UC-NRLF
172
i^csr^-m
i
LIBRARY
OF THE
University of California.
fl
>>
' 0u
{y^
/
"U y
V K
^^^
THE
PHILOSOPHY
OF
ARITHMETIC,
{Co7isldered as a Branch of Mathematical ScieJice)
AND THE
EJLEMEMTS OF ^LGEEMAs
DESIGNED FOR THE USE OF SCHOOLS,
AND IN AID OF PRIVATE INSTRUCTION, Rv JOHN WAT/K^FR-
PORMIRLY FELLOW OF DUBLIN COLLEGE.
" Would you have a man reason well, you must use him to it betimes, and ** exercise his mind in observing the connexion of ideas, and following them in ♦* train. Nothing does this better than Mathematics; which therefore, Ithinl', ** should be taught all those who have time and opportunity : not so much to make " them Mathematicians, as to make them reasonable creatures,^*
Locke's Conduct of the Understanding,
D UBUN:
PRINTED BY R. NAPPER, 29, CAPEL-STREET.
Sold by DuGDALE, Dame-Street; Keene, College-Green; Mahon, and Porter, Grafton-Street ; Mercier, and Parry, Anglesea-Street.
1812,
0 1\%
tir Mr. Walker gvoe$ private instructions to individuals^ or parties of six persons^ of either sex^ in the Subjects of this Treatise^ in the Elements of Geometry^ in Astronomy^ and the other Mathematical Branches of Natural Philosophy ; as "well as in the Greek and Latin Classics,
No. 73, lower Dorset-Street,
TO
Mss. AGNES CLEGHORN^
As to a Lady who is well ^uali£e4
To estimate the Execution of the following Worker
AND WHO EVIDENCES, BT HKR XXAMPLEy
That superior intellectual Endowments,
Improved by more than ordinary Acquirements!
IN LITERATURE AND SCIENCE,
Are perfectly consistent
With the Retiredness of the Female Character^ With its attractive Graces,
AKD WITH THE MOST EXEMPtAJlT SISCSARGe OF
DOMESTIC DUTIES 5
This Treatise is respectfully inscribed By her faithful
And much obliged Servant
THE AUTHOR^
84721
PREFACE.
►•e*©I^I®®«e«
Arithmetic is one of the two great branches of Mathematics; and, when scientifically treated, needs not fear a comparison with her more favoured sister. Geometry, either in precision of ideas, In clearness and certainty of demonstration, in practical utility, or in the beautiful deduction of the most interesting truths.
In the order of instruction, Arithmetic ought to take precedence of Geometry ; and has, I conceive, a more necessary connection with it, than some are willing to allow. " Number," as Mr. Locke remarks, *' is that which the mind makes use of, in measuring '' all things that by us are measurable.'' And I question whether the doctrine of ratio in Geometry has not been needlessly obscured, by a vain attempt to divest it of numerical considerations. Upon this sub- ject I have elsewhere expressed my views more at lar^e.
A Eut
VI PREFACE.
But as generally taught, Arithmetic has been degraded from the rank of Science, and converted into an art almost mechanical ; useful indeed in the compting-house, but affording more exercise to the fingers than to the understanding. It is commonly taught by persons, v^ho are rather expert Clerics than men of Scieiice^ and are themselves strangers to the rational principles of the most common operations which they perform. The absurd questions current among them about the product of money multiplied hy money ^ &c. afford a sufficient exemplification of this remark. Thus, while there are few things which chil- dren are more generally taught, than the technical art of calculation, perhaps there are few things of which men are more generally ignorant, than the Science of Arithmetic : and this ignorance indeed is betrayed by their contempt of it, as a branch of study beneath a scholar.
Yet, when rationally taught, it affords perhaps to the youthful mind the most advantageous exercise of its reasoning powers, and that for which the human intellect becomes most early ripe : while the more ad- vanced parts of the science may try the energies of an understanding the most vigorous and mature. — Re- duced also to a few comprehensive principles, and di- vested of that needless multiplicity of various littles^ by which the subject is commonly perplexed, — the knowledge . of it may be communicated with unspeak- ably
PREFACE. Vll
ably greater facility and expedition j and, when once attained, will not be liable (as at present) to be soon forgotten.
To present Arithmetic in that scientific form, Is the object of the following treatise ; which, it is hoped, may prove beneficial to the young of both sexes, and not uninteresting to some of more advanced age.
The scientific principles of common Arithmetic are so coincident with those of Algebra, (or Uni- versal Arithmetic) that — to persons acquainted with the former — the Elements of the latter offer no serious difficulty. Of the Elements of Algebra therefore I have given such a view, as may open that wide field of science to the Student, and enable him at his pleasure to extend his progress, by the aid of emy of the larger works extant on the subject.
Having designed this work for the Instruction of those, who come to it most uninitiated in Science, I have aimed at giving a clear and full explanation of the most elementary principles : I have endeavoured to be familiar and plain, yet without departing from the rigidness of demonstration. How far I have succeeded •in this attempt, other judges must decide. I shall think myself compensated for my labour, if it should prove, in any degree, the occasion of rescuing the Science of Arithmetic from general neglect, and
of
Vni PREFACE.
of introducing this branch of Mathematics into the system of liberal education.
Mr. Locke's remark, which I have prefixed as a motto to this treatise, is well worthy of attention. It is not so much the intrinsic dignity of Mathematical Science^ — nor even its extensive application to the most im- portant purposes of civil society, — that recommends it as an object of general study. In its influence on the mental character and habits, it possesses a still stronger claim for adoption into the course of general education. No study so much, as that of Mathema- tics, contributes to correct precipitancy of judgment ; to promote patience of investigation, clearness of con- ception, and accuracy of reasoning ; to communi- cate the power of fixed attention, and closeness of thinking.
These are habits universally important ; and to be formed in early Hfe. Nor is it necessary, in order to derive these benefits from mathematical studies, that we should pursue the study to any great length, or become profound Mathematicians. Here it is of much less consequence how far we proceed, than that we make ourselves fully masters of the ground — as far as we proceed; — that whatever we learn, whether little or much, should be learned thoroughly. A smattering of half information about a variety of sub- jects, is calculated to excite that vanity and presumption
of
PREFACE. IX
of knowledge, which is repressed by a radical acquaint- ance with the most elementary principles of some one science*
May I be allowed to express my opinion, that some degree of mathematical knowledge is no less useful to females, than to the other sex ; and importantly adapted to counteract the tendency of an ^ucation, which too often enfeebles the judgment, while it excites the imagination? Indeed it is with satis- faction that I perceive that absurd and illiberal pre- judice rapidly giving way, which would shut the door of solid information against those, on the formiation of whose minds so much of the welfare of society must depend.
In bringing this Volume through the press, I have encountered difficulties, which might not be expected to occur in a City< — the metropolis of iREi^AND, and the seat of a learned University. Some of those diffi. culties have been such, as necessarily make the price of the work higher, than is generally affixed to Vo- lumes of an equal bulk : — though it may be remarked that, if a little more of the modern art of printing had been employed, the Volume might easily have h^^Vi s\yelled to twice its present size, without any increase of the matter.
Notwithstanding
S .PREFApE*
Notwithstanding much pains bestowed on the cor- rection of the press, I have to intreat the indulgence of the reader for the following errors ; some of which escaped my eye, and others have been generated after the passages had undergone my last revision*
ERRATA.
Page
8. line 21. for " difference between 28 and 5," read^ " difference between 23 and 5." 14?. 1. 19. ready " engage'*
15. 1. 23. for " 3681 and 108, ''read, " 3681 and 1080"^
16. 1. 15. for " § 61 and 62," read " § 62 and 63." 22. 1. 34. read, '^ exercise."
34. 1. 16. for " 5 times 9 is 6," read, " 5 times 9 is to 6."
35. 1. 27. for " is equal to a" read, '* is equal to c." 47. 1. 9. read^ «« whosp sum is 1^ or \\t'*
77. Ex. 9. The last term should be " ^ax^:'
81. last line, for « — 2>axy," read ** — 'iaxy" r
83. 1. 2. for <« ^3/*-— y" read «' xf-^y^''. ibid. 1. 4.
for " j/^", rmt/, <<j/V* ibid. 1. H. for *' ;n/," rea^,
" ;n/\"
87. 1. 25. for - =:cZ, and - =f, r^«^, — = c, and
m 7)1 m
- =: d. ibid. 1. 28. for " aj = cfw, and 6 = cm,'' read^ m
*^a=cm, and b=zd7n."
88. 1. 4. for " da or c6," rm^, «« c« or £/5.". ibid. 1. 22.
for «* last," r^«^, " least." 90. 1. 18. and 20. for " lOOth." read, " lOOOth." 105. 1. 8. from bottom, for << |," read « f"
119.
errata;
Page
119. 1.2. for ± , read, zl. 2x 5
- ** 9 *'
155. I 2. from bottom, for <*4,*' ready -
156, 1. 25. for «« 57," r^fl^," « 27.'*
157 1. 5. from bottom, and last line, for << £60^" read,
<« ^100 " 162. 1. 11. for " §284," read, <« §290." 165. 1. 18. for ^« 4056," read, '« 7056." 168. 1. 2. from bottom. After <« +7." add, <* +6 + 5 + 4<."
THE PHILOSOPHY
OF
arithmetic;
^c.
CHAP. L
Nature and Principles of the Arabic Niimeral 'Notation, Its Advantages above the Greek and Roman, Insensibility to the Magnitude of high Numbers, Duodeciinal Notation,
1. THE first tiling in the subject of* this treatise, which claims our attention, is our^jtresent method of numeral., nQtatiojx j or the method employed For designating numbers ( by the aid of written characters. Faj;:J[t, as well as some / other most important improvements in Arithmetic, we^ are. indebted to the Arabs. It was brought by the Moors ]; into^^ain j, and John of Basingstoke, archdeacon of Lei- S cester, is supposed to have introduced it into England ( about the middle of the 1 1 th centur3^ It is one of those ( inventions, of w^hich we often enjoy the advantages, with- out duly estimating their importance. Simple, ingenious, and highly useful, it is yet so familiar to us from our child- hood, that it fails of engaging our attention, or exciting our admiration.
2. We may be impressed however wdth a conviction of \ its ingenious simplicity, if we reflect on the endless va- / rieties and indefinite magnitude of numbers ; and then \ observe that we are enabled, by the aid of only ten cha- (
B racters
)
( 2 j
facters (t"he nine significant figures and the cypJier) tcf designate any numbers whatsoever with the utmost facility and distinctness ; and this, in a form which subjects them most conveniently to arithmetical computation. The im- portant utility of the contrivance it may be sufficient for the present to illustrate by the following remark. Most children of a very young age can with ease multiply or di^ vide the number 67,489 by the number 508. But let the same^niimbers be expressed by the Roman method of no- tationT \nuci\ prevailed in Europe before the introduction \ of the Arabic, thus — Ixvii.cccclxxxix and dviii ; — a man / will be puzzled to perform either operation. The Greeks / employed a numeral notation similar to the Roman : and it is truly wonderful how their mathematicians (even with the aid of some mechanical contrivances) surmounted the difficulties, which they had to encounter in their arith- metical calculations ; while we know that they were en- gaged in some of a very long and complicated nature.
3. Yet when we examine the fundamental principle of the Arabic notation, it becomes a matter of surprise that the invention was not of earlier discovery ; for it proceeds on a principle extremely simple, and one that must have been employed in all ageSy whenever there was a practical occasion of* counting any very large number. We may illustrate the principle by supposing that we had to count a great heap of guineas. It is^ plain that unless we employ some check on our numeration, we shall be very apt to lose our reckoning, and get astray as we advance. What then is the most obvious method of securing accuracy in our reckoning ? Is it not to count by tens, or some fixed number, beyond which we never shall proceed ? Thus when we have reckoned ten guineas, we may lay them aside in one parcel ; and proceed to count another parcel of ten. But to prevent the number of these parcels from accumulating so as to lead us astray, whenever we have counted ten such parcels we may make them uji into a rouleau, containing therefore ten times ten guineas, or one hundred : and whenever we have ten such rouleaus, we may combine them into one set, consisting of ten hun- dred, cv a thousand, guineas : and so on. And by this^ simple contrivaJice it would never be necessary to reckon beyond the number ten. Now it is precisely upon this principle that we proceed in designating numbers by the
Arabic
( 8 )
Arabic notation. The several columns of figures, from the rijght hand column, are the compartments in which we dispose the several combinations of ten. The first column on the right hand is the place for all odd unk^sr^Jjelo^^ the next to it on the left hand, or second column, is the place for aU^^parcels of Jen, below ten such the
third column, for a3Lj2ar£3[§_ of a Jmndj^^^^ (or ten times ten) below ten : the fourth, for all parcels „o£^gtihQUgand (or ten hundred) below ten ; the fifth, for all parcels of ten thousaiid- below^ien i the sixth, for all parcels of a himdredjllj^^ (or ten times ten thousand) below ten : the seventh, for all^parcels of ten hundi'ed thoi\sand (or a million) belgjBLJLeilj ^^'
4. Thus by the help of the nine significant figures and ) the cyj^he'r we are able to designate all numbers however \ great ; and this, wlide ea,ch of the figures (called the ten/ digits, from the Latin wore! signifying a ^/7^6^r) always x^gsl tains tlie_sa^^ For example, in)
the two numbers 57. and 570, the character 5 denotes in each the number five, and the character 7 the number seven : but in the former th6 5 standing in the second co- lumn designates five parcels of ten each, or fifty ; but in the latter, where it stands in the third column, it desig- nates five parcels of a hundred each, or five hundred : and in the former, the 7 standing in the right hand column designates seven units ; but in the latter, standing in the second column, designates seven tens, or seventy. And thus we see that the cypher, though it denote that there is 710 number belonging. to its column, yet must be written ; in m;der to bring the significant fig]y|i:eg into .their proper pkces. If therefore I want to express the number Jour 7nillion and sixty-eight tJiousand andjifty-three j the seventh column being the place of millions, the character 4? must be followed by six figures ; and the fourth column being the place of thousands, the characters 68 must be, followed by three figures : and thence I conclude that besides the significant figures 4, 68, and 53, a cypher must be inter- posed between the two latter, and aitother cypher between the two former : thus — 4068053.
'5. To facilitate numeration, we commonly mark off by a comma every period of six figures, commencing from the right hand, and often semi-periods of three figures. And as the name of a inillioii is given to ten hundred thou-
B 2 5»and,
( * )
gand, so ten hundred thousand millions are called a billion ; the place of which therefore commences at the thirteenth column. In like manner the names of a trillion^ qiiadriU lion^ &c. are given to ten hundred thousand billions, tril- lions, &c.
6. But here it is to be observed, that the facility vv^ith which we can designate the highest numbers, and perform every arithmetical calculation on them, has occasioned an insensibility to the enormous magnitude of the numbers of which we speak* One billion is very easily mentioned, and easily designated by art unit followed by twelve cyphers : thus— 1,000000,000000. A child ako can multiply or di- ( vide that number. But perhaps the reader will be sur- / prised at the statement that there is not one billion of se- ") conds in thirty thousand years : though there/ be 60 seconds in every minute, 60 minutes in every hour, 24 hours in every day, and in a solar year 365 days 5 hours 48 mi- nutes and about 48 seconds. At that calculation, the precise number of seconds in 30,000 years is only 946707,840000 ; or above 50 thousand millions less than one billion. So that the number of seconds, which have passed since the creation of the world, is considerably less than the fifth part of one billion. In fact it is only by some such considerations that we can form any conception of numbers so immense. '^ 7. From the view we have taken of the Arabic notation, \ it is plain that a cypher, wherever it occurs, increases ten- ( fold the value of every figure standing on its left hand 5 but does not affect the value of the figures standing on its right hand. It appears also that the several columns may be conceived to be headed with their respective titles, as par^ nels of a thousand each, of a hundred, of tens, &c.
8. If the reader revert to the illustration adduced in § 3. he may observe that, instead of counting the heap of guineas by tens and combinations of tens, we might as well count by twelves and combinations of twelves ; or by any other fixed number sufficiently low. And to the numera- tion by twelves, for instance, a notation similar to the Arabic may be applied, only introducing two new characters to designate the numbers ten and eleven. Then the figures 10 would denote the number twelve; for the 1, standing in the second column, would denote one parcel of twelve : I and the figures 203 would denote the number two hundred »nd ninety-one 5 for the 2, standing in the third column,
would
(
( 5 )
would denote two parcels of twelve times twelve each, that
is, two hundred and eighty-eight, a >-w^*ij«<^'-^"^ /o*^ <»h^»4^ ^-^m^
9. And certainly if, this duodecimal notatlQlL haihfifi|;i -s y originally adopted^ and the language accommodated to it / ' By alfording distinct names for the several combinations of twelve, it_ would JiM£--pcitss£S&ed ^. .consider able ad va ntage above the feg'wa^ notatiQn, which proceeds by combino^- ^ itions often. ^^rlSe number twelve admitting fouy dLj % visors, (namely 2, 3, 4, 6) while the number ten can.J;iB *
jevenly .ciiyidecl. only by 2 and 5, we should be much le^ frequently involyed in, fractional remainders than at pr^j, sent. And if all the divisions of measures, weights., coin§, / 6cc. ran in the sa^me duodecimal progression, the practical k advantages would be very great.
10. But it appears from the structure of ajl known lan- guages that numeration by tens has been adopted by all na- / tions in all ages, rather than numeration by twelves, or any \ , other number. And, thi3 is. obviously to b^ accounted foj* { from the natural circumstance of the niipiber of our fingers ; { the fingers being in the origin of society the readiest instru- ment to assist numeration, and still indeed frequently em- ployed for th^t purpose by the rude peasantry. So that we may conclude that if natui'e had fiirnished men with twelve^ fingers instead of ten, the duodecimal numeration would have been a^ general* as the decimal now is; and lan- guages would h^ve abounded as much with, names for the combinations of twelve, as they now do with names for the combinations of ten.
1 1 . Observe that any two or more successive digits of a \ number may be considered as a number of the same deno- \
mination with the last^thgits Thus in the number 2345, -^ the digits 34 may be considered as 34 tens j the digits 23 j ^ 23 hundreds, &c.
CHAP. II.
Addition and Std)traction, ReasonsOf proceeding from Righi to Left, Methods of Proof Examples for Practice* Signs +, — , =.,
1 2, ON eiddition and subtraction little need be said. They
are thg^jtwo, fundamental operations of Arithmetics^ hito'
whicSaU others^ may be resolved. For vSateve/arTth-
~" "' ' ' """'''"" ''" ■^^•'•' -'•"""^■■•■'*^-^'--^- njetical :
( 6 )
/ metical operation we perform, tlie change made on the^ \ given number must be either an increase or diminution of /it, that is, an addition to it or subtraction from it. And ) accordingiy we shall find that multiplicntion and division t^ are but abridged methods of addition and subtraction.
13. In addition we want to find the total amount*" oF*^^ ¥eii«-l given numbers ; in subtraction, to find the difference between two given immbers, or the number remaining after taking the less from the greater. To perform either ope- ration, it is necessary that the learner should be able to assign the sum of any given number and another not ex- ceeding nine, or the difference between them.
1 4'. In addition we successively take the sum of the ^digits standing in each column, and combine those sums /into one total. The j;eason of commencing from the right \ hand column, or place of units, ^nd proceedings from right ]^ieft, ~ls fHat we may carry jonjlig combination of jthe />unis'^c^ Thus, in
Adding together 5019 and 293, the sums of the numbers standing in the several columns are 12 units, 9 tens (or 90) and 7 hundreds, or 700, Now adding the one ten con- tained in the 12 units to the 9 tens (the sum of the second column) Ave have 10 tens, or 1 hundred ; which added to ;. the 7 hundreds (the sinn of the third column) gives 8 hundreds ; and these combined with the 2 units in the sum of the first column give 802 as the total. By pro- ceeding from right to lefi^ we are saved the trouble of writing the sums of the several columns separately, and afterwards comhinih^ them by a second addition. We write down under each column the right hand figure of its sum, and carfy the other figures to the next co- lumn. But the .same_resjihjyvill be obtained by repeated additions proceeding from left to riglit^^or taking the sums oT thej?olumns_in_an^prd^ And in this way the young scholar may advantageously be made to prove his work.
15. In arranging the numbers which we want to add it is obviously needfid, that the digits of the corresponding columns of each number should be disposed in line exactly under each other : as it is necessary, in adding pounds, shillings and pence, to avoid placing a number denoting pence in the column appropriated to the* numbers de- noting shillings. And the scholar ought to be exercised in the due arrangement of the numbers for himself, and not have them given him arranged by the teacher.
16. In
( 7 )
16. In subtraction, tlie number which is to be subtracted from the other is called the subtrahend / the number from which the subtraction is to be made, the minuend. If we have to subtract 346 from 579, it is plain that we may subtract the units tens and hundreds of the subtraliend suc- cessively from the units tens and hundreds of the minuend* ; and that the sum of the. remaindei's 233 is the remainder ,. sought. Arid m such a case^it matters not whetiier we pro- "■ ceed from left to right, or from right to left. But if anyv digit of the mijfuend be less than the digit in tlie corre- / sponding column of the subtrahend, for instance if we have to subtract 279 from 5^6^ as we cannot subtract 9 units from 6 units, nor 7 tens from 4 tens, we may sup-l pose the minuend resolved into the parts 16, 130, and 400 : ) and then subtract the 9 units from 16 ; the 7 tens from( 13 tens ; and the two hundreds from 4 hundreds. And, . tbju^^yiien. any^di^it -of . the mirmend is less, tliatn the cor- responding digit of the subtrahend, conceiving a unitpre- nxecTto ft and performing Jjj^e.^^kti'action, when we pro- ceed to the next columii w:Q^ijjayjfe tp conceive tlie next digit of the minuend less by 1, on account of the one whicli Jias \ T^eenalreacTy borrowed from it. But it aflPords the same re- / suit in practice, to conceive the next digit qf the subtra- hend increased by one, and the digit of the ouRtraliend un- altered ; as it obviously gives the «ame remainder to sub- tract 8 from 14, as to subtract 7 from 13. And hence appears the reason of what is called the carriage in . sub- traction ; and the reason of proceeding from right to left : ( though the same result may be obtained by repeated sub- \ tractions proceeding from left to rights The carriage in subtraction may be accounted for on another principle, namely, that if the two numbers be equally increased, their difference will remain unvaried. Thus, in subtract- ing 19 from B6, when we take 9 from 16, we may con- ceive that we have added 10 to the minuend, and there- fore must add 10 also to the subtrahend.
17. Besides the same attention to the arrangement of the numbers as is necessary in addition, the scholar ought to be exercised in performing the operation of subtraction whether the subtrahend be above or below the minuend.
18. The remainder found being the difference betuten the" given numbers, or the number by which the minu«|d exceeds the subtrahend, it is plain that adding the re- mainder
( 8 •;
inainder to the subtrahend must give a total equal to the minuend : or that subtracting the remainder from the mi*- nuend must give a remainder equal to the subtrahend. This affords two methods of proving subtraction. And in addition if we subtract any one of the numbers from the total, the remainder must be equal to the sum of all the other numbers.
19. The sign -|- interposed between two numbers denotes that the numbers are to be added : the sign — interposed between two numbers denotes that the latter is to be subf tracted from the former. These signs are technically called j)his and minus, from the two Latin words signifying more and less. Thus SS-f-S (read 23 phis 5) denotes the sum of 23 and 5. And 23 — 5 (read 23 ijiimis 5) denotes the remainder subtracting 5 from 23. The sign = interposed between any two numbers or sets of numbers denotes an equality between the number or set of numbers on the one side and on the other side of that sign : and such a statement is called an equation. Thus 23-1-5 = 28, and 23 — 5 = 18 are equations, denoting that the sum of 232^and 5 is equal to 28, and that the difference between "^ and 5 is equal to 18. ""
20. We shall have such frequent occasion for these signs and terms, that the young Arithmetician cannot too soon become familiar with them. A little patient explanation and illustration will soon make a child as familiar with them, as with the Arabic characters : and it is ridiculous to think how many have been deterred from attempting the study of Algebra, by the mere formidable appearance of its out-works, a number of strange symbols and terms, which they do not understand. But every thing the most simple is obscure till it is understood ; arid every term is alike unintelligible, till its meaning is explained,
21. In the following questions for exercise in addition and subtraction, the sum or difference of the numbers«-is to be supplied by the scholar after the sign of equality,
1 Ex. 5209 + 726 + 30874 =
2 Ex. 5,678093 + 23,456789 + 908 + 4321 + 86 =
Let the answ'ers to these examples be proved by sub- tracting the numbers successively from the total -, or by subtracting any one or more of them from the total, and comparing the remainder with the sum of the rest ; or by adding two or more of the given numbers separately, an^
the^
( 9 )
their sum to tlie rest; or by repeated additions of the digiti in the several cohimiis proceeding from left to right.
3 Ex. 3456—508 = 4 Ex. 987654—109345 = Let the answers to these examples be proved by adding the subtrahend to the remainder ; and by subtracting the remainder from the minuend.
5 Ex. A man has five apple trees, of which the first bears 157 apples, the second 264, tjie third 305, the fourth 97, and the filth 123. He sells 42 3 apples ; 186 are stolen. How many has he left for his own use I
6 Ex. Out of an army of 57,068 men, 9503, are killed in battle ; 586 desert to the enemy ; 4794 are taken pri^ soners; 1234 die of their wounds on the passage home; 850 are drowned. How many return alive to their owu country ?
7 Ex. A man travelling from London to Edinburgh went the first day 87 mile^, the second day 94 miles, the third day 115 miles, and going the fourth day 86 miles he was within 12 miles of Edinburgh, What is the distance betwee^ London and JCdinlpurgh ; ai:id how far from the latter town was the traveller at the end of the third day ?
8 Ex. A man at the beginning of the year finds himself w^orth .£'123,078. Li the course of the year he gains by- trade .^8706; but spends in January .5^237, in February ^301, and in each succeeding month as much as in the two preceding, What was the state of his afF^]irs at the end of the year?
Chronology will furnish the teacher with an indefinite variety of examples.. But it is to be observed in general, that pains should be take^i to give the child a clear concep- tion of the terms employed in p, question, before he is called to solve it : and that the first illustrations of the use of Arithmetical rules should be borrowed from the objects with which the cliild is most familiar, and proposed in low numbers. The great advantage of an early application to Arithmetic is the exercise which it affords to the thinking iiiculty. And when a child is taught practically how to solve a question, the meaning of which he does not clearly understand, instead of any benefit accruing, a mental habit the most injurious is contracted, of resting in indistinct conceptions, and mistaking sounds or signs ibr knowledge^ Here patience and judgment in the teacher are especially needful.
CHAP,
( 10 )
CHAP. III.
"Nature and Principles of Multiplication. Sign X . Me-^ thoch of Proof Abbreviated Methods. Powers. Qjies- iions foi^ Exercise.
22. MULTIPLICATION is but an abridged method of addition, employed where we have occasion to add the same number repeatedly to itself. Of the two numbers multiplied together, and called by the common name of
factors^ the midtip>licand is that number which we want to add repeatedly to itself; and the midtiplier expresses the number of times that the former is to be repeated in that addition. The sum required is called the product. Thus, by the product of 6 multiplied by 4 we are really to un- derstand the sum of four sixes, or 6 + 6 + 6 + 6. The mul- tiplication table, which is supposed to be committed to memory, furnishes us with all the products as high as 12 times 12, or the sum of 12 twelves: and the rule of Mul- tiplication teaches us how to derive the higher products, where the factors (either or both of them) exceed twelve.
23. The product of any two numbers is the same, which- ever of them be made the multiplier. For instance, if we multiply 8 by 5 we sh^ll have the same product, as if we multiply 5 by 8, I have known many smile at the attempt to prove this, conceiving it so self-evident; as neither to admit nor require proof. But they are imposed on by their familiarity with the fact. It is by no means self-evident that the sum of 5 eights must be the same with the sum of 8 fives, or that 8-}-8-f-8-}-8-f 8 = 5 + 5 + 5-|-5-}-54-5 + 5 -f5 5 which is the meaning of the proposition. However it admits a very easy proof from the following illustration. Suppose 5 rows of 8 counters regularly disposed under each other. Whatever way we count them, the total amount of the number must be the same. But counting them one way, we liave 5 times eight ; and counting them another way, it is plain that we have 8 times five counters. It is obvious that a similar proof would be applicable to any higher numbers.
24. The sign of multiplication is X , or a St. Andrew's cross, interposed between the factors ; and is to be care- fully distinguished from the sign of addition -}-. Thus 12 X 8, or 8 X 12, denotes the product of 8 and 12.
25. The
( 11 )
25. The product of any two numbers is equal to the sum of all the products obtained, by multiplying all the parts, into which either is divided, by the other, or by each of the parts into which the other is divided, Thus, if we suppose 8 divided into the parts 4, 3, and 1 ; the product of 5 times 8 will be equal to the sum of the three products, 5 times 4, 5 times 3, and 5 times 1. And if we suppose the multiplier 5 also divided into the two parts 3 and 2 ; the product of 5 times 8 will be equal to the sum of the six products obtained by multiplying each of the three com- ponent parts of the multiplicand by each of the two com- ponent parts of the multiplier* The truth of this will ap- pear very plain, by employing the same illustration that was adcluced in the 23d section, In the 5 rows of 8 counters, aptly representing 5 times 8, let us suppose, first, two lines drawn downwards dividing each row of eight counters into the three parts 4, 3, and 1, It is then plain that the whole set of 5 times 8 counters is divided into three sets of 5 times 4, 5 times 3, and 5 times 1. Then supposing a line drawn across and dividing each row of 5 counters into 3 and 2, it is plain each of the 3 former sets will be divided into two, 3 times 4 and twice 4 ;• 3 times 3, and twice 3 ; 3 times 1 and twice 1 : so that the sum of these 6 sets is equal to the one set of 5 times 8 counters. This proof is exhibited to the eye iii the subjoined scheme.
0000 0000 0000
0000 0000
000 000 000
000 000
And it is plainly applicable to any other numbers, divided into any parts whatsoever. Thus, if we suppose 17 broken into the four parts, 6, 5, 4 and 2 ; and 9 broken into the three parts, 4, 3, and 2 ; the product of 9 times 17 must be equal to the sum of each of the twelve products obtained by multiplying each of the four parts of the mul- tiplicand by each of the three parts of the multiplier: that is 17x9=24 + 20 + 16 + 8 + 18 + 15+12+6 + 12 + 10 + 8 + 4. With the principle brought forward in this section the student cannot be too familiar ; as it is the foun- dation both of common multiplication and Algebraic, as nvell as fruitful in the most important inferences.
26. If
( 12 )
S6. If our multiplier be the product of any two knoMTi numbers, we may employ a successive multiplication by the factors, of which the multiplier is the product. Thus, if we want to multiply any number by 54, we may mul- tiply it by 9, and that product by 6 : for 6 times 9 being 54, when we first find a number that is 9 times the multi- plicand, and then multiply that numV/er by 6, our product must be 6 times 9 times, or 54 times the multiplicand.
27. It appears from § 7 and § 25. that the product of any number multiplied by 10, 100, 1000, &c. is obtained at once by annexing one, two, three, &c. cyphers to the multiplicand qn the right hand. Thus, the product of 327 multiplied by 1000 is 327,000: for each digit of the multiplicand is increased in value 1000 times. And com- bining the principle of the last section, it is plain that if our multiplier be 20, 300, 4000, &c. we may obtain the product by annexing one, two, three, &c. cyphers, antj then multiplying by" 2, 3, 4, &c. Thus 4296x700 = 429,600 X 7.
28. From the principle stated in § 25. it is manifest that \te can find the product of any two numbers : for however great the factors, they may be broken into parts not ex- ceeding 12, the products of all which parts are furnished by the multiplication table. But when the factors, either or both of them, exceed twelve, the most convenient part|» into which we can conceive them broken are those indi- cated by the digits. Thus, if I want to, find the product of 537 multiplied by 9, I conceive the multiplicand di- vided into the parts 7, 30, and 500 j and the product is by § 25. equal to the sum of the three products 9 times 7, 9 times 30, and 9 times 500 j each of which the multipli- cation table furnishes. For 30 being 3 tens, 9 times 30 must be 27 tens, or 270; and 9 times 5 hundreds must be 45 hundreds, or 4500. The product sought therefore must be the sum of the three products, 63 + 270 + 4500, that is, 4833. This addition of the successive products, by proceeding from right to left in taking the parts of the multiplicand, we are able to perform mentally, without writing the whole of each product separately.— Now if I want to find the product of 537 multiplied by 69, I sup- pose the multiplier also divided into the two parts 9 and 60 ; and having found the product of 9 times the multipli- cand, I proceed to find the product of 60 times the mul^.
tiplicand
( 13 )
tiplicand by § 27. writing the latter prodlict (32220) iincter the former, preparatory to the addition of the products; It is plain therefore that the cypher annexed to the mul- tiplicand for multiplying by 10 must stand in the colunm of units, and be preceded by the digits expressing the pro- duct of 6 times the multiplicand. But as that cypher will make no change in the subsequent addition, it is com- monly omitted; taking care however to place the next digit in the column of tens. In like manner if my mul- tiplier were 469, after having found the two former pro- ducts, I proceed to multiply by 400, supposing two cy- phers annexed to the multiplicand and then multiplying by 4, and writing this product (214800) under the second, preparatory to the addition of the three products. — The young Arithmetician should for some time be made to write the cyphers standing on the right hand of the successive products, that he may be convinced of the reason of the rule, which directs us to recede one figure towards the left hand in writing the several products obtained in multiply- ing by the successive digits of the multiplier.
29. The child should be taught to prove the accuracy of his work in multiplication by addition, so far as to con- vince him that the one is but an abridged method of per- forming the other ; and by resolving either or both factors into other parts than those indicated by the digits.
30. We have noticed the reason of proceeding from the right hand to the left of the multiplicand. But it is gene- rally indifferent in what order we take the digits of the mul- tiplier: and it will sometimes aiford a convenient abbre- viation to depart from the usual order. Thus, if our mul- tiplier be 945, instead of obtaining the product sought by three distinct products, two will be sufficient by commenc- ing from the left hand of the multiplier ; since having found the product of 9 times the multiplicand, 5 times that product will give us the product of 45 times the multiplicand. § 26. But when this method is employed, it is plain that the cyphers, which are usually omitted, ©ught to be expressed.
31. We have seen (§ 27.) the facility with which mul- tiplication proceeds, when the multiplier consists of a sig- nificant figure followed by any number of cyphers. Now if our multiplier be within twelve of any such number, we may avail ourselves of a convenient abbreviation. For
instance
( 1* )
instance if our multiplier be 4989, we obsei^ve by inspec- tion that this number is within 11 of 5000. If then I take 5000 times the multiplicand, and subtract from that product 1 1 times the multiplicand, the remainder must be 4989 times the multiplicand j or must be the product sought. Such abridged methods of operation are useful for exercising youthful ingenuity : but ought not to be pre- maturely introduced. Rational theory, going hand in hand with practice, will soon make the student expert in discerning various advantages which may be taken. For example, if we have to multiply 123,456789 by 107988, the multiplier being within 12 of 108000, and 9 times 12 being 108, we may first find 12 times the multiplicand, and subtracting that product from 9000 times that pro- duct will give the remainder 13,331851,730532 for the product sought. But in general it is useless to occupy the learner's time in arithmetical operations on numbers so high, as scarcely ever occur in real practice. A much more advantageous exercise is to enSge him in operating on low numbers mentally, without committing them to paper ; for instance to find the product of 25 times 36. This is calculated to form a habit of fixed attention, and to strengthen the mental powers.
32. The product of any number multiplied by itself is called the square of that number, or its second power. The original number thus multiplied is called the square root of the product. Thus 64 is the square of 8 : and 7 is the square root of 49. If the square of any number be mul- tiplied by its root, the product is called the cube, or third power, of the original number. Thus, 64X8, or 512, is the cube of 8 : and 8 the cube root of 512. And if the cube be multiplied by its root, the product is called its
fourth power : and so on. Those powers of any number are often represented by annexing to the right hand of the root, and somewhat elevated, the figures 2, 3, 4, &c. which are called indices of the powers. Tlius 8 * expresses the square of 8 ; 2 being the index of the power. And 15"^ expresses the fourth power of 15, or 15x15x15x1 5.
33. Among the inferences flowing from the principle laid down in § 25. we may here proceed to state two, for which we shall have frequent occasion in Algebra. The square of any number is four times the squaj-e of its half. Thus 8 " = 64 5 and 4 ^ = 1 6. But 64 = 16X4. That this
must
( IS )
must be so is evident from § 25. For 8 being botli mul- tiplicand and multiplier, may be divided in both factors into the parts 4 + 4 : and the product 8X8 must be equal to the sum of four products, each of which is 4X4, or the square of 4.
34. Again, the square of the sum of any two numbers is equal to the sum of their squares together with twice their product. Thus, the sum of 4 and 3 is 7 : and the «quare of 7 is equal to the sum of the squares of 4 and 3, together with twice the product of 4 and 3 : that is 49 = 16-|.9_[-24* This in like manner immediately appears by supposing the multiplicand and multiplier 7 divided into the two parts 4 and 3 ; or into any other two parts. On this principle, as we shall hereafter see, depends the ex- traction of the square root, and the reduction of qua- dratic equations in Algebra. It is of such frequent use, that the student cannot too soon become familiar with it : and it will afford a good 'exercise to calculate, without the pen, the square of any number within 100, by resolving the number into the two parts indicated by the digits. Thus, a child may be led to find the square of 69 men- tally, if he only know that 60 X 60 is 3600, that twice 54 is 108, and can add mentally 3681 and 1080 -
35. Any product is said to be a multiple of either factor ; and either factor is called a siibnmltiple of the product. Thus 96 is a multiple of 8 or of 12, because 12 X 8=96 ; and 8 or 12 is a submultiple of 96.
36. In the following examples of multiplication let the young student write the required product after the sign of equality, =. And let him observe that, as 234X89 de- notes the product of 234 and 89, so 234-f 6 X 89 denotes the product obtained by multiplying the sum of 234 and 6 (that is 240) by 89 ; and 234 + 6 X 89 + 11 denotes the product obtained by multiplying the sum of 234 and 6 (or 240) by the sum of 89 and 11, or 100. In like man- ner 234+6+7 X 89+11 is the same thing as 247X100: and that product is equal (§25) to the sum of the six pro- ducts 234X89,+ 6 X 89 + 7X89 + 234X11 + 6X11 + 7X11. Again 234 X 5 X 7 denotes the product obtained by multiplying 234 by 5, and that product by 7 ; and is the same (§ 26.) with the product of 234 X 35. Again let it be observed that 10^ (or the 5th power of 10, see § 32.) denotes the product of 10 x 10 X 10 X 10 x 10.
1st. Ex.
( 16 )
1st. Ex. 1+24-3 + 4. X 5 + 6 + 74-8 + 9 =
2d. Ex. 2X3X4X5 X 6x7x8x9 ==
3d. Ex. 9 + 8 + 7 + 6 X 5 + 4 + 3 + 2 + 1 =
4th. Ex. Wliat are the cubes (or 3d. powers) of the num- bers 4, 5, 6, 7, 8, 9, and 10?
5th. Ex. 35 + 45 + 9^ =
6th. Ex. 3453 X 100^ =
7th. Ex. 123456789 X 9988= (See § 31.)
8th. Ex. 7539 X 60054= See § 30.)
9th. Ex. How much does the square of 48 exceed the square of 24 ? (See § 33.)
10th. Ex. How much does the square of 57+28 (or 57 + 28]*) exceed 57* + 28* ? (See § 34.)
For the method of proving raidtiplication, independently of division, see § 29. (For other methods see § 61 and 62.) Questions for exercise in the practical application of mul- tiplication will be found in Chap. VI.
CHAP. IV.
Nature and Principles of Dlvisio?t» Sign ^. Division of a smaller Number by a greater. Methods of Proof Qiies- tions for Exercise.
37. DIVISION, in the primary view of it, is but anx abridged method of subtraction. Here we enquire how often one number, called the divisor^ may be subtracted from another number called the dividend. The quotient expresses the number of times, that the divisor may be sub- tracted from the dividend, or is contained in it. Thus, when I divide 96 by 12, the quotient is 8 : for I may sub- tract the divisor 12 from the dividend 96 just 8 times. This might be ascertained by performing the successive subtractions, and reckoning the number of them : but is at once discovered by the multiplication table, which in- forms me that 96 is equal to 8 times 12, and therefore con- tains 12 in it exactly 8 times. If I divide 103 by 12, it is plain that after subtracting 12 from 103 eight times, there will remain 7 : so that the quotient is still 8, but with 7 fox a remainder. ( See J 4 3 . )
38. When
( 17 )
38. When one number is contained in another a certain number of times exactly, without leaving any remainder, the former number is said to measure the latter. Thus, 12 measures 96, but does not measure 103. The numbers 8 and 12 measure 24 ; 8 being contained in it exactly 3 times, and 12 exactly twice.
39. We often express division by writing the dividend above the divisor with a line interposed between them.
Thus -r- expresses the division of 84 by 7 : and the fol-
lowing symbols — =12 express therefore that the quo-^
tient of 84 divided by 7 is equal to 12. The symbol -^ also is sometimes employed to express division, the di- vidend standing on the left hand of it, and the divisor on the right. Thus, 42-4-6 is another way of expressing the
. » 4 42
division of 42 by 6, as well as — .
. . 6
40. If any quotient be made the divisor 6f the same di- vidend, the former divisor will be the new quotient, and the same remainder (if any) as before. Thus, dividing 103 by 12, the quotient is 8 with the remainder 7. Now if we divide 103 by 8, the quotient must be 12, leaving the same remainder. For the first division shews that the divi- dend contains 12 eight times and 7 over. Therefore it must contain 8 twelve times and 7 over ; 8 times 12 and 12 times 8 being equal. {§ 23.) And thus also it is manifest that if any product be divided by either of the factors, the quotient must be the other factor : and that any dividend may be considered as the product of the divisor and quo- tient, with the remainder (if any) added.
41. In the view of division which has been hitherto proposed, the divisor .must be conceived not greater than the dividend ; else it would be absurd to enquire how often it is contained in the dividend. But there is another view of division, closely connected with the former, in which we may easily conceive the division of a smaller number by a greater. When we are called to divide 96, for in- stance, by 12, we may consider ourselves called to divide 96 into twelve equal parts, and to ascertain the amount of each. The quotient, found as before, is a number of that amount, or the twelfth part of 96. For since 96 contains in it just 8 twelves, it must contain just 12 eights ; and therefore the quotient 8 is the twelfth part of 96.
C And
( 18 )
And thus universally the quotient may be considered as that part, or submultiple, of the dividend which is de- nominated by the divisor; as the divisor may be con- sidered that part, or submultiple, of the dividend which is denominated by the quotient. (Hitherto I suppose the divisor to measure the dividend,) Thus, dividing 64- by 4 the quotient is .16 j for subtracting 16 fours from 64- there is no remainder. Therefore 4 is the sixteenth part of 64 5 and 16 is the fourth part of 64*
42. Now though it would be absurd to enquire how often 12 may be subtracted from 7, and therefore any di- vision of 7 by 12 is inconceivable according to that view; yet it is not absurd to enquire what is the twelfth part of 7, or to speak of dividing 7 by 12 according to the latter view. For instance, I might have occasion to divide 7 guineas among .12 persons equally, or into 12 equal shares: and then it is plain that each person must get the twelfth part of seven guineas. The quotient, or twelfth part of
7, may be represented by the notation — : (§ 39;) and the
child ought to be familiarized to this notation, previous to his entrance on the doctrine of fractions.
43. Let us now revert to the example of division intro- duced at the close of § 37. the division of 103 by 12, The quotient we saw is 8, but leaving a remainder of 7. Therefore 8 is not exactly the twelfth part of 103 : for if I were dividing 103 guineas equally among 12 persons, after giving each of them 8 guineas there would be 7 guineas over : which 7 guineas I should proceed to divide equally among them ; that is, I should give each of them the twelfth part of 7 guineas in addition to the 8 guineas he had received, in order to make the division accurate*
Therefore the, twelfth part of 103 is exactly 8^ — L • or 8
and the twelfth part of 7. And so, wherever there is a remainder on a division, the student should be taught to correct the quotient by annexing to it that' Temainder di- vided by the divisor.
44. As to the practical method of performing division, the grounds of it are obvious from § 37. Let us first sup- pose that our divisor does not exceed 12 : for instance let it be required to divide 51 12 by 8, We immediately know from the multiplication table that 8 may be subtracted 600 times from the dividend, but not 700 times; since 600
times
( 19 )
times 8 (or 8 times 600) is 4800, but 8 times 700 is 5600, a number greater than the dividend. Subtracting there- fore 4800 from 5112, there remains 312; and this one subtraction saves tlie trouble of 600 distinct subtractions of 8 from the dividend. We proceed now to the re- mainder 312, and consider, from the multiplication-table, what is the greatest number of times that 8 is certainly contained in it, or may be subtracted from it : and v^^e immediately know as before that 8 is contained 30 times in 312, but not 40 times ; 30 times 8 being 240, but 40 times 8 being 320, a number greater than 312. Subtracting therefore 30 times 8, or 240, from 312, there remains 72 : in which remainder we see that 8 is contained just 9 times. Thus we have ascertained that from 5112, 8 may be sub- tracted 600 times, 30 times, and 9 times ; or in all 639 times ; which number is therefore the quotient, and the eighth part of 5112. If our dividend were 51 19 it is plain that the quotient would be 639 with the remainder 7 : and
therefore that the eighth part of 5119 is 639 -f -. In
o
practice, w^e perform the successive multiplications and subtractions mentally, as we proceed ; attending only to that part of the dividend, which ascertains the successive digits of the quotient, and writing only those digits. But the learner ought to be exercised for some time in per- forming the operation at large, as I have described it ; that he may be grounded in the rational principles upon which the practical contractions rest.
45. Let us now suppose that our divisor exceeds 12 1 for instance, that we have to divide 27783 by 49. We may at once conclude that the quotient must be less than 700, as 700 times 40 (or 28000) would exceed the divi- dend, and therefore much more 700 times 49. But the dividend does not contain the divisor even 600 times ; for though 600 times 40 (or 24000) is less than the dividend, yet 600 times 4^) will be found greater than the dividend. ( Nothing but practice can make the student quick in per- ceiving this ; and he may for a time have the trouble of trying numbers in the quotient, which he will find to be too great.) Subtracting therefore 500 times the divisor, or 24500, from the dividend, there remains 3283 ; from which we subtract 60 times the divisor, or 2940. In the remainder 343 we find that the divisor is contained just 7 times. So that the entire quotient is 567. In such in-
C 2 stances
( 20 )
stances of what is called long division, it is necessary to write the successive remainders. But after the student has been grounded in the principles of the operation, it will be expedient that he should perform the subtractions with- out writing the successive products ; subtracting the several digits composing them as he proceeds with the multiplication,
46. Thus it appears that we are enabled by the multipli- cation-table to determine the successive digits of the quo- tient from the left hand. But although the order of pro- ceeding which we have described be the most convenient, I would have the young Arithmetician practised in resolv- ing the dividend differently, and proceeding on similar principles, but in another order. Let us again take the last example, to illustrate my meaning. In dividing 27783 by 49, we first took 27000, a component part of the di- vidend, and finding that it contained 500 forty-nines and 2500 over, we incorporated the latter with 783 the other component part of the dividend^ and proceeded in like man- ner to find the other component parts of the quotient. But the same result must be obtained by commencing with the latter component part of the dividend 783. Dividing it by 49 the quotient is 15 with the remainder 48. Adding that remainder to the other part of the dividend 27000, we may proceed in like manner to ascertain how many times 49 is contained in their siim, by commencing with the compo- nent part 7048. The quotient will be 143 with the re- mainder 41. And adding the remainder to the 20000 which has not yet been divided, 49 will be found to be contained in their sum 20041 just 409 times. Now the sum of the three quotients, 15-|-14'3-f-409, is 567 as be- fore. And thus the student may be taught to prove the accuracy of his work in division, not only by multiplying the divisor and quotient, (§ 40.) but also by resolving the dividend into any two or more parts, dividing each of them by the given divisor, and adding the quotients.
47. If the given divisor be the product of any two or more known factors, the quotient may often more expe- ditiously be obtained from successive divisions by those fac- tors. Thus in the last example, 49 being 7 times 7, if we divide 27783 by 7, and again divide the quotient 3969 by 7, we shall have the result 567. Perhaps the child might here be advantageously introduced to the principle, for which we have such constant occasion in fractions, that tlie 7th. part of the 7th. must be the 49th. part, &c. (See
c. viii.
( 21 )
c. viii. § 8.) The principle admits such clear and familiar illustration, that I think any child who is capable of learn- ing division may be convinced of its truth. But for esta- blishing the present rule in division the following princi-*^ pies also ought to be employed, and will be sufficient.
48. The given dividend 27783 is 7 times the first quo- tient ; and the first quotient 3969 is 7 times the second quotient. Therefore the given dividend is 49 times the second quotient ; or 567 is the 49th. part of the given di- vidend. For (putting a, b, and c for three numbers) if ij be 8 times Z>, and b 6 times c, then a must be 48 times c. Or thus :
49. The uumber 7, being 7 times less than 49, must be contained in the dividend seven times oftener. But 7 is contained in 27783 just 3969 times. Therefore 49 must be contained in it the 7th. part of 3969 times : or the quo- tient sought is the 7th. part of 3969.
