27 above; * and the remainder 76, to which we have no other TULGAR rRACTIONS. 89 number to set down from the divisor, gives the nu- merator of tlie proper fraction remaining, ■^\%^ as a division that cannot be executed with our present means.* In the manner the reasoning has been carried, in this example, every other more complicated case is to be executed ; it is therefore expected that it will suffice to introduce into the practice of this method. CHAPTER III. Of Vulgar Fractions. § 37. We have seen already, in § 11, and at the end of Division, that fractions are unexecuted divi- sions 5 we have also seen, that in consequence of this, they consist of two parts, corresponding to the two parts or numbers engaged in a division : their form, or the manner of writing them, we have seen to arise naturally fmm the division, when a number remained ultimately in the dividend, which was smaller than the divisor, or the number by which it should be divided ; we have there already observed, that this constituted a proper fraction^ while every division whatever, exprcNsed in the same form, was an improper fraction, as it would naturally be called, from its still containing the di- visor a whole number of tijues. Tlie number above the horizontal line, (as seen in § 34,) which corresponds to the dividend, is called the numerator of the fraction ; and the number be- * The proper fractious^ are still i)urpoHely here repiesented as unexerutable divisions, because the pre«"edins; operations Id whole numbers, do not furnish any mean- for such a divipion. We shall afterwards show, how these values may be expressed, either ex- actly or approximately, by a continued division, and an extension of the decimal system, below the unit. 40 VUXGA» FRACTI0X9. low this line, corresponding to the divisor, is called the denominator of the fraction ; thus considering the first as indicating the number of parts taken, and the second as indicating the vajue of the parts, or giving the name to theparts. By this means any fraction may evidently he represented as, or rather these considerations show it to be actually the pro- duct of a whole number into unity, divided by an- other number ; and this latter part must be con- sidered as characterizing a particular kind of quan- tity, in the same manner as the different places of figur^^s characterize units, tens, hundreds, and so on : we thus evidently have (expressing the above reasoning according to the forms and signs adopted) for an example, where 7 in the numerator, counting the parts, and 18 the denominator, showing these parts to be eighteenths of the unit. And the value of these parts may evidently be as much varied as the num- bers themselves ; therefore they have not, like the numerical system, one necessary and uniform law of connexion. § 38. From these considerations of the principles and nature of fraction, the following three funda- mental propositions for the arithmetic of fractions, naturally follow : Proposition i. M many times as the fiumerator of a fraction is made larger or smaller^ the denominator remaining unchanged^ so many times the value of the fraction is made larger or smaller. For, by multiplying the numerator by any num- ber, there are as many times more parts taken as this number indicates, and in dividing it by any number, there are as many times less parts taken, as the number indicates ; in the first case, therefore, the value of the fraction is as many times larger, and TUIiGAR FRACTIONS. 41 in the second, as many times smaller, as the numher used in the multiplication or division indicates. 13 X 7 7 Example, = 13 X according to the 18 18 same reasoDing as in the preceding §. 7:97 And = : 9, according to the same. 18 18 Proposition ii. ^s many times as the denominator of a fraction is made larger or smaller, the numerator remaining unchanged, so many times the value of the fraction is made smaller or larger. For, the ilc!Jominator being the number by which the unit is divided, as many times as this rmmber is multiplied, so many times the unit is divided into more parts ; and thereiorc, the parts becoming as many times smaller, an equal number of them re- presents a value as many times smaller ; that is to say, the value of the fraction is as many times smaller, and inversely, when the denominator is di- vided by a number, the unit is divided by a number as many times smaller than this divisor indicates ^ therelore, the parts become as many times larger, and the value of the fraction becomes as many times larger ; all under the supposition : that an equal num- ber of these parts be taken before and after the operation. 7 7 Example. — is 13 times smaller thnn be- 18 X 13 18 cause the 7 is divided by a number 1 3 times larger than 1 8 ; 1 r we have, 7 X 13 times smaller than 18 X 13 4 # 42 rWLGAR FRACTIONS. 1 7 7 7 X , and = — is 9 times larger than 18 18 : 9 2 7 , because the 7 is divided into parts 9 times larger : 18 ' 1 1 or, we have, 7 X 9 times smaller than 7 x — 18 2 Proposition hi. When the numerator and deno- minator of a fraction are both multiplied or divided by the same number, the value of the fraction remains unchanged. This is an evident consequence of the combina- tion of the two preceding propositions, vy^hich show the effect of the multiplication and division upon the numerator and the denominator, to be exactly op- posite, and therefore, when performed with the same number, they exactly compensate each other ; that is to say : as many times as the value of tlie frac- tion becomes larger or smaller^ by the multiplication or division of the numerator of the fraction, so many times it becomes again smaller or larger, by the mul- tiplication or division of the denominators* 7X9 7 7:9 Example. = — =s 18X9 18 18:9 where the mutual destruction of the effect, of the two operations, is self-evident. The two first propositions solve directly all multiplication or division of fractions by whole numbers, in a double manner ; for we have, evident- ly, every time, the choice between two operations, each of which may, according to the case, present a preference in application. The third proposition will evidently furnish us the means to reduce fractions from one denomina- tor to certain other ones, in order to obtain the fractional parts expressed so as to be adapted to ccr- I VULGAR FRACTIONS. 43 tain purposes in tlie operations of arithmetic, with- out changing their value. § 39. The investigations of § 37, have shown fractions to be eciuivalent to the product of a whole number into certain quantities expressed in parts of the unit ; when thus representing quantities of different values or kinds, they have different denomi- nators ; their numerators therefore cannot be taken into one sum, or difference, without previous appro- priate changes. By the tlurd of the foregoing pro- positions, we have obtained means to make such changes, without altering the value of the fractions. The aim of such a change, must evidently be to ob- tain the same denomination for both, or all the fractions, whose sum or difference is desired. We have seen in multiplication, that it is in- different which of the two factors is multiplier or multiplicand, this shows that equal denominators may he obtained for two fractions, by multiplying the denominators together ; if therefore, the numera- tors of the two fractions are also multiplied, each al- ternately by the denominator of the other, the value of the fraction will remain unchanged, according to the third proposition above ; and if more frac- tions are concerned, considering the first result as one, and operating upon it in conjunction with ano- ther, exactly in the same way as before, and so on to the end, a result is evidently obtained, that ap- plies to any number of fractions. This furnishes us with the following general rule. To reduce fractions to a common denominator ; mul- tiply the numerator and denominator of each fraction by all the denominators except its own ; then all the fractions will have the same denominator, and the numerators will be such that the value of the frac- tions will not be changed. 44 TriGAR TRACTIONS, 7 3 Example, and reduced to the same denominator 15 14 7 X 14 3X15 will give, by the above and — — ; 14 X 15 14 X 16 98 45 or , and ; 210 210 Being given to reduce to the same denominator ; 12 3 7 » ~" » » 2 3 5 8 we evidently obtain step, by step, the following results : 3 2X2 2x3 3X2 > 3 4 or, ~ 6 and — ; 6 from these and the third 3x5 4X5 3X6 6 X 6 ' 20 6x5 18 from these and the last, or, ~ ; 30 30 30 15 X 8 20 X 8 18 X 8 30 X 7 30 X 8 30 X 8 30 X 8 30 X 8 120 IriO 144 210 or, ; ; 240 240 2!i) 2)0 Here quantities of the same kind, are evidently ob- tained, say equal parts of the unit, only in differ- ent quantities j for, according to what has been seen abovey these fractions might be thus written : 1111 120 X ; 160 X ; 144 X ; 210X ; 240 240 240 240 VULGAR FRACTIONS. 45 § 40. It is evident from the above, that fractions cannot be reduced to any denominator indiscrimi- nately, as the new denominator must be a multiple or a quotient, of the former denominator. If it should become necessary to take whole num- bers under the same consideration, it will easily be judged, from what has been said, that they must be considered as having the denomina- nator, 1, and such indeed they are, for the unit is their measure as to quantity, like any other denomi- nator in a fraction, 34 1 Example. 34 = — = 34 X — ; 1 1 For every whole number whatsoever, must be considered as multiplied by 1, really to be a quantity : if it was mul- tiplied by 0, it vvoiild be said not to beat all, asO, denotes the absence of all quantitv ; and if multiplied by any other number, the product would be another number. 5 41. The continued multiplication of all the de- nominators evidently leads into large numbers, both for the numerators and tlie denominators, which it is desirable to avoid wherever possible ; this will be the case when some of the denominators are pro- ducts of the same number with different numbers, or have what is called, common factors^ these are therefore not necessary to be repeated in the con- tinued product of the denominators, which furnishes the new denominator, as tlie above example already shows, where 2 and 8, are products of 2, the first by 1, the second by 4. The following problem and its solution, which will best be explained immediately by an example, will lead to this result. Problem. To find the smallest number which will be divisible by several other given numbers. Solution* Write the numbers after each other, 46 VI716AB FRACTIONS. as 3 ; 4 ; 9 ; iO ; Z\ ; 35 ; 12 1 , 4 . 3 , 10 , 7 , 36 , 4 3 1 , 1 , 3 , 10 , 7 , 35 , 1 4 1,1,3,2,7,7,1 5 1,^,3,2,1,1,1 7 take any one number, w hich will divide several ol' these numbers without remninder, and divide these numbers by it, write the divisor, here 3, on the other side of a vertical line, the quotient under each ot* the numbers; write also all the other numbers, that are not divisible, down in the line, (as shown here in thesecond line of figures.) with the quotients, and the other numbers pi-oceed as before ; here we find the common divisor 4, and the third line of numbers is obtained, this line is reduced by the divisor 5, and the fourth row of figur-es is obtained, and the opera- tion is continued in the same wav, until no common divisor is found; as in the fifth line of the example. The continued product, that is here 3X2x7X5X4x3 = 2520 will be the smallest number divisible without re- mainder by all the given numbers. The units of course disappear in the multiplication, as they do not augment the product ; they indicate the num- ber of reductions obtaified by the operation, with- out which the continue, smaller than cB; between this remainder and the former divisor, or the gD, and the cB, the same reasoning takes place as be- fore, their greatest common measure could only be the^Z>, itself; dividing therefore the cB, by the gD, and suppose it is contained exactly twice in it, or gD = cf=fB, and leaves no remainder ; then this gD, that is the last divisor, will be the great- est common measure possible, of the two numbers represented by AB, and Ci>; (it may be observed, that this operation is to be continued, as long as a remainder is obtained by these successive divisions.) JBecausethe^D, measures the cBy without remain- HEDirCTlOJfl^ OF FSACTIOWS. 59 der, and this cB measures the Cg, the ^D mea- sures also the CD ; th e Ac being a multiple of CD, is therefore also measured by gD, and the other part of ABf namely cB, being also measured by it, the whole AB is measured by gD, which measures also CD; therefore, it is their common measure, and as we have always proceeded by the greatest num- her, which possibly could divide the two numbers successively given, it is also the greatest common measure, as was required. If no divisor is found, except the unit, the two numbers have no greater common measure ,• that is, they are prime to each other. To apply this to numbers, let the fraction given be the following : 46 — ; dividing the denominator by the numerator, the las*, 153 being always the smaller number in a proper fraction, (of which alone there can here be question, because an improper fraction must first be reduced, by dividing b}' its denominator) we make the result of 163 18 — = 3 -| — that is : we obtain the first quotient 3, and the 46 45 18 remainder — ; this fraction inverted for the similar di- 45 45 9 rision, and the division executed gives — = 2 H ; or 18 18 9 the quotient 2, and the remainder — ; which treated as 18 18 above, gives — = 2, as last quotient ; and proves the 9 last divisor 9, to be the greatest common measure^ for evidently, the 9 ia 18 twicej in 46 fire times j in 163/ 60 KEDUCTION or FRACTIONS. seventeen times ; or we obtain for the operation of the greatest reduction of the fraction, 45: 9 5 = — ; which can be no farther divided, or re- 153 : 9 17 duced. In the habitual mode of writing, this example would $tand thus : 45)153(3 18)45(2 9)18(2 § 50. If by the foregoing process, no number is found dividing any one of the remainders, succes- sively resulting, without a remainder, except the unit, the numbers are said to have no common mea- sure; or they are what is called prime numbers to each other 9 and the fraction is not exactly reducible into smaller numbers. It is however evident, by the foregoing process : that the successive division has always approached nearer and nearer to the real value ; that the re- mainders have become successively smaller, and we might say, in respect to a given case, always less important. If the above operation had been interrupted at any one of the steps, it is clear, that the part neg- lected, would have been sl fraction of the last subdi- vision, made by the division of the last (juotient by the last remainder ; therefore, so much the smaller, the farther this division has been carried. Con- sidering this fraction as only an unit, having the last quotient for a divisor, and the preceding quo- tient as the whole number of these quantities, to which this fraction belonged, we shall have these reduced to the improper fraction of that denomina- Hon, by multiply ing this number with the denomi- .tor, and adding the numerator, that is the unit ; KEDUCTIOX OF TRACTIONS. 61 and tlic same process continued to the beginning, through all the quotients obtained, adding always the numerators last obtained, will ultimately give the numerator and the denominator of a fraction, approaching to the fraction which is intended to be approximated, as near as the divisions executed w ill admit ; that is with the neglect of only that fraction of the last subdivision which has been neglected, as ahove stated ; for if the division liad been carried, fully to the end, we would evidently have obtained tlie full value of the fraction, as shown above. And, as it appears by the order in which the division has been made, namely, the inversion of the numerator into a divisor, and the denominator into a dividend, the last number resulting from this continued mul- tiplication will be the denominator of the approxi- mate fraction, and that Immediately preceding will be the numerator. This operation may be expressed by the following rule : Divide the denominator by the numerator, and the last divisor by the remainder _,• always marking the quotient, as far as the approximation is intended to be carried ; (as in finding the greatest common mea- sure ;) then from the place where the operation is in- terrupted, make the continued product of all the quo- tients, adding unity to thefrst product, and afterwards always the last previous result, until the first quo- tient is armed at. The last number resulting will be the denominator of the approximate fraction, and the one immediately preceding it, the numerator. Example. Let the fraction 3^,^% ^^ approximated : the successive division will give, 98216\367459(3 f 72814)98215(1 25401)72814(2 22012)25401(1 3389)22012(6 1678)3389(2 33)1678(50 The success! i'e approximations will be : 6 6£ KEDUCTION OF FRACTIONS. 1st approximation. = 3 : 1, or the fraction | ^successive quotients, 3, 1, 2 3 2d „ < continued products, 1 1, 3, 2, 1 or — ( n f successive quotients, 3, 1, 2, 1 4 3d " < continued products, 16, 4, 3, 1, 1, or -^ I 15 i 3, 1, 2,1,6 27 4th" { 101,27,20,7,6,1, or ( 101 ( 3, 1, 2, 1, 6,2 58 5th'' { 217,58,43,15,13,2,1 or ( 217 and so on for any subsequent approximation. If the above division had been carried on to the last divisor, or unity, as the numbers are prime to each other, the original fraction would have been obtained again, thus : 3 I 1 j 2 1 1 I 6 |2|50 11|5ll|l|2j 367459|98215|72814j2540l|22012i3389|l678|33 | 28 | 5 j 3 | 2 | 1 the upper line being the successive quotients, and the lower the continued products, with the addition of the preceding number*, (or last numerator.) Suppose the following fraction, (to give one more ex ample,) which represents the numbers expressing the diameter and the circumference of a circle, 100000000 314169265 The division gives as follows : lOOOOOOOO'jS 14159265(3 ^ 14159265)100000000(7 I 885145)14159265(15 5307815 882090)885l45(t 3055)882090(288 27109 26690 2250)3055(1 805)2250(2 640)805(1 165 &c. ^ DECIMAL FRACTIONS < 65 Ut approximaiion. 3,1 or - 3 2d 3d 4th Approximation. i 3 , 7 ,15,1 ) 355, 113, 16, 1,1 J 3 ,7 i 22 , 7, 1 i 3 , 7 ,15 i 333, 106 , 15, 22 1 or lOG 333 113 or 355 5th Approximation, ^ 3 , 7 , 15 , 1 ,288 32650 102573 , 32650 , 4623 , 289 , 288 , 1 or 1Q2573 &c. CHAPTER V. ^K* Of Decimal Fractions, § 51. In explaining the decimal system of oiir usual arithmetic, we have seen ; that every figure designates a quantity ten times greater, when it stands one place farther to the left hand from the unit, than when in the place preceding it,, and therefore conversely, the figure in the next place I to the right hand is ten times smaller than the same figure in the next place to the left of it. If we continue this reasoning helow the unit of j whole numhers, and after having marked that place by a (,) and give denominations to the parts of unit, according to the same system, we shall get succes- sively for the resulting places, the denominations of tenth parts, (or j\), hundredth parts, (or,^o^), thou- sandth parts, (ory^Vo)» and so on, to any part or subdivision, however minute, of the decimal sys- tem J as for instance, 3,45672 would be 3 units^ 64 DECIMAI. FRACTIONSi 4 tenths, (or VV), 5 hundredths, (t|o)» ^ thou- andths, (loVo)? 7^ ten thousandths, 2 hundred thousandths, or the whole would be, 4 5 6 7 2 3 _1- __ J j j L_ 10 100 1000 10,000 100,000 where it is evident that the writing of the denomi- nators can be spared, because the successive di- minution of value of the places is known by the system ; therefore the usual, and easiest, way of reading these fractions is, after having mentioned the wiiole numbers, to mention the (,) and read the subsequent numbers simply as tbey occur, leaving the denominations out, as understood from the prin- ciples of the system. We have hence an easy mode of expressing any fraction in the same system as the whole numbers, either in full or by approxima- tions to any desiralile extent. § 52, AVe have already seen that proper vulgar fractions result from a remainder oi a division smaller than the divisor, as a mere expression of the unexecuted or unexecutable division ; by de- cimal fractions we are, on the contrary, enabled to express these quantities, by continuing the division, according to the same law that is used in the sys- tem of numbers ; and all the difference between op- erations with these quantities, and those with whole juimbers, will evidently consist in pointing out the place where the whole numbers end, and these fractions begin ; all the rules which will be found hereafter, for the mechanical execution of the four rules of arithmetic in decimal fractions, will therefore merely relate to the determination of the place of the decimal mark. To continue the division that is required to ob- tain the decimal fraction, after the number in the dividend has become smaller than the divisor, it becomes necessary, to reduce the remainder to the same kind of unit which will follow in the quotient. DECIMAIi FRACTIONS. 65 and as this will be ten times smaller^ the dividend ^vill represent in it a ten times larger number ; to do this we have only to multiply it by ten, which is done by the simple addition of a (0) on its right hand side, which will enable us to continue the di- sion ; as this takes place at every step, it is only required to repeat it also at every step, as in the following example : Let the division be continued, 34721 = 46,5058964, &c. 763 4201 3860 4500 6850 7360 4930 3520 468, &c. Here, after having obtained 45 as a whole number, instead of expressing the fraction ^|| as a vulgar fraction, an has been added to the remainder 386, which pre- sents 3860, and is again divisible by the divisor 763 ; at this place therefore, or after 45 in the quotient, the dis- tinctive mark or (,) was placed, and the division con- tinued, as in common numbers, with the constant addi- tion of a 0, to every remainder, to make the continued division possible ; at the remainder 45, the addition of one 0, making a dividend still smallerjthan the divisor, the quotient was 0, as in any other division, and the addition of a second 0, making 4500 in the dividend, gave the quotient 6, proceeding in all such cases as has been shown in common division ; the division, which here stops at the seventh place of decimals, might perhaps continue in this case without ever closing. § 53. By the same principle we can express any vulgar fraction in fractions, either terminated, or continued as far as desired, or solve the FiioBLEM. To reduce vulgar fractions into ded- 6 # tit) DECIMAL TRACTIONS. dmal fractions. Put an 0, in the place of Ithe unit in the quotient^ with a (,) after it, and an 0, at the right hand side of the numerator, divide as iti com- mon division f adding to every remainder an 0, and C07itinui7ig this division as far as desired, or until re- curring numbers occur. 3,0 Example. — =0,42857142 7 20 60 40 50 10 30 20 6 Placing the in the quotient, and in the numerator^ the division is continued as in common numbers, with the constant addition of a (0) to the remainder, until we again meet the same quotient, 42, and remainders 2 and 6, which had been obtained at first, which are called re- curring numbers, and indicates that the same series of numbers would repeat themselves ; which may therefore be done, as fat as required, without calculation. Such se- ries of recurring numbers are called circulating decimals. 1 As an other example : reduce - = 0,3333 3 10 10 Here evidently we constantly obtain from the very begin- ning, the same quotients and remainders ; the calculation ne^ed therefore, not be continued. Of this kind are a number of other fractions, called repeating decimals. In the following example, we obtain a complete expres- sion in decimals, 10 — =0,125 ; which terminates the division of itself. 8 20 40 1 DBCIMAI FRACTIOXS. 67 § 54. ADDITION OF DECIMAL FRAC^ TIONS.