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I 

« 

I 



I 



COURSE 



OF 



MATHEMATICS. 



IN THREE VOLUMES. 



COMP08BD FOR 

THE USE OF THE ROYAL MILITARY ACADEMY, 

BT ORDBR or HIS LORDCHIP 

THE MASTER GENERAL OF THE OM)NANC£. 



BY 

CHARLES HUTTON, LL.D. F.R.S. 

J^TB PROFESSOR OF MATHEMATICS IN THE BOTAK. 

MILITARY ACADEMY. 



THE SIXTH EDITIOtr, 

XHLAROID AilD CORRBCTBI^ 



toL. I. 



LONDON: 

PRINTEB FOR F, C. AND J, RIVINGTON; O. WILKXB AN9 f, 
ROBINSON ; J. walker; G. ROBINSON; LACKINOTON, ALLBNf' 
AND CO.; VERNOR, HOOD, AND SHARPE ; 6. KSARSLBY ; 
LONGMANy HURSTy REES, ORME, AND BROWN ; CADELI AN1I 
DA VIES ; J. CUTHELL ; B. CROSBY AND CO. ; J. RICHARDSON ; 
J. M. RICHARDSON; BLACK^ PARRY^ AND I^^NOS^URY ; QAIA 
AND CURTIS ; AND J. JOHNSON AND CO. 

1811. 



' * »• J 



I , 



• « 



\ 



V •■ 



I * 



' t * 



' • • • F 



w >'^ *<r* 



T. Dttviwp, Lombard 8tice<» 









X 



PHEFACE. 



A SHORT and Easy Course of t&e Mathematical Sciences 
h^s* long been ^considered as a desideratum for the use of 
Students in the different schools of education: one that 
should hold a middle rank between the more voluminous 
^nd bulky collections of this kind, and the mi^e abstract 
% and brief common-place forms, of principles and. memo* 

fandums. 
«^ For Ung experience, in all Seminaries of Learningji ha$ 
j> shown, that such a work was very much wanted, and would 
>l prove a gre^t and general benefit ; as, for want of it, re- 
4iourse has always been oblig^ to be bad to a number of 
other books, by different authors; selecting a part from one 
and a part from another, as seemed most suitable to thef 
purpose in hand, and rejecting the other parts— a practice 
which occasioned much expence and trouble, in procuring 
and using such a number of odd volumes, of various forms 
and modes of composition; besides wanting the benefit of 
uniformity and referencei which are found in a regular series 
of composition. 

To remove these inconveniences, the Author of the pre- 
sent work has been induced, from time to time, to compose 
various parts of this Course of Mathematics; which the 
experience of many years' use in the Academy has enabled 
him to adapt and improve to the most useful form' and 
quantity, for the benefit of instruction there. And, to render 
ti^t benefit more eminent and lasting, the Master General 
of the Ordnance has been pleased to give it its present form, 
Ky ordering it to be enlarged and printed, for the use of the 
. Roayl Military Academy. 

A 2 As 



IT PREFACE. 

As this work has been composed expressly with the inten- 
tion of adapting it to the purposes of academical education, 
it is not designed to hold out the expectation of an entire 
new ipass of inventions and discoveries: but rather to collect 
^d arrange the most useful known principles of mathe- 
matics, disposed in a convenient practical form, demonstrated 
in a plain and concis'e way, and illustrated with suitable ex- 
amples; rejecting whatever seemed to be matters of mere 
curiosity, and retaining only such parts and branches, as have 
a direct tendency and application to some useful purpose in 
life or profession. 

It is however expected that much that is new will be found 
in many parts of these volumes; as well in the matter, as in 
the arrangement and manner of demonstration! throughout 
the whole work, especially in the geometry, which is" ren- 
dered much more easy and simple than heretofore ; and in 
the conic-sections, which are here treated in a manner at 
once new, easy, and natural ; so much so indeed, that all the 
propositions and their demonstrations, in the ellipsis, are the 
very same, word for word, as those in the hyperbola, using 
only, in a very few places, the word sum, for the word differ^ 
ence: also in many of the mechanical and philosophical parts 
which follow, in the second volume. In the conic sections^ 
too, it may be observed, that the first theorem of each sec- 
tion only is proved from the cone itself, and all the rest of 
the theorems are deduced from the fint, or from each other, 
in a very plain and simple manner. 

Besides renewing most of the rules, and introducing every- 
where new examples, this edition is much enlarged in several 
places; particularly by extending the tables of squares and 
cubes, square roots and cube roots, to 1000 numbers, which 
will be found of great use in many calculations ; also by the 
table of logarithms at the end of the first volume, and of lo- 
garithms, sines, a(nd tangents, at the end of the second 
Yolume ; by the add;ition of Cardan's rules for resolving cubic 

equations I 



PREFACE. V 

equations; with tables and rules for annuities; and many 
other improvements in different parts of the work* 

Though the several parts of this course of mathecnatics are 
ranged in the order naturally required by such elements^ yet 
students may omit any of the particulars that may be thought 
the least necessary to their several purposes; or they may 
study and learn various parts in a di&rent order from their 
present arrangement in the book, at the discretion of the 
tutor. So, for instance, all the notes at the foot of the pages 
may be omitted, as well as many of the rules ; particularly 
the 1st or Common Rule for the Cube Root, p. 85, may well 
be omitted, being more tedious than useful. Also the chaf^ 
ters on Surds and Infinite Series, in the Algebra : or these 
might be learned after Simple Equations. Also Compound 
Interest and Annuities at the end of the Algebra. Also any 
part of the Geometry, in vol. 1 ; any ojf the branches in 
vol. 2, at the discretion of the preceptor. And, in any of 
the parts, he may omit some of the examples, or he may 
give more than are printed in the book ; or he may Tery pro- 
fitably vary or change them, by altering^the Aumbera oc» 
casionally.— -As to the quantity of writing ; the author would 
recommend, that the student copy out into his fair book no 
more than the chief rules which he is directed to learn off by 
rote, with the. work of pne example only to' each rule, set 
down at full length : omitting to set down the work of all thff 
other examples, how many soever he may be directed to 
work out upon his slate or waste paper. — In short, a great 
deal of the business, as to the quantity and order and maimeri 
must depend on the jtt4goient of the discreet and {Hmdeiit 
#utor or director* 



CONTENTS 



OF VOLUME I. 



GENERAL Pr^mmaty Prtnci^ 



P«f» 



ARITHMETIC. 


/ 


Ihtaiiim and Nnmeratwn • • . • . 


4 


Rsman Notation . • • • * • 


7 


AMUim 


8 


jfuitraetion ....... 


11 


J/ukiptfcation • . 


.13 




18 


^Bedmtion . . ^ 


2S 


' Gmpaund JMMon • . 


92 


■ Subtracit&H .... 


S< 




M 




^o 


-— Dioision . . . . « 


41 


Getien Bule^ or Stde of Three • . • . 


44 


Compound Pfopofpfion . . . . , 


49 


Vtdgar Fractions 


51 


Beduetion of Vulgar Fractions . . . , 


54 


Addition of Vulgar Fractions . . . , 


62 


Subtraction of Vulgar Fractions 


62 


MtdttpKcaiim of Vulgar Fractions . . 


63 


DMsiou of Vulgar Fractions- . . . 


64 


i?Ei& g/" 7%retf in Vulgar Fractions 


65 


Decimal Fractions 


66 


Addition of Decimak . . . ^ . , 


67 


Subtraction of Decinuils 


68 


Multiplication of Djscimals 


ib. 



C0N1!ENI3. 


«n 




r»€» 


Dwision of Decimals . • « . 


70 


JMuction of Decimals • • 


. . 75 


Mute of Three in Decimals • 


T« 


Dmsdecimals ... 


. . TT 


Iwfolution 


•» 


Evolution ...... 


«• 


To extract the Square Soot 


«1 


To extract the Cube Root . . , 


«5 


To -extract any Root whatever 


«8 


Table qfJPawers and Roots 


«0 


Ratios f Propor turns, and Progressions , 


. , . 110 


Arithmetical Proportion • • • 


111 


Geometrical Proportion 


lie 


Musical Proportion . • • . 


i» 


fiUasosh^fj or Partnership , 


ft. 


Single Fellowship . • . . 


ISO 


Btmble Fellowship . . . . 


m 


Smple Interest . . • . . 


IS* 


Vmpound Inttrest . • « . 


. . . U7 


dU^gatim Medial 


U9 


^Hkgatwn Alternate . » - . . . 


• 


Jbh^/^ Position . . . . . . . 


iS5 


Double Position 


1S7 


Practical Suestions • . . < 

• 


140 


LOGARITHMS. 


• 


i^nitim and Propertks vfJJDgari^nm 


r . . 14S 


To 'Compute Logarithms . ■ , ■ 


: . . 149 


Description and Use cf Lagarkhim 


. « . i» 


Mnltiplieation-by LogariU^ms . 


157 


iiivisum-by Logarithms . . 


. . .158 


involution by Logarithms , 


159 


iSvoluiion h/ Logarithm* . . . 


kW 



«m 



CCKNTENTS. 



ALGEBRA. 

Definitions and Notation 
Addition . . • . 
Subtraction . . 

Multiplication . . 
Division . . . . 
Fractions . , . 
Involution * 
Evolution • . . 
Surds 

Infinite Series 
ArUhmetical Proportion . 
Arithmetical Progression 
PUes of S/iot or Shells 
Geometrical Proportion . 
Simple Equations . . 
Suadratic Equations . 
Cubic and Higher Powers 
Simple Interest 
Compound Interest - . 
Annuities • . . • 

GEOMETRY. 

Definitions • • • . . 

Axioms 

, Remarks and Theorems , 

Of Ratios and ProportionS'^Definitions 

Theorems 

Of Planes and Solids-^Definitions 

Theorems . • . . ; 

Problems . 

Application of Algebra to Geometry 
, , Problems . • . . . 

Table qf Logarithms . . . • 



Paftf 
161 

165 
170 

171 
174 
178 
189 
198 
196 
203 
208 
210 
213 
218 
28(1 
23d 
247 
256 
257 
260 



265 
271 
ib. 
S09 
318 
326 
328 
S43 
359 
360 
S66 



COURSE 



OP 



MATHEMATICS, 4-0. 



GENERAL PRINCIPLES. 

UANTITT, or Magnitude, is any thing that mil 
admit of increase or decrease ; or that is capable of any sort 
of calculation or mensuration : such as numbers, lines, spacey 
time, motion, weight. 

2, Mathematics is the science which treats of all kincls 
of quantity whatever, that can be numbered or measured.-— 
Th^ part which treats of numbering is called Arithmetic; 
and that which concerns measuring, or figured extension, is 
called Geametfy^-^These two, which are conversant about 
multitude and magnitude, being the foundation of all the 
other parts, are called Pure or Abstract Mathematics; be- 
cause they investigate and demonstrate the properties of ab- 
stract numbers ^d magnitudes of all sorts. And when th^se 
two parts are applied to particular or practical subjects, they 

.eonsdtute the branches or parts called Mixed Mathematics.--^ 
Mathematics is also distinguished into Speculative and Prvnv 

Jica/c viz. Speculative, when it is concerned in discovering 
properties and relations ; and Practical, when applied to 
practice and real use concerning physical objects* 

\, Vo^L B 3.to 



2 GEigERAL PRINCIPLES. 

3. In Mathematics are several general terms orprinci^es; 
such as. Definitions, Axioms, Propositions, Theorems, rro- 
blems. Lemmas, Corollaries, Scholiums, &c. 

4. A Definition is the explication of any term or word in a 
science ; showing the sense and meaning in which the term 
is employed. — Every Definition ought to be clear, and ex- 
pressed in words that are common and perfectly well under- 
stood. 

5. A Proposition is something proposed to be proved, or 
something required to be done ; and is accordingly either si 
Theorem or a Problem* 

6. A Theorem is a demonstrative proposition; in which 
some property is asserted, and the truth of it required to be 
proved. Thus, when it is said that. The sum of the three 
angles of any triangle is equal to two right angles, this is a 
Theorem, the truth of which is demonstrated by Geometry. 
—A set or collection of such Theorems constitutes a Theory. 

7. A Problem is a proposition or a question requiring 
something to be done ; either to investigate some truth or 
property, or to perform some operation. As, to find out the 
quantity or sum of all the three angles of any^triangle, or to 

di'aw one line perpendicular to another. A Limited Pro» 

hlem is that which has but one answer or solution.- An Un^ 
limited Problem is that which has innumerable answers. 
And a Determinate Problem is that which has a certain num- 
ber of answers. 

S. Solution of a Problem, is the resolution or answer given 
to it. A Numerical or Numeral Solutiony is the answer given 
in numbers. A Geometrical Solution^ is the answer given by 
the principles of Geometry. And a Mechanical Solution^ is 
one which is gained by trials. 

9. A Lemma is a preparatory proposition, laid down in 
order to shorten the demonstration of the main proposition 
which follows it. 

10. A Corollary y or Consectary^ is a consequence draW|x 
immediately from some proposition or other premises. 

11. A Scholium is a remark or observation made by some 
foregoing proposition or premises. • 

12. An Axiom, or Maxim, is a self-evident proposition ; 
requiring no formal demonstration to prove the truth of it ; 
but is received and assented to as soon as mentioned. Such 
as, The whole of any thing is greater than a part of it ; or. 
The whole is equal to all its parts taken together: or. Two 
quantities that are each of them equal to a third quantity, 
are equal to each other. / 

13. A 



GENERAL PRINCIPLES. S 

Id* A Pastulatey or Petition, is sometliing required to be 
donC) which is so easy and evident that no person will hesi- 
tate to allow it. 

14. An Hypothesis is a -supposition assumed to be truei in 
order to argue from^ or to found upon it the reasoning and 
demonstration of some proposition. ^ 

15. Demonstration is the collecting the several arguments 
and proofs, and laying them together in proper order, to 
«}iow the truth of the proposition under consideration* 

16. A Direct, Positive, or Ajffirmative Demonstration, h 
that which concludes with the direct and certain proof of the 
proposition in hand. — This kind of Demonstration is most 
satisfactory to the mind ; for which reason it is called some- 
times an Ostensive Demonstration. 

17. An Indirect, or Negative Demonstration, is that which 
shows a proposition to be true, by proving that some absur- 
dity would necessarily follow if the proposition advanced were 
false. Thrs is also sometimes called Reductio ad Absurdum; 
becausie it shows the absurdity and falsehood of all supposi- 
tions contrary to that contained in the proposition. 

18. Method is the art of disposing a train of arguments in 
a proper order, to investigate either the truth or falsity of a 
proposition, or to demonstrate it to others when it has been 
found out. — This is either Analytical or Synthetical. 

19. Analysis, or the Analytic Method, is the 2rt or mode 
of finding out the truth of a proposition, by first supposing 
the thing to be done, and then reasoning back, step by step, 
till we arrive at some known truth. — ^Thi$ is also called the 
Method of InventioHy or Resolution; and is that which is com- 
monly used in Algebra. 

20. Synthesis, or the Synthetic Method, is the searching 
out truth, by first laying down some simple and easy princi- 
jdes, and pursuing the consequences flowing from them till 
we arrive at the condusion. — This is also called the Method 
rfCon^sjfion; and is^ the reverse of the Analytic method, as 
£bis proceeds from known principles to an unknown conclu- 
sion ; while the other goes in a retrograde order, from the 
thing sought, considered as if it were true^ to some knpwn 
principle or fact. And therefore, when any truth has been 
foupd out by the Analytic method, it may be demonstrated 
by a process in the contrary order, by Synthesis. 



B 2 ARITH- 



t * 3 



ARITHMETIC. 

jljLRITHMETIC is the art or science of numbering ; be- 
ing that branch of Mathematics which treats of the nature 
and properties of numbers, — ^When it treats of whole num- 
bers, it is called Vulgar^ or Common Arithmetic; but when of 
broken numbers, or parts of numbers, it is called Frmctions. 

Unity J or an Unitj is that by which every thing is called 
one ; being the beginning of number ; as, one man, one baU^ 
one gun. 

Number is either simply one, or a compound of several 
units % as, one man, three men, ten men. 

An Integer^ or Whoh Number^ is some certain precise 
quantity ofunits ; as, one, three, ten.— These are so called as 
distinguished from Fractions^ which are broken numbers, or 
parts of numbers ; as, one-half, two-thirds, or three-fourths. 



NOTATION AND NUMERATION. 

Notation, or Numeration, teaches to denote or ex- 
press any proposed number, either by word$ or characters ; 
or to read and write down any sum or number. 

The numbers in Arithmetic are expressed by the following 
ten digits, or Arabic numeral figures, which were intitxlucea 
into Europe by the Moors, about eight or nine hundred 
^ years since ; viz. 1 one, 2 two, 3 three, 4 four, 5 five, 6 six, 
7 seven, 8 eight, 9 nine, cipher, or nothing. These cha* 
racters or figures were formerly all called by the general 
name of C^hers ; whence it came to pass that the art of 
Arithmetic was then often called Cohering. Abo the first 
nine are called Significant Figures^ as distinguished from the 
cipher, which is of itself quite insignificant. 

Besides this value of those figures, they have also another, 
which depends on the place they stand m when joined toge- 
ther; as in the following table : 

Units 



NOTATION AND NU»ffiRATION. 

S 9 

Si ^ ' 






o 



CO 



« »3 r^* a J rS S*3 ,2 JS « 

&€. 98765432 1 

98 7 65 4 3 2 

9 8 7 6 5 4 3 

9 B 7 6 .5 4 

9 8 7 6 5 

9 8 7 6 

9 8 7 

9 - 8 

9 

fl 

Here, any figure in the first place, reckoning from right to 
left, denotes pnly its own simple ralue ; but that in the 
second place, denotes ten times its simple value ; and that in 
the third place, a hundred times its simple value ; and so on : 
the value of any figure, in each successive place, being always 
ten times its former value. 

Thus, in the number 1796, the 6 in the first place denotes 
only six units, or simply six ; 9 in the second place signifies 
nine tens, or ninety ; 7 in the third place, seven hundred ; 
and the 1 in the fourth place, one thousand : so that the 
whole number is read thus, one thousand seven hundred and 
ninety^six. 

As to the cipher, 0, though it signify nothing of itself, yet 
being joined on the right-hand side to other figures, it in- 
creases their value in the same ten>fbld proportion : thus, 5 
signifies only five ; but 50 denotes 5 tens, or fifty ; and 500 
is five hundred ; and so on. 

For the more easily reading of large numbers, they are 
divided into periods and half-periods, each half-period con« 
sisting of three figures ; the name of the first period being 
units; of the second, millions; of the third, millions of 
millions, or bi*millipns, contracted to billions : of the fourth, 
millions of millions of millions, or tri-millions, contracted 
to trillions, and so on. Also the first part of any period is so 
many units of it, and the latter part so many thousand?. 

The 



6 ARITHMETIC. 

The following Table contains a summarj ef the whole 
docUine. 



Periods* QuadrilL; Trillions; Billions; Millions; Units. 



th. un. th« un, th. un. th. un. th. un. 



123,456; '78»,098; 765,432; 101,234; 567,890. 