50. But when this method is employed, we must care- fully attend to the management of the remainders. Thus, dividing 5689 by 42, the quotient is 135 with the re- mainder 19 : and if we employ a successive division by 7 and 6, the first quotient will be 812 with the remainder of 5, and on dividing that quotient by 6 we shall get the quotient 135 with the remainder of 2, But this 2 is to be considered as 2 sevens, or 14 ; which added to the former remainder ffives 19 for the true remainder, as before. The reason of this will be plain from considering that by the first division we find that the dividend contains in it 812 sevens: so that any remainder on dividing that 812 must be regarded as of the denomination sevens. This may be made quite clear to the youngest student by suppposing that we wanted to divide 53 guineas by 12 ^ that is, to find how many sets of 12 guineas are contained 53 guineas. Dividing 53 by 4, we find that it contains 13 sets of 4 guineas each and one over. Every three of this quotient will make a parcel of 12 guineas ; and now to find their Iiumber, dividing 13 by 3, the quotient is 4, (four parcels of 12 guineas) and 1 over. But this 1 is plainly 1 set of 4 guineas : which added to the former 1 guinea gives 5 for the remainder and 4 for the quotient. Hence appears the reason of the rule, which directs us to multiply the re- mainder on the second division by the first divisor, and add the product to the remainder on the first division.
ihe
( 22 )
The same thing will appear from the doctrine of frac- tions. . ^< 51. Any number is divided by 10 — 100—1000 &c. by putting off as many digits from the right hand of the di- vidend, as there are cyphers in the divisor. The digits thus cut off express the remainder, and the remaining digits of the dividend the quotient.,,, jThu§, dividing 234567 by 1000, the quotient is 234 >vith the remainder 567. This is manifest from § 40, since the dividend is equal to lOOQ times 234 with 567 added to the product. Hence it is plain that if our divisor consist of any significant figures folio vved by any number of cyphers, we may employ the method of division described in the last section. Thus if we want to divide 234567 by 7000, we may divide first by 1000 and then by 7 5 and the quotient will be 33 with the remainder 3567. For when we divide 234 by 7, the re- mainder of 3 is in fact 3 thousands, and is to be added to the first remainder 567. And we shall have the same re- sult (though not so expeditiously) if we first divide by 7' and then by 1000,
52. When the given divisor is a submultiple of any of those last described, we may often abridge . our work by multiplication. Thus if I have to divide 1234 by 25, I know at once that the quotient is 49 with the remainder 9. For 25 is the fourth part of 100, which is contained in the dividend 12 times with the remainder 34. Therefore {§ 49.) the dividend must contain 25 four times as often, that is, 48 times with the same remainder 34. But in this remainder 25 is contained once and 9 over, In hke man- ner, 75 being the fourth part of 300, I know at once that the 75th. part of 1234 is 16 with the remainder 34, or
that = 16 -{--^ . Some other abbreviations of division,
75 75
less commonly known, I shall point out in the following chapter. They may evercise the ingenuity of the student, and are calculated to develope very curious properties of certain numbers.
Examples for practice in division may be had jfrom all the examples of multiplication at the end of Chap, III. In the following examples let the student supply the quo- tient after the sign of equality = .
1st. Ex. 123456789 --9000 =
2d. Ex. 987654321-125= (See § 52.)
3d. Ex. 3933 -f- 19 =
4th, Ex. 31464-7-19= Let
Let the student observe in the twoiJlili *jtu)iUf5Tes that the dividend in the 4th. being 8 times the dividend in the Sd. the quotient in the 4th. is 8 times the quotient in the 3d.
5th. Ex. 3496 -^ 19=^
Here the dividend being the 9th. part of the dividend in the 4th. example, the quotient also is the 9th. part of the quotient in the 4th.
6th. Ex, 31464 -f- 133 =
Here the divisor being 7 times the divisor in the 4th. the quotient is the 7th. part of the quotient in the 4th. ' 7th. Ex. 180918-3933=:
8th. Ex. 180918^437 =
Here the divisor being the 9th. part of the divisor in the 7th example, the quotient is 9 times the quotient in the 7th.
9th. Ex. 5907^9^ =
1 0th. Ex. 9^+P -^ 7 ^ ==
Besides the methods of proving division already pointed out, another method will be assigned in the next chapter. j 62 and 63c
CHAP. V.
Methods of ahhreviaied Opet^ation, and of proving Division^ continued. Properties of the Numhey^s ^^ 9, 11, 4'^.
53. WE may arrive at the required quotient in division, by substituting for the given divisor any other whatsoever, either greater or less than the given one. To exhibit this, \ shall first employ a number greater by 1 than the given divisor. Suppose fqr instance we have to divide 796 by 19. Dividing it by 20 the quotient is 39 with the remainder of 1 6. I say then that the required quotient must be 4 1 with the remainder 17. For I have found that 20 may be sub- tracted from the dividend 39 times : but for every time that I have subtracted 20 instead of subtracting 19, I have sub- tracted 1 too much; that is, I have subtracted in all 39 too much. Hence we may infer that the dividend besides containing 39 nineteens with a remainder of 16, contains ^Iso 39 units more : in which 39 there are 2 nineteens and
1 over.
( 24 )
1 over. Therefore the dividend contains in all 41 nine* teens (39 + 2) with the remainder of 17 (16-f-l). Or, to give another illustration of the principle upon which this method proceeds ; suppose we divide 796 guineas equally first among 20 persons : they will each get 39 guineas and the 20th. part of 16 guineas. But now finding that we were wrong in making the division among 20 persons, and that it ought to have been made only among 19, we take
, one person's share from him and divide it equally among the rest-r so~that each shall now get for his share 41 gui- neas and the l^^th. part of 17 guineas. Thus if we have to divide 1234 by 99, we may know at once that the quo- tient is 12 with the remainder 46. For dividing by 100, the quotient is 12 with the remainder 34 : but having thus subtracted 12 units too much, they must be added to 34 for the true remainder. And if the dividend be 12345, the correction will be made by adding 1 to the first quotient and 24 to the remainder j inasmuch as 123 contains 99 once and 24 over : so that the quotient sought is 124 with the remainder 69.
54. Now suppose we have to divide 123456789 by 99, Substituting 100 as our divisor, the quotient is 1234567 with the remainder 89. If we knew what number of times 99 is contained in that quotient, and with what remainder, the necessary correction would be made by adding that number to the quotient, and that remainder to the former remainder. Now this would be ascertained by dividing the first quotient by 99 : but in place of this we may again substitute a division by 100, the result of which is to be similarly corrected. And thus continually dividing each successive quotient by 100, the sum of all the quotients and sum of all the remainders will furnish us with the true quotient and true remainder. Thus we have the quotient, by mere addition, 1234567 + 12345 + 123-1-1 = 1247036. But the sum of all the remainders, 89-f 67-f 45-f 23-f- 1 = 225, containing 2 ninety-nines and 27 over, we add the
2 to the quotient: so that the true quotient is 1247038 with the remainder 27. At any period of the above pro- cess, when we see how often 99 is contained in the last quotient, we may discontinue the division by 100 and com- plete the corrections at one step.
55, Hitherto we have supposed that the substituted di- visor exceeds the given divisor only by 1. But let us now
suppose '
( 25 )
suppose that we have to divide 1234 by 95; We may with equal facility conclude that the quotient is 1 2 with the re- mainder 94. For dividing by 100, the quotient is 12 with the remainder 34, But for every time that we have sub- tracted 100 instead of subtracting 95, we have subtracted 5 too much 5 that is, we have subtracted in all 60 too much, which 60 is therefore to be added to 34 the former remainder. (It is easy also to apply to this case the illus- tration adduced in § 53.) In like manner if we have to divide 1234567 by 7988, substituting 8000 which exceeds the given divisor by 12, we have the quotient 154 with the remainder 2567 to which remainder if we add 1848 (154 X 12) we shall have 4415 for the true remainder. It can scarcely however be advantageous to employ this me- thod in practice, if the given divisor be much less than the substituted, which it is convenient to employ ; and if the number of digits in the quotient be more than those in the divisor.
56 » Hence it appears that if 9 measure the sum of the digits of any number, it will measure the number ; and that the remainder left on dividing any number by 9 must be the same with the remainder on dividing the sura of its digits by 9. Thus 234, or 378, is evenly divisible by 9, because the sum of the digits 2 -f 3 -f 4, or 3 -|- 7 -f 8, is so. For if instead of dividing 378 by 9, we substitute con- tinued divisions by 10, the series of quotients will be 37 + 3 + 0, and of remainders 8 + 7 + 3 ; which latter sum containing ji^.t 2 nines, we carry 2 to the former quotients, and infer that the exact quotient is 42 without any re- mainder. In like manner it appears that the remainder on dividing 12345 by 9 must be 6, as that is the reihaindor on dividing the sum of the digits 15 by 9. And thus it is evident that any numbers written with the same digits, in whatever order, will give the same remainder on being divided by 9. A similar property of the numbers 99, 999, &c. may in like manner be inferred ; only taking the digits by pairs, by threes, &c. from the right hand. Thus 12345 divided by 99 must give the remainder 69 ; because 45 + 23 + 1=69: but 14652, or 15246 must be evenly divisible by 99, since 52 + 46 + 1=99. Another demon- gtration will be found in § 59 and 60. for the property of the number 9.
57. Let
( 26 )
57. Let us now consider how division may be performed by the substitution of a divisor less than the given one. Suppose we have to divide 123456 by 101. Substituting 100 as our divisor, the quotient 1234 with the remainder 56 is manifestly too great. For every time that we have subtracted 100 instead of subtracting 101, we have sub- tracted 1 too little, that is we subtracted in all too little by 1234, which cont^iins 101 twelve times with the remainder 22 ; as will appear on dividing 1234 by 101. The correc- tion therefore would be made at once by subtracting 12 from the first quotient 1234, and 22 from the first re^ mainder 56 i which gives 1222 for the quotient sought, with the remainder 34, But instead of ascertaining the correction at once by dividing the first quotient by 101, let us again substitute a division by 100 ; and subtracting the quotient 12 and remainder 34 from the :first quotient and first remainder, it is now plain that we have subtracted too much : and therefore the next correction must be made by addition. And thus when we successively employ a di- visor less than the given one, our successive corrections must be made by alternate subtractions and additions ; as we first subtract too much, then add too much, &c. Whereas when we employed a divisor greater than the given one, all our corrections proceeded by addition, as we were successively adding too little. If our substituted divisor be less than the given one by more than a unit, it appears as before that each quotient must be multiplied by the difference. Thus the quotient of 12345^6 divided by 5012 is 24 with the remainder 3168 : for dividing by 5000 the quotient is 24 with the remainder 3456 ; from which remainder subtracting 288 (24x12) we have the true remainder.
58, Hence we may infer a property of the number 11, which shall be demonstrated from other principles in § 61. namely, that any number must be evenly divisible by 11, if the sum of the alternate digits from the last and the sum of the alternate digits from the penultimate be equal, or their difference evenly divisible by 11. Thus, 190817, or 718091, is evenly divisible by 11, since 11 measures the difference between 7 + 8 + 9 and 1 + 0 + 1 . For if, instead of dividing by 11, we should investigate the quotient by successive divisions by 10, the successive digits would be the remainders ; and these would be to be subtracted and added alternately.
59. The
( 27 )
59. The property of the number 9 stated in § 56. maj be thus easily demonstrated. If from any number the sum of its digits be subtracted, the remainder must be evenly- divisible by 9. For instancej if from the number 345 we subtract 12 (3 + 4 + 5) the remainder 333 must be evenly divisible by 9. For the number 345 may be considered as made up of 100 threes, 10 fours, and 1 five. Let us now succesively subtract the digits, and observe the remainders. Subtracting 5, the remainder is 100 threes and 10 fours* Subtracting 4, the remainder is 100 threes and 9 fours. ' Finally subtracting 3, the remainder is 99 threes and 9 fours. But 9 must measure this number, as it is plain that it measures each of its component parts. And so putting the letters «, 6, r, d, &c. for the digits of any number, *1000«+ 1005+ lOc-^-d must be a just expression for any number written by four digits, that is, within 10,000. And if from this we subtract the sum of the digits a-^-b +C+C?, the remainder, 999<2 + 995 + 9c, must be evenly divisible by 9, inasmuch as 9 measures each of its com* ponent parts,
60. It immediately follows from the last section that the remainder on dividing any number by 9 must be the same with the remainder on dividing the sum of its digits by 9^ For instance, 345 divided by 9 must give a remainder of 3, since that is the remainder on dividing 12 (3 + 4 + 5) by 9. This is manifest from considering that 345 is equal to 333 + 12 ; of which parts we have just seen that the for-* mer (333) must be evenly divisible by 9, and therefore the only remainder on dividing the whole by 9 must be that which occurs on dividing the latter part 12 (or the sum of its digits) by 9. And in like manner the same property is demonstrated to belong to the number 3.
61. By a similar process of reasoning it appears, that if from any number we subtract the sum of the alternate digits commencing from the last, and add to it the sum of the alternate digits commencing from the last but one, 1 1 must measure the resulting number. For let a number consist- ing of 4 digits be represented as before by 1000«+1005 + 10c + t/. Subtracting cZ and 5, there will remain 1000a + 99Z>+ 10c. Now adding c and cf, the resulting number, 1 00 1 « + 99Z> + 1 1 c, must be evenly divisible by 11 . Hence given any number it is easy to know what the remainder must be on dividing it by 11. For instance, if the given
number
* In this notation 1000 a stands for 1000 timet a ; 999 a for 999 times a j &c.
( 28 )
number be 9 1827 » or 72819, the remainder must be 10; for subtracting 2 1 (the excess of the digits to be subtracted above those to be added) the remainder is evenly divisible by 1 1 : therefore the remainder on dividing the whole by 1 1 must be that which occurs on dividing 21 by 1 1 . But if the given number be 9182, the digits to be added ex- ceed the digits to be subtracted by l^, that is ll-f3: whence we may infer that the given number wants 3 of being evenly divisible by 1 1 ; or that dividing it by 11 there will be a remainder of 8. We might enlarge upon other curious properties : — (for instance, if 1 1 measure any number consisting of an even number of digits, and consequently measure also the number consisting of the same digits in an inverted order, the sum of the digits in each quotient must be the same,)— but as they are of little practical importance, we shall rather pass to a?f useful me- thod of proving multiplication and division.
62. Multiplication may be proved thus : divide both the factors by any number and (neglecting the quotients) mark the remainders ; divide the product of those remainders by the same number and mark the remainder. This re- mainder must be the same with the remainder on dividing by the same number the product of the given factors. For instance, 648 X 23 = 14904. Now dividing 648 and 23 by 7, the remainders are 4 and 2 ; whose product 8' divided by 7 gives 1 for the remainder : and 1 must also be the re- mainder on dividing 14904 by 7 ; which may be easily de- monstrated from the fundamental princijile of multiplica- tion. For breaking the factors into the parts 644 -f 4 and 21 -f 2, the former parts of each are evenly divisible by 7, and therefore also any multiples of those parts. Now the whole product 14904 is equal {§ 25.) to the sum of the four following pix)ducts 644 X 2 1 , and 4X21, and 644 X 2, and 4X2. Of these 7 measures the three first ; and there- fore the only remainder on dividing the whole product by 7 must be that which occurs on dividing the product of the remainders 4 and 2.
63. On this principle depends the common method of proving multiplication by what is called cashing out the nines* It is in fact nothing but an application of the number 9 as a test, just as in the last example we applied the number 7 : and the only advantage of the fbriiier is that we can ascertain the remainders without performing the divisions
by
( 29 )
by 9. It appears that if the remainder on the supposed product of the factors be not the same with that on the product of the remainders of the factors, we may conclude with certainty that there is an error in our work. But we cannot be equally certain that the work is right, if the re- mainders be the same. There is however a strong proba- bility of it : which will amount to a moral certainty, if, after applying 9 as a test, we also apply 7 or 11 . I would recommend the latter, from the facility with which the di- visions may be performed, or the remainders calculated by §61. It is to be observed that although any number may be employed as a test, yet there are some which would afford little or no evidence of the correctness of the work. For instance the application of 2 or 5 would only ascer- tain the correctness of the last digit of the product ; all numbers ending with the same digit giving the same re- mainder when divided by 2 or by 5, It is evident that the same method of proof is applicable to division, consider- ing the dividend mimis the remainder as the product of the divisor and quotient.
64<. We know by inspection whether a number may be evenly divisible by 2 or by 5, as the former measures all even numbers, and the latter all numbers ending with 5 or 0 ; and those alone. We have seen also that it is easily ascertained whether a number be evenly divisible by 3 or 9 or IL The number 4< measures all numbers ending with 2 or 6, preceded by an odd digit ; or ending with 4, 8, or 0, preceded by an even digit : or in short all numbers whose two last digits are evenly divisible by 4. Whether 6 measure a given number may be determined by observ- ing whether it be evenly divisible by 3, and end with an even digit. If any number evenly divisible by 9 or by 3 end with 5, it must be evenly divisible by 45 or by 15. For dividing it by 9 or by 3 the last digit of the quotient must be 5, and therefore that quotient must be evenly di- visible by 5. In like manner every even number that 9 measures must be evenly divisible by 1 8. Every even num- ber which 1 1 measures must be evenly divisible by 22 ; and B5 must measure every number ending with 5 oy 0 ajid evenly divisible by 11 .
CHAP.
{ 30 )
CHAP. VL
Practical Application of Multiplication and Division. Qjiestions for Exercise,
^5, HERE the great object of a rational teacher should be, not to furnish the child with rules of operation, but to employ his reason in investigating the rules. From the lirst initiation of the youthful student into multiplication and division, he ought to be led to the practical use of these operations by familiar questions involving low num- bers. For instance, he may be called to find how many apples are wanted in order to give 4 apiece to 1 6 persons ; or called to divide 96 apples equally among 4 persons. And, — instead of learning what is called KeductioUy ascend- ing and descending, as distinct rules^ — as soon as he can multiply and divide by 4, 12, and 20, he is capable of finding the number of farthings in ^£> and the number of pounds in 1920 farthings. Such tables as are needful for solving the following questions, will be found at the end of the volume.
Ex. 1 . How many miles does a nian travel in 6 days, who goes 87 miles a day ?
Ex 2. A man travels 465 miles in 5 days, and an equal distance each day. How many miles does he go in one day ?
Ex. 3. How many hours in 365 days ?
Ex. 4. How many weeks in 5824 days ?
Ex. 5. A man spends 1 85, a day. How much does he «pend in the whole year "i
Ex. 6. How much per day may a man spend, whose annual income is <3^1314 .?
Ex. 7. Supposing that a standard pint contains 9216 grains of wheat, how many grains in one gallon ; and how many in one bushel ?
Ex. 8. Supposing that one acre of land produces 30 bushels of wheat, how many acres would be necessary to produce 1844670 bushels ?
Ex. 9. How many farthings in .^8738 : 2 : 8 ?
Ex. 10. How many pounds, &c. in 16777215 farthings ?
Ex. 11, How many inches in 25 English miles .'*
Ex.12.
( 31 )
fex. 12. How many bank notes, 8 inches long, would reach round the earth, supposing the distance to be 25000 miles ?
Ex. IS. How many seconds are in a solar year ; or 365 days, 5 hours, 48 minutes, 48 seconds ?
Ex. 14. How many seconds are in a lunar month, or 29 days, 12 hours, 44 minutes, 3 seconds ?
Ex. 15* How many Julian years (of 365 days 6 hours) would exceed an equal number of solar years by 7 days ?
Let the young student observe that this question amounts to the enquiry how many times 11' 12^' (that is, 11 mi- nutes and 1 2 seconds) are equal to 7 days ; and that the answer may therefore be obtained by dividing the number of seconds in 7 days by 672, the number of seconds by which one Julian year exceeds a solar. But the three last questions ought not to be proposed to a child without ex- plaining the meaning of the terms employed in them : — that by a lunar month we mean the time which intervenes between one full moon and the next ^ by a solar year, the time which intervenes between one vernal equinox and the next ; and by a Julian year, the time which Julius Caesar in his regulation of the calendar assigned to the year, reckoning 365 days in ordinary years, but 366 days in every fourth, or leap year ; which gives the average length of the Julian year 365 days 6 hours.
Ex. 1 6. How many English miles are equal to 1 1 Irish ?
Ex. 17. How many pounds, &c. in 680314 grains ?
Ex. 18. How many grains in 59 lb. 13 dwts. 5 gr ?
Ex. 19. How many tons, &c. in 4114201 drams?
Ex, 20. How many drams in 35 ton 17 cwt. 1 qr. 23 lb* 7 oz. 13 dr. ?
Ex. 21. An old lady observed that she had been for 52 years taking 2 oz. of snufF weekly, and that the snufF cost at an average 5d. per oz. What weight of snuff had she consumed, and how much had it cost her, reckoning the years Julian ?
Ex. 22. Her husband remarked, that he for the same period had drank 1 quart of claret daily, and that the average price had been 35 guineas a hogshead. How much wine had he consumed, and what had it cost him ?
CHAP.
( S^ )
CHAP. VII.
Doctrine of Ratio — direct — inverse — comjpound. Method of finding a fourth 'proportional. Abbreviations, Ques*^ tions for Exercise.
6Q4 WE have already remarked, that when any num- ber is multiplied by another j the product is called a mid- tiple of the multiplicand ; and the latter is called a sub- mtdtiple of the product. Thus, 54 is a multiple of 6, and 6 a submultiple of 54 5 because 54 is equal to 9 times 6. Thus again, 2 or 3 or 6 or 9 is a submultiple of I8* (Sub- multiples are otherwise called aliquot parts.) Now when two numbers are multiplied each by the same number, the products are called equi-mtdtiples of the respective mul- tiplicands ; and the latter are called equi-^subMultiples of the products. Thus, 18 and 24 are equi- multiples of 3 and 4, or 3 and 4 equi-submultiples of 18 and 24 : because 18 is 6 times 3, and 24 is 6 times 4.
67. By the ratio of two quantities we mean their jrelative magnitudes, or the magnitude of one in comparison of the other. Thus, although the absolute magnitude of a mile and 12 miles, is much T»««te greater than that of an inch and a foot, yet the relative magnitude, or ratio, of the two former is just the same with that of the latter : or in other words, a mile is just as small a space in compa- rison of 12 miles, as an inch is in comparison of a foot.
68. A ratio is written by the aid of two dots interposed between the terms c^ the ratio ; of which the former is called the antecedent, and the latter the consequent. And the ratio is called a ratio of greater or of less inequality ^ according as the antecedent is greater or less than the con- sequent. Thusj 3 : 5 expresses the ratio of 3 to 5 ; in which 3 is the antecedent, and 5 the consequent ; and the ratio is a ratio of less inequality. But 7:5 is a ratio of greater inequality. The ratio of 5 to 7 is called the re- cipyocaly or inverse of the ratio of 7 to 5.
69. The ratio of any two numbers is the same with the ratio of any equi-multiples or equi-submultiples of those numbers. This is an important principle of very extensive application : and its truth will appear most manifest on a little consideration. Thus, if we take the ratio of 3 to 5,
and
( S3 )
and multiply both temis of it by 7 : the products 21 and 35 are equi-multiples of 3 and 5 ; and the ratio of 3 to 5 must be the same with the ratio of those products, because it is evidently the same with the ratio of 3 times 7 to 5 times 7. Or, to take another instance, is it not evident that the ratio of 9 to 6 is the same with the ratio of 900 to 600, or of 90 to 60 (i. e. 9 tens to 6 tens) or in short of 9 times any number to 6 times the same number ; that is, the same with the ratio of any equimultiples of 9 and 6 ? And is it not equally evident that the ratio of 9 to 6 is the same with the ratio of the third part of 9 to the third part of 6, that is of 3 to 2, or of any other equi-submultiples of 9 and 6 ? This indeed, if it were needful, might be de- duced by necessary inference from the former ; inasmuch as 9 and 6 are equi-multiples of 3 and 2, or of any equi- submultiples of 9 and 6 ; and therefore in the same ratio with them.
70. The equality or identity of two ratios is denoted by four dots interposed between the ratios. Thus, 9 : 6 : : 3 : 2 denotes that the ratio of 9 to 6 is the same with, or equal to, the ratio of 3 to 2 ; or, as we commonly more briefly express it, that 9 is to 6 as 3 to 2. Such a series is called a series of proportionals, or by one word, borrowed from the Greek language, an analogy. The first and fourth terms of such a series (i. e. the antecedent of the first ratio and consequent of the second) are called the extremes : the second and third terms (i. e. the consequent of the first ratio and antecedent of the second) are called the ineajis. If the antecedent of the second ratio be the same with the consequent of the first, the terms are said to be in continued proportion. Thus, the numbers 3, 9, and 27 are in continued proportion ; because 3 : 9 : : 9 : 27.
71. If any two ratios be equal, it is plain that their reciprocals must be equal ; that is, that the consequent of the first ratio is to its antecetleiit as the consequent of the second ratio to its antecedent. Thus, since 9 : 6 : : 3 : 2, we may iftfer that 6 : 9 : : 2 : 3. For if 9 be as much greater in comparison of 6, as 3 is in comparison of 2, it follows that 6 is as much less in comparison of 9, as 2 is in com- parison of 3.
72. Again, from any analogy we may infer that the first antecedent is to the second antecedent as the first conse- quent is to the second consequent. Thus,i ince 9 : 6 : : 3 : 2,
I) we
( 3* )
^e may infer iKat 9 : 3 : : 6 : ^. Fbr the two given ratios cotild hot be equal, unless 9 were just as mitcn gri^ater in c6itr,parison of 3, as 6 is in compai'ison of 2. lliis may ^Iso be demonstrated from § 74. for the fourth propor- tional either to 9, 6 and 3, or to 9, 3 and 6 indifferently,
is . To state the two last inferences generally, putting
the letters a, b, c, d for any four proportional numbers, since a : b : i c : c?, we inay infer that b : a : : d : c ; and that a : c : : b ': d. . The former inference is called invei^sion ; the latter alternatim, or permutation,
73. Again, from any given analogy we may infer that any equimultiples or cqui-submultiples of the antecedents bear the same ratio to their respective consequents : and that the antecedents bear the same ratio to any equi-mul- tiples or equi-slibmultiples Of their consequents. Thus, since 9 : 6 : : 3 : 2, w^e may infer that 5 times 9 is 6 as Jl times 3 to 2 ; or that the fifth part of 9 is to 6 as the fifth part of 3 to 2. For it is plain that if we increase or di- minish the correspondent terms of equal ratios propor- tionally\ the resulting ratios must still be equal. And from the same principle it appears that if we increase or diminish corresponding terms of each ratio by adding to them or sub- tracting from them Ihe other terms, the resulting i-atios must be equal : or in other words, that the sum or difference of the terms of the first ratio is to either of its terms as the sum or difference of the terms of the second ratio is to its correspondent term. For then correspondent terms of the equal ratios are increased or diminished propor- tionally. Thus, from the analogy 9 : 6 : : 3 : 2 we may in- fer that 15 (the sum of 9 and 6) is to 6 as 5 (the sum of 3 and 2) to 2, &c. Or generally, from the analogy a :b : : c:d we may infer that a-\-b :b : : c-\-d : d ; or that a:
a-\-b : : c : c-^-d y where the sign -{- denotes the sum or dif- ference of the terms between which it is interposed. — The inferences drawn in this section may be demonstrated also from the principles of § 76 and 77.
74. If we have given the three first terms of an analogy we may find the fourth, by taking the product of the se- cond and third terms, and dividing that product by the first. Thus, suppose we want to find a fourth proportional to the numbers 3, 4, and 6 j that is, such a xi umber that
the
{ 35 )
/
tlie ratio of 3 to 4 sliall be the same with the ratio of 6 to the fourth number found. Multiply 6 and 4, and divide their product 24 by 3 : the quotient 8 is the fourth pro- portional sought. The truth of this result is evident in the present instance, 6 the antecedent of the second ratio being twice 3 the antecedent of the first ; and therefore the ratio of 3 to 4 must be the same with the ratio of twice 3 to twice 4, that is, of 6 to 8. But suppose the three given terms are 3, 4, and 5. The fourth proportional is found by the same process : divide 20, the product of the given means, by 3 the first term ; the quotient 6 and 4 (or 6 and the third part of 2) is the fourth term sought : which we thus demonstrate. By the principle laid down in § 69. the ratio of 3 to 4 is the same with the ratio of their equi- multiples 5 times 3 to 5 times 4 : or again, is the same with the ratio of the equi-submultiples of the latter, the third part of 5 times 3 to the third part of 5 times 4. But the third part of 5 times 3 is 5. Therefore 3 is to 4 as 5 to the third part of 5 timea 4, that is to the quotient arising from dividing the product of the given means by the first term.^-Let us now employ a general notation for exhibiting the same proof. Let the letters «, b, c, and ^ represent any four proportional numbers, of which we have given the three first, but want to find the fourth a^, I say x i»
equal to f , that is to the product of b and c divided by a»
a
Forby §69.«;6:;aXc:^'Xr, or :: ^^ : ^JiE. But
a a
c-
is equal to^>^(§ 40. latter part). Therefore a :b : : c :
a hXc
Q. E. D.
' a
75, Although the preceding demonstration involve no principle, but what must be sufficiently evident to a con- siderate mind, yet it may be satisfactory to some that ano- ther demonstration of the same thing should be exhibited. Let us then again suppose that we want to investigate a method for finding a fourth proportional to 3, 4, and 5. We know that 3 is to 4 as 1 (the third part of 3) to the third part of 4 ; or as the equi-multiples of the latter terms, 5 times 1 , that is, 5 to 5 times the third part of 4. Thus we are landed in the same result as before 5 for 5
i) 2 times
( 86 )
times tiie third p?trt of 4, and the third part of 5 times 4 are equivalent, as the former must be three times less than 5 times 4, and therefore equal to its third part. This will be more fully shewn, when we come to the doctrine of fractions,
76. In any analogy the product of the extremes is equal to the product of the means. This immediately follows from what has been last demonstrated : since either ex- treme is equal to the product of the means divided by the other extreme* For instance; 5:7 t: 10: 14. But we have seen that 14 is equal to the quotient arising fi-om di- viding the product of 7 and 10 by 5. Therefore {§ 40.) multiplying 14 by 5 must give a product equal to the pro- duct of 7 and 10. Or generally, putting the letters a, by c, and d, for any four proportional numbers, we may infer that aXd =z bXCi In like manner it appears that, if three numbers be in continued proportion, the product of the extremes is equal to the square of the mean. Thus 4 is to 6 as 6 to 9 : and the prodiict of 4 and 9 is equal to the square of 6.
77. We may also infer that, if two products be equal, their factors are reciprocally proportional / that is, that the multiplier of one is to the multiplier of the other, as the multiplicand of the latter to the multiplicand of the for- mer. Thus, the product of 2 and 28 is equal to the pro- duct of 7 and 8 : whence we may infer that 2:7:: 8 : 28. And generally, employing letters to denote numbers, if aXb^a^Xy, we may infer that a : x : : y : b. For if to the three numbers a, ^, and y we find a fourth propor- tional, it must by the last section be such a number that the product of it and a^ shall be equal to the product of X and y ; that is, it must be equal to b,
78. In any multiplication, unity is to either factor as the other factor to the product. Thus, the product of 6 and 5 is 30 ; and 1 : 6 : : 5 : SO, This immediately appears either from the last section, or from § 69. inasmuch as 5 and 30 are equimultiples of 1 and 6, and therefore in the same ratio.
79. In any division, the divisor is to unity as the divi- dend to the quotient. Thus dividing 36 by 4 the quotient is 9 : and 4 : 1 : : 36 : 9. This appears from § 77. and from the principle that the dividend is always equal to the pro- duct of the divisor and quotient.
80. When
( 37 )
80. When we say that one quantity js directhj as another quantity, it is to be understood that the one increases or diminishes in the same ratio in which the other increases or diminishes. But when one quantity increases in the same ratio in which another diminishes, or diminishes in the same ratio in which the other increases, we say that the one is inversely as the other. For example, if I pur- chase cloth at 205. per yard, the amount of the cost de- pends upon the quantity purchased as in the first case, and is therefore said to be directly as that quantity. But if I have to ride a certain distance, the time requisite depends upon the speed employed as in the second pase, and is therefore said to be inversely as that speed.
81. In multiplication, the product is directly as either factor when the other is given, or remains unvaried. Thus if I multiply 7 first by 3 and then by 5, the products 21 and 35 are as 3 to 5. (§ 69.) But in division, the quo- tient is directly as the dividend when the divisor is given ; and inversely as the divisor when the dividend is given. Thus if J divide 24 and 27 by 3, the quotients 8 and 9 are in the ratio of 24 to 27. (§ 69.) But if I divide 24 first by 3 and then by 6, the quotients 8 and 4 are in the ratio of 6 to 3. (§ 77.)
82. Hence whenever any quantity so dependiSi upon two others, that it is directly as each of them when the other is given, it must vary in the ratio of the product of two numbers taken proportional to those two quantities. Thus the distance to which a man rides depends upon the time for which he rides and the speed at which he rides,, so as to be directly as either of them when the other is un- varied. If therefore A ride for three hours, and B for five Jiours, and A ride twice as fast as B, the number of miles which A rides must be to the number of miles which B rides as 6 : 5, the products of the numbers which are pro- portional to their times and speed. But whenever any quantity so depends upon two others, that it is directly as the first when the second is given, and inversely as the second when the first is given, it must vary as the quotient obtained by dividing the first by the second ; that is, di- viding numbers taken proportional to these quantities. Now if I ride a journey, the requisite time so depends on the distance which I have to ride and the speed which I employ, \i is directly as the distance, and inversely as the
( 58 )
Speed. If therefore A lias to ride 50 miles and B 40, and A ride twice as fast as B, the time in which A performs his journey must be to the time in w^hich B performs his, as V to V, that is as 25 to 40, or 5 to 8.
83. Any two products are said to be to each other in a ratio compounded of the ratios of their factors. Thus the ratio compounded of the ratios of 2 : 5 and 7 : 3 is the ratio of 14 : 15. Hence the ratio compounded of two equal ratios is, the ratio of the squares of the terms of either ratio. Thus the ratio compounded of the equal ratios 9 : 6 and 3 : 2 is the ratio 81 : 36, (9^ : 6^) or 9 : 4 (3^ : 2-). For since 9 : 6 : : 3 : 2, it follows (§ 73.) that multiplying both antecedents by 3 and both consequents by 2, 27 : 12 : : 9 : 4 ; or multiplying both antecedents by 9 and both consequents by 6, that 81 : 36 : : 27 : 12. But the ratio 27 : 12 is by definition the ratio compounded of the ratios 9 : 6 and 3:2. And thus it appears that, if any four numbers be proportional, their squares are propor- tional.
84. Hence also it is evident that the ratio compounded of any ratio and its reciprocal is a ratio of equality. Thus the ratio compounded of the ratios of 9 : 6 and 6 : 9 is the ratio of 54 ; 54, i. e. a ratio of equality.
85. Again, any ratio being given us, we mayconceive any number whatsoever interposed between its terms, and the given ratio as compounded of the ratios of the ante- cedent to the interposed number, and of the interposed number to the consequent. Thus the ratio of 9 : 6 may be considered as compounded of the ratios of 9 : 2 and 2 : 6, For 9 is to 6 as twice 9 to twice 6, which (by § 83.) is the compound ratio mentioned. In like manner wx may con- ceive any two or more terms interposed, and the given ratio compounded of all the ratios taken in continuation. Thus, we may conceive the numbers 2, 5, and 7 inter- posed between 9 and 6 ; and the ratio of 9 to 6 will be compounded of the ratios of 9 to 2, 2 to 5, 5 to 7, and 7 to 6. For 9:6::9X2X5X7::6X2X5X7. (§ 69.)
86. From what has been said it is manifest, that the problem of finding a fourth proportional to three given numbers will frequently admit of an abbreviated solution, by substituting lower numbers. For in the first place if the two first terms, or terms of the given ratio, admit of being divided evenly by the same number, we may substi- tute
( . 39 )
ute for them the resulting quotients, as being in the same ratio. Thus, if it be required to find a fourth propor- tion to 27, 63, and 21, solving t^he problem at large ac- cording to the rule laid ch^wn in § 74. we should have to take the product of 63 and 21, and then divide thaf pro- duct 1323 by 27, which gives the quotient 49 as the fourth proportional required. But 3 and 7 being equi-submultiples of 27 and 63 are in the same ratio ; {§ 69.) and operating with these lower numbers we find the same result. But secondly, whenever the first and third terms admit of being evenly divided by the same number, we may substitute the resulting quotients : for tho^e equi-submultiples of the given antecedents must be proportional to the given consequent and the consequent sought. (§ 73.) Thus in the last ex- ample, after reducing the question to |;hat of finding a fourth proportional to 3, 7, and 21, I may substitute for the first and third of these numbers their 6qui-svibmultiples 1 and 7 : for putting jc for the fourth proportional soiight, inasmuch as 3 : 7 : : 2 1 : .r, the third part of 3 must be to 7 as the third part of 21 to j:. And thus we at once arrive at the same result as before, that the number sought i^ 49. 87. Let it be required to find a number, to which a given number shall be in a ratio compounded of two or more given ratios. The ratio compounded of the given ratios is (by definition) the ratio of the products of their respec- tive terms. Therefore this problem resolves itself into that of finding a fourth proportional to three given terms. Thus, if. we want to find a number to which 6 shall be in a ratio compounded of 9 : 5 and 15 : 36, "it is the same filing as if we were required to find a niimber to which 6 shall be in the ratio of 9^15 : 5 X 36. But it is plain that |3oth terms of this ratio are divisible by '9 and by 5, and that we may therefore substitute the ratio of the resulting
quotients 3:4; so that the number sought is _ or 8.
Hence it appears that, in solving this problem, if ante- cedent and consequent of either the same or different ratios admit of being evenly divided by the same number, we may substitute the resulting quotients : and tliat we therefore ought not to take the products of the corresponding terms of the ratios which we want to compound, till we have in- spected them for the purpose of ascertaining whether they be capable of being thus reduced ; nor till we have com- pared
( 40 )
pared the antecedents of the given ratios with the given antecedent of the ratio whose consequent we seek. For in the last instance, after reducing the question to that of finding a fourth proportional to 3, 4, and 6, the term3 may be reduced still lower by substituting for 3 and 6 their equi-submultiples 1 and 2. And thus a question, at first involving very high numbers and appearing to re- quire a very tedious operation, may frequently admit a solution the most brief and facile.
88. The rule (§ 7 4. J for finding a fourth proportional is commonly called the Rule of Three ; because we have three terms of an analogy given us to find the fourth. It may more justly be called the 7ide of proportion. Its very extensive practical application will be shewn in the 1 3th. Chapter. Meanwhile the young student may exercise him- self in the principles of this chapter by solving the follow- ing questions ; and may easily increase the number of the examples, at pleasure, by substituting any other numbers. Besides investigating the answer by performing the re- quisite operations of multiplication and division, I would strongly recommend that he should be accustomed to ex- hibit it by the aid of the symbols denoting those operations. Thus, if it be required to find a fourth proportional to the numbers 23, 24, and 25, the answer mav be expressed
bv^X-11
^ 23
Ex. 1. Find a fourth proportional to 15, 40, and 24 } Ex. 2. The two first and the last terms of an analogy are
17, 9, and 234. What is the third term ?
Ex. 3. The first and the two last terms of an analogy are
18, 126, and 17 What is the second term ?
Ex. 4. What two numbers are in the ratio compounded of the ratios of 7 to 3, 4 to 5, and 11 to 13 ,^
Ex. 5. What two numbers are in the ratio compounded of 7: 3, and 6: 14 .?
Ex. 6. What two numbers are in the ratio compounded of 17: 3, 3: 14, and 14: 16.?
Ex. 7. What is the ratio compounded of 17 : 3, and 6: 34?
Ex. 8. From the analogy, 7 : 25 : : 21 : 75, what equa- tion may be derived ?
Ex. 9. From the equation 12x7 = 14X6. what ana- logy may be inferred ? ^' ^ CHAP.
( ^1 )
CHAP. VIIL On the Nature of Fractions.
89. IF we divide any one whole tiling, a foot, a yard, a pound, &c. into tliree equal parts, any one of them is one third of the whole ; written thus — \, If we take two of them, we take two thirds of the whole, or 4» Such ex- pressions are called fractions ; the number ^bove the line is called the numeratm^ of the fraction, and the number below the line the denominator, A 'projpcr fraction is that whose numerator is less than its denominator. If the nu- merator be equal to the denominator, or greater, the fraction is called improper,
90. The denominator always denotes the number of equal parts, into which the whole thing, or integer, is conceived to be divided. The numerator denotes the num- ber of those parts, which are taken in the fraction. Thus the fraction y intimates that the integer is divided into 7 equal parts, and that we take 3 of those parts in the fraction.
91. Hence any improper fraction whose numerator and denominator are equal, such as ^, ^ &c. is equivalent to the one integer which we suppose divided into equal parts. For if we divide a pound, for instance, into 7 equal parts, and txike 7 of those parts, we just take the whole pound, neither more nor less. On the other hand it is manifest that ^, or any proper fraction, is less than the whole; and that 4> or any improper fraction whose numerator is greater than its denominator, is greater than the whole. Observe, that we consider and speak of the whole thing divided as 07ie integer, whether it consist of a single pound, foot, yard &c. or^of ever so many pounds, feet, yards &c.
92. According to the view which has hitherto been given of any fraction, such as 4? we consider it as two thirds of one. But there is another view also, which it will be useful to attend to. It may be considered as the third part of two. This view arises immediately out of the for- mer ; for inasmuch as the third part of two is twice as great as the third part of one, it must be just equal to two thirds of one. In like manner the fraction ^ may be indiflerently considered either as three sevenths of one, or as the se- venth part of 3 : the latter being three times greater than
the
( 42 )
the seventli part of one, and therefore just equal to three sevenths of one. Thus any fraction may be considered as a quotient, arising from the division of the numerator by the denominator. And hence the fractional notation is commonly employed to express division.
93. The value of any fraction varies directly as th,e nu- merator and inversely as the denominator. This appears at once from what has been last said, compared with § 80 and 8 1 . The same thing also will appear from the first view given of a fraction, when we consider tliat if a whole thing be divided i^to a given number of equal parts, the greater the number we take of those parts the greater is the quantity we take and in the same ratio : but the greater the number of eqi^al parts into which the whole thing is di- vided, the less is any one of them, or any given number. Thus I is greater ihan \'m the ratio of 7 to 4. But ^ is less than 4 ii^ the ratio of 3 to 5. Therefore :^ is to 4- in a ratio compounded of 3 : 5 and 7 : 4 (the direct ratio of the numerators and inverse ratio of the denominators.) that is, as 21: 20, • ^
94b. Any fraction is to 1, as the numerator of the fraction to its denominator. Thus 4 is to 1 as 3 to 7. For 1 is equal to 4*. (^ 91.) But 4 is to ^ as 3 to 7. Here and throughout the subject when we speak of 1, it is to be understood in the sense explained at the end of ^ 91.
95. The value of any fraction will remain unaltered, if we multiply or divide both its ttirms by the same num- ber ; that value depending altogether on the ratio of its terms, and not their absolute magnitude. Thus the frac- tion ^ is equal to the fraction 4 or A or i^, &c. and the fraction ~ is equal to the fraction i. For comparing, for instance, the fractions ^ and -^, in the latter the whole thing is conceived to be divided into 10 times as many equal parts as in the former ; each of which therefore is 10 times less than each of the former i and consequently if we take 10 times as many of them as of the former, we shall take just the same quantity of the whole. Apd thus, the twelfth part of a foot being an inch, -^-^ of a foot is 6 inches ; but that is just equal to half a foot, or to the fraction 4. The principles laid down in this section are so simple, that by a few familiar illustrations a very young child may be made to comprehend them ; yet upon these simple prin- ciples the whole doctrine of fractions depends.
96. Hence
( *3 )
t)6. Hence we see how we may easily bring a given frne- tion to lower terms, if its numerator and denominator be capable of being divided evenly by the same number. As any number which evenly divides another is said to meamre it ; so a number which evenly divides two or more num- bers is called a common meamre of them. Numbers which admit no greater common measure than unity are said to be prime to each other : and if the terms of a fraction be prime to each other, it is in its lowest terms ; as we cannot bring it to any equivalent fraction of lower terms. Thus the fraction |. is in its lowest terms ; and the fraction % may be brought to its lowest terms by dividing both numerator and denominator by 2 : for the equal fraction y consists of numbers prime to each other.
97. Hence also it is easy to bring a given fractien (sup- posed to be in its lowest terms) to an equivalent one of another denominator, provided that other be some mul- tiple of the given denominator. Thus, if it be required to bring ^ to an equivalent fraction whose denominator shall be 18 : we observe that in changing the denominator from 6 to 18 we multiply it by 3 ; and therefore to maintain the equality of the two fractions, we must multiply the numerator by 3, tso that the required fraction is 4|. And if it be required to bring the same fraction ^ to another whose denominator shall be 162, we only want to ascertain by what number 6 must be multiplied in order to give the product 162, that we may multiply 5 the numerator by the same number. This is ascertained by dividing 162 by 6 ; and we thus find that 5 X 27 is the required numerator. Thus also ^ may be brought to a fraction whose denomi- nator is^ 15 ; because 15 (though not a multiple of 6) is a multiple of 3 the denominator of the equal fraction y.
98. To bring a given fraction to its lowest terms, it is only necessary to divide both its terms by their greatest common measure, that is by the greatest number which evenly divides them both. Thus, if we be given the frac- tion f i, it is plain that both its terms are evenly divisible by 3, or by 7, or by 21. Bui of these common measures 21 is the greatest, and will therefore give the smallest quo- tients : so that the lowest terms of the fraction are 4. But if the terms of the given fraction be high numbers, we may be unable to ascertain by inspection whether they be prime to each other j or if not, what number is their greatest com- mon
( ^4 )
0ion measure. We proeeed therefore to state and demon- strate the method of discovering this.
99. Divide the greater number by the less : if there be no remainder, your divisor is the greatest common mea- sure, inasmuch as no number greater than itself can mea- sure, the less of the two given numbers. Thus, if the two given numbers be 12 and 96, 12 must be their greatest common measure ; for it measures 96, and no number greater than 12 can measure 12. But if there be a re- mainder on the fnst division, then divide your last divisor by that remainder 5 and so on, till you come to a remain- der which v/ill measure the last divisor. Thi& remainder is the greatest common me?,sure of the two given numbers j and therefore if jou find no such remainder till you come to 1, the giveft numbers are prime to each other. Thus, if the two given numbers be 182 and 559; dividing the greater by the less we find the quotient 3 and the remain- der 13: then dividing 182 by 13, we find tlie quotient 14, and no remainder, I say then that 13, the remainder which measures the first divisor, is a common measure of 182 and 559, and their greatest common measure. First, it is a common measure of them ; for it measures 182, and therefore 3 times 182; and therefore 3X 182 -f 13, or the sum of 3 times 182 and 13. But that is equal to 559j^ as we saw by the first division. Therefore it is a com- mon measure of 182 and 559. — But secondly, it is their greatest common measure. For suppose any greater num- ber, for instance 17, to be a common measure of 182 and 559. Since it measures 182 it must also measure 3 times 182: and since 559, it measures 3 X 182 -f- 13, which is equal to 559. Inasmuch then as it measures both 3 X 182 and 3 X 182 + 13, it must measure 13 ; that is, a number greater than 13 must measure 13 : which is absurd. There? fore 13 is the greatest common measure of 182 and 559, Q. E. D.
100. Let us propose the same proof in a general manner, putting letters for the numbers. Let the numbers, whose greatest common measure we want to find, be represented by the letters a and b, of which a is the less : and dividing ib hy a let the quotient be represented by x and the re- mainder by c. We may infer that b =a:Xa + c. Then dividing a by r, let the quotient be 1/ and the remainder d. We may infer that a =j/ X c -}- d. Then dividing c by d,
let
( 45 )
let the quotient be % without any remainder. I say d k the greatest common measure of a and b. For since it measures c, it must measure 2/Xc + d, that is a. And since it measures a and c, it must measure a:Xa -f c, that is b. But if we should suppose any greater number than fZ to be a common measure of a and Z>, since it measures « , it must measure ^ X « ; and since it measures both :rXa and a;Xa + c (i. e. b.) it must measure c, and therefore j/Xc. And since it measures both i/Xc and yxc + d ( i. e. «) it must measure d ; that is a number greater than d will measure d : which is absurd. And in like manner^ if there be a remainder of 1 on the last division, we can prove that 1 is the greatest common measure of a and b ; that is, that the numbers a and b are prime to each other- Ex. 1. Whatis the value of the fractions y, y, y, and V ? (See § 102. next chapter.)
Ex. 2. What fractions are equal to 54-, 2-|., 7^, and 124 -
Ex. 3. Express 7, 8, 9, and 10 by fractions whose de^ nominators shall be 4, 5, 6, and 7.
Ex. 4. What is the ratio of 4 to 4- ? of 4 to tx ? and of ^ to 1 .?
Ex. 5. Express ^, 4, and -J, by equivalent fractions whose denominator shall be 18.
Ex. 6. Bring the fractions ^VV, tVVtj and t^ to their lowest terms ?
Ex. 7. Bring the fractions y, ^, and | to a connnon de- nominator ? (See § 104.)
Ex. 8. Also the fractions -f, tt, and in- ?
Ex. 9. What is the ratio of 4 to -^ ? of | to -j-V ? and of -J to I- ?
Ex. 10. A. and B. got legacies. A got .5^750; and his legacy was yths of B's. What was B's ?
Ex. 11. What is the greatest common measure of 153 and 493 ? of 336 and 1645 ? and of 133 and 459 ?
CHAP, IX.
On Addition and Subtraction of Fractions,
101. WHEN the fractions whose sum or difference we want to find have the same denominator, the method of
nerforminsf
- ( 46 )
performing tliose operations is as obvious, as the addition or subtraction of integers. For it is as plain that the Bum of two ninths and five ninths is seven ninths, and that their difference is three ninths, (i. e. that 5-f-|^ = -J and tliat -J- — ■J = i) as that the sum of two shillings and five shillings is seven shillings, and their difference three sliillings. Ninths in the former case, and shillings in the latter, are but the denomination of the numbers, which v,e add or subtract: and in place of the fractional notation, the column in v/hich the numbers 2 and 5 stand might be headed with the denomination ninths, as it is commonly with the denomination shillings.