*— P^acc the numbers under each other, in such a manner that the units may stand under the units, and all the numbers, at equal distances, to the right, or to the left of the units, may fall under each oilier ; then add them as in common numbers, begin- ning at the right hand figure, and place the decimal mark in the result, under the decimal mark of the given numbers. This rule is evident, from the ahove considera- tions ; for the numbers heing all of the same system, the carrying of the individual sum will be the same as in whole numbers, and each kind of number will thereby stand in its proper place ; therefore, also, the decimal mark will not change its position. If any one of the given numbers should have no whole numbers, in order to avoid all ambiguity, the place of the unit must be filled with a 0. Example. To make the sum, or execute : 3,4612 + 2,134891 3,4612 21,34891 ^4,81011 Write them as stated above ; add as in common numbers ; place the decimal in the sum under the decimal of the parts. Example of several numbers ; 13,76094 0,3809673 142,012 0,39052 156,6444273 § 55. SUBTRACTION OF DECIMALS Place the numbers under each other, as in Addition, (placing the smaller below,) and subtract as in com- 68 DECIMAL FRACTIONS. mon suhtradmi, beginning at the right hand figure^ and place the decimal mark in the difference under the decimal mark of the two given numbers. This rule is evident from the simple considera- tion : that the subtraction being the same as that of whole numbers, equal kinds subtract from equals, the borrowing goes according to the same princi- ples, and the place of the decimal mark does not undergo any change. Examfle, 3,490864-2,74962, write thus : 3,490864 2,74962 0,741244 3.490864 There being no number to subtract from the first 4, it is unchanged in the remainder, and the other numbers follow, as in common subtraction. The proof of this subtraction is evidently per- formed in the same way as in whole numbers, by adding the remainder and the subtracted number, which should give for their sum the number from which the subtraction has been made, as in the ex- ample. If the number to be subtracted has more deci- mals than the number from which it is to be sub- tracted, it is evident : that the vacant places must be considered as having each an 0, by which the sub- traction of the lower number is made by constant borrowing towards the left, until we reach to the first significant decimal. 0,9832 0,4986735 9,4846265 0,9832000 ^P DECIMAI rEACTIONS. 69 Here for the subtraction of the first 5, the supposed lending from before has given 10, and the remainder be- came 5 ; then the next lending being again from before, but one borrowed from the 10, leaving only 9, the 3 is subtracted from 9, remainder 6 ; in like manner in the next we obtained 9 — 7 = 2; and then the first number . above, namely the 2, appeared as diminished by 1, and (he next subtraction is made from 11, by borrowing again as in whole numbers : the rest is evident § 56. MULTIPLICATION OF DECIMAL FRACTIONS. From the principles of Decimal Fractions it is evident : that their multiplication can, in itself, have no other rule than that of whole num- bers. In respect to the value of the resulting deci- mals it is easy to observe, that as they are fractions with the denominator 10, each of its preceding number, they keep this quality in the result, and therefore present always in their product the pro- duct of these quotients, which, though not express- ed as a denominator, is indicated by the place as- signed to the decimal mark. This principle evi- dently causes the decimal mark to recede one place for every time that it is multiplied by a decimal, and in the multiplication of decimals, distant from each other, as many places as both decimals to- gether indicate ; because, the product of any tens, hundreds, thousands, &c., into any other tens, hun- dreds, thousands, &c., will be the unii followed by as many O's as are contained in the supposed denomi- nators of the two fractions together. This proves therefore, the following rule for multiplying decimal fractions. Multiply the factors together, as in whole numherSf (did cut off as many places of decimals, from the right hand side towards the left, as there are decimals ill both the factors, and place there the decimal mark. To execute this multiplication, it is best to begin by the unit, and multiply by the other numbers 70 DECIMAL TRACTIONS. to the left, and then to the right, advancing the re- sult in the first case to the left, and receding with the results of the decimals towards the right side, in the second case, in that order in which the mul- tiplication with whole numhers in the same ranks would indicate ; the first result, that is the result of the unit, will by this process, at once determine the place of the decimal. Example. Multiply 3,476 x 6,82 ; thus : 3,476 6,82 17,380 2, 7808 7962 20, 24032 Here the multiplication by 5, as units, has given to every number in the multiplication the same place as it occu- pied before ; the following numbers having receded to- wards the right, according to the rank they naturally have already, occupied their proper place, and we have obtained in the result 6 places of decimals, exactly as many as both factors have together, and as the rule above prescribes ; (for which therefore it might form a a practical deduction or proof.) Let the following examples be executed, to illustrate further the practical application and its consequences. 36, 462 0, 4937 172,7892 66,32 0, 6378 0, 0624 14,6808 3, 28068 109356 256164 1036, 7352 8639, 460 61,83674 3, 465784 0, 032668 10766 21512 0, 03355872 17,9963524 9731,488624 DECIMAIi raACTIONS. 71 The first example above gives a result that makes the product recede one place in the decimals, because the first multiplier is a decimal, this determining the decimal place, all the others follow by themselves, the results re- ceding always one place, and ultimately giving as many places of decimals as there are in both the factors taken together. When in the second example the multiplica- tion by 6, as unit, was performed, each number resulting kept the place it occupied in the multiplicand, the deci- msd place being determined by this ; the next multiplica- tion with 5, or properly 60, gives the advance towards the left, according to common multiplication of whole numbers, the two decimals being made to recede one place farther to the right, each tends to keep the final re- sult in its proper order, the number of decimals are thus determined by the mere addition, and are conformable to the above rule. 6 6 In the third example : the into the — ; gives, evi- 100- 10 30 dently, : which assigns to this result the place 1000 seen in the example; the others follow by themselves ; but if we had not attended to this, the rule above would give the coincidence with the result obtained, as may be easily seen. § 57, It is most generally needless to calculate to the^full extent of all' the decimals of both factors, the one or the other factor commonly indicates the number of decimals to which it is intended to carry the accuracy, and we may dispense with the smal- ler decimals, so much the rather, as they will at all events not be absolutely accurate, if the decimal fractions are not themselves exact, in consequence of the absence of the smaller decimals, that would influence the places taken beyond the lowest that is given in one or the other factor, for this reason it is desirable to hav£ an easy, and exact way of ex- 75 DECIMAI. FRACTIOTV-S- editing this abridged multiplication, which is as follows : Multiply by the greatest iiumher of the multvplier first, and determine the place of the decimal ; (as in the pre- ceding rule ;) then mark this number, and also the lowest decimal of the multiplicand^ then take the next lower number of the multiplier, and multiply all the multiplicand by it, taking from the product of the de- cimal marked off, only the part which is to be carried forward to the next place, using the ten nearest to the result, write the product under the foregoing, so that the first figure to the right comes under the first figure of the foregoing product; thus continue as long as there are figures in the multiplier, always marking off one figure in the multiplicand for each factor of me multiplier, and making the addition of the carry- ing as before; the decimal mark will take its place ac- cording to the determination of the first number used as a multiplier. Example. The first of those given above, executed according to this rule, will show that in this manner re- gularity is insured, and the deviation from the full result obtained above will not extend farther than the last place of figures, or rather only the next after it, as it does not differ a whole unit of that place. 36,452, 0,4937 14,6808 3,2807 1094 17,9964 4 The first line in the product of — into the multiplicand. 10 DIVISIOX OF DECIMAI FRACTIOXS. 73 In the second line we had to carry for the product of 9x2= 18, which being nearer to 20 than to 10, the tens to be carried were two. So that when we afterwards made 9 X 6 = 45, we added 2, making 47, the 7 being placed, the 4 carried, and the operation continued as in common numbers ; the rest of the operation is exactly the same with respect to the remaining numbers of the multipli- cand ; for in the third line we had 3x4 = 12, and the 3 X 52 from before, giving 2 to carry, rather than only 1 , we added 2, that is, we made 1 2 -|- 2 = 1 4, then continued 3 x 6=18, and 1 carried, = 19, and so on : the addition is as before. More examples will occur in the practical part. § 58. DIVISION OF DECIMAL FRAC- TIONS. From the manner the origin of Decimal Fractions has been deduced, it has already been seen : that the decimals began whenever the divisor was larger than the dividend, or which is the same thing, when an became necessary to be added ; in other words, when it became necessary to recede towards the right farther than the quantities of the dividend furnished numbers of the A^alue of the divisor ; as many such steps, therefore, as it may become ne- cessary to make until a figure, actually significant, can be obtained in the quotient, as many O's will precede it, the of the unity place being counted as the first; or the place of the first^significant deci- mal will be that indicated by the number of steps it was necessary to recede. It will be most easy in this place, and will furnish us with the clearest method of accounting for the necessary steps, to proceed from the relation of the unit in the dividend and the divisor, as a leading priiiciple for the determination of the decimal mark ; and in cases where no whole number is obtained in the quotient, it will be entirely referable to common division continued to decimals, as employed by us 7 74 DIVISION OF DECIMAL FRACTIONS. in elucidating the principle of decimal fractions. The rule resulting will therefore be this : Begin the division as in whole numbers , and place thejirst decimal resulting , as many places to the right of the decimal mark, as it has been necessary to re- cede from the first figure in the divisor, to obtain in the dividend a number sufficiently large to be divided by the divisor. Example, Divide 3, 46921 = 0,80173587 + &c. 4,327 7610 32830 25410 37750 31340 1059 + &c. Here, it is evident, that 4 not being contained in 3, whatever may be the following decimals, the divisor can- not be contained in the dividend, and there is no whole number in the quotient, therefore a is written in the place of the unit ; the next receding in the dividend giving a significant figure ; this falls in the first place of decimals. And now the division being performed, as in whole numbers, and the added always when there were no numbers more to be taken down from the dividend, the division can be carried as far as may be desired, for the decimals determine themselves without any further care. But it is evident here again : that the division to a greater extent than is warranted by the numbers given, will not give the full accuracy, when the decimals are not determined ones, and only approximations ; for the O's set down should evidently always be the figures which would have followed in the dividend ; and the products to be subtracted, should be affected by the lower deci- mals which are missing in the divisor. The two Examples that follow, will show more of this application. DIVISIOX OF DECIMAL FRACTIONS. 70 4,27G9 0,00042769 = 0,007537+&c. = 0,007537+ .j67,432 0,0567432 [&c. 3 048760 3048760 2116000 2116000 4137040 * 4137040 165016 165016 These ex.imples, showing both cases of whole num- bers and decimals, with a denominator exceecling the numerator, are both equ.il applications of the rule found above : and their result is th^^ same, as the one is evidently the product of* the other, by a multiple of 10 ; in both cases it was necessary to recede three steps, to obtain a signitir;mt numlter in the quotient; therefore, the first numher in the quotient is rn the third place of decimals ; or, we might say, it was three times impossi- ble to divide ; therefore we have three O's, the Oof the unit place being counted, as of course, because the first time the division was not posj-ible, was that of whole units. In the same way as we have found in vulgar fractions : that one fraction may be contained in another fraction a whole number of times, as well as one whole number in another ; so, of course, this also takes place in decimal fractions, as in the two followinir examples, which again present the same quotient, as the two last. 452,9673a . 0,045296738 =29,580770 + &c. = 29,580770 + Szc. 15,3129 0,0U1531J9 146 7093 1467093 V, 89328 889328 1 236U30 1236S30 1179!'.0 n79U0 1078930 1078930 70270 + &:c. 70270 + fcc. These examples first admitted of division by whole numbers, because 15 is evidently contained in 452 a whole number of times, which we found to be 29 : then the decimals began ; these two whole numbers are evi- dently easily determined, as the divisor has only two pla- ces of figures, and the dividend 3 in the corresponding places of both examples ; and these are already twice divisible, before recourse is had to the decimals from the 76 DEXOMINATE FRACTIOlSrS. Jividend, for the subtraction of the products of the whole numbers of the divisor into the quotient. Remark. If it is required to perform any one of the four rules o^' arithmetic, between decimal and vulgar frac- tion?, the decimal fractions are to be considered as whole numbers, paying, of course, due attention to the place of the decimal mark ; this will therefore need no special explanation. But it will, in many cases, be most conve- nient to reduce the vulgar fraction into a decimal frac- tion, and then proceed upon the principles of decimal fractions. This being therefore a subject depending on the judgment of the calculator, the principles of which have been explained sufficiently, it will not need here to be treated separately. CHAPTER V. Of Denominate Fractions, § 59. I take the liberty of calling all those subdi- visions of an accepted unit that have received par- ticular names, and which properly form fractions of this unit, with a certain conventional denomina- tor, that is therefore always understood. Deno- minate Fractions ; such are all the subdivisions of measures of any kind; of length, surface, or solid- ity, weights, money, time, &c. In order to make use of these fractions in arith- metic, it is necessary to know their conventional denominators, or to be able to say how many units of each subdivision make a whole, or a unit of a higher subdivision; as for instance, the general di- vision of pound (money) into shillings, pence, and farthings ; where the general habit is that 1 shil- ling = ^\ of a pound ; 1 denier or penny = /^ of a shilling; 1 farthing = ^ of a penny. Or, in the other manner of expressing it, £1 := 20s; Is = 12rf; Id = 4/. DENOMINATE FRACTIONS. 77 Of these subdivisions, old habits, unconnected in their origin, and therefore devoid of system, have introduced such a variety, that it is necessary to have tables in order to recall them to memory; such tables will be placed at the end of this book, to which 1 shall add tlic approximate or full decimal expression of the unit of each subdivision in the other, and in the wliolc, as it is evidently possible to express them all in decimal fractions, either ex- actly or approximately. Certain signs have been given to all these subdivisions, to abridge their notation ; these will be learnt from the tables. In thus stepping aside from the simple theory of a system to a mere practical habit, we shall soon feel what an advantage it would be in all transac- tions where quantity is concerned, to have a regu- lar and unique system for them all ; but the attempt, so often made, has always been frustrated, by the unwillingness of men engaged only in their private concerns, to all mental motion or exercise, not di- rectly advancing their private aims. Similar sys- tems had been in use in common arithmetic, before the adoption of the decimal system of numeration, the advantages of which soon expelled them from theoretic arithmetic. It is evident that the difficulties to be vanquished in this part of arithmetic, consist only in the atten- tion that is required to be paid to the effect of the irregular system of subdivision, which determines the principles of what may be called carrying from one denomination to the other ; the rules discovered hereafter, therefore, chiefly refer to this operation ; they will not need any proof, as they have only the arbitrary subdivisions for their principle, and for* their aim^ to facilitate the several processes, '^i'hey will therefore be given simply, witli a few exani- jdes for illustration, their proof, as far as arithme- tic principles are concerned, lying always in the 7 # TS DENOMINATE FRACTIONS. principles of calculation already explained; and their combination will be reserved for the practical part of this treatise. §G0. ADDITION OF DENOMINATE FRAC- TIONS. Rule. Jf^riie the numbers of each deno- mination nnder each other, distinguishing them by points ; add them as whole numbers, beginning at the most nght hand figures, and carry from one deno- mination to the other, according to the value of the subdivision, in parts of the next superior quantity, that is, by dividing the sum obtained by the deno- minator of the fraction, indicated by the subdivision. Example — in feet, inches, and tenths of inches. 12 in. = I f. affords the principle of carrying from inches into feet ; the mode oi carrying the tenths of inches being as explained in decimal arithmetic. To add 12 f. 7,6in. -fSf. 4,9 in. -f 2 f. 11,2 in. f. in. 12. 7,6 3. 4,9 2. 11,2 18. 11,7 Here tlie sum of inches being 23, 12 are taken away to carry as one unit to the feet ; there remain 1 1,7 inches; the rest is exactly like the addition of whole numbers. Example in weight, of pounds Troy ; the subdivisions of which are, 1 lb. = 12 oz ; 1 oz. = 20 dwt ; 1 dwt. = 24 gr. lb. oz. dwt.gr. To add 7. 10. 14. 12 19. 6. 17. 14 6. 11. 15. 19 36. 5. 7.21 Adding the first column to the right, or of grains, what is over 24 gives 21 to set under this denomina- tion, and 1 dwt. is carried to the next denomination, or DENOMINATE FRACTIONS. 79 Jvvt. column. This second culumn being added, gives 47 dvvt. = 2 oz. + 7 dwt ; therefore the 7 dwt. are placed under that column, and 2 oz are carried to the ounces ; the ounces added, with the carrying, give 29 oz. = 2 lb -|- o oz ; the latter placed under ounces and the 2 lb. carried give ultimatel}', by the addition ot' the last column, the number of whole units of pounds 3C; then the whole sum is txpressed. These examples may suffice for the present, as more will appear in the practical part. § bl. SUB IRACTION OF DENOMINATE FRACTIONS. Rule. JFriie the denominations of the subdivisions of the quantity to be subtracted under the same denominations in the quantity whence they are to be subtracted^ and siibtract in each column the lower from the upper ^ beginning at the lowest denomination; and in borrowing from a superior denomination^ give to the unit borrowed the value it has in the lower suhdivision in which it is used. This rule is evident froin the simple inversion of what is directed to be done in addition, and is ana- logous to the rule in the decimal system, as is evi- dent. These subtractions evidently admit of proof, by the addition of the remainder to the number subtracted, as is the case in all subtractions. Example in feet, inches, and tenths, f. in. 17. 7,3 8. 9,6 8.1 9,7 17. 7,3 Example in pounds T^03^ lb. oz. . dwt •g'"' 22. 7. 6. 6 14. 9. 16. 12 7. 9. 9. 17 7. 6. 80 DENOMINATE FRACTIONS. The borrowing in the hrsi example, from the feet to the inches, h.is given 12 -f 6 in. = 18 in from which 9 taken left 9, the decimals having been IreHted as usual, then 16 — 8 ft. = 8 ft. gave the remainder of the whole feet. So also in the second example, ive had first, by borrowing to the Jiraiiijj, 24 + 5 ~ 12 = 17 (grains) ; then in the second column 20 -f- 6— 16 = 9 pennyweights, in the third column 12+6 — 9 = 9 ounces, and lastly 21 — 14 = 7 poiHKSg, the borrowing having been made throughout accordmg to the dictates of the arbitrary sub- division, nnd the numbers from which the units were successively borrowed, having always been diminished by a unit § 62. MULTIPLICATION OF DENOMI- NATE FRAi 'ilONS. From the two different ways in which it has been seen that denominate fractions may be compared, namely, as units, of which a certain number form another unit, or as iVactions of the preceding, or rather the highest unit, with a certain denominator understood, it may be inferred : that the multiplication of them can be executed in two different ways. The first, using the given numbers as imits, w ill form fractions, of a denomination adapted to the conventional system of subdivision, which mode is exactly analogous to what has been done in decimal fractions. This is often called Cross Multijdicatiun, The second consists in using the lower units that are given as fractional parts of the whole, and takes the products of the multiplier into the multiplicand as such, distributed in parts that are best adapted to the easy division of the multiplicand, and ex- presses the results in the same unit, and its subdivi- sions ; the final result is obtained by the addition of the products j this method is usually called Practice. In the application of multiplication to this species of quantities, we are limited by their nature to those which are capable of producing things really exist- ing in nature ; as lineal measures into each other, ^^whicli produce surfaces, and these again into lineal measures, which produce solids. Or to such as arc of a different nature from each other; and their relative possibility of being compared witii each other renilers them fit to give a result existing in nature, as for instance, money into weights, or measures, of any kind, where only the numbers or quantities are usetl, and the things themselves arc considered as capable of representing each other, that is : conventionally comparable. But such quantities as money into money, weight into weight, being incapable of producing any pos- sible result, cannot be objects of this or any calcu- lation ; and to the latter only the second method is conveniently applicable, because the resulting infe- rior fractions become of an irregular computation, if cross multiplication was applied. § 63. To multiply by the first method, or Cross Mtdtiplicationf we have the following rule, grounded upon the general principles of fractions. Multiply all the columns of the multiplicand succes- sively by all the numbers of the columns of the multi- plier, beginning at the right hand figure^ and carry ^ in each passage to a higher columiu according to the value of the subdivisions made use of ; place the results of equal quantities under each other ; their sum in the result will give the whole quantities and the subdivisions, according to the same scale as the preceding subdivisions ; that is, the denominators be- coming equally products cf the denominators indicated by the subdivisions. Example, To multiply 7 ft. 2 in. X 6 ft. 5 in. ft. in. 7. 2 6. 6 2. 11. 10 43. 46. 11. 10 82 DENOMINATE FRACTIONS. Cy muUiphing here 2 in. into 5 in. we have properly 2 6 10 made — x — = ■ — ; therefore we have obtained a 12 12 144 subdivision of (he unit one degree lower thnn those em- ]>loyed, iis in decinrial fnictions ; therefore, also, in writ- ing; the rcj^ult, this has been removed one step more to the rij^ht. In making 5 35 11 7 ft. X 5 in. = 7 X - ft. = — == 2 H , 12 12 12 ^ve have obtained first twelfths, and then, by reductiou to whole numbers, 2 whole quantities, and — of a foot; the 12 2 12 result of 2 in. X 6 ft. = 6 X — = — = 1, hasgiren, bj 12 12 the same principles, one foot to carry to the next result ; then, by the uiiihiple of the feet into the feet, ivith this addition, we have 6 X 7 -j- 1 = 43 feet ; the final sum is obtained as in addition, but presents an inferior subdivision one degree lower in the same scale of subdi- vision that is used, or twelfths into twelves ; thence the 11 10 above result is = 45 -^ ■— + • 12 144 If we multiply again (for example the above result) !)y a lineal dimension, we shall obtain a solid, expressed in the same system of subdivision ; thus ; ft. in. 1. 45.11.10 5. 7 26. 9. 10. 10 229. 11. 2 25G. 9. 0. 10 Mere we again obtain, by the very same process as above. DENOMINATE FRACTIONS. 83 a denomination still lower in the same scale of subdivi- 9 10 sion, or = 256 -\ 1 . 