Numeration is the reading of any number in words 
that is proposed or set down in figures ; which will be easily 
done by help of the following rule, deduced from the fore- 
going tablets and observations — ^viz. 

Divide the figures in the proposed number, as in the sum- 
mary above, into periods and half-periods ; then begin at the 
left-hand side» and r^ad the figures with the names set to 
them in the two foregoing tables. 

EXAMPLES. 

Express in WQr4s the fbllo\ving numbers ; viz^ 



34 

96 

180 

304 

6134 

5028 



15080 

72003 

109026 

483500 

2500639 

7523000 



13405670 

47050023 

309025600 

4723507689 

274856390000 

6578600307024 



Notation is the setting dgwn ii| figures any number pro- 
posed in words ; which is done by setting down the figures 
instead of the words or names belonging to them in the sum- 
mary above ; supplying the vacant places with ciphers where 
any words do not occur. 

EXAMPLES. 

Set down iii figures the following numbers ; 

Fifty-seven. 

Two hundred eighty- six. 
Nine thousand two hundred and ten. 
Twenty-seven thousand five hundred and ninety-four. 
Six hundred and forty thousand, four hundred and eighty-one. 
Three millions, two hi^ndred sixty thousand, one hundred 
and six. 

Four 



NOTATION AND NUMERATrON. 



Four hundred and eight millions, two hundred and fifty-five 
thousand) one hundred and ninety-two. 

Twenty-seven thousand and eight millions, ninety-six thou- 
sand two hundred and four. 

Two hundred thousand and five hundred and fifty millions) 
one hundred and ten thousand, and sixteen. 

Twenty-one billions, eight hundred and ten millions^ sixty- 
four thousand) one hundred and fifty. 

Of the Roman Notation. 

The Romans, like several other nations, expressed their 
numbers by certain letters of the alphabet. The Romans 
used only seven numeral letters, being the seven following 
capitals: viz. I for one; Y for Jive; X for ten; lu for fifty; 
C for an hundred; D for five hundred; ^ for a thousand* 
The other numbers they expressed by various repetitions and 
combinations of these, aft:er the following manner : 



1 = 

2 = 
S = 

4 = 

5 = 



6 

7 

8 

9 

10 

50 

100 

500 

1000 
2000 

5000 

6000 

10000 

50000 

60000 

100000 

1000000 

2000000 
&c. 



I 

II 

III 

mi or IV 

V 

VI 

VII 

VIII 

IX 

X 

L 

C 

Dor ID 

M or CIO 
MM 



As often as any character is re- 
peated, so many times is its 
value repeated. 

A less character before a greater 
diminishes its value. 

A less character after a greater 
increases its value. 



V or IDD 

VI 

X or CCIOO 

Lj>r IDOD 

LX 

C^or CCCI0D3 

Mor CCCCIODDD 

MM 

&c. 



For every 3 annexed, this be- 
comes 10 times as many. 

For every C and O, placed one 
at each end, it becomes 10 
times as much. 

A bar over any number in- 
creases it 1000 fold. 



ExPLA- 



$ ARITHMETia 

• • • 

Explanation of certain Characters* 

There are various characters or marks used in Arithmetic, 
amd Algebra, to denote several of the operations and proposiF- 
tions ; the chief of which are as follow : 

+ signifies/Zi/x, or addition. 
— - •«■ minus, or subtraction. 
X or . - multiplication. 
-5- - - division. 
: :: : - proportion! 
= - - equality. 
\/ - - square ropt. 
-J/ • - cube root, &c. 

^ - - diffl between two numbers when it is not 
known which k the greater. 

Thus, 

5 + Sj denotes that 3 is to be added to 5. 
.6 <— 2, denotes that 2 is to be taken from 6. 
'7 X 8, or 7 . S, denotes that 7 is to be multiplied by S. 

5 -r 4, denotes that 8 is to be divided by 4. 
2 : 3 : : 4 : 6, shows that 2 is to 3 as 4 is to 6. 

6 + 4 = 10, shows that the sum of 6 and 4 is equal to 10. 
V^S, or 3i, denotes the square root of the number 3. 
^5, or 5% denotes the cube root of the number 5. 

7^, denotes that the number 1 is to be squared. 
8^, denotes that the number 8 is to be cubed. 
&c» 



OF ADDITION. 



Addition is the collecting or putting of several numbers 
together, in order to find their sum, or the total amount of the 
whole. This is done as follows : ^ . ^ 

Set or place the numbers under each other, so that each? 
figure may stand exactly under the figures of the same value, 

iha^ 



ADDITION. ft . 

thatisi units under units^ tens under tens, hundreds under 
hundreds, &c. and draw a line under the lowest number, to 
separate the given numbers from their sum, when it is found. 
-~Then add up the figures in the column or row of units» 
and find how many tens are contained in that sum.-— Set 
down exactly below, what remains more than those tens, or 
if nothing remains, a cipher, and carry- as many ones to the 
next row as there are tens. — Next add up the second row, 
together with the number carried, in the same manner as the 
first. And thus proceed till the whole Is finished, setting 
down the total amount of the last row. 



To PROVE Addition. 

• 
First Method. — ^Begin at the top, and add together all the 

rows of numbers downwards; in the same manner. as they 

were before added upwards ; then if the two sums agree, it 

may be presumed the work is right.— -This method of proof 

is only doing the same work twice over, a little varied. 

Second Method. — ^Draw a line below the uppermost number, 
and suppose it cut off. — ^Then add all the rest of the numbers 
together in the usual way, and set their sum under the num- 
ber to be proved.— r-Lastly, add this last found number 
aad the, uppermost line together ; then if their sum be the 
same as' that found by the first addition, it may be presumed 
the work is right.— This method of probf is founded on the 
plain axiom", that " The whole is equal to all its parts taken 
together.*' 



» 



Third Method.'-^Add the figures in 
the uppermost line together, and find example i. 
how many nines ^e contained in 
their sum. — Reject those nines, and 3497 g 5 
set down the remainder towards the 6512 .S 5 • 
right hand directly even with the 8295 ^ 6 

figures in the line, as in the annexed © — 

example. -r-Do the same with each 18304 g 7 

of the proposed lines of numbers, set- -^ • ^ — 

ting all these excesses of nines in a co- W 

himn on the right-hand, as here 5, 5, 6. Then, if the excess 
of 9^s in this sum, found as before, be equal to the excess 
of 9's in the total sum 1 8804, the work is probably right.-* 
Thus, the sum of the right-hand column, 5, 5, 6, is 1^, the 
excess of which above 9 is 7. ' A^o the sum of the figures in 

the 



10 ARITHMETIC 

the sum toul 18304, is 16, the excess of which above 9 is 
also 7 J the same as the former^. 



2. 

12345 


OTHER EXAMPLES. 
3. 

12345 


4. 

12345 


67890 
98765 
43210 
12345 
67890 


67890 

9876 

543 

21 

9 


876 

9087 

^ \ 56 

234 

1012 


302445 


90684 

•I 


23610 


290100 


78339 


11265 



302445 90684 23610 



* This method of proof depends on a property of* the number g, 
which, except the number 3, belongs to no other digit whatever ; 
namely^ that '^ any number divided by g, will leave the same re- 
mainder as .the sum of its figures or digits divided by 9 :** which 
may be demonstrated jn this manner. 

Demonstration. Let there be any number proposed, as 4658. 
This, separated into its several parts, becomes 4000 + 600 + 50 
+ 8. But 4000 = 4 X 1000 = 4 X (9«9 + 1) = 4 X 999 -J- 4. 
In like manner 600 = 6 xgg + 6', and 50 = 5 X 9 + 5. There- 
fore the given number 4658 = 4 X 999 +44-6x99 + ^ + 
5 X9-P5 + 8=:4X999 + <5X99 + 5X9 + 4 + 6 + 5 
+ 35 and 4058 -r-p = (4 X 999 -f 6 X 99 + 5 X 9 + 4 + 6 
+ 5 + 8) -r- 9. But 4 X 999 + 6x99 + 5 X9i8 evidently 
divisible by 9, without a remainder 5 therefore if the given num- 
ber 4658 be divided by 9ji it will leave the same remainder as 
4 + 6 + 5 + 8 divided by p. And the same, it is evident, will 
hold for any other number whatever. 

In like manner, the same property may be shown to belong to 
the number 3 ; but the preference is usually given to the number 
9, on account of its being more convenient in practice. 

Now, from the demonstration above given, the reason of the 
rule itself is evident ; for the excess of 9*s in two or more numbers 
being taken separately, and the excess of 9's taken also out of the 
sum of the former excesses, it is plain that this last excess must be 
equal to the excess of 9's contained in the total sum of all these 
numbers 5 all the parts taken together being equal to the whole. 
T his rule was first given by Dr. Wallis in his Arithmetic, 
published in the year l657* 

Ex. 



SUBTRACTION. 1 1 

'Ex.5. Add 3426 J 9024; 5106; 8890; 1204, together. 

Ans. 27650* 

6- Add 509267; 235809; 72920; 8392; 420; 21; and 9, 

together. Ans. 826838. 

7. Add 2; 19; 8l7; 4298; 50916; 730205; 9180634^ 
together. Ans. 9966891. 

8. How many days are In the twelve calendar months? 

Ans. 365. 

9. How many days are there from the 15th day of April to 
thie 24th day of November, both days included ? Ans. 224. 

40. An army consisting of 52714 infantry*, or foot, 5110 
horse, 6250 dragoons, 3927 light-horse, 928 artillery, or 
gunners, 1410 pioneers, 250 sappers, and 406 miners : what 
is the whole number of men? Ans. 70995. 



OF SUBTRACTION. 

• > 

SuBTKACtiON teaches to find how much one number 
exceeds another, called their drfferencty or the remainder^ by 
taking the less from the greater. The method of doing which 
is as follows : 

Place the less number under the greater, in the same man« 
ner as in Addition, that is, units under units, tens under tens, 
and so on ; and draw a line below them. — Begin at the right 
hand, and take each figure in the lower line, or number, from 
the figure above it, setting down the remainder below it,— 
But ifthe figure in the lower line be greater than that above 
it, first borrow, or add, 10 to the upper one, and then take 
the lower figure from that sum, setting down the remainder, 
and carrying 1 , for what was borrowed, to the next lower 
figure, with which proceed as before; and so on till the 
whole is fiqished. 



* The whole body of foot soldiers is denoted by the word Iri" 
fantry; and all those that charge on horseback by the word Cccvo/r^. 
— Some authors conjecture that the terra infantry is derived fi-om 
a certain Infanta of Spain, who, finding that the army commanded 
by the king her father had been defeated by the Moors, assembled 
a body, of the people together on foot, with which she engaged 
^and totally routed the enemy. In honour of this events and to 
distinguish the foot soldiers, who were not before held in much 
estimation, they received the name of Infantry, from her own 
title of Infanta. 

To 



12 



ARITHMETIC. 



To psovs Subtraction. 

Add the remsunder to the less number^ or dut which b 
just above it; and if the sum be equal to the greater or upper- 
most number^ the work is right*. 



1. 
From 5386427 
Take 2164815 



£XAMFL£S. 

2. 
From 5386427 
Take 4258792 



S. 
From 1234567 
Take 70297S 



Rem. 3222112 



Rem. 1127635 



Rem. 531594 



Proof.5386427 



Proof. 5336427 



Proof. 1234567 



4. From 5331806 take 5073918. 

5. From 7020974 take 2766809. 

6. From 8503602 take ^74271. 



Ans^. 257888. 
Ans. 4254165. 
Ans. 79291 3 U 



7. Sir Isaac Newton was- bom in the year 1642, and he 
died in 1 727 : how old was he at the time of his decease ? 

Ans. 85 years. 

8. Homer was bom 2543 years ago, and Christ 1810 years 
ago: then how long before Christ was the birth of Homer ^ 

Ans. 733 years. 

9. Noah's flood happened about the yeai^ of the world 1656, 
and the birth of Christ about the year 4000: then how long 
was the flood before Christ? . Ans. 2344 years. 

10. The Arabian or Indian method of notation was first 
known in England about the year 1150: then how long is 
it since tp this present year 1810 ? Ans. 660 years. 

1 1 . Gunpowder was invented in the year 1 330 : then how 
long was this before the invention or printing, which was 
in 1441 f Ans. 1 1 1 years. 

1 2. The mariner's compass was invented in Europe in the 
year 1302: then how long was that before the discovery of 
America by Columbus, which happened in 1492.^ 

Ans. 190 years. 



* The reason of this mediod of proof is evident; for if the 
diflerenoe of two numbers be added to the less^ it must manifestly 
siake up a sum e^ual to ibe greater. 

OF 



ikrLTlPLICATION. 



IS 



OF MULTIPLICATION. 

» 

Multiplication is a compendious method of Addition^ 
teaching how to find the amount of any given number when 
repeated a certain number of times; as, 4 times 6, which 
is 24<. 

The number to be multiplied, or repeated, is called the 
Mub^licand^'-^The number you multiply by, or the number of 
repetitions, is the Mubipiier.^^Aj^ the number found, being 
the. total ambunt, is called the Productr^Aho, both the 
multiplier and multiplicand are, in general, named the Termi 
or Factors. 

Before proceeding to any operatiops in this rule, it is ne« 
cessary to learn off very perfectly the following Table, of all 
the products of the first 12 numbers, commoidy called the 
Multiplication Table, or sometimes Fythagoras's Table^ from 
its inventor. 

Multiplication Table. 



1 

2 


2 


3 


4 
8 


5 6 


7 
14 


8 
16 


9 


10 


11 


12 


4 


6 


10 
15 
20 
25 

30 
35 


12 
18 


18 


20 


22 


24 


S 


6 


9 


12 
16 


21 


24 


27 


30 


33 


36 


4 


.8 


12 


24 
30 


28 


32 


36 


40 


44 


48 
60 


5 


10 


15 


20 
24 
28 


35 


40 


45 


50 


55 


6 

7 


12 


18 


36 


43 


48 


54 


60 


66 


72 


14 


21 


4^ 


49 


56 


63 


70 


77 


84 

96 

108 


8 


16 


24 


32 


40 


48 


56 


64 
72 


72 


80 


B8 
99 


9 


18 


27 
SO 


86 
40 
44 


45 
50 
55 
60 


54 


63 


81 


90 


10 
U 
12 


20 


60 
66 


70 

77 


80 


90 


100 


110 


120 


22 
24 


33 


88 


99 


110 


121 


132 


36 


48 


72 1 84 


96 


108 


120 


132 


144 



T0 



14 ARITHMETIC. 



To multiply any Given Number by a Single Figure, or by am 
^^ ' Number not nJf than I'i. * ^^ 

■ « 

* Set the multipliei^ under the units figure, or right-hand 
pl;(ce, of the multiplicand, and draw a line below it.*— Then, 
beginning at the right*hand, multiply every figxire in this by 
the multiplier.^ — Count how many tens there are in the pro- 
duct of every single figure, and set down the remainder di- 
rectly under the figure that is multiplied ; and if nothing 
remains, set down a cipher. — Carry as many units or ones as 
there are tens counted^ to the product of the next figures | 
and proceed in the same manner till the whole is finished. 

EXAMPLE. ' 

Multiply 9876543210 the Multiplicand. 
By - - - - 2 the Multiplier. 

19753086420 the Product. 



To multiply by a Number consisting of Several Figures. 

\ Set the multiplier below the multiplicand, placing thenv 
as in Addition, namely, units under units, tens under tens, &c. 
drawing a line below it. — ^Multiply the whole of the multi- 
plicand by each figure of the multiplier, as in the last article; 

setting 



^ The reason of this rule is the same as for 
the process in Addition^ in which 1 is car- 
ried for every 10, to the next place, gra- 
dually as the several products are produced^ 
one after another, instead of setting them 
all down one below each other, as in the an- 
nexed example. 



5§78 
4 


■ 




32 

280 

2400 

2000O 


8X4 

70 X 4 

600 X4 

5000 X 4 



22712 =5678 X 4 



f After having found the produce of the multiplicand by the first 
figure of the multiplier, as in the former case, the multiplier is 
supposed to be divided into parts, and the product is found for the 
secQud figure in the same manner : but as this figure stands in the 
place of tens, the product must be ten times its simple value ; and 
therefore the first figure of this product must be set in the place of 

tensi 



MULTDPLICATIOI?. IS 

setting down a line of products for each figure m the multi- 
ptier, so as that the first figure of each line ma^c stand straight 
under the figure multiplying by. — Add all the lines of pro- 
ducts together, in the order as they stand, and their sum will 
be the answer or whole product required. 

To PROVE Multiplication. 

There are three different ways of proving Multiplication, 
which are as below : . . 

First Method. — Make the multiplicand and multiplier 
change places, and' multiply the latter by the former in the 
same manner as before. Then if the product found in this 
way be the same as the former, the number is right. 

Second Method. — *Cast all the 9's out of the sum of the 
figures in each of the two factors, as in Addition, and set 
down the remainders. Multiply these two remainders 
together, and cast the 9's out of the product, as also out of 



tens ; or, which is the same things directly under the figure multi- 
plied by. And proceeding in 
this manner separately with all 

the ^ures of the multiplier, 1234567 the multiplicand, 
it is evident that we shall mul- 4567 

tiply all the parts of the mul- — 

tipdicand by all the parts of 8641969= 7 times the mult, 
the multiplier, or the whole of 7407402 r= 60 times ditto, 
the multiplicand by the whole 6172835 r= 500 times ditto, 
of the multiplier : therefore 493826S =4000 times ditto. 

these several products being _ 

added together, will he equal 5638267489=4567 times ditto. 

tothewhole required product; ■■ ■■ 

as in the example annexed. 

* This method of proof is derived from the peculiar property of 
the number 9, mentioned in the proof of Addition, and tha reason 
for the one may serve for that of the other. Another more ampla 
demonstration <^ this rule may be as follows : — Let P and Q denote 
the number of 9^s in the factors to be multiplied, and a and 6 what 
remain ; then 9 P+a and 9 Gl+6 will be the numbers themselves, 
and their product is (9 P X 9 Gl) + (9 P X A) + (9 Gl X a) -f 
(a X h) i but the first three of these -products are each a preiCise 
number of 9's> because their factors are so, either one or both : 
these therefore being cast away, there remains only a x 6; and if 
the 9's also be cast out of this, the excess is the excess of 9*s in the 
total product : but a and h are the excesses in the factors them- 
Bcjves^ and a x 6 is their product ; therefore the rule is true., 

the 



16 



ARITHMETIC 



the whole product or answer bf the questioni reserving the 
renuinders of these last two, which remainders must be eauai 
when the work is right. — Ifote^ It is ccMnmon to set the lour 
remainders within the four angular spaces of a crossj at in the 
example below. 

Third 3/^^.-- Multiplication b also very naturally 
proved by Division ; for the product divided by either of the 
factors, will evidently give the other. But this cannot be 
practised till the rule of Division is learned. 



Muk. 3542 
by 6196 

21252 
34878 
3542 
21252 



21946232 Product. 



EXAMPLES. 



Proof. 




or Mult. 6196 
by 3542 



12392 
247S4 
30980 
18588 

21946232 Proof. 



OTHER EXAMPLES. 