102. If the sum of the numerators exceed the common denominator, it is easy to ascertain what integral or mixed number it is equal to, by dividing the sum of the nume- rators by the denominator. Thus the sum of |, ^, and •J is V** ^ut since f is equal to 1 (§ 91.) V must be equal to 2 ; and therefore ¥ to 2|. In like manner in the ad- dition of pounds shillings and pence, if the sum of the numbers standing in the column of pence exceed 12, we divide it by 12, the number of pence in one shilling, &c. And in fact the operations, which the child is taught in addition and subtraction of what are called divers de- nominations, are all really fractional operations ; 8 pence, for instance, being ^-^ of a shilling, and 8 shillings ^ of a potmdi And in the case of half-pence and farthings, even the fractional notation is introduced*
103. In subtraction, if the fractional part of the subtra- hend exceed the fractional part of the minuend, w^e com- bine with the latter a unit borrowed from the integral part <3f the minuend ; and therefore have to conceive the right- hand integral digit of the minuend lessened by one. Thus, in subtracting 2 J from 15|, since we connot sub- tract ^ from I, we subtract it from 1+^, i. e. from y. The remainder is ^: and we have then to subtract 2, not from 15, but from 14. But in place of this, carrying 1 to the subtrahend we subtract 3 from 15. See § 16.
104. If the fractions which we are required to add or subtract have different denominators, we must first bring them to equivalent fractions of the same denominator ; and then proceed as before. Thus, if we have to find the sum or difference of f and ^, it is necessary to bring them both to the same denominator, Now we can bring | tQ
an
( 47 )
an equal fraction of any denominator, which is a multiple of 5, and ^ to any which is a multiple of 9. (§ 97) But the product of 5 arid 9 being a multiple of both, we may bring the two given fractions to the denominator 45. And in doing this, in order to niultiply both terms of the fraction by the same number (i. e. in order to keep it of the same value § 95 ) we must multiply the numerator of each fraction by the denominator of the other ; when they , become^ and If-, whose sum is ^ Ji»f ^i^, and their .^*" difference -^. Hence, let there be ever so many fractions, ^f'' of ever so various denominators, to be added, the reason is plain of the common practical rule, to take the product of all the denominators for a common denominator, and then to multiply the numerator of each fraction by all the denominators except its own : inasmuch as it is by these that we have multiplied its denominator.
105. But it is desirable to keep the terms of our fractions, as low as possible ; and we may often find a number less than the product of all the denominators, which is yet a multiple of them all, and will therefore answer for a com- mon denominator. Thus, if we have to add y, ^, and ^; 36 the product of 9 and 4 being also a multiple of 3 will be a common denominator, and the fractions become •jzi riy ^^^ tI- I forbear at present from bringing for- ward the rule for finding the least common multiple of two or more given numbers; (see Chap. 18.) as it would be hard to make the demonstration of it clearly intelligible without a little knowledge of Algebra.
Ex. 1 . What is the sum of -|, 4, and -J- .?
Ex. 2. What is the sum of |, |, ^, and -J .?
Ex. 3. What is the excess of -J above | ? and of -J above ^ •'*
Ex. 4. What is the difference between the sum of y-f J -f T and the sum of | + |4-i-i ?
Ex. 5. A man left a legacy of 10,000^ among three sons, so that the eldest should have 4 of it, and the second 4- of it. What proportion of the leg^acy did the youngest receive ?
CHAP. X
On Multiplication and Division of Fraction^, 1 06. FROM what has been said in the first section of th« preceding chapter, it is evident that to multiply any
Iraction
( 4S )
fraction by an integer we need only multiply its numerator by the integer : and that any fraction will be divided by an integer, by dividing its numerator by the integer, whenever the integral divisor measures the numerator. For it is as plain that 3 times -^ is 4-^, or 2-i^, and that the third part of -^ is w, as that 3 times 9 is 27, and that the third part of 9 is 3.
107. But if we want to divide -r^ by 3, we cannot ob- tain the quotient by this process, as 3 does not measure 7. The third part of 7 is 2 and y • so that the third part of 7 tenths is -^ and one third of a tenth. But we have still to enquire what fi-action is equal to one third of a tenth, or what is the quotient of ^ divided by 3. In such cases therefore we employ an operation always equivalent to the division of the numerator, namely the multiplication of the denominator. And accordingly the third part of to i« ■Yo' For if we suppose any whole thing divided first into 10 equal parts, and then into 30 equal parts, the latter being 3 times as many as the former must each of them be 3 times less than each of the former ; and therefore 7 of them must be 3 times less than 7 of the former : or in other words -o is the third part of t^. See § 93'
108. Thus the universal rule for dividing a fraction by an integer is, to multiply its denominator by the integer* And whenever we have to multiply a fraction by an in- teger which measures its denominator, the product is ex- hibited in lower terms by dividing the denominator by the integer, than by multiplying its numerator. Thus, 3 times I is by the one process ~ ; by the other ^ : results which we know are equal from § 95.
109. From the methods of multiplying and dividing a fraction by an integer, it is easy to pass to multiplication and division by a fraction. To multiply by a fraction, multiply by its numerator, and divide the product by its denominator. To divide by a fraction, divide by its nu- merator and multiply the quotient by its denominator. Thus, to multiply 4 by |-, multiply 4- by 3 ; and divide the product V by 4 : the quotient i| is the product sought. For the multiplier I being the fourth of 3, ( § 92.) the first product V (obtained by multiplying 4 by 3) is 4 times too great : and therefore its fourth part must be the true pro- duct sought. In like manner, if we have to divide 4 by ^^ dividing by 3 the quotient tt is 4 times too small, as we
have
( to )
liiivc employed a divisor four times too great : and therefore the true quotient must be 4 times -rV or 4t.
1 i 0. Ilence appears the reason of the practical rule com- monly given for multiplying a fraction by a fraction ; namely, take the product of the numerators for the nu- merator of your product, and the product of the denomi- nators for the denominator of your product. It appears that the latter operation is in fact a division of the fraction, in order to reduce the product to its just amount. Another proof of the operation may be derived from the principles laid down in § 78. and § 94. For if we want to multiply 4 ^^y T? unity must be to the multiplier as the multiplicand to the product. But 1 : | : : 3 : 2. Therefore 3 : 2 m i to the product, which fourth proportional must be obtained by multiplying -|- by 2, and dividing the product bv 3. (§74.)
111. The reason is equally evident of the practical rule commonly given for dividing a fraction by a fraction ; namely to multiply by a fraction the reciprocal of the di- visor. For it appears by comparing the operations, that to divide | by -f is the same thing as to multiply 4 by 4, Another proof of the operation may be derived from the principle laid down in § 79. For if we have to divide -| by 4 the divisor must be to unity as the dividend to the quotient. But ^ : 1 : : 2 : 3. Therefore 2 : 3 : : -f to the quotient, which fourth proportional must be obtained, by multiply- ing 4 hy 3 and dividing the product by 2.
112. The same things are at once applicable to the multiplication or division of an integer by a fraction. The product of 7 multiplied by | is V> or 5|; the same as the ))roduct of 1 multiplied by 7. The quotient of 7 divided by I is V > or 9|, the same as the product of 7 multiplied by 4- Any integer indeed may be conceived as an im- proper fraction, whose denominator is 1. And here let it be observed that whenever our multiplier is a proper fraction the product must be less than the multiplicand ; and whenever our divisor is a proper fraction, the quo- tient must be greater than the dividend. For when we talk of multiplying any thing by i, we really mean taking- three fourths of the multiplicand ; as when we talk of multiplying any thing by 1, we piean taking the mul- tiplicand once. But ^ (or any proper fraction) being less than one, three fourths of the multiplicand must be less
E than
( 50 )
tljcin the whole multiplicand. On the other hand in Ji-' vision, the less the divisor is the greater must be the quo- tient. Now if we divide any number by 1 , the quotient is equal to the dividend. Therefore if we divide by a proper fraction^ the quotient must be greater than the dividend. It is plain that since 1 is contained in 7 seven times, ^ (which is less than 1 ) must be contained in 7 more than seven times.
113. Observe that if we multiply any fraction by its denominator the product is the numerator integral. Thus, the product of f multiplied by 5 is 3, of 4 multiplied hj "7*1 4 '^ i^ ^5 ^^' ^^^' I X 5 is in the first place V 5 but to re- ***♦#« (luce this to its lowest terms, we should divide both terms : tL« : by 5, when the result will be ^ or 3. But w^e may save /^ the trouble both of the multiplication and division, the •* Hatter just undoing the former. The same thing indeed at once appears from considering I as the quotient of 3 divided by 5, ( § 92.) and from the principle that the product of the divisor and quotient is the dividend. It is evident also that any integer may be brought to a frac- tional form of any given denominator, by taking for the numerator of our fraction the product of the integer and given denominator. Thns, 7 is equal to V? to V5 &c. It is plain indeed that since there are 5 fifths in 1, there must be 35 fifths in 7.
Ex. 1. ^X5= ? 4-f-5=? x'^X3= ? A-^3= ?
ii<x. z. -^x^— r _-r-T— 1 ttX-t— - r T-r-r-y — r
Ex. 3. ^X9= ? ^VX27= ? 2X4= ? 2-T-4= ? Let the examples of multiplicatioii be proved by division^ and V. V.
Ex. 4. What fractional part of 3 is ^ds. of 4 ? — On this question let the student observe that ^ds. of 3 must be les&^than ^ds. of 4 in the ratio of 3 : 4 ; and therefore the fractional part sought of 3 mu&t be greater than ^ds. of it, in the ratio 4 : 3. Whence the following analogy, as 3 1 4:r4: ^. And accordingly ^ths. of 3 = 2|; and ^ds. of 4 = 2f
Ex. o. Wliat fractional part of 7 is^ 'ths. of 5 ?
Ex. 6. A man spent :^th. of a legacy in 5 months ; ^ds^* of the remainder in 7 months ; and then had £95 left. What was the amount of the legacy ?
Here observe that, when he had spent ^ of the legacy, |tlis. were left. And when he had spent ^ds. of this, he had spent in addition i of the whole: for | of =A = 4.
Ex,. 7.
( 51 )
Ex, 7. A man devised ^ds. of his fortune to his eldest son ; -f ds. of the remainder to his younger ; and the rest to his widow. The elder son's share exceeded the younger's by £150, How much had the widow? Here we are told that yds. — |ths. of the fortune (or ^ths- of it) amounted to 750^ ; whence we find the whole fortune, of which the widow had ^th.
CHAP. XI.
071 the Nature of Decimal Fractions,
11 4, After the doctrine of vulgar Fractions has opce been mastered, decimal fractions can present no difficulty to the student. It is only necessary to take a clear view of the notation employed in them. In decimal fractions we use no other denominators than 10, J 00, 1000, &c. and those denominators are not written^ but intimated by the position of the decimal jpoint ; for we understand as many" cypliers following a unit in the denominator, as there are dii^its standing on the riorht hand of the decimal point. Thus the decimal fraction .7 is equivalent with the vulgar fraction -^ ; the decimal .037 with t4^» And it appears that in decimal notation we write only the nu- merator, but have the understood denominator intimated by the decimal point : and that to write the vulgar frac- tion -r^Vcr decimally, we need only to omit the denominatof and to prefix to the numerator the decimal point followed by two cyphers — thus .0037. It is necessary to prefix two cyphers, in order that four digits may stand on the right hand of the decimal point, as there are four cyphers following the unit in the denominator ; and in order that the digits of the decimal fraction may yet express the nu- merator given.
115. From what has been said it appears, that annexing one or more cj^hers to a decimal fraction on the right hand makes no change in the value of the fraction, inas- much as for every cypher annexed both numerator and denominator are increased ten fold: but that prefixing one or more cyphers on the left hand, decreases the value of the fraction ten fold for everv cypher prefixed, inasmuck
'E2 as
( 52 )
«s tlie understood denominator is increased so many fold without an}' change in the numerator. Thus the decimals .7, .70, .700, &c. or their equivalent vulgar fractions t^o, x^, ■ro°6% &c. are all of the same value {§ 95.) : but .07 is ten times less than .7, .007 one hundred times less than .7, since T^^ is the tenth part of t^, and 7^0 the hundredth part. {§ 107.) And in general any decimal fraction is mul- tiplied by 10, 100, 1000, &c. by removing the decimal point one, two, three, &c. places towards the right hand ; or divided, by removing it towards the left. Thus if I have to multiply the decimal .37 by 10, the product i» J5.7 ; in which the 3 is integral, and only the 7 affected by the decimal point. For 10 times -r^ is 44 {§ 108.) or 3^ i. e. 3.7. Whereas 24.37 (in which the numbers on the left hand of the decimal point are integral) is divided by 10^ by bringing another digit under the decimal point, removing it one place towards the left ; which gives the quotient 2;437. For 2W-^ is equal to ^^ (24 being equal to \%%° : the tenth part of which is 441^ i. e. 2.^^; or 2.437.
116. In short there is no operation on decimals, which the student may not investigate by performing the same operation according to the notation of vulgar fractions ; and then expressing the result decimally according to the simple nde of decimal notation. I shall proceed how- ever to exliibit this investigation briefly in the following chapter : after ]:)remising the method of bringing any vul- gar fraction to the decimal form.
117. Suppose then we are required to bring the fraction 4 to the decimal form, or to find a decimal equal to the vulgar fraction ^. This is in flict to bring 4 to an equal fraction, whose denominator shall be some power of 10. Now we know that i is equal to |^, i-g^? iM-h &c. or that annexing an equal number of cyphers to both numerator and denominator will not change the value of the fraction. But instead of 4 followed by any number of cyphers, we want that the denominator should be 1 followed by some number of cyphers. This change can be effected no other- wise than by dividing some of the former denominators by 4 : and then to maintain tlie value of the fraction un- altered we must also divide the innnerator by 4, Let us accordingly divide both terms of the fraction i^^ by 4 ; and it becomes t^, a fraction capable of being written
decimally
( ■ 55 )
ilecimally ^ and the decimal sought is .75. Hence appeal t-^ the reason of the practical rule, to annex any number ol* cyphers that may be necessary to the numerator, and di- vide by the denominator, and point off Irom the quotient as many decimal places as you have annexed cyphers. By this process in fact botl| terms of the given fraction are multiplied by the same power of 10, and divided by the denominator : and thus the value of the obtained fraction, which is capable of decimal notation, is the same with that of the given onc>. 1 hu§ 4 ^ .5 i -^,2 ; ^V = -0^ - ^or
118- But in reducing vulgar fractions to decimals, we shall frequently find that, continuing the division ever so iiir, we can nevci" arrive at an exact quotient, but shall at length come to g. reniainder the vune v/ith the given niune- rator or one of the former remainders ; and therefore frou\ that recurrence the same digits must continually recur in the quotient. (See § 123.) And this indeed must always be the case, except when the denonjinajtor of the given fraction reduced to its lowest terms i^ 2 or 5, or some power of these numbers, or sonje product of their p.owers. Thus in reducing y to a decimal, we find it equal to .333, &c. and i = .lll, &c, ^^zzAGG, ^c. tVt ==423123, &c. Such are called interminfite decimals., and are said to cir- culate through the figure or figures which continually recur. A method of calculating tlie vulgar fraction, which will produce any given circulating decimal, shall be assigned and demonstrated in another part of this work. (See c. 20^)
Ex. 1. JLxpress t?^, -rf-o, t%V» two decimally.
Ex. 2^ What vulgar fractions are equal to the dcciiji-als .75, .075, .024, .0015? ' ^
- Ex. 3. Express |, 4t, -rs, \i decimally ?
Ex. 4. Express |, 4> iy ii decimally ?
Ex. 5. Multiply .0015 by 1000 ?
Ex, 6. Divide .75 by 1000 ?
CHAP. XIL
Arithmetical Operations on Decimals^
119. SUPPOSE we have to find the sum or difference ©f the dechnals .07 and .834. They are equivalent to the Tulgar fractions x^o and -^^^^ ; which must be brought to
thy
( 5i )
the same denominator, before we can find tlieir sum or dif- ference ; and then become -r^wu and ^-^% : whose sum is ^9_o^^^ or .904, and their difference t¥^, or .764. Now
.834 if we write the given decimals thus — .07 — so as that the decimal points shall stand in line, we may understand a cypher after the 7 on the right hand, as it will make no change in the addition or subtraction ; and then proceed- ing to take the sum or difference, we shall have the same results. And here we see that one advantage of decimal above vulgar fractions consists in the facility, with which they are brought to the same denominator. The decimal point of the sum or difference must also stand in line with the decimal points of the fractions, which we add or sub- tract ; so that any digits standing on its left hand are in- tegral. Thus if we add .9643 and .8, the sum is 1.7643. For brought to the same denomination in vulgar fractions the given fractions are -i^^%- and t^^% ; the sum of whose numerators is 17643. But the improper fraction 44lw is equal 1 44M, that is to 1.7643.
120. Suppose we have to multiply .04 by .3 ; that is, the vulgar fraction t|^ by tV : the product is -i4^^, or .012. The number of cyphers in the denominator of the product being necessarily the sum of the cyphers in the denomi^ nators of the factors, the denominator of the product must be intimated by pointing off as many decimals in the pro- duct of the numerators (multiplied as integers) as the suin of the decimal places in both the factors. The same rule applies, where one of the factors is an integer, or one or both mixed numbers. Thus the product of 1,2 multiplied by .8 is .96. For 1.2=44, and .8 = ^: the denominator of whose product is 100 ; and this product is intimated by pointing off' tv/o decimal places from the product of their numerators 96. In like manner the product of 12 mul- tiplied by .8 is 9.6 : but of 12 multiphed by .008 is .096.
121. Hence the i:ule of division is obvious. For since the dividend is always the product of the divisor and quo- tient, there must be as many decimal places in the divi- dend as the sum of the decimal places in the divisor and quotient. And if the given dividend have fewer decimal places than the divisor, we make tlie luimber equal by an- nexing decimal cyphers to the dividend on the right hand, which we have seen cannot alter the value of the dividend.
(5115.)
( 55 )
■\§ 115.) Thus dividing 1.2345 by :05 gives the quotient 24.69 : for we must point off two decimal places from the quotient, that the decimal places in the dividend may equal the sum of those in the divisor and quotient. But 1.2345 divided by .0005 gives the g 469 integral : and 123.45 di- vided by .0005 gives the quotient 246900 integral : for we must annex two cyphers to the dividend in order to make the number of its decimal places equal to those in the di- visor, and then there can be no decimal place in the quotient. It is plain that, in calculating the number of decimal places in the dividend, we must take into account every decimal cypher, which we have occasion to annex to the remainders for -continuing the division-. And that if the number of digits in the quotient be less than the num- ber of decimal places requisite in it, we must supply de- cimal cyphers on the left hand, Ihus dividing .25 by 4 integral gives the quotient .0625^ For since there is no decimal place in the divisor, there must be as many in the quotient as in the dividend ; and we have occasion to annex two decimal cyphers to the dividend in order to get a com- plete quotient. The truth of all these results will likewise appear by expressing our decnnals as vulgar fractions. Thus .25 decimal is -^^ or tV^^ * ihe foui*th part of which is T^U^ i. e. .0625.
122. In division of integers, when the divisor does net measure the dividend, it is common to continue the di- vision decimally, annexing cyphers to the remainders, and pointing off as many decimal places from the quotient as we have annexed cyphers. For by this operation we in fact reduce to the decimal form the vulgar fraction which is part of the quotient. Thus in dividing 25 by 8, wcliave seen that the real quotient is 3^^, of which the fractional part may be turned into the decimal .125.
123. But here let the student observe, that it cannot be requisite for any practical purpose to continue this process as far, as might be necessary in order to obtain a perfectly accurate result. Thus if I di^ ide 63 by 29, and continue the annexation of decimal cyphers, I find the quotient
2.17241379310344827586206896551, &c. the circulation of the same decimal digits not commencing till the 29th. place of decimals. But it would be quite useless in practice to continue the process so far. The three first decimal digits give us the fractional remainder
within
( 56 )
within less than Wg^th. part; (for we find it that it k somewhat more than tWo, but less than ttoo) : the four first, within less than To,^o^th. &c. — That the fraction ■^^^ reduced to the decimal form jnust at length circulate, will easily appear, if we consider — l«t. that it cannot produce a terminate decimal, since there is no digit which mul- tiplying 29 can give a product ending with a cypher : — 2ly. that some one of the remainders must at length re- cur, since each remainder must be less than 29, and can- not be either 10 or 20; so that there are but 26 possible remainders.
Ex 1. What is the sum of 20.05 + 1.5 + .005 ?
Ex. 5. What is the difference between 3.75 and 375 ?
Ex. 3. What is the product of 375 X .5 ? of 3.75 X •05? and of 3.75 X 10.5?
Ex. 4. What is the quotient of 3.75 -v- 5 ? of 3.75 4- .15? and of 375 -f- .15 ?
CHAP. XIII.
Practical Aj^pUcation of the Rule of Frojwrtion,
124. IF I can purchase 4 yards Cloth for £2 : 1 5s, and
want to know what quantity I ought to get, at the same
rate, for £2 : I2s. it can be ascertained by the rule of
proportion. For the quantities purchased at a given rate
must be directly as the prices paid : therefore 4 yards, the
quantity purchased for £2 : 155. must be greater than the
(juantity purchased for ^2 : I2s. in the same ratio in which
the former sum of money is greater than the latter, or in
the ratio of 55s. to 525. or of the abstract numbers 55 : 52.
Therefore 55 : 52 : : 4 yards to the quantity sought:
which fourth proportional is found (§ 74.) by taking the pro
duct of the second and third terms and dividing it by the
4 X 5*^
first; oris ~, or 3f|- yards, that is, 3 yards 9 inches
55
and somewhat more than one third of an inch.
125. This example may serve to ilhistrate the following general rule for solving all such questions. 1st. Place as the third term of your analogy that given quantity, which
is of the same denomination with the thing sought. Thus,
\ 57 )
wwrvEKsn
^n the last example, the question being what quantity of i.loth can I get, the given quantity of cloth, or 4 yards, ituist be the third term of the analogy. 2ly. Consider from tlie nature of the question whether the answer must be more or less than that given quantity ; and accordingly state the other two given terms in a ratio of less or greater inequality. Thus, in the last example, as it is plain that the answer must be less than 4 yards (that is, that 4* yards must be to the quantity sought in a ratio of greater inequality) the two given suins of money must be stated in a ratio of greater inequality ; or the greater must be made the antecedent. Sly. Having thus stated your terms, if the two first be mixed, or fractional, numbers, bring them to the same denomination ; and then, altogether disregarding their denomination, proceed to find a fourth proportional by the rtile given in § 74-. availing yourself of any such abbreviations as the numbers admit. See § 86. Thus, in the last example, we brought both the sums of money to the denomination shillings, and then disregarded their denomination, as it is only the ratio of the numbers that is concerned,
126. We may now form another question to prove the correctness of our work in the last : viz. If I pay s82 : 1 2.^,^ for 3rf yards of cloth, what must I pay, at the same rate, for 4 yards ? Here the thing sought being a sum of money, the given sum. of money £2 : i^s. must be the third term of the analogy. And as the answer must be a greater sum of money, the two given quantities of cloth must be stated in a ratio of less inequality, that is, as 3rf to 4. These terms, brought to the same denomination 55ths. become VV and Vt whose ratio rejecting the common denominator is that of the numbers 208 and 220. So that, as 208 : 220 :: £2 : 125. to the sum sought. The two first terms being both divisible by 4, we may substitute for them tiie ratio of the quotients 52 : 55 ; and we may then sec by inspection that the fourth proportional sought is 55S' or £2: 15, since there are 525. in £2 : 12.
127. Let us apply our rule to another example, such as is commonly proposed as a question in the ! ule ofThreci?iverse. If a mason can build a wall in 6 days, working 7 hours a day, how many hours a day must he work in order to build it in 5 days ? It is plain that he must work a greater jmmber of hours each day ; and tlvcrefore the fourth term
cf
( 38 )
of the analogy must be greater than the third term, 7 hours : and bence the two first terms must be stated in a ratio of less inequality, thus — as 5:6:: 7 hours to the number of hours sought. The answer therefore is y-^ or Sf hours ; that is 8 hours and 24 minutes. The truth of this may be proved by forming another question in which this answer shall be one of the given t^rms, and any one of the former given terms shall be the term sought. Thus : if a mason, working 8 hours and 24 minutes a day, build a wall in 5 days, how many hours a day must he work in order to build it in 6 chiys ? or — in how many days shall he build it working 7 hours a day ? or lastly — if he build it in 6 days working 7 hours a day, in how many days shall he build it working each day 8 hours and 24 minutes ? And thus whenever a question has been solved by the rule of proportion, the student may be profitably exercised in forming three other questions adapted to prove the truth of his answer : since we can find any one of the four terms of an analogy from having given the three others.
128. Those, who have learned Arithmetic according to tlie common systems, will perceive that I wholly dis- regard the distinction introduced in them between the Rule of Three direct and inv^^se. It is perfectly useless : and like all useless distinctions it is calculated only to perplex the learner and to render a simple subject complicated. They will also perceive that I place that as the third term of the analogy, which is commonly stated as the second. The common order never could havp obtained such a cur- rency, as to have been admitted even into some treatises written by men of science, unless Arithmetic had been degraded from the rank of science. Unimportant as the difference may appear to some in practice, the vulgar ar- rangement is mischievously calculated to conceal from view the princi})les of ratio, on which the solution proceeds : and is intrinsically absurd ; as absurd, as if we spoke of tlie ratio between such heterogeneous quantities as 5lbs. of beef and 3 bars of music.
129. Hitherto we have supposed cases, in which the question is affected only by one given ratio : but there may be two ratios, or ever so many, concerned in the question. For instance: if 3 masons w^orking 7 hours a day build a wall in 6 days, how many hours a day must 4 masons work in order to build it in 5 days i Here, if we consider only
the
{ 59 )
the decreased number of days, 7 hours would be less than the answer in the ratio of 5 ; 6, And if we consider only the increased number of masons, 7 hours would be greater than the answer in the ratio of 4 ; 3. Therefore 7 hours is to the answer in a ratio compounded of 5 : 6 and 4 ^ 3, that is, in the ratio of 20 : 18 {§ 83. 87.) or of 10: 9. But 10 : 9 : : 7 ; 4^. Therefore the answer is GyV hour5, or 6 hours and 18 minutes. The generp.1 rule therefore for sol vino- all such questions is this : — 1st. deter- mine the third term of the analogy as before. 2ndly. Con^ sider how the answer would be affected by each of the ratios separately, and arrange the terms of each ratio ac- cordingly, by the rule before given. 3rdly, Multiply the third term by the product of all the consequents and di- vide by the product of the antecedents. But here much trouble may frequently be saved by observing whether the terms of the given ratios may be reduced to lower, according to the rule given § 87.
130. That the student may be the more thoroughly convinced of the justice of the principles, on which we have proceeded in the solution of this question, let it be observed that the question might be resolved into two ; first to find how many hours a day the same number of masons should work in order to build the wall in 5 days ; and secondly, after having found this, to find how many hours a day 4 masons should work in order to build it in the same number of days. The first question would be solved by the analogy — as 5 : 6 : : 7 : "^^ 5 and the second question by the analogy — as 4 : 3 : :' V ^^ the answer. And thus we see that the answer would be obtained by multiplying 7 hours by the consequent of each of the given ratios, and dividing by the antecedent of each.
131. Let us now apply our rule to a question involving three distinct ratios. If a family of 13 persons spend ^64 on butcher's meat, in 8 months when the meat is Gd- per lb. how much (at the same rate) should a family of 12 persons spend in 9 months, when the meat is Giper lb ? Here .^64 is to the sum sought in a ratio compounded of the direct ratios of the number of consumers, the times of consumption, and the prices of the meat per lb. that is, in a ratio compounded of the ratios of 13 to 12, 8 to 9, and 6 to 6i. But the last ratio being the same with that of 12 to 13, the terms of the first and last ratios may be
^ erased
{ 60 )
erased (§ 87.) and therefore as 8 : 9 : : j^64 to the answer, which is known by inspection to be £1^,
1 32. Considering the different questions, to which have liitlierto apphed the rule of proportion, any person of common sense must see the absurdity of conceiving them solved by different rules ; — must see that it woukl be ab- surd to talk of the question § 1 24. as solved by the ride of clcdhy the question J 126. by the rule of masovry, &c. Yet this absurdity would not be a whit greater than that, which perva(5es all the common systems of Arithmetic, in presenting to the student as distinct rules the Rule of Interest^ of Exchange, of Felloiscshij) &c. &c. All these are but different applications of the one Rule of Pi'opoi'tion : and any student, acquainted scientifically with the prin? ciples oi^ proportion^ needs only to have the meaning of the terms employed in these different subjects distinctly explained to him, in order to be able to solve every ques- tion that can occur in them. Wc shall proceed to exem- plify this in a few instances.
183. After explaining the meaning of the terms Interest
and per cent, pe?^ annum — if it be asked, At 5 per cent.
per 4inn. what is the interest of .^275 : 10 for '6\ years?
it is plain that we are given the interest of 100.^' for 1
year, in order to find the interest of ^^75 : 10 at the
same rate for 3^ years. The third term pf the analogy
therefore must be the given interest £^ ; and this must
be to the interest sought in a ratio compoundi3d of the
ratios of the principals and times, that is in a ratio com-r
poumled of 100: 2754 and of 1 : 3|, or of 200 : 551 and
of 2 : 7, that is in the ratio of 400 : 385?. The answer
, - ^ . 3857X5, 3857 ^, '. ^.^ . _ ^
therefore is -' or , that is ,€48 : 4 : 3. in
400 80
this manner, though often not the most expeditious, the learner, ought for some time to calculate all questions in interest ; and to prove his answer by such questions as the following : At what rate per cent, per annum, will the in- terest of ^275 : 10 for 34 years be ^48 : 4 : 3 ? or. At 5 per cent, per annum, what principal will gain at 48 : 4 : 3 interest in 34 years? or — in what time will ^275: 10 gain ^48 : 4 : 3 interest ? And in some of those forms I liave known persons, who have been for years calculating interest by the common technical rules, quite at a loss how to set about the solution j while children rationally
taught
( 61 )
laught for a very few months have found no difficulty in the question.
1S4, But wherever tho rate of interest is 5 per cent, per annum, the calculation is greatly facilitated by observing that S£ being 100.<f. this is at the rate of a shilling for every pound : so that we at once know that at this rate the interest of ^^275 : 10 for 1 year is 275,v. and 6r7. or ^'13 : 15:6: which sum therefore multiplied by 3 4 gives the interest required. And when the interest is 6 or 4 per cent, per annum, it is often convenient to calculate it as at 5 per cent, tlien adding or subtracting a 5th. part. Various other advantages may be taken in particular ca- ses, which are better left to the ingenuity of the student to discover.
135. As to discount It is but a species of mterest; in the calculation of which however mercantile practice is at va- riance with scientific theory. If I hold a bill for ^100 which will not be due for 31 days to come, and want ready money for it, it is plain that the person who should give me £100 in cash for the bill would be a loser of the in- terest for 31 days : and that he is therefore entitled to de- duct part of the amount in cashing the bill for me. But it is as plain that if he retain the full interest upon €100 ibr 31 days, which is the mercantile practice, he retains too much and gives me too little : for he charges me with interest not only upon the principal which he advances, but also upon the interest which he keeps in his own hands. He ought equitably to give me the principal, which put to interest for 31 days would amount to Ji^lOO.
136. The calculation ot Exchange maybe sufficiently illustrated, by considering the exchange between Great Britain and Ireland. A British shilling, or \2d, is equi- valent to 1 3^. Irish currency ; therefore ^Os. British to ^1:1:8 Irish; and €100 British to ^108 : 6 : 8 Irish. Now Exchange is said to be at paVy or at 8y, whenever, I can get ^100 British for ^108 : 6 : 8, or ^108^, Irish. It is said to be above or below /xyr, when ^\e promiuni to be paid is more or less than at this rate. For instance. Exchange is said to be 9i, when for ;€100 British I must pay ^'109 : 5 Irish. The meaning of the terms being thus explained, all calculations are easy by the rule of propor- tion. For example : — At par what is the value in Irish currency of ^275 : 10 British ? The amount in Irisli cur-
( 62 )
rency must be greater, and in the ratio of 13 : 12. There- fore as 12 ; 13 :: £275: 10 to the answer. And this answer may be fomid most expeditiously by adding to ^275 : 10 its 12th part. On the contrary Irish money may be changed into British at par by subtracting its 13th part. To calculate the amount in British currency of £215 : 10 Irish, exchange being 9\, it is plain that the analogy must be — as 109^ : 100 : : £215 : 10 to the answer,
137. In calculations of Felloisoship we are called to di- vide the profits of trade among several partners equitably, according to the time each has been in the trade and the capital he has invested in it. If they have had equal ca- pitals in the trade and for the same time, it is plain that the profits must be divided equally between them. And universally each partner's share of the profits must be in a ratio compounded of his capital stock, and of the time it has been employed in the trade : for supposing either of these circumstances to be the same with all the partners, their shares will be directly as the other. The problem therefore resolves itself into this — ^To divide a given num- ber into parts that shall be in any given ratios, or propor- tional to any given numbers: for instance, to divide 100 into 3 parts that shall be as 10, 8, and 7. Now 10 + 84-7 = 25 j and it is plain that the proportional parts of 100 must be greater than 10, 8, and 7, (the parts of 25) in the same ratio in which 100 is greater than 25, that is in the ratio of 4 : 1. Therefore the parts required are 40, 32, and 28» And universally the sum of the given num- bers which assign the ratios of the parts is to the number to be divided, as the several given numbers to the propor- tional parts required. Now let us suppose that three part- ners, A, B, and C have had capitals of ^^2000, ^3000, and <3€4000 in trade for 12, 9, and 7 months; and that at die end of the year they have to divide between them a profit of .^2133. Their capitals are as 2, 3, and 4 ; their times as 12, 9, 7: and ^€2133 is to be divided between them into parts in the compound ratio of those numbers, that is, as 24, 27, and 28, the sum of which numbers is 79. Therefore as 79 : 2133 (or as 1 : 27) : : ^24 to A's share, : : sB21 to B's share, and : : ^-28 to C's share.
138. Although it be not the design of this treatise to enter into the minutiae of practical Arithmetic, as applied
to
( 63 )
to mercantile transactions, yet I must not dismiss the sub- ject without pointing out the application of the rule of pro- portion to another matter of frequent occurrence — t/ie equa- tion of payments. If A owe B £15 payable in 5 months, and <fel2.5 payable in 7 months, it is inqiiired at what time he should pay both sums together, without loss to either debtor or creditor. Now if the sums were equal, it aj)- pears obvious that the time sought must be exactly the middle period between the two times of payment, or 6 months: for thus each would lose the interest of one pay- ment for a month, and gain the interest of an equal pay- ment for the same time. But the sums due at the different times being unequal, it appears as obvious that A must withhold the payment of the ^75 for a longer time than he anticipates the payment of the ^125, in order to make^ the interest gained and lost equal ; and that, in the ratio of 125: 75, or of 5:3. We have only then to divide in that ratio the interval of 2 months (the distance between the two given dates of payment) and add the greater part, \~ months, to 5 months, in order to find the equated time of paying both sums : for thus the interest of £15 with- held for one month and a quarter is equal to the interest of ^125 anticipated in the payment by J;ths. of a month. Now if A should owe B a third sum, suppose £Sl payable in 9 months, having combined the two former into one sum of ^200 payable in 6-5: months, it is plain that by a similar process we may find the equated time of payment of the three, dividing 2 1 months (9 — 6^) into two parts in the ratio of 87 : 200, and adding the lesser part to 6\ months: which gives the equated time for the payment of the three sums together 7tVt months, or what m^ay be considered in practice 7 months and 3 days. Now let the student calcu- late the interest of ^75 for 2^4t months, and of .s^l25 for ^Vt of a month, and £S1 for I^t months : he will find the third, lost by A and gained by B, exactly equal to the »um of the two first gained by A and lost by B.
139. This operation however, which we have hitherto described at large in order to shew the scientific principles, would be altogether too tedious for mercantile practice: and it fortunately happens that it admits a most convenient abbreviation. Let us now return to the operation, by which we found 6j months as the equated time for payment of the two first sums. We first proceeded to divide 2
months
{ 64 )
months {'7'— 5) in the ratio of 125 : 75, Now this is done by the following analogy: (§ 137.) — as 1 25 -f 75:2 :: 125
to the greater part, which is therefore — ^ , But
^ ^ 125 + 75
125 X 2 is equal (§ 25.) to 125 X 7 minus\2oX5, So that the
fourth proportional maybe thus expressed, — ! ZL_2i_ ,
We then added this fourth proportional to 5 months. In
order to perform that addition let us bring 5 to the same
denominator with the fourth proportional, and it becomes
125X5 + 75X5 ^.^ i r .i . r .• i
1 .. JN'ow addniff those two fractional ex-
125 + 75 ^
pressions, the sum of their numerators is plainly 75 X 5 plus 125 X 7 ; (for on account of the subtraction of 125 X 5 in the one numerator and its addition in the other, that part must disappear) tliat is, the sum of the products of each pajmient multiplied by the time when it is payable : and the denominator, 125 + 75 is the sum of the payments. And thus we arrive at the following practical rule : — mul- tiply each payment by the time when it is due, and divide the sum of those products by the sum of the i^ayments ; the quotient is the equated time of payment sought. Ac- cordingly proceeding by this rule to find the equated time of the three payments proposed in the last section, the
answer is '[^±^l^<^^2^, or £i±?^J:^ 75+125 + 87 75 + 125 + 87 »
or = 7 oVt > ^s before.
287
1 10. I am aware that some have questioned the mathe- matical accuracy of this calculation, on the principle that a person paying money before it is due can justly be con- sidered as losing only the discount^ which is less than the interest. According to this idea the calculation is somewhat unfavourable to the creditor. But I confess that the principle upon which it is controverted appears to me palpably erroneous. If I owe .^'100 payable in three months, and have the money to pay it immediately, must it not as reasonably be supposed that I can gain the interest of £\00 by delaying the payment till it become due, as it is supposed that my creditor will gain the interest by my paying him immediately ? And if I have not the money, XmX wish to raise it for immediate payment, suppose by
issuingr
( 65 )
issuing my note for 3 months, is it not equally plain that I must lose more than the discount of j^lOO for 3 months ? For even according to the theojy of discount (reckoning- interest at 5 per cent, per annum) I must issue my note foK-o-^lOl : 55. in order to receive immediately ^^100. And is not this just the same thing as if I borrowed ^100 for 3 months at 5 per cent, interest, in order to make imme- diate payment to my creditor ? But according to the mer- cantile practice of discount I must issue my note for a still larger sum. It is not however worth while to pursue the discussion of this subject further. Those who know how much it has been contested will not wonder at my having said so much ; and will be most ready to pardon me, if my ideas should be found incorrect.
Examples for practice.
Ex. 1. If I of a yard of cloth cost Ss. Sd, what will 9 yards cost at the same rate ^
Ex. 2. At the same rate, how many yards should I get for .£4^ : 19?
Ex. 3. If 7 horses eat a certain quantity of corn in 9 days, how many at the same rate will eat it in 7 days ?
Ex. 4. If 75 workmen finish a piece of work in 12 days, in what time will 1 5 workmen finish it ?
Ex. 5. A mason having built 4 of a wall in 6 days, at the wages of 3^. 6d, per day, his employer agrees to pay him for the remainder at an increased rate of wages, in proportion as he shall increase his dispatch : and he finishes the wall in 2 days more. How much per day is he to re- ceive ? — Observe here that, if he had continued to work at the same rate, he would have taken 3 days to finish the wall, as 4 of it remained to be built.
Ex. 6. If a man walk 7^ miles in 2 hours and 10 minutes, how many miles will he walk at the same rate in 3 hours ?
Ex. 7. At 5 per cent, per annum, what is the yearly in- terest of ^725 : 15 : 6 ?
Ex. 8. — at 4, 6, and 4i per cent, per annum ?
Ex. 9. Of what principal is ^27 : 10 the yearly interest, at 54. per cent, per annum ?
Ex. 10. What is the commission on goods bought by a factor to the amount of ^576 : 15 : 8, at 24- per cent. ? — Commission is an allowance of so much per cent, made to a factor for buying or selling for his employer. Brokerage
F is
( 66 }
is a similar allowance made to a broker, for assisting a merchant or factor in buying or selling goods.
Ex. 11. On what amount of goods is the brokerage *83 : 5 : 11|, at -|- per cent. ?
Ex. 12. At what rate per cent, per annum will the Interest of .^100 for 5 years and 2 months amount to ^.^24 : 10: 10?
Ex. 13. At 41 per cent, per annum, in what time will the interest of ^100 amount to <s€34 : 16 : 8 ?
Ex. 14i Divide 79 into 5 parts that shall be in the ratio of 2, 34^ 5, 64, and 8 ?
Ex. 15. Five partners A, B, C, Dj and E joined in trade at the beginning of the year, putting in the respec- tive capitals of ^200, ^350, ^500, ^650, and £800. Their joint profit at the end of the year was ^790. What are their respective shares of it ?
Ex. 16. A. went into trade at the beginning of the year with a capital of ^2576 : 10. On the 1st. of March he took B. into partnership with an equal capital : and on the 1st. of June they took C. into partnership with an equal capital. The joint profit at the end of the year is £1725, How is it to be divided between them ?
Ex. 17. Exchange being iat par what is the amount in British currency of ^€217 : 15 : 6 Irish ? and in Irish cur- rency of ^217 : 15 : 6 British .?
Ex. 18. Ditto, Exchange being 9|j and Exchange being 104^?
Ex. 19. If A. can mow a field in 5 hours, and B* can mow it in 7 hours, in what time can A. and B. together mow it ? — On this and similar questions let it be considered that, if A. and B. worked with equal dispatch, they would together do the work in half the time that one of them would require to perform it alone : and if B.'s dispatch were twice as great as A.'s, they would together perform it in the third part of the time, which A. would require to perform it alone ; for A. and B. together would then be equivalent to three A.'s* Now according to the terms of the question B* working slower than A. in the ratio of 7 : 5, A. and B. are not equal to two A.'s, but only to A-f T of A. So that V (or 14) is to 1, or 12 is to 7, as 5 hours to tlie time sought.
Ex. 20. If A. can mow a field in 5 hours; and A. andB. to- gether can mow it in three hours, in what time can B. mow
it
( 67 )
it alone ? Here it is plain, from the observations on tlie last question, that 5 : 3 : : A-f B : B. Therefore (§ 72.) 5 — 3 i. e. 2 : 3 : : A : B. But A/s dispatch is as 5. There- fore 2 : 3 : : 5 to B.'s dispatch.
Ex. 21. If 9 bushels of corn serve 7 horses 10 day^, how many bushels at the same rate will serve 20 horses 21 days?
Ex. 22. At the same rate, how many horses will eat 27 bushels in 3 days? And in what time will 21 horses eat 18 bushels?
Ex. 23. If a family of 19 persons expend i^235 in S months, how much at the same rate will a family of 12 persons expend in 5 months ?
Ex. 24. If 96 men working 9 hours a day for 10 days can dig a trench 400 yards long, 3 wide, and 2 deep, in how mahy days at the same rate can 108 men working 7 hours a day dig a trench of 175 yards long, 4 wide, and 3 deep ?
Ex. 25. At 41^ per cent, per annum, what is the interest of £61 5' 15 for 7 years and 11 months ?
Ex. 26. At what rate per cent, per annum will the in- terest of ^^1025 for 3 years and 5 months amount to :C175 : 2 : 1 ?
Ex. 27. At 5 per cent, per annum, what principal will gain ^^350 : 4 : 2 interest in 10 years and 3 months ?
Ex. 28. At 4|: per cent, per annum, in what time will the interest of 5^375 : 10 amount to £^ : 15 ?
Ex. 29. A. began trade on ths 1st. of January with a capital of ^'1000 ; and on the 1st. of March took in B. as a partner with a capital of ^1 500 ; and on the 1st. of Ma^ they admit C. as a partner with a capital of ^^2725. The joint profit at the end of the year is ^1896. What are their respective shares ?
Ex. 30 Three graziers. A, B, C, hold a piece of ground in common, for which they are to pay £75 a year. A. on the 1st. of January puts in 12 sheep, on the 1st. of March 8 sheep more, and on the 1st of June draws 10 sheep. B. on the 1st. of January puts in 15 sheep, on the 1st. of February draws 6 sheep, and en the 1st of July puts in 1 2 sheep more. C. does not put in any sheep till the end of one month, and on the 1st. of February puts in 14^; on the 1st. of April 4 sheep more; and on the 1st. of August draws 9 sheep. How much ought each to pay of
F 2 the
( 68 )
the rent at the end of the year ? — On this and similar questions in fellowship, where the capital of any partner varies during the partnership, let the student observe that the sum of all the products obtained by multiplying each cajpital by the time it has been employed must be propor- tional to his share in the partition of the profit, loss, &c. Just as we have seen that if A. had grazed 10 sheep for 12 months his share would be jtistly represented by 10 X 12^ or 120 ; so when he grazes 12 sheep for 2 months, 20 sheep for 3 months^ and 10 sheep for 7 months, his share riiust be represented by 12x2 + 20X3 -f- 10X7, or 154.
Ex. 81. A. owes B. £25 to be paid in 1 month j ^30 to be paid in 2 months ; j£45 to be paid in 3 months ; and J815 to be paid in 4 months. What is the eqtiated time for paying the whole ? i. e, when should he pay him £l 15^ so that it should be equivalent with the several distinct pay- ments at the time specified ?
Ex. 32. A. purchases goods froni B. on the 15th. of January to the amount of ^^275 : on the 1st. of February to the amoimt of ^125 : and on the 10th. of March to the amount of ^^312. He is allowed 3 months credit on each purchase : but wishes to give B. a bill for the whole amount at 31 days after date. When should it be dated ?
CHAP. XIV.
Origin atid Advantages of Algebra. Algebraic N'otatiom Dejinitions*
141. ALGEBRA is to be considered as but another method of Arithmetical computation, much more exten- sively applicable than the common, and much more pow- erful : while its fundamental principles are so coincident with those already stated, that no one who has made him- self master of the former part of the subject can find any serious difficulty in the Elements of Algebra, so far as they are pursued in this treatise. The great advantage, which modern Mathematicians possess above the ancient^ consists in their acquaintance with this art ; which came to tis originally from the Arabs, according to the testimony^ of Lucas de Burgo, who first published a treatise on it in
ItaliaiJr
( 69 )
Jtaliaa in the year 1494. That the Greek mathcmaticitin*, bur masters in Geometry, were ignorant of Algebra, is certain ; from their having in vain attempted to solyc a problem, which with the aid of this science would have presented i?o serious difficulty. Yet it is not to be doubted that men so acute, and so conversant about numbers, must often unknowingly have epiployed a kjnd of Algebraic in- vestigation ; as it is common at this day to observe shrewd accountants, whp have never learned Algebra, yet pursu- ing the solution of more complicated questions by a chain of reasoning perfectly Algebraic t while they labour indeed under much incojivenience and disadvantage from their unacquaintance wijli the notation and systematic rules of the art. Diophantus, a most ingenious mathematician of Alexandria, who lived in the fourth century, made won? derful advances in this- method ; insomuch that he is con- sidered by some as the inventor of Algebra : — ^liow justly, j shall not stop to inquire. It was certainly not from him/ |)ut from the Arabs, that we derived tl^^ art.
142. Algebra is also called Universal Arithmetic^ from ijts employing general symbols instead of particular num- bers, and affording us conclusions wMch form universal theorems. Thus putting the letters a and h for any twa numbers whatsoever, a-^-b expresses their sum, or the number produced by adding the number represented by h to the number represented by a .\and fl^-— 5 represents their difference, or the number produced by subtracting b from a. J^ow if we add a — ^ to a-^-b algebraically, we shj^U find' (as will appear in the next chapter) that the aiifiount is twice a : and if we subtract a — b from a-^-b we shall find that the resulting number is twice ^. And thus we are ihr- nished with these general principles — that, if to the sum of any two numbers whatsoever we add their difference, the amount is twice, the greater member ; but if from the sum we subtract l;he difference, the remainder is twice the less. (These principles might be stated still more gene- rally : but to do so at present would involve the student^ preftiaturely in the consideration of positive snd negative quantities.) Let the student try this in any numbers what- soever, and he shall find it true : but he might often per- form the same operations in common Arithmetic, adding for instance the difference between 19 and 5 (14) to their 4um 24, or subtracting tlie former from the latter, without
observing;
( 70 )
observing even in that particular case that the sum was twice 19, and the difference twice 5 Whereas in the j^ame Algebraic operations the results are obtained in a form, which at once presents those principles in their most universal extent to the attention
143. Fvom what has been said § 23. and 24. it appears, that the product of any two numbers represented by a and h may be expressed oy axb or bxa\ but it is more fre- quently and briefly expressed by ab or ha, writing the letters which denote the factors in continuation, without any sign interposed between them. Thus xyz, or zyx, or i^xz expresses the product of the three factors de- moted by the letters x, ?/, and z. In like manner 3a ex- presses three times a, or the product of a multiplied by 3 ; ^nd Ixy expresses seven times xtJu In such forms of ex- pression the numbers prefixed to the letter or letters is called the numeral coefficient ; and when no other numeral coefficient appears, 1 is understood to be prefixed.
144 According to what has been observed § 39. the di- vision of <z by b may be expressed thus a-i-b ; or (as i§
morfe usual) fractionally, thus -^. Therefore — — ex-
b 3.r
presses the quotient arising from dividing ^mn by Sx. And
if we want to express :^ths. ofx algebraically, it is — ; for
' ■ - 4
this expresses the 4th part of three times x, or ||;hs. pf once X. See § 92. And if w^ be called to find a fourth proportional to three numbers represented by a, b, and c,
be the fourth proportional will be justly represented by >
' • ' ■' ■ a '
for this expresses the quotient arising from dividing the product of the given means by the given extreme. See § 74. And if we have this analogy a : b : : c : d, we may infer the equation ad — be, or from the equation ax-=^by we may infer the analogy a : b : : y : x. See § 76. and 77.