12 12 X 12 X 12 § 64. Multiplication of Denominate Fractions as Fractional Parts of the Whole^ or Practice. The nature of this operation, as stated above, evidently leads to the rule. Multiply the whole and fractional parts of the multi- plicand by the whole numbers of the multiplier ; then distribute the fractional parts of the multiplier into such as are most easily taken; take such parts of the whole and fractional part of the multiplicand as will be indicated by them, and add all these parts for the final result. By this operation the products of the different fractions, distributed for the convenience of the ope- ration, being partially taken, the proof of the rule lies in the simple multiplication of fractions, and this is only repeated ; it is generally convenient to divide the fractions of the multiplier so that the smaller subsequent parts are again fractions of the first. For the purpose of comparison with the other mode, we shall use the example already given, ft. in. 7. 2 6. 5 43. 2. H =z f 4 0. H = I 4 X i • = 1 m. ^ 45.111 Here, after multiplying the 7 ft. 2 in. by 6, the whole ^Timber, giving 12 in. -|- 42 ft. = 43 ft. the 5 in. ofthe mul- )Uer are divided into 4 in. =4 ft, and 1 in. = i x 4, in. 84 DEXOMINATE FRACTIONS. or 1^ X I ft, the i x 7 ft. giving 2 ft. and 1 ft. remaining, which gives 12 in. to add to the 2 in., the I of the (12 + 2) in. = 14 in. being taken, gives 4^ in., as in the second line ; the second fractional part being ] of that, or 2 of 2 ft. + 42 in. = 28| in., the I of which is 7 -\ 3X4 = 7 +i in., gives the third line ; the addition of the pro- ducts is easily understood from the addition effractions, as I + I = f + i = f , and the rest is like the addition of denominate fractions. Let this example be continued, as was done in the for- mer case. ft. in. 45.1 If 5. 7 229. Ill 111 = 1=6 S Q7 1 — 1 = 6 in. } „ . |x^=lin.r-- 256. 9^'^ We have 5 x | = V *= ^ -f i, for the product of the whole number 5 into the fraction 9 then 5 x 11+4 = 59 in. = 4 ft. 11 in., from the whole into the inches, with the carrying, the rest then as whole numbers, or 45 X 6 -+- 4 feet. For the 7 in. we take 6 in. = i ft, and J in. = i. X ^ ft. ; therefore h x (45 ft. llf in ) giving the second line easily ; and the third line being J- of that, presents 2_2 ft. = 3 ft. + %« in. ; the second part or 48 in. + 11 in. . = 9 in. + I ; and the fractional part of 6 ' " this, making in ^\ as the upper fraction is }2, which add- 11 " 71 ed to , gives = ^. 6 X 12 12 X 6 The addition is executed in the fractional part by re- ducing them all to the denominator 72, which is their t DEN^OMINATE FRACTIONS. 85 )mmon multiplier ; the whole inches are then carried J the next addition, which is executed as shown in its place. If any such denominate fraction of any kind is to be multiplied by a whole number, it is evident, that nothing is required but to multiply each of the parts by this number successively, in the above order ; carrying according to the principle of the given arbitrary subdivision, exactly as was done with the first, or whole number above. This is evidently admissible with any subdivision, and needs no separate explanation, as it ha;? actually been already given. § 65. The two preceding modes of performing the multiplication of denominate fractions being evi- dently cumbersome when applied to great calcula- tions, and when the fractional parts, or lower deno- minations, are not easy aliquot parts of the whole, it will be often most convenient, to reduce the factors to whole and decimal fractions, by the me- thods taught in their proper place, and then to pro- ceed by multiplication of decimals ; for this purpose the fractions would be marked with their divisors, according to the habitual subdivision. Example. Let the preceding question be executed in- this way, and by abridged multipHcation of decimals ; wo, obtain as follows : 7 + ^2^ == 7^1666 . . . . : 6 + /^ = 6,4166G Performing the abridged multiplication thus : 6,41666 7,16666 44,91666 64167 38500 3850 385 38 4 45,98610 S6 DEKTOMIN-ATE FRACTIONS, As both fractions are repeating, having a continued repeti- tion of the 6, the products of these have been inserted, as long as they have any influence in the numbers pre- served, by repeating the first product of 6, receding every time one place more to the right ; and in the last numbers of the products always carrying from a product of a previous 6 ; this has been performed throughout, by augmenting the last figure one unit. The second multiplication, treated in the same way, will, when executed, give the following Example. 5 -f ^2 = 6,58333, the series of 3 be- ing again continued. 45,9861 6,6833 229,9305 22,9330 3,6789 1379 138 14 1 256,7566 &c. To compare these three results together will be mofev easily done by reducing the two former ones to decimals also ; thus we obtain by § 60, 9 10 256 + - -} =: 256,75578 &c. 12 1728 9 6 by §64, 256 ^ H = 256,75578 &c. 12 864 by the decimals above, = 256,7556 The difference of nearly two units in the fourth decimal, or the ten thousandth parts, is owing to the neglect of the last decimals of the factors, which of course gives a DENOMINATE FRACTIONS. 87 smaller result, but which is in most cases sufficiently ac- curate ; a farther extension of the decimals would of course cause a nearer approximation to the other result. § 66. DIVISION OF DENOMINATE FRAC- TIONS. The remark made in relation to the mul- tiplication of this kind of fractions, upon the incon- veniences of the operation, and its being applicable, generally speaking, to lineal dimensions only, ap- plies still more forcibly to their division; in all other cases the quotients are not required in the same denomination of subdivisions ; and in cases where the divisor and the dividend are of a dilFerent kind of quantity, they are in reality impossible in nature. But in the case of lineal dimensions, which produce by the first multiplication superficial mag^ nitudes, and in a second solids, as both objects really exist in nature, it may sometimes be desira- ble to have a quotient given in the same denominate fractions, or subdivisions. In all cases, therefore, where such a division oc- curs in the course of a calculation, the nature of the quantities concerned are left out of consideration, and the quotient inquired into is considered as a mere number. To do this two ways present them- selves ; either to reduce every different denomination of the numbers concerned to the lowest denomina- tion of denominate fractions contained in them, and divide the whole numbers resulting ; the quo- tient will give a result in units of the whole ("not of the subdivision) employed, because it expresses the value of the general fraction expressed by the divi- sion, which is itself independent of the subdivision employed in the calculation. Or the denominate fractions may be reduced into decimals of the whole quantity, as seen in the pre- ceding §, and the division of these will give exactly the same result, as the reduction to the lowest deno- mination ; because the quotient resulting gives also ^^ DENOMINATE ABACTIONS. liere the actual value of the fraction expressed by the division. § 6r. To perform a Division of Denominate Frac- tions, As this may be desired, in lineal dimensions^ it will he proper to give an appropriate general rule ; as follows : Find the whole number which, multiplied into the divisor, will give a product nearest below the divi- dend, and divide by it as in common division, only minding the transfer or carrying from one denomina- tion to the other, according to the principle of the deno- ininate fractioji; then reduce the remainder to the next lower denominatioii, and multiply the product by the denominator of the denominate fraction ; reduce also the whole divisor to the next lower denominate fraction, and divide the last obtained dividend by it; the result will give a number expressed in this next lower denomination ; thus continue fo the end of all the desired subdivisions* The reason of this rule is evident in its first stej), from common division of whole numhers ; in the second, and following steps, the multiplication of the remainder, after reduction by the denominator of the denominate fraction, is necessary to give by the division a result expressed in units of this lower division ; in the same manner as for decimal frac- tions, a was added to every remainder, to pro- duce a quotient of the next lower rank of decimals^ because this produced a multiplication by 10. Example, To divide ft. in. 17^9^ = 2ft. Sin. +5\ 7. 10 15. 8 ^. 1 = 25 in. multiplied by 12 DENOMINATE TRACTIOKS. 89 300 in. 12 = 3 in. + — Divisor reduced = 94 94 282 12 The first quotient found here, or the feet, is 2 ; then the remainder, 2 ft. 1 in., is reduced, and multiplied by 12, to give the next denominate fraction, by the division with the reduced divisor, or 94, which gives 3 in., and the fraction which is here left, but could be reduced again to twelfth parts, as the next subdivision, by the same opera- tion as the inches were obtained, if desired. It must be observed here : that in the dividend the 9 were treated as twelfth parts of the square foot, which are not inches cubic ; if they were such, they would present the deno- minator 12 X 12 = 144, as may easily be judged, by reflecting upon the multiplication shown above, or be- cause 1 ft. = 12 in. gives 1 ft. square = 12 in. X 12 in. = 144 square inches. To execute the same Operation or Division btj the two other Methods, The process of reduction to the lowest denomination, or to whole and decimal num- bers, is evident from the principles of division taught for the two cases, in their proper places. The nbove example would stand in them as follows : 1st. By reduction to the lowest denomination, ft. in. 17^% 213 = = 2, 2659 ft. = 2 ft. 3, 1908 in. 7. 10 94 this last by multiplying the decimals by 12, the denomi- nator of the denominate fraction. 2d. By the whole numbers, and the value of the sub- divisions in decimals, 17-^ 17 75 — ^ = — '- = 2, 2660 = 2 ft. 3, 192 in. 7. 10 7, 8333 le first result will be somewhat too small, on account 8# 90 DEJSrOMINATB FRACTIONS. of the discontinued division, the second somewhat too large, on account of the discontinued series of 3 in the divisor, which, being smaller, leaves the quotient to be- come somewhat larger. The whole of the examples in denominate frac- tions, and particularly the latter ones, show : that the calculation of denominate fractions properly be- longs to the applied or practical part of arithmetic, which is intended to be treated in the next chapter ; it has however appeared proper to treat of their principles here, considering them as fractions of a particular nature, as these considerations tend to illustrate the general view of fractions; to enter more minutely into details is however the province of the practical part, where more examples will ap- pear, and where it will become evident to every attentive peruser of this work : that the proper un- derstanding of the principles of arithmetic will sug- gest to him in every case the ideas which will lead him to the most judicious, accurate, and short way to execute calculations, implying such detail cases. PART II. PRACTICAL APPLICATIONS OF THE POUR RULES Oy ARITHMETIC. CHAPTER I. General Principles of the Application of the Four Rules of Arithmetic, § 68. In the previous chapters have been deduced from the first principles of the combination of quan- tity and numbers what are called the Four Rules of Arithmetic, and they have been applied to the dif- ferent forms in which quantities are presented, namely : as whole numbers of units, or as parts of the same ; and these latter expressed either by their general relation to the unit, as in Vulgar Fractions, or by a continuation of the decimal system below the unit, as in Decimal Fractions, or as arbitrary subdivisions under the name of Denominate Frac- tions. The preceding part may therefore be considered as the theory of the four rules of arithmetic ; it will already present the solution of a great num- ber of the questions arising in common life from daily intercourse or occupation. Though this ap- plication might be made by the teacher, it may not be improper, particularly for such persons as should wish to undertake the study of arithmetic by them- selves, to give a few leading ideas to guide them in the proper choice of the rule for a certain given case, together with some examples. ^ 9^ PEACTICAI. APPLICATIONS OP THE 5 69. Under the head of Addition will come : all such questions, where quantities of the same kind are to be counted together, as has been seen to be the origin of this first rule. It is of course impossi- ble, to add quantities of different kinds together un- der any denomination than as mere things ; and this remark, simple as it is, may escape in certain cases. We have seen, for example, in fractions, that we were compelled to make such changes in the deno- minators as produced the effect, of reducing the quantities which are of a different kind, on account of their being different parts of the unit, to quanti- ties of the same kind, or denomination, before they could be added. That all tliis applies equally to Subtraction, is evident from the principle, that it is only the oppo- site operation of Addition. § 70. So for example. A farmer, making the enumeration of his live stock, may add them either under their different denominations, as different kinds, or in sum total. Suppose, therefore, a farmer had his live stock distri- buted in different lots of ground, as follows : In the door yard are 3 cows, each with a calf, 2 horses, and 4 pigs. In the meadow he has 4 oxen, 6 cows, and 3 young horses. In the field, a flock of 35 sheep, 5 cows, and 4 calves. He lets run in the woods, 9 pigs, 7 cows, 4 young oxen, and 2 horses. We may ask here, first, the sum of all his live stock, which will comprehend all what is abot^e under one sum : thus : FOLK BUIES OF ARITHMETIC. 2 horses 3 cows 3 calves 4 pigs 4 oxen 6 cows 3 horses 35 sheep ' 5 cows 4 calves 9 nigs 7 cows 4 oxen 2 horses 93 Oxen. Calves. Horses. Pigs. ; 4 8 2 4 4 4 3 2 9 8 7 13 91 heads of live stock. 2d. We may ask how many of each kind, and then we shall have to separate the quantities above, in this manner ; Cows. Oxen. Calves. Horses. Pigs, i Sheep. 3 4 8 2 4 35 6 5 7 21 Other examples of Simple Addition may be the fol • lowing : For a certain undertaking in a village, seven men agree to give each as much money as he has in cash in pocket; John has ^47; Peter gl21; James g30 ; Rich- ard g79 ; Francis gl07 ; Frederic gl92 ; and William J305 ; how much stock do they bring together ? The addition gives : $ 47 121 60 79 107 192 305 ^901 94 PRACTICAX APPIICATIONS OP THE How much is the whole banking stock in New-York, (he stocks of the chartered banks being as follows Bank of New- York, $ 930, 000 Manhattan Bank, 2, 050, 000 Merchants' Bank, 1, 490, 000 Mechanics' Bank, 2, 000, 000 Union Bank, 1,000,000 Bank of America, 2, 000, 000 City Bank, 2, 000, 000 Phoenix Bank, 600, 000 United States' Bank,a 35, 000, 000 Frankhn Bank, 500,000 North River Bank, 500,000 Tradesmen's Bank, 600, 000 Chemical Bank, 600, 000 Fulton Bank, 600, 000 Examples of this kind are too easy to require the inser- tion of more. § 71. Examples of Subtraction. 1. Francis has 35 head of cattle on his farm, and his neighbour James 84 ; how much has the one more than the other? James' cattle 84 Francis' cattle 36 Difference 49 which James has more. 2. A man going into account with himself finds his whole property amounts to J 18406, and that he has .$10509 debts ; how does he stand ? Property ^ 18406 Debts 10509 Difference ^ 7897 clear property left. 3. In a year there are 365 days ; of these 62 are Sun- ^lays ; how many working days are there in a year ? Ans. 313 days. 4. A man in business bought, during the year, goods to the jynount of ^106409, and sold to Jhe amount of FOUR BUIES OF ARITHMETIC. 95 j?59879 ; taken at the same price or estimate, what amount of goods has he left ? Ans. §46530. § 72. The applications of Multiplication occur in every case where one of the quantities occurs as often as indicated by another number, which forms the multiplier, as is the case, for instance, in all purchases, profits, interest, at a certain rate for the adopted unit of the things bought, sold or lent, or in general, whenever the same thing or quantity is repeatedly taken. 1st Example. John buys 12 peaches at 3 cents a-piece ; how much has he to pay ? Ans. 36 cents. 2d. 48 head of poultry bought at 6 cents per head ; how many cents to pay ? Ans. 240. 3d. The yefg: has 365 days, every day 24 hours ; how many hours in a year ? Ans. 8760 hours. 4th. How many minutes are there in a month of 31 days, the day having 24 hours, and the hour 60 minutes ? Ans. 44640 minutes. 6th. A merchant bought 56 bales of cotton goods; 15 of them held 21 pieces, 29 held 28 pieces, and the rest 25 pieces each ; for each piece he pays g3 ; how much must he pay for the whole ? ^^^J $4281 to pay. I 1427 pieces of cotton goods. The numbers indicating the quantity of pieces in each bale are to be multiplied by the number of bales respect- ively ; the sum of these results gives the whole number of pieces, which being multiplied by 3, the price of each piece, gives the final result. 6th. A merchant bought 963 barrels of flour ; on weighing them, he finds their average weight 202 lb. and that the barrels average 7 lb. weight each ; how many pounds of flour has he ? Ans. 187,785 lbs. Note. The subtraction needed in tbe above example from each barrel, is what in commerce is called tare ; the remaining weight is what is called neat weight. Tare is usually determined by an approximate valuation, in each particular kind of package, according to certain ha- bitual, and even local, rules. It is sufficient to know 96 PRACTICAL APPIICATIOXS OF THE these, to execute any example of mercantile calculation relating to what is called tare, as they form a subtraction upon agreed principles. 7th. The sum of ^6500 is lent out at interest, for three years, at 6 per cent, simple interest, annually ; what will be the whole amount of that interest in three years ? The interest per cent being evidently a decimal frac- tion, in the place of |the hundreds, or second decimal ; the whole operation of any interest calculation for the year, consists in multiplying the capital given with the corresponding decimal fraction, and for more years, to multiply this product again by the number of years required ; thus the above consists in the execution oi the following multiplication : 6500 X 0,06 X 3 = 390 X 3 = gll70. It is evident also from this, that all transactions of commission, brokerage, exchange, notes, drafts, stock, &c., which are grounded upon a certain per centage of premium, or discount, are exactly of the same nature, and determine a decimal fraction by which the amount is to be multiplied, as in this example. 5 72. Division applies in ordinary business to all cases, where any quantity of things is to he divided among an equal number of persons, or in an equal number of lots or parts ; the quotient will give the share of each person, or the quantity of things in each lot or part. It will therefore also apply to find the price of a single piece of a thing, of which^ a large number has been purchased, for a certain price ; as in the following examples : 1st A father, having six sons, leaves among them a property of §76590, to be shared equally among them j how much will each son get? Ans. g 12765. 2d. The provision of an army in bread is 90567 lb. : it is intended to distribute the whole to the soldiers, to save separate transportation ; there are 10063 soldiers j how many pounds will each have to carry ? Ans. 9 lb. 3d. The expenses of paving a street, 500 feet in length, amount to ^1000 ; the amount is to be distributed among rOUR BUIES OF ARITHMETIC. 97 the owners ol" the Hcijoining lots, each having a lot of 25 feet ; how much wili each lot, or owner, have to pay ? Ans. g50. 4th. A merchant bought 109 bales of calico, for ihe total amount of $12232 ; he finds that 40 bales contain each 30 pieces, 60 contain each 25 pieces, and the rest contain 32 pieces each ; how high does each piece stand him ? Ans. $4. 5th. How many days are there in 24480 minutes, each day having; 24 hours, eaf h hour 60 minutes ? Ans. 7ds. 6th. How many days will it take a man to travel 946 miles, if he travel .36 miles per day ? Ans. 27 ds. Upon the principle!^ here shown for the four rules in xvhole numbers, it will be easy for the teacher and scholar to form abundance of examples for practice, and to solve those given at the end. § 73. It will be proper here to draw the attention to a general principle, which will always guide ns in the use of multiplication, as applied to any purpose of life, or even of science; namely : it ex- presses always by the two factors a certain cause and a certain repetition of the same, which may be best represented by time, for this is the measure of repetition of effects in nature, as we have seen it to be, for instance, in calculations of interest, &c. | the result of these factors, giving the product of the numbers, represents e jually well the effect produced by the cause, represented by the one factor acting a certain time, which is repi-esented by the other factor. So we may represent the multiplication and its results as the product of cause into time, being equal to their effects (the great law which is to be exactly filled in every explanation or investigation of a sub- ject of natural philosophy.) Instead of time, we may also call that factor power, and the other the object acted upon by this power ; the ideas of cause anti time however, always apply equally well ; as for instance, a man having certain means to do a thing. .9 98 PRACTICAL APPLICATIONS OF THE and using them so much, or so many times, would be the same thing in respect to the effect; and so in all similar cases. As we have shown in multiplication, that its application included all the cases where a cause acting a certain time, or number of times, pro- duced a certain effect, So division may, with equal propriety, be considered as decomposing the effectf Mo its cause and time; the one of these being given, besides the effect. Thus we evidently find: that if a certain work had been done, by a certain number of men, in a certain time, the work expressed in numbers representing the effect, tlie time of the work, or the number of men, being giv- en, the number of men, or the time of their work, may be found, by dividing the effect by the number of men, or the time they worked. In like manner: if the interest btaintd upon a sum of money in a certain number of yeas be given, the yearly amount of it (that is the cause) will be given, by the quotient arising from the division of the w hole amuunt by the number of years, and vice versa. CHAPTER II. Application of the Four Kules of Jlrithmetic to all iiinds of ^uestions^ iuvoLviug Fractions of either kind, § 74. In most of the circumstances, where calcula- tion is to be applied in common lite, the given quan- tities either contain certain fractions, or denominate subdivisionsof the unit, which have been called De- nominate" Fractions, or they often lead to such by division, as has been seen in its place. The calcu- lator must determine by tlie aid of pj-oper reflection upon any given case, and by liis knowledge of the principles or theory of arithmetic, in what manner FOim RULE 3 OF ARITHMETIC. 99 it will be most easy, and, according to the aim of the calculation, most accurate, to obtain the result. Practice gives great facilities for this determina- tion ; in the instructions lor performing it, only general considerations, or priniriples, can be pre- sented, and examples that may serve as an intro- duction to it ; this is the aim of the present chapter. It may, for instance, be reatiily inferred from a comparison of the operations in Vulgar and in De- cimal Fractions, that in complicated additixms of numbers, involvitig vulgar fractiojis whose denomi- nators are not simple, or commensurable, (that is, the one a multipl(* of the other,) the reduction of the fractional part into decimal fractions, before they are added, is peculiarly advantageous. In subtraction the same is the case, in a less de- gree however, on account of the circumstance of there being only two fractions that can possibly be engaged in one operation. Denominate fractions present no difficulty, in either addition or subtraction^ more than common numbers, except the attention that is necessary in the carrying, or borrowing, from one denomination to the other, but are from that circumvStance, and their irregular progressions, far less convenient than decimal fractions. From this circumstance the decimal system derives great advantage, and has for that reason been introduced at least in the coins of the United States. The reduction of whole numbers into fractions will never be needed in addition; when it may be required in subtraction, the application of the prin- ciple of borrowing one single unit, and reducing the same into the required fraction, as in the addi- tion of whole numbers and decimals, will be the most advantageous and shortest method, wherever 4he reduction of the vulgar fraction into a decimal fraction does not present greater advantages. 