Multiply 
Multiply 
Multiply 
Multiply 
Multiply 
Multiply 
-Multiply 
Multiply 
Multiply 
•Multiply 
Mukiply 
Multiply 
Multiply 
Multiply 
Multiply 
Multiply 



123456789 

123456789 

123456789 

123456789 

123456789 

123456789 

123456789 

123456789 

123456789 

302914603 

273580961 

402097316 

82164973 

7564900 

8496427 

2760325 



by S. Ans. 

by 4. Ans. 

by 5. Ans. 

by 6. Ans. 

by 7. Ans. 

by 8. Ans. 

by 9. Ans. 

by 11. Ans. 

by 12. Ans. 

by 16. Ans. 

by 23. Ans. 

by 195. Ans. 

by 3027. Ans. 

by 579. Ans. 

by 874359. Ans. 

by 37072. Ans. 



370370367. 

493827156* 

617283945. 

7407407S4. 

864197523. 

987654S12. 

1111111101. 

1358024679. 

1481481468. 

4846633648. 

6292362103. 

78408976620. 

248713373271. 

4380077100. 

7428927415293* 

102330768400. 



COHTIUG- 



MULTIPUCATION. 11 

Contractions in MoLTtrucATioM. 

I. Jf^im there an Ciphers in the Factors. 

If the ciphers be at the right-hand of the numbers ; mul- 
tiply the other figures only, and annex as many ciphers to 
the right-hand otthe whole prodocti as are in both the fac* 
tors.— When the ciphers are in the middle parts of the mul* 
tiplier } neglect them as before^ only taking care to place 
the first figure of every line of products exactly ttnd«r th* 
£gure multiplying with. 

EXAUPLES. 

Mult. 9001635 Mult. 3901S0400 

by • 70100 by - 406000 

■■III- ■■■■■II .^ ■ ■ » — — — .— ^ 

9001635 2844S284 

63011445 15628816 



631014613500 Products 158632482400000 



S. Mukiply 81503600 by 7080. Ans. 572970308000. 

4. Multiply 9030100 by 2100. Am. 18963210000. 

5. Multiply 8057069 by 70050. Ans. 56439768345Q. 

n. When the Multiplier is the Proiiict of two or more Numbers 

in the Tabk; then 

^ Multiply by each of those 'parts separately, instead pf 
the whole number at once. 

EXAMPLES. 

1. Multiply 51307298 by 5Q^ or 7 times t. 

61307298 

7 



359151086 

' 8 

2873208688 



f The reason of this rule is obvious enough $ for any number 
mttltif^lied by the component parts of another^ must give the same 
product as if it were multiplied by that number at once Thus, in 
tbe 1st example, 7 times the product of b by the giv^n number^ 
ifnakes 5Q times the same number, as plainly as 7 times 8 makes a6. 

Vol. I. C 2.Mul- 



It: ARITHMETIC 

2. Multrpl7 3!7045d2 -by 86. Aus, .1141365512. 

S. Multiply. 2S753804 by 72. Ans 2142273888* 

4. Multiply 71283^8 by 96. Ans. 684i52:3326. 

^. A|i*hiply 160430800 by 10$, Aiis. 11^26526 \QO. 

,«, Multiply 61835720 by 1320. Ans. 81623150400. 
. 7* There was aa army composed of 104 "^ battalions, each 
consisting q£ ^00 men j what was the number of men con* 
tained in the whole? Ans. ^2000* / 

•. 9f A conToy of ammnnkion f bread, consisting of 250^ 
waggonst and each waggon containing 320^ loaves, hzvinfr^ 
been intercepted and taken by the enemy ^ what is the num- 
ber of loaves lostA Ans. 80000. 



tsam 



OF DIVISION. 

' Division is ,a kind of compendious ^lethod of Subtrac* 
tion, teaching to find how often one number is contained in 
another, or may be taken out of it : wiiidx is the same thing. 

The number to be divide is called the Dividend. — 
The number to divide by, is the Z>;wflr.4r— And the number 
of times the c^vi^nd contain^, the divisor, is called the Quo* 
tient. — iSometiiqes thfire b g Rnnaitider left^ after the division 
i^^nished. 

The usual manner of placing the terms> is, the dividend in 
the. middle^ haying.the cU visor en the left hand, and the<}uo* 
tient on the right, each separated by a curve line j as, ta 
divide 12 Iq^ 4, the quotient is 3, 

• Dividend ' ' ' - l2 

Divisor 4) 12 f 3 Quotient; 4 subtr, 

showing that the number 4 \& 3 times — 

contained i9 12, t)E may be S times 8 

subtracted out of it, as m the ipargin. 4 subtr* 

X RuU. — Having placec^ the divisor — 

before the dividend, as above .direct- 4 

ed, find how often the 'divisor is con- 4 subtr* 

tained in as many figures of the divi- -— 

dend as are just necessary, and place the 

number on the right in the quotient. — 

Mul- 

*yi^^— W^ I ■ ■■ I H ■ II ,1 ^ ■ ■ I I III I I ■■ fcl^M^I—— *^^ll ■ 

* A battalion is a body of foot, censisting of 500, or 600, or 700- 
men, more or less. 

f The amitoufiitioii bread, is that which b provWedfor, ami ifi^ 

'triboted to, the soldiers j t!^ u^u^ allowanee being- a leaf of ^ 

pounds to evety soldier, once in 4 days. ' ' ■> 

- Xlta this way the dividend b resolved into parts, end by trial i* 

*- - - ■ '• - ibdnd 



Multiply the divisor by this number, and set the product 
tinder the figures of the dividend before^mentioned.-*-*Sub« 
tract this product froro that part of the dividend undi^ which 
it stands, and bring down the n^xt figure of the dividend, ox* 
more if necessary, to join on the right of the temainder.—Di* 
vide this ntunber, scr increased, in the same n^aiinera;^ before | 
and so on till all the figures are brought down and used. 

• 
If. B, If it be necessary to bring down more figures than 
one to any remainder, in order' to make it as large as the 
divisor, or larger, a cipher must be set in the quotient for 
every Bf^ate so brou^t down more than one. 

To PRQvt Division* 

* Multiply the quotient by the divisor; to this product 
add the jpemainder, if there be any ; then the sum will be 
equal to the dividend when the work is right* 



found how often the divisor is contsllned in each of those patt8> one 
idler another^ arranging the several figureS'Of the qaotieutone after 
another^ into one numben 

When there is no remainder to a division, the^ quotient is the 
whole and perfect answer to the question. But when there is a re- 
mainder^ it goiea so much towards another time> as it approaches to 
the divisor : so» if the remainder be half the divisor^ it will go the 
half of ^ time iQore) if the 4th part of the divisor^ it will go. one 
fourth of a dq^e more ; and so on. Therefore, to complete the 
quotient, set the remainder at the end of it, above a small line, and 
toe divisor below it> thus forming a fractional part of the whole 
quotient. 

* This method of proof is plain, eoough : for since the quotient 
is the nimiber of times the dividend contains the divisor, the quo- 
tieni multiplied by the divisor must evidendy be equal to the 
dividend. 

There are also several other methods sometimes used for proving 
IHvision, some of the most useful of which are as follow : 

Stccftd Me/W.-*Snbtract the remainder from the dividend^ end 
4ivide what is left by the quotient ; so shall the nei^ quotient £ro9i, 
this last divisi9n be equal to the former diviaor, when the wofk is 
tight. 

T^r4 Method, — Add together the remainder and all die pro- 
ducts of the several quotient figures by the divisor^ according to the 
order is which they stand ia the work $ and the sum will be 
^ual to the dividend wl^en the work is right. 

C 2 EXAM- 



2« 



ARrrHMETIC. 



EXAMPLES. 



I. Quot. 

3} 1234567 (411522 



12 



mult. S 



2. Quot. 

37 ) 12345678 (333666 

111 37 



3 
3 


1234566 
add 1 


4 

S 1 


1234567 


9 •■ 

15 

15 


Proof. 


6 




7 
6 




Rem. ' 1 


\ 



2335662 
1000998 
rem. 36 



124 

135 . 

1 1 1 1234567$ ^ 



246 Proof. 
222 



247 

222 



258 
222 

Rem. 36 



3. Divide 73146085 by 4. 

4. Divide 5317986027 by 7. 

5. Divide 5701^6382 by 12. 

6. Divide 74638105 
7* Divide 137896254 

8. Divide 35821649 

9. Divide 72091365 



Ans. 1 828652 Ij^. 
Ans. 7597122894*. 
Ans. 47516365^. 
Ans.20l7246yy. 
Ans. 1421610^. 
Ans. 46886fJ^. 
Ans. I386I3WT. 
Ans. 80496|U?-^ 



by 37. 
by 97. 

by 764. 
by 5201. 

10. Divide 4631064283 ^ 57606. 

11. Suppose 471 men are formed into ranks of 3 deep, 
what is the number in each rank? Ans. 157. 

12. A party, at the dbtance of 378 miles from the head 
quartersi receive orders to join their corps in 18 days : what 
number of miles must they march eadi dajr to obey their 
orders? Ans. 21. 

13. The annual revenue of a gentleman being 88330/; 
kow ^uch per day is that equivalent to, there being 365 days 
in the year ? Ans. 104A 

CoNTHACTlONS IN DIVISION. 

. There are certain contractions in Division,, by which the 
operation in particular cases may be performed in a shorter 
jmanner : as follows : . . 



I. Divp 



DIVISION. 



« 



!• jPivision if any Small Number^ not greater tlian 12, vaxj 
be expeditioualy performedi bj multiplying and subtracting 
mentallyy omitting to set down the work, except only the 
quotient inunediately below the dividend. 



3) 56103961 
Quot. 18701320J. 



EXAMPLES. 

4) 52619675 



5) 1379192^ 



6) 38^72940 7 > 81396627 3 ) 23718920 



9 ) 439819^2 1 1 ) S76 14230 J2 ) 27980373 



-«■ 



»■ ■ ' U 



n. * When ethers are annexed to the Divisor i cut off those 
ciphers from tt, and cut off&e same number of figures from 
the right*hand of the dividend; then divide with the remain- 
ing figures, as usual. And if there be any^ thing remaining 
after this division, place the figures cut off from the dividend 
to the right of it, and .the whole will be the true remainder; 
otherwise, the figures cut off only will be the remainder. 

EXlMPi.ES. 

1. Pivide 3704196 by 20. 2. Divide 31086901 by 7100. 
2,0) 370419,6 71,00)310869,01 (4378ff 
284 



Ol- 



Quot. 185209ii 



268 
213 



556 

599 
568 



SI 



■y^ 



3. Divide 



* This method is only to avoid a needless repetition of ciphers^ 
which wQuld happen in the coQimoa V9y. And ihe truth of the 

principle 



B2 AlUTHMETICt 

3. Divide 7380964 by 23000^ Am. ilpii^^^ 

4. Divide fli04109 fay 580a Am. 397if^. 

nif W7>en tie Dhuistnr is the ex^ct Product -rf two or moh 
jf the small Numbers not grettter than 12 : * Divide by eadl 
of those numbers Separately^ iiutead of the whole divisor sit 
pnce. 

N. B» There are commonly several remainderi in work*- 
ing by this rule, one to each division', and to find the true w 
whole remainder, the same as if the division had been per-r 
formed ^U at opce, proceed as follows; Multifdy the. last 
remainder by thf preceding divisor, or last but one| and to 
the product add the preceding remainder; multiply this sum 
by the n^xt .preceding divisor, and to the product add the 
nex^ preceding remainder i and so pn, till you have gone 
backward through all the divisors ^4 remainders to the firsts 
As in the example following : 

EXAMPILES. 

1. I^ivide 31046895 l^y 56 or 7 times d. 

7 ) 810^835 6 the iastremf 

tsxdttf 7 precpd^ iHvisof^ 



p ) 4435262--1 iirst rem. 

; 48 

554401 — fi second ren^. ^lid> 1 the I'st rem* 



Ans. 55440744 43 'whole rem. 



TS 



principle on which it is foande^, is evident ; for, pi^ti^Dg off the 
fame number of ciphers^ pr figures, from each> is the same at 
dividing each of them by 10, or lOO, or 1000, &c. according to 
the number of ciphers cut off; and it is e'^rdent, that as ofien as 
f be whole divisor is cpn^ined in the whole dividend, so often must 
^ny part of the former be containcji in a like part of the latter. 

*' This fdlows from the second contraction in Multiplication^ 
being only the cpnverse of it; for the half of the third part of any 
thing, is evidently the same as the ^ixth part of the whole ; and 
00 of ^ny other numbers.— The reaspn of the method of finding 
the whole remainder from the .several particular ones, will best 
appear from the- nature of Vulgar Fraptipps. Thus^ in the firs^ 
example abpvfs, the- first remainder being 1, when the divisor i^ 
7| makes j- ; this must be added to the second remainder, 6^^ 
making 6f to the flivisor 8^ or to be divided by 8. But €f =: 

r— ^ = yi and thi^ divided by 8 gives ^^=:|^.. 

2. P^vide 



r 



«. Divide 7014596 by 72. Ans. 97424f^ 

3. Qisrid^ .5 130652 by .132. . . Ans^ SSiea^ZyV 

4. Divide 83016572 hj 2ii9. Ans. 945502^. 

. IV. Common Divisign may be pftfirmed more condse/jj 
by omitting the several products, and setting down only the 
remainders; namely, x multiply the divisor by the quotient 
figures as before, and*^ without setting down the product^ 
subtract each figure of it from the dividend, as it is produc^ds 
mlvrays remembering to carry as many to tbft next figure as 
wbre borrowed before. 



[ EXAICPIES* 



U Divide 3104679 liy $33. 

«33 ) 3 104679 ( 3727^V» 
6056 
2257 
5919 
88 

«. Divide 79165288 by 238* . Ans^ 332627^^^. 

8. Divide 291S7062 by 5317. . . Ans. 5479mf. 
4. Divide 62015735 by 7803- - Ansu 7?47f||f 



OF REDUCTION. 

Reduction is the changing of numbers firofcn oot name 
or denomination to another, Hithout altering their valqe.-— 
This is chiefly concerned in reducing' money^ weights^ and 
measures. . 

When tilt numfiers are to be reduced from a higher name 
to a lovrer, it is called Reduetkn DescenSngi mst wheii^ 
contrarywi^, from a lower name to a higher, it is Redudkn 
AscenHngm 

Before proceeding to the rules and qtresttc^fs of Rednctim, 
it will be proper to set down the usual Tables of mdns^^ 
vireights, and measures^ whick ar^ as follow : \ 

bf 



f4 ARITHMETIC 



(y MONET, WEIGHTS, amd MEAStTRES. 



Tables of Moi^et*, 



. 3 Farthings 
4 Farthings 
)2 Penc9 
20 ShilUngs 


=s I Halfpenny f 
^* 1 Penny d 
«; I ShilUng / 
= 1 Pound £ 


4 

48 

S60 


S5r 1 / 

== 12 =^ 1 
=; 240 = 20 


£ 
= I 


PENCE TABLE. 


fHltLINGS TABLE, 


1 


d 


i d 


S 




l/ 


1 


20 


is 1 8 


1 


is 


12 


• 


30 


— 2 6 


2 


— 


24 




40 


-^3 4 


3 


-«* 


36 




30 


- 4 2 


4 


«.» 


48 




60 


— 5 


5 


— 


60 


1 


70 


— 5 10 


6 


-.^ 


72 


' 


80 


— 6 8 


7 


«^ 


84 


i 


90 


—-7 6 


8 


--. 


96 


' 


100 


— 8 4 


9 


^^ 


lOS 




HO 


— 9 a 


10 


-~ 


120 


1 


120 


-, JO 


11 


— 


132 






1 








Trot 



Gold. 



* £ denotes ponndi , « shillings, and d denotes pence. 
^ denotes i farthing, or one quarter of any thing. 
\ denotes a halibenny^ or the half of any thing. 
^ denotes 3 farthings, or tliree <|uarter8 of aoy thing. 

The full weight and value of the English gold and silver coin, 
i^ as here below : 

Weigh. 

dwt gr 

a 16| 

1 8^ 

I The usual value of gold is nearly 4/ an ounce^ or 2<i a grain | 

an4 that cf ^silver is oearlv ^s an ounce. Alsoj the value pf any 

quantity of gold^ is to the value of the s^me weight of standard 

iiiver^ nearly as t. 5 to I « or more nearly as 15 and 1-1 4th to I. 

Pure gold, free from mixture with other metals, usually called 

\ $Dt gol4» U of so puie 9 Datur^> that it wiU endure the fire 

•' vitfeQut 



Vahe. 

£ » d 
1 1 
10 d 



A Guinea 

Half-guinea 

Seven Shillings 7 

Quarter-guineaQ ^ 3 



SlLVEH. 


Value. 


Weight. 




9 d 


dwt gr 


A Crown 


5 


19 84 


Half-crowa 


% 6 


9 l^i 


ShiUing . 


1 O 


3 21 


Sixpence 


Q 6 


. 1 ^^i 



24= 1 oz 
480=r 20= 1 A 
5760=240 = 12= r 



TA5US5 OF W^QHTS. 15 

Trov Weight*. 

Grains - - marked ^r \ gr dnvt 
24 Graim make i Pennyweight dwt 
20 Pennyweights 1 Ounce «z 

1 2 Ounces 1 Pound lb 

By this weight are weighed Gold, Silyer, and Jewels* 

Apothecahies' Wjeight. 

Grains - - marked gr 

^0 Grains make 1 Scruple sc or 9 

3 Scruples 1 Dram dr or 3 

.8 Dr^mi. 1 Ounce w; or t 

12 Ounces 1 Pbund lb or jl, 

" gr sc • 

20 = 1 ir 
60 ass 3 = 1 «5 
480 = 24 = 8 = 1 lb 
. 5760 = 288 = 96 = 12. =3 1 

This is the same as Troy weight, only having some dif- 
ferent divisions. Apothecaries make use of this weight in 
compounding th^ir M^icines i but they buy and sell their 
Drugs by Avoirdupois weight. 

AvoiR- 



widiout wasting, though it be kept contigually melted. But silver, 
not having the purity of gold^ will not endure the fire like it : yet 
£ne silver will waste but a very little by being in the fire any 
moderate timej whereas copper, tin, lead, &c. will not only 
waste, but may be calcined, or burnt to a powder. 

Both gold and silver, in their purity^ are so very soft and flexible 
(like new lead, &c.), that they are not so useful, either in coin or 
otherwise (except to beat into leaf gold or silver), as wnen they 
are allayed, or mixed and hardened with copper or brass. And 
though most nations dlifer, more or less, in the quantity of such 
allay, as well asHn the samis place at different times, yet in £ng- 
land the standard for gold and silver coin has been for a long time 
as follows — viz. That 2*JL parts, of fine gold, and 2 parts of copper, 
being melted together, shall be esteemed the true standard for gold 
coin : And that 1 1 ounces and 2 pennyweights of fine silver, and 
. 1 8 pennyweights of cop|)er^ being melted together, is esteemed 
the true standard for silver coin, called Sterling silver. 