145. The square of a may be e^^pressed by oa ; its cube or 3rd. power by aaa ; its 4th. power by aaaa, Sec, ($ 143.) But they are more frequently denoted by indices or expo- ne?ifs of the powers, thus, a'', «% «% &c. (See § 32.) And if I want to multiply any power of a by any other power, suppose the 7th. power by the 5th. power, the pro- duct will be the 12th. power, Or «'*, its index being the
sum
( 71 J
«um of the indices of the factors ; as is evident from § 143. hy performing the operation according to the other notation. And as powers of the same root are multiphed by adding the indices of the factors, it is plain that they may be divided by subtracting the index of the divisor
from the index of the dividend. Thus ^~ = a^, — The
square root of a^ , or that number whose square is «, is de-
noted thus, Va, or by the radical sign alone, Va ; the
cube root, or that, number whose cube is a, thus Va, &c. Quantities with the radical sign V prefixed are called surds. We otherwise write such surds by the aid of frac- tional exponents, of which the denominator indicates the
root intended ; thus, a^, a^, aT^ &c. And according to this
notation «t expresses the cube root of the square of a, or that number of which a^ is the cube. See Chap. 22.
146. We may here notice the facility with which many fractional expressions in Algebra may be reduced to lower
terms. For instance, — ::i-may be at once reduced to
: for ^ and j/ being factors of both numerator and de-
minator, I may divide them both by ^3/ ; but this is done at once by erasing a^ from both. For as the mere annex- ation of any letters expresses Algebraically the multipli- cation of the numbers which they represent, so the mere withdrawing of any letter must be equivalent to division by tliat letter. Thus if I want to divide abc by Z>, the quotient must be ac 5 since acxb — abc. (See § 40 )
Thus again J the fractional expressioii ■^—~- = ^— > as will
appear by writing the given fraction in the longer notation
xxyz^ ^^^ dividing both nunxerator and denominator by 2>xyyz the common factor xyz.
147. A vmculum^ or line drawn over several terms of a compound quantity, i^ designed to give precision to the Algebraic expression. Thus a ■\- 6 X c denotes the mul- tiplication of the sum. of a and 6 by c; whereas a-f 6xc (without the vinculum uniting the terms a^b) might be understood as denoting the sum of a and the product of
h and
( 72 )
b and r, or a-^hc. In like manner cy^a^h expresses the multiplication of c by the difference between a and b ; whereas without the vinculum it mighi express the same thing as ca — &. And a — b\^ expresses the square of the difference between a and b ,• whereas a — b^ would express the difference between a and the square of b. In place of the vinculum we often employ the mark of a parenthesis. Thus [a^b) -r-x expresses the division of a-\-b by .r.
148. Propositions concerning the relative magnitude of quantities we commonly express in Algebra by equations. (See § 19.) Thus to express algebraically that a exceeds ^ by 7, we employ the equation a-=^h-\-l^ or the equation a — 7 = 6, or the equation a — 6 = 7; any of which, ac- cording to the import of the notation as already explained, will be found to express the given relation between a and b. To express that half of a is less than two thirds of b by 4, we
may employ this equation, --.-|-4 = -^, But more of this
hereafter.
1 49. The observation* in this chapter may be considered ns the grammar of Algebra ; and it is very desirable for the student to make himself expert in such exercises as the following. — Putting the letters x and ij for any two numbers, express algebraically 1. the addition of twice y to three fifths of ^,- 2. the subtraction of half .r from twice 1/ ; S. the multiplication of their sum by their difference 5 4. the quotient from dividing 25 by their difference j 5. the quotient from dividing their sum by three times X/ 6. the subtraction of the square root of 3/ from the cube of K ; 7. that the product of their sum aiid difference is equal to the difference of their squares 5 8: that the square of their sum exceeds the square of their difference by four times their product On the other hand let the student exercise himself in interpreting such algebraic expressions as
the followmg 1, X'\-y — x — j/= 2^/ ^ 2. -r- -f 2^ = 4j/ — ^> 3. .r+j/|-Xj:— I =10^^; 4. ^i^^:^II^-f 7 = 20—^; 5- 1/ x
^^yr=,^J^; 6. ^ 4-3/ X x-^y = x^'-^y-' ; 7. ^HKyh— 4arj/ = ^^r5 8. ^^|T = 5.
CHAP.
( V3 )
CHAP. XV.
Positive and Negative Qjiantities. Algebraic Addition and Subtractio7i»
150. Ever}^ quantity in Algebra is said to ha positive or negative according as it is affected with the sign pins or hiinuSf -I- of-—: and whenever a quantity %has not either of these signs prefixed, the sign -f is understood, and the quantity is said to be positive. Thus 5, or -f-^* is positive ; but — 5 is negative Positive quantities are other-* wise called affirmative Some mathematicians, in treat- ing this subject, have involved it in much perplexity, and plunged themselves into extravagant absurdities ; talking of — 5 as a quantity less than ?iotki?ig, &c. to the disgrace of the science. But the student is to observe that- — 5 de- notes just the same number as +5, but with the additional consideration that the former is to be subtracted, wt^ile the latter is to be added.
151. The simplest illustration of positive and negative quantities may be derived from a merchant's credits and debts. Five pounds are the same sum, whether it be due to him, or he owe it to another ; but in the one case it may be considered as positive 36.% for it is an addition to his property ; and in the other as iiegafive £5^ for it is a subtraction from his property. And if the sum of his debts exceed the sum of his credits by .£1000, the state of his affairs may be represented by — lOOO^a^, and undoubtedly is worse than if he had nothing and owed nothing. In such a case indeed, the man is often said even in mer- cantile lano^uao^e to be inimis one thousand. Whereas if tlic sum of his credits exceed the sum of his debts by 'j^ 1000, the state of bis affairs may justly be represented by -f- fOOO^. These opposite signs then, without at all affecting the absolute magnitude of the quantities to which they are prefixed, intimate {he additional consideration that those quantities are in contrary circumstances. Many other illustration? might be employed. Thus, if jt, or -fir, denote the force ^dth which a body is moving in a certain direction, — x vv'ill denote an equal force in the contrary direction. ' But for younger students, I think it more expedient to confine their attention to the familiar illustration first adduced. Wlieu we talk of quantities of contrary affections^ we mean quantities of which one is positive and the other negatiyc. Aud by the signs we mean
the signs -{" ^^^ — •
152. Let
{ v* )
152. Let us now consider the addition and subtraction of positive and negative quantities. And is it not plain from what we have said, that to add or subtract either kind of quantity must give the same result, as to subtract or add the same quantity with the contrary sign, or of the contrary aifectiqn ? Thus, to add — 5 is the same thing as to subtract + 5 : for is it not the same thing to add a debt of ^5, as to subtract a credit of ^5, or to take away £5 of po^itivQ property ? On the other hand to subtract — 5 must be the same thing as to add + 5 : just as it is the same thing to take away a debt and to add a credit of the same amount, or to give the person so much positive property. If a merchant's credits exceed his debts by 3g'5000, and the state of his affairs be therefore + 5000, it M'ill just produce the same change in thcmj whether I cancel a debt of .2^1000 which he owes me, or another give him ^1000, In either case alike the state of his affairs must become -f- 6000. Hence if we have to add 4*3 tq -f-^? the sum must be +8 ; but if — 3 to — 5, the sum must be — -S : just as the sum of two credits of £'i and £5 is a credit of ^'8 ; but the sum of two debts of £^ and £5 must be a debt of £S, Again the su^n of + 3 added to — 5 must be — 2 ; and the sum of — 3 added to -[- 5 must be 4-2 : just as if a merchant be minus £5Q00 (that is, if }ie owe c^'5000 more than he is worth) and I give him i^3000, the state of his affairs becomes— ^2000 j but if the state of his affairs have been plus .^5000, (that is, if }ie be worth .^5000 more than he owes) and there be then added to him a debt of .^3000, the state of his affairs becomes +2000. And thus we see that in the addition of numbers of the same affection, (both positive, or both negative) the suin of the numbers with the common sign is the sum sought : but that in the addition of numbers of con- trary affections, (one positive and the other negative) the difference of the numbers with the sign of the greater is the sum sought.
153. Algebraic quantities are said to be like, when they consist of the same literal part, that is, are written with the same letters find having the same exponent. Thus, 2x and — Sx are like quantities ; also — Satj/ and 4.v?/ ; also xy and —-2x^1/ ; also V:cy and 3 Vxy'^ or x^\^ and S^\^' But 2x and 3j/ are 7inlike quantities , as aiso xt/ and x""^. From what has been said and from the import of the signs
4- and
( 75 )
-|-and — , it is plain that unlike quantities can be adde4 only by annexing them together with their proper signs. Thus the sum of x and y is x -j-j/ ; but the sum of x and — -2/ is X — ;y, or — ^-{-x ; the addition of — 7/ being the same thing as the subtraction of -f 3/.
154. But like quantities may be further added by an in-^ corporation of them into one sum j and the rule for their addition is now most simple. Add their numeral co- efficients according to the rule given at the end of J 152, and annex the common letter or letters. Thus the sum of Sx and ox is 8x ; the sum of — 3x and — 5x is — 8^ 5 the sum of Sx and — 5x is — 2x ; the sum of — Sx and 5x is 2x ; the sum of x^^ and — 3x^i/ is — 2x^i/, For in the last example, since there is no numeral coefficient ex- pressed to the former quantity, w^e must understand the coefficient 1 ; and since there is no sign prefixed, w^e must understand the sign -{-. Then whatever quantity x^i/ re- present, since the sum of -j- 1 and — 3 is -^2, it is plain that the sum of + l;c^j/ and — 3^*3/ must be — -2x^7/, And thus the rule for adding like algebraic quantities, or in- corporating them into one sum, is — take the sum of the numeral coefficients if they be of the same affection, pre- fixing the common sign ; or the difference of the co- efficients if they be of contrary affections, prefixing the sign of the greater j and in both cases annex the common literal part.
155. We have seen how to add simple Algebraic quan- tities, or those which consist of but one term. Compound quantities are those which consist of several terms, and called binomml if consisting of two terms (as the expression a;^ — y^) ; trinomial if consisting of three terms, as the ex- pression ^*- — 2xy-\-y^., Compound quantities are added, by adding separately the parts that are like and the parts that are unlike, according to the rules given in tlie two last sections. Thus the sum of the last binomial and trinomial exhibited i$ 2x^ — 2xy, If we have many quantities to ad4* let them be arranged as in the following example, placing like quantities under each other ; and added according to the rule.
5 Vrt6 — ale — 125c 4. h^ 3 Va6 + 2>ahc — ^hc — 2h^ 7 V"^ -h Sahc 4- nic — 3Z>* — ^"06 — ^iobc — he — W' -f air' — a^h
Total . U ilb * —I \bc — 86^ + ah'' — a^-h
Observe
( 76 )
Observe that in algebraic operations we commonly pro- cecd from left to riglit : and that when the leading term of any quantity is positive, the sign -f- is seldom prefixed. Now to incorporte any of tbe like terms in the preceding example into one smii, suppose the several sets of abc^ we^ take the sum of all the positive terms and the sum pf all the negative terms distinctly 5 an^ then incorporate these two sums. But the sura of -|- 5 <^hQ and -|- 3 abc is -f 8 ahc ; qiid the sum of — 7 abc and — ahc is — 8 ahc ; so that we have to add -^Sahc to — 8 abc j and their sum is 0, since the difference of their coe^cients is nothing. Though such an example as the preceding is often proposed to the (Student, for the purpose of exercising him in the rules of addition, yet it is very rarely indeed that any such occurs in actual practice. The student who is expert in stating the sum of any two numbers, whether of the same or con- irary affections, cgn find no difiiculty in algebraic addition.
156. The rule of Subtraction is simple, and obvious from the, principle mentioned in the beginning of § 152, Ciuppose the sign of the subtrahend changed to its contrary ; (that is, if it be positive, suppose it negative, and if ne^* gative, suppose it positive :) then, instead of subtracting^ add it to the minuend. Thus, if from -f 2a 1 want to subtract +2h) tjic remainder is 2a — 2b, ^The terms here being unlike cannot be further incorporated.) But if I subtract — 2^, the remainder, or result, miist'be 2a-f-26; it being the same thing ('§ 152.) to subtract— 2/; and to add v-j- 2h, Thus again, it is evident that subtracting So: fron> 10<r the remainder is Ix'i but this is also the sum of — 3^ added to IOj:. But 3«r subtracted from — IOj-, gives for the remainder — 13^, the sum of — Sx and — 10^. Any longer example can now present no difficulty, for instance—^
From 5Vab-- ahc -f I2bc + b''.
Take 7 V~ab + Sahc -f Ibc -— 3^^ -f ah'' ■— a^h
Remainder — 2 s^'ab — 6abc -f- 5hc -|- 4Z>* — 06* + a^'b And accordingly if to this remainder the subtrahend be nddcd, the sum' will be the minuend : or if the remainder, be subtracted from the minuend, we shall have the sub- trahend as the result.
157. The student should observe, that in Algebra we commonly talk of subtracting a greater number from a ie»s : as in the leading terms of the preceding example we
subtract
( 77 )
suiatract 7 from 5, and that by adding — 7 to 5. And it appears that in the general expression a: — 2/, if a: denote a quantity less than 3/, the value of the expression a: — 1/ is negative; just as 7 — 5=4-2, but 5 — 7= — 2i
158; But it may be objected, *' is not the subtraction of ** 7 from 5 an unintelligible operation ? and is the art of ^* Algebra only an art of jugglery^ to enable us to do ** strange things, without our understanding what we mean ** by doing them ?" It must be acknowledged that the science has been too often disfigured by writers, who have but it forward in some such form *, and have seemed to forget that to talk an unintelligible language is to talk non^ sense. But there is a sense in which we may easily com^ ptehend the subtraction of 7 from 5 ; namely by consider- ing 5 as equivalent with the compound expression 7 — 2. Now from thi§ binomial 7 — 2 we may subtract 7 ; and the remainder is evidently — -2* Thus again, if the state of a merchant's affairs be + 10,o60c>^, he may lose or have sub- tracted from him 15,000^, and the state of his affairs be- comes — 5000c^ ; so that — 5 justly expresses the remainder on subtracting 15 from 10^ or from the equal binomial 15 — 5. Ill like manner subtracting — 15 from — 10, or from the equal binomial — 15 4-5, the remainder must be 4-5: and since 4-10 = 25 — 15, subtracting — 15 from 4- 10 inust give the remainder 4-25; while subtracting 4-15 from — 10 (or from its equal — 254-15) must give the re- in ainder — 25;
In the following examples let the questions in additioii be proved by subtraction ; and v. v.
Ex. 1 . What is the sum of 5a; and 3rV ?
Ex; 2. Of —5a: and — 3« ?
Ex. 3. Of — ^5;v and Sx ?
Ex. 4. Of 5x and ~3a;J
Ex. 5i Of 3;y 4- 5 and 3a; — 5 ?
Ex. 6. Of 3x* — 2jn/ 4- y and — 5a;* 4- 5xj/ ^f-^B >
Ex. 7. Subtract 2 di/ — b from 2ay -{-b^
Ex. 8. 5a'' — U from — tz* -f- 86 ?
Ex. 9* a^ — 2,0" X 4. 3a;v* — x^ from a^ 4. 3«*a? 4-^i5^i^
* it was with rfegret ahd with surprise that 1 met with some instances of this in a late Edition of Eultr's Algebra, which has come into my hands since these pages were written. It is full time for such rbmrdities to be ex- ploded, as the multiplication of notliing by infinity, &c. &c. Sec Vol. I. p. 34.
CHAP
{ 78 }
CHAP. XVt.
Algebraic Multiplication,
159. WE have seen that the product of any two simpk Quantities, as x and j/, is expressed by xy or yx. But we have now to regulate the sign of the product. The practical rule is simple, viz. if the factors be of the same affection^ the product is positive ,• hut fiegativcy if the factor's be of contrary affections : that is, tlie product either of x Xj/, or of — X X — y is 4- ^j/; but the product of a: X — y or of — x Xy is — xy,
160. The truth of this rule is sufficiently evident from the nature of multiplication, where the multiplier is posi- tive. To multiply any quantity by x is in fact to add the multiplicand as many times as are represented by x, (§ 22.) Suppose X stand for the number 5, and the multiplicand be -f^> rejDresenting a positive quantity, suppose an ar- ticle of credit in mercantile accounts. The sum of that quantity added 5 times must be positive, or a credit of 5 times that amount. But if the multiplicand be — -j/, re- presenting a negative quantity, suppose a debt, then the sum of that quantity added 5 times must be negative, or a debt of 5 times that amount. And thus it is plain that -}- j/ X ;v rr -}- Ay , but — yXxz=i — xy,
161. Let us now consider the case where the multiplier is negative — x^ or — 5. And first, suppose we have to multiply -\-y by — 5. Some might be willing to conclude that the product must be —oyy from the principle that it is indifferent in multiplication which of the factors be made the multiplier ; and we have already seen that the product of — 5 multiplied by -\-y is — 5y, Others have drawn the same inference from the consideration, that the multipliers — 5 and +5, must give products just of contrary affec- tions ; and since the product of -\-y multiplied by -f- 5 is -}-53/, the product of -\-y multiplied by — 5 must be — ^5y, But although such arguments may render the conclusion probable from analogy, they do not amount to a convinc- ing proof satisfactory to the reason. This must be derived from considering what we mean by multiplying any thing by a negative multiplier. Now as multiplj'ing any thing by 4* 5 imports an addition of the multiplicand 5 times, so
multiplying
( 79 )
multiplying it by — 5 must import a subtraction of ,^lie mul-* tiplicand 5 times. But we have seen that the subtraction of +j/ is the same thing as the addition of — y: (§ 1^2.) and therefore to subtract +y five times, or to multiply +3/ by — 5, is the same thing as to add — y five times, or to multiply — y by +5; that is, the product must be — 5y, The same consideration leads us at once to a view of the principle, which has appeared mysterious to many ; namely, that the product of two negative quantities is positive- For instance, the product of — y multiplied by — 5 must be +5y, since the subtraction of — y five times is the Same thing as the addition of -\-y five times. . 162. After the multiplication of simple quantities, there remains no difficulty in the multiplication of compound. The principle on which it is performed is just the same as in common Arithmetic : (See § 25.) — multiply each part of the multiplicand by each part of the multiplier^ and add all the products thus obtained, (Proceed in the operation re- gularly from left to right of each factor, lest you should omit any of the products.) Thus the product of 2x-^$y multiplied by 5 is \Op(-{-lBy ; but multiplied by — 5 is — \Qx — I By. The product of 2a? + Sy multiplied by 5 — y must be the sum of four parts, namely 10a?+15j/ (or 2a:+3j/X5) and -^2a2/ — 3j/* (or 2a; + % X—^.) The product sought is therefore 10a?-|- 1% — 2 at?/ — Sj/^. If any of the products be like quantities, write them one under the other, to prepare for the addition : as in the following example.
Multiply
by
Product
163. If the student multiply x — y by a; — y, he will fijrid tiie product «* — ^^y+y'^i which is therefore the square' of the binomial x — y, of which consequently the product ex- hibited in the preceding example is the cube, or third power. And here we may see another instance of the na- ture and use of Algebra, or Universal Arithmetic. The binomial at— 3/ is a general expression for the difference, be- tween
|
/v3 cj^z^ _j_ ,^^3, |
-y' |
|
x^ — Sx^'y 4- Sxy'' ' |
~y |
( 80 >
tween any two numbers. If we take any two numbers, for instance 7 and 3, we may by common Arithmetic mul- tiply their difference 4 by itself, and the product 16 is the square of that difference. But here the product appears in a form which does not enable us to observe its relation with the factors. But performing the same operation al- gebraically, and comparing the product a;* — ^xy-\-y^ with Uie factors, we at once observe that the square of ^ — y consists of the sum of the squares of >: and y {j<^ -\-y^) minus twice the product of x and y ( — 2Arj/) : whence we are immediately furnished with this universal truth, that the square of the difference between any two numbers is equal to the sum of their squares mimis twice their pro-^ duct ; or is less than the sum of their squares by twice their product. (Thus 7— 3t' = 16 = 4.9 -L- 9— 42 = 58—42. In
like manner x-^y)^ (or the square of the sum of any two numbers) =i>?*4-2A?2/-f^^, or is equal to the sum of their squares plv^ twice their product ; as we have before ob- served. (§ 34.) Again if we multiply ^+3/ by x — y, we shall find the product at* — y^ ; for of the four products which compose it x^-\-xy — xy — 3/^, the second and third when added together disappear. But this presents to us the general principle that the product of the sum and dif- ference of any two numbers is equal to the difference of their squares. Thus, the product of 7-f 3 (or 10) and 7 — 3 (or 4) is 40 J but this is the difference between the square of 7 and the square of 3.
164. Since, according to the rule of the signs in multi- plication, the square of either -f ^ oi' — 3 is -f 9, no num- ber can be assigned for the square root of — 9 : and there- fore the square root of —9 is an impossible quantity. In
like manner V — a* is an expression that indicates an im- possible quantity. But the square root of a'' may be either +« or — a ; since either of these roots multiplied by itself gives -f-«* for the product. And therefore every positive quantity in Algebra is considered as having two square roots, one positive and the other negative.
Let th« student now employ himself on the following questions for exercise in multiplication.
Ex, 1. x-^-y X 2a = ?
Ex. S. x-i^y X -^ 2a =: ?
£x»3.
|
( |
Sl |
||||
|
tex. |
3* |
^—y |
-X. 2a- |
? |
|
|
E^. |
4. |
x-^y |
X —2ft |
= |
? |
|
Ex. |
5. |
\2ax |
+ 23/ X |
;v- |
-3^ |
= ?
Ex. 6. ;c^ — 3.v^j/ + Satz/^— j/^ X *?+^= ? Ex. 7. What is the 6th power of « + J ? Ex. 8. What is the 6th power of a — b ?
CHAP. XVIL
Algebraic Division: Resolution of Fractions into infinite
Series*
165. IF the divisor and dividend be simple quantities, and the divisor be not any factor of the dividend, the quotient is expressed fractionally. Thus, the quotient of
ab divided by .v is — : the quotient of x divided by — ah
is — 1- : and -> — ^ expresses the quotient arising from — ab a
dividing — V2 by a. And any quotient may be thiig
expressed.
166. If the divisor be a factor of the dividend, the quo- tient is obtained, as we have already observed, (§ 146.) by expunging that flictor from the dividend : and the sign of the quotient 7nust be -{-^ if the dividend and divisor be of the same affection ; but — , if they be of co7itrary qffectiotis / as is evident from the consideration that the dividend is the product of the divisor and quotient. Thus 2abc divided by b, or — 2abc divided by — b^ gives the quotient 2ac ; since ^ac X Z>, or — 2ac X — b, gives the product 2ahc. But — 2ahc divided by b, or 2abc divided by — 6, gives the quotient — 2ac ; since — 2ac X ^ = — 2abc, and -~-2ac X — b •=. 2abc, In like manner, if the divisor and dividend have any common factors, but others not common, the divisioa Is performed by expunging the common factors from both, and writing the remaining terms fractionally with their proper signs. Thus, a'^bc divided by — 2dc gives the quo- tient -^ : and — ?>axy^ SbxyzzZZ^^, This is in fact but
Gr reducing
( S2 )
reducing tlie oric^ional fraction ^ ^ to lower terms* ^ "^ 2,bxy ^
by dividing both numerator and denominator by Satj/.
167. If the dividend be compound, but the divisor sim- ple and a factor of each term of the dividend, the division is performed by expunging that factor from each term of the dividend, observing the former rule of the signs : for thus each part of the dividend is divided by the divisor.
For instance x^ — ^xy-^x-=.x — 2 3^: and ax — a~ a =
— x-\-\. And if the simple divisor have other factors not found in each term of the dividend, after expunging the common factors, the quotient is expressed fractionally.
Thus x^ — 2x y-7- Zxa z= ^ '^•j^ ; and ax — a-i ab= ^ "^ ,
or . In the first form of the quotient We have divided
both dividend and divisor hy^a ; in the second form by •^a. And here it may be observed, that in any fractional expression, or in any division, we may change all the signs of the numerator and denominator, or of the divi- dend and divisor, without altering the value of the fractioi> or quotient.
168. If the divisor be compound, the quotient is often most conveniently expressed fractionally. But not unfre- quently also we may obtain the quotient in a simpler form by an operation perfectly analogous to long division in numbers : only it is needful in the first instance to arrange the terms of both dividend and divisor according to th^ powers of some one letter. Thus if we have to divide :ixy^ — 3.v^j/-f a;.? — y^ by x — y, arranging the terms of the dividend according to the povrers of the letter ^, it be-^ comes x^ — 5x^y-\-^xy^ — 1/^» Now divide x^, the first term of the dividend, by x^ the first term of the divisor ; and set down the quotient ^^ as the first term of your quotient. Then multiply the divisor x — 2/ by /v^, the first term found of the quotient : and subtract the product ;v^—;>f*j/ from tlie dividend. The remainder is — 2x^tf-\-Sx7/^ — y^. In like manner divide — 2x^2/, the first term of this remainder, by X ; and set down the quotient — 2xy as the second term of your quotient : by which multiplying the divisor ^—-3/, and subtracting the product — 2a?^j/-|-2a;j/^, the se- cond remainder k a:j/*— ^^^. Finally repeating the opera-
tion=
( 83 )
tloh oil this remainder, the third term of your quotient is +?/^, the product of which and x — tj is xy^ — 3/*, which is equal to the last remainder : and therefore the quotient sought is x^ — 2x^-{-i/^y without any remainder* Let us now exhibit the work at large.
?c- — ?/) x^ — 3^*y-{-3.^j/* — 2/^ {x^ — 2^2/4-2/*
0 + ^J/'— y ^xtj —2/
169. We may prosecute any algebraic divisioti by this method, whatever be the terms of the dividend and di- visor, provided the divisor be compound. But obviously it must often happen, that we shall never arrive at an exact quotient without a remainder : but, as in the case of com- mon division, the exact quotient may be exhibited by an- nexing to the quotient the remainder divided by the di- visor fractionally (§ 43. ) ; and this may be done at any period of the division. For instance, taking the same di- vidend as in the last example, but the divisor ^+j/, we shall find the three first terms of the quotient to be fc^ — 4.^3/ -j- 7?/^, but with the remainder — 8?/^. Therefore
the quotient may be completed by annexing to it -^ : thus—
^'^+j/) ^^ — 3 ;v *?/-}- 3^3/ * — 2/^ («* — ^Jf^z+T?/* — -^ ■
x^ + x^ly * ^+3f
— 4^x^y + Sx2/^ — -j/^ — 4<x^i/ — 4^2/*
-f 7xy'^ — t/^
+ 7;vy + 7y —8^3
170. Accordingly if we multiply the three first terms of the quotient by the divisor /v+3/, and add — Sy^ to the pro- duct, we shall find the dividend. But instead of termi- nating the division at the remainder — Sj/^, we may con-
G 2 tinue
( 8^ )
thiue the same process of divisron as long as we please: only let the stuclent recollect that any fraction is multiplied by an integer either by multiplying the numerator or di- viding the denominator ; and on the other hand is divided by an integer either by multiplying the denominator or dividing the numerator. (Sec § 106. 107. 108.) Let us liow continue to divide the last remainder — 83/^ by ^•-i-i/'
|
+%* |
|
|
^ |
|
|
+8y + 8y |
|
|
X |
X- |
|
x^ |
+ 8^
171. We need not continue the process of division fur^ ther ; for it is now manifest by what law the series pro- ceeds, namely that the signs of the terms are alternately jplus and 7nimis, and that each successive term is produced by multiplying the last term by y and dividing by x, or
multiplying the last term by •^. Such a series is called an
X
htfinite series, because it may be continued without end : and at any period of it, in order to complete the true quo- tient, we must discontinue the series, and annex the last remainder divided by the divisor. And by this method of actual division we may resolve any fraction into an infinite series : for even if the given denominator be simple, we may consider and express it as the sum or diiFerencc of
two numbers^ Thus let ~- express any fraction, we may put
c+l for 5, and performing the actual division of er by
f + 1
I
£4-1 we obtain the series -, + -3 ;> S^cTlvlie^ the
c c^ c^ c'*^
law of continuation is obvious. Or, if we arrange the
terms of our divisor differeritly and divido ahy l+c, the
scries becomes a — ac + ac ^ — ac ^ &c. Or if we put c — 1 for ^,
the terms of the series will be the same, but all positive.
1 72.0 The doctrine of infinite series is very interesting in
its nature, and important in its application : but in this
elementary treatise we caunot enlarge on it furj;her, thapi
^ simple illustration of it by another example. Let it l^e
f equired to resolve ^i^~ into an infinite serie^.
f^-^)l+l (1 + -^ + -^^ ^^^ -f26
-f26^ d
|vrpw suppose a=zlO and Z>=: 1, then -i-.= y = 1,222, &q.
(§ 118.) or l+T^o+T§^ +1^0-5 &c. the terms of which
ilifinite series coincide in value ^yith the terms found by
the Algebraic division. And as we say that ^ is equal to
the circulating decimal .222, &c. so the Algebraic fr^ictioi^
25 . iV 1 • c . • 2Z> 2b'' 2b^ Q j^
J- is equal to the innmte series 1 1 — 5-, occ. It we
a-^b a ^ a^ a\
terminate either the decimal or the algebraic series at any
period, we must annex the last remainder divided by the
given divisor in order to obtain the exact value of the
25 2Z> 26* given fraction. Thus | = .2.22 + g^, and — ^ — — |- — -
^ a — o a a*
173. In resolving the^ fraction ^i- , if we tJisGontinue
a — b
the division at the first term of the quotient, we find that the complete quotient is 1+ 26-7- a — 6; which points out ^his general theorem, that dividing the sum of any two
jiiinab^'S
( 86 )
numbers by their difference gives a quotient greater by 1 than the quotient of twice the less divided by the difference. Thus the quotient of 7 + 3 -v- 7—3 = 2i ; and 6 -^ 4 = U.
Each of the examples in multiplication at the end of the last chapter will afford the student an exercise in division :
and let him resolve into infinite series the fractions ;
1 — a
1 X oc-\-y X
i-{-a' X — y' X — y' 9
CHAP. XVIIL
Algebraic Operations on Fractional Quantities, Method of finding the least common Multiple,
174. EVERY rule here is exactly the same with that for the corresponding operation in common Arithmetic. After referring the student therefore to chap. 8. 9. and 10. it is only needful to illustrate the several operations by ex- amples. Let it then be required to add - to -. Here the
fractions having the same denominator, we add their nu-^ merators and subscribe the common denominator. There-
fore the sum required is -— L-. And in like manner
y y
is the remainder, subtracting - from -. But if the frac-
y y .
tions to be added or subtracted have different denomina- tors, they must (in order to incorporate the sum or dif- ference into one fraction) be brought to equivalent frac- tions of the same denominator : and the product of the several denominators must always afford a common de- nominator, to which they may all be brought. (§ 104.)
m X a xb , ay xb -\- ay rr«, ^ x xh n
1 hus -4--=-— -{ — 'Zzz —L-^, 1 hat - zr -— appears from y ^ b yb yb yb y yb
the consideration that the value of a fraction remains un- altered, if We multiply or divide both numerator and de- nominator by the same quantity. Thus again, - — ^-zz
J 75. Since
( 87 )
175. Since any fraction is multipliei! by an integer, ■either by multiplying the numerator or dividing the de- nominator by that integer, it follows that - Xa-=. — ; and
li±fx^ = ^±^or=:^^l±^. Andthuslx7/ = ^;for^^ ay y ay y
=-. (See § 113.; And since any fraction is divided by
an integer, either by dividing the numerator or multi- plying the denominator by that integer, it follows that
X X ^a-\-ax X-^-x a4-ax
--~a = — ; and —^ — -r-«= — i— , or = — ^ — -,
y ^y y y ^y,
176. in Algebra as in common Arithmetic, to multiply
by a fraction we multiply by the numerator and divide by
the denominator : and to divide by a fraction we divide by
the numerator and multiply by the denominator, or (which
amounts to the same thing) we multiply by the reciprocal
p , . J. . ,T-i X ^a ax ^ X a hx
oi the ffiven divisor, 1 hug _ x - = 7- ; and ----- = — ;
y 0 ^y y ^ ^y
and :^X-^=:1 ; since ^=1. y X xy
177. Let the student recollect that any integral ex- pression may be brought to a fractional of any given de- nominator y multiplying the integer by that denominator,
(§113.) Thus, « = 1« = (^^±^ = '^". Therefore f^±^
a
178. We may now propose and demonstrate the rule for finding the least common multiple of two or more numbers. And first, let any two numbers a and h be given, and let m be their greatest common measure ;
(§ 98. 99.) and let^='S^ and'^^X Then I say that the
product of c, dy and m is the least common multiple of a and Z», And 1st. it is a common multiple of them 5 for ^ since -et^dm^ and ^Muwr, it is plain that both a and h mea- -^-^ ^ sure cdm. But 2dly. it is their least common multiple ; ^^ /^ for let any other common multiple n be assumed, and let ya = w, and xh = w. Then yaz=:xb ^ and therefore x :.y : :a:b (J 77.) But a : b: ,'c:d (J 81.) Therefore^ ; y::c : d.
Now
( 88 )
Now m being the greatest common measure of a and ?, it is plain that c and d arc the lowest numbers in that ratio* Therefore x is greater than c^ and y greater than d. There- fore 7ja or xb (that is 7i) is greater than rf», or e^, that is than m. Thus, if I want to hnd the least common mul- tiple of 15 and 20, I bring those numbers to the lowest terms in the same ratio, 3 and 4, by dividing them both bj their greatest common measure 5 : and the product of 3, 4, and 5 (or 60; is the least common multiple of 15 and 20. (See § 181.)
179. Any other common multiple of a and b must also be a multiple of m. For suppose that w is a common mul- tiple of a and b and not measured by ??z, but that m is con- tained in 71 X times, leaving a remainder t/, less than ;??. Then nzzxm -\-y. Now since both a and b measure m, they must measure xm ; and by hypothesis they measure 7?, or xm + 2/. Since then they measure xm and xm -f ?/, they must both measure y ; and j/, a number less than W2, will be a common multiple of a and b i which is contrary to the hypothesis.
180. Now suppose three numbers given, a, b, mule; to find their tea common multiple. Let m be the least common raultiple of a and b. . Let ?i be the least common multiple of 111 and c. Then I say tha^ n is the least comr mon multiple of a^ b, and c. For since (as we have just shewn) any common multiple of ^ and b, must also be a multiple of w, it is evident that any common multiple of a, b, and c, must be a common multiple of w and c -, and therefore n the least common multiple of the two latter must also be the least conmion multiple of the three for- mer. It is plain that, how many numbers soever be given, we can find their least common multiple by a similar process.
181. What has been demonstrated in § 178. may per- haps appear more clearly, if proposed in the following form. Let a and b represent any two numbers prime to each other, and therefore the lowest in the same ratio. Then ab, their product, must be their least common mul- tiple y for if there were any lower the quotients of it di- vided by a find b. (~ ar^d-j would be numbers less than ci
and b, and in the same ratio : which is absurd. Now let ma and mb represent any two numbers not prime to each
other,
( s<> )
otLer, of which m is the greatest common measure, and therefore a and b the lowest numbers in the same ratio. Then mab must be the least common nmltiple of ma an4 mb ; for if there were any less, the quotients of it divided by 7nb and rua would be less than the quotients of 7nab di- vided by the same, i. e. than a and b j and would be ia the same ratio : which is absurd.
Ex. 1. What is the sum, and what the diiFercnce, of
the two fractions — Jt^ and ^^^ ?
Ex. 2. ... Do. of the two fractions — ^ — ^and ?
Ex.3. J^X-^
Ex.4. _J-_x^/+5= ? y + 5 -^ *
Ex.5.
A?-f-j/ ^ y
A? .V-
~J/
Ex.6. -j — =r
Ex. 7. Find the least common multiple of the number^ }&, 20, 25 and 35?
CHAP. XIX.
Arithmetical Progression.
^82. QUANTITIES are said to be in Arithmetical pro-^ gressioUy when they increase or decrease by a common dif- ference. Thus, the series of natural numbers, 1, 2, 3, ^, 5, &c. increasing by the common difference 1 j the series 7, 10, 13, 16, 8^c. increasing by the common dif- ferences; the series 19, 15, 11, 7, 3, decreasing by the common difference 4. It will be sufficient to consider the constitution and properties of an increasing series ; as every thing said upon that kind will be easily applicable to the other : for by taking the terms of an increasing series in the contrary order we shall have a decreasing series.
183. Now if we put a for the first term of such a series and d for the common difference, the increfising series in
Arithmetical
( ^0 )
Arithmetical progression must be justly represented by a, a-\'d, a -{-2d, tt-f-3</, &c. For as the second term of the series is generated by adding the common difference to the first term, and is therefore a-\-d, so the third term is generated by adding the common difference to the second term, and is therefore a-\-2d: and so on. Hence it is manifest that any term of such a series consists of the first term j)l2is the common difference multiplied by a number one less than the number of that term. For instance, the lOOthc term must be the sum of the first term and 99 time^ the common difference : just as the second term is the sum of the first term and once the common difference. And universally if we put n for the number of the term, the nth, term of such a series must be a^dii — ^^; for dn — d = 71 — IXd, And thus, if we have given the first term and common difference, it is easy to find any proposeci term of"^ the series. For instance, let it be required to find the lOOth. term of an increasing series in Arithmetical pro- gression whose first term is 1 2 and the common difference 3 ; that is, of the series 12, 15, 18, &c. The 100th. term /'^^ must be 12-f 999X3, that is 12 + 2997 = 3009.
184. In any such series the sum of the extremes (that is, of the first and last terms) is equal to the sum of any txm terms equallj/ remote from the extremes ; for instance, of the second term and last but one, or of the third term and last but two, &c. For whatever pair of terms equally remote from the extremes you take, one of them must be just as much greater- than the first term as the other is less than the last ', and therefore their sum must be just equal to the sum of the first and last, Thus, in the series a, a-^-dj a-\-2d, a-^-Sd, a-\-^d, a-\-5d, consisting of 6 terms, the sum of the extremes, a and a-^-Sd, is 2a -^-Sd: but the. same is the sum of the 2d. and 5th. terms, or of the 3d., and 4th. terms. Or in the numerical series 5, 7, 9, 11, 13, 15, the sum of the first and last is 20, which is equal to 7 + 13 or to 9+11. And in like manner it is evident, that if the series consist of an odd number of terms, the sum of the extremes is equal to twice the mean, or middle term. Thus the 7th. term of the last series is 17 j and 5 + 17 = 22 = 11 X2, or twice the fourth term.
185. Hence it is easy to find the sum of all the terms of such a series, by multiplying' the sum of the extremes by
half
( 91 )
half the number of terms in the series. For let the series consist for instance of 6 terms, all the terms may be com- bined into 3 equal pairs of terms, the sum of each pair being equal to the sum of the extremes ; and therefore 3 times any one of these pairs must be equal to the sum of all the terms. Thus the sum of the series 5, 7, 9, 11, 13, 15 is equal to 20X3 = 60; and if continued to ano- ther term, the sum of the series is 22x31 = 77. And thus, if we have given the extremes and the number of terms in the series, we can at once find the sum of the se- ries. For instance the sum of the natural numbers from 3 to 100 inclusive is 103 X 49 = 5047 ; for it is plain that the number of terms in the series is 98,
•186. Hence if we have given the first term, the common difference and the number of terms we can easily find the sum of the series ; since we can find the last term {§ 183.) by adding to the first term the product of the common difference and a number less by one than the number of terms. Thus let the first term of a series in Arithmetical progression be 3, and the common difference 4 ; the 17th. term of that series must be 3-f.4x 16 = 3H-64 = 67 ; and the sum of the series continued to 1 7 terms must there- fore be 3 4-67 X V = 70 X 84 = 595. And universally, let a be the first term, d the common difference, n the num-^ ber of terms. Then the last term must be a-^-dXn — 1 jzza-^-dn — d; and the sum of the series must be ^a-^-dn-^d
X - = — — — i which IS thereiore a general express
sion for the sum of any series in Arithmetical progression. When the common difference is equal to the first term,
this expression becomes And when the number
of terms also is equal to the first term, the expression
becomes — ■ — , 2
187. As we may find the last term of a series from hav- ing given the first term, the common difference, and the number of terms j so we may find the common difference from, having given both extremes and the number of terms : namely, by subtracting the first term [a) from the last, {a-\-dn — d) and dividing the remainder {dn — d) by ?2 — 1. Thus, let it be required to constitute a series of 8 terms in
Arithmetical
( 9'^ )
Arithmetical progression, whose first term shall be 3 and tlie last term 30. The common dillerence must be V =3^. Accordingly the series js 3, 6^, 104, 1^4, 184> 22^, 26y, 30. Or if, insteaci of the last term, we are given the
smn of the series 132, dividing that sum ( — ^^ — j
by 4 (i j ^nd subtracting 6 {2a) from the quotient, th<j
remainder divided by n — 1 affords the same result.
188. If there be a series of 3 numbers in arithmetical pro-» gression, which may therefore be represented by «, «-f-<^, a-l-2c?5 the product of the extremes (a^'\-2ad) evidently is
less ihan the square of the mean, a-\-d\ (or a* + ^ad-t- dl ) by the square of the common difference. But we have seen (§ 76*) that the same product is equal to the square pf a geometrical mean betv/een the same extremes. Thus in the arithmetical series 2, 10, 18, the product of 2 and 18 is less than the square of 10 by G^, the square of the couit mon differ.ence 8. But in the geometrical series 2, 6, 18, the product of 2 and 18 is equal to the square of 6. And an arithmetical mean must always be greater than a geome- trical between the same extremes.
Ex. 1. What is the 17th. term of the Arithmetical series 5, 9, 13, &c. ? And what is the sum of the series ?
Ex. 2. What is the common diiSerence, and what is the sum of the Arithmetical series, whose first term is 5, and the 10th. term 15 ?
Ex. 3. The common difference of a decreasing Arith- i^ietical series is 3| y the first term 12^. What is the lOtlu term ?
Ex, 4. If I spend os. in the first week of the year, and each succeeding week Is. more than in the preceding, how iTiuch shall I spend in the whole year ?
Ex. 5. If 100 eggs be laid in a right line, 1 yard asunder, and a man be placed at a basket 1 yard from the first egg, in what time can he put the eggs one by one into the bas- ket, suppcfjing him to go at the rate of 5 Ei]gHsh miles au hour, including ail delays ?
Ex. 6. What is the sum of the even jiumbers from 2 to iOOO inclusive?
Ex. 7, Do. of the odd numbers from 1 to 999 inclusive ?
CHAR
( 93 )
CHAP. XX.
Geojnetrical Progressmt,
189. TERMS, wlijch are in c5ontinued proportion^ (§ 70.) or each of which bears the same ratio to the next, are said to be in geometrical progression. As in arithmetical progression the terms of a series liave a common difference^ so in geometrical they have a common ratio* Thus 2, 6^ 18, 54, &c. are in geometrical progression, since each term is 3 times the preceding : and 3 is called the de?iomi- nator, or expoiient^ of the common ratio. So again, 3, 6, \2, 24, &c. where the denominator of the common ratio i% 2 ; each pair of adjacent terms being in tlie ratio of 1:2. And it is plain that any siich series may be continaed by multiplying the term last found by the denominator of the common ratio. If therefore the first term of the series be 3, and the denominator of tlie ratio 2|, the series will b^ 3, 3x2i, 3x2ix2t, &c. or 3, 7i, 18|, &c. And as the 3rd. term is the product of 3 and the square of 2i-, so the 4th. term must be the product of 3 and the cube of 2i ; the 5th. tefm the product of 3 and the fourth power of Sf.
190. Thus we see that any term of a geometrical series is the product of the first term and that poxver of the' detw* 7ni?iator of the common ratio whose index is less by 1 than the number of the term : just as we have seen that any term of an arithmetical series is the su?n of the first term and that multiple of the common difference whose coefficient is less by 1 than the number of the term. For instance, in the geometrical series S^ 6, 12^ 24, &g- the 24th. term must be the product of 3 and the 23rd. power of 2, or 3X2^^; as in the arithmetical series 3, 5, 7, &c. the 24th. term is 3 +23 X 2. In finding the 23rd. power of 2, to avoid the tediousness of successive multiplications by 2, we square the 5th. power, which gives us the 10th. and square the 10th. power, which gives us the 20th. This multiplied by the cube of 2 gives us the 23rd. power. Thus2^ = 325 2»° = 32X32=1024; 2=»° — 1024 X 1024 = ] ,048576 ; and 2*3 == i ,048576 X 8 = 8,388608.
191. We may now easily express such a series alge- braically. Putting a for the first term, and d for the cle- nominator of the common ratio, the 2nd. term must be ad ;
the
( 9^ )
tlie Srd. term ad.'' ; the 4th* term ad^^ &c. And let n bd the number of terms in the series , then the index of d in the last term must be n — 1 5 that is, the last term must be ad""^^. Any geometrical series therefore is justly repre- sented by «, ad, ad^, ad^ ad"*"^. And the product of
the extremes is evidently equal to the product of any two terms equally remote from the extremes ; as is true of the mms of the terms in an Arithmetical seriesi^
192. Let it now be proposed to find the sum of such a series, continued (suppose) to 5 terms ; and put s for that sum. We know that s = a-\-ad-{-ad^ -{-ad^-]-ad^ i and if we multiply these equals by d, the products must be equal. But the product of s multiplied by d is sd; and the pro- duct of a-\-ad-{-ad^-\-ad^ + ad'^ multiplied by d h ad-\-ad^ •{■ ad"^ -\- ad^ + ad^ , Now, since subtracting equals from equals the remainders must be equal, if we subtract 5 from 5f/, and the value of 5 from the value of 5^?, we shall have equal remainders. Let us perform the operation, and ob- serve the result. Thus —
From sd=ad'{-ad^-Yad}-\-ad'^^ad^ Take s — a-\-ad-^ad^-\-ad^-^ad'^
Remainder sd — s = ad^ — a
In this operation the student will observe, that fo subtract 5 from sd we annex it to sd with the sign — ; and that in subtracting the value of s from the value of sd, all the terms disappear except the last term of the minuend ad^ and the first of the subtrahetid a, which is therefore sub- tracted from ad^ by annexing it with the sign- — . There- fore we are certain that sd — s = ad.^ — a : and now if we di* Tide both of these equals by d — 1, the quotients must be
1 4.1. 4. ' sd — s ad^ — a -n ^ sd — s
equal; that is, =— . Jout =5, as appears
d — 1 d — 1 d — 1
by performing the division, or by observing that d — i
X 5=56? — s. Hence it follows that 5= II_ : and there-
a— -1
fore the sum of the series, a, ad, ad^, ad}, ad^, is found
by continuing it to one term more, (or multiplying the last
term ad"^ by d) subtracting the first term a, and dividing
the remainder by a number less by 1 than the denominator
of the common ratio. And universally, whatever be the
number
{ 95 )
ii limber of terms, s=:a-{-ad-{-ad^,,»'\-ad'-„^ ; and multlpU'-' ing both sides of that equation by d, sd-=ad-\-ad^ -^ad^^ ^i^-^ad"; and from these equals subtracting the former equals, 5^ — 8=^ ad" — a ; and dividing these equals by d — 1, ad'
I93. Thus we see that the sum of any geometrical series is found by the following rule : — multiply the first term by that power of the denominator of the common ratio whose index is the number of terms in the series ; from this pro- duct subtract the first term ; and divide the remainder by the denominator of the ratio minus 1. For instance, let it be required to find the sum of the series 2, 6, 18, &c« continued to 8 terms. The denominator of the common
ratio is 3 ; therefore the sum of the series is
3—1
" -^l- = 6560. When the denominator of
2 2
the common ratio is 2, since 2 — 1 = 1, we are saved the trouble of the division. Thus the sum of the series 3, 6^ 12, &c. continued to 10 terms is 3x2^^ — 3, or 3 X 1024^ — 3 = 3069. The same calculation is obviously applicable to the sum of a decreasing series, as 54<, 18, 6, 2, by taking the terms in an inverted order ; or always subtracting the least term from the product of the greatest and the deno- minator of the ratio considered as a ratio of less inequality* [And this method is less apt to perplex tiros, than the con- sideration of -f as the denominator of the ratio. By in-
terting the series the sum is ; in the other me-
thod (y — 54) -J y. Tlie two expressions are equivalent •
for in dividing by y we should multiply the diviclend by 3 ^
and divide by 2 ; so that the expression becomes . ,
in which fractional expression both numerator and deno- minator being negative, the value is positive and the same
.. 54 X 3—2 1 with _^— .J
194. From the nature of the Arabic notation it is evi- dent that any number written by a repetition of the same digit, as 3333, or 77777, may' be considered as the sum
of
( 96 )
6i a geometrical series, in which the denominator of t*i^ common ratio is 10: for 7777 = 7 + 70 + 700 + 7000. Ami accordingly the sum of this series calculated according to
the rule given in thef last section, or ~-^^^, is 7777«
J?