100 PRACTICAL APPLICATIONS OP THE The preceding remarks, in addition to what has been said upon the application of the two first rules of arithmetic, may suffice in this place ; particu- larly as in multiplication and division they again naturally occur, and receive their explanation and application, in a manner still more instructive, than when treated alone. A few examples placed here for exercise may therefore suffice ; they will be expressed by the signs of the ope- ration, when eiven simply, in order at the same time tt afford an opportunity of exercise in their use. 1st. Execute =17-1-4+ V +3 + ^+9+1 = 2d. " I 4-1 +5-4- ^^2+^ + }^ + 6 + ^ = 3d. " tV- H-3 + 1 + 7 - f.+ 6-^\ = 5th. '' 4,65080906 + 0,0070606 + 1 14,604091 4- 0,985 4- 406,307506 + 3000,040907 = 6th. " 6,04097062 - 5,908986072 = 7th. " 3,4091 - 3,064723 -f- 5,08(»9701 - 2,90806£ -I- 101 ,01980-67.520998 + 3,05 - 0,0672 = 3lb. " 13 ft. 7,5 in. -|- 4 ft. 6,3 in. + 16 ft. 0,5 in. 4- ft. 7,13 in. = 9th. *' 6 ft. 9,2 in. -}- ft. 1 1 ,6 in. - 6 ft. 4,25 in. = 10th " 3lb. 7oz. 2dwt. 7gr. -|- 6oz. 7dwt. 19gr.' = nth " 107 lb. 4 oz. dwt. 6 gr -f 7 oz. 6 dwt. 2 gr. 4- 5 dwt. 19 gr 4 6 lb oz. 7 gr. - 106 lb. 10 oz. 16 dwt. 20 gr. = Question 1 . A farmer thrashed grain seven days ; the 1st day 12| bushels »> 2d 18i ;' 3d 241 »j 4th 30f it 5th 32f a 6th 44| 5) 7th 15i He paid his help in grain ; to one man he gave 3i busk y^ W i^^ FOUR RULES OF ARITJIWTIS* 101 lo another 2^ ; and he returned to t;is.neighboar vh^t h<} had last borrowed of hira to go tc rriHlVwhich ''■Vas !J|. bushels ; how much grain has he left ? Ans. IGSy^jbush. 2d. A man goes out to collect payment of bills ; he pays also his own debts in going his round ; thus he gets from James g77, 65 ; from William gl05, 37|; then he pays his grocer ^98, 12^ ; going on he gets payment from P. Jones ^3U7, 62i ; and from J. Johnson he gets g692, 875 (= 87| cts. ;) now he thinks himself able to make a payment on his house of (with interest) ^856, 626, and pays his tailor yet^28, 375 ; how much has he left when he gets home? Ans. §200,4 § 7b, We have seen that the multiplication of whole numbers alone, and of fractions alone, pre- sents no diihculty, while the mixture of both, as we have given an example, by reducing the denominate fractions to fractional parts, with small denomina- tors, equivalent to them, has shown an operation, which we would gladly have exchanged for one on the same principle as the decimal system. Still it will always depend on the judgment of the calcula- tor, to which mode he shall give the preference, if his data are partly given in fractions ; because these are often, even more generally, in no complicated proportions, as for instance, i; \j {\ i; |^ =; and the like; and particularly when only one factor lias a fraction, the operation may be easily enough performed, to permit the like reductions to be avoid- ed, which, may, for instance, in J; ^; |; and the like, lead to interminate decimals. In this, therefore, the judgment of the calculator must decide, and it is very improper to bind one's self to any single peculiar mode 5 reflection will lead to a calculation easy and accurate, w hile a mere mechanical process will, wlien a mistake occurs, cause embarrassment. What we would here advise is, good order in all calculations; that any example, however compU- 9# (02. PHACTICAX- APPLICATIONS OP THE c&ted^ .be written . distinctly and regularly, in the order- in- which it proceeds, accompanied by the signs of the operations that are appr<»priate, whene- ver the operation itself might not declare it dis- tinctly. All this is nothing else but the necessary and well-known principle, that every thing must be done with reflection and order, if it is to succeed. The mode of proceeding will most likely be best elu- cidated by a few examples, of different kinds, accompa- nied by appropriate reasoning. 1st Example. Seventeen packages of goods, each weighing 72^ pounds; what is the total weight ? Ans. 1231 lb, 2d. If one hundred weight of wool is bought at 4U.\^ •lollars, what will 17f hundreds cost? Write this example thus ; 17. 6 4 40. i »_ «. m-m 680. 30. 5. H 716. H The first product is obtained by the multiplication of 17 X 40; then f x 40 gives the second line; then mul- tiplying the 17| by i, or what is the sanje thing, taking the third part of it, the 17 gives 6. and the remaining whole quantity 2 ; which reduced to fourth parts, gives | to be added to the fraction f, jiiving y ; which are to be' divided by 3 ; and give the fractional part ji ; the sum of all the three products, is the product of the whole numbers, and fractions, into each other, as required. 3d. To give an example requiring more fractional operation, let the following multiplication be given : ^V I-OVA RULES OF ARITHMETIC. lOS B 34. 1 following the operations as before, (the opera- 19. | tioDS shall here be denoted by the signs.) . 9 X 34 = 306 10 X 34 = 340 Decomposing \ (1^X^=9. i the fraction I \ f X 19 = ^ 19 x i = 4. f and the fractiot. ^ « v r.^d O. 1^ = ) xhl ^ml = e! || [id the fraction ? » . . /«^ , ,\ ) ,u^^ intoX + i + xl?X(34 4.,)-j.he tinioii-i-rt>^ ^ •' - 104 PRACTICAI. APPLICATIONS 0¥ THE Here we may evidently P' oceed either by Denominate Fractions, and Practice, or Cross Multiplication, or by Decimals ; for the latter the exwmple presents the great- est facility, on accouni ot the fractions being easily redu- cible ; it shall be made nere in the three ways, parallel to each other, for the contents of each piece, the rest being in simple quantities, and the ruoney of decimal division is in all cases best suited for decimal multiplication. By Cross Multiplication. 46. 6 1. 3 11. 46. 7. 6 6 58. 1. 1. 3 6 14. 58. 6. 4. 6 6 72. 7. 10. 6 By Deci- mah. 1,25 1,25 1. 25 250 625 1, 5625 46, 5 % 3750 62, 600 78125 By Practice. ft. in. 1. 3 1. 3 1. 3 -1 1. 46. 6 46. 23. 1. 0. 6 in. ft. 3 =6 = i 72. n 2 '^2 4 72,65625 The cubic coritents of each piece being, in decimal fractions, 72,65625 feet, the 12 pieces give 871,875 cubic feet, and thet?e, at I2i cents or 0, 125 dollars, bring the amount by multiplying 871, 874 x 0,125, or, what is the same thing, 87 I, "^874 X ^ = 108, 984375 dollars ; for which would in actual practice be given, Ans. $108, 98i ^7&. The Division of quantities, expressed in whole numbers aud fraction;d parts, either vulgar or denominate, when the dividend only has fractions, ran be made as in common numbers ,* after having divided the whole number, the remainder is reduced into a fraction of the same denominator as tke fraction given, and the fractional part jbeing added, I VOVn RULES OP AKITHMETIC. 105 the division is continued, and gives fractions of the same denominator; so may be continued as far as- desired for the intended aim. For decimal fractions, the directions given § 58 may suffice. When both dividend and divisor aie numbers mixed with fractions, it will be found tlie most satisfactory, tlierefore most generally the easiest, to reduce the whole numbers of both to the denomi- nator of the fraction annexed to each ; and wri ing the resulting fractions, execute the division as shown in its place. When the divisor and dividend have both deno- minate fractions, we have seen (§ 67) that the divi- sor in that shape is unwieldy and disadvantageous, and therefore we have there shown two methods applicable with nearly equal advantages, to which "we therefore refer. In the practical application, therefore, either the one or the other of these modes will be cho- sen, according as it presents t e greatest advanta- ges ; for it is evidently useless to raise difficulties in a practi< al work of any kind, to have the plea- sure or glory of solving them. The object here will therefoi'C only be : to present such questions as are soluble by division, either alone, or com- bined with the preceding rules ; for it must have been observed, that it is a general principle in arithmetic : t.' trary to principles ia exact science. i GEOMETRICAL PROPORTION. 119 • §86. The principle now deduced, and proved, givfes all the consequences, which are so useful in the application of proportions to practical calcula- tion ; namely : that in a geometrical proportion, all those mutations are admissible^ which do not alter the principle, that the product of the two extreme terms is equal to the product of the two mean terms, ♦ Therefore we can make all the changes shown above, in relation to the example before used. From 12:3 = 16:4 1 , rp ,. ^middle }, U2:16=3:4 1st. Transposing the ^ ^^^^^^^ ^ terms ^ ^.3 ^ ,^. ^^ 2d. Changing antecedents into consequents 3:12= 4: IG These are all evident, from the simple principle ; that the products of two quantities are the same, whichever of the two be the multiplier, or multipli- cand ; that is, because 3 times 4 is the same as 4 times 3^ as well known ; or any two other numbers ^ they all equally present: 3 X 16 = 4 X 12 = 48. 3d. Multiplying by the same number either both antecedents, as 2X12:3 = 2x16:4 or both consequents, as 12:2X3=16:2X4 or all the terms, as 2X12:2X3=:2xl6:2x4 The results must evidently preserve the principle of equality of products of extremes and means; be- cause in every case the same multiplier is contained in each product ; for, though tlic first product appa- rently presents other numbers, the identity of their '•osult reduces them to the same principle. 4th. Dividing in the same manner as before will give .) the same order ; V^ : 3 = V : 4 12:1 = 16:1 12*3 . n • 4 ¥ • 2 3 • 2 To which the same reasoning applies as to the multi- ISO GEOMETRICiX PROPOIlTIO]!f. plication ; and it is proper to make this division in all cases, where the data of a proportion are compounded of numbers having common measures, in the terms forming the numerator and the denominator of the final result. We can also compose and decompose the geome- trical proportion by its antecedent and consequent terms, in such a manner as to obtain the proportion between their sum or difference with the antecedents or consequents, or between these sums and differ- ences themselves, which furnishes an additional means of calculation for a number of practical cases. 6th. Thus we obtain from our example the following results of mutations ; viz : ^ By adding the antecedent and consequent and compar- ing them with the antecedents : 12 + 3 : 12 = 16 -f 4 : 16 By comparing the same with the consequents : 12 4- 3 : 3 = 16 + 4: 4 By comparing the differences of the antecedents and the consequents with the antecedents : 12 — 3 : 12 = 16 - 4 : 16 By comparing the same with the consequents : 12 - 3:3= 16 -4 :4 By comparing these sums and differences themselves : 12 + 3 : 12 - 3 = 16 4- 4 : 16 - 4 All these compound proportions have necessarily the property of giving equal products of the ex- treme and the mean terms, because they always- contain only a different combination of the factors, giving equal products, exactly as in the simple proportions All the mutations under Sd, 4th and 5th, again admit of course the exchange of the places of the extreme and the mean terms, which the original proportion admits. Any one of these mutations is to be applied either to disengage one of the quanti- GEOMETRICAL PBOPOKTION. 121 ties contained in a given proportion, or whenever it can lead to an abridgment of the statement ; and it will be found that in proportions apparently com- pound they often lead to the final solution, without its being necessary to have recourse to both the multiplication and division of the terms themselves, only the one or the other of the operations remain- ing to be performed ; that is, the one or the other term is reducible by it to unity ; the future applica- tion will show their use by examples in given cases. 5 87. If we have two geometrical proportions, they may be multiplied together, or divided the one by the other, term by term, with equal correctness of conclusion ; for it is the same as multiplying two equal fractions by two other equal fractions, the pro- ducts of which will again be equal ; therefore, ac- cording to the principles first deduced, the products of the extreme and mean terms will again be equal. For example, let the two followiDg proportions be thus composed ; viz : 18 : 6 = 12 : 4 or the fractions Y = ¥ and 15 : 3 = 25: 5 " " V = ¥ Multiplying the proportions term by term, or equal fractions by equal fractions, we obtain : 18X15 12X25 18X16:6x3 = 12 X 25 : 4X5; or— == . 6x3 4X5 where the product of extremes and means gives 18X15x4x5 = 6X3x12X25 5400 = 5400 and by reducing the fractions, by means of their common measures : 15 = 15 In like manner, by division, we would obtain from the foregoing 18 6 12 4 18 X 3 12 X 5 16 3 25 5 6X15 4 x 26* 11 122 GEOMETRIC AX PROPORTION. giving, by products of extremes and means, 6 X 12 4 X 18 24 24 3 X 26 6x15 26 25 or as fractions, f = !• all equally leading to identical results. Supposing, therefore, six terms in these two pro- portions given, in any manner, the t\^o remaining terms may be determined from them. And in ge- neral : as many proportions as are given, so many unknown quantities may be determined by them. This is is also the principle of what is called in arithmetic the Compound Rule of Three, It may be carried to any length, by further combination upon the same principles ; when it is carried through a number of proportions, to determine only one un- known quantity, it is called the chain rule. The application of both, and their extensive utility, will be shown in their proper places. The proportion may also be multiplied into itself term by term ; and thereby may be obtained, from the proportions of lineal dimensions, the proportion of the superficial dimensions corresponding to them, or the squares. By the products of three such equal proportions term by term, will be obtained the pro- portion of the solids having the same lineal dimen- sions for their sides, or the cubes. Thus would, for instance, be obtained : From the simple proportion 18:6 = 12:4 the square, 18 X 18 : 6X6 = 12X12 : 4X4 or 328 :36 = 144 : 16 the cubes, 18x18x18:6x6X6 = 12x12x12:4X4X4 dr 5904 : 216 = 1728:64 § 88. It may readily be conceived : that in geo- metrical proportions a continuance may take place,. GEOMETHICAI* PBOPOKTIOW. 1^3 Hs well as in the arithmetical ; that condition may be again expressed by the equality of the two middle terms ; as follows : 16:8 = 8:4; which gives 8 X 8 = 16 X 4 as the products of extremes and means. The middle term is called the geometrical mean. To this every property applies that belongs to general proportion ; it therefore admits all the changes heretofore shown. The product of the two mean terms, being compounded of two equal factors, presents what is called a square number; comparing it by this to the rectangular surface which would have all its sides equal, and showing the reduction of a rectangular figure of two unequal sides into a square. 5 89. Such a proportion may evidently be conti- nued by either increasing or decreasing numbers as well as an arithmetical one ; producing quanti- ties having a common factor, which is called the common index^ or constant ratio ; and the progres- sion or series resulting from it is called a geometric eal progression or series. In the increasing progres- sion the common index is a whole number, and in the decreasing one It i» trrid^ntly a frartimi • it cop- responds likewise, as in the ratio itself, to the quo- tient arising from the division of two successive terms. The following is an example of such a progres- sion or series : 64 : 32 = 32 : 16 = 16 : 8 = 8 : 4 = 4 : 2 = 2 : 1 = 1-i = l'l — 1.1 — 1. ^f 2 2*4 4*"8 f'TF This is also usually written omitting the signs of equality, and the terms are separated by the sign of addition (-f ) instead of the sign of division ( : ), be- cause this notation is better adapted to the use made of these series in higher calculations, where they are of great utility ; the above series may then be written thus : ^24 EVLE OF THREE. 5 = 64 -f- 32 + 16 + 8 +4 + 2 + 1 -f i+f +|4-tV+ ^G- Every subsequent number being here the half of the preceding one, the common index ot the series is = ^ ; or any one of the numbers multiplied by I will produce the number immediately succeeding it. It is proper here to drop this subject for the pre- sent in order to take it up in a later part of the work, when we shall investigate its consequences and practical applications. CHAPTER IV. Rule of Three, § 90. In the preceding chapter we have found ; that geometrical proportion is the same with the equality of two fractions, and that the products of its extreme and mean terms are equal. We proceeded in the demonstration thus : the numerator and deno- minator of the two equal fractions were multiplied each by the denominator of the other, equal denomina- tors being obtained by it, the conclusion was thai tlie numerators arp. also equal. If, instead of multiplying both factors by the de- nominators mutually, we multiply only one in nume- rator and denominator, the equality will evidently remain, because the value of the fraction so multi- plied does not change. Thus we obtain from the proportion 12 : 3 = 16 : 4 or, expressed as a fraction, y = V* by multiplying the first fraction, in numerator and denominator, by the denominator of the second, 12 X 4 16 3X4 4 and by operating equally upon the second fraction* mXTLE «* THBBE* 125 12 3 X 16 i 3 3X4 In botli cases the two fractions having one of the factors in the denominator equal, the same prin- ciple applies to this equal factor, as to the equal denominators, according to what is known of the principles of fractions ; they therefore compensate each other in this equality, and we obtain 12 X 4 y the first: == 16 3 16 X 3 and by the second : 12 = _— — 4 That is : we obtain one of the terms expressed by the three others ; and this in such a manner, that the product of either ecctremes or means being made, and this divided by the one mean or extreme, the result gives the other mean or extreme. As we have seen : that the mutations allowed in geometrical proportion admit any one term to be made either extreme or either mean, under the corresponding mutations of the other terms, we can generally, by dividing any one of the products by one factor of the other, obtain a result equal to the other factor of that product. Thus we would deduce from the above proportion all the following results ; viz : 12x4 16X3 12x4 16X3 3 '4 ' 16 '12 This is the complete principle and mode of per^ forming, what is called, the rule of three, from the circumstance that three quantities, or numbers, are used to determine a fourth. If therefore any ratio between two known qiian*^ 11=^ 126 RCXE OF THREE. titles is said to be the same as (or equal to) the ratio between one other known quantity and an unknown one, the above principle gives the determination of this unknown quantity by the above process, adapt- ed to the given case or question, and any of the mutations shown in the preceding chapter can be applied to it, as may be required. § 91. We will now, authorised by the foregoing proofs, make the application of the principles of geometric proportion to the practical irperations of the rule of three. As it will often be necessary to act upon the unknown quantity as if we knew it, in order to make such of the above demonstrated muta- tions as may be required, we shall here introduce the method so advantageously practised in universal arithmetic, namely, to denote the unknown quantity by a letter, and choose for that always one of the last letters of the alphabet, as x, y, &c. ; and when we shall have this letter alone on one side of the sign of equality, we have seen from what has already been said, that the unknown quantity is determined by the combinations of the known ones presented on the other side of this sign of equality ; that is, the number obtained by them will be the value of this unknown quantity ; this is but a small extension of the use of signs to denote the operations of arithme- tic, which has been introduced in the very begin- ning, and found so useful in expressing distinctly the operations of arithmetic. Though it is evidently indifferent in which of the four places of the proportion the unknown quantity stands, a habit prevails, of stating the proportion so that the unknown term occupies the fourth place ia the proportion ; we shall follow it, wherever the combinations do not present reasons for another ar- rangement. Isi Example. To determine the unknown quantity in the proportion 1 6 : 7 == ( 9 : a- r RULE OF THREK, 127 The product of the two mean terms divided by the tirst extreme will, as proved above, give the value of x, or the other extreme, which is the quantity sought ; thus 7 X 19 133 13 =a; = --=8H = 8, 8666 + &c. 15 15 16 which, placed in common examples, as has been fully ahown in multiplication and division, stands thus : 19 7 133 13 = 8 -f ~ 15 15 13 or by continuing the division into decimal fractions : 133 = 8,8666 -f &c. 16 130 100 100 10 when the division continued would evidently give a con- tinued succession of the 6. Thus therefore, the fourth term, or x, is deter- mined ; and any other proportion, or rule of three, the terms of which are ever so great or complicated, may be solved by the same operations, performed upon the respective numbers. 2d Example. Suppose that 7 men mow 37 acres of meadow in a certain time ; how many acres will 27 men mow in the same time ? Here we have given : the ratio between the tneji em- ployed, to which, by the nature of the subject, the ratio between the acres of meadow, mowed by each number of men respectively, must be equal ; of this only the number if acres mowed by the 7 men is given, and the number I" acres that can be mowed by 27 men is the quantity i28 KUIE OF THREE. sought, which we have agreed to designate first by a letter, as x. If therefore we make the number of men correspond- ing to the number of acres given, the first antecedent term of the geometric proportion, the second number of men will be the first consequent, or second term of the pro> portion ; the antecedent of the second ratio, that is, that of the number of acres mowed in each case, must be the 37 acres ; as corresponding to the work of the number of men forming the antecedent in the first ratio ; the num- ber of acres corresponding to the number of men, whose work it is intended to ascertain by the operation, here our re, must therefore be the consequent of the second ratio, or the fourth term of our proportion. This gives therefore the statement : Men. Men. Acres. Acres. 1 : 21 = 37 : x And by the operation shown above, and deduced before^ we obtain : Acres. Acres. 21X31 6 X = = 142 -}- - = 142,714 &c. 7 7 where the decimal fractions are evidently carried far enough for any practical purpose in the case. I have been thus long and detailed in this first exam- ple of the application of geometric proportion to the rule of three, to show the details of the reasoning which must guide in the statement of a practical question ; that I may be allowed in future to suppose them known, and that 1 may have to explain only the peculiarities which may occur in other cases, in the same manner as I here suppose the arithmetical operations of multiplication and division as sufficiently explained in the first example. The scholar will now observe : that in performing the arithmetical operations, the things or objects, which the numbers represent, do not enter into the consideration, and that the numbers alone are treated, as indicative of the relation of these things in regard to quantity, according to our first definition of quanti- I BVXE OF THREE. 129 ty; for, what would a product « *f men into acres of land represent in nature ? But the- divisif»n made again by a number rep- esenting men, may be considered as compensating, in a maimer similar to that ot the equal factors in the numerator* and denominator in a frac- tion, which compensate each other ; and there then remains, we might say, t!»e denomination of acres in the numerator, to give tlie denomination to the result. This is exactly analogous to what has been said at the beginning of this part of aritlimetic ; that the ratio only of the two things of the same kind is taken, as the principle that determines the ratio of two other things, which may be of a nature com- pletely different from the two first. We shall in general find, in all results of calculations relating to objects of different kinds, that the denomination of the result is that of the kind of quantity or things which appear in it in an odd number of terms, and that those which appear in an even number of terms act as mere numbers, giving no denomination to the quantity of the result. This remark, which is here very simple, becomes of greater importance in higli- er calculations, and is in all cases an indispensable property of an accurate result. 