♦ The original of all weights used in England, was a grain or 
com of wheat, gathered o^t of the middte of the ear, and, being 
W^Jl dri^d^ 32 of them were to m^e one pennyweight, 20 penny- 
weights 



/ 



t6 AR|THM£71C. 



Atoiildupois Weight. 



DraiQS 


• 


- 




naarked (ir 


16 Drains 


make 


1 Ounce 


f 


• - • 


OZ 


16 Ounces -^ 


• • 


1 Pound 




• • • 


a 


t9 Pounds - 


. • 


1 Quarter 
1 Hundred 




• - . 


9^ 


4 Quarters - 
2(X Hundred 1 


. . 


1 Weight ^ 


nvt 


RTeight 


iTon 


p» 


- - 




Jr ' 


'OZ 








^ 


16 = 


1 


a 








256 = 


16 = 


= 1 


i^ 




1 


7l6« = 


448 = 


= 28 =s 


1 


cwt 




28^72 = 


1792 = 


s H2 = 


4 


s 1 


tm 


573440 3e 


35840 = 


= 2240 8= 


80 


= 20 s= 


1 



By this weight are weighed all things of a coarse or drossy 
nature^ as Com, Bread, Butter, Cheese, Flesh, Grocery 
Wares, and somt liquids ( also all Metak, except Silver and 
Cold.. 

OZ dtift gr 
N§if^ that la AToirdupcns at 14 U 154 Troy, 
las - - =s 18 54 
Ur - - «= 1 34 





LoN^ Measure. 


* 


• 


3 Barley*coms 


make 1 Inch 


*> 


Li 


12 Inches 


« 


1 Foot 


» 


' Ft 


3 Feet 


• 


1 Yard 


• 


rd 


6 Feet 


■■ 


1 Fathom -/ 


•■• 


Fth 


S Yards and a 


half 


1 Pole or Rod 


«» 


PI 


4b Poles 


* 


1 Furlong - 


J- 


Fur 


S Furlongs - 


«• 


1 Mile 


• 


Mile 


3 Miles 


m 


1 League - 


• 


Lea 


694 ^1^ nearly - 


1 Degree - 


. 


Dfg at ". 



• 

wdghts one ounqe> and 1^ ounces one pound. But in later times, 
it was thought sufficient to divide Hhe same pennyweight into 
24 equal parts, still called grains, being the least weight now in 
common use ; and from tb^ce the vest are computed, as in the 
Tables above. 

In 



TABUS «fM&A9C7K£8. ft 



In 




2% 












12 


SSI 


1 




rd 




- 




36' 


sa 


3 


s= 


1 




fl 


■ 


198 


=r 


1<54 


' zs 


5i 


=: 


1 


Fur 


^7920 


=: 


660 


=: 


220 


ss 


40 


= I 


13960 


ss 


6230 


^^ 


1760 


ss 


320 


s: 8 








ClOTH MB4SyKB. 





Mik 



2 Inches and a ijuart^r make 1 Nail <- * Nl 

*4 Nails r - sr 1. Quarter of ^ Tard Or 

3 'Quarters .r - » 1 EU Flemish - £-F 

4 Quarters - ^ ^ 1 Yard - * Td , 

5 Qu;qpters r «. * 1 Ell English - EE 
4 Quarters 14 Inch - 1 £11 Scotch ^ ^ £S 



» 

144 Squarefochetmake 1 Sq Foot - R 

9 Square Feet r 1 SqTard - rd 

90i Square Tafds - 1 Sq Pol^ - Pok 

40 Sq^are Poles - 1 Rood * Rd 

4 Roods -pi Acw f jfcr ' 

SqLic ^Ft 

; 144 « i SqTd 

1296 = 9=1 SqPi 

39204 .s 272^ s 30f r? 1 Jtd 
1568160 s: 1O890 b 1210 s 4Q « i >/rr 
6272640 = 4^^6Q b 4840 == 160 » 4 ^ i , 

By diis measure^ Land, and Husl|aadme& and Oardenen? 
irork are measured; also Artificers' work, ancl^ ^s Board, 
Glass, Pavements, PlaMering, Wainscoting, Til|j9g, Floorilig, 
and every dimension pf laigth smd breadth only; 

. When three dimensions arjs concemed| namely, length, 
breadthy aapd depth or thkkness, it is called cubic or solid 
measure, which is used to measure fllmbery Stone» &c. 

The cubic or s^lid Foot, which is 12- inches ia length and 
)>readth and thickness> contains 1728 cubic or sblici inches, 
u^d {27 solid feet make one solid yard. 



29 



ARnHMETia 



DrY) or Cork Measchb. 



1 Quart 



< 1 Gallon 
- 1 Peck 



f Pifits make 

2 Quart* - 1 Potde 

2 Pottles 

2 Gallons 

4 Pecks 

5 Bashels 
S Quarters 
2 Weys 




■■ « M 



Bushel 
Quarter 



1 

1 

1 Wey» Load> or Ton 

1 Last - 



Gai 
Pec 
Bu 

Wej 

JLaU 



Pts 

' % 

16 

64 

512 

2560 

5120 



Gal 
1 

2 =x 

8 = 

64 = 

S20 = 

640 =: 



PiC 
1 
4 

32 
160 
S20 



1 

40 

80 



Or 
J 

5 

10 



: 1 ZdRlf 
: 2 =: I 



By this are measured all dry wares> aS) Corn> Seeds, Rooft% 
Fruits, Salt, Coals, Sand, Oysters, &c. 

The standi Gsdlon dry-measur^ cpntauis268| cubic ot 
solid inches, and the Cdm or Winchester husl^el 21 50|- cubic 
inches j for the dimensions of the Winchester bushel, by the 
Statute, are 8 inches deep, and 1 8-^ iiv;hei wide or in diameter. 
But the Coal bushel must be 19^ inches in dlanneter; and 36 
bushels, heaped up, make a London chaldron of coals> the 
weight of which is ^ 1 5 6 lb Avoirdupois* 



AhZ and Beeil Measure ^ 



2 Pints make 

4 Quarts - •• 

36 Gallons ^. -w 

1 Barrel aud a half 
f Barrels m 

2 Hogsheads ^ 
2 Butts; <* -r 

Pts Of 

2 =; I Gat 

8 =; 4 =?, 1 

28^ = 144 =: 36 

482 23 2li5 = 54 

864 2= 432 = 108 



1 Quart 
1 Gallon 
1 Barrel 
1 Hogshead 
\ Puncheon 
1 Butt 
\ Tw - 



Gal 



Bar 

Hhd 

^ Pim 

fiuu 

Tut\ 



Bar 
: 1 






mi 

1 Sm 

2 i= 1 



NitCji The Ale Gallon contains ^82 cubic or soSd Lichee. 

Win* 



TABLES ot MEASURES amo TIME. 



2» 



Wine Measure. 

•• 




2 Pints make 


1 Quart 


Qt 


4 Quarts - *• 


1 Gallon 


Gal 


42 Gallons - . >- 


1 Tierce 


Tter 


63 Galkms or it Tierces 


1 Hogshead - 


Hhd 


2 Tierces - 


1 Puncheon - 


Pun. 


2 Hotheads 


1 Pipe or Butt 


Pi 


2 Pipes or 4Hhds 


I Tun - 


Tun 


Pti Qt 


V 




2 = i Go/ 






« = 4= 1 : 


tter 




9J6 = 168 r: 42 = 


I Hid 




504 = 252;= 63 = 


lt= I P«/» 




672 = 336 = ^4 = 


2 = lt= 1 Pi 


1 


1008 :;= 504 = 126 = 


3=2= lt= I 


Tim 


2016 =: 1008 = 252 = 


6=4=3 =a 


= 1 



/ 

.Notit By this are measured all Wines, Spirits, Strong- 
Waters, Cyder, Mead, Perry, Vinegar, Oil, Honey, &c. 

The Wine Gallon contains 231 cubic or solid inches. 
And it is remarkable, that the Wine and Ale Gallons have the 
same propprtipn to each other, as the Troy and Avoirdupois 
Pounds hare^ that is, as one Pound Troy is to one P4!>und 
Avoirdupois^ so is one Wine Gallon to one Ale Gallon* 



Of TIMiE. 



60 Seconds or 60'' make 


• 


1 Minute - 


Mot' 


^ Minutes 


- 


1 Hour 


Hr 


24 Hours - 


- 


i Day 


Da* 


7 Days * - - 


- 


1 Week . 


ni 


4 Weeks - - - 


• 


1 Month - 


M, 


13 Months I Day 6 Hours,) 
or 365 Days 6 Hours ) 


1 Julian: Year 


Tr 

4 


■ Sec Min 




, 




60 =: 1 


Hr 


, 




. 3600 = 60 = 


I 


■ -^"^ „. 


• 


S6400 = 1440 = 


:■ 24. 


=s -4 IVk 


*• 


604800 =: . lOOSO =s. 


168 


= 7 = 1 


M» ■ . 


1 2419200 =: 4.0320 = 


672 


= 28 = 4 .: 


= 1 ! 


31557600 == 5250QQ ^ 


8766 


= 365t;=i 


llrfar. 


V 






Or 



( 



y 



m ARITHBIEnC* 

Wk Da Hr Mo Da Hr 
Or 52 I 6 = 13 1 6 = 1 Julianrear 

Da Hr M Sec 
But 365 6 48 48 =r 1 Sflcir rear! 



ItULES FOR REDUCTION. 



^ trim tie Numben are to he reduced from a Higher Denond^ ' 

nation to a Lower : 

Multiply the number m the highest denomination by as 
many as of the next lower make an integer, or I, in that 
higher ; to this jH'oduct add the number, if any, which was 
in this lower denominatioir before, a^ set down the aihonnt. 

Reduce this amount in like manner, by multiplying^ it by 
as many as of the next lower make an integer of this, taking 
in the odd parts of this lower, as before. And so pnroceed 
through all the denominations to the lowest ; so shall die 
number last found be the value of all the numbers which, 
were in the higher denominations, taken together*. 

EXAMPLE. 

1. In 1234/ Idx 7i^ how many farthings I 

lid 
1234 15 7- 
20 



24695 Shillings 
12 



296347 Pence 
4 



^ Answer 1185388 Farthings. 



* The reason of this ru)e is very evident; for potmda are 
brought i^to shillings by maltiplying them by 20 ; ^hilltngs into 
pence« by multipljrtig them -bj 12; and pence into ^rthings^ by 
multiplying by 4 5 and tKe reverse of this nile by Division. — And 
the same^ it is evident^ will be true in the reduction of numbers 
consisting of any denominations whatever. 

ll.When 



RULES foil REDtJCnCMJ. St 

IL When tie Number $ are to be reduced from a Lower .Deftomt" 

nation to a Higher : 

m 

Divide the given number by as many as of that denomt'* 
nation make 1 of the next higher, and sec down what 
nemainsy as well as the quotient. 

Divide the quotient hj as many as 6f this denominatbn 
make 1 of the nett. higher } setting down the new quotient, 
and remainder, as before. / 

Proceed in the same manner through all tike denomina*- 
tions, to the highest ; and the quotient last fimnd, ti^ether 
with the several remainders, if any, will be of the same vahif 
s the first number proposed. 

V 

EXAMPLES. 

2. Reduce 1185388 farthings into pounds, shillings, and 
pence. 

4) 11853*8 



1*1 I I 



12) 296347^ 



■ t ■' I !■ * 



2,0 ) 9469,5 /-^7rf 



1S^4/ 14^ *!d 



< ■ »»■»» 



3. Reduce 24/ to farthinjgs. Ans. 23(Hai 

4> Reduce 3375^7 farthings to pounds, &c. 

Ans. S6U I3i 6^. 
5. Kow many futhings^nre in 3€ guineas? Ans. 36S89« 
€. In 36288 farthings how many guineas I Ans. 36. 

7. In 59 lb 13 dwts 5^ how many gnttis ? Ans. 340157* 

8. In 8012131 grains how many pounds, &c.? 

Ans. I390ib 11 oz 18dwt t9gr 

9. In 35'ton I7cwt I qr 23 lb 7oz i3dr how many drams? 

Ans. 20571005. 

10. How many barley-cornSx will rqadb round the earth, 
supposing it^ according to the best calculations, to be 25000 
miles ? * Ans. 4752000000. 

11. How inany seconds are in a solar year, or 365 days 
5 hrs 49 min 48^ sec ? Ans. 3 1 556928. 

12. In a lunar month, or 29 ds 12 hrs 44 min 3 sec, how 
many second 2 Ans. 2551443. 

COM. 



St 



ARTTHMETia 



COMPOUND ADDITION, 

CoMPOiTND ApDiTiON shows how to add or collect seTeral 
numbers of different denominations into one sum. 

RijL£.*-Place the numbers so» that those of the same de^* 
nomination may stand directly under eauch other, and draw z 
line below them» Add up the figures in the lowest denomi- 
nadon, and find, by Reduction^ how many units, or ones, of 
the next higher denomination are contained in their $um.-~-» 
Set down the remainder below its proper column, and cany 
those units or ones to the next denomination, which add up 
in the same manner as be£bre«r-*Proceed thus through all the 
denominations, to the highest, whose sum, together with tht 
several remainders, will give the answer sought. 

The method of proof is the same as in Simple Addition. 

EXAMPLES or MONET» 



I. 




2. 




s. 




4. 




/ J 


d 


/ X 


d 


/ / d 


/ 


/ 


rf 


1 13 


3 


14 1 


5 


15 11 10 


J3 


14 


8 


a 5 


lot 


8 19 


2t 


3 14 6 


5 


10 


H 


6 18 


7 


7 8 


It 


23 6 2| 


93 


11 


6 


2 


H 


21 2 


9 


14 d 4t 


7 


5 





4^ 


3 


7 16 


8f 


15 6 4 


IS 


2 


S 


n 15 


4i 


4 


3 

V 


6 1^ 9^ 





18 


7 


%9 15 


H 




' 


• 




• 




32 2 


. • 












39 15 


H 








/ 






. 5. 


6. 




. 7. 




8- 




/ / 


d . 


/ s 


d 


/ s i 


/ 


J- 


d 


H 


n 


37 15 


s 


613 2t 


472 


15 


3 


8 15 


3 


14 12 


H 


7 16 8 


9 


2 


2t 


62 4 


7 ■ 


17 14 


9 


29 13 10 J 


27 


12 


6t 


4 IT 


& 


23 10 


9t 


12 16 2 


370 


16 


2t 


23 


H 


8 6 





1 b^ 


13 


7 


4 


6 6 


7 


14 


H 


24 13 


6 


10 


■5t 


SI 


lOj 


54 £ 


ii 


5 10|. 


30 





111 

» 

1 




\ 














• 




EXAH<^ 



/ 



COMPOUND AjDOmON. 



' aa 



Exam. 9. A nobleman, going cmt of tovn, is informed by 
his steward, that his butcher's bill comes to 197/ ISj '\\d: his 
baker's to 59/ .'J/ 2|^; his brewer's to 85/; his wine-mer- 
chant's to 103/ 13x; to his corn-chandler is due 75/ %d; \o 
his tallow-chandler and cheesemonger, 27 i IBs 11|</; and 
to his tailor 55i Ss B^d; also for rent, servants' wages,, aqd 
other charges, 127/ 3/; Now, supposing he would take 100/ 
with him, to defray his charges on the road, for what sum 
must he send to his banker ? Ans! 830/14/ 6^. 

10. The strength of a regiment of foot, of 10 compani^, 
and the amount of their subsistence*, for a month bf 30 days, 
:flu:cording to l)he anaeKed Table, are r^uired i 



Numb. 


Rank. 


Sul^istence for 


- a Montih. 






* / # 


d 


1 


Colonel 


27 





1 
1 

7 

11 


Lieutenant Colond 
IVfajor 
Captains 
Lieutenants 


19 10 
17 5 
78 15 
57 13 





* 


9 

1 

\ 

1 


JEnsigns 
Chaplain 
Adjutant 
Quarter-Master 


40 10 

7 10 

4 10 

, 5 5 








1 

1 

30 
30 

2P 
2 


Surgeon 

Surgeon's ]\{ate 

Seijeants 

Corporals 

Pruipmers 

Kfes 


4 10 

4 10 

45 

30 

20 

2 













390 


Private Men 


292 10 





5Qir 


Total 


656 iO 






■*■**• 



* Subsistence Money, is the money paid to the soldiers weekly 
which is short of their full pay, because their clothes^ accoutre-* 
ments, ^c. are to be accounted for. It is likewise, the money ad- 
vanced to officers till their account are made op*J||^<^^ is com- 

*monly once a year, when they are paid their arrearir^lhe follo)w- 
ing Table shows the full pay and subsistence of each rank on the 

^English establishment . 



VoLiL 



D 



DAILT 



34 



ARITHMETIC. 





•- 


• 


o o 


CO ^ O o c 


> 00 




'• 


n 




»-* p^ 






"2 


£ 


O) QD 


-* (o t^ lo -i* 


»o »o "O 




k4 

C3 


g 


•H • 


-H 


^^ »«^ 





O 


-It »- 


^ o o o o 

t 


oooo 






o ^ 


O CO o o o 


<oo 




•£ 


'3S 

9 


o -• 


CO ^ ^ -^ M 


, 121.^ 




1 c/> 


»H »-« 


o o o o o 


.000 




b:. 




O GO 


CO 00 




Cfl 


rt 










-E 


o. 


1 1 


1 12 2 1^ 


00 1 c» »o 










O O O 


» 00 




• 


o ^ 


CO co<o o ^ 


1 00 




1 


en 

'J 

9 


^4 


•H ^2 :2 1 ^ 


' ^ 1 0)«0 


• 


<^ - 


-H ^ 


^ O O c 


> .0 00 


ft 

b3 




• 


o o 


oooo O 





y 


• 

00 


£. 


CO -H 


O "t O -H 1 -H 


11*1 


Lb 


•TS 




—1 r-l 


•-I ^i* '1 ^ 


U. 


S 


g 








c 


3 

o 


(s« 


^ -H 


p- ph o o c 





q 


• 


O CO 


CO O '- <N ^ 





c 


^ 

a 


4>< 

in 

1 


t^ CO 


O QD C^ X 11^ 


1 l'°l 


CO 

y5 


w i 


»H f— 


oooo o 





c 






•H lO (X) 00 00 p 


) 00 10 
1 »o Ci 1 »o 


u 




- o 


O O O O CO 


1 

00 


C 




' 






> 


it 

o rz 


ooooo oooo 





< 


^ •* O t^ "o 1 o t^'o *^ 1 1 1 


12 l-^ 


>- 


O w 


c<i-«^oo oooo 


'o 


< 


u .* 


o o 


CO iSiO O O C 


1 -* 9 


e 




1— 1 

2 I« 1 




4 ^^ 1 




Q^ 


"^ fH 


O OOOO o 


1 00 00, 






o o 


O OOOO 


00. 




2^ ?: ^ 


12^ 1 


r-i »f5 O C^C» I • 


1 CS 1 CO GO 




""-^^^S 


mn •^ 


^ OOOO 


00 


- 




• "'-N . . 