. , 50000 — S o„„„ Q ■ And so --±=3333^ &ci
195. It is observable how rapidly numbers increase ir? geometrical progression. One billioti is the 13th term of a decuple progression whose first term is unity : and we have already noticed (§ 6.) the enormous magnitude of that number. The inventor of the game of Chess, which is played on a board divided into 64 squares, is said to have been offered by an Eastern Monarch any reward he might desire. He desired only 1 grain of corn for the first square of the board, 2 for the second, 4 for the third ; and. so on in geometrical progression to the 64th. square; But it was found that riot only all the corn in his majesty's do- minions would not be suflicient to pay him, but not all that could be produced in 8 years on the surface of tlie terra- queous globe, if it were all arable land, and under culti- t^ation. The number of grains demanded was 2^*^ — 1. We have already seen (§ 190.) that 2^'^= 1,048576 and 24°= 1,048576^ = 10,9951 1,627776; 2^^=1048576^ ; and 2*54 = 1048576^ X 2+ = 1,1529^1,504606,846976 X 16 = 18,446744,073709,551616, or less than 18 trillions and a half. Now supposing a bushel of corn to contain 600,000 grains, (i. e.. Supposing a standard pint to contain 9375 grains) and supposing an acre of land to produce in d year 30 bushels of corn, it would require one billion of acres to produce 18 trillions of grains. But the wliole surface of the terraqueous globe amounts to little more than the 8th. of 1 billion of acres*
196. Let us now suppose a decreasing series in Geome- trical progression, for instance 2, 1, 4 J T> &c. The22d,
term of tin? series must be - — or : and therefore
2*^ 1048576
the sum of the series = 4 --. And if we continue th-e
series to 66 terms, the siun must be 4 ^-: that is, less
than 4 by a jfraction so small that, although subject to
iiumerical
( 97 )
numerical calculation, it baffles all conception. But there is no limit to our power of continuing the series ; and the further we continue it, the nearer must the sum approach to 4 : while, continued ever so far, the sum of all the terms never can exceed 4. For if we continued the series to 1000 terms, the sum would be 4 minus a fraction whose numerator is 1 and the denominator the 998th. power of 2. Hence we may say that 4 is the exact sum of that series continued in infinitum : by which we mean that, let the se- ries be continued ever so far, the sum of all the terms ne- ver can exceed 4 ; and that it may be continued so far a% that the sum shall exceed any number ever so little less than 4, or that is less than 4 by a fraction ever so small. In like manner the sum of the infinite series 3, 1, -f, |, &c. is % or 44. For by § 193. the sum of the finite series 3, 1, -f, -^is (9 — ^) -r- 2 : and let the series be continued ever so far, the sum would be found by subtracting the last, or leasts term from 9, and dividing the remainder by 2. But when the series is considered infinite^ or continued without end, there is no least term to be subtracted, and therefore
tlie sum is f. And universally let «, -, -^, &c. repre- sent a decreasing infinite series. The sum of that series
is _^^ . For, continued to n terms, its sum is ( ax — )
^—1 \ X-')
-i- (a- — 1 ). But if the series be continued without end, there is no fraction to be subtracted from ax*
197. I have generally observed, that on the first dis- cussion of this very curious subject there remains in the^ mind a suspicion of some latent fallacy in the reasoning. But let us bring its accuracy to a particular test. We know that the vulvar fraction 4» turned into the decimal
'&
form, produces the circulating decimal .666, &c. (JUS.) Now this circulating decimal is in fact the sum of an in- finite decreasing series in geometrical progression ; for it is equal to t^ + xIo+two, &c. (See § 114.) Let us then calculate its value according to the principles of the last section. The greatest term is -^ ; the denominator of the common ratio 10. Therefore the sum of the series = (-/^XlO)~10 — 1=§. But this fraction being equiva* lent to }, we have a confirmation that the principles are
H just
( 98 )
just, which we have kid down for calculating the sum of an infinite decreasing series. Thus again ^ = .222, &c. and ^+T^, &c. =(tVX10)-^9 = |/ In like manner .999, &c. =1.
198. Upon these principles we can easily find tlie vulgar fraction, which' produces any given circulating decimaL For instance, .212121, &c. =t?o^+t^W> &c. where the denominator of the common ratio is 100. Therefore the sum of the series is Ji=y'y: and accordingly yy reduced to the decimal form produces the given circulate. Let it be required to find the vulgar fraction, which shall circu- late through the ten digits in regular order. The deno- minator of the ratio being the tenth power of 10^ the sunt
^ , . .^1234-567890 1&7174210
of the series is = ,-
9999999999 1111111111
199. Upon the principles brought forward in § 196. we may detect the sophism, by which Zeno pretended to prove that the swift-footed Achilles could never overtake a tor- toise, if they set out together, and the tortoise were at first any distance before Achillesv <*" If," said he, « the tor- *« toise at setting off be a furlong before Achilles,: though- ** the latter runs 100 times faster than the tortoise crawls,. *< yet, when he has run a furlong, the tortoise will be the *< 100th. part of a furlong before him: and when Achilles. *« has advanced that small space^ the tortoise will still be << before him by the lOOth^ part of it, and so on for everJ^ Now it is very true that, if we take the spaces or times de«^ creasing in that geometrical ratio of 100: 1, we cannot assign among them — (how far soever we continue the pro- gression)— any one, at which Achilles will have overtaken the tortoise. But it is altogether false, that the sum of those spaces or times will be an infinite quantity, as is im- plied in Zeno's conclusion : for the sum of the infinite se- YiQs I, 74^, 8tc. is exactly VV ^^ ^irV* -^^^^ accordingly that gives us the precise spot where Achilles will overtake the tortoise : for when he has gone VV^^s* of a furlongs the tortoise, moving 100 times slower, will: have gone ^'pth, that is, they will be just together. And this affords ano- ther confirmation,, to prove the truth of q\w calculation of the sum of an infinite decreasing series*
Ex. 1 ^ What is the 8th. term of the Geometrical series- 4, 12, 36, &c. and what is tlie sum of the series ?
Ex. 2,
( 99 )
Ex. 2. What is the 9th. term, and what is the sum, of the Geometrical series «, a*, a^, &c. ?
Ex. 3. What is the sum of the decreasing series 18, 6, 2, &c. continued iji irifinitum P
Ex. 4. Do. of the decreasing series a^, a^, a^^ &c. ^
Ex. 5. What vulgar fraction will produce the circulating- decimal .102102, &c. ?
Ex. 6. If a man spend 1 farthing in the first week of the year, and each succeeding week twice as much as in the preceding, how much will he spend in the whole year ?
Ex. 7. In how many minutes after 6 o'clock will the mi- nute hand of a watch overtake the hour hand ?
Ex. 8. If two men at opposite points of a circle set out at the same time and in the same direction, with velocities that are as 7 : 6, how many times must the quicker go round the circle before he overtakes the slower ?
Ex. 9. If a courier ride at the rate of 6 miles an hour, and in -|: of an hour after he has set out a second courier be dispatched to recall him, and ride at the rate of 74 miles ani hour, at what distance will the second overtake the first ?
CHAR XXL
Extraction of the Square Root.
200. TO extract the square root of a number is to find a number, whose square is the given number : and the multiplication table enables us to assign the root of any square number as far as 144. Many fractional numbers may have their square roots assigned with equal facility. Thus the square root of |t, or V^^^ is ^, because 4 X | = |-f. And the square root of -5^ is | ; for, although we cannot extract the square root of 8 or of 18, they not be- ing square numbers, yet — = :J=|p : so that before we conclude that the square root of a fraction cannot be ex- actly assigned, the fraction should be brought to its lowest terms.
201. It is equally easy to assign the square root of any simple Algebraic quantity, which is a perfect square.
/1 2 a* 2«
Thus V^a^zz^a-^ for SaXSa:=i9a^ : and y/^Y^^i^*
H 2 for
( lOO )
^'*^i* TTiTTT = KTi ' and Va^b^ z=ab; for a6 Xah = a^b^, From
this last example we may observe, that the square root of any product is equal to the product of the square roots of its factors. Thus 4X16 = 64, and V64 = 8 = 2 x 4=r V4 X VI 6. And hence it follows, that the product of any two square numbers' must be a square number j for its square root is the product of the roots of the two factors.
202. The operation by which we extract the root of higher square numbers proceeds on the princi})le that the square of the binomial a-j-6 is a^ -\-2ab-\-b^^ and that if we divide the two latter terms 2ab-\-b- hy2a-\-b the quotient is b. Now suppose we want to find the square root of 5476. We know that the square of 70 is 4900, and that the square of 80 is 6400. J herefore the square root sought is less than f 0 but more than 70. Considering therefore the root sought as a binomial (a+^j of which we now know" one part {a) we subtract the square of 70 (4900) from 5476. The remainder 576, corresponding with ^ab-^-b^, must contain twice the product of 70 and the other part plus the square of the latter ; and therefore if divided by twice 70 (140) plus the other part must give that other part for the quotient. And thus we find that the second part of the binomial root is 4 ; for 140 -f- 4 X 4 = 576. The root sought therefore is 70 -j- 4, or 74.
203. Let it now be required to extract the square root of 225625. We know at once that the root sought must be greater than 400 and less than 500 : for 400* = 160000, but 500* = 250000. Subtracting therefore 400* from 225625, there remains 65625 ; which contains indeed 800 (400 X 2) above 80 times, but does not contain 800 + 80 (or 880) so often as 80 times. The remaining part of the root therefore is less than 80, but more than 70 ; for multiplying 870 by 70 the product 60900 is less than 65625 by the remainder 4725. We have now however ascertained the second of the three digits of which the root must consist ; and only want to find the last which stands in the place of units: for the root sought is above 470, but below 480. If then, considering 470 as the first part of a binomial root, we subtract its square from the proposed number 225625, the remainder divided by twice 4'70 plus the last digit of the root must give that last digit for the quotient. But we may save ourselves the trouble of
squaring
( 101 )
squaring 4^70, observing that the subtraction of its square?
has been ah'eady performed. For 470^ = 400^ +70* -j-
800 X 70. Now in our first operation we subtracted 400*
from the given square; and 60900 which we subtracted
from the remainder is 870 X 70, that is 800 X 70 + Yox'io^,
If therefore we divide the last remainder 4725 by 940
(twice 470) 2;/z^5 the last digit of the root, the quotient
4725 must be that last dio^it. But — 1^ = 5. Therefore the root ^ 945
nought is 475. And in like manner we find that the square root of 6953769 is 2637 ; for the third remainder in the operation is 36869 : but that is the remainder after sub- tracting 2630^ from the given number; and therefore divided by twice 2630 plus the 4th. digit of the root must give that 4th. digit for the quotient : — just as the quotient of 2ab + b^ divided by 2a-\-b is b. And so, let there be ever so many digits in the root, they may be successively discovered.
204. In practice, we begin with the ^rst or the Uw first digits of the proposed square, according as the number of its digits is odd or even ; and subtracting from it the square number next below it, (afforded us by the mul- tiplication table) writing its root as the first digit of our root, we annex to the remainder the next pair of digits in the proposed square. And so on, successively dividing all the digits of each completed remainder, except the last digit, by twice the digits of the root found ; and thus ascertaining the next digit. Then annexing that digit to our divisor, we multiply the completed divisor by the digit of the root last found, and subtract the product from the last completed remainder. Let us annex the operation performed at large, and according to the abbre- viated method^ that a comparison of them may make their identity manifest.
. . . ,/
6953769/ 2000 6953769\2637 4000000V + 600 4
4000X29753769 +30 46")295 + 600/2760000 +7 276
5200\ 193769 523)1937
+ 30y 156900 1569
5260\ 36869 5267) 36869
+ 7; 36869 36869
( 102 )
205. It is plain by inspection of these two operations that the only difference between them is, that in the shorter method we neglect writing the cyphers, and attend only to the significant figures concerned in each part of the process. But let us trace the several steps of the operation in that example. The number of digits in the proposed square being odd, we first attend to the single digit on the left hand, 6. The square number next below it is 4, whose root 2 we write as the first digit of our root ; and subtracting 4 from 6 there remains 2 ; to which we annex the two next digits of our proposed square, 95. Theif dividing 29 by twice 2, or 4, we might conceive that 7 should be the next digit of the root. But 7 times 47 being more than 295, we fix upon 6 as the next digit of the root; and annexing it to 4 we subtract 6 times 46, or 276, from 295, and to the remainder 19 we annex the two next digits of the proposed square 37. Then doubling 26, or adding 6 to the last divisor 46, we observe that 52 is contained in 193 three times. Therefore writing 3 as the next digit of the root, and annexing it to 52, we subtract 3 times 523, or 1569, from 1937, and to the re- mainder 368 we annex the two last digits of the proposed square, 69. Then doubling 263, or adding 3 to the last divisor 523 ; and observing that 526 is contained in 3686 seven times, we write 7 as me next digit of the root, annex it to 526, and subtract 7 times 5267 from 36869, when nothing remains : sp that the proposed number is a com- plete square whose root is 2637, The proposed number is commonly pointed off by pairs of digits from the right hand, to ascertain the pairs which are to be annexed to the successive remainders, and whether we are to begin with the first or the two first digits on the left hand.
206. To explain the reason of the rule, by which we determine whether we are to begin with the first digit of the proposed number or with the two first digits ; let it he observed that the number of digits in any square cannot exceed double the nvunber of digits in the root, and cannot fall short of that by more than 1. Thus if there be 3 digits in the root, there must be at least 5 in the square, and there cannot be more than 6 : if there be 9 or 10 digits in the square there must be 5 digits in the root. For take the greatest number consisting, for instance, of 3 digits,
namely
( 103 )
namely 999. Its square must be less than the square of 1000, that is, less than 1000000. Therefore the number of digits in the square of 999 cannot exceed 6. Now take the least number written with 3 digits, namely 100 ; and its sqnare consists of 5 digits. By the same mode of reasoning it is manifest that, if the root begin with any digit except 1, 2, or 3, (whose squares consist of a single digit) the square must consist of twice as many digits as the root. Since therefore the square proposed in the last example, 6953769 consists of 7 digits, the root must consits of 4 digits, and its first digit must be less than 4. Therefore we begin with inquiring the nearest square number to 6, not the nearest to 69.: for this would give 8 for the first digit of the root
207. If we find any remaindej' after the last subtraction, we conclude that the proposed number is not a complete square ; but by annexing decimal cyphers in pairs, and thus continuing the process of extraction, we may approx- imate to the root at pleasure. In such a case it is evi- dently impossible ever to arrive at the exact root ; since there is no significant digit whose square ends with a cypher : but we may approach nearer it than any assign- able difference. Thus, if we desire to find a number which shall be nearer the root than by the millionth part of unity, we need only continue the process of extraction to 6 places of decimals, for which purpose we must have annexed 6 pairs of decimal cyphers. For even if the root could cir- culate from that in 9's, the remaining part would only be equal to t>o-5-5-o^o^. But the root in this case can never circulate : for the value of every circulating decimal may be exactly assigned in a finite fraction {§ 198.) and we have seen that the exact root of such a number as we have supposed never can be assigned. We annex the decimal cyphers in pairs, because for every digit in the root after the fii-st there must be two digits in the square. If the proposed number be partly integral and partly de- cimal, we must point off the integral part distinctly, and make the number of decimal places even, by annexing a cypher if necessary. Thus in extracting the square root of 27.345, the first digit of the root is, not 1, but 5.
208. From what has been said it appears that we may cither express the square root of 2, for instance, as a
surd.
( 104. )
surd, — thus VS or 2^, or else proceed to extract it within any degree of accuracy that may be required : — thus.
2.00(1.414213 1
24)1.00
281) 400
2824)11900
28282).60400 28284 l)Ts"83600 2828423)T0075900
1590631, &c.
Kow the square of 1.41 is 1.9881 or within .0119 of 2. The square of 1.414 is 1.999396, or within .000604 of 2 j and so on.
209. The square roots of compound algebraic squares are extracted exactly in the same manner ; first arranging the terms of the proposed square according to the powers of some one letter. For example let it be required to ex- tract the square root of
a4__4^3^ ^ 8^33 ^ 4Z>4(a*^2fl5— 26^
2a^— 2aZ>) -~4a^^ + 8«/^H4^>+ — 4tt3Z>-f-4«^Z>*
— 4a^Z>^ + 8flZ>3^4Z»4
And accordingly if we multiply the trinomial a^ — 2ah — 26* by itself, the product will be the proposed quantity. It is manifest in this example that the second remainder has been found by subtracting the square of a* — ^ah from the given quantity: for a^ — 2ab\^ =:a^ — 4«^64-4«^Z>*.
210. We may here remark that 4 times the product of any two numbers differing by unity, plus 1, gives the square of their sum. For let a represent the less ; then c-fl will represent the greater; and 2a-\-l their sum.
But 2a-f lp=4a* + 4fl4-l; and 4a* + 4^5 = 4 X « X«4-l.
Thus
or ^rar*
{ 1G5 )
Thus 10 + 91»=361=4.X90+1. And if we add to any number its square 4-^5 the sum must be a square number: for « "^ -f « + ^ is the square of a-\-i. Thus 9 + 8 1 + ^ is the square of 94 or \^. Lastly the sum of any two numbers differing by unity is the difference of their sqiiares. For a + lV—a^-^a+l,
Ex. 1. Extract the square roots of 6889? of 38416? and of 3 ?
Ex. 2. Extract the 4th. root of 4096 ? Since jp^= Va;'* and a:= V^^, it is plain that the 4th. root sought must be the square root of the square root of 4096.
CHAP. XXII.
Fractional and Negative Indices. Calculations of Surds.
211. IT is evident that the square of a* is a^^ and that the square of a^ is a^ ; since aaa X aaa = aaaaaa=:a^. In like manner the cube of a"^ is a^ ; since aaXaaXaa=z aaaaaa = a^. And putting 7i for the index of any power of «, the square of a"" is aj^", its cube a^% its fourth power a^""^ &c. So that a" is raised to any power by only mul^ tiplying its index n by the index of that power. It fol- lows that a" is the square root of a^'\ the cube rootofc^", the fourth, or biquadrate, root of a^" &c. So that we may express any root of a given quantity by dividing its index by the denominator of that root : just as the cube root of a^ is «, or «% and the cube root of a^ is a^. For 4=1, andf|-=2. Hence the origin of expressing roots by fractional exponents : for thus the square root of a*
is justly expressed by a"", its cube root by a^, &c. In like
J. manner the square root of a^ is a^ ; the cube root of a*
is a"^, &c. And universally, putting n and m for any num^
m
bers whatsoever, the nth. root of a*" is «". And this mode of notation has many advantages above the expression by
n
the radical sign Va*".
212. Since
( 106 )
212. Sincei = |: = |, &c. and | = J = |, &c. therefore a*
(or Va)=:a^z=:a^, &c. and a"^ (or Va)=d^=za^^ &c. it follows that, as we can bring any two numbers integral or fractional to fractions of the same denomination, we may easily reduce any two quantities to equivalent expressions of the same radical sign. For instance, let it be required
to bring a"^ and 6* to the same radical sign: we have only to reduce the fractional indices y and t to equivalent frac- tions with the same denominator, and the expressions be-
come a'^ and b ^ or aaaa^ and hhh^ ^ or V a'^ and V 6^. Now we have observed (§ 201) that the square root of any product is equal to the product of the square roots of its factors: whence it follows that VaX VhzzVab. And universally the product of the ni\\, roots of any factors is equal to the nth., root of the product of the factors : or
n n w ,
\^a X Vbzz Vab* A similar principle must evidently be applicable to division V«~-VZ>=: /^. And since we can
transform any two given surds of different radical signs into surds of the same radical sign, it is plain that we can thus express the product or quotient of any two given
surds under one radical siffn. Thus Va X VbzzVa^'br
for Va—a^'"^ and Vb:=ibnm.^ but V oT X VZ>"= V a^'b"",
J
In like manner 3 -^ V 2 = ^ — .
213. Let it also be remembered that powers of the same quantity are multiplied by adding their indices, (e. gr. a^Xa^—a^) and divided by subtracting the index of the
divisor from the index of the dividend, (e. gr. -_=fl^)
Now suppose we have to multiply Va hya: the product may be expressed by prefixing fl^ as a coefficient, thus, aVa, But since a= Va*, the product may also be expressed by Va^ ', for VaX Va^zz Va^, But we may at once arrive at the same conclusion by adding the indices of the factors
X JL I ^
a- and a 5 for a^ X « =«^, since i-f 1 =1. In like man- ner
( 107 )
ner Va^'XVa, or «^X^*=ra^, or Va"^, since |+i=J, Thus also -^=a^"~^=«' and :^=a^— ^=:at
214. Now we know that - = 1. But it may also be ex»
pressed by a'-*, or a'^. And in like manner ~, or _, may be expressed by a"' ; since 1 — 2, or 0 — 1, = — I, And
1 —2 1 — ^ — n
thus — =<x : -— = a , &c. Thus we see that a is a
«
a"
just expression for the reciprocal of -p or of «". W^e iiave observed that the product of any quantity and its re-
ciprocal IS 1 : e. OT. — X - = — = 1. And accordingly a^ X
n m mn °
^-" = a° = l.
215. Though we cannot add or subtract surds by incor- poration, unless they have the same irrational part, and otherwise must denote the addition or subtraction by the .sign -f- or — ; yet it often happens that unlike surds may be transformed into like by resolving one or both of them into a rational part and an irrational. Thus V2 and V8 are unlike surds, and their sum or difference is a/8z±= V2. But since V8 = V4?X V2, and ^4 = 2, therefore V8 = 2\/2 : and 2\/2 and V2 being like surds may be incor- porated ; their sum being 3^2 and their difference ^^2.
Thus also V24+v'81=2v'3-f 3V3 = 5v^3. And uni-
versally V<2".a;:±:V 6" J7=«=±:6 X \^.r. It is plain that the product of any two quadratic surds which are like, or may be transformed into like surds, must be rational. Thus V2X V8 = \/16 = 4: and V a'x'KVh''x—abx. Other- wise the product of any two quadratic surds must be irra- tional. And as we may sometimes take one part of a given surd from under the radical sign and prefix it as a rational coefficient ; so, whenever we have a surd with a rational coefficient, we may bring it under the radical sign : since
n n
aV x-zzV aJ'x*
216. Any fraction with a binomial denominator, one or both of whose terms is a surd, may be transformed into an
equivalent
( 108 )
equivalent fraction whose denominator shall be rational j upon the principle that the product of the sum and differ- ence of any two quantities is equal to the difference of their
squares. (§ 163.) Thus the fraction ^g^^^> by mul- tiplying both numerator and denominator by V3 — V2,
becomes g^^ ^^ . ^ ox aV 2>—a V 2. For when we mul-
3 — 2 tiply V3+ V2, (the sum of VS and V2) by ^3 — ^2, {their difference}, the product must be the difference of their squares. And in like manner, if the denominator consist of three or more parts, we may by successive mul- tiplications render it rational, e. gr. Let the denominator be 2+ V2 — Vai we may consider ^2 — Va as one term, and 2 as the other term composing the denominator ; and if we multiply both numerator and denominator of the frac- tion by 2 — V2 — Va, the new denominator 2 + 2^/2^; — a \\\\\ have in it but one irrational term, since there is but one irrational term in the square of the binomial surd V 2 — V a. And now considering the new denominator as consisting of the two parts 2 — g, and 2\/2a, if we mul- tiply both numerator and denominator by 2 — a — 2v^2a, the denominator must be the difierence of the squares of 2 — a and 2V2a, or 4 — l2a-\-a^. And thus the irrationahty is removed from the denominator to the numerator. Pur- suing this process the student will find that the fraction
^ 5V'2 '
V ^-^ =:W2', and accordingly 4 + ^2X3 — 2v/2 = 8
— 5V/2.
217. The square root of any binomial a-:±=h may justly be
representted by the following expression, /?d_^JZI^
*y
2 a — A/a^ — b""
_ 2 2 a ^2Vb
; for the square of this expression is
, that is a:±ib. By performing the operation
the student will find that the square of that binomial is what we have assigned ; and from the following conside- rations he may be convinced that it must be so. We know that the square of any binomial is composed of the sum of
the
( 109 )
the squares of its parts, plus or minus twice the product of the parts. Now the parts of that binomial surd are squared by throwing olF the radical signs prefixed to them; and.
therefore their squares are — ■ and ^,
.2a and the sum of these two quantities is -~- = a. Again let us
consider what must be the product of the two binomial
surds [j^l±^^ and ^a-Va^-h\ n ^m be
found by taking the product of the numerators and the product of the denominators, and prefixing to each the radical sign V , Therefore the denominator of the pro- duct must be V 4, or 2. But since the numerators are the sum and difference of the same quantities a and V a^ — 6*, their product must be the difference of the squares of those quantities: (§ 163.) that is the difference between a^ and a^—h^^ which difference is b^. Therefore the numerator of the product of the two binomial surds is Vb^ or b, and
their product is ~ j and twice that product is b. And thus
we see that the square of the assigned binomial surd must
be ar=^b. Let us exemplify the truth of this in numbers.
We know that V'10-f.6 = 4. But it is also equal to
/l0 + VT00=36+ /lO-VlQO^it . for ^"looIISS
= V^ 64 = 8 : and therefore /lO + \/l00— 36 = /Ts
= %/9 = Sj and . /lO-V 100—36 = . /§ _ ^ , ^ ,
Again V"T0=6 =2 = yi0 + Vj00^_^/l0-V^10Q"^
218. This mode of expressing the square root of a bi- nomial has its principal use in some binomial surds, which often occur in practice. (See § 238.) For instance if we want to express the square root of az^Vb^ it may be de- signated
( HO )
signated by prefixing tlie radical sign to the binomial sUrd ;
thus Vadt:\/b, or a=i=:\/b\'^' But whenever tt*—^' is a square number, let us put r for the square root of that number, and we may express the square root of the given
binomial by y^/ ^+~^ JZL, Thus the square root
ofll+6v^2(orll + i/72)isy/iii^^ + ^—
■V49
2 ' "V 2
= ^/ii±2+. /iilzZ = \/94-V2 = 3H-v'2: which isa V 2 ^ 2
simpler expression than Vll-^6V2. Again ^7-1-2^/6 may be more simply expressed, since 49 — 24 = 25, a square
number. Therefore the square root of 7 + 2 V 6 = j^ _i_.
2 = 7+2^6.
= V 6 + 1 , or 1 + V 6. And accordingly 1 + V ef
Ex. 1. ^^^X V^*= ? V2a''a:XVSaV= ?
3 3
Ex. 2. V^^~Va:*= ? 1/3^3^5 -^V2«*^= ?
Ex. 3. ^^3/Xa^= ? J^^X^ ^= ?
Ex. 4. >V^ 125j;=S=V^ 4jr = ? V SOo-^j/rtA/ 20jc4j^ ?
3 Ex. 5. Reduce the fraction to an equivalent
fraction with a rational denominator ? Ex. 6. Also the fraction
l-^Vx — V^
Ex. 7. What is the simplest value of V9 + \/45 ? Ex. 8. ...of V 19—1/261?
CHAP. XXIIL
jReduction of Algebraic Equations ^ Simple and Qjuadratic»
219. TO reduce an equation is to discover the value of the unknown quantity in it, which has been represented
( HI )
by one of the final letters of the alphabet. Tims, if we
have proposed to us the equation 5x — 34 = 57 + — ; we may
3 by a very short and easy process discover what number PC stands for. Now, according to the import of the Alge- braic symbols as already explained, the proposed equation expresses this fact, that the subtraction of 34 from 5 times the number represented by x gives a remainder equal to the sum of 57 and ^rds of the number represented by x. And therefore whenever we shall have ascertained the value of /v, this property must belong to the number found ; so that if we substitute the number found for Xy in each ex- pression where that letter occurs in the proposed equation, the amount of the terms at one side must be equal to the amount of the terms at the other side of the equation.
Thus, by reducing the equation 5x — 34=57+-—, we
3
shall find that ;v = 21 : and the truth of this result will ap*
2x pear by substituting 5 X 21 for 5x ; and y X 21 for — . For
3
5X21 = 105; and 4X21 = 14: but 105—34 = 71; and
2a? 57 + 14 = 71. Such an equation as 5x — 34 = 57H is
called a simple equation, because the unknown quantity x does not rise in any term of it beyond the 1st. power.
220. The process of reducing such equations depends upon the following simple principles; that if to equal quantities we add the same or equal quantities the sums will be equal ; or if from equal quantities we subtract the same or equal quantities the remainders will be equal ; and that if we multiply or divide equal quantities by the same number, the products or quotients will be equal. From the former of these principles it follows, that we may transpose any term of an equation from one side of it to the other, changing its sign. Thus in
2a; the proposed equation 5x — 34 = 57 + — , we may bring over
3 34 from the left side of the equation to the right with the
sign + ; and infer that 5;v = 57 + 34 + -^. For this is in
3
fact an addition of 34 to both sides of the equation ; the
sum
( li2 ) sum of — 34j ajad -f-34* being 0. But we may also bring over — from the right side of the equation to the left with
the sign — ; and infer that 5x — -1 = 57+ 34 = 91. For
this is in fact but a subtraction of -1 from both sides of
- 3
the equation ; since _ _ = 0. From the same principle
3 3
it follows, that we may at pleasure change the signs of all
the terms at both sides of an equation. Thus from the
equation 24 — 2x=: — 10, we may infer that — 24+2^ (or
2x — 24) = 10: for this is in fact but a subtraction of the
affirmative terms, and an addition of the negative to both
sides of the equation.
221. Lot us now take the equation 5x — !L. = 9l; in
which we have brought over to one side of the equation all
the terms in which x (the unknown quantity) occurs, and
have only the amount of known numbers at the other side.
We may now infer, that 3 times the one side is equal to 3
times the other side of the equation. But 3 times the bino-
2x . mial 5x — ^ is equal to 1 5x — 2x = 1 3a; : for when we mul-
2x tiply the fraction ~ by its denominator 3, the product is
the numerator ^x integral. (§ 113.) Therefore 13a; = 91 X 3 = 273. And now we may divide both sides of this equa- tion, 13a: = 273, by 13, and infer that the quotients will be equal. But the quotient of 1 3x divided by 1 3 is a^ j which
273 is therefore equal to - — = 21. And thus we have ascer- 13
tained the value of x ; and the reduction of the equation
5x — 34 = 57+ — is completed. Let us exhibit the steps, 3
■which we have taken in one view.
9st!
5a;--34 = 57 + 1^ 3
+ 34-^?f ^ 3
XS
5;v— ^' = 57 + 34 = 91 3
{I5x—'2x = ) 13;c = 9lX 3 = 273
;v = 273-~13=21
222. The
( H3 )
222. The marks on the left hand of the derived equa- tions denote the operntion, by which each equation is de- rived from the preceding; 1st. the addition of 34 — -i
to both sides, or the transposition of those terms with their signs changed : 2ndly. the mukiplication of both sides by 3; 3rdly. the division of both sides by 13. And it may be useful to the student at first to adopt that practice, of marking in the margin the operation by which he proceeds to derive each equation ; although this will afterwards be- come unnecessary. In the first step of the preceding ex-
ample, both the terms 34 and^ are transposed by one ope- ration ; and ever so many terms may be transposed at once, only taking care to change the signs. But for a time it may be better for the student to transpose the terms one by one.
223. After the first step, we might have completed our
reduction by one inference, observing that 5^ — 1- is the
product of 5 — I X x. If therefore we divide both sides by 5 — I, that is by 4j or y, we shall at once have the equa- tion ^ = 9 1 -r- y = Vy = 2 1 . But in the second step of the reduction, as exhibited at the end of § 221. the student should well observe, how an equation may be cleai'ed of any fractional expression, by multiplying both sides of the equation by the denominator of that fraction^ And let there be ever so many fractional expressions in an equa-, tion, they may be all removed either successively, or by one operation. For instance, if we have this equatioiT
L-^-L-f- — = 5, successive multiplications of both sides by
2, by 3, and by 4, would remove the several fractions,
producing successively the equations ;c-f- — H — 1 = 10, and
(3^ + 4a? i. e.) 7;v+L^ = 30,and(28;t+18;tfi.e.)46A: = 120.
Hence it is plain that the same result must be afforded by one multiplication of both sides by the product of 2, 3, and 4, or by 24. But it will answer the same purpose, and keep our numbers lower, to multiply both sides l)y 1 2 the
I least
( 11* )
least common multiple of 2, 3, and 4 : for each of the three fractional expressions might be brought to an equivalent fraction of tliat denominator. Multiplying then both sides of the given equation by 12, we derive this equation (6^+8.r-{-9^ i. e.) 23a' = 60: whence dividing both sides by 23, we obtain the value of .v, namely Ar = |y=:2i4. But to this value we might have arrived at once, by dividing
both sides of the given equation by 44-T+i» For -+-7-
2 3
+~=4 + T+iX^^- Therefore x=i5 ^T+¥H=z5^A'
224. The rule therefore for reducing any simple equa- tion of this kind may be thus proposed : Bring over by transposition to one side of the equation all the terms, in which the unknown quantity (whose value you are inves- tigating) occurs ; and all the other terms to the other side. Then divide both sides of the equation by such a divisor as, if multiplied by the unknown quantity, would give the former side for the prbduct. Let us exemplify this rule by
2.V other instances. Let the given equation be 3a? -f --1 + 24
= 49 — 2^:*. Now by transposition we infer that (SA^-f --
3
2a* + 2a? i. eO 5^-f -^=49 — 24 = 25 : and dividing both sides \j
of this equation by 5 + t> we find that ^ = 25 -r- 5y = 25 -^ y = 4^4=4'TT» And accordingly if 4tt be substituted for x in the original equation, we shall find the resulting num- ber the same on both sides. That the divisor, which will give on one side x for the quotient, is 5-f--y, appears from the consideration that this divisor multiplied by x gives for
2x the product 5^v -}-_-. And if there be ever so many terms 3
on one side, in each of which x (or the letter denoting the
unknown quantity) appears as a factor, it is easy from the
principle proposed in § 167 to assign the divisor which will
give X for the quotient. Thus if both sides of the equation
±_^4:x Jl -. = 1 be divided by 14-4 — 4 — |, the quo-
tient on the left side will be x, and on the right side the value of ^ in a known number.
225. i^gAin,
(' 115 )
225, Again, if wie:procced to reduce the equation -—1^
5
4- 20 = 20 4- ^—^?~ according to the rule proposed in die
beginning of the last section, we must observe that . ^"^
5
is the same thin<? as -; and that — ^1_ is the same
^55 7
thing as -^4-~. So that after the necessary transpositions
Sx 2.V
the equation will be -j-— -y = t4-t > which gives ^r=44-|
^j. — -5-=|f-^||=4T. Observe also, that wherever the same quantity stands on both sides of an equation with the same sign, (as -f 20 in the last proposed example) it may be expunged from both sides. For this is only a subtrac- tion of the same quantity from two equals. But let us ex- hibit the same equation reduced, by first clearing it of fractions, (after expunging the -f 20 from both sides) and let the student observe the correspondence of the opera- tions, aud sameness of the results.
X5
X7
4.28— 10.V
-r-ll
7 2U' — 28=:10.v-f 15 lLv = 43
11
12 3
226. If we be ffiven such an equation as -4 1-- = 4"
XXX
where -x appears not as a factor but as a divisor, it is evi- dent that multiplying both sid(js by x will bring it to the other form; giving the equation (14-24-3 i. e.) 6 = 4a., But if in an equation the unknown quantity x appear in one term as a factor and in another term as a divisor, we shall find produced a quadratic equation, in which x will rise to the second power or square. Thus, if the given 1 X
<*quation be -4--=: 4, the multiplication of both sides by v
I 2 £fivfs
( 116 )
gives the quadratic equation 1 + .-^=4a'; the niethc^ of
reducing which we shall deliver in the 231st. and follow- ing sections^
227. On the contrary many equations that appear in the form of quadratic, cubic, &c. may be easily brought to the form of simple equations. Thus the quadratic equation
5a' ^ — — = 7^, by dividing both sides of it by x^ becomes 3
5,v — r =71 And the cubic equation Bx"^ + ~ *7;v^ — 4^*,
3 3
2 by dividing both sides by x^ ^ becomes 5»v4-- = 7.y — 4.
And here we may observe that, if all the terms in an equa- tion have any common factor or divisor, we ought in the first instance to divide both sides of the equation by that common factor, or multiply both sides by that common di- visor : that is, the .cominon factor or divisot ought to be expmi^ed from, all the terms. .,- ,
228. If the unknown quantity in any term of an equa- tion be affected with a radical sign, we may free the equa- tion from irrationality, iind often bring it to the form of a simple equation by bringing that term to one side, and then raising both sides to sucli a poweir as will make t|iat term rational. Thus if we be given the equation V x — 3 = 7. we 'first infer by transposition that V^^lOj, and then squaring both sides we have ;^ = 10^ = 100. The ground of this inference is obvious ; namely, that if two quantities be equal, their squares, cubes, &c. must be equal. And
3
thus from the equation Vx — 3=7, we may infer that ;^ = 1 03 =r 1 000 : and from the equation w^ — f- 5 = 7,
= 2, and then I = 2* = 4;
2 2
whence x — 3 = 8, and ^ = 11, If we have the equation V b-\-x-=z\-\'V x^ two such operations will be necessary. For first, squaring both sides, we have o -^x r=:\ ^^W x ■{ x '^ whence, expunging x from both sides 5 = l-f2V^: in which equation there is but one surd, to be removed as be- fore, ihus, 2V'^ = 5 — l=4j or Vx'=z2'y and squaring both sides, ^ = 4.
229. As
I
p
( IIJ )
229. As such equations are r^duc^d upon the principle that the squares of equal numbers, are equal, so an equa- tion in which the iinknown quantity appears, in every term where it occurs, in its second power, may be reduced upon the principle that the square roots of equal numbers are equal : and such equations may as reasonably as the former be reckoned simple. Thus if we have the equation x^ ' ' - -
-- — 2== 10, after reducing it to the form ^^ = 10 + 2 X 3
= 36, we infer that the square root of one side is equal to the square root of the other, that is, x = 6. And from the equa-
tion 2^* = 40 — — we infer first that =:4fO : then that
2 2 '
^* =40-^-4= 16 : and lastly, by extracting the square root of each side, that a' = 4.
230. Upon just the same principle, if one side of our equa- tion be the perfect square of a binomial, of which x is one term, we may arrive at the value of x by extracting the square root of both sides. For instance the square of ;v -|- 3 is x^ + 6x -f- 9 : and therefore if we have this equation x^ -}- 6x 4-9=25, w^e may infer that a? + S=V25 = 5, and there- fore that Ar = 5-^S = 2. Or if we have tHe equation x^-^6x 4- ,9 = 25, yye may infer that .v— 3 = 5, and therefore that a; = 8; since x^ — -6Ac-f- 9 is the square of a; — 3. And here let the student recollect, what has been shewn in § 34. and 163. that the square of any binomial consists of the sum of the squares of each term of the root, pkis or minus twice their product, according as the terms of the binomial root are connected with ihe signs -|- or^ — . Such an equation as ;v^ =25 is called a pure quadratic, the unknown quantity appearing only in the second pow^r. But if in another term it appear also -in the first power, as in the example ^^z±=6^-j-9±=25, the equation is called a mixed, or affected quadratic. Simple, quadratic, cubic, &c. equations are oiherwise called equations of the first, second, third &c. degree. • -
231. Now suppose the equation a- :^6 a; = 16 were pro- ppsed to us: it is plain, from what we have selsn in the liist section, that by only adding 9 to each side we shall have an equation reducible by the mere extraction of the square root. And that operation of adding 9 to each side is called completing the square ; for by that addition
we
( 118' )
we render one' side the complete square of the; binomial root Pi±i/,
232. Every mixed quadratic equation may be reduced by a similar process. Suppose we are given /v^ -}-3^= 18. Let us consider x^-^-Sx as the two first terms of the square of a binomial root, . whose first term is a\ Now I say that the other term of the root is 1, and that the square will be completed by adding, to both sides the square of I, or f. For 3a: is the double product of ^ and the other term of
the binomial root : therefore-^ is the simple product of
the two terms of the root ; that is, .r and ^ aa'e the terms of the binomial root. Accordingly .v* -f- 3.x -f fis the com- plete square of jr-fl. And adding^, also to the other side of tire equation, we have cT^ + SAr-f-f = IS-f f = V* And now extracting the square root of both sides, (which is always the operation to be employed after completing
the square) we have a-'-f~=: / — =1-, and therefore #?
.222'
233. But in reducing mixed: quadratic equations, we must often employ some other steps to prepare for com- pleting the square. And the steps previously necessary are sufficiently obvious, when we consider what object we propose ; namely to arrive at an equation of which one side shall be the complete square of -a binomial, whose first term is .r. At that side .v^ must stand in the first place, affirmative, and without any coefficient dilfereiit from unity. It may always be made affirmative, if ne- cessary, by changing the signs of all the terms in the equa- tion: (§ 220.) and, it may be divested of any coefficient different from unity by a division or multiplication of both sides. In the second place must stand at that side, with its proper sign, the term in which jt appears in its simple ])ower ; which term is the double product of ^ and the other part of the binomial root. Now when we have brought these two terms to one side, (which may ahvaj^s be done by transposition) and tlie remaining terms to the other side of the equation, we are prepared for completing the square. And it is completed by adding to both sides the square of half the coefficient of .v in tlio second term.
234j. Let
( 119 ) 234. Let us now exhibit all the necessary steps in another example. Let the proposed equation be 6 — :cz=Jl — ^_§/ .//i-d^
which appears simple, but will produce a quadratic ; and is. thus reduced.
X 5
^3
' 5 + 25
V + 5
6x-^x^=:l5'--
2x^
^x
Sx
-6a— —15
^*— 10a^=— 25
^^ — 10a^4.25 = — 25 + 25 = 0
^-^5 = 0
xz=5
Here the student will observe that after the second step, in which we have brought to one side all the terms invol- ving the unknown quantity, we then change the signs of all the terms on both sides, in order to make ^* affirmative, and place that term first in which x^ appears. As this term has the coefficient j-, we next throw off that coefficient by dividing both sides by | ; and are then ready for com- pleting the^ square. Now the second term being 10.r, the square is to be completed by adding such a number, that lOjr shall be twice the product of the terms of the binomial root. Whence it is plain that 5x is the product of those terms ; and x being one of them, 5 must be the other ; the square of which therefore, or 25, we add to both sides. Lastly, 10;^ or the double product of the parts being negative, the binomial root must be a; — 5, not x-^5» 235. The student should now make himself expert at the process of completing the square, by taking a variety of examples in which x in the second term shall be affected with various coefficients. Thus, let x^ -^xzz^l ; the square will be completed by adding ^ to both sides ; for the bino- mial root must be x + 4? the coefficient of x in the second
the binomial root must be
term being 1.
Let .V'— 4 = 14, 3
( 120 )
be A' — ^; and tlierefore the square will be completed by
addins: — to both sides. This ffives us x^ — __4- — =: H.
-f — = , and X = ^ / = — ; 23 bem": found bf
^36 36 6 V 36 6 "= ^
23 extraction to be the square root of 529. Hence a' = _^
5 28 + -=— r='^f* ^^^ since every positiye quantity has two
square roots, (§ 164.) one positive and the other negative,
y529 ^— ^ may be
23 23
either — or — — And the latter value will mve x=: 6 6 . fe
— f--= — — = — ^* And accordingly, if in the given
equation .r* — —- = 14 we substitute for a; either 4y or — 3, 3
we shall find the result 14. And hence in reducing qua- dratic equations, we may commanly arrive at two distinct values for the unknown quantity : of which more in the next chapter.
236. If after having prepared our quadratic equation for completing the square, we put a for the coefficient of x in the second term on the left hand, and b for the number on the right hand, it is plain that x'^z±zax zz2±zb will be a general expression for such an equation so prepared. And every affected quadratic equation may be brought to that form. To complete the square we add to both sides the
square of -^ ; whence we have the equation x'^z^ax-\-—-=i
'=^h-\-. — : whence by extracting the square root of both 4
slides we have ^:rfc:?=:=t:^^z±zZ>-f^ an4 therefore x •=.
'^^TV^^sJ =t:^-f-^. With this general expression for the
^4
value of the unknown quantity in an affected quadratic equation, the student ought to make himself very familiar.
And
fe
( 121 ) And in It let him observe the nature and ground of th^ ambiguous sign =p prefixed to -. If the binomial root in
the former step be .sr-f--, then upon transposing - it be- comes negative. But if the binomial root be x — _, then-
becomes affirmative in the assigned value of a*. Let him also observe that the ambiguity of the sign prefixed to the
surd ^^:±rZ>-|-~-, arises from the circumstance that the
square root of any number may be indifferently affirmative
or negative. (§164.) In that surd, the term — is always
affirmative, as it is the term which has been added to both sides for the purpose of completing the square. The term ' h is affirmative or negative, according to the sign it has annexed in the given equation x^z±zax-=.z±ib.
237. A biquadratic equation, or an equation of the 4th. degree, may be reduced just as a quadratic, if the unknown, quantity x appear only in the 4th. power or only in the 4th. and 2nd. powers of it. Thus if ^'"'• = 81, the value of x is found by two extractions of the square root : for ^^ = V81 = 9, and a; == V 9 = 3. And if -v^ — ^x"- = 36, let us substi- tute^ for ;v* in that equation, and it becomes y^ — %=
36, which is an affected quadratic^ and affords 3/=—
=t:y36 + !f=^db /i^ = £-!^=+9 or-~4. Now V ^ 4 2 V 4 2 2
having the value of y or ^^, a second extraction gives us the value of x^ since a'=:V'j/= v'9=:=±:3; For as to the expression V — 4 it is an impossible or imaginary quan- tity. (§ 164.) It is plain that any equation, in which the index of the unknown quantity in one term is half of its index in the other term, may in like manner be treated as an affected quadratic.
238. If we here employ the same general notation as in
§236. then y, or a'-, ==p^^:±:y=±=&+~; and there- fore Kzz dby^ |^=±=^y cfc ^ 4- ^. Now let us put c for the
Vi
alue
( 122 ) ralue of z±:h-\~ — : and the expression for the value of x in
4y
the supposed biquadratic equation becomes=±=:,^/fzt=V<:. But we have seen in § 218. that this surd is capable of being expressed more simply, whenever — — c is a square number.
239. Let the student exercise himself in reducing the following equations ; of which the first seven are examples of simple equations, but those at the end of § 240, of quadratics.
Ex. 1. 5x — 8 = 3^ + 20.
Ex. 2. Sx + — = ^ + i'. 3 2
Ex.3. -^4-12 = 7+12. 5x X
Ex.4. \/251+Ar^ — S=x.
Ex.5. 5x''—\2x — \1x-^^x\
Ex. 6. Vl2 + A=2-f Vx,
'la
Ex. 7. Va'+ Va-fAr:
V a-\'X
Here the letter a denotes any known number, and \\vg ob- ject of the reduction is to find the value of ^, which de- notes the unknown. The steps by which the reduction may be completed are — 1st. X V a-\'K (that is, multiply both sides by Vfl-j-^v, in order to remove the irrationality from the denominator) 2ndly. — a — x (that is, subtract a-\-x from both sides, or transpose, in order to have the ^urd alone at one side) 3rdly. square both sides ; 4-thly. — x^ (that is, subtract x^ from both sides, or expunge -f-^") : when we get the simple equation ax-=.a^ — 2ax^ or 3fl;Af =a*,
midx=:_. And this is 2i geneixd value for .v, whatever
number be substituted for a in the given equation. Thus,
if V.v-^\/3+a:==— ====== (where we have substituted 3 for
V 3 -}" a:
10
a) then A- = 1=1. If V x -in \^ 6 -\- x z=. -p==^ \ (where we
have substituted 5 for a^) then.v = 4=ly. And accord- inoly if we calculate the value of each side of the equation
hy
( 1^3 ) hy substituting | for .v, we. shall find the amount of each side to be Vl5, For then v^^ + V5 + A'=r^ --hw^
20 3
^^ • -^^'^ (§; 215.) = -^-^rrrVSX V5=:V15. And
V3 '^ ' a/3-
10-i-: /_=:—-_ = ,-- = — ^ = \/5 X V'S
\/5+x V 3 V20 2v^5 V.
z=:Vl5, Let the student exercise himself m similar cal- culations, substituting 7, 8, &c. for a.
240. Let us now change the numeral coefficient of a in the numerator of the fraction on the right side of the pro- posed equation ; and observe how tlie value of x will vary*
3a
Thus, let Vx+Va -\-a:=: — ■rz=z=, Ileilucino' this cqua-
Va-\-x ^ ^
A* ft —_ __ A*fi
tion, we find jc = — -. And \^ V x-{-V a->rx:=. — ==, then 5 Va-i-x'
Tj^^ find X = — We now perceive the law of variation in
the values of jr. For having successively employed 2, 3, and 4 as the coefficients of a in the numerator, we have found cr equalto the product of « multiplied successively by the fractions 4-, Y^ ''^^^ t > of which fractions the nume- rators are the squares of 2 — 1, 3 — 1,4 — 1 ; and the de- nominators are 2x2 — 1, 2x3—1, and 2 x 4 — 1. And
Ba
accordindy if V x -^V a'\-x=L—-^==^ we sjiall find x:=za ^ '' Va-\-x
^ 5__1 \ 16a ^ ^ ^ . ^ ,
X 1-=: . But we may at^once arrive at a general
■ fdrniula. for the value of x, by reducing^ the equation
\/ ^Ji^\/a^x^~ > '^^ which 7n the literal coefficient
V a-^-x denotes any number whatsoever, integral or fractional.
For we shall find x:=za+ — From this formula we
2m— 1
may easily calculate the value of x, whatever numbers be substituted for a and ma. Thus let V x + V 9 + x
12 ^ jji 1 1*
:, then m =z^-y m—\\ =|, 2m — lz=j,-
V9+^' ^' ' '' '' 2;;z— 1
{ 124 )
"-i i J
= 1^1=: tV, and^=:fljX.|^-J-=A = T- And accord-
jngly if j- be substituted for x in the preceding equation, the value of each side will be found to be Vl5- These observations might be pursued farther: but enough has been said to call the attention of the student to the advan- tages of employing Ut^^al equations, in which we designate hy letters known, as well as unknown, quantities.
Ex. 8. ^*-~40 = 2a:— 5. "'
Ex. 9. 7^—^^ + 5 = 1 1 4- 2t.
Ex.10. SA'^f 7 = !f+4r*— 21, 5
Ex. 11. ^rrV'jr + e.
■Ex,12. _.+ _=_^_.
Ex. 13. ^-!ff! + aa: + — = V«^* + 5ca;— 15. 2« 2a
Here after having prepared for completing the square by brinoriiior the equation to this form, ;v* 4- , ^^'^ ^ ^ -^^^
— ^ : we consider the second term (the fractional tri-
liomial in each part of which x appears) as the product of X multiplying — _I!I — ^ '^ ; the half of which therefore,
or ^ — IZ—iZL , is the other part of tjie binomial root, Ha"^ — 3wj ■
whose square we w^ant to complete. The square will there-
fore be completed by adding to bpth sides — r— j —
— / 30g 5ac — g^ — b
Sac + «^ -f-^ 1 4a^ — 3w
CHAP. XXIV.