3d Example. My neighbour bought 372, 45 acres of land for ^720,6, but I can dispotse of only §215, 5 for that purpose ; how much land can I purchase at the same rate ? The ratio of the money is here given, and the ratio of the land purchased by it must of course be the same ; we have therefrom the statement : §720, 5 : §215, 5 = 372, 45 acres : x acres. This proportion can be reduced to simpler numbers by dividing corresponding terms by 5, which is a common factor ; it is therefore proper to do it ; thus it becomes: 144, 1 : 43, I = 372,45 : x 43, 1 X 372, 45 tvhich gives x = = 111,422 acres. ISO HULE OF THREE. Here it is evidently most proper to proceed altogether by decimal fractions, in which also the answer fits best. 4th Example. If 57 lb. 7 oz. of spices be bought for ^17, 25, what must I pay for «7 lb. 16 oz. 7 dwt. ? Here the ratio of the spices is given, and the quantities contain denominate fractions ; we would have to divide the second by the first, which is, as shown above, a very inconvenient operation ; we may therefore either reduce the \yeights to the lowest denomination of the denominate fractions, which is the pennyweight, and then proceed as in whole numbers, or reduce thej^e denominate fractions to decimal fractions of the pounds We have seen above that the first is the most convenient, when we do not foresee that the denominate fractions will ^ive short and determinate decimals ; we shall therefore proceed by this reduction ; thus we obtain for the two first num- bers, 57 and 87 12 12 114 174 577 87 10 691 oz. 20 1054 oz. 2U 13820 dwt. 21080 7 21087 dwt. by multiplying first the pounds by 12, to reduce them to ounces, then adding the ounces given, then multiplying by 20, to reduce to pennyweights, and adding the penny- weights given ; thus we obtam the statement : 13820 : 21087 = 17,25 : x Dividing by 5, to reduce : 2764 : 21087 = 3, 45 : a; 3,45X21087 which gives : x s= __— _ = J26, 32 2764 or 6 Divid which ing by 5, i gives 1 and x-^ and the whole sum SUXE OF THBEE. 131 icimal fractions resulting we slop at the 32 cents, no mills coming after ; further accuracy would be useless. 6th Example. A sum of money being shared between John and James in the proportion of 9 to 4, it results that John has §15 more than James; what were the shares of each ? and what was the whole sum shared ? The proportion stated from the above data stands thus : JohrCs. James*. 9 : 4 = X -{- \o : X Subtracting the consequents from the antecedents, and comparing with the consequents, we obtain : 9 — 4:4 = i:-f- 15 — xra; : 4 = 15 : a; : 4 = 3 : a; X = 12 = James' share ; 15 = 27 = John's share ; = 39 has been shared. 6th Example. Two merchants make a joint stock : they contribute in the proportion of 14 to 5; the differ- ence between the full shares is §504 ; what was each individual's share, and the whole stock ? ( Stock 2064 which is obtained by exactly the same process as above. 7th Example. Three merchants make a joint stock ; the first puts in a certain unknown part of the capital, the second 2000 dollars more, and the third 3000 dollars less, than the first ; the ratio of the shares of the second and third is as 9 : to 5 ; what are all the individual shares, and the stock itself? If we call the share of the first, which regulates the whole question, x, we shall have the statement thus : 9 : 5 = X -f 2000 : x - 3000 Comparing the difference between the antecedents and consequents with the same consequents, we obtain : 9 - 5 : 5 = a; -f 2000 - a; + 3000 : x ~ 3000 er 4:5 = 5000 : x - 3000 132 aULE OF THREE. Dividing the antecedents b}- 4 ; 1 : 5 =s 1250 : x — 3000 whence 5 X 1250 = a; - 3000 6250 = X — 3000 That is, §6250 is the share of the third. The share of the first is therefore = g 9250. " " second " = 11250. And the whole stock = 26750. 8th Example. A bankrupt leaves clear property ^84421, 26 ; his creditors are as follows ; viz : Jones for $ 5629 Williams 14207 Rufus 592 King 29768 Eldridge 120352 What dividend in the hundred, or proportional part, can be paid, (under the supposition of equal concourse,) and what will each creditor get for his share ? Here the ratio of the sum of the debts to the clear pro- perty will be the constant ratio, »which will give the rule for the division ; each claim forms the second antece- dent, or what is the same thing, the first term of the second ratio. Or, the fraction arising from the division of the property by the sum of the debts, which may be most easily expressed in decimals, will be a constant multiplier for each of the individual debts, and the shares will be the product of this fraction by the amount of the 84421,26 42210,63 claim. Thus = = 0,495 will 170548 85274 be a constant multiplier for each of the claims, which will give the shares as follows : Of Jones, §2786, 355 Williams, 7032, 466 Rufus, 293, 04 King, 14735, 16 Eldridge, 59574,24 § 92. In many cases in nature, and the common intercourse of life, the things whose ratio is com- pared, augment, the one in the same ratio as the J RtriiE OF THRBE. 133 ther diminishes, and inversely; as for instance, the more men are about a work, the less time it will require to do it ; the quicker a man walks, in proportion to another man, the less time he will require to go through a certain space ; and so in many other cases in nature. That is to say : the ratios (of these things, or the results) are inversed. Tiierefore, in all such cases, the ratio of the two given terms of the same kind is also to be inverted in the statement of the proportion, and then the ope- ration of the rule of three is to be executed with this inverted ratio, in the same manner as above with the direct one; this operation is evidently grounded on the nature of the things, or the ques- tion ; as in the following examples. 1st Example. I have a meadow, which 6 men usually mowed in 17 days; but, the season being precarious, I wish to have it mowed in 3 days ; how many men must I employ ? Evidently the shorter the time, the more men I must employ, ^o the ratio of the men is the inverse of that of the time ; and as this latter ratio is given, I must write it inversely ; thus the statement becomes : Days. Men. 3 i \1 = G : X or 1 : 17 = 2 : a; giving a: = 2 X 17 = 34 men ; and so many men must be employed to do in 3 daj'S the work of 6 men in 17 days. 2d Example. Two men, starting at the same time, ride a certain distance ; John travels at the rate of 6^ miles an hoar, and Peter 7| an hour ; Peter arrives after 20 hours 20 minutes ; when will John arrive ? The ratio of the time of arrival is evidently the inverse of that of the speed, or number of miles made per hour ; therefore the statement must also be inverted ; thus : Miles. H. Min. 61 : 7f = 20. 20 : x 12 134 RUXE OF THREE. Fractions occurring here, they must be reduced ; but 20 minutes being a third of an hour, and the fraction -^ oc- curring in the first term, we may take advantage of it to shorten this operation thus : reducing the whole num- bers to fractions upon this consideration, we obtain : Multiplying by 3 19 : V = 61 : ^' The fraction of the second term may be left unreduced. and the result written thus : 31 X 61 1891 X = = = 24, 88157 hours. 4X19 76 As 60 minutes make one hour, every tenth of an hour is 6 minutes ; the decimals of hours are therefore reduced to minutes by multiplying by 6, and remarking that the result of the tenths gives the units of the minutes, or the denominate fraction of 60 parts, or ^^^, the above becomes thereby 24 h. 52,8492m. The same subdivision reaching to the seconds, the same reduction will reduce the deci- mals of minutes into seconds jjnd decimals of seconds j thus: 24 h. 62 m. 53,652 s. = time of arrival of John. 3d Example. In a besieged place the garrison consists of 2000 men ; in a retreat bOO throw themselves into it, to escape the enemy ; the provisions of the place were, sufficient for the former garrison for 250 days ; how long will they last the increased number of men, at the same rate of daily allowance ? Of course the greater the number of men, tlie less time the provisions will last, and that in the inverse ratio of the orijjinal to the augmented garrison ; thus we have the statement : Men. Men. Days. 2000 + 600 : 2000 = 250 : x or 2600 : 2000 = 250 : a; Dividing by 2000 : 1, 3 : 1 = 250 : a; 250 This gives x = = 192, 3 days; 1,3 BXJLE OP THSBE. 135 ♦hat is : the provisions will leave a small remainder after 192 days, as we obtain only three tenths of a day over. 4th Example. A father, leaving a property of ^76743, makes the regulation in his will, that it shall be divided between his two sons in the inverse ratio of their ages ; the one is 12^ years old, the other 16 and 4 months ; what will be the share of each ? In this question the inversion consists only in the con- dition of the disposition itself, namely : that the age of the one shall determine the share of the other mutually ; and the sum of the ages forms the antecedent term of the (comparison or) ratio, given for the proportional ?hare of each in the whole amount ; we have therefore, expressing the months as twelfth parts of the year, the following statement : 12-|-/_-|-i6-f-^- : 12-hy6_ c=z g76743 : sh.of the older; ?2-f ^2 + 16+^ : 16-1- y\ c= §76743 : »' younger; or, by successive reductions of each, which will be easily followed : 12 -^ I + 16 -I- I : 12 -i- I = §76743 : and 12 + 1 -h 16 -f. f : 16 -1- f = 76743 : or 28 -h f : 12 -M = jj >j and 28 -h f : 16 -H f = it a or 'P'-V = ?> 3 5 and If 3 . ^B ^ )> SJ lastly. 173 : 76 = 9) il and 173 : 98 = J) M giving the share? . ^^ X 76743 _ of the older 5 ^^^ §33270, 086 98 X 76743 ■' younger = = §43432,913 173 which produce again the full property within one mille, lost by effect of the interminate decimal fractions. 136 COMPOUND BUIE OF THREE* CHAPTER V. Compound Rule of Three, § 93. From the principle explained in $ 87, we derive, as is tliere stated, the Compound Rule of Three; where several proportions being given, which all concur in the determination of an un- known quantity, the product of the different propor- tions term for term being made, the same principle, of the equality of the products of the extreme and mean terms, takes jjlace, as in simple proportion, and the same arithmetical process gives the means of determining the unknown quantity. It is neces- sary, of course, to pay proper attention to the na- ture of the ratios given, in respect to whether they are direct or inverse, and to make the statement of each accordingly. As the operation in itself has already been ex- plained in § 87, and as we shall immediately explain a simple and general principle, by which all such compound influences and effects as produce a compound proportion, or, what is called the com- pound rule of three, can be calculated with the greatest ease, whatever may be tlieir complication ; we will here only apply it to such examples as have for their first ratio units of different denomina- tions, and form thereby what in mercantile calcula- tions is called the Chain Rule, This comprehends the finding of the equivalent of exchange, weight or measure, of two places, by means of the given ratios of intermediate places, when the direct ratio is not known. This operation will exemplify still more strikingly the remark made above, in relation to the compensations of the denominations in the multiplications and divisions, resulting from the operations of the rule of three. COMPOUND RXJIiE OF THREE. 137 Example 1. li 60lb. weight at Paris, make 601b. at Amsterdam ; 451b at Amsterdam, 601b in New-York ; how many pound of New- York make 720 lb of Paris ? Multiplying tliese proportions, term for term, we ob- tain the compound proportion by the products, as below : P. Am. 1 : 1 = 60 : 50 A. N.Y. 1 : 1 = 43 : 50 N.Y. P. 1 1 = ^ : 720 P. A. N.Y. Am. N.Y. P. IXl X 1 : 1 X 1 X 1 =60x45Xa; = 50X60x720 60X60X720 1 : 1 = a; : 60x46 or X = 666^666 by equality of products of extremes and means. The products of the un'ties of the first ratios, give the ratio of unity to the product of the second ratios ; the denominations in the first ratios are all compensated, as observed before, and we obtain, by dividing in the se- cond compound ratio by the numbers multiplying the a', the proportion. Example 2. A merchant of Petersburg has to pay in Berlin 1000 ducats, which he wishes to pay in rubles by the way of Holland ; and he ha>" for the data of his ope- ration, the following proportional values of moneys, viz. that 1 ruble gives 47,5 stivers ; 20 stivers make 1 florin ; 2,5 florin make I rix dollar, Hollandsh ; 100 rix dollars, Hollandsh fetch 142 rix dollars Prussian ; and finally, 1 ducat in Berlin is 3 rix dollars Prussian ; how man}-^ rubles must he pay ? This gives the following statement: 1 rubl. : 1 St. =47,6 : 1 1 St. ? fl. = 1 : 20 1 fl. ■ Ird.h. = 1 :2,6 1 rd.h; Ird pr= 142: 100 1 rd.pr : 1 due. = 1 : 3 1 due. : Irubl. = X : 1000 12* 138 COMPOUND RULE OF THREE c By the sarae process, as in the former example, is ob- tained : 1000 X 3 X 100 X 2,5 X 20 •r = . = 2223,87 rub. 47,5 X 142 § 94. In the activity which nature presents to us, as well as in all our actions, we observe this prin- ciple: that the product of any cause into the time of its action is equal to the effect of it. Or, the product of any means whatsoever, into the time of tlieir action, or the power which acts upon them, or the conven- tional law of their action, produces a determined ef- fect ; that is, it is equal to it. Thus we have seen, that a capital loaned on interest renders as the product of the rate of interest into the time ; that a man's la- hour is the result of the product of his strength (or power) into the time he exercises this strength (or power.) In all this therefore, we see nothing but the simple multiplication of certain (actors, and their product ; as has been quoted in the remarks to § 71 and 72. In the same manner as products in arithmetic may be tlie result of a continued multipli- cation, so may an effect in nature be the combined product of a number of causes, means, powers, or times; and the effect itself may be represented by a combined product; as occurs, for instance, in high- er mechanics, where these quantities often appear as multiplied by themselves, or in the square, cube, &c. § 95. If we now^ consider the relation of two such effects ; that is to say, their ratio to each other, we find, as we have done in simple numbers, that: the same ratio must take place between the j)roducts of cause into time (as it will be siuqilest to call that by a general name) as that existing between the effects. We have now foj; some time made use of letters to denote quantities, before we knew the numbers which would correspond to them: we shall here COMPOTND UUIE OF THREE. 139 extend the advantage derived from it, in order to present this idea at one glance in its full connex- ions, and with the arithmetical operations connected with it. For that purpose we shall designate the objects of calculation, or the quantities of them, by their initial letters, and call the cause = C^ the time = T I for one of the objects ; the effect = E) and for the other, which is coin])ared to it in the compound proportion, we sliall call the same objects by the corresponding small letters, as : the cause = e the time = t the effect = e AVe then obtain, by the principles stated already in the remarks to § 71 : C X T = E; and c Xt = e and for the proportion arising from this, in a man- ner exactly similar to wliar lias! been done in com- mon numbers, we obtain the statement : C X T: c X t = E : c which corresponds, as simple products expressed by their factors and their results, to a statement similar to C T c t E € 3 X 4 : 7 X 9. = 12 : 63 It evidently follows from this, by the division of the corresponding terms of the proportion, that we have also : E c E e C : c = — : - and T : f = — : - T t C c And in numbers, also : J 40 COMPOUND RUIE OF THREE. 12 63 12 63 3 :7 = •— : — and 4:9=—:—. ^ 4 9 3 7 § 96, As we have seen in the preceding applica- tion of geometric proportion to the rule of three, that whatever term of the proportion be unknown, if the three others are given, this fourth is deter- mined by the principles of the proportion ; so in the present case, whatever may be the quantity unknown in such a compound rule of three, whether a cause, a time, or an effect, or a part of the one or the other of them, this quantity will be determined by the others, and obtained by tlie appropriate mutations of the proportion, or the operations of arithmetic resulting from it. By this consideration and process all the compli- cation, often resulting from combinations of direct and inverse proportions, in a compound rule of three, w^hich are apt to lead young calculators into mis- takes, are avoided, because every quantity, in any way concerned, is by its nature placed as factor in its proper place, by the simple reflection of its act- ing as eitlier cause, time, or effect. It may be easily seen that it will solve with ease questions upon combined actions of capitals during different times, as well in interest, as in shares of profit or loss, that is, in partnership, in complicated * The teacher who will take the trouble ti> speak with his scho- lar upon this principle, or ihe attentive reader, who willromparo it with the circuoistances that surround him, will have no diffi- culty in explaining this simple idea ; its correctness and gene- rality will prove a *reat facility to the intelligent arithmetician. My own experience ha? proved to me th'.t it meets no difliculty with boys of about 12 or 14 years, as scholars usually are, when in common schools they are thus far advanced in arithmetic, and that they made the statements appropriated to it very readily, and with peculiar satisfa-.tion. It furai?hes the best exercise of the mind for the appropriate application of common arithmetic. The examples which follow are worked out, and will, I hope, lead the way lo its proper and easy application. I COMPOUND KUIE OF THREE. 141 questions upon combined works, and all similar cases, as the following examples will show. Example 1. A capital of §6200 produces in 6 years, at '"7 per cent. §2170, amount of interest ; what will a capital of §9300, at 4 per cent, produce in 9 j'ears ? Here the statement is extremely simple, thus : C XT c xt=E:e G200 X 0,07 X 5 : 9300 X 0,04 x 9 = 2170 : x This proportion may evidently be much reduced. 1st by dividing by 100, it becomes, 62 X 0,07 X 5 : 93 X 0,04 X 9 = 2170 : x Dividing by 2, 31 X 0,07 X 5 : 93 X 0, 02 x 9 = 2170 : x Dividing by 70, i 31 X 0,001 X 5 : 93 X 0,02 X 9 = 31 : re Dividing by 31, 0,001 X 5 : 93 X 0,02 X 9 = 1 : x 93 X 0,02 X 9 16,74 Giving, X = = = §3348 6 X 0,001 0,005 That is, the capital of §9300, at 4 per cent, produces, in 9 years, ;gf3348 interest. Example 2. A capital of ^9500, at 6 per cent, inter- est, annually produced ^4560, m 8 years, at what rate of interest must a capital of ;^ 12000 be lent out, which shall render §4800 in 5 years ? Statement : 9500 X 0,06 X 8 : 12000 x 5 X a; = 4560 : 4800 Reducing as above, by dividing the first by 500, and the second by 40 ; 19 X 0,06 X 8 : 24 x 5 X a; = 114 : 120 Dividing the first and third term by 6 ; 19 X 0,01 X 8 : 24 X 5 X a; = 19 : 120 Dividing the first and third by 19, and the second and fourth by 24. 0,01 X 8 : a; X 5 = 1 : 6 142 COMPOtTND RULE OF THREE. Dividing the second and fourth term by 6, and executing the multiplication indicated in the first term, we obtain : 0,08 : rr = 1 : 1 Or, the rate per cent. = a; = 0, 08 or 8 per cent. Thus the simple reductions of the proportion given, has furnished the result. It is evident, that if we had at the first outset of this and the preceding example, ex« pressed the term in which x is, by the other three, we would have reached the same results by the compensa- tions in the numerator and denominator, and the factors of X with the opposite numerator. Example 3. Two men, in partnership, contribute as follows : A puts in 7521 dollars, which he withdraws af- ter 5 years and a half B puts in 9772 dollars, which act in the company during6 years, before which time the ac- counts cannot he settled. It is required to determine the share of each in the general result of all the operations, (which are taken together,) amounting to a net profit of 15472 dollars ? The sum of the products of the stocks into the times of their acting, are here to be compared to each single product of stock into the time of its acting, as cause and time ; the whole benefit evidently represents the efiect, corresponding to the whole stock, and its time of action. Thus we obtain the two following statemens : 7521 X5,6+9772X6 : 7521 X5,5 = 15472 : share of A 7521X5,5+9772x6 : 9772x6 = 15472 : share of B Or 99997,5 : 41365,5 = 15472 : share of A And 99997,5 : 58632,0 = 15472 : share of B Here we evidently obtain, as in the case of a bank- rupt, treated in a former example, a constant fraction from the third term divided by the first, with which the second, or the product of the stock into the time of each partner is to be multiplied, to obtain his share in the pro- fit ; or we have ; 15472 Theshareof A = X 41365,5 == 99997,5 COMPOtrWD RULE OF THREE. 15472 The share of B = X 58632 = 99997,6 Example 4. If 9 men working 6 days, at the rate of 8 hours per day, can build a wall of 152 feet long, and 9,5 feet high, how many days must 16 men work, at the pate of 10 hours each day to build a wall, 295 feet long, and 17,5 feet high? Example 5. If 180 men, working 6 days, each day 10 hours, can dig a trench of 200 yards long, by 3 yards wide, and 2 yards deep, how many days will 100 men take to dig a trench of 360 yards long, 4 wide, and 3 deep, by working 8 hours in a day ? This gives the following statement, in which the effect is a compound product, because the trench has the three dimensions of length, breadth, and depth. The reduc- tions which it admits, will here be made without men- tioning them, under the supposition that the preceding examples have shown the principle of them ; y being taken for the unknown days. 180 X lOX 6 : 100 X 8X y =200x3X2 : 360x 4 x *" 18 6: X8 X y = 10 : 36 9 X 3 : 2 X y = 1 : 3,6 27 : 2/ = 1 : 1,8 2/ = 27 X 1,8 = 48,6 days. Example 6. A hare is 50 leap? before a greyhound, and he takes 4 leaps while the greyhound takes 3 ; but 2 greyhound's leaps are equal to 3 hare's leaps ; how many leaps must the greyhound make to overtake the hare ? This, as it appears a standing question in all books on arithmetic, is well adapted for an example in this case. The proportion of the leaps as given, are : In time ; hare's leap : hound's leap = 4:3 In length ; " : " =2:3 The compound ratio of them, or the product of cause into time, which determines the efiect, is therefore : hare : hound = 8:9 144 GEJfERAL APPtlCATIOX OF If we call the distance the oound has to run = x, in hare's leaps, (as the determined distance is given in this kind of quantity,) the hare's ron in the same time will be X — 50 in the time they both run ; these two circum stances of the data give the following statement, a; : a; — 60 = 9 : 8 By comparing the antecedent with the difference be- tween antecedent and consequent, we obtain : a- : 60 = 9 : 1 X = 9 X 60 = 450 hare's As the hare's leaps are | of the hound's, this distance will require 300 hound's leaps ; so many therefore, he will have to make to overtake the hare. CHAPTER VI. General Application of Geometric Proportion* § 97. When two or more proportions are given, two unknown quantities may be determined by means of the mutations of these propoi-tions ; and the determination of the one by the other, appropri- ating the choice of the operations to the given case^ in such a manner that, by whatever operation the quantity sought is involvetl with other given quanti- ties, these become disengaged by performing the con- trary operation ; this is grounded upon tlie principle of arithmetic stated in the beginning, that each ope- ration (or rule of arithmetic) has its opposite opera- tion ; and this is the principle used in all the reduc- tions that have been made in the proportions in the preceding sections, to obtain, or render easy the ob- taining of, the results. 1st Example, Two numbers are in the ratio of 2 : 3 ; when each is augmented by 4, they are in the ratio of 5:7: what are these numbers ? GEOMETRIC PROPORTION". 