' 


>•••,••• 






--§ : ; 


k 4 
















> • • • t^D • * • 

> . , , u • • • 




< 




1 

> < 
» 1 

> 4 


1 












* '-< 


, . / -3 . . . 

. . S3 gco . . . 
fc S S • h^^ • 




• 


3 c 

13 "oJ <i> r 


> O o ^ "r pj O • • ^ 


^ »L C t <^ CO fe 






5 o«Jc 








U U — cs 


1 « T3 ,^ ,«^ S ^o .9 a 3 


► 2 (3 eg pq vj 



I ° 

o 

•> 
01 

e 
o 

'5 

a 
s 

o 
;> 

o 



"2 

o 



so 

c 



o 



a 

■i 



'^ a 

^ S 

•»••■ 

Cfa o 

S w 

Of " 



1*^ 

3f^ 



.J 



.tXAMfLSS 



COMPOUND ADPITION. S3 

EXAMPLES OF WEIGHTS, MEASURES, t(e. 



' 


TROY 


WEIGfiT. 






APOTHECARIES* WEIGHT. 


f 


K 


2. 






3. 




4. 


lb 


oz dwt 


oz dwt gr 




lb 


OZ 


dr sc 


oz dr sc gr 


17 


3 15 


37 9 3 




3 


5 


7 2 


3 5 1 17 


, 7 


9 4 


9 5 3 




13 


,? 


3 


7 3 2 5 


O 10 7 


8 12 12 




19 


6 2 


16 7 12 


^ 9 


5 


17 7 8 







9 


1 2 


7 3 2 9 


i76 


2 17 


5 9 




36 


3 


5 


4 1 2 18 


i23 11 12 


3 19 




5 


8 


6 1 


36 4 1 14 




AVOI«l>UPOIS WEIGHT. 








T.ONO MEASURE. 




5. , 


6. 








7. 


. 8. 


lb 


oz dr ' 


♦ cwt qr 


lb 


1 


mis fiir pis 


yds feetinc 


• 17 


10 13 


15 2 


15 




29 


3 14 


127 1 5 • 


5 


14 8 


6 3 


24 




19 


6 ^9 


12 2 9 


12 


9 18 


9 1 


14 




7 


24 


10 10 


27 


1 6 


9 1 


17 




9 


1^37 


54 1 11 





4 


10 2 


6 




7 


3 


5 2 7 


6 


14 10 


3 


3 




4 


5 9 


23 5. 


- 


CLOTH 


MEASURE. 








LAND 1 


MEASURE. 




9. 


10. 








11.. 


12. 


yds 


qr nls 


el en qrs 


nls 




ac 


ro p 


ac ro p 


26 


3 1 


270 1 







225 


3 37 


19 16 


13 


1 2 


57 4 


3 




16 


1 25 


270 3 29 


9 


1, 2 


18 1 


2 




7- 


2 18 


6 3 13 


217 


3 


3 


2 




4 


2 9 


53 34 


9 


1 


10 1 







42 


1 19 


7 2 16 


55 


3 1 


4 4 


1 




7 


6 


75 23 




WINK 


E^EASURB. 


— 


- 


ALE and BEER MEASURE. 




13. 


14 


• 






15. 


16. 


t 


hdsgal 


hds gal pts 




hds 


gal pts 


hds gal pts 


13 


3 -15 


15 61 


5 




17 


37 3 


29 43 5 


8 


1 37 


* 17 14 


13 




9 


10 \5 


12 19 7 


14 


1 20 


29 23 


7 




3 


6 2 


14 16 6 


25 


12 


3 15^ 


1 




5 


14 


6 8 1 


. 3 


1 9 


16 8 







12 


9 6. 


. .57 13 4 


72 


3 21 


4 3.6 ' 


6 




8 42 4 


5 6 



D2 COM- 



■"S6 ARfTHMETlC. 



COMPOUND SUBTRACTION. 

« 

Compound Subtraction shows how to find the difl&r- 
0nce between any two numbers of different deaomin^tio^Sr 
To perform which^ ob^'erve the following Rule : 

* Place the less number below the greater, so that the 
parts of the same denomination may stand directly tinder 
each other ; and draw -a Kne below them. — Begin at the 
right-hand, and subtract each number ,or^part m the lower 
line, from the one just above it, and set the remainder 
.straight below it. — But if any number in the lower line be 
greater than that above it, add as many to the upper number 
is make 1 of the next higher denominatibn v then take the 
lower number from the upper one thus increased, and set 
down the remainder. Carry the unit borrowed to the next 
number in the lower line ; after which subtract this number 
from the one above ity as befofe; and so proceed till* the 
whole is finished. ' Then the several remainders, taken to^ 
jcther, will be the whole dffference sought. 

The method of proof is the same as in Simple Subtraction. 



iEXAMPLES OF MONET. 

I. - 2. 3. . 4. ^ 

I s i I s d I s d t s d 

From 79 17 8|. 103 3 2J- SI 10 J I 254, 12 

Take 35 12 4f 71 12 5^ 29 13 S^ 37 9 4^ 



Rem. 44 5 4Jr 31 10 8^ 

■ ' ' ■ * 

Proof 79 17 8| 103 -^ 2^ 



5. Wh^t ia the difFerence between 73/ 5|^/ and 19/13/ lOrff 



Ans. 53/ 6* 7^. 



* The red-son of this Rule wiM easily appear from what* has hcenc 
said in Simple Subtraction; for the borrowing depends on the 
same principle, and -is only different as Hie;numberstoi>e^b<' 
Uacted: are of difiete&t deApmination^. 



COMPOUND SUBTRACTION. 



93 



Ex 6. A lends to B 100/, how much, is B. in debt afier^ 
has taken goods of him to the amount of 73/ 12/ 4Jr//* 

Ans. 26/ Is 7i(L 
, 7, Suppose' that my rent for half a year is 20/ I2/9.an<l 
ihat I haye laid out for the land-tax 14« 6di and for. severs^ 
repJlirs 1/3* 3 J^, what have I tppay of my half-year's rent ? 

Ans. 'l8/ 14j 2id. 
8* A trader^ failing, owes to A 35/ Is 6d, to B 91/ 13/ id^ 
to C 53/ 7id, to D 87/ 5s, and to E 11 1/ 3/ 5^ When 
this happened, he had by him* in c^sh 23/ 7/ Sd^ in wares 
53/ 1 1/ lOj^, in household furniture 63/ 171 7^, and in re- 
coverable book^-debts 25/ 7/ 5d. What will his creditors lose 
by him, suppose these things delivered to them ? 

. Ans. 2121. 5s Sid. 

EICAMPLES OF WEIGHTS, MEASURES, iS^C. 



TROV WfilGUT. 



1. 

lb ozdwtgr 
Prom 9 2 12 10 
Take 5 4 6 17 



2. 
lb oz dwt gr 
7 10 4 r7 
3 7 16 12 



Al'OTHECARIES WfilOHT- 

3. 
lb oz dr scr gr 

73 4 7 14 
29 5 3 4 19 



Rem. 
Proof 



AVOIRDUPOIS WEIOHT. 



LONG MEASURE. 



4. 


5. . 


;6- 


7. 


C qrs lb 


lb oz dr 


m fu pi 


yd ft in 


From 5 0.l7 


71 5 9 


14 3 17 


96 4 


Take 2 3 10 


17 9 18 


7 6 U 


72 2,9 



Rem. 
Prbof 



^LOTH MEASURE. 



8. . 9. 

yd qr nl yd qr nl 

From 17 2 1 9 2 

T^e 9 2 7 2 1 



LAND MEA817R£* 
10. 11. 


ac ro p 
17 1 14 
16 2 8 


ax: ro p 
57 1 16 
22 3 2d 



Rem. 
Proof 



WINE 



38 


ARITHMETIC. 


WINE 


MEASURE. 


ALE and BE 


12. 
t hdgal 
From 17 2 23 
Take 9 1 36 


13. 
hdgal pt 
5 4 
2 12^ 6 


14. 
hd gal pt 
14 29 3 
9 35 7 



Rem. 
Proof 



IS. 

hd gal 
71 16 
19 7 


pt 

5 
1 






- 



^ 




DRY MEASURE. 


From 
Take 


la 
9 
6 


16. 17. 
qr bu bu gal pt 
4 7 13 7 1 
3 5 9 2 7 



TIME. 



Hem. 

Proof 



-r*~ 



LP ™ ' * 



18. 19. 

mo we da ds hrsmin 

71 2 5 114 17 26 
17 1 6 72 10 37 



20. The line of defence in a certain polygon being 236 
yards, and that part of it which is terminated by the curtain 
and shoulder being 146 yards 1 foot 4 inches; what then was 
the length of the f^ce of the bastion? Ans. 89 yds 1 ft S iq. 



"TT" 



COMPOUND MU^LTIPLICATION, 

Compound Multiplication shows how to find the 
amount of any given number of different denominations re^ 
peated a certain proposed number of times \ which is per** 
formed by the following rule. 

Set the multiplier under the lowest denomination of th^ 
multiplicand, and drjiw a line below it. — Multiply the num- 
ber in the lowest denomination by the multiplier, and find 
how many units of the next higher denomination are con- 
tained in the product, setting down what remains. — In like 
manner, multiply the number in the next denomination, and 
to the product carry or add the units, before found, and find 
how many units of the n^xt higher denomination are in thi^ 

, , f^mowit, 



COMPOUND MULTIPLICATION. 39 

amount, which carry in Hke manner to the next product, 
settiiig down the overplus.*^Proceed thus to the highest de- 
nomination proposed : so shall the kst product, with the se« 
Verat remainders, taken as one compound number, be the 
whole amount required. — ^The method of Proof, and the 
reason of the Rule, are the same as in Simple Multiplication. 

EXAMPLES OF MONEY. 

i. To find the amount of 8 lb of Tea, at 5s 8 Id per lb. 

i- d 

• 8 



jf 2 5 8 Answer. 

/ / d 

2. 4 lb of Tea, at Is Sd per lb. Ans. 110 8 

3. 6 lb of Butter, at 9^4 per lb. Ans. 4 9 

4. YlbofTobacco, at li 84rfperlb, Ans. 11 11^ 

5. 9 stone of Beef, at 2^7 trf per St. Ans. 1 10 

6. 10 cwt of Cheese, at 2/ i7« KWper cwt. Ans. 28 1 8 4 

7. 12 cwt of Sugar, at 3/7/ 4rf per cwt. Ans. 40 8 

CONTRACTIONS. 

I. If the multiplier exceed 12, multiply successively by its 
component parts, instead of the whole number at once. 

EXAMPLES. 

1 . 15 cwt of Cheese, at 17^ 6d per cwt» 

/ / V 

17 6 

S 



2 12 6 
5 



13 2 6 Answer. 



/ X d 

2. 20 cwt of Hops, at 4/ Is 2d per cwt. Ans. ^7 3 4 

3. 24 tons of Hay, at 3/ 7/ 6d per ton. Ans. 810 O 

4. 45 ells of cloth, at is 6d per ell. . Ans. 3 7 6 

Ex. 5. 



4a ARITHMETRi 

t 

Ex. 5. 63 gallon J of Oil, at 2/ %d per galh Aas* 7 
$. 70 iKirrels of Ale, at l/^j per t^urel. Ans. 84 
*7. 84 quarters of Oats, at 1/ 12/ 8//per qr. Ans* 187 

8. 96 quarters of Barley,at l/3x4rf)per qr. Am. 112 

9. 120 days' Wages, at bs 9^ per day. . Ans; 34 
10. 144 reamsof Paper, at 13/ 4^^ per ream. Ans. 96 

II. If the multiplier cannot be exactly produced by the 
multiplication of simj^^ numbers, take the nearest number to 
it, either greater or less, which can b^ so produced,' aAmuI-^ 
tipiy by its parts, as before. — ^Then multiply the giveii mul- 
tiplicand by the difference between this assumed number and 
the multiplier, and add the product tq that before found, 
>rhen the assumed number is less than the multiplier^ bul^ 
subtract the same wh^n it \& greater. 



/ 


4 


1 


9 








41 











10 


G 













EXAMPLES. 


f.' 


■?^ 


y?irds of Cloth, at 

/ 

• 


3/ Ot^ per yard. 
/ d 









3 


f 







15 


H 








, 


5 

f . 




3 


16 


H 




# 


§ 


3 


oi . 






19 


li ■Answer. 



/ s d 

2. 29 quarters of Corfi, sit 2/ 5/ S^d pet qr. Arts. 65 12 lO^ 

si 53 loads of Hay, at 3/ 15/ 2^ per ibad. Ans. 199 3 10 

. 4. 79bushelsofWh^at,atll/5|.rf*^er bush. Ans. 45 6 lOf 

5. 97 casks of Bepr, at 12/ 2^?peif cask. Ans. 59 O 2 

6. 114 stone of Meat, at 1 5/ 3|tf per stone. Ans. 87 5 74. 

EXAMPLES OF WEIGHTS AND MEASURES. 



1. 

• 

lb oz dwt gr 
28 7 14 10 

4 


2. 
lb oz dr sc 

2 6 3 2 

• 


8 


cwt 
29 


3. 

qr lb oz 

2 16 14 

12 


<^- 


• 


■ """V ■ 


> 





•♦►•'■ 



COMPOUND DIVISION. 41 



4. 






5, 






6. 


ttds fu pis 


yds 


yds 


qfs 


na 




ro po 


22 5 . 29 


6 
4 


126 


3 


1 

7 


28 


3 27 
9 



T»" 



V. 



tuns hhd gal pts 
^0 2 26 2 

3 



8, 


9. 


we qr bu pe 


mo we da ho min 


24 2 5 3 


172 3 5 16 49 


6 


10 



«•■ 



"W" 



*?iP 



COMPOUND DIVISION, 

CoMi^oUND Division teaches how to divide a number of 
sj^veral denominations by any given number^ or into any 
number of equal pjarts ; as follows : 

Fla^e the divisor on the left of the dividend^ as in Simple 
Division. — ^Begin at the left-hand^ and divide the number of 
the highest denomination by the divisor^ setting dgwn th^ 
quotient in its proper place.-«-If there be any remainder after 
this division, reduce it to the next lower denominattony 
which add to the number, if any, belonging to that denomi- 
nation, and divide the sum by the divisor. — Set down a^a 
tlfis 4dot}ent| reduce its remainder to the next lower deho-* 
mination again^ and so on through all th^ denominations tq 
Isfae last. 

r 

EXAMPLES OF MONBY. 

I. Divide 237/ 8/ 6d by 2. 

lid 

2 ) 231 8 6 



;^118 14 3 the Quotient. 



2. Divids . 



4a ARTTHMEnC- 

I s d t s d 

^i Divide 4S2 12 \\ by 3. Ans. 144. 4 a|. 

3. Divide 507 3 5 by 4. Ans. 126 15 lOf 

4. Divide 632 7 61- by 5. Ans. 126 9 6 

5. Divide 690 14 S^J: by 6. Ans. 115 2 44- 

6. Divide' 705 10 2 by 7. Ans. 100 15 8^ 

7. Divide 760 5 6 by 8. Ans. 95 O 8:^ 

8. Divide 761 5 7| by 9. Ans. 84 11 %\ 

9. Divide 829 17 10 by 10. Ans. $2 19 ^\: 

10. Divide 937 '8* 8|byll. Ans. 85 4 5 

11. Divide 1145 11 4^ by 12. Ans. ^^ 9 3^. 



CONTRACTIONS. 

\p If the divisor exceed 12, find what simple numbers, 
multiplied together, will produce ity^and divide by them se-> 
parately, as in Simple Division, as below. 



' . £XAMPJ^£5. 

I. What is Cheese per cwt, if 16 cwt cost 25/ 14j ^i? 

Isd 
4) 25 14 8 



4) 6 8 8 



j^ 1 12 2 the Answer. 



/ 



2. If 20 cwt of Tobacco come to 



: 20 cWt ot lobacco come to 7 An « a in a 

150/ 6/ 8i, what is that per cwt ?i ^^* ^ * 

3. Divide 98/ 8j by 36. Ans. 2 14 8 

4. Divide 71/ 13/ \Odhj 56. Ans. 1 5 7-5- 

5. Divide 44/ 4 j by 96. Ans. 9 2t 

6. At 3J / lOx per cwt, how much per lb } Ans. 5 7^ 

IL If the divisor cannot be produce4 by the multiplication 
of small numbei-s, divide by the whole divjisor at once, after 
the manner^of Long Division, as follows. 



EXAU- 



COMPOUND DIVISION. 43 



EXAMPLES. 



1. Divide 59/6/ Sid by 19. 

I s d I s d 
19 ) 59 6 3| (3 2 51 An^. 
57 



2 
20 

46 (2 
38 



8 
12 





99 (5 

95 




• 




4 
4 


. 






19 (1 


■ 


t 


2, 

3. 

4. 
5. 


Divide 39 14 5| by 57. 
Divide 125 4 9 by 43- 
Divide 542 7 10 by 97. 
Divide 123 11 2t by J 27. 


/ s 
Ans. 13 
Ans* 2 18 
Ans. 5 11 
Ans. 19 


Hi 

3 
10 



EXAMPLES OF WEIGHTS AND MEASURES. 

1. Divide 17 lb 9 oz dwts 2 gr by 7. 

Ans. 2 lb 6 oz 8 dwts 14gn 

2. Divide 17 lb 5 oz 2 dr 1 scr 4 gr by 1 2. 

Ans. lib 5 oz 3 dr 1 scr 12 gr. 

3. Divide 178 cwt 3 qrs 14 lb by 53. Ans. 3cwt Iqr 14lb. 

4. Dividie 144 mi 4 fur 2 po 1 yd 2 ft in by 39. 

Ans. 3 mi 5 fur 26 po yds 2 ft 8 in. 

5. Divide 534 yds 2 qrs 2 na by 47. Ans. 1 1 yds 1 qr 2iia. 
6.- Divide 7 1 ac 1 ro 33 po by 5 1 . Ans. 1 ac 2 ro 3 po. 
1. Divide 7 tu hhds 47 gal 7 pi by 65. Ans. 27 gal. 7 pi. 

8. Divide 387 la 9 qr by 72. Ans. 3 la 3 qrs 7 bu. 

9. Dividij 206 ino 4 da by 26. Ans. 7 mo 3 vire 5 ds. 



44 ARITHMETIC. 



The golden RULE,' or RULE OF THREE. - 

The Rule of Three teaches how to find a fourth pro- 
portional to three numbers given : for which reason it is 
sometimes called the Rule of Proportion. It is called the 
Rule of ,Three, because three terms or numbers are given, to 
find a fourth. And because of its great and extensive use- 
fuhiess, it is often called the Golden Rule. This Rule is 
usually considered as of two kinds, namely. Direct, and 
Inverse. 