On the Forms and. Boots of Qjiadratzc Equations, Method of exterminating the Second Term,
24^1. WE have seen (§ 236.) that all mixed quadratic equations may be brouglit to this form x''z±zaxz:zz±:b. But
of
( 125 )
©f these four varieties, w*+a^= — ^> may be disregarded, as it is manifest that the value of x^-}-ax cannot be negative unless when x h sl negative quantity ; and this form there- fore really coincides with x^ — axzz — b. We distino-uish therefore only three forms of quadratic equations ; the first x^-^-axi^b ', the second a* — ax = b; the third x^ — a.r =
242. The first of these forms when reduced gives ^=: ^=*^>\/^+~-— 2^ in which ^5+1- being necessarily a
greater quantity than j or ;^-7' the value + y^-ffl
- — is necessarily affirmative ; while the other value
— j^ h + T"~o i^ necessarily negative. Tlie second form
(when reduced gives x = ziz/b -f ^ + 2 : and here also the value of — ^^^ + ^-f- is necessarily negative, as the negative part of it exceeds the affirmative j while the other \ value +>w/^+— +~ is evidently affirmative.
243. But in the third form ^^^■^ax-=z—^j where reduc- ; tion gives us x=.z±Zf^ — b-\- h~> ho\h valiies oi x will
'- - a*
;■ be impossible, or imaginary, if b exceed — : for then the
' value of — 5 4"— will be negative, and its square root
■ *
— h-\ — , an impossible quantity. (§ 164.) If & = — then it is plain that the two values of ^ coincide, and be- come = ~ , since the expression =i= w^ — b -{ — -becomes = 0*
But if b be less than — , then both the values of a' mu5t be 4
affirmative, since the value of — b -{ is affirmative, and
4
in
( ^^6 )
in the expression — /w/— -^+ — +~ the iiQgatlve part is
less than the %affirmati ve.
244. By the root, or roofs, of ^ti' equation We mean the vahie, or vakies, of the imkno^'n quantity. Andwehavi* seen that every quadratic equation has two roots. For this we have hitherto accounted from the ambiguous sign of every square root, ©ut the same thing may be illus- trated from other principles. If we bring all the terms of on equation to one side by transposition, we shall have 0 at tlie other side. Thus, the quadratic equation of the first form, .^^-f 4-4^ = 21, may become, by tlie transposition of 21, x^' + ix — 21=0. And the roots of that equation ai'e therefore the numbers, which substituted for a give 0 as the value of the trinomial ^ ^ -f 4^* — 2 1 . Now if we mul- tiply any two binomials, each of which has x for the first term, and for their second terms quantities into which x does not enter, we shall have a trinomial product, whose first term is^*, the second term tlie product of x and the sum of the second terms of the binomial factors, and the third term the product of the second tcnns of the binomial factors. Thus the product of x-^a multiplied by x — b, or x^-\-aX'-^bx — ^b, ^may be considered as a trinomial, by considering ax — bx as one quantity; and we see that it is the product of x and a — b ; while the third term — ab is the product of a: and — b. If tlien we put ':±::-s for the sum of any two quantities denoted by a and b, and p for their product, then the product of x:-±za multiplied by x-=±ib, must be justly represented by x''z±zsxz±zp -j—the general formula for a quadratic in which all the terms are brought to one side, and therefore 0 standing on the other side.
245. Hence it appears that any such quadratic may be considered as generated by the multiplication of two such binomials xz±za and x:±ib. But their product v/ill become equal toO, if either of the factors be equal to 0 ; that is if A'^z^tt, or irz^n^. So that there must be two values of*' in the quadratic x'^z±zsx:±-p = Q, or two roots of that equa- tion. And we have seen that the eoeftieient of x in its second term is the sum of those roots v/ith their signs changed, and that its third term is their product. In lilce manner the quadratic equation x^ ^'4<x — 21=0 must have two roots, whose sum, when we change 'their signs, is 4-4
and
i
( 127 )
and tlieir product — ^2 1 . And accordingly reducing thai equation, we have ^ = dt:v'21 + 4' — 2^ ==t:5— -2 ; that is 4-3 or -—7 : and multiplying the binomial factors x — 3 and ^-f-7, their product is the given trinomial ^*-f 4;v'— 21, in wliich -f ^ the coefficient of the second term is the sum of — 3 and +7, and the third term — 21 is their product* If ;v'=-f3, then the binomial factor oc — 3 is equal to 0, and therefore the trinomial; as it must also \i x-=. — 7, and therefore a' -f- 7 = 0.
24?6. A quadratic of the first form, ^*-j-5^ — jp^ will be generated by the multiplication of the binomial fax:toi*s x-\-a and x — 5, if the sum oi ■\-a and — 5 be affirmative: that is if flj, the negative root, exceed ^, the affirmative. But if the sum of those roots be negative, that is, if a be less than Z>, the equation generated will be of .the second form ;v* — sx — jp. And thus also we see, that in the first and second forms one of the roots must be affirmative and the other negative. But it is plain that a quadratic of the third form, a* — sx-\-jp^ cannot be generated but by the niultiplication of such binomial factors as x — a and x — h ; for thus alone the product of the two roots will be affirma- tive and at the same time their sum negative. Hence also it is plain that in this form both the roots must be affirma- tive, when they are possible. We saw in § 243. that when
(in the equation A^* — 5^ +_p) jp exceeds — , both the roots
4*
are impossible ; and that appeared from the impossibility of the square root of a negative quantity. But the same thing also appears, and more satisfactorily, from the considera- tion that s is the sum of two numbers, whose product is ^. For it is impossible that the product of any two numbers should exceed the square of half their sum : which may be thus proved.
247. Let « and h represent any two numbers; then the square of their difference a — by will be represented by a^ — 2a5-[-6^ ; ^which must be positive in its value, whe- ther the value of a — h be positive or negative. (§ 164.) Therefore the negative part lah cannot exceed the affirma- tive a^ -\-h^ \ and adding 2ah to both, 4«^ cannot exceed a'--\'1ah-\-h'' ; that is, four times the product of a and h cannot exceed the square of their sum : and therefore their product cannot exceed the fourth part of the square of
their
( 128 )
ttieir sum, or the square of half their siim. — Otherwise, if we put a for the smaller of two numbers and d for their difference, a-\'d will represent the greater. But the pro- duct of a and a-\-d is a'' -{-ad: and their sum is 2a -{-di
Therefore half their sum is a+ _, the square of which is
d^ d^ d^
a*+«£?+ — * But a^-\'ad is less than fl^^-f-flrcZ-f _^ by — :
4 _ 4 4
that is, the product of any two numbers is less than the square of half their sum by the square of their difference. If the two numbers be equal, that is, if d vanishes, then the product becomes equal to the square of half their sum, or to the square of either number* But in no case can the former quantity exceed the latter.
248. The second term of any affected quadratic equa- tion may be exterminated, and the equation may therelbre be brought to the form of a pure quadratic, by substituting for A' in the given equation y 7ninus or phis half the co- efficient of ^ in the second term, according as the sign of that term is plus or minus. Thus suppose we be given the
affected quadratic A' ^ + «\v — J = 0» Let us substitute j/ — ?
a^ a^
fbr;^. Then ►v* =3/* — ay-^- — ; tmd axz=aTj — — . There*
4 2
ft * a^
fcre A?*-f fl:A-=j/* and^*-l-«^ — ^=3/" — — — ^ = 0. But
4 4
a^ a^ .
^' 6 = 0, or 3/*= — + b, is a pure quadratic, which
4 4
gives ^ = =t=>y/ -J- 4- ^» And since we supposed that x =7/
▼ 4
— -, it follows that x =r =tr W^ — -f 6 — - ; the very same va-
lues that we arrive at by completing the square. If our given equation be a'* — 3a — 5 = 0, then substituting 3/ + 1 for A*, we have J/* — V=0, or j/ = =t:V^/} and x ==±=Vy 4-1*
CHAP.
{ m )
CHAP. XXV.
Reduction of two or more Equations^ involving several unknowJi Quantities*
249. IF we have given two sdmple equations, involving two unknown quantities, for instance .x'-f-j/ = 7, and a' — ^ = 2|, we may derive from them an equation involving but one unknown quantity, the reduction of which will afford us its value, and thence the value of the other unknown quantity. There are always three methods, by which this may be efTected. For 1st. we may find from each of the equations an expression for the value of one of the un- known quantities, and then state the equality of those ex- pressions in a new equation, involving only the other un- known quantity. Thus, given the equations ;v -f-j/ = 7 and .V — -y = 2y, from the former equation we find x = 7 — -y ; and from the latter x=z2j -\-y. Therefore 7 — y = 2y -^y j which
equation reduced gives y = — -—I = 2y. Now by substitut-
ing 2-f for y in either of the given equations, we find the value of X, Thus since x -^y = 7, it follows that .v -f- 2 j = 7, which gives x-=z^ — 2y = 4|.
250. Or, 2ndly. finding from one of the given equations an expression for the value of one of the unknown quan- tities, we may substitute that expression in the other equa- tion for that unknown quantity, and so derive an equation involving only the other unknown quantity. Thus from the equation ;<-f ?/=7, we have x-=.1 — y. and substituting 7 — y for Pi in the equation x — y = 2y, we have 7 — y — ij (i. e. 7—23/) = 2-f . Therefore 2j/= 7— 2| = 4f , and y^2\i which affords us also the value of x as before.
251. Or, 3rdly. when we have the same unknown quan- tity appearing in one term of each equation, and affected w^ith the same coefficient, we may by subtracting one equa-^ tion from the other (if the signs of those terms be the same) or by adding the one to the other (if the signs be contrary) exterminate that unknown quantity, and derive an equation involving only the other. Thus in the equa- tions x-^y-zzl and iV-^j/=:2y, subtracting x — y from x-\-y
K the
( 130 )
the quantity x disappears, and the remainder 2y must be equal to the remainder obtained by subtracting 2|, the va- lue of X — y, from 7, the value of x-\-y\ that is, 2?/ = 4^. Or, adding x — y tp x-^-y, the quantity y will disappear, and the sum 2x miist be equal to the sum of 7 and 2\ y that is '2a^ = 94. *- ' -'^ --^-ir
252. With this method, as being the most generally ex- peditious and convenient, the student ought to make him- self very familiar, and expert at the preparatory opera- tions, which are often necessary. Thus, if we have the
equations 2X'\-2>yzzlB and 3.v — _:i= 12, and want to exter- minate X, we must prepare for the subtraction of one equa- tion from the other by giving x the same coefficient in both. This might evidently be done by dividing the former by 2, and the latter equation by 3 : for then the coefficient of x in both would be 1. Or it might be done by multiplying the former equation by 3, and the latter by 2 : for then the coefficient of x in both would be 6. But it may at once be
3 done by multiplying the former equation by -, or the latter
ji
2 by -. By the one process the coefficient of x in both wiH 3
be 3, since 2a'X| = 3^; and by the other process will be 2, since 3^ X T = 2'*'' Thus again, if the coefficient of .v in one of the given equations be 5 and in the other 7, mul- tiplying the former equation by -, or the latter equation
b
by -, will give the same coefficient in both. Or, if the
coefficient of x in one or both equations be fractional, it is equally easy to determine the number, which multiplying either equation will make the coefficient of x in it the same as in the other. For it resolves itself into this question —
what number multiplying ^ will give - for the product ?
The number required must be — - (or _-^_), since any
ad \ d 0/
pinDduct divided by either factor gives the other factor for
the quotient. Or thus, since any number multiplied by ita
reciprocal
( 131 )
reciprocal gives 1 for the product, i x - X ^ must equal
bad ^
rx'-, that is must equal-..
} , d . .^ . ,. j^,.,.- ;^ , d
253. If then we have given us the two equation
5f -f- J^=t?^^ and Jl^^-ri^^ to &d the values of iv and ?/: o a ' > ^ a ^p a ^
we are to remember that the first object is to derive from
the two given equations another equation involving onlv
one of the unknown quantities And this may be elfected
by the first method described in § 249. thus. From the
first of the given equations, we find that K-=zh* — ~Ji. (For
=zab—^ ; whence, multiplying both sides by -, xzzb^
— — ^. j And from the second of the given equations,
■v = 1 — ^—^ . ( For ^ Ei ~ — ^ ; whence, multiply in oj both b^ \ a a b x-.'o
sides by-, x — l- — ^ /.) Therefore equating the two va-
lue3 of Xy we have i>^ :^=1 -^,- which equation in^
volves only the tiffkhown quantity ?/, whose value is found
by reduction; namely 3/=^ 74^"^ ' ^"^ substituting
for ?/, in either of the given equations, this its value, wo have an, .^equation inyolying only the unknown quantity x^
which by reduction gives ^ = --ZI — _^. — Or, pursuing the
second, method described in § 250. we derive from either of the equations, suppose fi-pm the first of them, an expres-
b^ y sion fo the value of.*, Wamely x^b^ ^, and substi- tute this expression for t>c in the second of the given equa- tions ; whence we have the equation i_— -!^-f ^zt^, in-
a a^ b a volving only the unknown quantity ?/, and affording by re- duction the same value for y as before. — Or lastly, pursu- ing the^third method described in § 252. we may multiplv
K 2 botU
( 132 )
7 *
both sides of the first given equation by — , in order that X may be affected with the same coefficient in both ; when
we have — •{ — r ='-* • froi^i which subtracting the second a a^ a ^
of the given equations we have -r-^ — ^z=i— _. And
^ a^ baa
this reduced gives the same value of 3/ as before. For, mul- tiplying both sides by a^Z>, we have b'^y — a^y=.a^b^ — a^b^ :
whence, dividing by h^ — a^^ we have ?/ = . —
If in the given equations we put ^ for -, they will become
^^J^dyzna^d, and c?^ + -, = 6/ 5 and we shall have ;v = d d
d^-^a^'d'' , a^d^'-^d'' ^ .^ ^.c^ h ,
— ; , and v= — ; • ^r II we put d tdr -w and ©
^4___i ^ ^4_i ^ a ^
for dby we shall have /v= , — , and yzs.^JLZZ^^^ — After ' ^4_i -^ ^4_l
thus reducing the literal equations, let the student substi- tute any numbers whatsoever for a and b^ and prove the truth of the literal formulae for the values of yi and 3/, by calculating their numeral values according to them. It is plain that if a = 1 , the values of x and y must coincide, and
d^ become -~ . And if flf = ^, the value of x vanishes, or
'a*-|"l
is =0.
254. If three equations be given us, involving three un- known quantities, we may by rtiethods very similar suc- cesssively ascertain the value of each. Thus, if we be given the equations x + j/ = 5, and at + ;:;=: 7, and ?/ -f- 2; = 8 ; equating the expressions for x afforded by the two first of these equations, we have 5-^yz^l — z*, from which and the third of the given equations 3/-fs = 8, we find z-=.5y. and J/ = 3 : and therefore x = 2. Or, if we be given the equations A;-f?/-f-2;=:6, and ^-j-2y+32;= 10, and ?>x-\-^2y — ;2=12; subtracting the first of these equations from the second, we have j/-|-22:=:4; and subtracting the third of them from three times the first, we have 3/4-4^ = 6. But from the two equations 3/-]- 22r = 4 and 3/-}- 4:^ = 6, we find as before 2;=!, and 3/ = 2. And substituting these num- bers
( .1313 )
bers for y and z in any of the given equations, we find a; = 3. Thus the student will observe, that when we are given three equations involving three unknown quantities, we proceed to derive from them two equations involving only two unknown quantities : which we then reduce by the rules before laid down. And in like manner if we be given four equations involving four unknown quantities, we may always derive from them three equations involving only three unknown quantities, and so on.
255. But here let it be remarked, that the number of given independent equations niust always be equal to the number of unknown quantities, else the values of these cannot be ascertained. Thus, if we be given the equation
A? = 8 — -^y it is impossible from this alone to determine the
values of either x or ?/ : for we may suppose either of them to be any number "johatsoever^ and may then find by reduc- tion a numeral value for the other which shall answer the condition of the given equation. Or if, along with that equation, we be given the equation 2a? -{-J/ =16, it will not at all assist us in the discovery of x and y : for the latter equation may be derived from the former, by doubling both sides, and transposing y -, so that it affords us no information in addition to what: the first gave us. There- ^ fore we say that the given equations must be independent, /^j^^ ^ But neither ought their numoer to exceed that of the un- known quantities. For if we have given us, for instance, three equations involving only two unknown quantities, we have seen that the value of these quantities is absolutely determined and may be ascertained from any two of the given equations. The third equation therefore, if not de- ducible from the former, must be inconsistent with them. Thus, if we be given x-\-y=.^^ and a;— 3/ = 2, and 2>:
-J--^ = a i from the two former equations we find that
a: = 5, and3/ = 3. Therefore 2a; +1= 10-fU=llT; and if
in the third given equation a—\\\^ it is an equation de- ducible from the two former ; but if a be a number greater or less than 11^5 the condition expressed in that equation is inconsistent with the conditions expressed in the two former.
CHAP.
( 134 )
CHAP. XXVI.
The Application aj' Algebra to the Solution of Arithnetkal
Pwhlems,
256. WHEN ail arithmetical question is proposed, to be solved algebraically, the first thing to be done, after clearly understanding its terms, is to express the con- ditions of it in the symbolic language of Algebra. And here, in the first place, v/e represent the number or num- bers which we want to discover by some of the final letters of the alphabet ; and then we express in the form of an Algebraic equation w^hat ■ we are told in the question about each of these unknown numbers, (See § 148. and 149.) After we have thus accurately translated the proposed question into the language of Algebra, no more difficulty can remain to the student who is acquainted with the doctrine of the two last Chapters; since by merely re- ducing the given equations the value of the unknown quantities is discovered. — Thus, let it be required to find such a number, that multiplying it by 3, and dividing it by 3, the former product shall exceed the latter quotient by 3 : or in other words) to find a number, whose third partis less than three times the numbei* by 3. Let us put
»: for the number sought. Then -- expresses its third part j
and 2)X expresses three times the number. Now we are
told that ~ is less than 3.v by 3, which is to be expressed by
an equation. But the equation 1 -f 3 = 3^, or tlie equation
3a' — 3 = -, or the equation Sx — - = 3, accurately expresses-
what we want j for the first expresses that adding 3 to. -- the sum is equal to Sx ^ the second expresses that subtract- ing 3 from 3a? leaves a remainder equal to * ; and the third
3
expresses that subtracting ~ from ox leaves a remainder
equal
( 135 )
equal to 3 : all which are propositions equivalent with each other, and with the conditions of the question. It now only remains to reduce any of these equations, according to the rules already given. Thus, from the equation 3x
— - = 3, dividing both sides by 3 — y, or by -f- (that is, mul-
tiplying both sides by 4-) we find Ar = 3 X-| = |; that is, we discover th.at the number required is |. And accordingly three times f , or W exceeds the third part of |, or |, by Yj that is by 3. [If we propose a question perfectly similar to the last, only substituting the number 4 for the number 3, we sliall find the answer to be 44* And we may obtain a general formula for the answer to all suqh. ques- tions, by putting a for any number whatsoever, and in- quiring what number is that, which multiplied by cr, and divided by «, gives the former product exceeding the latter quotient by a ? For then by the terms of the question ax
= a : whence we have xzz — . I
a ' a*— 1 J
257. Again, if it be required to find two numbers, whose difference is 5, and the third part of their sum is 7 : we may put x for the greater of the numbers sought and 3/ for the less. Then the equation x — 7/=z5 expresses what we are told of their difference, that it is 5 ; and the equation
•■ -^ = 7 expresses what we are told of their sum, that its
third part is 7. And here let it be recollected that, where there are two unknown numbers, there must be two equa- tions afforded us by the terms of the question, in order to ascertain them. (See § 255.) Now reducing the equations
X — y=-o, and "J"-^ = 7, by any of the three methods de- scribed in the last chapter, we may find the numbers sought. Thus, from the second of those equations we may infer that ;v-f 2/ = 21 ; and from this equation subtracting the first, we fin4 that 22/= 16, and therefore j/ = 8 ; or adding to it the first equation, we find that 2^ = 26, and therefore ^=13. So that the numbers sought are 1 3 and 8 : and accordingly their difference is 5, and the third part of their sum is 7.
258. In like manner, if we be required to find two such numbers, that two thirds of their sum shall be equal to six times their difference, and two thirds of their product shall
be
( 136 )
be equal to six times the quotient of the greater divided by the less : putting x for the greater and y for the less, their
sum is x^y^ and two thirds of this sum is — ZllZ. Their
3
difference is x — -j/, and six times this difference is 6*- — 6y :
and by the terms of the question — IL^ — Qx — 6y, Again,
O xy their product is a-j/, and two thirds of it is tl-il : the quo-
3
X
tient of the greater divided by the less is -, and six times this quotient is ~^ : and by the terms of the question -1^
==~ Nothing now remains but to reduce the two equa-
^ o
tions "^"^"^ -^=6^ — 6v, and:l^=r_!; from the latter of
? . . ^ -^
which, dividing both sides by a?, (see § 227.) we infer that
-i?=- ; and thence that 2^ = 18, and 3/* = 9, and j/ = 3. Then substituting 3 for j/ in the first of the given equations
Qx -i- 6
it will stand , "^ = Gat — 18 ; whence 2x + 6 = 1 8x — 54,
and 16^ = 60, and xz^^=y. So that we find the greater of the numbers sought to be 5 J, and the less to be 3. And accordingly two thirds of their sum, 6 J:, is equal to 6 times their diflerence, |ths. for -^rds of V is V, and 6 times |ths. is V : and two thirds of their product, V> is equal to six times the quotient of the greater divided by the less, :^ths. for ^rds. of V is V°j -'ind 6 times Jths. isV°« [Although
the second of the given equations, -—■= — > seem at first to
3 y
involve two unknown numbers x and y, yet from the dis- appearance of X, we may infer that y alone is really con-, cerned in it, and is equal to 3. And accordingly, i^ any number whatsoever be multiplied by 3, and divided by 3, two tliirds of the product must be equal to 6 times the quotient j suice two thirds of 3 times a is 2a, and 2a
_6a 1 "" 3 J
259. But
( 137 )
259. But it often happens, that a question apparently involving two unknown quantities may be treated most ad- vantageously, as if it involved only one. For after de- signating one of them by one of the final letters of the al- phabet, we may , express the other by the aid of this letter and some given number. Thus, if it be required to find two numbers such, that their difference is 7, and their ratio that of 5 to 3 5 putting x for the less, it is plain that ^-|- 7 is a just expression for the greater; so that we need not introduce another letter to designate the greater. And now, having derived this expression for the greater, A'-f 7, from one of the things told us in the question about the two numbers, we proceed to express algebraically the other circumstance told us, namely that the ratio of the greater to the less is that of 5 to 3. But this is expressed thus (§ 70.) x-^1 : A— : 5 : 3. But from this analogy (§ 76.) we may derive the equation, 3.v-f21 =5^* ; w^hence we have 2xz=.2\^ and therefore x-=. V ; and x-{-l (or the greater of the two numbers sought) = V -f-7 = V. Accordingly, the difference between V and V is 7 ; and their ratio (§ 93.) is that of 35 : 21, or 5 : 3. Which of the two unknown quantities we shall employ the letter originally to designate, is often indifferent : but in general it is more convenient to employ it for designating the smaller of the two, and thence to derive an expression tor the greater.
260. In like manner, if it be required what tw^o numbers they are, whose difference is 5, and the difference of their squares 45: putting x for the less, x-\'5 expresses the greater; whose square is ;v*-f IOa'4-25. Now we are told that the difference between this and x^ (the square of the less) is 45. That is, 10a' + 25 = 45; whence we have 10a; = 20, and A- = 2; and therefore A'-f 5 = 7. (See also the 5th of the Questions for Exercise.) Accordingly, the dif- ference of 7 and 2 is 5 ; and the difference of their squares (49 — 4) is 45. If we investigate a general solution for all such questions as the last, by putting a for the given dif- ference of the numbers, and 6 for the given difference of their squares, then x designating the less of the two num- bers, x-\-a expresses the greater; from whose square, a* -i-2aA:4-a*, subtracting A-^ we have 2ax-\'a^ ■=zh: whence
2ax^h — a* ; and a'=-ZI — . [As long as x and a are any 2a
positive numbers, it is plain that b must exceed a*, else the
value
( 138 )
value of —^ — would be negative: that k, it appears tliat
tlic difference of the squares of any two numbers must ex- ceed the square of their diffference, And from the equa- tion 2axz=:b — a', it appears that the difference of the squares of any two numbers exceeds the square of their difference by twice the product of the less and difference. — *rhe Geometrical Student may with advantage compare many such Algebraic results with the principles in the se- cond book of Euclid's Elements.]
261. If it be required to find two numbers whose sinn is 10, and the diffh'ence of their squares 40 : putting x for the less, we may express the greater by 10 — Xy according to the first of the given conditions ; and then the second con- dition is expressed by the equation, 10 — x\ — ;v'^ = 40, that is, 100 — 20^^ = 40: whence we have 20.v = 60, and ^ = 3. Therefore the greater, or 10 — x\ is 7. Or putting a for the given sum, and b for the given difference of the squares, we have a^ — 2ax z=ih^ and thence ^ax-=za^ — 6, and x
= — H— . And from this literal notation we are furnished
la
with the general theorem, that the square of the sum of any two numbers exceeds the difference of their squares by twice the product of the smaller and the sum. Or if we put X for the greater and a — x for the Jess, then we have
2ax — a^ = b ; and thence ^ax = «* -|- ^, and x = — ZL. ;
2a
which equations afford the general theorem, that twice the
product of the greater of any two numbers and their sum
is equal to the square of their sum plus the difference of
their squares. Thus let the numbers be 8 and 5 ; their sum
is 13, its square, is 169 ^ the difference of the squares of 8
and 5 is 39 5 and 8z=i5!±52, and 5=i~Z±?. ButJtho'
26 26
we have given these methods of solution, there is a much
better and readier solution of this problem, by dividing the
difference of the squares by the sum of the numbers : for
the quotient is the difference of the numbers. § 163.
262. Let us now investigate what two numbers they are, whose sum is 12, and their product 33|. Putting x for either of the numbers, the other is represented by 12 — x ; and therefore their product is 12 — xXx^ or 12.v — x^y whose
amount
( 139 )
amount we are told is 33|. Therefore 12^- — ^-— 3S|, or x^ — 1 '2x = — 33 J. This is a quadratic equation of the third ibrm; which induced, by the rules given in the 231st. and
following sections, gives A* = rt:V — 33^+36 + 6 = =!=;^-
-f-6 = 6=fc:lf, that is, 7t or 4t : which are the numbers that t^olve the problem ; for their sum is 12, and their product ^— , that is 33Ji And universally putting s for the sum of tv/o numbers, p for their product, and x for either of them > the other is expressed by 5— vV, their product by sx — x^ ; and tlie equation sx — a'^=^, when reduced, gives x =
— — T + -• (For then a* — s>: ~ — p j and completing
the square, /v* — sx-\ = — — p; and extracting the root
of each side, x — -- z=:z±Zj^/ — — ■^; ; whence x-=:.z±i^ — — p 4-|-. j We have seen (§ 243.) that both these values will
be impossible if p exceed — - ; and accordingly it is im-
possible that there should be any two numbers, whose pro- duct exceeds the square of half their sum. (See § 247.) If
^•=. — , the numbers sought are equal, and each of them
half the given sum. The student may exercise himself in observing the varieties in the solution of this problem, w^hen the given sum, or product, is riegativey or both of them.
263. But w« have now to remark that the same problem ma-y be solved, without the introduction of a quadratic equation. For, if we subtract 4 times the given product , from the square of the given sum, the remainder must be equal to the square of the difference between the numbers sought. (For let a and b stand for any two numbers, the square of « + &, their sum, is a^ ■\'2ah-\-b^ ; and subtract- ing from this ^ah, or 4 times their product, the remainder is a^ — 2ab-{-h^, But this is the square of a — Z>, the dif- ference of the numbers. See § 163.) Hence therefore in the proposed problem we know, that 12- — 4 X 33^ is equal to the square of the difference between the numbers sought ;
that
( 1*0 )
tliat is, that the square of their clifFerence is 144—135, or 9; and therefore their difference is \/9, or 3. So that the problem resolves itself into that of finding two num- bei*s, whose sum is 12, and their difference 3. Or gene- rally, putting s for the given sum, p for the given product, and d for the difference between the numbers ; s^ — 4fpz=.d^, and therefore =1:1 v^.s'^ — ^2)z=.d, But given 5 j the sum of two numbers, and d tlieir difference, the greater %
s d f / S ^ e
of the numbers =~H — = --f^/ — — », and the lessr:-
d_s / '2~2~'V
— — J) ; the same expressions which we ar- rived at (§ 262.) by the reduction of the quadratic.
264. Let it now be required to find two numbers, whose difference is 4|, and their p7-oduct 25^. Putting x for the less, ^4-4y will express the greater: and their product
1 3.V therefore is Af-f-4|XA-, or x^ -] — -1. But by the terms of
the question A-^-f -7—= 25|^. Now the reduction of this
quadratic of the first form w^ill give us the value of ^, and therefore of -y + '^T' Thus: — completing the square by adding the square of V to both sides, we have.
, , 13^ , 169 „.y , 169 1089 121 3 36 ''36 36 4
rrr, r 13 /^^^ ^ ^
Therefore .v-f =>>/ -^--^
, 11 13 33 IS 20 _,
and xz= ^_ = — — — = — = 34
2 6 6 6 6^
And therefore ^4.4|=7|. So that the numbei*s sought are 7-f and 3^: whose difference accordingly is 4|, and their product 25|. If we adopt the negative value for the square root of ~, the resulting numbers will be the same, but negative. But though w^e have exhibited the most ob- vious solution of this question, as producing a quadratic equation, yet it appears from the observations in the last section, that it may be more expeditiously and elegantly solved, by adding 4 times the given product to the square of the given difference : which affords us the square of the sum, and therefore the sum. Thus, the square of the
given
• ( 141 )
given diiFefence is L£5 ; and 4 times the given product is
^. Therefore ^'-\-'i^, or ^p, is the square of the sum ; whose square root therefore, or V , is the sum of the numbers sought. Therefore V + V» or 7y, is the greater of the numbers 5 and V — y, or 3f, is the less. Uni- versally, let d be the given dilference, p the given pro- duct : then d^ + ^p is the square of the sum, and therefore =±z \^ d"" 4- ^p is the sum- Hence the numbers sought are
x/d'-\-'^p-{-d Vd\-ir±P^
2 ' 2 ' *
265, Let us now inquire, what two numbers they are
whose sum is lOj-, and the sum of their squares 61^. If we
21 put ^ for either number, the other must =1- — .v, whose
44'1 square is — - — 2lx-{'X^ : to which adding «*, we have the
sum of the squares 2x^ — 2lx-\- = 61:^= . This af-
fected quadratic the student may proceed to reduce ; and
he will find .=£1^ /^^ + !ii = ^-: /!! = ^
7 28 14
=±=~. So that the numbers sought are — and '■ — ; that is, 4 4 4"
7 and 3f . But this problem also we may solve with more simplicity and elegance, by proceeding at once to investi- gate the differerice of the numbers. Now, if we subtract the given sum of their squares from the square of their giv^en sum, the remainder must be twice the product of the num- bers : (since a-{'b\' — a^ -{-b" •=z2ab) and we have seen that subtracting 4 times the product (or twice this remainder) gives the square of the difference. Thus in the present in-
^ 441 245 196 ,n - ^ - ^i. i ^ c .\
stance, — = ., or 49, is twice the product or the
4 4 4 ^
numbers; and therefore -^ — — 1^ — f = — ) is the square of
their difference ; which difference therefore is --. The half
of this added to half the given sum affords us the greater, and subtracted from half the given sum affords the less. See remarks on the 7th. of the questions for exercise. [And
universally
( 1*2 )
universally let a be tlie given sum of the numbers, h the given sum of their squares: then a~ — h is twice the pro- duct of the numbers ; and therefore a^ — 2a ^ — 26, or 26 — cr*, is the square of the difference. So that the greater of the a
numbers is ^-f- ^26 — a% and the loss is -■ — ^26 — a'^.
And it appears that the square of the sum of any two num- bers cannot exceed twice the sum of the squares; (else V26 — a"- would bean hnpossible quantity. § 164.) and that if tiiese two quantities be equal, the numbers must be equal ; for then V^Imi^ ::rO.]
*2Q6, In like maimer, given the difference of two number^ 3, and the sum of their sq2ia7^es 29, we may proceed to in- vestigate the sum of the numbers, instead of solving the question by a quadratic equation. For subtracting the square of the given difference from the given sum of the squares, we have twice the product of the numbers : which added to the sum of the squares gives us the square of the
sum: since c'-+6* -{-2flr6 = fi-f 6|\ Thus 29 — 3^=20;
and 29 + 20 = 49, the square of the sum; which sum is
7 4-3 therefore V49, c5r 7: and the numbers sought —7— and
, that is, 5 and 2. [And universally putting a for the
given difference of the numbers, and h for the given sum of their squares ; h — a * is twice their product ; and there- fore h-\'b — a*, or 26 — c^, is the square of their sum; and V'26 — a^ is their sum. Whence the numbers sought are
and v-2^-"'-'^.-] 2 J
2
267. Let it now be required to find two numbers, whose "product is 24 and the sum of their squares 73. Adding twice the product to the sum of the squares, we have the square of the sum ; which is therefore 73-f48 = 121. And subtracting twice the product from the square of the sum, we have the square of the difference ; which is therefore 73 — 48 = 25. Whence we have the sum 11, and the dif- ference 5 : so that the numbers are — i- and — ^^, or 8
2 2
and 3. [And universally, putting a for the given product,
and
( 1*3 )
and b for the given sum of the squares -, h-\-2ah the square of the sum, and b — 2a the square of the diiference.
Whence the numbers are ~- , and
; \^b-\-2a^Vb—2a^ ^^^ j^ appears that twice the pro- duct of any two numbers cannot exceed the sum of their squares ; and cannot be equal to it except wlien the nujn- bers themselves are equal.]
268. But though we have given this solution, as the most facile and scientific, the student ought to exercise himself in the other ntethod of solving the question by a quadratic equation. Thus, putting x for either of the
24 numbers, the other will be represented by ^, The sum
576 of their squares therefore is *•* + _—, which by the term?
of the question is equal to 73. Let us now reduce the equation
.- + ^ = 73 ^* Therefore ^^4^576-73;^* (See § 237,)
and ^^^—7 3 A- ^=--576
7S Therefore ^c^— 73^;^ + --
* ^ 73
= —576 + —
2
, . ^ ^„ , , 5329 -^^ , 5329 3025
that IS a;4 — 73^*4--^— -=—576 + -——=,
4 4 4
rvx. V a '73 . /3025 .Sc>
1 hereiore x * = =t:^ / = =±=--
2 V 4 2 ^
and ^^=Z£=fci^ = 64or 9 2 2
Therefore xz=. V64 or V9 j that is, 8 or 3.
269. But let US now in like manner solve the general problem — To find two numbers whose product is a., and the sum of their squares b. Putting x for either of them,
the other is expressed by -. Then
( 144 }
4 4
2 V 4
o V 4
v,*y^.-
Now in § 267. we found the general expression for x to be,
flr = — — [ . 13 ut It appears irom § 217. and
218. that tlie two expressions are equivalent. For in the
binomial -r±=^^/ — — a*, the square of the rational part
minus square of the irrational is equal to «^, whose root is a. Therefore, as we have shewn in § 218. the square
root of that binomial is equal to ^^ - '^ ztz/^V ^—^^
See also § 238.]
270. If it be required to find two munbers, whose jt?rc>-
duct is 8 J, and the diffh'ence of their squares 6 : putting a-
for the greater, the less will be expressed by S-^-^-Xy that is
, 35 , . 1225 r^. f , 1225 ^
by — , whose square is . ihereiore a* — :; — r=6;
^ 4;v' ^ 16^=' 16>*
J . 1225 ^ ^ 1 ^ ^ , 1225 TT
and x'^ =6x^1 and x^ — 6x^zz~ . Hence com-
16 16
1995 1369 pleting the square, x"^ — 6 a; * + 9 = -^ + 9 =r ; and ex- tracting the root, x * — 3 = / = — . Therefore x^zz
37 , ^ 49 , /49 7 , 35 35 ^^ ^.^
— -f3= — ; and^rr / — = -; and — = — = 2t, So 4 4 V 4 2 4a: 14
that the numbers sought are 3i and 2\. [But let us now
pursue the same investigation generally, putting a for the
given
C U5 ) given product, b for the given difference of the squares, x for the greater of the numbers sought, and therefore - for the less. Then
X
Therefore x^ — a^=zbx* and x'^—-bx^=d''
Therefore x^^-bx^ +tlz=z^^a^
4, 4j
and' ^»— ^rrdtV^H-rt* therefore x^=z -dtz. / — -j- a *
and
^=^^1-^1:+"^
But this expression for the value of x cannot be simplified as in the last problem ; for if we attempt it, we shall be involved in the impossible quantity V — a*.]
271. If it be required to find two numbers, whose sum or differ eiice is «, and the sitm or difference of their square roots is b ; putting x and y for the square roots of the num- bers sought, the numbers will be represented by ** and j/*. So that the problem resolves itself into that of finding tvv'o numbers, whose sum or difference is ^, and the sum or difference of their squares is a : the solution of which we have seen in ^260. 261. 265, 266. If we be given the product of two numbers equal to c, and the product of their square roots equal b^ the conditions are insufficient to ascertain the numbers, since the product of their square roots must be equal to the square root of their product, (§ 201.) and the two conditions given are therefore not inde- pendent. See § 255. If we be given the sum or differejice of two numbers =a, and the "product of their square roots z=b; this resolves itself into the problem of finding two numbers whose sum or difference is given, and their pro- duct: (see § 262. 263. 264.) since the given product of their square roots is the square root of their product, and its square is therefore the product of the numbers sought.
L If
(146)
If we be given tlie proditct of two numbers, and the sum or difference of their sgiiare roots ; tbe square root of the for- mer being the product of their square roots, this also is the same thing as if we were given the sum or difference of two numbers and their product, to fmd the numbers. And in Hke manner we may find two numbers whose sum or difference is given and the product of their squares,
272; Hitherto we have exemphfied th^ apphcation of Algebra to questions purely numerical, and in which the Algebraic expression of their conditions is very obvious. But when more than abstract numbers are concerned in the problem, the translation of it into the language of Al- gebra will often exercise the ingenuity of the student. For instance let the following question be proposed — A gentle- man, mounting his horse, was asked by a schoolmaster, what o'clock it was ? He replied, / 7nust be at a friend's house in the country against 5 o'clock : 7iow if I ride at the rate of 1 0 miles an hour, I shall have 5 minutes to spare ; hut if at the rate of 9 miles an houry I shall he 8 minutes too late. What was the hour ? Here we are told in fact, that the time it would take to ride a certain distance at the rate of 10 miles an hour is 13 minutes less, than the time it would take to ride the same distance at the rate of 9 miles an hour : but we are not told either of these times, nor the distance. Let us put x for the distance or num-* ber of miles, which the man had to ride. Now will not
— and 1- be just expressions for the times, in which he 10 9 ** ^
would ride that distance at the rate of 10 and of 9 miles
^n hour ? (Thus, if a man has to ride 50 miles, and ride
at the rate of 10 miles an hour, he will ride it in 5 hours,
or ~. If he ride at the rate of 8 miles an hour, he will 10
ride it in — of an hour, that is, in 6 J hours.) Biit we
8
are told that the former time, — , is less than the latter,
►V . ] 3
1, by 13 minutes^ that is, by — of an hour : which is ex-
9» ^ . ' * 60
.V X 13
pressed by this eq 11 ationc ^ — = — . And this reduced
* -^ •* 9 10 bO
give?
( 1*7 )
13 117
gives us *' = — X 90 = -— -, or 19|. So that the distance he
had to ride was 194 miles: and—, or the time in which
. he would ride this distance at the rate of 10 miles an hour, is '-i^^ of an hour, that is 117 minutes, or an hour and 57 minutes. It must therefore have wanted 2 minutes of 3 o'clock when he was setting out, since we are told that at this rate of riding he would arrive at his destination 5 mi- nutes before 5 o'clock. And accordingly riding 1 91 mile^ at 9 miles an hour, it would take him 4^ of an hour, or two hours and 10 minutes : and setting off 2 minutes be- fore 3 o'clock he would not arrive at his destination till 8 minutes past 5. Or, we may put x for the interval (in mi- nutes) between the time of his setting out and 5 o'clock : then X — 5 and a' -f 8 are the times he would take to ride the same distance at the rate of 10 and of 9 miles an hour ; which times must be as 9:10. Therefore 10;v — 50=:9a;' -f-72. And thus we find more directly that it wanted I2S? minutes, or two hours and two minutes, of 3 o'clock, when he set out.
273. From the solution of this question the student may observe, that a problem apparently very complicated, and at first view seeming to present inextricable difhcuities, may yet admit the shortest and most easy solution from the fa- cilities afforded us by Algebraic notation. But let him also observe the care and attention requisite in forming Alge- braic expressions for the quantities concerned in the pro- blem, and stating the equations which its conditions afford. The slightest error here must affect all our subsequent ope- rations, and lead us astray. Thus, for instance, we wanted to express by an equation, that the time represented by
.— is less by 13 minutes than the time represented by -. If
X X
we attempted to do that by the equation ^z=. 13, we
. should be involved in a completely false result. For y-.
and - express time in the denomination of hours ; while 1 3
expresses time in the denomination oi'mintites. This error
L 2 therefore
( 148 )
therefore must be avoided by bringing them both to the same denomination ; that is, by expressing 1 3 minutes as
a fractional part of an hour, ^, or by bringing — and ~
to minutes, that is, multiplying them both by 60. For the
equation — =r 1 3 expresses the same thmg as ^
-^ — = — — A^irain let it be observed that, instead of pro- 10 60 ^ ^
ceeding immediately to investigate the thing which the pro- blem requires us to find, it is often necessary, and oftener convenient, to pi'oceed to the investigation of some other quantity, upon which the determination of that thing de- pends. Thus we were required to find what o'clock it was, when the man was setting out : but from the nature of the question it appears that this must depend upon the distance he had to ride ; which distance therefore (in the first me- thod) we proceed to investigate. — Lastly let it be observed, that the utmost precision is necessary at the commencement in fixing the import of the lettei*s x, 3/, &;c. or determining for what quantities they are designed to stand : and that this must be distinctly recollected at the conclusion, when v/c have reduced our equation It is therefore expedient, that the young Algebraist should for some time mark in writing the designed import of each letter, and of each Algebraic expression, which he employs. The following question will exemplify the importance of this rule.
274. A man, being askdd his age, replied — Ten years ago I Xi:>as eight times as old as my so?i : and if we both live, till he be twice as old as he is now, I shall then be twice as old as he. What are their present age's ? Here are six dif- ferent quantities, any one of which we might proceed to investigate, and for each of which we ought to have ex- pressions : — the present ages of the father and son j their ages ten years ago ; and their ages hereafter, when the son shall be twice as old as he is now. But all these are so connected by the terms of the question, that the determi- nation of any one of them will detetmine all the rest. Now the first thing we should do is — not to look for an equation prematurely — but to fix on expressions for all the quan- tities concerned in the question. Thus — let
( 149 ) .
^ arson's age 10 years ago. Then a; -f 1 0 = his present age. and 2;v-f-2()=his age when twice aa old as now. But 8^ = father's age 10 years ago. Therefore 8>?-f- 10=father's present age. [as his son.
and 4;r-f-40=his age when he shall be twice as old
The last of these expressions l^as been formed by doubling the third of them. But we may have another expression for the father's age at that time, by adding to his present age the same number of years, which we added to the son's present age for expressing his age at that time : as it is plain that the father and son must be older then than they are now by the same number of years. Now wc doubleq the son's present age ; that 13, we added ^-|- 10 to his pre- sent age. So that adding A'-flO to the father's present age 8a;-|-10, we have 9a'4-20 for another just expression of his future age, when he shall be twice as old as his 3on. Now equating these two expressions for the same age, 4a: -f- 4-0 = 9;^' -f- 20, and reducing the equation, we have ^==4. That is, ten }^ears ago the son was 4 years old, and the father 32: and therefore the son is now 14, and the . father 42. Accordingly, when the son shall be 28, the fa- ther will be 56, or twice as old as his son. — We shall now propose various other examples of Arithmetical Problems, and exhibit their Algebraic solution.
275. What fraction is that, isohich "isoill become equal to 1 1)1) adding 3 to the numerator, but equal to i by adding 3 to the denominator ? Putting x for the nun^erator, and y for
the denominator of the fraction, we are told that -3-1 = 1,
y
and that — r-r = — From the first of these equations y^i^ 4 ^
.v4-3=rj/, and ^ ==^ — 3. From the second, >y='^ "^ -.,
Therefore 3/— 3 ^^il^ ; and 4j/— 12 =3/4. 3 j and 3j^= 15 ; 4
and J/ =5. Therefore x {or y — 3) = 5 — 3 = 2. And the fraction required is f. Accordingly adding 3 to the nu- merator, it becomes f or 1 ; and adding 3 to the denomi- ^'' nator, it becomes 4 or |. [If we now generauize thfs pro- I blem, thus — To Jind two 7iwnberSy x a}id y, such that adding
a
( 150 )
a ^d X tlie quotient of x-^a, divided by y shall he m ,- hut adding a to y the quotie?it of :x. divided by y-\-SLshaU be
i?? then we have the equations n
A '*" —-^^
y+a n
From the first of which x = 77?^ — a and from the second x zz , -^ ■ —
Ihereiore jwt/ — a;=— :i-J
and mny—an = ?»i/ + w^<z
and mny — my = wa + an
J ma 4- an
and yrr— — ^
7«/2 — 7n
tr>^ r / \ 7na-\-an ma-\-a rp, f»
Therefore .v (or my — a)=^ — I — — fzr: ! — . Ihefra^-
n — 1 n — 1
tion required therefore is — J— h — — ; which reduced
n — 1 mn — m
to its lowest terms (multiplying both dividend and divisor
by w — 1, and dividing them both by a) becomes m-\-l
m 4- 11 m^ 4- m . j xi • • ^ • c ^\
H ■ — = -^—i — . And this IS a general expression lor the
m m-\-n
value of -. And accordingly the two equations — "r^^^-r
y ^^ ^ m-iru
zzmy and — — =— , give the same value for a : namely
m-^-ii-^-a 71 *
a = mn — m* Thus assuming for m and n any numbers what- soever, suppose 7 and 5, the fraction — III—, or — , is such ^^ 74-5 12
that adding 7x4 (or 28) to the numerator, it becomes €qual to 7 ; but adding 28 to the denominator, it becomes equal to f.]
276. A merchanfs property consists of goods ^ hills^ and cash. The value of his goods is equal to the amoimt of Ms bills ajid cash together : the amount of his cash is equal to twice the amount of his bills and half his goods together : and
if
( 151 ) -^^^dhlSSS^
if he had. not lost the third part of his goods hy a fire^ the amount of his property "would have been j^ 12,000. What is the value of his goods y of his bills, and of his cash ? — Putting A? for his bills and y for his cash, ^+3/ will express the present amount of his goods, according to the first of the conditions j and by the second of the conditions
j/ = 2^ 4- ■•^« But X -f- ?/, the present amount of his goods, being 4 of their former amount, (as we are told that he had
lost y of his goods by fire) x -\-y X -1-, or — ^ — , will ex- press their former amount, or what would be their amount were it not for the loss by fire. By the third of the condi-^ .;, , 3 a: 4- 3?/
^ tions therefore ^4-?/+ — -L-i^= 12000. So that we have :... 2
now two equations given us for finding ^ and y. By the first of them y=:5x ; and this value substituted for y in the second, gives us 15^=12000. Therefore ■** = 800, the amount of his bills, and 3/, or 5x, =4000, the amount of his cash; and .v +3/ = 4800, the present amount of his goods. But he had lost by fire 2400 : and were it not for this, his property would have been 800 + 4000 + 4800 + 2400, that is ^12,000.
277. A person buying a set oj books was asked 45. a vo-^ Iwne : but finding that he had not enough of money by 3^. to pay for them at that price, he cheapened them to 35. Sd, a volume ; afid after paying for them found he had 6s. 4^d, left. How many volumes were there ? — Putting x for the number of volumes, we are furnished with two diiferent expressions for the money, which he had. For at 4^. a vo- lume, the cost of the books would be 4^, and his money therefore w^as 4x' — 3. But at 3^. M, a volume, the cost of
1 \x the books is 3y X ^, or ; and another expression there- fore for his money is + 6y. (See § 273.) Therefore
3
equating these two expressions for his money, we have 4.V-- 3=ii^ + 6|: whence 12a--9 = 11;v+19 ; and ;^ = 28.
Accordingly 28 volumes at 45. would cost 1125. and his money was 35. less than this sum, that is 1095. Now the cost of the 28 volumes at 35. M. a volume was 1025. 8<^.
and
( 152 )
and nfter paying for them at this price, he had left 6s. ^d, •^ — But we may more expeditiously arrive at the value of *', by observing that we are given" S5. + 65. 4^. or 9-js, as the difference of the cost of x volumes at 45. and at 35. 8^. a volume : the difierencc of which prices is 4^. or \s. per vo- lume; and therefore the whole difference of cost is justly
expressed by | X a*, or ^. Therefore - = 9y j and x =z
28, as before.