145 Denoting the one by x, the other by y, we have the first statement : 2:3 = X : y And as the fourth term is equal to the product of the two mean terms divided by the first, we have slso : 3Xx 2:3 = x : 2 3Xx that is, y = 2 The second proportion, by using this result, will be stated thus : 3XX 5 : 7 = X + 4 : + 4 Multiplying the second ratio by 2 : 5:7 == 2x + 8:3Xx + 8 By subtracting antecedents from consequents : 5 : 2 = 2x + 8 : X Subtracting consequents from antecedents twice c 3 : 2 = X 4- 8 : X 1 : 2 = 8 :x whereby a; = 2 X 8 = 16 And y by the first proportion, placing the value of x, J ast found, in its place : 2 : 3 = 16 : y or 1 : 3 = 8 : y whence 2/ = 3 X 8 = 24 2d Example. A father being asked how many sons and daughters he had, answered, " If I had two more of each, I should have three sons to two daughters, and if I had two less of each, I should have two sons to one daugh- ter ;" how many sons, and how many daughters, had he ? This evidently furnishes two proportions, one stated by the sums of the numbers sought and 2, and the other 13 146 GENERAL APPXTCATIOIf OF by the difference between the numbers sought and t in the other, as follows : Calling the number of the sons = x ; That of the daughters = y : a- 4- 2 : 2/ H- 2 =; 3 : 2 x~2:?/~2 = 2:l From these proportions are obtained, by steps ground- ed upon the principles of proportion, demonstrated in § 86, the following buccessive results : From X -\- '2 : y -\- 2 = 3:2 x-^2:x-\-2 — y— 2 = 3: 1 X -\- 2 : X -y =z 3 : I In like manner, from ar — 2:y — 2 = 2:1 a; - 2 : ^ - 2 - j^ + 2 =^ 2 : 1 X — 2:x-.y = 2:l Dividing these two results term by term, as hy § 86 : x-{-2 3 : 1 = -. : I a: — 2 2 or a:-f2:a:_-2=--3:2 From this x + 2 + x ^ 2 : x + 2 -^ x J{^ 2 = b : \ or 2 ^ : 4 == 5 : 1 and X ',2 =. b\\ a: = 2 X 5 = 10 the number of sons. Though this determines the number of daughters, if we place this value in eitfaer one of the first proportions, and then determine the y^ as in the foregoing example ; still it is evident that both x, and ?/, are dependent upon the data in exactly the same manner ; 1 will therefore also determine y by a similar ai>propriate process, as it will be a good ex.i -pie to show the princijiles of this use of proportions in determining quantities in general. We made the first term, containing ^, our standing term ; we shall have now to make the second terro> containing ?/, the standing term of the operation. Thus we have from the first proportion: GEOMETRIC PROPORTIOy. 147 r-h2-?/-2:y + 2 = 1 :2 or X — y : y -\- 2 = 1 : 2 And from the second proportion : a: — 2 — y4-2:y— 2 = 1;1 X — y : y — 2 = 1 : 1 Dividing these two proportions term by term, as be- lore, we obtain : y + 2 = 1 y-2 I or y — 2:2^4-2=1:2 tBy sum and difference ; 2/ + 2 + y-2:2/-t-2~»/ + 2 = 3:1 or 2 y : 4 = 3 : 1 y : 2 = 3 : 1 giving y = 2 X 3 = 6 for the number of daughters. 3d Example. I asked my two neighbours, John and Peter, how many head of cattle each had ; Peter, thinking to puzzle me, says, '* Our cattle, taken together, are to what John has more than I, in the ratio of 3 to 2 ; and if we multiply the two numbers of our cattle together, that product will bd to all our cattle in the ratio of 5 to 3. I find how much each of them has in the following way. Calling John's cattle === x and Peter's cattle = y the first proportion given furnishes me the statement X -^ y : X — y = 3:2 and the second, X -^ y : X . y = 3:6 By addition and subtraction of the first proportion Is obtained ; X'{-y'\^x-'y:x+y-x + y = 5: 1 or 2a: : 2y = 6 : 1 X : y s= 5 1 thence «-f 2/:y = 6 1 and x + y : a: = 6 • d 148 GENERAL APPLICATION OP Diriding the second proportion given by either of these, term for term, I get : x-i-y X .y 3 1 x + y y 6 2 a:-^1/ X . y 3 1 X -^ y X 6 2 tha^-is, 1 : a: = 3 : 30 = 1 : 10 and 1 : y = i : 2 giving a; = 10 ; y ~ 2 So I find John has 10 head of cattle, and Peter appears lo be richer in puzzles than in cattle, which he did not like to tell me. 4ih Example. A, B, and C, in a joint speculation, gain, and give only the following account of the quantity each gained : the product of the gain of A into that of B is equal to gl200, that of A into that of C = gl800, and that of B into C = §2400 ; what was the gain of •ach ? This example will show, that an equality of products as is given here expresses a geometric proportion equally as well as an equality of fractions or ratios; for by the de- composition of these products into the extreme and mean terms of a proportion, we obtain the three proportions : a : 40 == 30 : y X : 60 = 30 : z y : 40 = 60 : z Dividing the first by the second, term by term, we obtain : X 40 30 y X 60 30 z or 60 : 40 = r : y Dividing this proportiott by the third, term for term.: 60 4a" ■ z y V 40 60 r GEOMETftlC PROPOHTIOIf. 60 :y = z Xz : 60y 60 : z = z : 60 z = 60 DiFiding the first and third, term by term : X 40 30 y y ,40 60 z X : y =s 30 2r : 60 ^ X : 1 = z : 2 Multiplying this by the second, term for term : xXa;:60 = 30X2':2e a: : 30 = 30 : a; a; = 30 Dividing the second by the third, term for term : a: 60 30 z 149 y 40 60 z 40 ar : 60 2/ ==' 30 : 60 4a; : 7/ = 3 : 1 a; :i^ = 3 :4 riding this by the first, term for term : X y 14 X 40 10 y or 40 : y = y : 40 2/ = 40 This example, expressly chosen for its simplicity, may suffice to explain the principle. 13^ PART IV. EXTENSION or ARITHMETIC TO HIGHER BRANCHES AND OTHER PRACTICAL APPLICATIONS. CHAPTER I. Of Square and Cube Roots. § 98. When in a multiplication the two factor* are equal, the product is called a square ; because it corresponds to what would be produced in nature by laying off the quantity which these numbers repre- sent, in any unit of lineal measure, in two directions perpendicular to each other; and completing the figure by two equal lines, drawn perpendicular at ^ the end of these ; as, for instance, taking 4 feet and laying them off upon AB, and also upon AC, and then drawing BD, and CD, at equal distances, again perpendi- cular to AB, and CD; ABCD will be a square, representing the square of A, E F that is, 4 X 4 = 16. (ICIIi The product of any two numbers may A be represented in the same way, by two ij lines perpendicular to each other, divided into equal parts, and completing the rec- tangular figure, having its opposite sides equal ; as here the figure EFGH, "^ " So we may, when we have sucli a sur- face, or product, given, and one of the sides, find the other side by division, as is evident from the second figure. But when the figure is a square, as \± BitrARB AND CUBE ROOTf. 151 in the first case, we can find the two equal sides of it by a peculiar process, which is called the extract (ion of the square root; the principle of which it is now intended lo explain. For this purpose it is necessary to investigate what a product is composed of, by decomposing each factor into two parts, not unlike the method we have used to show ihe propriety of the principle of carrying in multiplication; namely, we divide the number into two parts; thus, for instance, we would write 14 as 10 -f- 4 ; or merely consider it as so composed, and by multiplying tlie number into itself under that form, keeping each individual result separate, we shall obtain the following pro- cess and results : 14 14 4X4 4 X 10 10 X 10 -f 4 X 10 10 X 10 -f 2 X 4 X 10 + 4 X 4 100 + 80 + 16 = 196 ITiat is, we obtain by the product of the units 4X4 = 16 ; by the product of the unit of the mul- tiplier into the tens of the multiplicand, 4 x 10 = 40, and the same again by the product of the tens of the multiplier into tlie units of the multiplicand ; then lastly, by the product of the tens, 10 X 10 ~ 100. Tliis gives, by the addition, three distinct pro- ducts, viz : 1st. The square of the first part, that is, the product of the first part into itself, here 10 X 10. 2d. Twice the product of the two parts into each other, here twice 4X10, or2X4XlO. 152 SI^UAKE AND CUBE ROOTS. 3(1. The square of the last part, or the liiiits, licre =i: 4 X 4. '■'''' '*^-'"' " s "■ •. In making the cUvision of the number acfcrtrding to our decimal system of numeration, they follow the same order in magnitude as here stated. We find also by the ins])ection of tliis result, as we know besides by the multiplication table, that the product of the units can iiitluence two places of figures, namely, units and tens, and cannot influence the third ; the same is the case with any of the subse- qjient numbers, each influencing only tfie rank tvhich it occupies, and t!)e next higher rank ; this gives the principle, by wliich we may know in any number, of how many numbers the square root will be composed, namely : by dividing it into as many pairs of figures, from tlie rig]»t hand side towards the left, as it will admit ; the immbor of these divi- sions, will be the number of figures of the square root. As tlie extraction of the square root of a number will again be the opposite of tlie elevation to the square, the above operation must be executed in an inverted order to extract the square root, as in di- vision the inverse order of the multiplication has been followed. The operation of raising to a power is also called. Involution, and the extracting of the root. Evolution, In order to denote in an abridged manner the jnultiple of a number by itself, the idea will readily occur, to write the number only once, and to indi- cate the number of factors intended, by jflacing a small number at the top and to the right hand of the number, corresponding witli this number of factors ; and 103 = 10X10; i 03 = ioxiO):<10; and so for any other. To indicate the extraction of the root the sigh ^fOv an extended r, is written before the num- ber; as v/ 196, denotes the square root of 196; if SCtUARE AND CUBE ROOTS. 15$ other roots are to be extracted, the number corres- ponding to the degree of the root is written in the -/, as ^; ^; and so on; hut a much better method is, to continue the same manner of notation as in raising numbers to their powers, expressing the roots in their corresponding fractions, so that ^ 196 = (196)*,- ^196 = (I96)i; and so on in higher degrees. § 99. In Evolution the first step will therefore be, as in divisinn, to find that number which, mul- tiplied into itself, will give the product nearest be- low the most left hand number; this square being subtracted, the remainder must furnish the two other products ; as the second of these is the larger, if wc multiply tiie number found before by 2, and divide the remainder by it, we shall have a number as quotient, near the second, or next following num- ber ; with which we shall then have to execute the two products, indicated by the above result of such a multiplication. lit Example. Let the above number be chosen to ei- tract the square root ; to explain tl^§ direct inversion of the operation, or to execute ' V' '/ ~ ^ 1196 = 10 -f- 4 = 14 First square 10 X 10 = 1 00 Remainder = 96 ;Divisor 2 X 10 =a 20) in 96 ; 4 times (20 -I- 4) x4 = 96 No remainder 00 The number divided off by 2 from the right hand shows that the root has two places of figures ; so the first will be in the tens, and the number in the second division being l,the square root of which is also 1, the first •quare will be 10 X 10 = 100 ; the root 10 being writ- 154 SCtr\TlE AXD CUBE ROOTS. ten, the square = 100, is subtracted from the whole 196 ; the remainder, 96, being written, the divisor, which shall serve to tind the other number, will be the product 2 X 10 = 20 ; which being found to go 4 times in the remHinder, Tis written in the root, and being also added to the 20, tlii; sum ot both is multiiilied by 4 again, as we have found it to be factor in both the two last terms ; the product of this = 96, written under the remainder 96, being exactly equal, gives the 14 as the square root of 196 in return. 2d Example, Let it be given to extract the square root of a number of more than two places of figures, as 13456. Dividing the jiumber off as before directed, we find, that the root must have three jdaces of figures, or, the first figure will be in the hundreds ; thus we obtain the fol- low mg process ; v/1 100X100 J 'H) Divisor 2 X 100' Product 2X100-1- 10 multiplied by 10 Remainder Divisor = 2X110 (220-1- 6) X6 =5 56 = 100-^10+6 = 116 34 oolilyo 66 c^uotient = 10 00 56 56 quotient = 6 00 It will easily be conceived, that here, as in division, every number does not give a whole number for a radi- cal, because every number is not the product of another number multiplied into itself. "We may evidently in large numbers, by way of abridgment, take only the two next numbers down, as in division, arid consider the former as a ten, in relation to this number taken down, and proceed thus to the end, or to any desired number of places of decimals; for the process, as first mentioned, will proceed in this case according to th# ^^^^H S^UABE AND CUBIS HOOTS. 155 ^^^^stem, exactly as if it was a mere division continued to decimals, only the mode of making up the successiye products Avhich are to be subtracted being different. Therefore, also, the evolution of a number with decimal fractions is exactly the same as the evolution of whole numbers, whether it have an exact root or not. But it must be remarked, what the principles upon which decimal fractions arc grounded might easily suggest, namely : that the partitioning into pairs must again begin from the unit, botij to the left and to the right; there- fore, if there be an odd number of decimal places, a must be placed to the right to make up the pair, Avhich, as is well known, does not cJiajige the value of the fraction. The following two examples will suffice to give a cor- rect idea of it, and lead to the practice of this operation. To execute ^ 1419,7064, being a number with a decimal fraction: writing it partitioned off as before, the foUowins results : First square Divisor 2 X 30 (60 -i- 7) 7 Dirisor 2 X 370 (740 ■\-^)^ Divisor 2 x 37, 6 (7520 -f ^) 8 9 19, 5 19 4 69 50 44 6 B 6 7C 24 =37,68 quot = 0224 00 00 quot = G 24 quot = 8 ^ Here the process is evident, from the ei^reSsions placed opposite to each number ; the number obtained is always augmented by a 0,and multiplied by 2, to form the divisor, which from the remainder gives the next figure ; this is considering it as the ten of the following number ; the quotient added, and the sum multiphed by 156 5(tUAIl£ XTUD CUBE ROOTS. it, gives the product to be subtracted ; and the remainder is to be treated as before. Exactly in the same manner the following example gives v^ 2 ; it is here placed without any further indica- tion, in order to give room for study. ^21 = 1,41421356 4* &c. 1 1 oc 9i ) ! 4 2 00 81 .1 1 1 19 12 00 96 1 -1— 6 5 04 65 64 t 38 28 36 28 00 41 10 8 1 07 48 59 00 52 69 7 1 1 59 41 06 31 42 13 00 25 -I 17 64117 75 00 ^5 99. From the preceding we have only a short and easy step to make, by means of reflections grounded upon the principles just used to explain the extraC" tion of square roots, in order to determine the prin- ciples upon which a quadratic equation is solved; that is, to furnish the means to determine an un- known quantity, which, in a combination with oth- ers, would be multiplied into itself, or what, as wc have stated above, is said to be squared. To make SCtVABE AfTD CUBE HOOTS. 15T tbetjxplanation more simple, we may use two means, which taken in conjunction will, I hope, leave to the attentive student of this hook no difficulty. I wish to introduce this here, although unusual, because its absence would leave us in the subsequeni: parts, when we shall treat of progressions, without the means of finding, or satisfactorily explaining, the solution of certain questions arising from them ; for I have proposed to myself, never to lead the student blindfold over any step ; while at the same time I wish to give him all the means of calcu- lation in arithmetic, that he may desire, in a man- ner satisfactory to a reflecting mind. We have before decomposed the number, of which w^e wished to show the different products forming the square, into two parts, and have there shown, that tlie square number resulting was composed of the sum of the squares of the two parts, and twice the product of the two factors into each others we there decomposed the 14 into 10 and 4 ; we chose this division on account of its direct appli- cation to the extraction of the square root of a number written in our usual decimal system, but Any division will do the same thing. If in the annexed fi- ^ _^ gure, of 14 subdivisions on each side, we di- vide the sides into 9 and 5 parts, the result will be exactly the same ; we shall have tlie square d9a9 =■ 9 X 9 = 81 ; the product of 5X9 twice, on each side of this square, in 9abBf and 9acC, and the small square ubDc = 5X5, which toge- J ther will fill up the large square JBDC^ and summing up these products.* 14 ; M ■ M M^ I I I I I I I I I I I I I I M I I M I I I I I I I I I M f B 158 SC^lTABE ANi) ttlBM ROOTS. obtained by the multiplication as above, we of course f 81 "^ exactly as by the othei divi- [ 90 I sion. In like manner any other obtain -^ 55 ^ division would give the same I I sum. L 196 J As, therefore, a square number can be decomposed in any two parts, so as to obtain from it two smaller squares, and twice the product of the two parts into each other, we are allowed to consider any square number to be thus composed. We have seen in the very beginning, that in aritlimetic we have always two operations, exactly opposite to each other, the one always compen- sating the effect of the other. We have seen in treating of proportions, that, when the same operation was executed on both sides of the sign of equality, the results were again equal, and therefore the principle of equality still subsisted ; or, what is the same, that equal operations performed upon equal quantities do not destroy the equality; by this means we were enabled to (obtain solutions of questions, or, what is the same) determine unknown quantities, variously involved by other known ones. If now, in application of these principles, we con- sider an unknown quantity in any manner involved, which appears in any one or more of the parts, multiplied into itself, that is, in the square, and in other parts simple, we are, by the principle last shown, authorised and enabled to separate the square from all other numbers, or quantities ; and we can consider it thus insulated, according to the explained principles of the division of the square, as representing the square of the first part or sub- division of the number, or of the square. To apply this to an example, we must again give our unknown quantity a designation, and treat it as if we knew it, until it comes to stand alone on one side of the sign of equality, which gives tlie solu- I SttTTARE AND CTJBE BOOTS* 159 tion, by indicating that it is equal to the result ol the combination represented by the known quanti- ties on the other side of the sign of equality. Then the terms multiplied by the unknown quantity must be considered as representing twice the product of the first term into the second, or, in that case, of the unknown quantity into the known ones. The half of tliis factor being squared will represent the smaller square ; (or in general the other square needed to complete the entire square;) by the addi- tion of this square on both sides of the equality, a square number is obtained, of which the square root can be exti'arted by the rules given, or, what is in this case equivalent, which can be expressed by the given numbers. The quantity sought for is there- fore known from it. Example. Suppose we had given, by the result of a calculation, a combination of quantities which have the following form; 480 = 3x2 -f 36x 160 = x3 + 12x 196 = xa + 12r-l-36 v/ 196 = X -1- 6 14_ 6 = 8 =x having the x in the square multiplied by 3 ; this must first be disengaged, by dividing both sides by 3 ; this gives the second line ; then the 12, multiplying the simple x, represents the product of 2 into the second part of the subdivision of the whole square ; therefore its half, or 6, is the side of this second square, when x is the side of the other, because the 12x, or 2X6Xx, must re- present the double product of the two parts, like 9 abB -f 9acC. If, therefore, we square the 6, and add it to both sides, by which the equality is not changed, we shall have on the right hand side a full square, in which the X is the side of one of the lesser squares, and the other is known ; thus the third line above is obtained ; the two parts, into which the square appears divided. 160 S(IUARE AND CUBE HOOTS, are therefore x and 6, which will together be equal t& the square root of 196 ; this gives the fourth line Ex- tracting the squire root of 196, gives 14, and if the 6 is subtracted on both sides, gives the value of x as in the last line, for the final result. The operations needed in consequence of the above principles are therefore the following. 1 . Write the given quantities in such an order ^ that the parts containing the unknown quantity stand all on one side of the sign of equality ^ and those hav- ing none but known qua7itities on the other side, 2. Arrange it so : that the square of the unknown quantity multiplies at once all the quantities which it has to multiply, and do the same with the quantities that multiply the unknown quantity simply. 3. Disengage the square of the unknown quantity of all its multipliers, either whole or fractional, by dividing every term of the equation by them. 4. Make the square of the half of the factors which multiply the unknown quantity in the simple form, and add this square to both sides. 5. Extract the sqvxire root of tJiat side of the equa- tion which has no icnknown quantity, and write on the side of the unknown quantity the root of this un- known quantity and of the square added. 6. Subtract the part added to the side of the un- known quantity from the square root of the determin- ed number of the other side. 7. The result will be the value of the unknown quantity sought. These general principles will include all cases- that may occur. § 100. For the cube, or the product of three equal factors, which corresponds in nature to the solid, we have to multiply the product, which has been obtained for the square, once more by the first quan- tity ; in order to show what different parts it is composed of, the aboTc mode of separating the fac I S(tUJLR£ AND CUBE BOOTS* 161 tors is to be preserved, because it will show how the products are to be made in the extraction of the cube root. For this purpose the same example^ which has served before, will again be made use of. We have obtained in ^ 98, by 14 X 14, or 142. the result lOX 104-2 X 10X4+4X4 which being multiplied by 10x4 gives 10X10x10+2X10x4x10+4X4x10 + 4X10x10 +2X4X4X10+4X4>r4 10X10xlO+3X10X10x4 + 3x 10X4X4 + 4X4x4 = 103+3x10 X4 + 3X10X42+43 = 2744 = 14^ It will be observed, that this product is composed of the cube of 10,* three times the square of 10 into 4 ; three times the product of 10 into the square of 4 ; and the cube of 4. Or, generally, the cube of the first part, and three times the product of the square of the first part into the second ; then three times the product of the first into the square of the second part ; and lastly, the cube of the second part. Tliese products are therefore to be formed out of the parts of a cube the root of which it is intended to extract. It will again be observed here, that, with refer- ence to the subdivision of the cube in the order of our decimal system, the second term will be the largest after the first, as it contains the double -square of the first, as the largest factor which may occur after the cube of the first; it forms therefore the leading part, or factor, to find the second part, as in the extraction of the square root. It will also appear, that, as we had to divide off the number into pairs of figures in the square, here it will be necessary to divide off the number 14* i65 StttTAEE AND CUBE ROOTS^. every three figures, from the right hand side towards the left, because the product of a number of two figures into one of one figure may give three figures in the result. With these results, and the principles which arise from them, for the converse operation, that is, the extraction of the cube root, we shall be able to exe- cute this operation properly. 1st Example. The above resulting number, 2744, being given, to extract the cube root, which is indicated thus : First cubic root taking oflf 1 ^ -Remainder 1 Divisor = 3x10x10 = First term =300x4= 1 Second term =3X10X4X4 = Third term =4X4x4 = Sum of the three terms = 1 Subtracted from the remainder == 744 = 14 744 300 quot— 4 200) 480) 64) 744 000 The only number which cubed will not exceed 2 is 1 , taking away this cube gives the remainder 1744 ; forming the triple product of the 10^ = 300 ; this in common di- vision would go 6 times in 1744 ; but there must here be room for the subtraction of the products indicated above, and it will be found that only 4 will admit that : thereby we form the 3 terms placed under, as indicated j the sum of which is equal to the former remainder, and subtracted leaves 0, giving 14 the exact cube root of 5744. 