The Rule of Three Direct is that in which more requires 
more, or less requires less. As in this; if 3 men dig 21 yards 
. of trench in a certain time, how much will 6 men dig in the 
same time ? Here more requires more, that is, 6 men, which 
are more than 3 men, will also perform, more work, in the 
same tihie. Or when it is thus: if 6 men .dig 42 yards, how 
much will 3 men dig in the same time ? Here then, less re- 
quires less, or 3 men will perform proportionably less work 
than 6 men, in the same time. In both these cases then, the 
Rule, OK the Proportion, is Direct j and the stating must be 

thus. As 3 : 21 :: 6 : 42, 
or thus,' As 6 : 42 1 : 3 : 21, 

But the Rule of Three Inverse, is when nlore requires less, 
or less reouires more. As in this i if S m^o dig a celtain 
quantity or trench in 14 hours, in how many hour^ Will 6 
men dig the Kke quantity ? Here it ii evident that 6 men, 
being more than 3, will perform an equal quantity of work in 
less time, or fewer hours. Or thus : if 6 men perform a 
certain quantity of work in 7 hours, in how many hours will 
'3 nien perform the same ? Here less requires more, for 3 
men will take more hours than 6 to perform the same work. 
In both those cases then the Rule, or the Proportion, is 
Inv^i^ ; and the stating must be 

thus. As 6 ? 14 :: 3 : 7^ 
or thus. As 3 : 7 :: 6 : 14. 

And in all these statings, the £:nirth term is found, hj 
inukiplying the 2d and 3d terms together, and dividing the 
product by the 1 st term. 

^ Of the three given numbers ; two of them contain the 
supposition, and the third a demand. And for stating and 
working questions of these kinds, observe th^ following ge- 
lieral Rule :.• 

Stat^ 



RULE OF THREE. 



*» 



State the qt^stion, by setting down.in astraigbtline the 
three given numbers, in the following mannery'viz. ao that 
the 2d term be that ntunbea* of supposition which is of the 
same kind that the answer or 4th term is to be ; 'making the 
other number of supposition the 1st term, and the demanding 
number the dd term, when the question is in direct propor- 
tion ; but contrariwise, the other number of supposition the 
Sd term, and the demanding number the 1st term, when th^ 
question has inverse proportion. 

Then, in both cases, multiply the 2d and Sd terms to* 
gether, and divide the product by the Ist, which will give 
the answer^ or 4th term sought, viz. of the same denomma- 
tion as the second term. 

Nate^ If the first and third t^ms consist of diffinrent deno^ 
minations, reduce them both to the same : and if the second 
term be a compound number, it is mostly convenient to re- 
duce it to the lowest denomination mentioned.^-^If, after di« 
vision, there be any remainder,, reduce, it to the next lower 
denomination, and divide by the same divisor as before, and 
the quotieilt will be of this last denominatioh. Proceed in 
the same manner with all the remainders, till they be re* 
duced to the lowest denomination which the second admits 
of, and the several quotients taken together will be the an- 
swer required. 

N^te also. The reason for thtf foregoing Rules will appear, 
when we come to treat of the nature of Proportions.— Some- 
times two or more statings are necessary, which may always 
be known from the nature of the question. 

* EXAMPLES. 

i. If 8 yards of Cloth cost 1/ 4^, what will 96 yards cost i 

yds 1 s yds 1 s 
As 8 : 1 4 *: : 9€ : 14 S the Answer. 
20 



24 
96 



144 
216 

8) 2304 



2^0) 28,8* 



jf 14 8 Answer* 



£x. 2s 



4^ ARITHMETrC% 

Ex. 2. An engineer having rabed 100 yards of a certain 
work in 24 days with 5 men ; how many men must he enw 
jploy to finish a like quantity of work in 15 days ? 

ds men ds men 
As 1 5 : 5 ': : 24 : 8 Ans. 



15) ^20 ( 8 Answer. 
120 



3» What will 72 yards of cloth cost> at the rate of 9 yards 
for 51 12j ? Ans. 44/ 16/. 

4. A person^s annual income being 146/; how much is 
that per day ? Ans. 8/. 

5. If 3 paces or common steps of a certain person be equal 
to 2 yardS) how many yards wiU 160 of his paces make ? 

Ans. 106 yds 2 ft. 

6. What length must be cut off a board, that is 9 inches 
broad, to make a square foot, or as much as 12 inches in 
length and 12 in breadth contains ? Ans. 16 inches. 

7. If V50 men require 22500 rations. of bread for a month; 
how' many rations will a garrison of 1 200 men require ? 

• Ans. 36000. 

8. If 7 cwt 1 qr of sugar cost 26/ 10/ 4^; what will be the 
price of 43 cwt 2 qrs ? Ans. 159/ 2s. 

9. The clothing of a regiment of foot of 750 men amount- 
ing to 2831/ 5s; what will the clothing of a body of 3500 
men amount to? Ans. 13212/ 10/; 

10. How many yards of matting, that is 3 ft broad, will 
cover a floor that is 27 feet long and 20 feet broad ? 

Ans. 60 yards. 

1 1 . What is the value of 6 bushels of coals, at the rate of 
\l I4fs 6d the chaldron ? Ans. 5s 9d. 

12. If 6352 stones of 3 feet long complete a certain quan- 
tity of walling ; how many stones of 2 reet long will raise a 
like quantity ? Ans. 9528. 

13. What must be given for a piece of silver weighing 
73 lb 6 oz 15 dwts, at the rate of 55 9d per ounce ? 

Ans. 253/10/0|rf. 

14. A garrison of 536 nien having provision for 12 months; 
how long will those provisions last, if the garrison be increased 
to 1 124 men ? Ans. 174 days and -tttt' 

15. What will be the tax upon 763/ 15x, at the rate of 
35 6d per pound sterling ? Ans^ 1 33/ 1 3/ If ^. 



16. A 



RULE OF THREE. 47 

16* A certain work being raised in 12 days, by working 4 
liours each day ; how long would it hjive been In ra^ising bf 
%Krorking 6 hours per day ? . Ans. 8 days, 

IT. What quantity of com can I buy for 90 guineas, at the^ 
raje of 6s the bushel ? Ans. 39 qi^s 3 bu. 

18. A person, failing in trade, owes in all 977/; at which 
time he has, in money, gobds, and recoverable debts, 420/ 6s 
S^d\ now supposing these things delivered to his creditors, 
lidw much will they get per pound ? Ans. 8x 7 j€f.> 

« 1 9. A plain of a certain extent having supplied a body of 
SOOO horse with forage for 1 8 days ; then how many days 
would the same plain have supplied a body of 2000 horse? 

Ans. 27 days. 

20. Suppose a gentleman^s income is 600 guineas a year, 
and that he spends 25/ 6d per day, one day with another ; 
' ho'w much will he have saved at the year's end ? 

Ans. 16^/ I2s 6d. 

2i . What cost 30 pieces of lead, each weighing 1 cwt 121b, 
at the rate of I6j W the cwt ? Ans. 27/ 2s 6d. 

22. The governor of a besieged place having provision for 
54 days, at the rate of 1-^ lb ofbread ; but being d^irous to 
prolong the siege to 80 days, in expectation of succour, in 
that case what must the ration ofbread be ? Ans. l-^lb. 

23. At half a guinea per week, how long can I be boarded 
for 20 pounds ? Ans. 38^?^ wks* 

24. How much vrill 75 chaldrons 7 bushels of coals come 
to, at the 'rate of 1/ 13i 6^ per chaldron ? 

Ans, 125/ 19x0|J. 

25. If the pormy loaf weigh 8 ounces when the bushel of 
wheat costs 7s 3rf, what ought the penny loaf to weighi|lhen 
the wheat is at 8/ ^dP Ans. 6 oz 15t^ dr. 

26» How much a year will 173 acres 2 roods 14 poles of 
land give, at the rate oi Ills 8d per acre? 

- > Ans. 240/ 2s 1^^. 

• 27. To how much amounts 73 pieces of lead, each weigh- 
ing 1 cwt 3 qrs 7 lb, at 10/ 4j per fother of 1 9^ cwt? 

Ans. 69/ 4j 2i/ 111 q. 
/^ 28. How many yards of stuff, of 3 qrs wide, will line a 
' cloak that is ^ yards in length and 3t yards wide ? 

Ans. 8 yds qrs 2| nl. 
'29. If 5 yards of cloth cost 14/ 2d, what must be. given for 
9 pieces, containing each 21 yards 1 quarter? 

^ Ans. 27/ \s \Q\d, '; 

30. If a gentleman's estate be wortt 2i07/ 12/ a year j 
what may he spend per day, to save 5C0/ in the year ? 

Ans. 4/ Ss l-^Sd. 
31. Wanting 



M ARTTHMETia 

31. Wanting just an acre of land cut off from a piece 
^which is 13^ pdes in hreadthj what length must the piece be? 

/ Ans 11 po 4yds 2ft O^f in. 

32. At 7s 9f «/ per yard, what is- the value of a piece of 
doth containing 53 ells English 1 qu. Ans. 25/ lb/ 1^. 

33. If the carriage of 5 cwt 14 lb for 96 miles be 1/ 12/ 6ds 
Jhow far may I have 3 cwt 1 qr carried for the same money i 

Ans. 151 m 3 fur ^rrt^* 

34. Bought a silver tankard, weighing 1 lb 7 oz 14 dwts i 
what did it cost me at 6/ 4^ the ounce ? Ans. 6/ 4/ 9 V« 

35. What is the half year's rent of 547 acres of land, at 
1 5s 6d the acre ? Aas. 21 1/ 19j SJ. 

36. A wall that is to be built to the height of 36 feet, was^ 
raised 9 £set high by 16 men in 6 days; then how many men 
must be employed to finish the wall in 4 days, at the same 
rate of working ? Ans. 72 men* 

37. What will be the chargfe of keeping 20 horses for a 
year, at the rate of 14^^ per day for each horse ? 

Ans. 441/Oj lOrf. 

38. If 18 elk of stuff that is i yard wide, cost Z9s6d'} 
what will 50 ells, of the same goodness, cost, being yard wide-^ 

Ans. 7/ 6s :i||</. 

39. How many yards of paper that is 30 inthes wide, will 
hang a room that is 20 yards in circuit an^ 9 feet high? 

Ans. 72 yards. 

40. If a gentleman's estate be worth 384/ 16/ a year, and 
the lapd-tax be assessed at 2/ 9id per pound, what is his net 
aftnuar income? Ans. 331/ 1/ 94^. 

41 . The circumference of the earth is about 25000 miles i 
at what rate per hour is a person at the middle of its sur£a€e 
carried round, one whole rotation being made in 23 hours 
56 minutes ? Ans. i044-ry^ miles. 

42. If a person drink 20 bottles of wine per month, when 
It costs 8/ a gallon j how many bottles per month may he 
drink, without increasing the expense, when wine costs 10/ 
the gallon? Ans. 16 bottles. 

43. What cost 43 qrs 5 bushels of corn, at 1/ 8/ 6d the 
. quarter ? Ans. 62/ 3/ S^d. 

44. How many yards of canvas that is ell wide will line 
50 yards of say that is 3 quarters wide? Ans. 30 yds. 

45. If an ounce of gold cost 4 guineas, whaC is the valfae 
of -a grain? Ans. 2-yT,it 

46. if 3 cwt of tea cost 40/ 12/; at how much a pound 
must it be retailed, to gain loi^ by the whole ? Ans.3y|^^j. 



CX)MPOUN» 



C « 3 



COMPOUND PROPORTKHf- 



Compound Proportion ^ows how to resolve such ques- 
tions as require two or more statings by Simple Proportion ; 
and these may be either Direct or Inverse. • • 

In these questions, there is always given an odd mimber of 
terms, either five, or seven, or nine, Sec. These are distm-. 
guished into terms of supposition, and terms of demand, 
there being always one term more of the former *than of the y^ 
latter, which is of the same kind with the answer sought. 
The method is thus : 

Set down in the middle place that term of supposition 
which is of the same kind with the answer sought.— -Take 
one of the other terms of supposition, and one of the demand- 
ing terms which is of the same kind with it ; then place one 
of them for a first term, and the other for a third, according 
to the directions given in the Rule of Three. — Do the same 
with another term of supposition, and its corresponding de- 
manding term ;, and so on if there be more terms of each 
kind ; setting the numbers under each other which fall all on 
the left-hand side of the middle term, and the same for the 
others on the right-^hand side. — ^Then^ to work 

By several Operations, — ^Take the two upper terms and 
the middle term, in the same order as they stand, for the first 
Rule-of-Three question to be worked, whence will be found 
a fourth term. Then take this fourth number, so found, for 
the middle term of a second Rule-of-Three question, and the 
next two under terms in the general stating, in the same 
order as they stand, finding a fourth term for them. And so 
on, as far as there are any i|iumber$ in the general Stating, 
making always the fourth number, resulting from each simple 
stating, to be the second term in the next following one. 
So shall the last resulting number be the answer to the 
question. 

By one Operathfi, -^Multiply together all the terms stand- ^ ^t^o^^^^Ur 
ing under eajch other, on the left-hand side of the middle I ^^eSL^ 
terni} and, in like manner, multiply together all those on the > ^^^^JgSul, 
right-hand side of it. Then multiply the middle term by \ ^f^iim^ 
jfthe latter product, and divide the result by the former pro- «^ ^^^^^-^^ 
'duct } so shall the quotient be the answer sought. ^t^ij?^'*^ 

Vq|.. I, % l&^^AMPtES 



50 



ARITHMETIC. 



EXAMPLES. 



1. How ms^y men can complete ^ trench of 135 yards 
long in 8 da^y when 16 men can dig 54 yards in 6 days ? 

General Stating. 

yds 54 : 16 :: 185 yds 
dfiys 8 6 days 



432 



810 
16 

4860 

81 men 



432 ) 1 2960 ( 30 Ans. by one opcratioq. 
1296 ^' ' 







The same hy two Operations. 



1st. 
As 54 : 16 :: 135 : 40 

16 
/ 

8IQ 
135 

54) 2160 (40 
216 



2d. 
As 8 : 40 : : 6 : 30 
6 



8 ) 240 ( 30 Ans. 
24 

O 



2. If 100/ in one year gain 5/ interest, what will be the 
interest of 750/ for 7 years ? Ans. 2621 "iq^, 

3. If a family of 8 persons expend 200/ in 9 months ; Jxow 

much will serve a family of 1 8 people 1? months ? . 

^ Ans. 300/: 

4. If 2*7s be the wages of 4 men for 7 days ; what will be 
the wages of 14 men tor ip days ? Ans. 6/ 15/. 

5. If a footman travel 130 miles in 3 days, when the days 
are 12 hours long; in how many days, of 10 hours each, , 
may be travel 360 miles ? Ans. 9|| days. 



"i 






VULGAR FRACripNS, 54, 

# 

Ex. 6. If 120 bushels o£ com am s^rve 14 horyet SS daysi 
how inany days will 94 bushels serye, 6 horses ?, . • • ^ 

Ans. 102^ days.., 

7. If 3000 lb of Ijeef serve ^0 mta^ IS Jays ^ how jiiany 
lbs win serve 120 men for 25 days ? >t)s.\ l764lb 1114.02^ ' 

8. If a barrel of beer be sufficient to last a family of 8 per-. 
sons 12 days ; how many barrels will be .drank by i6 persons, 
in the space o^ year? - . ..AftSf.6P|. barrels*^ 

9. If 180 men, infe d^ys^ of 10 hours ttcj^i, can dig a. 
trench 200 yards long, 3 wide* and 2 deep Zip hpw many, 
days, of 8 hours long, will 100 mth dig % treyh of 36 yards . 

^Jong, 4 wide, and- 3 deep ? Ans. '^""*"* 



9*" 




•"; -c. /•■.■.■; ■••■I , 

OF VULGAR FRACTIONS, 
part, 




with a line bet\^een them : 

_,_ 3 numerator i , . , . - « - , *• 

Thus, -- , > , which IS named 3»fourth§. 

4 denominator J > 

The Denominator, or number placed below the line, shhws 
how m;^y equal ps^ts the whpl6(rqj|(i^f^it^ i^divid^qiflo ; 
and it represents the Divisor in D,iVjision«-t^An4 tH^/^^IMfifr 
tor, or numbeir set ^oye the line« shows, how m^y.^JFjtti^^ - 
parts are expressed by the Fraction; l^^ing the r^|n^^^^;|4ff ' 
after division. — Also, both these i^i^Jser^ ^c, in gen^r^ 
named the Tjerms of the Fracjt;iort. '> » ^ r! ; »t 

Fractions' are either. Proper* ImjMroper, Siajpjte>'jQf)aw 
'pounds or Mixed. v . ; '^^ . f 

A Proper Fraction, is when the numerator is less than the 
denominator ; as, 4/ or f , or j-* ^c* 
y An Improper Fraction, is when the numerator is equal to, 
or exceeds, the denominatoj* ; as, j.,' Qrr^,.or ^s &c. V / 
' A Simple FractiQjSy is a single exprei^ion, deiaodng any 
nuini>er of parts of the integer \ asi^,.er i^ 

A Compound Fraction, i$ the fir^Kririon of a fraction^ (r 
several fractions connected with the word ^ bet W^0n theii^; 
as, j^ of I, or -f of ^ of 3, &c* 

A Mixed Number, is composed of a whyle nipnb^ miA^ 
fraction together; at^.3^, or I24, &c. - 

E2 Sr A whole 



Hk 



A Krliote* nt hit^br htunlier may bef expressed like t frftc** 
tion, by writisg 1 btelo^'^it, ia a denominator ; so 8 is fs <^ 
4is4i&c, 

A fraction denotes division ; and Its valtie is equal toliie 
qaotient obtained by dividing the nuinerator by the dtno^ 
ininator : so' '-^ is ecjud to'B, and V is eqnal to 4. 

Hence tlien, if the nomerator be lesfi than the denominator, 
the talne of the fraction is less tiian 1 . But if the numerator 
be the samfe iH the deno^ninator;,' the fiction is just eqUa! 
to 1 / And If the numerator be greater than the denraU^ 
natorj the ffattioji is greater than J^ 



tssst 



REDUCTION OF VULGAR FRACTIONS. 

REDi7CTX0N of Vujglt FractionS| is the bringing them 
out of.one form or denomination into another } commonly to 
{)repagre.theip for the o|>erations of Additionj Subtractioni ^c^ 
of vl^ch there ^^ure several cases. 

1 
I 

4^ PROBLEM, 

1 . ^ . . i • i m 

ToApd the Great f St Ccmmcn Measure ofTtuo or more Numbers* 

T|rtE Cotirtnoti Measure of two or more numbers, is that 
trulxiber which will divide them both without remainder ; so> 
f is a-^omtnon measure of 18 and 24 ; the quotient of the 
Ibrm^ being 6, and of the latter S. And the greatest num« 
ber that will do thisy is the grejttest common measure : so 6 
is the greatest common measure of IS and 24 ; the quotient 
t>f tWe former being S, and ci the latter 4, whi^rh will not 
iHDtli diyidp furtJier. 

t^ there be two numbers only; divide the greater by thft 
tesfe \ l4ven divide the divisor by the remainder \ $md so on,^ 
dividing always the last divisor by the last remainder} t3l no- 
thing remains ; so shall the last diyisol^ of all be th^ greatest 
fimimon measure sought* 

When there are more than two numb^rs^ find the greatest 
Wttkfofsti miKisure of two of thom^ as bisfore ; then do the 
If^n^ for that cpmmpxi'^ineasure and another of the numbers} 

an4 



REDUCTION w VUWJAH FRACTIONS. It 

wd io oo» through «U the uMibers; so uriU the great^t^ com- 
mon mtwmt last ibund be the answfr/ . . 