278. A grocer^ having txo kinds of tea, which stand /lim ifi 85. and 7s. per lb. desires to mix them so, that the com- pound may stand him in Is. 2d. per lb. In ivhat pi'oportion must they be mixed P Put x tor the num.ber of ^^5. of the dearer tea in the compound, and y for the number of lbs. of the cheaper ; then x 4-3/ expresses the number of lbs.
in the whole compound, which at 75. 2d. or -^-5. per lb.
costs- — T — ^. But the part of the compound repre- sented by X costs Sx, and the part represented by y costs 7y. Therefore 8*- + 7j/=li^i±i2^; whence 48^ + 42j/
= 43a' + 4Sj/ ; and Sxzzy, Resolving this equation into an analogy (§ 77.) vi^e have x: y :: 1:5; that is, with every pound of the dearer tea 5lbs. of the cheaper are to be inixed; and so in proportion for any greater or smaller quantities. Accordingly mixing lib. of the dearer with 3lbs. of the cheaper, the cost of the whole 6lbs. is 85. + 355. or 435. : and this divided by 6 gives' 7|, or 75.' 2d. for the cost of the mixture per lb.- — The same result appears from common Arithmetical principles. If the teas were mixed in equal quantities, it is plain that the cost of the compound would be 73. 6d, per lb. or the cost of the compound would be found by dividing Is. (the difference of the prices) into two equal parts, and adding the half to the smaller price, or subtracting it from the greater/ If a smaller propor- tion of the dearer tea be in the compound, the cost of tht? mixture per lb. will be less, and would be ascertained by adding a proportionally smaller part of Is. to the price of the cheaper tea. Now we are told that the cost of the pbmpound is to be 75. 2d. per lb. that is, Is. the difference of the prices is divided in the ratio of 2 : 10, or 1:5^ ' ■ ' which
( 153 )
which therefore must be the ratio in which the quantity of the dearer tea is less than the quantity of the cheaper. And in like manner, if the cost of the compound was to be 7s. 7d, per lb, the quantity of the dearer tea in the mix- ture must exceed the quantity of the cheaper in the ratio of 7 : 5 5 or with every 5lbs. of the cheaper tea 7lbs, of the dearer must be mixedf. If the prices of the teas instead of 85. and 7^. were 8s, Sd. and 7^. 5d. the difference of the prices would be 10^. and in order that the compound should cost 7s. 7d. per lb. the quantity of the dearer should be to that of the cheaper tea as 2 : 8, or 1 : 4.
279. How much brandy at 8s. per gallon, and British spirits at 3j. per gallon, Jtiust be mixed together, so that in selling the compound at 9s. per gallon, the distiller may clear 30 per cent. ? Here, in the first place, the student ought to form a distinct conception of the meaning of the ex- pression, clearing 30 per cent. And if he set out without accurately understanding this, he would probably be in- volved in error. It does not mean, that on what he sells for ^100 he is to have a profit of £2>K), or that he is to sell for ^100 w^hat costs him but £7^ : but it means, that wiiat costs him ^100 he is to sell for ^130; and so in proportion on any other quantities. Instead therefore of proceeding to calculate the quantity of brandy and spirits in what shall cost .^100, and be sold for .j^l30 at 9^. per gallon ; we may advantageously calculate the quantities in what shall cost IO5. and be sold at the rate specified for 1 3^. Now putting X for the number of gallons of brandy in the compound, their cost is 8a; ; and putting y for the number of gallons of spirits in the coiBpound, their cost is Sy: and we have the equation 8^v-|-3?/= 10. But x-^i/ is the number of gallons in the whole compGund, and their selling price at 9s. per gallon is 9^ + % • and we have the equation 9^ -f-9j/=13» Reducing these two equations we find ^ = if and ?/ = tt. (For multiplying both sides of the first given equation by 3, w^e have 24-^4-9^ = 30 ; from which subtracting the second given equation we have 15a'=17, and therefore A' = 44 ; which number substituted for A' in either of the given equations affords us y = ^f.) These numbers afford the precise quantities of the brandy and spirits which would cost 10^. and at 9s. per gallon be sold for 135. Mixing them therefore in the ratio of H : if? or 51 : 14, the required profit will be had at that
selling
( 154^ )
selling price. — We might arrive at the same conclusion by common Arithmetic, from the principles stated in the last section ; first finding the cost of a gallon of the compound by the analogy 130 : 100 :: 95. : ^Sy or 6445. The dif- ference of the cost prices is 5s. and the excess of the cost price of the brandy above the cost price of the compound is i4^» t>ut the excess of the cost price of the compound above the cost price of the spirits is {\s, : from which we collect as before that to every 5 1 gallons of brandy 1 4 gal- ions of spirits are to be added.
280. Two couriers set out at the same time in contrary di^ rectioiis, 525 miles asunder. The one travels 40 miles the Jlrst dayy and every mcceeding day goes 4 miles farther than the preceding. The other travels 50 miles the Jirst day, and every succeeding day 5 miles less than the precedirig day. When mil they meet P It is plain that the principles of Arithmetical progression are applicable to this question j as the number of miles that each courier has travelled when they meet is the simi of a series in Arithmetical progres- sion, the terms of the one increasing by the common dif- ference 4, and the terms of the other decreasing by the common difference 5. Putting x therefore for the number of days at which they meet, this will also be the number of terms in each series. The first term in one series is 40, and the last term is 40 + 4 Xx — 1, or 36 4-4^. Therefore
X 76;^' + 4*
the sum of that series is 40 4-36-f 4;vX7;» or
The first term of the other series is 50, and its last term is 50 — 5 Y.X — 1, or 55 — 5x. Therefore the sum of this se-
. _ X 105-*' — 5a:*
ries is 5^-^-55 — bx X ^> or -^y * We now have ex- pressions for the distance, which each courier has travelled w^hen they meet ; and we are told that the sum of those distances is 525 miles, which gives us this equation :
181^—*-* -o-
=:52a
2
Therefore a-*— 181^: =— 1050
and .^-.181;c+!!!^=-1050-f2!2£l=:?£££i 4 4 4
Ti r ^81 _. /28561 .^169
Thereiore :< = =±:^ / ==t:
2 V 4 2
But
( 155 )
But the nature of the question marks that the positive value of the root cannot afford the answer. Adopting
therefore the negative, value, we have x^ — =«-•
^ 2 2 2
= 6. And accordingly calculating the distance that each
courier has gone in 6 days, we shall find the sum of the
distances 525 miles. — But we may arrive more expeditiously
at the equation ^ =525, by considering the com^i.
pletion of 525 miles by the two couriers, travelling at the rates specified, as equivalent with the completion of the same distance by one courier, travelling at a rate com- pounded of the two rates, that is, going 90 miles the first day, and one mile less every successive day. So that we have to find the number of terms in a decreasing Arith- metical series, whose first term is 90, the common dif- ference 1, and the sum of the series 525. Putting x there- fore for the number of terms, the last term is 90 — x — 1, or 91 — ^i and the sum of the terms is expressed by
90 + 911=^ X^, or iH^=f!. ^ 2 2
281. A company x<oanting to make up a confrihufioyi of
£S0» Jind that they must each pay £\. 65. 8c?. more^ than
if there "were three mm^e contributors. What is the number
in company F Putting x for the number in company, the
80 quota of each must be represented by — If there were
three more in company, the number would be a: + 3, and
SO the quota of each -* ^ow we are told that the for-
mer quota exceeds the latter by Ij^B. that is, 80 , ;^__50
Therefore i^L±i!f = 240 and A-^ + S^vzrlSO
Therefore ^* + 3a' + 2 = 180+_=— '
4? 4 4*
, 27 3 24 ,^
and ^ = ^-^- = -^=12
Accordingly
( 156 )
Accordingly the twelfth part of ^80 is £6. 135. 4^'. but the fifteenth part of ^80 is £5, 6s. Sd. less than the for- mer by j^l. 6s. 8d. — [If we generalize the problem, by putting a for the total sum to be contributed, b for the supposed additional number of contributors, and c for the
difference of the quotas ; then x =r ^ — + : and it
appears that the problem is impossible in fact, unless c
measure aby and unless — f- — be a square number.]
c 4>
282. What number is tJiat^ "mhick divided hy the "product of its digits gives 2 for the quotient ; and if '2.1 be added to the number y the digits mil be inverted P Here it is to be un- derstood that the number sought is written with two digits, or is less than 100; as may be collected from the latter condition. And let the student form a clear conception of the meaning of that condition ; nainely, that the sum of 27 and the number sought is a number written with the same digits, but in an inverted order. Now putting x for the left hand digit of the number and y for the right hand digit, we have seen (§ 59.) that the number sought will be expressed by 10^ +3/ j as the number written with the same digits inverted will be expressed by ZOy-f at, gut we are told that 10A;-f ?/ divided by xi/ gives 2 for the quotient; and that the sum of 10.v-f j/ and ^7 is lO^-f ^ ; that is
10^-f.y^o
1 0^ -f j/ -f- ar = 1 oj/ 4- ^
From thje latter of these equations we have i/=zx-\-3 ; and
substituting for ^ in the former equation this its value, we
, 11^ + 3 ^ , /49 , 5 7 , 5 ^
have — = 2: whence a- = ^ / - — — =r — ^--. = 3.
x''+3x V 16 4 4 4
Therefore 3/ (or A'-f 3) =6. So that the number required is 36. And accordingly ^^ = 2 ; and 36 -f 27 = 63.-.From
the general equation 1 0^*4-2/ -}-« = 10^4-'*') we may derive the equation a = 9j/ — 9^? ; from which we may infer the ge- neral principle that if to any number written with two digits, of which the left hand digit is less than the right hand digit, 9 times the difference of the digits be added, the
sum
|- ( 157 )
sum will he A number written with the same digits, but ki- verted. And in like manner, if the riglit hand digit of the number be greater than the left hand digit, subtracting from the number 9 times the difference of the digits will give a similar remainder. — In like manner, if there be a number written with three digits, adding to it, or sub- tracting from it (according as the right hand digit is less or greater than the left hand digit) 99 times the difference of the first and last digits, must give a sum or remainder written with the same digits, but inverted : as appears from the equation 100a-{-l6b-{-c=tix=zl00cf lOb + a. The stu- dent may pursue this investigation at his pleasure.
283. 2\vo partners A, and B, gained .j^140 by trade, A,'s monei/ was 3 months in trade, and his gain was ^60 less than his stock : and BJs money, which was £50 more than A.^s, was in trade 5 months. What were their respec^ tive stocks and profits ? This question differs from any of the common questions in Fellowship (§ 137.) only in this, that we are not told the stock of either partner, but must investigate their stocks as well as profits. Putting ;v for A.'s stock, X'{-50 will express B.'s stock ; x — 60 A.'s gain ; and therefore B.'s gain must be 140 — x — 60, that is 200 — x. But we know that their gains are in the ratio compounded of their stocks and times, or in a ratio compounded of the ratios of x : x-\-50y and of 3 : 5, that is in the ratio of Sx : 5a; 4- 250. So that we have the analogy 3^ : 5^+250 : : X — 60 ; 200 — X ; and thence the equation
600.V— 3.v^ = 5^*— 50a^— 15000
or 8.v^— 650;^' =15000
^, r , 325^,105625 15000,105625 225625
Therefore x^ =:: f- _ == ^ .
4 64 8 64 64
'- V 64 8 8 8
So that A.'s stock having been ^I06,^his gain was £^0 ; B/s stock was .€l50, and his gain ^^S&. Accordingly, calculating the division of the joint profit between them at those capitals and the given times ftliat is, dividing £l40 into two parts in the ratio of 10 X 3 : 15 X 5, or 2 : 5) we
shall find the shares £^0 and i^^ ^/^^ ^
234. Sold
( 158 )
284. Sold a piece of cloth for £^^>, and gained as much per cent, as the cloth cost me P What was the price of the cloth ? Putting X for tlic price of the cloth, the absolute profit is 24 — a*. Now this being the profit on what costs £x the profit on what would cost ^100 (or the gain per Cent* see § 279.) is determined by the following analogy ;
as ^ : 1 00 : : 24 — ;f : ^^00— ^OQ-^^ gut we are told that this
gain per cent, is equal to iv ; so that we have the equation ^_2400--100^.^^, ^^ ^ 100a— 2400. Therefore ;c^ + IOOa*
X
+ 2500^:4900; and ;v = 70— 50 = 20. Accordingly, £^^ profit on £ 20 is at the rate of 20 per cent,
285. A grazier bought as inany sheep as cost him' £Q0^ out of which he reserved 1 5 sheep ; and selling the remainder for 5^54, he gained 2s, a head by them. How many sheep did he buy? Putting x for the number bought, x — 15 ex- presses the number sold for j^54, and therefore ■- ex-
presses the selling price per head. But — expresses the
purchasing price per head ; and we are told that the former exceeds the latter by 2^. or £'^, Therefore we have
60 J__ 54
X 10 x—15
afsr\ I 540.V
600 + A? =
^ AT— 15
585^ + ;v*— 9000 = 540a?
flr»-}-45^' = 9000
a -i_ 4. -,.2025 onnn_i2025 38025 Pi^ 4" ^^^ -i = 9000 + =
4 4 4
X -- _._/
Accordingly 60 sheep (75 — 15) sold for £o^ give the selling price of 18^. p^r head: and 75 sheep bought for ^60 give the purchasing price of I65. per head.
286. What two numbers are they whose su7n is 8, and the sum of their cubes 152 ? Here if we employ the notation, which might probably first occur to the student, we shall
put
( 159 )
put .V and y for tlie numbers sought ; and we have the two equations ^-f-?/ = 8, and x^ -f-//^ = 1 52 : and our object must now be to reduce the cubic equation to one of a lower or- der. Cubing therefore both sides of the first equation, we have AT 3 + <^x^y -|_ '^xy^ -f-j/^ = 8^ = 512; and from this equa- tion subtracting the second of the given equations, we have 3>;*3/ + 3^y = 512 — 152 = 360. Now dividing one side of this equation by Sa'-I-Sj/, and the other side by its equal 8x3, or 24, we have xy=.\5 \ and the problem therefore resolves itself into that of finding two numbers whose sum is 8 and their product 15* (See § 262. and 263.) The numbers required are 3 and 5.
287. But we may frequently obtain a more facile and elegant solution for a problem, by employing for the num- bers sought a designation borrowed from the principle, that the greater of any two numbers is equal to half their sum 'plus half their difference, and the less equal to half their sum minus half their difference. (This appears from reducing the equations xJf-yzza^ and x — 3/ = 5. See also ^ 142.) Let us now resume the solution of the last pro- blem. We are told that 4 is half the sum of the numbers sought. Therefore putting x for half their difference, the greater will be expressed by 4 + ;v, and the less by 4 — x. The cube of the greater, or 4-[-A?p, is 64-f-48;^^-{-12A*-f-vV^, — The cube of the less is 64 — 48A^-f 12.v^ — x^. And the sum of these cubes is 128 4-24?'^* ; which sum we are told is equal to 152. Therefore 24^^^ = 152—128 = 24 5 and A'^ = 1 J and x-=z\. The numbers sought therefore are 4 4-1 and 4 — 1, or 5 and 3. — [Generalizing this solution by putting a for half the given sum, x for half the difference of the numbers, and h for the given sum of their cubesj the numbers sought are expressed by a-\-x and a — v, whose cubes are a^'\-Sa^x-\-Sax'^ -\-x^ and a^ — Sa^x-\-3ax^ — a'-\ But the sum of these cubes is ^a^ + Gax^, Therefore
2«^ -f 6^z.v* = b ; and Gax'' =zb — 2a^ ; and .v* = '~^
ha
Therefore . = ^^-^ 5 or ^ ^~-]
288. What two iiumhers are they, whose sum is 6 aiid the sum of their ^th powers 272 ? Putting x (as in the last sec« tion) for half their difference, the numbers nought are ex» pressed by S-f-^ and S—x, But 3+^|'^-f 3— Af = 162
( 160 )
-f lOSx* + 2^^ = 272. Therefore x"^ + 54^* = 55 : and com- pleting the square jc'^+ 54^:^4- 729 =784 ; and ^'* + 27 = 28. Therefore ^r^ = 1 ; and ^ = 1 ; and the numbers sought are 3+1 and 3 — 1, or 4 and 2.] — Universally putting a for half the given sum, and b for the sum of the biquadrates, a + a\^ 4- flr._j-|4 = 2^^^ + 1 2a* jc- + 2^^ = ^> : whence jr'^ + 6a*a:*
=r -— a^ 5 and ^^ + 6a^r* + 9^^* = - + 8a^. Therefore a: - 2 2
+ 3a^ = >y/- + 8a4; and ^ = a/ — 3a'+ /^ -f-Sa^— -
By the aid of a similar notation we can find two numbers, whose sum is given and the sum of their fifth powers. For the 5th. power of a-j-x is a^ -{- 5a'^a: -{- lOa^a;^ -{- lOa^a^^ •^5ai'^-\-x^ : and the 5th power of a — x is a^ — Sa'^x -f- 10a ^07* — 10a\r^4- 5a^'* — a:^. But the sum of these 5th. powers is 2a^ -f- 20a ^x^ -{- 1 Oax"^ z=b ; a biquadratic equation of that form which we can reduce as a quadratic.]
289. To ^nd four ?mmbers in Arithmetical 2^'>'ogressiony 'whereof the product of the extremes is 54, and that of the means 104 ? Putting x for the smaller extreme, and y for the common difference, the series is expressed by .r, ^ -f?/, x-\-^y^ and ^-f 3j/. The product of the extremes is ^* 4-3^3/ = 54 : the product of the means is J7*-f3jn/ + 2j/* = 104: from which subtracting the former equation, we have 23/* =50; and 3/* =25. Therefore the common dif- ference 3/= 5; and substituting this number for ^ in the equation x^ -f ^^n/ = 54, we have jt* -f- 1 5^' = 54 : wliich gives
15 , /441 15 , 21 _ o ^1 ^^1
us ^ = + / = — — H = 3. feo that the num-
2 V 4 2 2
bers sought are 3, 8, 13, and 18. — We see in th€ solution of this problem, that when four numbers are in Arith- metical progression the product of the means exceeds the product of the extremes by twice the square of the com- mon difference : as, if three numbers be in Arithmetical progression, the square of the mean exceeds the product of the extremes by the square of the common difference.
290. Given the smn of three ?mmbeis in Arithmetical pro- gression = 24, a7id the S2fm of their squares =210, to find the numbers ? Employing the same notation as in the last section, the numbers are expressed by x^ cc-\-y^ and ;r4-23/: and their squares by ^*, ^*-f 2:ry-f3^*> and x^ 4-4«^+^J/*' The sum of the numbers is 3.r-f33/=24:
the
( 161 )
the sum of their squares is Sjt* + 6xy -f %* = 210. Squaring the first of these equations, and multiplying the second by 3, we have 9^^ + 18a?j/ + 9j/*=576, and 9^-^ + 18^-3/4. 15j/* = 630. Subtracting the former of these equations from the latter we have 6y^ = 54>; and,y^ = 9. Therefore the common difference 3/ = 3 ; and substituting this number for T/ in the first of the given equations, we have 3^-f-9 = 24! ;
and a; = __ =5. So that the numbers sought are 5, 8, and
11. — In like manner, if we be given the sum, and sum of the squares, o? four numbers in Arithmetical progression, we have the equations 4Af-f 6y = «, and 4^* + 12a;2/-{- 14k/* = b. Squaring both sides of the former, and multiplying both sides of the latter by 4, we have 16^*-|-48a"?/4-36?/* = a% and 16a;" 4-48^ + 5%* =46. Whence 20j/" = 4Z>—«*. — In like manner if there he Jive terms in the series, we shall find 50y" = 56 — a*. If there be six terms in the se- ries, we shall find 105j/" =66 — a*. If there be seven terms in the series we shall find 1963/^ = 76 — «*. — In all this in- vestigation let it be remembered that a denotes the given sum of the terms, h the given sum of their squares,' and y the common difference of the terms. And we find that the coefficient of b is always the number of terms in the se- ries ; but the coefficients of 3/* are found to be successively 1^ 6, 20, 50, 105, 196, according as the number of tei*ms in the series is 1,2, 3, 4, 5, 6, or 7.
291. [We might now proceed to investigate the law of continuation in the series of coefficients of y^ ; so as to be able to calculate the coefficient of j/^, Avhen the number of terms in the series is 10, or any other assigned number ; and this without being at the trouble of discovering it by the same operation, by which we have ascertained the first seven terms. But as the investigation lies rather beyond the elementary subject of this treatise, and, if minutely de- tailed, would lead us too far away from our present object ; I shall content myself with pointing out to the curious stu- dent some of the steps and the ultimate result. Observing the series 1, 6, 20, 50, 105, 196, we find the differences of the successive terms to be 5, 14, 30, 55, 91. Observing this series, wc find the differences of its successive terms (called the second differences of the terms of the former series) to be 9, 16, 25, 36, or the squares of the numbers 3, 4, 5, 6 : so that in the series of the first differences 5>
M 14,
( 162 )
14, 30, 55^ 91, the first term 5 is the sum of 2* + P ; the second term U = 3^+2^4-l' ; the thh'd term 30r=:4*-|-3* -{-2^ + 1*; and so on. This may lead us to the constitu- tion of the series 1 , 6, 20, 50, &c. whose law of conti- nuation we investigate. Its first term is unity : its second term 6 = 2* + twice i\ Its third term 20 = 3" + twice 2' + three times 1\ Its fourth term 50 = 4* -}- twice S* + three times 2* + four tunes 1* : and so on. Now 50, the fourth term of that series, is the coefficient of 3/* when the number of terms in the Arithmetical series is Jive. (§ 284.) Suppose then that the number of terms in the Arithmetical series is 10. The coefficient of ^*, in the equation IO6-— a:^ =^5^% wiUbe the sum of the following numbers 9M-'^ X8*-f3 X 7" + 4X6* + By. 5'' + 6X4^ -f.7x3*-f 8X2*+9X 1* j or will be 825. And accord- ingly taking any series of ten terms in Arithmetical pro- gression, it will be found that 825 times the square of the common difference = 10 times the sum of the squares of the terms minu$ the square of the sum of the terms.]
292. [But we still need to simplify the calculation of the coefficient of 3/*^ Suppose then that the Arithmetical se- ries consists of 5 terms : and let nz=.5. We have seen that the coefficient of 2/* will be the sum of the following terms, ?;— 1|^+2 X n—2\^ -f 3 X n^Y + 4 X n—^\'' j that is, the sum of the following terms,
n"- — 27^4-1 2w*— 8;j + 8 3?i* — 18?? + 27 4«^-_32;i-f.64
10?i*— 60?i4-100
In this expression^ 10?z*-- ^0/?+100, the coefficient of rr- is the sum of 1 4-2 4-3 4-4, The last term 100 is the sum of 1^4-2^4-3^4-4 ^ The coefficient of n in the second term is 1*^2 4-2^^ 4-3^ X2 4- 4^ X 2 = 1^4-2^4-3^4-4* X 2. Now fi'om the doctrine of Arithmetical progression we can easily calculate the sum of any of the natural num- bers ascending from unity. We therefore only want to know a facile method oiP calculating the sum of their squares, and the sum of their cubes. The latter is easily calculated firom the following curjous property — thai the
sum
V
( 163 )
sum of the ctihes of any of the natural 7iiimhers C07nmenci7ig 'with unity is equal to the square of their sum : as in the pre- ceding instance 1-1-2 + 3 + 4=: 10 5 and 10* = 100 = 13 + 25 + 3^ + 4'^ And the sum of the squares of the terms of such a series is equal to the 6th. part of the sum of the highest term + three times its square + twice its cube* Therefore twice the sum of the squares is equal to the third part of the latter sum. Accordingly in the preceding
instance P + 2- +3^ + 4>X2 = ^+^+^"^= ^
= 60.]
293. [Let it be recollected that in the trinomial, 10;^* -^60w+100, (expressing the coefficient of j/* when the number of terms in the Arithmetical series isfve) the series of natural numbers from unity, of which 10 is the sum, 100 the sum of the cubes, and 60 twice the sum of the squares, is 1, 2, 3, 4; its highest term being one less than the number of terms in the series w^hose common dif- ference is y. Whatever therefore be the number of terms in this series, represented by n, the series of natural num- bers, from which tlie terms of the trinomial formula are to be determined, is 1, 2, 3,,.w— 1. Now the sum of this
is (§ 185.) n X =^- . Therefore the first term of
the trinomial formula is universally expressed by ~
I
2
X 72* = . The third term also is universally ex- pressed by ^ \ = — "^ i — J for we have remarked
in the last section that the sum of the cubes of 1, 2, 3..« n — 1, is equal to the square of their sum. In the second term of the trinomial formula (which term is to be sub- tracted from the sum of the first and third) the coefficient of n is by the last section universally expressed by ^Zn-+3 X >^|' + 2 X ^^l^^n^-^n- ^n .^ which mul- 3 3
tiplied by n gives Z — for the universal expres- sion of the second term in the trinomial formula. In or- der to subtract this from the sum of the two former, let ^ M 2 us
'{ 1^4< ) us bring them all to the common denominator 12. The two former become — and -__J : whose
surn IS — — . — -— — ■ f Irom which subtractmg
^^ ~^ It (the value of the other term) we have left
12. ^
n — y? £.^^, ^j^^ universal expression of the coefficient^ of j/^
in the equation 7^6 — «^=wj/^ = _ — ^ — -Xj/* > where j/ re- presents the common difference of any Arithmetical pro- gression, a the sum of the series, 7i the number of terms, and b the sum of their squares.]
294. [This investigation originated in the problem pro- posed § 290. to find a series in Arithmetical progression from having given us the sum of its terms, and the sum of tlieir squares. But we may now reverse the problem, and <?asily find the sum of the squares of the terms of any given Arithmetical' prpgsession. For from the last equation we
arrived at, ,ib-a^ J^^lz^, we &ndb='j!£=J!^
4-^. Therefore putting 5 for the sum of any Arithmetical
n
scries, d for thcGommon difference, and ii for the number
of terms, the sum of the squares of the terms is equal 19 __ . ^ ^
. Xn^ — n-\-^- — Thus, if the Arithmetical series 3, 5, 7*
&c. be continued to ten terms, the 10th. term is 21 ; the sum of the terms is 120 ; its square is 14400 j and there^
fore il = 1 440. But i^ = | ; and 7^^— ;i = 990. Therefore n 12
.__X?^^ — n = - — = 330. And the sum of the squares of 1 ^ ■ %j
the terms, 3^ + 5^ + 7^. .. + 21 % =330+1440=1770.]
295. To jiyid four nwnhcrs in Arithmetical iwogression the sum of U'hose squares shall be 214, a7id the continued product of the numbers 880 F Here putting x-r-^^ for the smaller extreme, and .v + 3?/ for the greater, from the na- ture of Arithmetical progression 2v will be the common difference 5 and the two means will be expressed by a- — ^
and
( 165 )
mid x-^2/: so that the Arithiretical series is ?( — 3j/, ^-^2/9 •*-fjy> an^ ^' + 3y. (If the student should attempt to ex- press the series by .v, x + ij, x-\-2y, .v + Sj/, he would find himself involved in considerable difficulties : and he may observe how the notation we have adopted tends to simplify the equations, which express the conditions of the pro- blem.) The sum of the squares of these four terms is
4A'M-_20y* = 214'. Their continued product, or ^ — 3?/
X x + ^X>:—y X x -f ?/, or^^^— 9j/" X .r^-~?/% is x"^ — lO^r^j/* + 9^/"^ = 880. From the former of these equations ^_10r-d03^ and therefore .4^ 107-^10^1- ^ 2 ' 2 1
^_Jl hZ_. Substituting for x^ and x^ in
the second of the given equations these their values de-
. , r ,x r ^ 1 11449— 2 140?/* + 100?/^ rived from the first, we have -' ^ ^.
4
_i2i22lzi22^4-9y = 880; whence 336j/4— 4280j/*
= 3520— 11449 = — 7929: and 2/4^-il£^=— 1?£?
-^ 42 336
\^_2643 ^^.,^f^^.^ ,_535y ^ 286225 ^ 286225_2643 , 112 -^ 42 7056 7056 112
119716 , , 535 _^ /119716 .346 ,
= : and ij^ — = ±b: /. — ^^ = z±: ; and
7056 '^ 8i V 4056 8i
, 535 346 189 9 rr.. ^ 3 , _
2/*= = = _. Therefore ?/= -; and 2?/, or
.^ 84 84 84 4 ^2' -"
. the common difference of the series, = 3. But we have
107— lOz/ 107— 22i 169 rp. n ■ seen that x ^ = -. jL. ~ —i = 1 hereiore .v
2 2 4
= / = -^ j and the Arithmetical series .v — 3j/, x — -y,
AT -1-2/, .v + 3j/, is 2, 5, 8, 11.
296. In pursuing the solution of this problem, I have retained the given numbers 214 and 880, which are the assigned values of the sum of the squares and continued product of the four nnmbers sought. And it is important that the student should acquire a readiness and accuracy in performing the numerical calculations thus occasioned. ' But except for promoting this object, it is much preferable
to
( 166 >
to substitute for the given numbers (when they are so large) some of the initial letters of the alphabet. Thus the given equations in the last problem may be stated generally, 4.a;*+20j/" =«, and x"^ — 10/v^7/^4-9j/'^ = &. — If instead of the sum of the squares and continued product, there be given the common difference and continued product of four numbers in Arithmetical progression, the series is found still more easily. For putting 2a for the given common dif- ference, the four numbers may be expressed as in the last section by ^ — ^a, x — a, A'-f-a, and ^ + 3a: the continued product of which terms is x"^ — 1 Oa^x* +9a'^z=ib. Whence, completing the square, we have^^^-lOa^^^-f- 25a^ =Z> + 16a^: and therefore x^ — 5a* = =±= VZ> + I6a^ ; and .r* = 5a*
z±z\/b-{' I6a^, Now if the given common difference be 3,
3
and therefore a=-^^ and if ^ = 880, then 5a^z±zVb+l6a'^
r:--=i= v'QSl = — =t:31 = (taking the positive value of the
root) . And therefore .v=:^ / =~, as before.
'4 V 4 2
We fix on the positive value of the root, because it affords
a positive square number, for the value of x^ : whereas the
45 79
negative value of V961 would eive a^* = .-. — 31=—- — » ^^44'
which is impossible. — It is to be remarked, that although we can find fozir numbers in Arithmetical progression from the data assigned in either of the last sections, yet to find three such numbers from similar data would necessarily in- volve us in a cubic equation ; the management of which does not come w^ithin the subject of the present treatise.
297. To find three numbers in geometrical progression whose mm shall be 26, a?id the sum of their squares 364 ? Putting X and 3/ for the two first terms, the third will be
expressed by •!-; and we have given us the equations
X
*'+^+~=«f and .v*+^*+-^ = ^. In the first of these equations transposing j/, we have x-\-^zza—7^ ; & sqiiar- ing both sides of this equation we have A-^+^y^-f-^rra*
'< ' X
^2ay
I
( 167 )
— 2^i/+j/*. Therefore x* -^^^ -{--L^zza* — 2^3/ =r (by the
second of the given equations) b. Therefore 3/= »
_ 676--36^_. 312 ^ g^ ^^^^^ ^^^ ^^^^^ ^^^^^ ^j^ ^^^ ^
52 52 *
or second term of the series is 6. Therefore the sum of the extremes is a — 6 = 20; and their product is 3/* = 36: so that we now have given us the sum and product of the extremes ; from which they are found to be 2 and 18; and the series required is 2, 6, 18.
298. To Jind four riumhers ill geometrical progression^ such that the difference of the extremes shall be 52, and the difference of the means 12 F Putting x for half the sum of the means, and a for half their given difference, the two means will be expressed by x — a, and x + a, (See§ 287.) Then from the nature of geometrical progression the smaller
extreme must be ^!^m^' , and the greater extreme l~t^' : for xz±ia : xzxza :: «•==:«: ^rlrS . Therefore the difference
I «f tlie extremes is -X-' ' =52 = ^: and multiply-
^ x^a x + a ^ '^
ing both sides of this equation by x — a and by x+a^ wft have
x^a\ 5 — X — a[^ zzbX x-^a" = bx^ — a'b.
But subtracting x"^ — 3;v-a-f3^«* — a^ (or x — a^) from x^ •\-2>x'^a-\'^xa^ -\'a^ (or x-\'ct^) the remainder is ^x'^a. + 2^3 = bx^^d'b. Therefore bx^ — 6x^a = 2a^ + a'b.
Whence .^ = ?^1±^ = t??±ll!2=!£2f : and ^=t 6— 6a 52—36 16
= 12. So that the two means are 6 and 1 8,
y
16 4. and the extremes 2 and 54.
[299. If we have the sum and product of any two num-s bers, we may thence derive expressions for the sum ofthmr- squares^ cubes, biquadrates^ &c. For putting x and ^ foy the numbers ; s for their sum and p for their product.— in the first place ^*-f?/^ = jf^-^2^, since the square of tho
sum
( 168 )
sum is equal to the sum of the squares -f- twice the product. — In the second place, multiplying the equation x^-k-^^ — 5^ — 2p by the equation a'-|-3/=.S ^^'^ have ^^+3/^ + ^3/* '\-yx'^ =zs^ — 2sp. But XT/'^ +3/'*'^ = •* +3/ X ^3/ =sXp, There- fore /v^-|-3/3 + 5j9 = 5^ — 25j9; and >;^-f3/^=5^ — Ssp. — In the third place, multiplying both sides of the last equation by the equation a' +3/ = 5, we have x'^ 4-3/-^ + xi/^ 4-3/-^^ = ^''^ —35^- But X2/ -\-yx 5 = x^ +3/* XX7/ = s' — 2p X ^ = s^p—^p 3. Therefore ^v^-f-3/'* + 5^j9 — 2p^=5^--.3s>i and ;v'^+3/'* = s* -^45'j»4-2jD^. — In like manner if we proceed to calculate the value of ^^+.^^> it will be found by multiplying the value of x^-\'y^ by 5, and subtracting from the product the value of x^-\-y^ multiplied by p ; whence x^ -{-7/^ = 5^ — 55^/>^-5.9/>^ And again multiplying this value of x^ +3/^ by s, and subtracting from the product the value of x'^-\-y'^ multiplied by p^ we have the value of x^-^-i/^zzs^ — Ss'^p -f 95^^* — 2p^. We may now remark on these expres- sions for the sums of the powers, 1st. that the signs of the terms are alternately affirmative and negative : 2ndly. that the number of terms is always half of the even number next above the index of the power ; (for instance, the ex- pression for the value ;v'° -{-3/*°, or for /v" 4-3/*% will con- sist of 6 terms, but for a-^* -f 3/^* of 7 terms). 3rdly. Putting 71 for the index of the pow^r, the first term of the expres- sion for the value /v" 4-3/" wdll be s", and in every succeed- ing term the index of s decreases bp 2, and the index of p increases by 1. 4thly. the coefficient of the second term is 71 ; and if ?i be an odd number, the coefficient of the last term also is 71 ; but if 71 be an even number, the coefficient of the last term is 2, and the literal part p with the index
- : 5thly. the coefficient of the third term is the sum of
71'' S yZ
the natural numbers from 2 to fi — ^2, or is : 6thly.
the coefficient of the fourth term is the sum of all the co- efficients of the third terms of the preceding powers from the last but 07ie ; the coefficient of the fifth term is the sum of all the coefficients of their fourth terms: and so on. Thus in the expression for the sum of the 12th. powers of AT and 3/, the coefficient of the third term will be 10 4- 9 4- 8 4-74-34-2, or the sum of the coefficients of the second terhis in all the expressions of the preceding powers from r-s^- the
( 169 )
the tenth : and the coefficient of the 6th. term will be the sum of the coefficients of the 5th. terms of all the pre- ceding powers from the tenth; that is, of the tenth,, ninth, and eighth powers, as it is in the sum of the 8th. powers of X and 1/ that a Jift/i term first appears. 1 lence we may derive the following expressions for the coefficients of the terms in the expression for the value of A;"-f z/". The co-
<?fficient of the third term is 7i X ; of the fourth term
2
IS w X X ; ot the fifth term is n X X X
iJ 3 2 3
, &c. &c. And thus calculating the value of ^^* -f-?/^*
it is found to be s'''—l2s"'j)i-54^s^p'''-ll2s^j)^ + lOBs'^^*
300. To find txiio mimlers, "whose jproduct shall exceed their sum hij 11, and the sum. of "whose squai^es shall he 58 ? Putting X for the sum and y for the product of the tw^o numbers, the sum of their squares is expressed by v'' — 2j/j as wx have seen in the beginning of the last section : so that we have the two equations y — .v =11, and k'^ — ^y-zzb^^ Adding twice the former equation to the latter, we have ;r"— 2a— 58 + 22 = 80. Therefore a'=V81 + 1 = 10; and ?/= ll-j-10 = 21. Having thus ascertained the sum of the numbers =10, and their product =21, we find that the numbers required are 3 and 7. (§ 263.)
[301, To find four numbers in geometrical progression whose sum shall be (a) 80, and the suin of their squares {h) 3280 r' Putting x and y for the two means, the extremes
will be expressed by — and ^L-, (For x\ y \\ y\^-^\ and
x^ \
y: X :: X : — ) Now putting s for the sum of the means
2/ ^ and p for their product, (which is also the product of the
extremes) we have the sura of tlie extremes, or — -f -^
y .V
=?flr — s. But by § 299. A-^+y =s^ — 2p ; and in like man- -ner the sum ot the squares of the extremes, or ^ — + ~, is
equal to the square of their sum 7ninus tw^ice their product, •that is,. =a — ^1 — 2p, Hence, adding the sum of the
squares
( 170 )
squares of the means to the sum of the squares of the ex- tremes, we have the equation s*-{-a — s\^ — 4^ = 3280 = ^. — Affain, from the equation — -{-•z^zza — 5, wehaveA'^-f-t/^
zza — sX^?/=« — sXp^ap — sp. But by § 299. -y^-f j/^--^? ~— 35??. Therefore s^ — Ssp^ap — sp^ and s^=:ap + 2spi
whence p= . Now substituting for v this expression
25 + «
of its value in the equation s*+a — ^gf— 4p = 5, we have
5*+a^f-— ^=5; thatis, 2s*— .25^+tr*— -i£L=5; ^ ' 25 + « 25 + a
whence, multiplying both sides by Ss + cr, we have — 2s* a '^a^;:z2sh-\-ab. Therefore 5 * -f L- =r — rZ_ ; & compl eting
the square s * + L. -| = fJZL -j . Whence we have
. = y£zi-^ + il^~ = y 1560 + 420i-~20i=!?-. 20i;=24. Thus we have ascertained that the sum of the two means is 24. But we have before found jp=-
2s + a
Therefore the product of the means = = ^
^ 48 + 80 128
= 108. Hence the means are found to be 6 and 18 ; and therefore the extremes 2 and 54.]
In the following questions for exercise, lest any diffi- culty should remain to the student, I have either referred to a preceding section where a similar question has been solved, or have exhibited the translation of the question into the language of Algebra. Yet I would strongly re- commend, that he should not apply to these aids, until he has attempted to solve the questions without them,
Qjiestions for Exercise*
1. What two numbers are they, whose sum is 7 and their difference 2\ ? (§ 287.)
2. Divide iB20 between A. and B. so that A. shall have lOs. 6d. more than B. ? (J 287.)
— s#
( 171 )
S. so that yrds. of A.'s share shall exceed |ths of B/s
by 6s. 8d. ? — Putting x for A.'s share and 20 — x for B/s,
we have the equation — HI— =~ : from which A/s
^ 3 4 3
share will be found 1 0^£. and B/s share 9^^£, But let the student receive a caution in the reduction of that equa- tion. After multiplying both sides by 12, it will stand — not Sx — 180—9*^ = 4, but 8x — 180 + 9^ = 4. For in the
fraction — — — the mark of division, or line separating 4
the numerator and denominator, acts as a vinculum on the terms of the numerator : and therefore after the multipli- cation by 12, we have to subtract 180 — 9x from Sx, that is, to add — 180 + 9a\
4. Wliat two numbers are they, whose ratio is that of 7:5, and whose sum is 7 ? — or — whose difference is 5 ? (§ 259.)
5. v/hose difference is 3, and the difference of their
squares 1 8 ? (§ 260.) Dividing the difference of the squares by the difference of the numbers the quotient is the sum of the numbers. (§ 163.)
6. whose sum is 3, and the difference of their
squares 5|? (§261.)
7. whose sum is 3, and the sum of their squares
6i ? (§ 265.) Putting ■*• for half the difference of the num- bers, the greater is expressed by 4-|-^, and the less by 4 — X, Therefore the sum of their squares is expressed by %-\-2x^-=z6i; which gives a?* = 1 : and therefore the dif- ference of the numbers is 2. This method of denoting two numbers is frequently of the greatest advantage.
8. whose difference is 2, and the sum of their
squares 13^? (§266.)
9. whose sum is 15, and their product 31| ? (§ 262.
263.)
10. . whose difference is 10, and their product 31 J ? (§ 264.)
11. whose product is 8|, and the sum of their
squares 17|-|? (§267.)
12. whose product is 18, and the difference of their
squares 27 ? (§ 270.)
' 13. whose sum is 4-5?^ (or difference ^i) ^"^ ^^^
sum of their square roots 2| ? or — the difference of their square roots | ? (§ 271.)
*■■■■■ whose
( 172 )
14. -vvliosc sum is ^yV (or their difference -f^) and
the product of their square roots 2 ? (§271.)
15. whose product is 4-, and the sum of their square
roots 2| ? or the difference of their square roots ^ (§ 271.)
16. whose sum is 5, and the product of their
squares 36? (§271.)
1 7. whose difference is 1 , and the product of their
squares 2^ ?
18. whose product is 7, and their ratio that of
7:4? (x:l ::7 : 4^.\
19. To find a fraction such, that if you add 8 to thp numerator it shall become equal to 2 ; but if you add th^ numerator to the denominator it shall become equal to f ?
(Putting - for the fraction, we have ^ =2, and — ^
= h)
20. To find a fraction which shall be to its reciprocal as 4 ; 9, and whose denominator exceeds its numerator by
3 ? (We have -^ : f±^ : : 4 : 9 ; and therefore -^ ^ ;v4-3 .V .v + S
4^^4.12
X
•)
21. A man riding from his own house to Dublin went at the rate of 7i miles an hour. Returning home he came at the rate of 6i miles an hour, and was 8 minutes longer on the road. What was the distance ? (§ 272.)
22. A.'s age is to B/s as 4 : 3 ; and three years ago it was as 3 : 2. What are their ages ? (§ 274.)
23. A man left in his will ^10,000 to be equally divided among his children. Three of them died before their fa- ther, and the survivors in consequence got ^750 a-piece more than they would have got, if all had lived. What was the number of children ? (§ 281.)
24. There are two silver cups and one cover for both. The first cup with the cover weighs 1 4^oz. The second cup with the cover weighs ^rds. of the first cup without the cover ; but without the cover weighs i of the first cup. What are the weights of each ? — (Putting x for the weight of the cover, we have 14 — x for the weight of the first
cup, and therefore 7 — - for the weight of the second cup.
Adding
( 173 )
Adding x to this, we have 7+- for the weight of the se-
cond cup and cover together, which we are told is yrds, of
1 4 — ^, that IS = -. j
25. A journeyman was engaged for 40 days, at the wages of 35. 6rf. a day for eyery day he worked ; but to forfeit 2s. 6d. for every day he absented himself. At the end of the period he received 4fj6, 6s. How many days did he work, and how many was he absent ? (Putting x for
the former number, the amount of his wages is -5. X x
7.V , , „,.„„. .5
= -^ ; and the amount of his forfeitures is -^s, X 40 — tc
,=_, — ZI — . This subtracted from .— mves a remainder ^ 2 ^
equal to 865. whence we have \2x — 200=172.)
26. A market woman bought a certain number of eggs at 2 a penny, and as many at 3 a penny : and selling them at the rate of 5 for 2c?. she lost 4^. on the whole. What number of eggs had she ? C Putting x for the number of
each sort, we have - + - for the whole cost, and 2^----, or ' 2 3 ' 2
— , for the whole selling amount ; which is less than --f ^ by 5 23
*•)
27. A person desiring to give 3d, a-picce to some beg- gars, found he had not money enough in his pocket by 8^. but giving them 2d, a-piece, he had 3d. remaining. How many beggars were there ? (§ 277.)
28. There is a fish whose tail weighs 9lb. his head weighs as much as his tail and half his body ; and his body weighs as much as his head and tail. What is the weight of the fish ? (Putting X for the weight of the body, the weight of
X V
the head is 9 + ~ ; and we are told that A=:9-}-'--f9 = 18
+!•)
29. A bill of j870. 12^. was paid in guineas and crown pieces : and the number of pieces of both sorts was 100. How many were there of each ? (Putting x for the number
of
( 17* )
©f guineas, 100—^ is the number of crowns. The amount of the former, at 21 5. is 21^; and of the latter at 5s, is 500 — 5x: so that 21^+500 — 5^^=14? 12, the number of shillings in j^70. 12s.)
30. A person bought a chaise, horse, and harness, for £60, The horse came to twice the price of the harness, and the chaise to twice the price of the horse and harness. What did he give for each ? (Putting x for the price of the harness, 2x is the price of the horse, and 6x the price of the chaise. But Gx-^^x+x^z 60.)
31. A» saves fth. of his income yearly. B. with the same income spends yearly £50 more than A. and at the end of 4 years finds himself :^100 in debt. What is their
income ? (Putting x for the income, — is A.'s yearly ex-
penditure, and therefore — + 50 is B.'s yearly expendi-
5
tare ,• which in 4 years amounts to — ~ + 200 : and this
5 exceeds 4a? by 100.)
32. To divide 36 into three such parts, that i of the first, 4 of the second, and ^j- of the third, may be all equal to each other, (Putting x for half of the first part, that part is 2.V, the second part Sxy and the third part 4:x, But 2Ar-f.3;c + 4A', or 9^ = 36.)
33. A footman, hired at the wages of S8 a year and a liver}', was turned away at the end of 7 months, and re- ceived only £2, 135. 4<Z. and his livery. What was its value ^ (Putting x for the value of the livery, 8+;v is the amount of what he should have received for 12 months service. Wliat he receives for 7 months service is 2\-\'Xy which therefore is to 8 4- a; :: 7 : 12.)
34. A hare is 50 leaps before a gi'ey hound, and takes 4 leaps to the greyhound's 3 : but two of the greyhound's leaps are as much as three of the hare's. How many leaps must the grej^liound take to catch the hare ? (Putting a? for the number of leaps taken by the hare before she is over- taken, it is plain that the greyhound must go over a space of ground equal to x-{-50 of the hare's leaps : and this he will do in a smaller number of leaps than x-\-50y and
smaller in the ratio of 3:2, that is in it of his own
3
leaps.
( 175 )
leaps. But the number of leaps taken by the hare is to the number taken by the greyhound in the same time as
4 : 3. Therefore x : ?^'"^^^^ : : 4 : 3 j whence ^ = 400 ;
and — i^ , or the number of leaps taken by the grey-
hound to overtake the hare, =300.) t
35. A person in play lost ^ of his money, and then won 35. after which he lost -f of what he then had, and then won 2s, lastly he lost 4 of what he then had, and found he had but I2s, remaining. What had he at first ? (Putting
9* for the number of shillings which he had at first, [-3
4
expresses what he had after his first loss and first winning ; |rds. of this, or ^+2 expresses what he had after his se-
cond loss, and therefore Tths. of ~-|-4? expresses what he
had after his third loss, or when he had I2s, left.)
36. A. gives to B. as much money as B. has already: B. returns to A. as much as A. has left : A. returns to B. as much as B. has then left ; and lastly B. returning to A. as much as A. has then left, it is found that they have each 165. How much had each originally ? (Putting >v for the number of shillings which A. had originally, and y for the number which B. had, their numbers after the successive changes are expressed by x — y and 2?/, %\' — 2y and 2>y — x^ 2x — 5y and 6y — 2x, 6x — lOj/ and lly — 5x, So that w^e have the two equations 6Ar — 10j/=16, and llj/ — -5^ =16; whence we find a? = 2 1 and j/ = 1 1 .)
37. What two numbers are they, whose sum is twice their difference, and whose product is 12 times their dif- ference ? (Putting X for the less, 3x must express the greater (as appears from the equation j/ 4-^ = 2j/ — 2x) and 3;v* their product. Therefore 3a?* =24^, and x=z8.)
38. What number is it (written with 2 digits) which is equal to 4 times the sum of its digits ; and to which if 1 8 be added, the digits will be inverted ? (§ 282.)
39. To find four numbers such, that the first with half the rest, the second with | of the rest, the third with ^ of the rest, and the fourth with -f of the rest may each of them equal 10 ? (Platting v, .v, j/, z for the numbers we
have
< 176 )
have t? 4- ;: > and x+ ^-^J , and I/+LZLJL:, and ^ i> /^
s -f ^X-Jt^, all equal to each other: Therefore subtract- 5
ing twice the first of those from 3 times the second, from 4 times the third, and from 5 times the fourth, we have 2a' — Vy and 3?/ — v, and 4^; — v, each equal to 0 ; and there- fore ^=-, 3/=-, » = -. Hence, substituting these values
for A-, y, Zy in the equation ^+' jf - = 10, we have t;+-
40. To divide the number 90 into 4 such parts, that if ^ the first be increased by 2, tlie second diminislied by 2, the third multiplied by 2, and the fourth divided by 2 ; the sum, difference, product, and quotient shall be all equal
to each other? (Putting x for the first part, a-|-4 will ex- press the second part, (for 3/ — 2=:x-{-2) and *^ + 1 the third
part, (for 2;y=A'-[-2) and 2a +4 the fourth part, since ^
= A'-|-2. And the sum of these four expressions =90)
41. If A. and B. together can perform a piece of work in 8 days ; A. and C. together in 9 days ; and B. and C. \\\ 10 days ; how many da3\s will it take eacli person to perform tlte same work alone ? (Putting a^ for the time in which A. would perform it alone, the times in which B. and
C would perform it alone are expressed by — '■ — , & ^
according to the two first conditions. (For x — 8 : 8 : : x :
Sa" 9** ; and x — 9 : 9 : : x : See remarks on questions
A—— o X——^
19'. and 20. page 66,) But B.'s time being -^i C'stime accoi-ding to the third condition is also expressed by (For — 10, or , is to 10 : :
80— 2x a;— 8 X — 8 x-^S
80
— . f Inereiore = ; wiience we nave a
.2x } 80— 2a' a-_9
22. A person
*80-
=:I4if, -^ = 17ii,and-^ = 23-rr.)
A' 8 K 9
( 177 )
43. A person bought a number of oxen for jf 80 ; and if he had bought 4 more for the same money, he would have paid £1 less for each. How itiany did he buy ?
80
(Putting X for the number, we are told that — exceeds
A"
by 1.)