2d Example. Extract the cube root of 994011992, or execute V mp> First remainder Dinsor = 3x9002 = 3X900^X90 = 3x900X90* = 903 -^ Sum of factors =» Second remainder c= Divisor = 3 X(990)2 = 3x9903X8 = 3X990X82 = 83 = Sum of factors = Third remainder = 430 Si^rABB AKB CUBB BOOTS* 1<>5 11992 = 9004-904-8 = 998 000000 On|992 000 quotient = 90 000) 000 > ooo) 000 992 300 quotient = 8 »,994|0 9003 = 729 265 sis 21 241 23 IP 23 700 870 729 299 712 940 522 190 23,712 'ool'oo 992 ~00 This gives a root of three places of figures, as indicated by the partition. The nearest cube root of the first division of the numbers on the left being 9, which in the third place is equivalent to 900, the cube being made and subtracted, leaves the first remainder ; the triple product of the square ©f it, taken as a divisor^ shows 90 as quotient, for the root. The products are now formed as indicated ; their sum being subtracted from the first remninder, leaves the second remainder, upon which the same process takes place as before, taking the whole of the root found as the first term ; and the sum of the products being equal to the last remainder, the Dumber given proves an exact cube of the number ob- tained as root. 3d Example. If the number is no exact cube, we may extract the approximate root in decimal fractions, as well as in the square root ; the number of O's to be added each time must of course be three, and the products are formed as required in the former example ; the pro- cess will go on, in other respects, as has been seen in the square root. To make this strikingly apparent, we will hex'e execute 3,2 . thus: 164 SClUAllE AND CUBE ROOTS. Remainder Divisor SXIO^ 3X102x2 3x10x22 23 Sum of factors . First remainder Divisor 3X1202 3X1203X5 3X120X62 63 Sum of factors Second remainder Divisor 3X1262 3x12602x9 3X1260X92 93 Sum of factors Third remainder Divisor 3X126902 3x126902X9 3X12690x92 93 Sum of factors Fourth remainder l,2699 + &c. 000 000 300 600*1 120 I 728 272 000 43 216 9 226 46 4 42 42' ~4 adding three O^s 200 000 000 126 876 68^ 187 303 491 383 476 279 3 282 iOO 000 adding three G's 500 500) 750) 729 ^ 979 000 adding 3 O'ir 300 700) 370 > 729) 799 021 624 718 069 778 242 201 Adding 3 0's,it would be continued as before. The place of the decimal mark is evidently again de- termined by the usual principle, namely : where it be- comes necessary to add O's to continue the opera- tion. § 101. We here see again, that the principles de- duced may lead to the solution of equations of the third degree, as this is called in higher calculations. p:B;OGR£SSIOfirS OB SEBIB9. 165 or to determine a quantity which appears as form- ed of three equal factors multiplied into each other ; but it is not the province of arithmetic to go into this inquiry ; because it requires operations, and pro- duces cases, which are reserved to be solved only in universal arithmetic, or algebra. It is evidently possible to produce the involutions of higher degrees in the same manner that has here been shown for the square and the cube ; but the evolution presents increasing difficulties as we pro- ceed, the possible combinations of different factors to the same ultimate result being evidently always more numerous, and thei^fore, also, the possible roots. Even in algebra there is not yet a general method found to solve such questions, and it steps entirely out of the limits of arithmetic to treat any thing relating to this subject. ► CHAPTER II. Of Progressions or Series, § 102. In mentioning (§ 84 and 89) continued proportions, and the progressions or series which result from their continuance, we referred to a future extension of the subject to the progressions or series, which are intended as the subject of the present chapter. According as the continued proportion is either an arithmetical or a geometrical proportion, we ob- tain by its extension to a greater number of qanti- ties : either an arithmetical or a geometrical progres- sion, or series; each of which has peculiar laws ; we shall here begin with the first. $ 103. A series of numbers which progresses in- creasing, or decreasing, by the same constant differ^ 16^ FROGKESSIOXS OR SERIES, ence, forms a continued arithmetical proportion^ or an arithmetical series. This principle is therefore the element of all in- vestigation in relation to the properties of this kind of series; according to it we shall be able to write all tlie terms successively, and therefore obtain the la\s of the mutual dependance of all the quantities con- cerned in it ; such a series (which we will caii equal to SJ will, for instance, be the following : 5 = 2-}-(2+3)-}-(2+2 X3)4-(2-f 3 X3)-i-(2-4-4X3)H-(2-f 5X3)+ &c. In the writing of these series the terms are joined by the sign +» which ma^^ equally serve to express tlie arithmetic proportion, as I stated at first, and the constant equality of the difference will become equally apparent by the subtraction of each term from its immediately subsequent term^ which gives liere the constant difference, 3. Considering the successive dependance of these terms upon each other, and comparing their value in relation to their distance from the first term, we observe that the constant difference makes its first appearance in the second term, and being afterwards found added in each subsequent term, it will in any term whatever be one less than the number of terms indicates, whether the series be increasing or decreasing. Thus we find it here in the sixth term added five times to the first term. This gives us the principle by which to determine any term, when the first term and the constant difference arc given. It will be of the greatest advantage in the exten- sion of arithmetic in this state of foi wardncss, to apply the use of letters to denote certain quan- tities, until tliey are determined, that we may ex- press our ideas clearly, fully, and briefly, by applying to them the signs of arithmetic which hare been taught in the beginning. We will there- I PROGKESSIONS OB SERIES. IST fore generally denote the quantities concerned in onr present investigation by proper letters ; thus : Xict the first term be designated by, or = a " constant difference by = d " number of terms of the series = n " sum of the series = S Thus we shall be able to express the property, which we have just found, of the value of any term, which we denote by ii, by term (n) = a -{- (n— \) d And the whole series extended to the term n, would be written thus, (omitting the intermediate terms :) 1st 2d (n-l)st nth .S = a4-(a + t/) .... (rt+(7i-2)d)+(a-f(n-l)d)+&c. Considering the Tith term, it is evident that if, of tlie three quantities concerned in it, and the whole value of the term itself, any three are given, the fourth may be determined from them, just as we determin- ed the fourth term in a geometrical proportion, not- withstanding tliat the law of their mutual depend- ance is very different. Example. In the above series we had o = 2 ; d = 3 ; let n denote the sixth term. We shall, by putting the values of the letters in their places, and performing the operations indicated, obtain the following : Value of the 6th term = 2 + 5x3 = 17 In a similar manner any other term would be obtained, as : The 21st term = 2 + 20 X 3 = 62 and so on. If we had 62 as the value of the term given, and the 6rst term, together with the constant difference, we would evidently obtain the number corresponding to the term, by subtracting the first term from the sum, and di- viding the remainder by the difference, then adding a imit to the quotient ; thus : 16$ FBOGRESSIOXS 611 SEIIIE9. 62-2 = 60 ; then V = 20. Adding 1 gives for- n «= 21. In like manner any other part can be found, by revers- ing the operations accordingly. § 104. The most frequent use of these series, and therefore the principal object of inquiry, is the de- termination of their sum by means of the three other quantities concerned in it. The principle of this determination is deduced from the nature of the se- ries, in the following manner. As we found in arithmetic proportion that the sum of the extremes is equal to the sum of the means, so it is evident that here the sum of the extremes is equal to the sum of any two terms equally distant from them, for the sum of every such pair of terms must contain the first term twice, and the constant diifer- ence an equal number of times, because these in- crease in numbers equally from the beginning onward, as they decrease from the end backward. In the above series we obtain : By the first and last or 6th term : 2 + 2 + 5X3 = 19 By the second and last but one, or fifth term : 2 + 3 + 2 + 4X3 = 19 By the third and fourth term : 2 + 2X3 + 2 + 3X3 = 19 And generally, by the first and nth term, we would obtain, adopting the expressions above used, the general value of any pair of terms : a -{- a -{- {n — 1) d Summing up all these pairs of terms, we would of course obtain the sum of the whole series. But there are as many pairs of terms as the number of terms divided by 2 ; therefore we may obtain the value of the whole series at once, by multiplying ( PKOGRESSIOX OB SERIES. 169 e value found above by half the number of terms i that is, in the above numbers : (2 -f 2 + 5 X 3) f = 57 And in the general expression in letters, or, as this is usually called, equation : n ^ = — (2 a -I- (« - 1) d) 2 In this general expression again there are only four quantities concerned, three of which being given the fourth is determined, by making such operations upon the above equation as will bring the quantity to be determined alone on one side of the sign of equality, as in this case the S, § 105. To determine any quantity in any way involved in such an expression as the above, which in general arithmetic is called an equation, the same principle is made use of as has been shown in pro- portion, namely, that all such mutations are allowed as do not change the principle, that after the change made, the quantities on each side of the sign of equality are again equal. This leads directly to the consequence, that we are allowed to perform any operation of arithmetic we may wish upon such an equation, provided we do the same on both sides. As we have seen above, that the operations, com- monly called rules of arithmetic, ai'e of such a na- ture, that two are ahvays opposite to eac h other, that is to say, the one will alv. ays evolve what the other has involved, or disengage what the other has engaged, we shall naturally in an operation such as is proposed always perform upon such an equation success ively all the operations which will disen- gage the quantity from all others, until it ulti- mately be found alone on one side of the sign of equality. 15 iro pro6ressio:n^ or series. We will therefore now apply these principles to the equation before us, to obtain successively expres- sions or equations for each of the quantities by means of all the others. 1st Problem, To find the first term of the series, knowing all the other parts, we would proceed thus : Taking the original equation n 8 = (2 a 4- (w - 1) r)" — 1 and C =^ p ; r (1 -f- ry- by the sin^ple rule of three. In substitutihg here, by way of example, the numbers found or given § 110, the abdv'e expression would stan€l thus : 0. 06(1,06)« Payme^nt g 800 = 6580, 2G6 — (1,06)8 — 1 (1,06)8 — 1 Capital $ 5580, 266 = 800 0,06 (1,06)6 The determination of the number of years that it will take to extinguish a debt by given yearly and equal payments, is another question that is beyond our present limits, for it is the same as that stated in «5 108, This subject is therefore dismissed, and it is expected that any student, who has applied himself to this exposition of the principles of this kind of calculation, witJi tlie necessary understand- ing of the general principles of arithmetic taught in this book, will find no difficulty in solving an} of tlie questions, which will appear at the end. upon this subject. AXLIGATIOS-* iS7 CHAPTER IV. Of Migation, or Mixtures of objects of different Value. § 113. In retail mercantile concerns it often oc- curs, that it is desirable to ascertain tlie proportional value of a mixture of things of different values which are given. Reflection upon what has been hereto- fore taught would point out the principle upon which such a proportional value may be determined. This value of the mixture being naturally a certain mean of all the component parts, this operation of arith- metic is usually called alligation medial. The quantity of each component part multiplied by the price of its unit (what is u>sually called its value) evidently gives the influence of this part up- on the general mixture. It might therefore be con- sidered generally as acting exactly in the same way as the product of cause into time. The sum of all these products evidently constitutes the whole. Thus we might say in any number of things mixed, CxT-f-cXi4-3X^-faXi = E the sum of all these uniting in the common effect = E^ If therefore the mean eff*ect, that is, the mean value of each individual thing,^ or unit, in the miiXture, is to be determined, this whole effect, that is, the sum of all the partial effects, is to be divided by the juimber of things mixed, or the objects acting in the general result. This expressed in the above form will give, considering C for the cause as the ob- ects and (the time) T as their value, the following general result : CX T + cXt-^d Xi-t-j XX Mean = ■ Ib8 ALLIGATIOIf. Example, Suppose that a number of men work at a certain work during a month, as follows, namely : 6 men work 15 days each ; 4 men work 19 days each ; 12 men work 20 days each ; and 10 men work 26 days each, during that time ; on how many days' work, on an ave- rage, can one calculate for each man in a month ? This gives : 6X15-1-4 X 19-1-12X20+ 10X26 13 Mean •= 20 H 6+4 + 12+10 16 In this manner it may evidently also be calculated, that in a number of workmen engaged in a work the oc- casional absences may reduce the amount of work which they would otherwise perform ; to the mere result of the product of the denominator of the above fraction into the quotient found, or the above workmen taken to- gether, would in a month have executed only the work W= 32 (20 + if) = 666 days ; or the amount of the numerator of the fraction, as is evi- dent ; instead of which, if they had all been present the whole of the 26 working days in a month, they would have produced the work = W = 26 X 32 = 832 days. § 114. When in such a composition it is desired to obtain a certain mean value of the objects mixed, or (as in the preceding example) a certain amount of work by means of objects of different value, (or, as above, men differently assiduous to their work,) it becomes necessary to determine the quantity of eacli individual ingredient, (or, as above, the quantity of each men of a certain assiduity,) to obtain the desired aim, that is, the price of the thing aimed at, (or the number of days' work desired.) This operatioji of arithmetic is usually called alliga- tion alternate. It is requisite that the quantity of objectsbelow the mean value must compensate for those above it; their products must therefore become inverted. In thus composing a mean with- out limitation of the quantity to be made up, or of AIXIGATION. 18d %ny of the parts given, it is evident that a number of solutions will be possible for each question, but that all will be multiples of each other. The prac- tical method used is the following. The different values being written under each other, the difference between one value above the mean and this mean is taken, and placed oppo- site one of the values below the mean; and alter- nately, the difference between this lower value and the mean is written opposite to the value above the mean; thus all the differences being taken, the numbers opposite to each value are added, and give the quantity to be taken of each of these respective values, the products of which into the values to which they are opposite will give a sum answering a compound as desired. And every equal multiple of all the parts will also give an equal multiple of the whole. (The parts compared are linked, to show the operation.) Example. A goldsmith having gold 15 carats fine, 19 carats, 21 carats, and 24 carats, wishes to make a mix- ture 20 carats fine ; how much of each has he to take ? 20 which gives 13X5-f. 19X 5 + 21x6 4-24X 6 = 20 X 22 = 440 or the whole mixture being 22, be it ounces, grains, or That it may, there must be in it 5 of the 15 carats gold ; 5 of the 19 ; 6 of the 21 ; and 6 of the 24 carats gold ; which evidently bears the proof of giving, when 20, the mean price, is multiplied by 22, ihe whole quantity mix- ed, the same result as is obtained by the sum of the indi- vidual products. § 115. If either the whole amount of the mixture, or any one of the parts to be mixed, is limited to a certain quantity, it becomes necessary, after the above operation, to take the ratio between the part 15— — 4+1=5 19— — 1 -f- 4 = 5 21 — 1+5 = 6 24— 5+1=6 *90 AtLlGATIOlr. given and its corresponding number in the above result, to make all the other numbers in the like manner proportional to their corresponding ones in the above result. Isf Example. If in the above the whole mixture was required to be 36, instead of 22, we should have to make the proportions (the 15 carats, or) 8,46 (" 19 " ) 22 : 36 = ('* 21 " ) 9,818 (" rirvc €X nil 24 " ) 9,818 2d Example. A goldsmith has silver 6 ounces fine, 10 ounces fine, and 20 ounces of silver 9 ounces fine ; how much of the two first must he add to the 20 ouucea of 9 ounces fine, to make a mixture 8 ounces fine ? 1+2 = 3 1 = 1 1 c= 1 This will give the ratio of the silvers ; now the silver at 9 ounces fine being determined at 20 ounces, the propor- tion formed from the ratio of the number found for that kind of silver, to the number limited for it, is that which must guide all the others, as follows : 1 : 20 = 3 : (silver of 7 ounces fine =) 60 1 : 20 = 1 : ( " 10 " =) 20 § 116. We shall now close these elements of arithmetic ; for to go into more complicated practical applications would exceed the proper limits of first elements, and may be much better treated algebra- ically. The regula falsi, or rule of false supposi- tion, both simple and compound, is intentionally omitted, the first because an attentive scholar of what has been here taught will not need it, but find in what he has learnt the better means to solve the question, the second because its operations belong more properly to algebra, so far as they actually lead to a determined result. § 1 1 7» A short retrospective view of what has been treated in these elements may not be misplaced* KETBOSFECTIVE VIEW. t^f I have dwelt at some length upon the very first elementary ideas of arithmetic, the notation or signs of the arithmetic operations, and the principles of the systems of numeration, because, as was there said, these first elementary ideas, if well under- stood, will be of the greatest utility in rendering every operation in arithmetic easy ; it is therefore to be wished, that the teacher extend them still more by some practice upon other systems of numeration resides the decimal system, and by familiarising the varied combination of the signs of arithmetic, the full value of these combinations being ultimately assigned. The same reasons dictated to me the detailed de scription of the four rules of arithmetic, which it is certainly proper to make easy, and satis- factory to the mind of the beginner, if he is ever to know how to apply them in their proper place. In treating vulgar fractions, I considered it obli- gatory upon me to proceed by exact mathematical demonstration, and to deduce them from their actual origin in an unexecuted division ; while in decimal fractions the whole of their principles will at once spring from the consideration of division conti- nued below the unit, according to the same sys- tem as above it. In considering all conventional subdivisions of the units of different kinds of quan- tities as denominate fractions, I found it possible to treat it with some system, which is not possible when each is treated separately. If I have deviated in these considerations from the usual method, I hope the clearness that results will excuse me. It appeared to me proper to bring the scliolars to tliis point by what might be called theoretical steps. The Second Part will afford the scholar the satis- faction of a useful and amusing application of the principles learnt before I considered it proper to devote a separate part of the book to this, in order to give the scholar the satisfaction of seeing how much he could do with the few elements he had learned J 92 RETKOSPECTnrB VIEW. before ; and it is to be hoped that every teacher will know how to relieve his scholar in an agreeable manner by this Second Part, and the questions which will be placed hereafter, or others of his own making. In the Third Part, treating of ratios and propor- tions, 1 considered myself both bound by true prin- ciple, and authorised by the progress of the scho- lar, to treat the subject as the beginning of the elements of the actual science of quantity ; the prin- ciples being so few and simple, the task appeared to me, only to lay them well open to the scholar, and to show him all their bearings and conse- quences ; a defective treatment of this part of arith- metic, cannot but destroy, instead of cultivating, the reasoning and understanding of the scholar. These reasons determined me to a more detailed application to examples fully worked out, as they both help to explain the principles, and make their application pleasant to the scholar. The use of letters to denote a quantity before its determination appeared to me proper to be intro- duced, and gradually to habituate the scholar to more general considerations in regard to quantity, not servilely attached to the figures of our system of numeration. After the steps made in the Third Part, I hope to need no excuse for the greater degree of generalisa- tion which has been introduced in the Fourth, except to say that it was done with the avowed intention of leading the scholar imperceptibly into the entrance of algebra. It is absolutely useless to teach these parts by rules ; no scholar ever remembers them ; and he, whose memory is mechanical enough for this, seldom knows where they are applicable. They are therefore useless to him; and to omit teaching properly the principles of these parts is an injustice towards the student of f^rithmetic, who wishes to prepare himself by it for higher studies. COLLECTION of QUESTIONS. I NtTMERATIOJT. Read the following numbers : Ist. 73,064; 6th. 94,070,790 2a. 101,070,101; 7th. 4,399,080,502 3d. 500,007; 8th. 100,010,007 4th. 90, 807, 060, 501 ; 9th. 7, 070, 409 5tb. 1,897,510,234; 10th. 1,902,010,571 ADDITION. Add the following- numbers : Ist. 1, 006, 052 + 70, 401 -|- 8, 040, 107 -I- 9, 080, 071, 402 =^ 2d. 17,040,109 -f- 50,201 -f- 701 -f 30 -f- 5. 000, 127 = 3d. 70904 + 398125 + 8079123 -f- 98162753 = 4th. 37 -f 90005 + 1009645 4- 309047 = .5th. 773 -I- 104462 -f- 34983 -f 81090406 = EXAMPLES IN MULTIPLICATION. 1. Seven boys have each twelve marbles; how many marbles have they altogether? 2. If 5 boys buy each half a peck of apples, and each half peck holds on an average 16 apples, how many apples have they altogether ? 3. A company of soldiers of 105 men with the officers, having all muskets, each weighing 5 pounds, and 2 pounds of ammunition, how much weight have they to carry altogether ? 4. A ton, ship's weight, is 2200 pounds; how many pound? weight will be in a vessel carrying 450 tons? 5. Twenty bales of cloih, containing each 27 pieces, of 2C yards the piece, how many yards are there in the whole? DIVISION. 1. I have 750 pieces of cloth, and can put no more than 13 pieces in a bale ; how many bales shall I have to make ? 2. A schoolmaster has 62 boys, and having a lot of 434 marbles 17 194 ctrESTioifs. Tvhich he wishes to distribute equally among his boys as arewarff^ how many will each of them get ? 3. If a man has an annual income of ^3555, how much can he spend per day ? 4. A man having two hundred and fifty miles to travel, and tra- velling 24 miles per day, how long will he be in performing the journey ? VULGAR FRACTIONS. ADDITION. 7 3 5 3 9 12 1. Add -H 1 1 { h— = 8 7 9 14 24 15 9 2 7 8 3 9 11 5 9 n 25 32 1 7 2 17 19 15 16 5 8 9 27 32 38 42 9 15 13 14 8 10 5 4. " - + -- + ---f- + - + --{- ~=- 13 19 21 27 23 34 18 SUBTRACTION. Make the diflference between the following fractions, added an® subtracted as indicated by the signs. 343 7 936 11 2 1. + - + + -4- 7 5 11 12 14 8 35 12 15 42325 5 7 114 7 9 14 15 8 12 16 18 5 3427 3 6 6 9i 3. .4. -f 4. 4-_ 8 5 9 15 14 21 25 26 3 TO FIND THE GREATEST COMMON MEASURE 24598 74844 2. 44226 150579 61047 3. 77373 XO riS3 THE 8VCCESSIVS ArPROXIMATmO BBACTI0K8* 794973 5967 1. 1674219 13843 38126 81097 2. : 4. 2, » 3. »» 4. r 5. »» $. V 7. « «. )> 516412 649321 DECIMAL FRACTIONS. REDCCTI0N TO DECIMAL FRACTIOUS. 1. Reduce 13h. 7m. into decimals of the day. 56d. 7h. into decimals of the year. 10, 5 inches into decimals of the foot. 5oz. 7 dwt. 3§:r. troy into decimals of the pound. 75 lb. 7 oz. into decimals of the cwt. avoirdupois. 