If it happen that the conunooi measure thus found' is If 
then the numbers are said to be incommensurabIp> Or uofi 
liif ing any comqion measure^ 

BXAM^L^. 

1. To find the griNitest common measure of 1908> 936, 
and 630. 

936) 1908 (2 So that 36 is the greatest common 

1872 measure of 1808 and 9S6* 

36 ) 936 ( 26 Hence 36 ) 630 ( 17 
72 36 



216 270 

216 252 



18) 36 (2 
36 



^p 



Hence then 18 is the answer required. 

2. What is the greatest common measure of 246 ind 372 ? 

Ans. 6* 

3. What is the greatest common measure of 324, 61 9, 
and 1032/ ' A»3. 12. 

CASE I. 

T$ Abbrevii^ or Reduce Fractions U their Lowest Terms* 

* Divide the terms of the given fraction by any numher 
chat will divide .them irithput z remainder; tl^n divide these 

guotients 



* Tluit dtviding both the terms of the fraction by th^sam^ 
namt>er^ whatever it be, will give another fraction equal to the 
former, is evident A^d when these divisions' are performed as 
if^n as can be dobe^ or when the common divisor is the greatest 
possible^ tbf», terms of the resuhing fraction must be ^he least 
possible. 

Note, 1. Aiqr number endlag widi aoevea ijiuQiher, or a cipher^ 
n divisible^ or can be divided, by 2. 
2. Any number ending with ^> or 0^ is divisible by 5, 

3, If 



si ' Ai^ITHMETIC. 

c^ucei^tits again in the same manner ; and so on, till it appears 
that Uiere is no number greater than 1 which will divide 

thetti} then the fraction will be in its lowest terms. 

• • * . » 

Or^ divide both the terms of the fraction by their greatest 
common measure at once, and the quotients will be the terms 
of the fraction required, of the same value as at first. 



EXAMPLES. 

1 . Rediuc^ a^ to its least terms.^ 

m-U- il = Tf =1 == ii the answer.^ 

Or thus : 

216 ) 288 * { 1 Therefore 72 is the greatest common 

216 taeasure; and 72; |4f =s ^ the An- 

swer, the same as before. 

72) 216 {3 
216 

2. Reduce 



3 . If the right-hand place of any numfier be O, the whole is di- 
visible by 10 3 if there be two ciphers, it is divisible by 100 j if 
three ciphers, by 1000 : and so on; which b only cutting offdiose 
ciphers. 

4. If the two right-hand figures of any number be divisible by 
4, the whole is divisible by 4. And if the thrqe right-hand figures 

be dtvisible by 8, the whde is divisible by 8. And so on. 

If 

5. If the sum of the digits in any number be divisible by a, or 
by 9, the whole is divisible by 3, or by g. 

e. If the right-hand digit be even, and the sum of all the digits 
be divisible by 0, then the vlhole is divisible by 6. 

7. A number is divisible by 1 1 , when the sum of the l st, 3d, 5th, 
. 3cc, or all the odd places, is equal to the sum of the 2d, 4tt, 6th, 

&c, or of all the even places of digits. 

8. If a number cannot be divided by some quantity less than the 
sqliarfe root of the same, that number is a prime, o.^ cannot be di- 
vided by any number whatever, 

g. All priine numbers, except 2 and 5, have either 1, 3> 7, org, 
m the place of units ; and all other numbers are composite, or can 
be divided. 

10. When 



REDUCTION OF VULGAR FRACTIONS. 55 

t 

^. Reduce f|^ to its lowest terms. Ans. ^* 

3. Reduce 441 1® ^^^ lowest terms. Ans. \. 

4. Reduce fyv ^^ ^^^ lowest terms. Ans. -I* 

CASE II. 

To Reduce a Mixed if umber to its Squivalent Improper Fraction. 

* MuLTiPLt the integer or whole number by the deno^ 
minator of the fraction, and to the product add the numera- 
tor ; then set that sum above the denominator for the frao^ 
tion required* 

EXAMPLES; 

1 . Reduce 23| to a fraction* 

23 
5 

115 Or, 

2 (23x5)+2 in , . 
rs — , the Answer. 

117 ^ ^ ' . 

5 

2. Reduce 12|- to a fraction. Ans. '^^ 

3. Reduce 14t^ to a fraction. Ans. Vtf . 

4. Reduce 183^ to a fraction. Ans. ^.t**. 



10. When numbers, with the sign of addition or subtraction be^ 
tween them^ are to be divided by any number, then each of those 

, 4i ,, , . . rr.i 10-1-8—4 
numbers must be divided by it. Thus — i =5 + 4'-2~7« 

11. But if the numbers have the sign of multiplication between 
% them, only one of them muSt be divided. Thus, 

10X8X3 _ 10X4X3 _ 10 X4X 1 _^ 10X2X1 _20^^ 

6X2 "~ 6X1 ■" 2X1 " 1X1 ""T*"" \ . 

* This is* no more than first multiplyitig atjuaritity by some 
number, and then dividing the result l»ck again by the samd : 
which it is evident does not alter the value 5 for any fraction re- 
presents i. divisiori of the numerator by the detiominatot". 

. ' CASB 



g$ ARITHMETIC 



CASE III. 

T9 Reduce an Improper Fraction to its Equivalent Wb^ or 

Mixed Number, 

* DiVfDE the numerator, by the denominariior, and the 
quotient will be the whole or mixed number sought. 

EXAMPLES* 

1. Reduce y to its equivalent number* 

Here y or 1 2 -r- 3 = 4, the Answer. 

2. Reduce y to its equivalent number. 

Here V or 15 -r- 7 = 2|, the Answer. 

3. Reduce V^^ to Its equivalent number. 

Thus, 17) 749 (44^ 
68 

69 So that ^ 3= 447V> the Answer. 
68 

1 

4. Ileduce l^ to its equivalent number. Alls. 8. 

5. Reduce ' ij* to its equivalent number. Ans. 54|4» 

6. Reduce *^* to its equivalent number. Ans. 17147* 

i 

CASE IV. 

To Reduce a Whole Nundfer to an Equivalent Fraction^ having 

a Given Denmmnator^ 

f MULTIPLT tlie whole number by the given denominators 
then set the product over the said denominator, and it will 
form the fraction required* 

/ 

* Tills Rule is evidently the reverse of the former ; and the 
reason of it is manifest from the nature of Common Division* 

f Multiplication and Division being here equally used> the re- 
sult must be the same as the quantity first proposed. 

EXAMPLES. 



REDUCTION op VULGAR FRACTIONS. S7 



EXAMPLES. 

I 

1. Reduce 9 to a fraction whose denominator shall he 7. 
• Here 9 X 7 s 63: then y is the Answer; 
For V = 63 -^ 7 = 9, the Proof. 
^ Reduce 12 to a fraction whose denominator shall be IS. 

Aas. Vt • 
3^ Reduce 27 la a fraction whose denominator shall be 1 U 

Ans. Vi^. 

CASE Y. 

To Reduce a Compound Fraction to an Equivalent Simple One* 

* Multiply ail the numerators together for a numerator, 
and aU the denominators together for a denominator, and 
they will form the simple fraction sought.' 

When part of the compound fraction is a whole or mixed 
number, it must first be reduced to a fraction bv one of the 
former cases. ; 

And, when it can be done, any two terms of the fraction 
may be divided by the same number, and the quotients used 
instead of them. Or, when there are terms that are conunon, 
they may be omitted, or cancelled. 

£XAMPL£S. 

1. Reduce |^ of |- of |- to a simple fraction. 

,- 1x2x3 6 1 , ^ 

**€^^:;: — :;; = :r: = rr> ^e Answer. 

2x3x4 24 4' 

Or, ^ — - — 2 ~ T* ^y cancelling the 2's and i*s. 

^ X Q X 4? 4 



* The truth of this Rule may be shown as follows : Let the 
.compound fraction be -| of j . Now j- of -f is | -r 3» which is ^\-j 
consequently f of -^ will be ^V X 2 or ij--, that is, the numerators 
are muluplied togetlier, and ulso the denominators, as in the Rule. 
When the compound fraction consists of more than two single 
ones i having tirit reduced two of them as above, then the resulting 
Action and a third will be the same as a compound fraction of t«vo 
jparts^ and so on to the last of all. 

2, Reduce 



58 ARITHMETIC. 

2. Reduce ^ of -^ of -{4 to a simple fraction. 
_, 2x3x10 60 13 4 , . 

""'" sTI^TTT = 165 = 33 = IT' ^^^'^ ^"'^"'••. 

2 X 3^ X jd 4 

^^» ^ — f — TT ~ 7T> ^^ ^^°^^ ^^ before, by cancelling 

the 3's, and dividing by 5's. 

S. .Reduce ^ of | to a simple fraction. Ans. 54* 

4. Reduce f of f of 4 to a simple fraction. Ans. -J. 

5. Reduce |. of -f- of Si to a simple fraction. Ans. {• 

6. Reduce 7^ of ^ of -J of 4 to a simple fraction. Ans. i. 

7. Reduce 2 and | of |^ to a fraction. Ans. |^. 

CASE VI, 

To Reduce Fractions of Different Denominators^ to Mquivatent 
Fractions having a CommoH Denominator. 

* Multiply each numerator by all the denominators ex- 
cept its own, for the new numerators : and. multiply all the 
denominators together for a common denominator. 

NotCy It is evident, that in this and several other (fperation^ 
when any of the proposed quantities are integers, or mixed 
numbers, or compound fractions, they must first be reduced, 
by their proper Rules, to the form of simple fractions* 

EXAMPLES. 

1 . Reduce 4-> t> and |., to a common denominator. 

1 X 3 X 4 = 12 the new numerator for 4. 
2x2x4 = 16 ditto f. 

3 X 2 X 3= 18 ditto f. 

2 X 3 X 4 = 24 the common denominator. 

Therefore the equivalent fractions are ^, -J^, and i^. 

Or the whole operation of multiplying may be best per^ 
formed mentally, only setting down the results and given 

fractions thus: 4, 4, J- = H> ih H = A» tV> tV, by 
abbl'eviation. 

2. Reduce y and ^ to fractions of a common denominator. 

Ans. II, 14. 



* This is evidently bo more than multiplying each numerator 
And its denominator by the same quantity, and consequently th6 
value of the fraction is not altered. 

3. keduce 



REDUCTION OF VULGAR FRACTIONS. v<9 

5. Reduce •§-, ^i and ^ to a conunon denominator. 

Ans. ^, IJ, 1^. 

4. Reduce i, 2^1 and 4 to a common denominator. 

AnQ 2 5 7 8 »ao 

Note 1 . When the denominators of two given fractions 
have a common measure, let them be divided by it ; then 
muhiply the terms of each given fraction by the quotient 
arising from the other's denominator, 

£/:• 2T and j^ =: -rrr ^^^ tttj ^7 niuhiplying the former 
5 7 by 7 and the latter by 5. 

2. When the less denominator of two fractions exactly 
divides the greater, multiply the terms of that which has 
the less denominator by the quotient. 

J?ar* f and -^ =: ^ and -^^ by mult, the former by 2. 
2 

3. When more than two fractions are proposed, it is some- 
times convenient^ first to reduce two o£them to a common 
denominator ; then these and a third ; and so on till they be 
all reduced to their least conunon denominator. 

£:e. I and J and I =;: I and J and | = ^^ and if and J^. 



CASE vir. 
To find the value of a Fraction in Parts of the Integer, 

Multiply the integer by the numerator, and divide the 
product by the denominator^ by Compound Multiplication 
and Division, if the integer be a compound quantity. 

Or, if it be a single integer, multiply the numerator by the 
parts in the next inferior denomination, and divide the pro- 
duct by the denominator. Then, if any thing remains, mul- 
tiply it by the parts in the next inferior denomination, and 
divide by the denominator as before \ and so on as far as 
necessary ; so shall the quotients, placed in order, be the 
value of the fraction required *i 



* The numerator of a fraction being considered as a remainder, 
in Division^ and the deriomitiator ^s the divisor, this rule is of the 
same nature as Compound Dtvisioni or the valuation of remainders 
in th« Rule of Three, before e&plained. 

EXAMPLES. 



tQ 



ARTTHMEnC. 



EXAMPLES. 



I. What is the f of 2/ 6s f" 

B J the former part of the Rule, 
2/ 6s 



5)9 4 
Ans» 1/ 16/, 9d 2\q. 



2. What Is the value of ^of 1/r 

By the 2d part of the Rule^ 
2 

20 



3 ) 40 ( 13x 4rf Am. 



1 

12 



3) 12 (4</ 



$• Find the value of -f of a pound sterling* Ans. Is 6d» 
4. What is the value of f of a guinea ? Ans. 4i 8i^ 

Sm What is the value of ^ of a half crown i Ans^ 1/ lO^^. 

Ans* 1/ 1 l{d. 

Ans. 9oz 12dwts» 

Ans* Iqr 71b* 

Ans. 3 ro. 20 po* 

Ans* 7brs 12mi&» 



6. What is the value of f of 4/ \0d? 

7. What is the value of * lb troy ? 
8* What is the value of i ^of a cwt ? 
9. What is the value of {-of an acre? 

iO.. What is the value of -^ of a day ? 



CISE VJII. 

I 

Tq Reduce a Fraction from one Denomination to another. 

^ Consider how many of the less denomination make one 
of the greater ; then multiply the numerator by that number, 
if the reduction be to a less name^ but multiply the denomir 
nator, if to a greater. 

EXAMPLES. 

2 . Reduce |^ of a pound to the fraction of a penny, 
f X V X V == *r =^ 't""* ^^e Answer. 



* This is tiie same as the Rule of Keduclion in whole numbers 
from ooe denomiiiatign to another. 



2. Reduce 



ADDITION OF VULGAR FRACTIONS. 6fr 

$• Reduce f of a penny to the fraction of a pound* 

I X A X ^ = ^, the Answer. 
9. Reduce i%/ to the fraction of a penny. Ans. V^« 

4* Reduce |q to the fribCtion of a pound. Ans. t^V?- 

5. Reduce y cwt to the fraction df a lb. Ans. V^ 

6. Re<htce f dwt to the fraction of a lb troy. Ans. -j^^* 

7. Reduce |- crown to the fraction of a guinea.. Ans. ^^. 
6« .Reduce ^ half-<:rown to th« fract. of a shilling. Ans. 44* 
9. Reduce 2s 6d to the fraction of a £• Ans. 4« 

10. Reduce 17/ 7rf S\j to the fraction of a £. 



*lmmmmiitmmmm^ttmmatagmmmmmmm*im 



ADDmON OP VULGAR FRACTIONS. 

tp the fractions have a common denominator; add all the 
liumerators together, then place the sum over the common 
denominator, and that will be the sum of the fractions 
required. 

* If the proposed fractional have not a common denomina- 
tor, they must be reduced to .one. Also compound fractions 
must be reduced to simple ones, and fractions of diderent 
denominations to those of the same denomination. Then 
add the numerators as before. As to mixed numbers, they 
may either be reduced to improper fractions, and so added 
with the others ^ or else the n-actional parts only added, and 
the integers united afterwards. 



* Before fractions are reduced to a common denominator, they 
are quite dissimilar, as much as shillings and pence are, and there- 
fore cannot he incorporated with one another, any more than these 
can. But when they are reduced to a common denominator, and 
made parts of the same thing, their sum, or difference, may then.be 
as properly expressed by the sum or difference of the numerators, 
as the sum or difference of any two quantities whatever, by the 
sum or difference of their iedividuals. Whence the reason of thQ 
B-ule is manifest, both for Additipn and Subtraction. 

When several fractions are to be collected, it is commonly best 
^rst to add two of them together tuat mrst easily reduce to a com- 
fifwn dptiomn^^^f then ftdd their ^um and a thirds and so on. 

EXAMPLES. 



» \ 



63 ARITHMETIC. 

EXAMPLES. 

1. To add ^ and f together. 

Here ^ + ^ = ^ = 1 4, the Answer. 
• 2. To add 4 and | together. 

H-i = 44 + l^=^=lT4,the Answer. 

3. To add 4 and 7i and 7 of |- together, 

4. To add* f and 4 together. Ans. If. 

5. To add ^^ and f together. ; Ans. IfJ^, 

6. Add y and -/^ together. Aiis. ^2^. 

7. What is the sum of -f and ^ and ^? Ans. 1tS4* 

8. What is the sum of | and ^ and 2^? Ans. S|^. 

9. What is the sum of 4 and ^ off, and 9,^ ? Ans. 10^^. 
10. What is the sum of |- of a pound and 4- of a shilling ? 

Ans. '|5,or 13i 10^24^. 
1 1 • What is the sum of 4 of fi shilling and -j^ of a penny ? 

Ans. Vt ^ or Id \^g. 

12. What is the sum of 4 of a pound, and |. of a shilling, 

and TT'of a penny ? Ans. 4^^j or Ss Id 145?* 



SUBTRACTiqjf^ OF VULGAR FRACTIONS. 

' Prepare the fractions the same as for Addition, when 
necessary 5 then subtract the one numerator from the other, 
and set the remainder over the common denominator, for the 
difference of thp fractions sought.* 



EXAMPLES, 

1 . To find the difference between 4 and 4« 

Here 4 "^ 1^ ^ 4 == t> ^^^ Answer. 

2. To find the difference between 4 and 4. 

T - V = T^ •- If = Ty» the Answer. 

3. What 



MULTIPLICATION of VULGAR FRACTIONS. 63 

S. what IS the difference between -^ and -f-^ ? Ans. ^. 

4. What IS the difference between ^ and -f^i Ans, y^. 

5. What Is the difference bel;ween ^ and ^Zj ? Ans. -rh- 

6. What is the diffl between 5| and ^ of 4^ ? Ans. 4-,^^. 

7. What is the difference between -^ of a pound, and J- of 
i of a shilling ? Ans. Vt -^ ^r lOx Id l^. 

8. What is the difference between f of 5^ of a pound, 
and I of a shilling ? - Ans. lU-il or V Ss \ 1 -.y. 



MULTIPLICATION of VULGAR FRACTIONS. 

* Reduce ^ixed numbers, if there be any, to equivalent 
fractions ; then multiply all the numerators together for a 
numerator, and all the denominators together for a denomi« 
nater, which will give the product required. 

EXAMPLES* N 

1 . Required the product of ^ and |.. 

Here t ^ i == t% ^ t> ^*^c Answer. 
Or^x J = 4x^ = i. 

2. Required t}ie continued product of ^ 3--, 5, and | of -fi 

,, ?f 13 ^ a 3 ISx 3 39 

Here y x -x ^ X "^ X y = -;^;3^ =- = 4^, Ans. 

3. Required the product of f and ■^. Ans. j^y. 

4. Required the product of -^ and -j^. Ans. -rV. 

5. Required the product of f , ^, and 44» Ans. ^*j. 



* Multiplication of any thing by a fraction, implies the taking 
some part or parts of the thing j it may therefore be truly expressed 
by a compound fraction; which is resolved by multiplying toge- 
ther the numerators and the denominators. 