43. What two numbers are they w^hose sum, product, and difference of their squares are all equal to each other ? (Since their sum is equal to the difference of their squares, dividing the latter by the former must give 1 for the quo- tient, which is therefore equal to the difference of the num- bers. § 163. Therefore putting x for the less, ^+1 is the greater, 2^+1 their sum, and a* -f x their product. Sq
^5 1 \ that ii^ •\-xz=.^x-\'\\ whence a;= — -+^*)
44. To divide 6 into two such parts, that their product may be to the sum of their squares as 2 to 5 ? (Putting ;*• and 6 — x for the parts, their product is Qx — a?*, and the sum of their squares is 2^* — 12^ + 36 : so that Qx — x"^ : 2a;* — 12;^ 4- 36 :: 2 : 5.)
45. To find two numbers whose difference is 3, and the difference of their cubes 117. (Dividing ;r5 — y^ \y^ x — ^, the quotient is x^ -^-xy-^if .)
46. To find two numbers w^hose difference is 15, and half their product is equal to the cube of the smaller num- ber ? ; Putting A? and x-^-XS for the numbers, we have
f ' »=:;y^, which is depressed to a quadratic by dividing
both sides by x,)
47. A person bought a number of sheep for 5^18. l55. and seihng them again at 305. a-piece, gained by the bar- gain as much as 3 sheep had cost him. What was their number ? (Putting x for the number, the amount of the sale was 30a^, and the profit 30^—375. The cost of each
sheep was-^, and therefore of 3 sheep was ,-. ^ = 30;^
X K
—375.)
48. What number is it (written with two digits) which divided by the sum of its digits gives 8 for the quotient, and if 5 times the sum of the digits be subtracted from it, the dicrits will be inverted ? ($ 282.)
° N 49. To
) W'
( ns )
^ ; 49. To firid ^ fttinrber written with 3 digits in AritH* Thctical progression, ftnd such that if divided by the sum of its digits the quotient is 59 ; and if 396 be subtracted from it, the digits will be inverted. (By the last of the conditions we know that 396 is 99 times the excess of the first digit above the last. See latter part of § 282. There- fore that excess is 4, and the difference of the series is 2, and the digits will be represented by x, x — 2, and x — 4? ; whose sum is Sx — 6, and the number written with those digits is expressed by 100^+10^ — 20 + .r— 4? =11 l-v— 24'»
Therefore ~ — =59.)
3a— 6 ^
50. ^Vliat two numbers are they, whose sum multiplied by the greater is equal to 77, and whose difference mul- tiplied by the less is equal to 12 ? (x*-|-ac?/ = 77, and xy-^^ = 12.)
51. To find a number such, that if you subtract it from 10, and multiply the remainder by the number itself, the product shall be 21 ? (10 — .vX^ = 21.)
52. To divide 24 into two such parts, that their product may be equal to 35 times tlieir difference? (24- — xxx = 24— 2a? X 35.)
53. A. and B. having 100 eggs between them, and selling at different prices, received each the same sum for his eggs. If A. had sold as many as B he would have received i8d, if B. had sold no more than A. he would have received only Sd, How many eggs had each ? (Putting x for the number of A.^s eggs, B's number will be 100— ;v. Now if A. had sold 100 — x at the price he got, the amount would have been 18c/.: therefore as 100 — x : x : : 18 :
the sum which A. received. In like manner the
100— «
1 l/^/^ o 800 — Sx . . ^
analogy, as x 1 100 — x : ; 8 : ^ gives a just expres- sion for the equal sum which B. received.. Therefore
I8x _800— A?\ 100 — x"^ X 7
54. One bought 120 pounds of pepper, and as many of ginger, and had one pound of ginger more for a crown than of pepper ; and the whole price of the pepper ex- ceeded that of the ginger by 6 crowns. How many pounds of pepper had he for a crown, and how many of ginger ?
(Putting
( 179 )
(Putting X for the number of pounds of pepper vvliich he had for a crown, the number of pounds of ginger will be Pf -f 1 : the number of crowns which the pepper cost will be
expressed by , and which the ginger cost by 5 the
former of which exceeds the latter by 6.)
SS, To find 4? numbers in Arithmetical progression, whose sum is 18, and the sum of their squares 86 ? (§290.)
SQ* A. sets off from Dublin to Belfast at the same time that B. sets off from Belfast to Dublin. Each travels uni- formly the same road : but A. arrives at Belfast 4 hours after they have met, B. 9 hours after they have met. In what time did each perform his journey ? (Putting ic for the number of hours in which A. performed it, x-\-S is B/s number. Therefore the part which B. performs in 9 hours A. had performed in a shorter time, and that in the ratio of .y -f 5 : A". Therefore A. had performed that part
9^ in ^ hours : and in 4? hours more he arrived at Belfast.
A'-f 5
9^ Hence we have x = h 4?. )
57. Wliat two numbers are they whose sum is 4^, and the 3um of their cubes 33| ? (§ 286. 287.)
58. — - whose sum is 5, and the sum of their 4th* powers 87 ? (§ 288.)
59. whose sum is 3|, and the sum of their 5th.
powers 39-fl ?
60. To find four numbers in Arithmetical progression, whereof the product of the extremes is 25, and the pro- duct of the means 494 ? (§ 289.)
61. To find 3 numbers in Arithmetical progression, whose sum is 9, and the sum of their squares 27|- ? (§ 290.)
62. the sum of whose squares shall be 84, and their
continued product 105 ? (§ 295.)
63. whose common difference is 3, and their con- tinued product 308 ? (§ 296.)
64. To find 3 numbers in geometrical progression, whose sum is 13, and the sum of their squares 91 ? (§ 297.)
65. To find 4 numbers in geometrical progression, whereof the difference of the extremes shall be 78, and the difference of the means 18 ? (§ 298.)
^^, To find 4 numbers in geometrical progression, whose sum shall be 15, and the sum of their squares 85 ? (^ 301.)
N 2 CHAP.
( 18(3 )
CHAP. XXVII.
On Pennutations aiid Comhinations*
302. THE doctrine of permiitaiion, or alternation^ teaches us to find all the varieties of order, in which any number of different things inay be arranged. Thus, tbe five first letters of the alphabet, {a^ b, c, </, e) may be ar- ranged in 1 20 different ways. For it is plain that any two of them, as a and b, may be arranged in two Avays, either ab or ba. Therefore I say that any three of them, as cr, by and Cf may be arranged in six (2X3) ways ; for beginning the arrangement with any one of the three, the other two may follow in two different orders : thus, abc and acb, bac and bcaj cab and cba. In like manner it appears that any four of them, as «, ^, c, and rf, may be arranged in 24« (2X3X4) different ways : for beginning With «, the other three may follow in 6 different orders ; and we shall equally have six different arrangements beginning wath b, or c, or d ; therefore in all 24 different arrangements of the four letters. And just in the same way it is manifest that the five letters, ff, by c, dy and ^, admit five tunes 24 different arrangements or permutations. And thus we see that the number of permutations of 5 different things is the con- tinued product of 5, 4, 3, and 2; or 120: — of 6 different things is 6X5X4X3X2 = 720 : and universally that the number of permutations of n different things is w X ?z — 1 y^n — 2, &c. X2 5 or is the continued product of all the natural numbers from 2 to n. And thus it will be fbund that on a set of 10 bells there may be rung 3,628800 changes. And if we suppose ten changes to be rung in one minute, it would require 252 days to rilig all the changes on 10 bells. But it will be found that all the changes on 12 bells could not be rung in 91 years.
303. Hitherto we have supposed all the terms, whose permutations we enquire, to be different. But let us now suppose that any of the terms are alike : for instance, let us enquire in how many different orders we may arrange the digits of the number 232234, among which six digits there are three 2.'s and two 3.'s. Here the rule for ascer- taining the number of permutations is this : calculate ag before what the number of permutations would be if all
* the
( 181 )
the terms were different ; then the number of permuta*
tions which each set of like terms would admit if they were
different ; divide the former number by the product of the
latter numbers, and the quotient will be the number of
permutations sought. Thus in the present example, six
different digits would admit 720 permutations : but of the
six given digits there are three 2.'s, and three different
terms admit 6 permutations ; there are two 3.'s, and two
720 different terms admit 2 permutations: therefore ~ — , or
^ 6X2
60, is the number of permutations which the digits of the number 232234 admit ; or, with these digits we may ex- press 60 different numbers. We shall proceed to exhibit the truth of this rule in a sufficient variety of instances, to establish it by induction.
304. If there be any number of terms all alike, aa three or four or five a's, it is plain that they admit of but i arrangement ; that is, the number of permutations which so many different letters would admit is - to be divided by itself on account of their being all the same. — If we have any number of terms all of which but one are the same, fhey will admit just as many different arrangements as the number of the terms. Thus, four a's and one 6, will ad- mit five permutations ; for we may begin or end with 6, or interpose b among the a's in three different places ; — haaacty aaaah^ abaauy aabaa^ aaaha. Now the number of permutations which 5 different letters admit is 5X4X3X2 z=120 ; but we find that on account oi four of the letters being the same, this number is to be divided by 4 X 3 X 2, that is, by the number of permutations which four different letters admit. — Again, if we have any number {n) of term§ all of which but two are the same, they will admit a num^ ber of permutations equal to ny^n — 1. Thus, four fli's^ one 5, and one c (or 6 terms, of which four are alike) will admit 30 (6X5.) permutations. For we have proved that the four a\ and the h admit 5 permutations : but in each one of these 5 arrangements (as aaaab) c may take 6 different positions, either in the beginning, or end, or four intermediate places. Therefore 6 times 5 must be the total number of permutations. But 6 different letters ad- mit a number of permutations equal to 6x5x4><:3X2: and OH account of 4 of the letters being the same we see;
th^t
( 1«2 )
that this number must be divided by 4 X 3 X 2, that is, by the number of permutations which 4 different letters admit. — A similar reasoning will establish the rule, where all but three of the terms are alike ; and in every case of this kind. — And we may hence infer the truth of the rule, where we have different sets of like terms : as if we have three a's, tv/o 6's, and one c. For on account of the three a\ being like terms, we have seen that the total number of permutations which six different letters would admit must be divided by 3 X 2 ; and that on account of the two d's being like terms, it must be divided by 2 : therefore on both these accounts together it must be divided by SX2X2.
305. We have hitherto in each permutation included all the given terms. But let us now enquire how many permutations may be formed, out of any number of given terms, in sets consisting each of some lower number : for instance, how many sets of 3 letters variously arranged we may form out of the 8 first letters of the alphabet. The number is 8 X 7 X 6 = 336 ; or is the product of the natural numbers decreasing from 8 to three terms. And univer- sally, let m be the number of different things given, and n the number to be taken at a time in each set, the number of different sets consisting each of n terms which may be formed out of m things is m X m — 1 X m — ^, &c. continued to n terms. Let us now establish the truth of this rule. And first suppose there be 8 different letters, and each per- mutation is to consist of 2 letters. Any permutation may begin with any one of the 8 letters, and this may be fol- lowed by any one of the remaining 7 letters. Therefore the number of permutations in all is 8 X 7, or mXm — 1. Then suppose that each set is to consist of 3 letters. It may begin, as before, with any one of the 8 letters, and this may be followed by as many different sets of 2 letters as can be formed out of the remaining 7 letters. But the latter number we have seen is 7 X 6. Therefore the num- ber of sets of 3 letters variously arranged which can be formed out of 8 different letters is 8 X 7 X 6, or m X m — 1 X m — 2. And just in the same way it may be proved, that the number of sets of 4 letters each, which may be formed out of 8 different letters, is 8X7X6X5 = 1680, or the product of the terms of the series m X m — i , continued to
4 terms :
( m )
4f terms : for beginning with any one of the S letters, it jnay be followed by as many different sets of 3 letters as can be formed out of the remaining 7 ; and this number is, by the last case, 7 X 6 X 5, — If the number in each set is to be only 1 less than the total number of given things, the number of sets will be the same with the number of permutations of the total number of things : or the num- ber of sets consisting each of m — 1 things, which may he formed out of m things, is the same with the number of permutations of m things,
306. As the permtitations of any given things are the dif- ferent orders in which they may be arranged, so the com" binations of any given things are the different collections which can be formed out of them, without regarding the order of arrangement. Here no two sets are to consist of precisely the same things ; but we do not consider a dif- ferent arrangement of the same things as a distinct com,' hination. Thus, let it be required to find how many com- binations of 4 different letters may be formed out of the first 6 letters of the alphabet- Each combination, as abcd^ admits 24< (4 X 3 X 2) permutations. (§ 302.) Therefore the total number of combinations must be the 24th. part of the total number of permutations of 4? letters which caa be formed out of 6 different letters. But this latter num- ber is 6X5X4X3. (§ 305.) Therefore the number of
combinations sought is — — ^ — ^ = 15. And universally
let m be the total number of different things given, n the number of them in each combination, the number of pcr^ mutations consisting each of n things which may be formed out of m things is the product of the terms of the series w, m — 1, &c, continued to n terms: and if this product be divided by 2X3X4... Xw (the number of permutations which n things admit) the quotient will be the number of combinations sought.
Qtiestions for Exercise^^
1. How many different numbers may be written with ^U the significant figures ?
2. How often may a club of 7 persons place themselves at dinner in a different order ?
3. How
{ m )
S, Hqw many different numbers may be written with two units, three 2.'s, four 3 's, and five 4.'s ?
4. How many numbers are there consisting each of four different digits ?
5. How many changes may be rung with 3 bells out of 10?
6. Out of the letters «, b, c, c?, e, x, y^ Zy how many different products may be obtained by the multiplication of two, of three, and of four factors ?
CHAP. XXVIII.
On the Binomial Theorem. Extractioii of the Cube and highei' Hoots^
507. WE have seen that the square of the binomial xdtza is x^z±=.2xa-{-a'' : and that its cube is x^z±iSx^a 4-3.\:a*zt:a^ If we multiply this by xdtzuy we shall have the 4th. power pf that binomial root, and shall find it to he x'^z±z4:x^a'{'6x^a^z:^4!xa^-^a'^. Multiplying this again hy x:±zay we find the 5th. power to be x^z±z5x^a-^lOx^a^ •=±z 1 Ox^a^ 4- 5xa'^=±za^. And in like manner the 6th. power of xz±za is found to be
x^zti6x^a'\'l5x^a^zi=20x^a^ + ISx^a'^zizGxa^ J^a^.
To find the higher powers by this process of continued multiplication would be very tedious ; and in the powers already ascertained there are obvious circumstances appear- ing, which encourage us to investigate the law of their generation : — so much so indeed that I cannot but wonder the discovery was not earlier made. For 1st. we may ob- serve that the number of terms in each series is one more than the index of the power : 2ndly. that in the powers of X — 1/ the signs are alternately plus and 77tijws ; while it is only in this circumstance they differ from the powers of x-\-2/: Srdly. that the first and last terms of each series are the correspondent powers of x and a ; and that in the in- termediate terms, consisting of combinations of a's and <z's, the powers of x continually decrease, and the powers of a increase, by unity ; so that in each term the sum of the
indices
( 185 )
indices of ^ and a is equal to the index of the power of the binomial : 4thly. that in all the powers the coefficient of the first and last terms is 1, and the coefficient of the second and penultimate is the same with the index of the power: 5thly. ^at the series of coefficients proceeding from left to right and irom right to left is the same. And 6thly. it may be remarked that the sum of the coefficients in any of the powers is equal to the corresponding power of 2. Thus in the square of x-\-ay the sum of the three coefficients is 4 = 2^ : in the cube of ^-|-^> the sum of the four coefficients is 8 = 2^, &c.
308. Thus it appears that the only thing remaining to be determined is — the coefficients of the intermediate terms between the second and penultimate. Returning now to the 6th. power of x-\-a^ the two first terms are k^ + Q>x^a»
The coefficient of the second term, 6, or y, or , is
the product of the coefficient of the first term multiplied by the index of x in that first term, and divided by the index of a in the second term. Now, in like manner, the coefficient of the third term 1 Sx^a^ is the product of the coefficient of the second term multiplied by the index of ^ in that second term, and divided by the index of a in the
third term. For — ^ — = 15, And again, 20 the coeffici-
ent of the 4th. term is equal ^o — ^~— , or is obtained by
multiplying the coefficient of the 3rd. term by the index of X in it, and dividing the product by the index of a iu the 4th. term. And this rule we shall find hold good in every other instance.
309. Let us now raise x-\-a to the 7th. power, according to the principles which we have noticed. The literal parts of the eight terms must be
The coefficient of the 1st. term must be 1 j of the 2nd. i term 1^ = 7 ; of the 3rd. term Z-^ = 21 j of the 4th.
^er.m?l^ = a5 ; of the 5th. term ^i^ = 35; of the 6th. 3 4
term
( 186 )
term £1^=21 ; of the 7th, term V-^=.1 ; of the 8th. 5 o
term ..-,^;:;= 1, But we need not have prosecuted the dis-
coveiy of the coefficients beyond the 4th. term ; as we have seen that the coefficients of the four latter terms must be the same with those of the first four in an inverted order. And thus we ascertain that the 7 th. power of ^-fa is
And this result will be found the same with that, which is obtained by multiplying the 6th. power of iV + a by ^-f-^» In like manner the 7th. power of x — a consists of precisely the same terms, but the signs of the 2nd. 4th. 6th. and 8th. terms negative.
310. We may now employ a general formula, putting n for the index of the power to which we want to raise the binomial Xr\-a, The wth. power of ^ + a will consist of n-^l terms ; of which the literal parts will be
The numeral coefficients, or (as they are called) the unci^
|
of the |
terms |
will |
be |
1 '^ |
71 Xn — 1 1X2 ' |
nXn — 1 Xn — 2 1X2XS ' |
|
n X n— |
iXw— |
-2 X n |
—3 3 |
. &c. |
And this |
is the celebrated |
1X2X3X4
binomial theorein discovered (or first brought to perfection) by Sir Isaac Newton : according to which the uncia^ or numeral coefficient of the ?wth. term will be ascertained by taking the continued product of the natural numbers de- creasing from n and continued for m — 1 terms, and di- viding that product by the continued product of the natu- ral numbers decreasing from m — 1 to 2, or to unity. The literal part of the m\\\. term will be a;"-''"+' X «'""' : and if the binomial root be a^— «, the sign of the m\ki, texm will be 7)mius or jplm^ according as m is an even or an odd num- ber. Thus in the 10th. power of ■** — ff, the literal part of
4 ,.^ CR . ^.10X9X8X7
^"^5 and its coefficient is .
4X3X2
= 210.
( 187 )
=210. Tlierefore the 5th. term of a- — «f* is +210*'*^^ j but its 6th. term is — 259..v^a^,
311. After having thus explained the rule, and exhibited its truth in a sufficient number of instances to establish it by induction ; let us now endeavour to investigate the rea- son, why things must be as we have seen they are. Now if we multiply together the 5 binomial factors, -*'+«, x-^-b^ A'-f-c, x-i-d, ^-fe, I say that the terms of the product must include every cornbrnation of 5 letters out of those 10, and no other combinations of letters. For if any one of those combinations, o^ xxbde, did not appear in the product, it is plain that one necessary term of it would be omitted : for the product may be considered as produced by multi- plying xJ^aX X 4- c by x-j-b X x^d X a- -f^ ; and it is plain that in the product of the two former factors xa: is a neces- sary term, and that in the product of the three latter fac- tors bde is a necessary term : tlierefore in the product of the five factors we muBt have the product of jca; multiplied by bde ; or a^a:bde is a necessary term. It is equally evi- dent that no combination of fewer letters than 5, nor of more than 5, can appear in the product. Let us now sup- pose the second term of each binomial factor to be the same, that is, that each of the 5 factors isx-i-a; it is plain that all the possible combinations of 5 letters which can be formed out of these are sij^^ viz. 1. the combination of five jT.'s ; 2. of four ^.'s and one a ; 3. of three ^.'s and tv\T> a.'s ; 4. of two jr.'s and three aJ's ; 5. of one .r and four a.'s ; 6. of five a.'s. And thus it appears that in the 5th- power of ^-f a, the number of terms must be 6, and that their literal parts proceed as we have described in § 307, the indices of a: decreasing by unity from the index. 5, and the indices of a similarly increasing. — Further, if each binomial factor be .v — a, (instead of x-^-a) then the sign of the second, fourth, and sixth terms must be 7ni?7us : since in these terms the index of a is an odd number, and any odd power of a negative root is necessarily negative. — The student will observe that all the same reasoning, which we employ for determining the j^A power of zrdtra, is equally applicable to any other power.
312. Let us now return to the continued multiplication
of the 5 binomial factors x-^-a., ^+b, a^H-c, a' -{-</, j: + ^.
* We see that the first term of the product will consist of a
combination
( 188 )
combination of 5 jr.'s, or wilJ be a:^. This will be followed by all the possible combinations of 4 ^,'s with some one of the 5 letters a, b, c, d, e. But is plain that the number of these combinations is Jive, These will be followed by all the possible combinations of 3 x\ with some two of the 5 letters a^ h^ c, c/, e» But the number of these is
= 10 ; for (by § 306.) this is the number of combina-
tions of 2 that can be formed out of those 5 letters. We shall next have all the possible combinations of 2 xJs with some three of the five letters a^ Z>, c, i1, e^ But the num- ber of combinations of 3 letters which can be formed out
of these 5 beino^ ' = 10, the same must be the num-
^3X2
her of those terms of the product in which only two .r/s are combined with three other letters. These in like man- ner will be followed by all the possible combinations of one X with some four of the other 5 letters : and it appears from the same principles of § 306. tliat the number of these
is . = 5. And lastly we shall have one combi-
4X3X2 '^
nation of the 5 letters ahcde. Now when the second term in each of the binomial factors is the same, or where all the factors are x-\-a^ the Jive combinations in which 4 a:.'s ap- pear become each of them x^a : and therefore 5x^a must be the second term in the 5th. power of 2r-{-«. The ten terms in which 3 jr.'s appear become each of them x^a^ : jind therefore the third term must be lOx^a^. And in like jnanner it appears that the three following terms are lOx^a^y Bxa'^y and a^. — By a perfectly similar process of reasoning, putting 7i for the index of the power, it appears that the first term of the wth. power of a; -fa is a"* j the
71 X J2 1
, second term Jix^'^^a-y the third term .r"--^^*, &c.
For in the third term, for instance, the literal part must consist of a combination of a number of a-.'s less by 2 than n with two ff.'s ; and the number of these combinations, or the numeral coefficient of the third term, must be equal -to the number of combinations of /w-o which can be formed
out of n things. But this by § 306. is ^^^^^~~^ Lastly,
wc have seen that the sum of the coefficients of the 5th.
power
( m )
power of ^4-« is equal to the number of all the diiFerent terms composing the product ofx + aXx-j-bx .v-f c X x^d Xx-\-e. But from the nature of multiplication the num- ber of terms in that product must be 2^ or 32. For the two first factors must give a product consisting of 4 (2*) terms ; and that multiplied by the third factor must give a product consisting of 8 (2^) terms; and the product of this multiplied by the fourth must consist of 16 (2"^^) terms; and this multiplied by the fifth factor must give a product consisting of 32 (2^j terms. — In like manner it appears that the sum of the coefficients in the nth, power of ^-f-^ must be 2".
3 1 3. We have thus strictly demonstrated the binomial theorem for raising a binomial ^ + « to any power, as far as we have hitherto applied it ; namely, where the index of the power is integral and affirmative. But what is most striking and importantly useful in this theorem is, that it is applicable also to those powers whose indices are frac^ tional or negative, Tliis part of the subject we cannot at- tempt to treat minutely in the present elementary treatise : but we shall just present it to the attention of the student by a few examples. Let it be recollected that the square root of ^-f-<7, or Vx-^-a^ may be expressed as the power
of ^ + «, whose index is i, thus (a'-J-a)* j and that the ex- pressions Af-', x^'^y &c. are equivalent with -, — , &c«
See Chap. 22. Now if we apply the binomial theorem for determining the power of a;-J-« whose index is f, we shall find produced an infinite series, which continually app7'oa:imates in value to the square root of ^-j-^* Accord- ing to the formula, or the principles laid down in § 310.
the first term of the series must be ^*, or V^v. The co- efficient of the second term must be i, and its literal part the product of a into that power of x whose index is \ — 1
= — |. But X *=r~--. Therefore the second term is V X
The coefficient of the third term must be
2Vx 2x
- f X — 1^2, that is — \ ; and its literal part the product of I' a^ into that power of x whose index is ^—2=; — ^ j that i&
( ISO )
J^ — or . Therefore the third term is
The coefficient of the 4<th. term must be - and of the fifth term must be tVx — 4-^-4 = literal parts of the fourth and fifth terms must be the pro- ducts of a^ and a'^ into those powers of x whose indices are ^ — 3 (= — i) and | — 4- (= — t) ; that is, must be
— and -^— =♦ or — ^ and f . Therefore the fourth
term is -| r- ; and the fifth term is — • ■■ . . : and so
on. It is plain that the series can never terminate, as the negative values of w — 1, n — 2, &c. continually increase: but the farther we continue the series the more nearly we approximate to the value of the square root of a -fa. Further, that we have not been led astray by any fanciful analogy in considering that square root as the power whose index is i, and applying the binomial theorem to expand that poAver into the form of a series, we may be convinced by proceeding to extract the square root, according to the rule given in § 209. For continuing that process, we shall find precisely the same series
'^2Vx 8v^ T6v^ r28vV7' ^' , . ax^x a^ V X , a^V X Ba'^s/x ^ •r, ^^ + ^--^^+16^-128^' &<^-
314. It appears from the latter form of the series tliat,
if X be a square number, all the terms of the series will
be rational. Suppose ;v = 4, and a=l: then x-]-a=z5;
Vxz=2 ; and all the powers of a=l. Therefore VS = 2
2 2 1 10
-4 -H , &c. Now the square of the
8 128 512 128X256 ^
tw^o first terms exceeds 5 by ^V • hut the square of the three
first terms is less than 5 only by the fraction ^^^. But
instead of seeking greater accuracy in our root by summing
up a greater number of terms in the series, it is better to
change our numeral substitutions for .v and a, by taking a
square number nearer to 5 than 4 is. Now the square of
2^ {the two first terms of the last series) or |J is only T^th.
greater
( 161 )
greater tlian 5. Resolving 5 therefore into |J — -^-^^ and
expanding the square root of this binomial into a series by
the binomial theorem, or the formula at the end of the
Q 2 last section, the two first terms of the series are , or
-=-= . Now this traction is so near the square
72 72 72 ^
root of 5, that its square exceeds 5 only by ^-^^ or is
true to the fifth place of decimals. And if we wish for
greater accuracy, it may be attained by resolving 5 into
leir 1 25921 I K A u
•-- *, or — . And as any number may
72 J 5184< 5184^ 5184 ^ ^
be divided into two parts, one of which shall be a square
number, it is plain that we may thus approximate to the
square root of any number whatsoever : tho' the facility
of continuing the process of extraction decimally makesf
it superfluous to apply the binomial theorem in practice to
this purpose.
315. But let us now by a similar process investigate the
X
cube root of ^+^> or {x-J^ay, Here the first term of the
X
series is ^^ ; and the coefficient of the second term is -f. The index of x in the second term is \ — 1 = — \ ; and the
itidex of a is 1. Therefore the second term is —2 — or
3
. The coefficient of the third term is-fx — |^-7-2r=
3^
— J 5 and its literal part is the product of «* into that power
of f( whose ihdex is \ — 2 = — |. Therefore the third term is
«* X Vx
Qy^z^' The coefficient of the fourth term is — -J X -—4
-j-3=:-/r J and its literal part is the product of a} into that power of ►v whose index is i — S = — |. Therefore the
fourth term is ^^ ^ ^ ...f . And so on. Now from this formula
i
(in
( 1^^ )
(111 which all the terms will be rational ir »• be a cube nlim-' ber) we may approximate to the cube root of any number whatsoever. Thus if we want to extract the cube root of 5, we must divide it into two parts, one of which shall be a cube number, and as near as we can obtain it in value to 5. Now the cube root of 5 evidently lyiiig between 1 and
3 17
2, and nearer to 2 than to 1, we may put V.v= — , and
. r. rr 1^1^ 5000—4913 87 rp, ,. ^ .
therefore a = 5 — — ~ - -.- == Then the 2nd,
JC 1000 1000
aXVx 87 _17 4913X3__ 87x17 493 term — -= X
Sx 1000 10 1000 3X49130 49130'
A A^i. 4^*u * \c 4.^ 17, 493 835214-493
And the sum ot the two first terms — + = JL
10 49130 ^ 49130 ,
rr^ : which exceeds the ti'ue root by less than ;00006.
49130 . . y ■ : (;;.
We might approximate still nearer at pleasure, either by
calculating the value of more terms of the series, or by
3 84014 putting Vx = — or = the nearly equivalent fraction |^|.
And in this manner we may approximate to the 4th. 5th. or any of the higher roots of any assigned number. But for this approximation to the cube root another and much more convenient formula will be assigned in § 3 1 9.
316. The binomial theorem ma}^ similarly be applied to the calculation of powers whose indices are negative. Thus
(x -f- «)—^ =r = _ — -I — — , Sec. this beinnf the series
into which the fraction is expanded by actual divi- sion. See Chap. 17. But we shall have the very same series, if we calculate the value of (/v-f ^)""' by the bino- mial theorem. For then the first term of the series must
be A?-' =-. The coefficient of the second term must be
X
-— 1 ; and its literal part the product of a into x"'^ or into — . Therefore the second term must be . The coeffi-
fl?* . ■ . . x^
rient of the third term must be _— ^ = 1 j and its literal
part
( 193 ) .«f^-**^^
f •■■Mill
part tke product of u^ into w**^, or into — . Hierefoi'e
■2,
the third term is H — j. The coefEclent of the fourth term
1 b^ g
must be = — !• And so on.
3
317. In like manner if we expand {xJ^^ay^ into an in-^
finite scries by the binomial theorem, the first term is
A-'^'ir — , The coefficient of the second term is — 2 j and
AT*
its literal part the product of a into ;«-*', or into —
Therefore the second term is 5. The coefficient of the
third term is =3; and its literal part is — :
so that the third term is -f - — . In like manner the fourth
and fifth terms ai*e found to be and --?• : and so on.
And universally expanding (x^a)--^ into an infinite series by the binomial theorem we find the series
* _1__^ na j.ny.n-^1 Xa* y?Xy?4-l Xy?4-2xa^ g^
AT""^^ 2^"+^ 2x3;f"+J '
And by this formula we may calculate the value of any 1 Jx+af
such fi'actiotis as -, or , or . — The
I
VA' + a V{xJira)^
truth of this formula may be thus established. Since any fraction multiplied by its reciprocal gives 1 for the product,
unity must be the product of — X (^r+a)". The latter
* In Elder's Algebra Vol. I. p. 179 (2nd. Ed. Lond. 1810) there is a material error in the delivery of this formula. In the numerators of the terras^ in- stead of «+l, «+2, &c. they are given n — 1, « — 2, &c. which neither corresponds wkti the result of the bin^n^al theorem, nor with the parti- cular cases before exhibited.
O by
( iU )
h^ the^ binomial theorem is equal to
..'+««.-'+^!f!=^x«'-^ &c.
Now if we multiply the terms of this formula for (*'+«)« by the terms of the formula for its reciprocal, we shall find the product of the two first terms to be 1, and the several products of the other terms successively destroying each other. Let us exhibit this in a trinomial of each formu- la, as it will afford a useful praxis to the student : and let him recollect that powers of the same root are multiplied or divided by adding or subtracting their indices.
T., , . , 1 na n^a^
Multiply — -r + — — r
by x"" 4- 7iax'^ +
|
1 |
X |
n^'a'' |
|
*~ |
n^ a^ n'^a^ x' 2x^ + w'^a* n^a"^ n^a"^ 2;^* 2x^ 4a?^ |
|
+
4^^
Thus all the terms have disappeared except 1, and the product of the two last terms of the trinomial factors : which would in like manner be destroyed by the following- terms, if we took another term of each formida.
318. The rule commonly given in the systems of Arith- metic for the extraction of the cube root directs to an ope- ration so extremely tedious and troublesome, that it is of little or no practical utility, It may be needful however to make a few remarks on the grounds of tlie operation. It depends upon the constitution of the cube of the bino- mial a-\-x, namely a'^-^Sa^ x-^ 3 ax^-^-x^. The cube root of the first term of this is the first term of the root ; and
3 times
( 195 )
8 time^ its square dividing the second tei'm, Sa^x^ give^ the second term of tlie root. If there be more terms thari two terms in the root, for instance if we have to extract the cube root of
—after determining the first term of the root x^, we divide the second terrii 6x^d by Sx^, The quotient 2xa is the se- cond term of the root. Now considering the two terms found, x^ -{-^xa; as the ascertained part of the root, we subtract the cube of that binomial, x^ -{■ 6x^a'{-l2x'^a^ -{•Sx^a^y from the given cube. The remainder is Sx'^a^ + {2x^a^y Sec, the first term of which we divide by Sx"^^ and the quotient a"^ is the third term of the root. And the extraction is complete, since the cube of x'''\-2xa-\-a'' is found to be just equal to the assigned cube. By a similar process we may proceed in the extraction of the 4th. root of any assigned quantity, (arranged according to the powers of some one letter) by taking the 4th. root of its first term for the first term of the root, and dividing the second term by 4 times the cube of this, for finding tlie second term of the root. Subtracting then the 4th. power of the two parts of the root found from the given quantity, we divide the first term of the remainder by 4 times the cube of the first term of the root for determining the third term of the root. And we may proceed similarly in ex- tracting any higher roots*
319. But to extract the cube root of 5, for instance, to 6 decimal places by such a process would be insufferably tedious : and we may effect the object with little compara- tive trouble by the following formula. Let a be any num- ber, w^hose cube root we desire to extract. Assume r^ a perfect cube, as near as may be to a, either greater or less.
Then — ^— =-X r =::Va nearly. Suppose we want to find
the cube root of 5 : we are in the first place to assume a perfect cube number as near as may be to 5 -, and the nearer we approximate to 5 in our substitution for r^ the more accurate will be the result of the formula. Now the cube root of 5 lying between 1 and 2, we might try |, 4> and ^, as approximations to its root : but of these i is the
O 2 nearest*
( 196 )
nearest. (For the cube of 4=y, less than 5 by V ; tKe ^ubeof 4— VtS l^s than 5 by -14 j the cube of i= VV>
more than 5 by i|.) Assuming then r=-, r^=: , a = 5
4 64?
= ; we have — -Xrrr- X-=^ = 1.709990,
64 a + 2r^ 1006 4 4024 '
which is the true root to the 5th place of decimals, and
exceeds the true root by little more than — l — . If in-
•' 5 oooo
stead of :J, we put |# for r (which is -^ less than ^, though its cube be gtill somewhat greater than 5) the same formula
would afford us V 5 -.^^^^H?! = 1.709976 ; which is ac- 39080721
curately true at least to the 7th, place of decimals. — Again,
to extract the cube root of 131, putting 7-^ = 125, r=z5f
, , 2624-125^- 387^. 1935 ^ n^Q>,. ,
iv-e have , — ~L X 5 = X 5 = = 5.07874 = nearly
131 -f 250 381 381 ^
the cube root of 131, being true to the 5th. place of de- cimals. And we may approximate nearer by putting
r=z — —^ or rather =— — . — But I would recommend that 38 100
the number originally assumed for r^ should be taken suf- ficiently near the given number, to prevent the necessity of a repeated operation. Thus instead of assuming r^ = 5^ = 125, which is less by 6 than the given number, let
us assume r=^, — , and therefore r'^ = —^ — — , which ex- 10 1000
ceeds 131, or — , only by 1 and we shall find the
' 1000 -^ ^ 1000
result of the formula in one operation to be v'lSlrr 5.0.78753, which is true to at least the 7th. place of de- cimals.
320. Having thus shewn how we may approximate at pleasure to the cube root of any assigned number not a perfect cube J I shall only add that the roots of perfect cube numbers, up to one billion^ may be ascertained with much facility in the following manner. We at once know the number of digits in the root, by pointing off the num- ber in periods of 3 figures from the right hand, and reckoning as a period the left hand digits thus cut off, whether they be one, two, or three. Thus, if the cube consist of 4, 5, or 6 digits, its cube root must consist of 2 digits ; if the cube consist of 7, 8, or 9 digits, its root
must
( 197 )
must consist of 3 digits; 'if the cube consist of 10, 11, or 12 digits, its root must consist of 4 digits. (The reason of this will appear just as we ascertained the number of digits in the square root, § 206.) The first period of the cube determines the first digit of the root to be tliat, whose cube is next below that period. The last digit of the cube determines the last digit of the root to be that, w^hose cube ends with that digit : for there are no two digits whose cubes end with the same digit. Thus if 300,763 be proposed as a perfect cube, we at once know that its root is 67, as the cube of 6 (or 216) is the nearest cube number below 300, and 7 is the only digit whose cube ends with 3. But w^ might otl^erwise detern^ine the first digit of this root to be 6 : thus — Subtract the penultimate digit of 7^ (or 343) from 6 the penultimate digit of the given cube. The re- mainder is 2. Then consider what anultiple of 7, the last digit of 3 X 7^, ends with 2 : and 42 being 6 times 7, this determines the 'penultimate digit of the root to be 6. Thus again 5,451,776 being proposed as a cube number, the first digit of its root is necessarily 1 , and the last 6 ; and the penuhimate digit is necessarily 7: for subtracting 1, the penultimate digit of 216 (6^) from 7 the penultimate digit of the given cube, the remainder is 6 j but 3X6^ ends with 8 ; and 7 or 2 is the only digit which multiplying 8 gives a product ending with 6. We fix npon 7, as the root sought is evidently nearer 200 than 100. — From the two last digits of the root being 76, we might determine that the first, or antepenultimate, digit is 1. For 76^ = 438976 : and subtracting 9, t^he antepenultimate digit of this number from 7 (or 1 7 ) the corresponding digit of the given cube, the remainder is 8; and 1 (or 6) is the only digit which multiplying 8 (the last digit of 3x6*) gives a product ending with 8. — Thus again, 3,086,626,816 being proposed as a cube number, the last digit of the root is 6j, and subtracting 1, the penultimate digit of 6^, from 1, the rem?4nder is 0 ; and 5 being the only digit which mul- tiplying 8 (the last digit of 3X6*) gives a product ending with 0, 5 must be the penultimate digit of the root. No\y 56^ = 175616: and subtracting 6 the penultimate digit of 56^ from 8 the corresponding digit of the given cube, the remainder is 2. Therefore 4 is the antepenultimate digit of the root, as 4 is the only digit which multiplying 8 gives a product ending with 2. But 1 is the first digit of 2ie root. Therefore the cube root sought is 1456.
TABLES
I
( 198 )
TABLES
L Of English Money,
4 Farthings = 1 Penny. 4 Pence = 1 Groat. 1 2 Pence x=. 1 Shilling. 55. = 1 Crown. 20 Shillings = 1 Pound Sterling. 21 Shillings = 1 Guinea. {6s. 8d. = 1 Noble. 105. = 1 Angel. 13s. 4^. = 1 Marl?. — In Ireland the va- lue of the Penny is le§s in the ratio of 13 : 12. — Scots Money is divided in the same manner as English ; but has one twelfth of its value. Thus a Pound Scots =15. 8d.
II, So7?ie Foreign Coins, or Denominations of Money
rediiced to English,
A Florin = Is. 6d, a Ducat = 9^, Sd. a Guilder (= 20 Stivers) = Is. 9d. a Rix-dollar (= 50 Stivers) = 45. 4fid, a Ruble (=100 Copecs) = 45. 6d, a Sol (=12Deniers^ = id. a Livre Tournois ( = 20 Sols) = 10c?. a French Pistole (= 10 Livres) = 85. 4c?. a Louis d'Or (=24 Livres) = l£. a Milre :::^ 5s. l\d. a Spanish Dol- lar (= 10 Rials) = 45. 6d. a Spanish Pistole (=36 Rials) = 1 Qs. 9d. a Sequin = 75. 6d. a Rupee = 25. 6d. a Gold Rupee (=4 Pagodas) = jBJ, 15s.
III. So7ne ancient Coins, or Denominations of Money ^
reduced to English.
Drachma (= 6 Oboli) = l^d. aMina (= 100 Drachmae) = j£3. 45. Id. a common, or Attic Talent (= 60 minae) = j^l93. 155. — (Note — the Mina and Talent are properly denominations of w^/V/z^.)-a Golden Stater ( = 25 Drachmae) = I65. l\d. A Denarius (= 10 Asses = 4 Sestertii) = 7-J:(^.
IV. English Weights — Avoirdupois.
1 Ounce = 16 Drams. 16 oz. = 1 Pound. 28 lbs. = 1 Quartci'.112lbs. ( = 4 Qrs.) = l Hundred. 20 Ovt. = lTon.
V. Troy
( 199 )
V. Trot^ Weight — -used for "weighing Gold, K^ilver, Jexvels, Silk, a?id all Liquors*
24 Grains = 1 Penny-weight (dwt.) 20dwts. ^l Ouncfe. 12 oz. = 1 Pound. — The following also used by Apothe- caries in compounding their medicines, 20 Grains = 1 Scruple. 3 Scruples = 1 Dram. 8 Drams = 1 Ounce. — Note, — the troy Pound is to the Avoirdupois Pound nearly as 88 : 107. The Troy Ounce is to the Avoir- dupois Ounce nearly as 80 : 73.
The Paris Pound = 1 lb. 3 oz. 15 dwts. Troy.
The Paris Ounce =19 dwts. 16i:gr. Troy.
The Roman Libra (=12 Unciae) = 10 oz. 18 dwts. 14 gr. Troy, nearly.
The Roman Uncia = the English Avoirdupois Ounce.
The Attic Drachma = 2 dwts. 1 7 gr. nearly.
The Attic Mina (=100 Drachmae) = 1 lb. 1 oz. 10 dwts. 10 gr.
The Attic Talent (=60 Minae) 67 1b. 7 oz. 5 dwts. Troy.
VI. Measures of Length:,
12 Inches = 1 Foot.' 3 Feet = 1 Yard. 2 Yards = 1 Fathom. B\ Yards = \ Pole. 40 Poles (= 220 Yards) = 1 Furlong. 8 Furlongs (= 1760 Yards) = 1 Mile. 3 Miles = I League.— The Irish Mile = 2240 Yards 5 arid therefore is to the English as 14 : 11,
The Roman Foot = llf Inches nearly.
5 Roman Feet = 1 Passus. 125 Passus = 1 Stadium. 8 Stadia (= 1000 Passus) = 1 Milliare: which was there- fore to the English Mile as 967 ; 1056 j or nearly as 23 : 25.
The Grecian Foot exceeded the English by nearly -j^ of an Inch. — The Persian Parasang = 30 Stadia.
A French League = 2^^ English Miles nearly.
A Toise = 6 French Feet, or 6f English Feet nearly.
A German Mile = 4 English. — A Russian Verst = ^ Do.
In measuring Cloth, &c. 2^ Inches = 1 Nail ; and there- fore 4 Nails = 1 Quarter of a Yard. 3 Quarters = 1 Ell Flemish. 5 Quarters = 1 Ell English. 4 Quarters, 1} Inch. = 1 Ell Scots.
Ill
( 200 )
In Land-measurinff, a Perch = 16i Feet in Length : of which 40 in Length and 4? in Breadth make an English Statute Acre = 43560 Square Feet = 4840 Square Yards rr 160 Square Poles = 4 Roods. — The Irish Acre exceeds the English by 2 Roods 1 9| Perches nearly. — The French arj^ent contains 1^ English Acre.
VIL Measures of Capacity-— for Liquids,
2 Pints =: 1 Quart. 4 Quarts = 1 Gallon. — In Ale and Beery 36 Gallons = 1 Barrel. H Barrel (= 54 Gallons) = 1 Hogshead. 2 Barrels = 1 Puncheon. 2 Hogsheads = 1 Butt. 2 Butts = 1 Tun.—In Winey Spirits^ &c. 42 Gallons =r 1 Tierce. It Tierce (=63 Gallons) = 1 hogs- head. 2 Tierces = 1 Puncheon, 2 Hogsheads = 1 Pipe.
2 Pipes = 1 Tun. — Note — the Ale Gallon contains 282 cubic Inches ; the Wine Gallon 231.
The Roman Cyathus =: -rV Pint, Wine Measure : the Hemina ( = 6 Cyathi) = ~ Pint : the Sextarius = 1 Pint : the Congius r= 7 Pints : the Urna = 3 Gallons 4t Pints : the Amphora == 7 Gallons 1 Pint.
The Attic Cyathus = -/^ Pint : the Cotyle = f Pint.
yill. Ihy Measure.
2 Pints =: 1 Quart. 2 Quarts = 1 Pottle. 2 Pottles :;= 1 Gallon. 2 Gallons = 1 Peck. 4 Pecks = 1 Bushel. 8 Bushels = 1 Quarter. 5 Quarters = 1 Wey, 2 Weys =r 1 Last. — Note— the Winchester Bushel contains 2250 Cubic Inches. ,
The Roman Modius z=z 1 Peck, or 2 Gallons.
The Attic Choenix = 1 Pint : the Medimnos = 1 Bush.
3 Quarts.
IX. Time.
60 Seconds = 1 Minute. 60 Minutes = 1 hour. 24 Hours = 1 Day. 7 Days = 1 Week. 365^ Days = 1 Juliiin Year — 52 Weeks, 1 Day, 6 Flours : — The Solar Year = 365 Days, 5 Hours, 4 minutes, 48 Seconds.
FINIS.
CONTENTS.
Page Chap. I. Nature and Princi-ples of the Arabic Nu^ meral Notation, Its Advantages above the Greek a?id Roman, Insensibility to the Magnitude of high Num^ hers* Duodecimal Notation, - • 1
Chap. II. Addition and Subtraction, Reason of jpro* ceeding from Right to Left, Methods of Proof Examples for Practice. Signs -^-^ — , =. - 5
Chap. III. Nature and Princijples of Midtiplicatio7i» Sign X . Methods of Proof Powers. Questions for Exercise - - - - JO
Chap. IV. Nature and Pi^inciples of Division, Sign -f-. Division of a smaller Number hy a greater ^ Methods of Proof, Qtiestioiis for Exercise - 1^
*Chap. V. Methods of abbreviated Operation^ and of proving Division^ continued. Properties of the Num^ hers 3, 9, 11, ^c. - - - 23
Chap. VI. Practical Application of Multiplication and Division, Qiiestioiis far Exercise - - 30
Chap,
262 <ibNTENTS^.
Page Chap. VII. Doctrine of Ratio — direct — inverse — compound. Method ofjtnding a fourth Proportional. Abbreviations.. Qiicstiojis for ILxerxise - - 32
Chap. VIII. On the Nature of Fractions - 41
Chap. IX. Addition and SubtMction of Fractions 45
Chap. X. Midtiplication and Division of Fractions 47
Chap. XI. On the Nature of Decimal Fractions - 51
Chap. XII. Arithmetical Ojferations on Decimals - 53
Chap. XIII. Practical Application of the Ride of
Proportion to Interest, Discount, Exchange, Fellow-
ship, Eqiiation of Payments, %c - -56
Examples for Practice - - ■* 65
Chap. XIV. Origin and Advantages of Algebra. Algebraic Notation., Definitions - -■ 6^
Chap. XV. Positive and Negative Qiiantities. Al- gehraic Addition and Sid)traction * - 72
Chap. XVI. Algebraic Multiplication * 78
Chap. XVII. Algebraic Division. Resolution of Fractions into infinite Series - - 81
Chap. XVIII. Algebraic Operations on Fractional Quantities. Method of fnding the least common Multiple ^ . ^ ^ ^ m
Chap. XIX. Arithmetical Progression - 89
Chap. XX. Geometrical Progression - • 93
Chap. XXI. Extraction of the Squoi^e Root - 99
* Chap. XXII. Fractional and Negative Indices. Cal- culation of Surds - - - - 105
Chap. XXIII. Reduction of Algebraic Equations^ Simple and Qiuidratic - * * 110
*Chap.
CONTENTS* 205
Page
*ChAP. XXIV. On the Forms and Roots of Qim* dratic Eqtmtions. Method of exterminating the se* cond Term * - . . 12^
Chap. XXV. Reduction of two or more EquationSf involving several unknown Quantities - 129
Chap. XXVI. Application of Algebra to the Solutio?i of Arithmetical Problems - - • 134«
Qjuestions for Exercise * - - 170
*Chap. XXVII. On Permutations and Combinations 180 *Chap. XXVIII. On the Binomial Tlieorem. Ex- traction of the Cube and higher Roots • - 134« TABLES . • . . . 198
CCj" The Chapters marked with an Asterisk may be omitted by the Student, in the first reading ; a* well as the passages included between Crotchets [ ],
The following Works, by tlie same Author, are sold by the Publishers :
The First, Second, and Sixth Books of Euclid's Ele- ments, demonstrated in general Terms ; with Notes and Observations.
A Commentary on the Compendium of Logic used by under-graduates in the University of Dublin : to which are subjoined An Address to a young Student on his En- trance into College : and A full and plain Account of the Horatian Metres.
Also, speedily will be published
An Essay on the following Prize-question, proposed by the Royal Irish Academy, WJiether and Jicm far the CuU twation of Science and that of Polite Literature assist or obstruct each other.
Cjf The seventh^ and lasty Volume^ of Mr. Walkf<r's Edition of Livy is in the Press,
i
f . *- > -v
i
m
kSw~
THIS BOOK IS DUE ON THE LAST DATE STAMPED BELOW
AN INITIAL FINE OF 25 CENTS
WILL BE ASSESSED FOR FAILURE TO RETURN THIS BOOK ON THE DATE DUE. THE PENALTY WILL INCREASE TO 50 CENTS ON THE FOURTH DAY AND TO $1.00 ON THE SEVENTH DAY OVERDUE.
Onr
^^m^
OCT A7iaa
73reb'50SS
m3L
><^^<^ 31953iU
RECEIVED
FEB13^69-10PM
i;qan dept^
SgWrONfLL
JAN 1 8 20(11
u.aiERKiLey
LD 21-100m-7,'39(402s)
WW^^^^:*,