2 ft. 5, 7 in. into decimals of the yard. 27 h. 5 m. 3 s. into decimals of the year. 17 cubic inches into decimals of the cubic foot. ADDITION. A grocer making an inventory, finds he has in cash $ 17> 52; in various liquors the amount of $ 215, 17 ; in soap, candles, and such articles, | 92, 54 ; in spices, | 107, 32 ; in salt fish and similar pro- visions, $ 49, 62 ; and in various small articles, besides the furni- ture of his store, in all $ 57, 84 ; what is the whole amount of his stock ? SUBTRACTIOW. t. Subtract as follows : 7, 0107605 — 4, 901979865 2. '» " 35,0964-34,9895602 3. " " 670,4801—669,94013 4. " " 0,04217 — 0,03948 5. " " 0, 9080706 — 0, 8950326 MULTIPLICATION. 1 . Bought I7f yards of cloth at $ 2, 65 per yard; how much is the amount to pay ? 2. Multiply 10,09562X7,8059 0, 00867X 9, 0472 9, 80604X0, 0976 301,0605X0,003908 7503,09706X0,0009801 DIVISION. 6,0453 36,45097 I. Divide = : 2. Divide 3. n 4. 5» 6. '> 6. »> 9^8106 0,00438 196 ftUBSTlONS. 52, 0096 3, 09042 2. Divide = ; 6. Di?ide = 6, 49502 95, 763 0, 00652 655, 3708 3. " == ; 7. '> - 3,4096 ' ' 942,01)7 0, 0043106 0, 04609 4. " = ; 8. 0, 09459 0, 000762 MIXED aUESTIONS IN DECIMAL FRACTIONS. 1 What do 5 pieces of cloth of 28i yards each, come to, at |3, 37^ per yard ? 2. One pound sterling is equal to $4, 444 ; (with continued de- cimals of 4;) how much is £975^, expressed iu dollars ? ^ns. 14335,55122. 3. A captain of a vessel has on board 706 packages, each mea- suring 1-8 of a ton ; 89 others, each measuring | a ton ; and 405 others, each measuring | of a ton ; how many tons of lading has he ? ^ns, 264| tons. 3. A captain has on board 170 bales, each paying freight |1,25 ; 305 packages, each paying 87| cents ; 230 tons of other goods, each ton paying $12, 62^; and 6 passengers, each paying .$78,60; how much does bis whole freight and passage money amount to? ji ns. $2Q54,12h 5. A raft contains 305 pieces of timber ; of these 120 are oak, 36 feet long and 16 inches square; 50 pieces of oak, 45 feet 6 inches long, and 18 inches by 14 inches on the sides ; 166 pieces of pine masts, reckoned at 2 feet 6 inches square and 60 feet long. The rest pine timber, 17 inches square by 50 feet in length. The oak timber sells at 45 cents per cubic foot ; the masts at 80 cents the cubic foot, and the pine timber at 15 cents the cubic foot. How much money will the whole raft come to in the sale ? 6. For plastering a wall the mason has to receive 21 cents per square yard (or the square of 3 feet each way, and containing there- fore 9 square feet ;) the wall which he has plastered is 13i feet high, and 22 feet long ; how much has he to receive for it ? J^ns. |6,93, And how many yards does the wall contain ? j^ns. 33 yards. 7. How many square feet front of brick wall can be built with 3600 bricks, the thickness of the wall being the length of two bricks, and the end of the bricks being four inches by two? Jins. 1000 feet square. 8. A merchant makes 16i per cent, upon merchandise that costs him |7, 65 ; how much will his profit amount to ? = 7650 XO, 165 = ^ns. $1262, 25, (according to the princi- ples of decimal fractions.) 9. The tare allowed upon a certain merchandise is 2i per cent. ; how much will it amount to upon 7355 weight ? (Expressed as ^bove) = 283, 875. (^VESTIONS. 19T DENOMINATE FRACTIONS. ADDITIOir. 1. Add J67 6s. 7d. -f £3 4s. lOd. -{- Ss. 4d. -|- £9 14s. lid, -4- £23 17s. 5d. 2. Add 3 lb. 4 oz. 17 dwt. 5 gr. -|- 15 dwt. 17 gr. -f 17 lb. 3 dwt. 4 gr. 4- 17 oz. 15 gr. -4- 31b- 12 dwt. 6 gr. 3. Add 6 yds. 2 ft. 3, 4 in. + 17 yds. 5 iu. + 22 yds. 1 ft. 11 in. -J- 62 yds. 1 ft. 9 in. -{- 34 yds. 10 in. -\- 69 yds. 2 ft. 9 in. 4. Add 7 miles 3 farlongs 17 yds. -j- 21 m. 1 fur. 30 yds. + S4m. 3 yds. 5. Add 24 bush. 3 pecks + 19 bush. 5 pecks + 18 bash. 2 pecks -{• 42 bush. 1 peck. SUBTRACTIOir. 1. A grocer had according to his last inventory 317 lb. 10 oz. of sugar; 561 lb. 4 oz. of coffee; 451 lb. 6 oz. tea; 15 lb. 3 oz. pep- per; 3 oz. 6 dwt. mace; 152 lb. rice; 17 gallons rum. He has sold since, 283 lb. 6 oz. sugar ; 341 lb. 7 oz. coffee ; 349 lb. 5 oz. tea; 11 lb. 8 oz. pepper ; 2 oz. 6 dwt. mace ; 5 gallons and 3 gills of rum ; 121 lb. 7 oz. rice ; how much has he left of each kind .' 2. A man has to travel 75 miles; he walks thejfirst day 20 miles 3 furlongs ; the second 18 miles 5 fur. 20 yds. ; the third 23 miles 7 fur. 50 yds. ; how much of his journey remains eyery evening to be performed .'' 3. William the Conqueror acquired the throne of England the 26th December, 1066, and died 8th September, 1087. ' His son William the Second, who immediately succeeded, died the 2d August, 1100. Henry the first succeeded, and died the 1 0th De- cember, 1 1 35 How long did each of them reign .•* 4. Three men, starting at the same time from one place, arrived at another determined place, the first after 10 h. 16 m. ; the second after 12 h. 42 m. ; the third after 15 h. 3 m. How much did each of them arrive after the other ? MULTIPLICATION. 1. Bought 27 lb. 5 oz. 16 dwt. of drugs at the rate of $9, 75 the pound; how much will be the amount ? 2. Bought three bales of cotton, the first weighing 1016 lb., the second 998 lb., the third 1093 lb., at 17^ cents the pound ; what is the amount to pay? 3. A room is 22 feet 5 inches long and 18 feet 9 inches broad ; how many yards of carpet will it need ? 4. A wall is 8 feet 7 inches high, and 65 feet 9 inches in circum- ference ; how many feet of plastering will be in it .'* 5. Required the solid contents of a wall 74 feet 6 inches long, 2 feet 9 inches broad, and 24 feet 4 iachee high .'' 17* ^98 ^ITEITIONS. 6. Required the solid contents of a box 5 feet 2, 5 inches lonff. 3 feet 5 inches broad, and 2 feet 5,8 inches deep ? 6. How many cubic feet of earth will fill a dock 205 feet longf 75 feet broad, and 8 feet 7 inches deep ? DIVISION. 1. If 87 lb 6 oz. of coffee cost $18, 38, what is the price of one pound ? 2. What is the price per pound of spices, when 34 lb. 7 oz. cost ^25, 82 ? 3. What is the length of a piece of timber 15 inches square, the cubic contents of which is 69 feet 6 inches ? 4. What must be the depth of a square vessel, 1 foot 3 inches one way, and 2 feet 2,5 inches the other way, that shall hold 4 feet 2,5 inches cubic measure ? 5. What must be one side of an area containing 2015 square feet, when the other side is 50 feet 7 inches ? 6. If a horse runs 8 times around a circus in 1 h. 45 m. 20 s^,. how much time will it need for each turn ? 7. A lumber merchant bought 6527 cubic feet of timber, in 321 pieces ; how much-did each piece averaa;c in cubic feet ? 8. A brick wall, two bricks' length in thickness, is 69 feet long and 26 feet high ; how many bricks does it contain, each brick being 8 inches long, 4 inches broad, and 2 in^thes thick, when laid ? PRACTICAL QUESTIONS FOR THE SECOND PART. 1. A purchase of goods that cost ;^765,25 was sold for ^973,52 ; what was the profit ? 2. A man has ;^8264,91 debts, and his property amounts to j^743l,80; how does he stand? 3- Three men buy land, the one 5,212 acres, at ^2,25 per acre, the other bought 281 acres for ^600, and the third bought as much land as they both, for ;^892 ; what had the first to pay, how much land did the second buy, how much land had the third, and at what price did it stand him ? 4. A brick, when laid in the wall, has 7,8 inches length, 3,{> breadth, and 1,8 inches thickness ; how many bricks will it take to build a wall two lengths of bricks thick, 25 feet long, and 36 feet high ? 5. At 6 per cent, interest, what must be the capital that will produce an income of ;g;500? 6. A man having $600 a year, how much may he spend a day to save ^200 in the year ? 7. What is the interest at 7 per cent, of ;g[12,450? 8. Upon 20 hogsheads of Sugar, of 850 lbs. each, what is the tare, at 3 lb. for every hundred weight ? 9. Three persons purchase together ^500 of stock, at 5 per cent, premium, which brings in 8 per cent, interest; how much must each pay, and how much yearly interest will each have for his share ? (QUESTIONS. 19§ iOi A house is to be plastered, at 21 cents per square yard. Now there has been plastered an entry 35 feet loug, and 1 1 feet 6 inches high on both sides, the 2 ends being given in, as compen- sation for the vacancies on the sides. Two roon;8 of the same height, in each of which, two sides of 20 feet long are reckoned full, and one end of 18 feet also reckond full, to compensate for the vacancies, the fourth side is given iu. Two upper rooms of 20 feet long, 14 feet broad, and 9 feet high, are reckoned in the same manner as those below, and one room has 14 feet by 16 feet 6 inches, which is considered as plastered all round. How much will the expense of the whole plastering be ? 11. Suppose the above entry and rooms were to be wainscoted with simple boards, at the rate of 1 1,23 for every hundred square feet, what would be the expense ? 12. A quantity of goods is bought for ^3,521, and sold at 15 per cent, loss, for what was it sold ? 13. A dock to be filled in, has 250 feet length, 95 feet breadth, and the perpendicular depth being 8 feet on an average, how many cart loads of earth are needed to fill it, at the rate of 7 cu- bic feet for a cart load ; and how much will it cost at 6 cents per load ? 14. How much will the glazing of a house cost, that has 28 windows, each of 24 panes of glass, at the rate of 13i cents for each pane ? 15. How many bricks are there in a wall two lengths of a brick thick, 20 feet long, and 38feet high, the bricks being of the dimen- sions stated in the fourth question ? 16. An old tower 40 feet square on the outside, has at first, a wall 10 feet thick for 20 leet of elevation, then for 36 feet the wall is 8 feet thick, then for 16 feet it is 5 feet thick, the outer sides be- ing perpendicular; how many cubic yards of stone are there in these walls, (neglecting doors and window opeoiwjjs) how much will the stones cost, at 22 cents the cubic yard, and how much will the building of the wall cost, at the rate of 29 cents for every cubic fathom ? What will be the weight of stones in it, the cubic foot being reckoned at 178 lbs.? 17. A carpenter has 6i cents pef cubic foot for hewing timber : now he hewed 25 pieces of 15 inches square, (on each side) and 36 feet long; 16 pieces of one foot each way, Hud 42 feet long; 28 pieces 18 inches by 20, and 26 feet long ; 12 pieces of 10 inches each side, and 32 feet long ; and 15 pieces of 8 inches? by 12 each side, and 18 feet long. How much money has he earned ? 18. Two rooms are to be painted all round, the height of which is 12 feet 4 inches, the length of one, 32 feet, and its breadth 24 feet ; the other, 18 feet 6 inches long, aud 16 , . S The child's age, 8 yrs. '^'^' I The father's age, 36 yrs. TO EXTRACT THE SaUARE KOOT. 1, To extract the Square Root of 1296 2. (( li (( « t( (( 7921 3. o {( u l( « (( 9899 4. tt (( -xac{iy equal to 5 miles? 7. How many eg*s will be neeueJ to lay 4 feet apart, to occa- sion a roan who has to pick thera uj* oiie by one, and bring them to a ba«ket 3 yards behind the firsf, to hiive to walk 2,75 miles ? 8. To find 7 arithmetic meaur between 6 and 46. 9. How many strokes does a clock strike in one whole day, by our common division of time, striking from 1 to 12. 204 IIUESTIOKS. 10. A man having to pick up 102 eggs laid in a row on the grouod at one yard from each other, and carry them in a basket at two yards from the first, while another has to walk a distance of three miles from the same place and back asjain, which of the two has the advantage in the distance to be walked through ? and how much ? Ans. The one who walks, has only 56 yards more to walk. 11. If one cent is placed on the first square of the chess board, and one more on each subsequent square, how many dollars will be upon the whole board ? Ans $20,80. GEOMETRIC PROGRESSION. 1. What is the sum of one hundred terms of the powers of 2 . 2. What is the sum of 30 terms of the powers of 3 ? 3. The first term of a geometric series being 20, and the ratio I, what will the 25th term be ? ALLIGATION. 1 A man in a month of twenty-six working days works 6 days at the rate of |1,15 a day, 5 days at 75 cent? per day, 3 days at $2 per day, ten days at $1,50 a day, and is idle the 2 remaining days ; at what rate per day does he earn, counting the whole of the 30 days in a month ? 2. What is the fineness of a mixture of 2 oz. of gold 23 carats fine, 7 oz. 22 carats fine, 9 oz. 17 carats fine, and 3 oz. 20 carats fine? 4. A merchant sold a quantity of cloth, namely, 150 yards at |3,75, which cost him $3 per yard, 720 yards at $5 the yard, which cost him $3,75 per yard, 305 yards at $7,50 per yard, which cost him $6,35 per yard, and 100 yards at $2,50, which cost him $2 per yard ; how much did he make per yard on an average ? 5. A mixture is to be made of silver, some of which has cost $1,10 per oz., some 97 cents per oz., and the rest 88 cents per oz. ; the mixture is to weigh 3 lbs. ; how much is to be put in of each, to make the intrinsic value of the silver just a dollar an ounce ? COMPOUND INTEREST AND ANNUITY. 1. A man having an income of $5000 a year, rHves one quarter of his income a year, which he puts to iiiierein.iual income, and one quarter of that sum in addition, which ho t ike^^ from his ca- pital bearing interest at 5 per cent. ; how many y^'ars will he be, in spending the whole capital itself, from the first beginning ? Q,€ESTlONS. 905 PROMISCUOUS QUESTIONS. 1. The sides of two square pieces of ground, are as 3 to 5, and the sum of their superficial content is 30600 square feet; what is (he \ea^\.h of the sides of each? ^ns. 90 feet and 150 feet. 2. Three young men entering into partnership, agree to make a common stock, to which each shall contribute in the ratio of the sum of the ages of the two other partners. A is 24 years old, B 27 years, and C 31 years ; what will be the share of each ? 3. A parcel of tobacco is sold, .some at 12 cents per pound, the rest at 15 cents per pound ; the proportion of the first to the latter, was as 4 to |, and the amount of the sale $380 ; how many pounds were there of each kind ? - V Of the first, 1500 lbs. '^^- \ Of the 2d, i:333j lbs. 4. A grocer bought cofiee, 3 bags of 80 lbs. at 21 cents per lb., 6 bags of 53 lbs. at i;4 cents per lb., and 9 bags of 90 lbs. each, at 18 cents per It)., and sold the whole together at 22 cents the pound ; what did he make by it .=• j^ns. $47,84. 5. What fraction is that, to the numerator of which, if 1 be added, it becomes ^, and if 1 be added to the denominator, it be- comes i ? Ans. y*3.. 6. The quick step of troops in marching, is 2 steps of 28 inches each in a second ; how far wid such troops travel in a day of eight hours? 7. The captain of a vessel, of which he owned ^, sold out the half of his share ; he had before the sale $350 annual profit from it, besides his wages ; how much remains to him annually after the sale of this part of his share ? 8. A draper sold from a piece of cloth, i at $5 the yard, one fifth at $4 per yard, and one sixth at $4,50 per yard ; by this he obtained $168 ; how many yards were there in the piece ? ^4ns. 60. 9. On the first of January 1793, a royalist in Europe agreed with a democrat, to pay him 3 cents per day, until the restoration of the Bourbons, on condition of his paying him one louis d'or on $4,44, every day after that restoration. Taking the first of Au- gust, 1814, for the day of the return of the Bourbons, how would their account stand on the first of January, 1827, omitting all in- terest ; and how, on calculating compound interest for every day from the epoch of these payments, to the first of January, 1827, at 5 percent, annually ? Without interest, the first would have paid $236,43. the second, $21760,44. 10. A merchant gains in trade such a sum, that $320 has the same ratio to it, as five times the sum has to $2500 ; what did he gain ? ,qns. $400. .11. Two brothers comparing their ages, find that the sum of 18 206 (^UESTIOXS. both ages is to that of the elder, as 19 to 7, and to 30, as 9 to the agf .f the other ; what are their ii'^cs ? ^ ( 21 ^n,.'^ 9. 12. The difference rif the sfdes of two square rooms j* to the side of the greater, as 2 to 6, and the difference of th*>ir square con- tent, is = 128 feet ; what are the sides of each of the^e rooais ? jins. U8. ( !4. 13. The profits of two men !n their work, are as 8 to 5, and the proJuct of the numbers expressins their profits is 360 : what was the profit of each ? ^ { A"24 •^"*- \ Jl5. 14. A merchant 2:aining $7500 in 6 years, wjth a capital of .$15000, what would he gain at the same rate in 11 years, with a capital of $21000.'' 15. It OIK man travels 52 miles in a Jay, walking 12 hours, and another 61 miles in 11 hours, what will be gauged in time, on euch, by sending both at the sime time to meet from two places 180 miles from each other, to exchange dispatches, instead of ppndin<. Years, 60 0,016606 HO H. I 3600 0,0002777 86400 0,0011674 0,. H. M. S. 365. 5.48.48 y. 1 CIRCULAR PARTS. Seconds. Minutes. Degrees. Circum- ference, 0,016666 j 360. 0,0002777 ^5, 01 6666 .1 1296000 0,00000077 x>1600 0, 0000462 360 0,002777 c. 1 S08 S U w o h4 » *j 9 If TABIES. c^ - ^* ^ §co ^1 J i CO CO CO S .-^ CO ^ CO CO o CTi cT 1 CO -* CO CO lO CO CO O <>* _4, CO r.00 O O ^ o o-^ o c2 p. 40 0,025 320 0,0031350 960 0,0010417 22144 0,0004516 so 1 2,75 0, 3636364 110 0, 0990909 880 0,0011361 2640 0, 0003787 60n96 0,0001642 ^ ''o- CO lO CO irfco o 220 0, 0045454 1760 0, 0005680 5280 0,0001893 121792 0,0000821 o'^ CO CO CO CO CO CO CO o-^ 16,5 0,0606061 660 0,0015151 5280 0,0001893 15840 0,0000631 365376 0, 0000274 1 CO « '-I o o o o 00 00 CO o 198 0,0050505 7920 0,0001263 63360 0,0000158 1 90080 0, 0000053 CO ^ o rt O CO o CO o ^ TABIJfiS< 209 I/} «3 S ■^:-. ■{]'■■ •i ^ -■••y i J a ^ 9^ •5 < -- G< o S coo CO CO ^ o c§ sS *^ o 8 § CO s; •* o ©< o o 1 i.- o liii o 1 >. '-« CO G^ O O GO CO o o o- " o '^' ° fi " 05 ^ O si o si CO O O O O o ©< CO o lO o CO o ^ o o J CO o o ii O CO O CO O ^ o CO o G^ O t- o &< o ! 18* nQ TABLES* CUBIC MEASURE. Inches. F..> Yards. Fathoms. IN. 1728 0, 0006787 F. 1 46656 0,00002143 27 0, 037037 Y. 1 376648 0,00000265 216 0, 0046296 8 0, 125 FTH. 1 CLOTH MEASURE. Inches. JVails. Quarters. Yards. Ells. IN. 2,25 0, 4444 NL. 1 9 0, 111111 4 0,25 1 36 0, 02*^77 16 0, 0625 4 0,25 Y. 1 45 0, 02222 20 0,05 5 0,2 1,25 0,8 E. 1 DRY MEASURE. PinU. Gallons. Pecks. Bushels. PT. 8 0,125 G. 1 16 0,0625 2 0,6 PK 1 64 0,015625 8 0,126 4 0,25 B. 1 Eight Bushels make a Quarter ; but as this is not used in any part of this country, any more than the Wey and Last, we have omilted them. TABLES. SIX CO g k; ^ i 40 as 1 •> CO CO e CO "% s ^ o o I CO Q CO lO CO m to > T-^ CO o ^ G^ • o:) rj« CO 1 o O CO CO O ^ ^i CO ,^ CO 05 G^ l> ^ O &» ex. kO CO G< O o O o O o^ o" d^ o o' "* ©^ &< o G^ CO CO Tf (N eo >0 CO I:- 00 2g r a . CO Oi ©< O) CO a -* Oi CO i-H CO G^ o « «> — £ O GO O lO o CO o o -8 O o o 1 o ^ o O O 1 o" o" o Habit alone determines, in different countries where these measures are used, to which purposes the two different measures of liquids are applied besides the two liquids of which they bear the name, and these habits vary from time to time. In the state of New- York, Beer measure is little u^ed, but the ordinary mea- sure for all liquids is Wine measure. u^ TABLES. BEER MEASURE. Pints Quarts Ua'iotis. ■ .arr ,s Ho^^sheinh Buiti, 2 P. 0,5 1 8 0,125 4 G 0, 25 . 1 288 0, 00347 144 j 36 0,00o 4 0,02777 B. 1 4<>2 n H ' r)4 1,5 1 HHD. 864 432 1 108 3 2 iB. TROY WEIGHT. (Used for gold, -liver, jewels, and retail dealing.) Grains. Pennyweights, \ Ounces. Pounds. GR. 24 0,0416660 DWT. 1 480 0, 0020833 20 0,05 1 5760 0,0001736 240 0,0041666 12 0,08333 LB. 1 APOTHECARIES WEIGHT, (Used in conij.iiuudit.^ ri, .-diciues.) Grains Scruples. Drums Ounces. Pounds. GR. 20 0,05 sc. 1 60 0,016666 3 0, 333 DR. 1 480 0, 0020833 24 0,041666 8 0, 125 OZ. 1 5760 0,0001736 288 0, 0034722 96 0,0104165 12 0,08333 LB. 1 TABLES. Sl<5 AVOIRDUPOIS WEIGHT . Drams Ounces. Pounds (^uortrs. \iwt. Tons. DR. 16 0, 0626 oz. 1 266 0,0039014 18 0, 06566 LB. 1 7168 0,0001396 448 0,0022321 28 0,0367143 an. 1 28672 179^' « », 0005680 112 0,00P0«78 4 0,26 nwT. 1 673440 35840 2240 0, 0004464 80 0,0126 20 0,06 TONS. 1 This kind of weight is used in every other case of mercanfile transaction, whether in the great transactions of general com' merce, or in the retail trade. oz. dwl. gr. 1 lb. Avoirdupois = 14. 11. 16 Troy. 1 oz. " = 18. 61 ♦« 1 dr. ♦* = 1. 3i '* Before the last law in England, of 1826, regulating weights and measures, the following were the cubic COHtents of the dilBferent measures of capacity ; viz : The Bushel, 2160| cubic inches = a cylinder 8 in. deep, 18,6 in. diameter. The Gallon, dry measure, 268| cubic inches. »* " for beer, 282 " " <* " for wine, 231 »' <* These two latter gallons have to each other the same ratio as the weights of Avoirdupois and Troy. By the law of 1826, The Bushel contains 2217,6 in. cubic. The Gallon '' 277,2 ** and is used indiscriminately for dry and liquid measure. The capacities are determined, not by measurement of the cubic contents, but by the weight of pure water at the temperature of 62° of Fahrnheit's thermometer contained in the vessels ; the bushel holding 80, and the gallon 10 lbs. avoirdupois. S14 OTABLES. a •- ^ SJ « ° M a t- SB * ^ a> 5 *J ♦J iS 6/5 C/2 ,S (^ 1 e 1 s c c 'J ■1 1 f Si f 1 cr! 1 '^ g£ i £ £ :2 ! — r-. ? i ^ ''^ S 1 ^ 1 £ V ^- : . . ^ • o -■ - : £ •£ =o ;l 5^ ;. Sis 3 ^ ij^ : : : 1 i i : ^^r-^?| i i. — : ! j ; • c2 a. - j: -3 c», :.- ■ i:^ ai 5? f- ci a? O ail ■8,- si |2 5^ rj J^ I' 3 S Value of Foreign Coins accoHmg to the Laws of the United Sfiites. Gold coins of G. Britaio aud Portugal are rated at ;JJ 1 for 27 grainj- " France »' " 27| '> »' SpaiQ " " 28J " TABLES. 215 Coiirtc of All 't Jam, Antwerp, ditto, Au^^burg-, ditto, Basil and i Zuric, \ Berliu, ditto, Boloj^na, Coustan- ) tinople, \ ditto, Copenhageu ditto, Frankfort ) OD Main, \ ditto, Genoa, < Geneva, Hamburg t £l Altona, S Lci!!zi;^& ) Dresden, \ Lfsbou, Leghorn, London, Milan, Naples ? New-York, Palermo, Petersburg, Spain, Stockholm, Turin, Venice, Vienna, 1 *? gtMS § or c«. rtc'^s ^"^ r 3 r 3 s: 10<3 r 3U0 ^ " rorg luO V JuO ^ jt; i r 3 r 300 cr 1 r 300 1 ^1 30. 100 g 100 s I 100 100 3 V iOO r 3 1 1: I 3 ^•■ , 3 ] (Q L i iC 1 r 1 r 1 1 & ( 1 r • > r 3 r -JUO \l 1 1 2-_ Denomtnatioti. Fr»ar«, . . Francs,. ., Fiaiics, . . P.anrN. . Florin Ct, Francs, Frrmcs, B;-«nco Prii&siau, Fraucs, Frniics, Piastre, Francs, Rix dollar,. Francs, Fraiirs, Francs, Piaster (ot .15s. h .b.) Livres courant, .. vlark banco, Fr-\iics, , Francs,. , Francs, P) istre(of8Reals) .ij sterling, Francs, Francs, Franc?, Lire iinue«-iale, .. Ducat (of 10 Car- lin.,) Franc, Friioc,. R^ible, Pistole(of 32 reals) Piastre (of 8 reals) Francs, Francs, Francs,. Frnno, Florin, lily. .70 5.->, ! 99,50 U7 25. ^9,50 78 56 102,5 2,90 ♦35,5 4,45 79 99,5 99,5 4,8< 166 190 24 ! 76 480 5,10 23 21 55 71irelo l.OS 4,20 0,1 8i' 46 4,40 15 3, 75 25 50 61 23 2,53 Denomination. Deniers groats. Ueniers groats. Francs. Florins courant. Centimes. Francs. R-ix dollars. 80 Francs. Sols. Piastres. Francs, Centimes. Francs. Rix dollars. Francs. Francs. Francs. Francs. Francs. Sols Lubs. Rix dollars. Rees. Francs. Francs. Pence sterling. Sols imperial. Sols Courant. Francs. Franc?. f^ollars. Grains. Francs, Francs. Francs. Shillings. Sols Piedmont. Ducats banco. Cruezers. Francs. END. TABLE OF CONTENTS. Introduction. Pcig^' Part I. — First Elements and Deductions of the Four Rules of Arithmetic. Chapter I. — Fundamental Idea of Quantity. Sys- tem of Numeration, ... I " II. — General Ideas and Notation of the Four Rules of Arithmetic, . . 13 " III. — Four Rules of Arithmetic in Whole Numbers, ..... 18 " IV.— Of Vulgar Fractions, ... 39 " v.— Of Decimal Fractions, ... 63 " VI.— Of Denominate Fractions, . . 76 Fart II. — Practical Applications of the Four Rules of Arithmetic. Chapter I — General Principles of the Application of the Four Rules of Arithmetic, . 91 " II.— Application of the Four Rules of Arith- metic to all kinds of Questions involv- ing Fractions of either kind, . 98 Fart III. — Of Ratios and Proportions. Chapter I. — Elementary Considerations of Ratio, 108 " II. — Arithmetical Proportion, . . 113 " III. — Geometrical Proportion, . . 116 " IV —Rule of Three, .... 124 " v.— Compound Rule of Three, . . 136 " VI. — General Application of Geometric Pro- portion, 144 Part IV. — Extension of Arithmetic to higher Branches and other practical Applications. C/jflp<«r I.— Of Square and Cube Roots, . . 150 " II. — Of Progressions or Series, . . 165 " III. — Of Compound Interest. Ideaof Annuity, 180 " IV.— Of Alligation, or Mixture of Objects of different Value, . . . 187 Collection of Questions, . . . , 193 Tables, . , 207 i 14 DAY USE RETUKM TO DESK FROM WHICH BORROWED LOAN DEPT. This book is due on the last date stamped below, or on the date to which renewed. Renewed books are subject to immediate recall. CL F m ! J A N 2 1966 6 > m&'o Lii m^Mm^iAm LD 21A-60m-3.'65 (F2336sl0)476B Genei. University o^ Berkele. - / YB 17369 ^ 9181 i^l ^}il6Z. ^57 THE UNIVERSITY OF CALIFORNIA LIBRARY