Note, A Fraction is best inultiplied by an integer, by dividing 
the denominator by it -, but if it will not exactly divide^ then 
multiply the numerator by it 

6. Required 



64 ARITHMETia 

6. Kequired the product ofi, jf^nd S* An$«> 1; 

7. Required the product of ^, ^, and 4^ Ans. 2^. 

8. Required the product of ^ and •§- of y. Ans. H^ 

9. Required the product of ^, and f of 5, Ans, 20» 

10. Required the product of | of j, and -J. of 3f . Ans. f J^ 

1 1 . Required the product of 3| and 4^. . Ans, i 4444* 

1 2. Required the product ef 5, |> f of f, and 4|. Ans. 2^. 



■««» 



DIVISION or VULGAR FRACTIONS. 

^ Prepare the fractions as before in MuhipKcation ; then 
divide the numerator bv the numeratorj and the denominator 
hj the denominator, it they will exactly divide : but if not» 
then invert the terms of the divisor, and multiply the dividenc) 
by h, as in Multiplication* 



ZXAUPtES. 

1. Divide V by f . 

Here V "^ 4^ = t = Ht ^7 the first method* 

2, Divide^ by ,5^. 

Here^^T^=:^x V=-f xi = V =-H- 

5, It is required to divide 44 ''T 4* Ans. f* 

4. It is required to divide -^ by ^. Ans. ^V 

5. It is required to divide \^ by -^. Ans. l-f. 

6. It is required to divide ^ by V • Ans. -^^ 

7. It is required to divide y^ by ^. Ans. -f . 
.8. It is required to divide 7 by ^ Ans. -Jt 



« 



^ Divison being the reverse of Multiplication^ the reason of 
the Rule is evident. 

Note, A fraction is best divided by an integer, by dividing the 
numerator by it ; but if it vyill not exactly divide^ then multiply 
the denominator by it* 



9. ft 



RULE OF THREE m VULGAR FR A CTIONS. fi^ 

•9. It is required to divide V^ t)y 3. • Ans. -^ 

10. It is required to divide 4 by 2. Ans. ,25^' 

11. It is required to divide 7 j. by 9^. Ans. ff . 

1 2. It is required to divide f ttf j- by f of 7f . Ans. Tfr- 



' * ' " 



RULE OF THREE in VULGAR FRACTIONS. 

Makb the necessary preparations as before directed ; then 
multiply continually together, the second and third terms^ 
Mtkd the first with its parts inverted as in Divi^ion^ for the 
answer** 

Examples. 

1 • If f of a yard of velvet cost | of a pound sterling ; vfhM 
Will 1^ of a yard cost ? 

S 2 5 S i S , , ^ 

y •• "5" •• li = T ^ 7^ ig =i'*^^«^^ Answer. 

^i What trill S^ot bf silver cost^ at 6s 4i an ounce f 

Ans. 1/ li 44rf. 

3. If T^ of a ship be worth 273/2/ 6d; what are ^r of her 
worth? Ans. 227/ 12/ Id. 

4i What is the purchase of 1230/ b^k-stbck, at 108| per 
cent.? Ans. 1336/ UW. 

5. What is the interest of 273/ 15/ for a year, at 3^ per 
cent.? Ans. 8/17/ ll|i/. 

.6. If I of a ship be worth 73/ 1/ 3/// what part of her is 
worth 250/ iOsP Ans. j-. 

7. What length thust be cut off a bo^d thit is 7^ inches 
broadi to tontain a square foot, or as much as another piece 
of 12 inches long and 12 broad ? ' An^ I844 inched. 

S, What quantity of shalloon that is ^ of a ydrd wide^ will 
line 9i yards of cloth/ th^at is 24 yards widei Ans. 31|^ yds. 

» . ^ . . ■ . 

, "* This is only multiplying the 2d and 3d terms Oogether^ and 
diidding the product by the firsts as in the Rule o^^ree in whote 
^umbera. >. 

Vol. h T ' 9. If 



66 ' ' AWtllMlETIC: 

9. If'tlie penny loaf weigli 6^<5K| when the prite bl 
w'heat is -5/ the bushel ; what CH^t it to weigh whe^ the 

wheat is 8s 6d the bushel? Ans. 4-^7- oz. 

10. How much in length, of a piece of land that is ll^^* 
poles broad, will make an acre of land, or as much as 40 
poles in length and 4 in breadth ? Ans. 13-^^ poles. 

11. If a courier perform a. certain journey in %5i days^ 
travelling 13^- hours a day ; how long would he be in per- 
forming the same, 'travelling only I'l-ru liours a day ? 

Ansi 40|44 days. 

12. A regiment 6f soldiers, tonsisttng of 976 mw, are to 
be iiew cloathed ; each coat to contain ^i yards of cloth 
that is i{ yzrft wide, and lined ifith shalloon ^ yard wide : 
how maily yaid$* of shalloon will line them ? 

• Ans. 4531 yds 1 ^ 2| nails. 



* .d' i /tiii; 



DECIMAL FRACTIONS. 



A Decimal Fraction, is that which has foi** its deno- 
minsttonaatmit (1), with as many ciphers annexed as the 
humet-atoi: ha^ places ; and it is usually expressed by setting 
down the numerator only, with a point before it, on the left- 
hand.- Thus, ^^ is -4, and -,^\ is •24,>and ^i-ttr is "074, arid 
^^>5*y^^ is '00124 ; where ciphers are prefixed to make up as 
many places as are ciphers in the denominator, when there 
is a deficiency* of figures. 

• A mixed nuipber is made up of a whole number with some 
decimal iiraction, the one being separated from the .other by 
a point.; Thus, 3 '25 is the same as S-^y or 4^. 
' Ciphers on the right-hand of decimals make no alteration 
in their value; for*4, or '40, or '400 are decimals having all 
the same value, each being == -/„, or |^. But when tliey are 
placed on the left-hand, they decrease the value in a ten-fold 
proportion: Thus, '4 is T^y* or 4 tenths ; but '04 is oiily -r^t 
or 4 hunciredfhs, and '004 is only -ri^j or 4 thousandths* 

The 1st place of deciinals, counted from the left-hand to- 
wards the^rightj j.s called the place of primes, or 1 Oths ; the 
2d is the place of seconds, or lOOths ; the 3d is the place of 
thirdsj or lOOOths ; and so on. For, in decimals, as well ag 
in whole numbers,- the values of the places increase towards 
the left-hand, afid decrease towards the right, both in the 

same 



« 

same tenfold proportion ; as in the following Scale or Table 
«f Notation. 









CO 






§ -9 .•i'.-!-.g,"S<'3-g;| iltM 






to 









ill I ^ •;■■*,■ .^-.i I ^ ^ § 






3 3 3-3 3 3 3 •3.3 • 3 3 3 3 






ADDtTlbN'oF DEfclMALS. 

S£t 't&e) ttic&bcfrs ttxubr each*oth«9 «ccdniiiig >to die faltte 
.of jAac >plac€js, like as ip^wljLple nipnb^K j. jp jwrb^^ta^tp tl^ 
tieqijiipgl .5^para^ing poinds will stand all exacriy ynder each 
other. Then, beginning at the ri;^t-hand, add u'p aH the 
columns of numbers as in integers ; and point off as many 
places, for decimals, as are in the greatest number of decimal 
places in any of the litres that are addled ; or place the point 
directly below all the other points. 

EXAMPLES. 

1. Ta add together 29-0146, and"3146*5, and 2109, and 
; . 29-0 146 

3146*5 : 



» J > • . 






2109- .. ,: . ' • , 
14-16 



5299*29877 ' the Sum. 



Ex.2. What is the sum of ^^6, 3 9 '2 13: 72014*9, 417, 
and 50S2? . ' • ' . 

3. What is th^ sutti qf .7530y 16^201, «-0142,;957*X3, 
^•72119 and 'OSOU*: 

4. What is the syrii pf 312-09, 3-57llj 7195-6, 7l*4»8, 
9739.215, 179, and -0027? 

F2 - SUBTRACTIOJI 



€i ARITHMETIC 



. SUBTRACTION or DECIMALS. 

Place the numbers under each other according to the 
value of their places^ as in the last Rule. Then^ beginning 
at the right-hand) subtract as in whole numbers, and point 
off the decimals as in Addition. 



EXAMPLES. 

1. To find the difference betwfeen Sl-IS and 2^13«. 

9r73 

2-138 



Aixs. 89*592 the Difference. 



q: Find the diff. between 1-9185 and 2*73. Ans. 0'81 15. 
'3. To subtract 4.-90U2 from 214*81. Ans* 209-9085&. 

4. Find the diff. between 2714 and 'QIS. Ans. 2713-084. 



MULTIPLICATION of DECIMALS. 

/ . • 

^ Place the factors^ and multiply them together the same 
as if they were whole numbers.— -Then point off in the pro- 
duct just as many places of decimals as there are decimals in 
both the factors. Bufif there be not so many figures in the 
product) then supply the defect by prefixing ciphers. 



* The Rule will be evident from this example :-«Let it be re- 
quired to multiply *12 by '36l ; these tumbers are equivalent to 
;^ and ^t}„ 5 the product of which is rUlhs = 04332, by the 
nature' (^ Notation^ which consists of as many places as there are 
ciphers, that is, of as many places as there are in both numbers. 
And in like manner for any other numbers. 

' ' • EXAMPLES. 



MULTIPLICATION of DECIMALS. 69 



EXAMPLES. 

1. Multiply •321096 
by '2465 

1605480 
1926576 
1284384 
642192 

" 1 W ' ■ I P 

Ans. -0791501640 the Product. 



2. Multiply 79-347 by 23-15. Ans. 1836'88305. 

3. Multiply -63478 by -8204. Ans. .520773512- 
A. Multiply -385746 by -00464. Ans. -00178986144. 



CONTRACTION I. 

To multiply Decimal/ by 1 with any number of Ciphers^ as by lO, 

or 100, or 1000, isTc. 

■ 

This is done by only removing the decimal point so many 
places farther to the right-hand,* as there are ciphers in the 
Qiultiplier ; and subjoining ciphers if need be. 

EXAMPLES* 

1. The product of 51-3 and 1000 is 51300. 
•2. The product of 2-714 and 100 is 

3. The product of -916 and 1000 is 

4. The product of 21-31 and lOOOO is 



CONTRACTION 11. 

To Contract tie Operation^ sq as to retain only as many Decimals 
in the Prodsjct,f^ may be thought Necessary^ tuhen the Product 
would naturally contain several more f^lfu:es^ 

Set the units' place of the multiplier under that figure of 
the multiplicand whose place is %\^ same as is to be retained 
for the last in the product ; aind dispose of the rest of the 
figures in the inverted or contrary order to what they are 
usually placed in. — ^Then, in multiplying, reject all the figures 
that are more to the right-hand than each multiplying figure, 
and set down the products, so tliat their right-hand figures 

may 



iof 



ARmiRtETlC. 



may fall in a column straight below each other; but observing 
to increase the first figure of every line with what would 
arise from the figures omittedy in this manner, namely 1 
from 5 to 14, 2 £om 15 to 24, 3 from 25 to 34, &c ; and 
the sum of all the lines will be the product as required^ com<« 
monly to the nearest unit.in the last figure. 



EXAMPLESi 



1. To multiply 27^14986 by 92.41035, so as to retain 
only four j^ace^ of decimals in the product. 

Contracted Way* Common Way. 

27-14986 » 27-14986 

53014-^9 92-41095. 



24434874 

542997 

108599 

2715 

81 

14 

2508-9280 



13 


574930 


81 


4.4958 


2714 


986. 


108599 


44 


542997 


2 


24434874 




250S-9280 


650510 


t 


• 



2. Multiply 480*14936 by 9-72416, retaining only four 
decimals in the product. 

3. Multiply 2490-3048 by -573286, retaining only five 
decimals in the product. 

4. Multiply 325-701428 by -7218393, retaining oidythre^ 
decimals in the product* 



DIVISION OF DECIMALS. 

Divide as in whole numbers; and point off in the quo- 
. tfent as many places for decimals, as the dedi^i^^l places in the 



dividend exceed those in the divisor*. 



l^ 



II 1 1 1 1 1 1 



♦ The reasort' of t^its. Rule is evident 5 for, since the divlsoi: 
multiplied by the quotient gives the dividend, therefore the num- 
ber of decimar places in the dividend, is equal to those in the"di- 
vi^orand quotfettt, taken together, by the nature of Mnltiplfca- 
<fe& 5 and consequently thequoti«at itBelf must cohtdfe ds many as 
i(he dividend exceeds the divisor* 



Another 



DIVISION ct DXCiMAJ^. 



%■ 



Another way to know the place {gr the decimal point, is 
this : The first figure of the quotient must be made to oc- 
cupy the same place, of integers or decimals, as doth that 
figure of the dividend which stands over the unit's figure of 
the first product. 

When the places of the quotient are not so many as the 
Rule requires, the defect is tp be supplied by prefixing 
ciphers. 

When there happens to be a remainder after the division'; 
or when the decimal places in the divisor are more than those 
in the dividend ; then ciphers may be aivi?xed ('p the divi* 
dend, and the quotient carried on as far as required. 



EXAMPLES. . 



1. 



nS) -48520998 (-00272589 
1292 
460 
1049 
1599 
1758 
156\ 



2. 



•2639) 27-00000 (102-3114 
6100 
8220 
3030 
3910 
12710 
2154 



J. Divide 123;70536 by 54*25, 

4. Divide 12 by -7854. 

5. Divide 4195*68 by 100. 

6. Divide -8297592 by -153. 



Ans, 2-2802, 

Ans. 15-278. 

' Ans. 41*9568, 

Ans. 5*4232. 



CONTRACTION I. 



When the divisor is an integer, with any number of ci-^ 
phers annexed : cut off those ciphers, and remove the deci--. 
mal point in the dividend as many places farther to the left 
as there are ciphers cut off, prefixing ciphers if need be; then 
proceed as before*. 



* This is no hjopc than dividing both divisor and dividend by* 
the same number, either 10^ or lOO, or 1000, &c, ^c^otrdibg to< 
the number of ciphers cut off, which^ leaving them in the same 
proportion, does not affect the quotienf . And, in the same way, 
the decimal point may be moved the same number of p^ces in 
both the divisor and dividend, either to the right or left, whetheif 
they have ciphers or not. 

tXAMPX-ES^ 



if ARITHMETIC. 

EXAMPLES, 

1. Dividp 45*5 by 2100. 

21-00) -455 (-0216, &C. V '^2 

35 
140 
14 



$. Divide 41020 by 32006. 
5. Divide 953 by 2 1600, 
4. Divide 61 hj 19000. 

CONTRACTION IJ. 

HbncEi if the divisor be 1 vith ciphers, as 10, 100, O]^ 
1000, &c : then the quotient will be found by merely mov- 
ing the decimal point in the dividend so many places farther 
to the left, as the divisor has ciphers j prefixing ciphers if 
need be. 

3EXAM?LES. 

So, 217^3 -r 100 ==2-173 And 419 -t- 10 = 

And 5-ll5 -r 100 = And -21 — lOpO = 

CONTRACTION III. 

When the^e are many figures in the djvisor; or when 
only a certain number of decimals arie necessary to be re- 
tained in the quotient ; then take only as many figures of 
the divisor as ^mW be equal to the number of figures, both ^l- 
tegers and decimals, to be in the quotient, and find how 
many times they may be contained in the first figures of the 
divia^nid, as usual. 

l(jpX each rjjn^aind^ b^ a ne^ dividend i^ and for ^very such 
<livi4fnd, Ipve oyt one figure mpr^ on the rightrhand side of 
tide divisoi: j' r^m^jmbering tQ parry for the increase ox the 
figures ciit p^, as in the ^ contractipn in Multiplication. 

Note. Wb^n there are not so many figures in the divisor, 
as are required to be in th^ quotient, begin the operation 
with all the figures, and continue it as usual till the number 
of figures ill the divisor be equal to those remaining to be 
found in the quoUent ; after which begin the contraction. 

EXAMPLES. 

1. Divide 2508-92806 by 92'41035, so as to have only 
four decimals in the quotient, in which case tJ^e quotient will 
contain six figures. 

Contracted. 



REDUCTION OF DECIMALS. 73 

Contracted, Common. 



PJ'4103,5)25O8'928,06(a7- 1498 
660721 



13849 
4608 

80 
6 



92-4 103,5)2508-928,06(27. 1 498 
66072106 
13848610 
46075750 
91116100 
79467850 
5539570 

2. Divide 4109-2351 by 230*409, so that the quotient may 
contain only four decimals. Ans. 17*8345. 

3. Divide 37' J 0438 by 5713-96, that the quotient may 
contain only five decimals. Ans. •00649. 

4.Divide 913*08 by 2137*3, that the quotient may contain 
fmly threie decimals. 



flEDUCTION OF DECIMALS. 

CASE I. 

To reduce a Vulgar Fraction to ks equivalent Decimal. 

« 

Divide the numerator by the denominator as in Division 
pf Decimals, annexing ciphers to the numerator as far as 
necessary ; so shall the quotient be the decimal required. 

EXAMPLES. 

1. Reduce ^ to a decimal. 

24=4 X 6. Thei£4) 7* 

W) 1*750000. 

•291666 &c. 



2. Reduce ^y and ^ and \<, \o decimals. 

Ans. '25, and '5, and *75. 

' 3. Reduce -^ to a decimal, Ans, '625. 

4. Reduce ^ to a decimal. Ans. -12, 

5. Reduce ttt *o * decimal. Ans. '03^59. 

6. Reduce ^:^ to a decimal. Ans.. '143155 &c. 

CASE 



74 . ARITHMETia 



CISB II. 

To find the Value of a Decimal in terms of the Inferior Den^ 

minationSm 

Multiply the decimal by the number of parts in the 
next lower denomination ; and cut off as many places for a 
remainder to the right-hand, as there are places in the given 
decimal. 

Multiply that remainder by the parts in the next lower 
denomination again, cutting off for another remainder as 
before. 

Proceed in the same manner through all the parts of the 
integer ; then the several denominations separated on the left- 
hand, will make up the answer. 

Notey Tliis Operation is the same as Reduction Descending; 
in whole numbers. ' • 



exa'mples. 

1. Required to. fin,d the value of ^775 pounds sterling* 

•775 
90 



/ 15-500 
\2 



d 6-000 Ans. 15/ 6i: 



2. What is the value of '625 shil ? Ans, 7^/. 

3. What is the value of -8635/.^ Ans. 17/S-24A 

4. What is the value of '0125 lb troy ? Ans. 3 dwts,.^ 

5. What is the value of '4694 lb troy ? 

Ans. 5 oz J2 dwts 15*744 gr. 

6. What is the value of '625 cwt ? ^Ans. 2 qr H lb» 

7. What is the value of -009^43 miles ? 

^ Ans. 17 yd 1 ft 5-93S48 inc. 

8. What is the valu^f -6875 yd? Ans. 2 qr 3 nls. 
.9. What is the value of '3375 acr ? Ans. 1 rd 14 poles* 
10. What is the value of '2083 hhdof wine? 

Ans. 13-1229 gal. 



iCASE 



REDUCTION OF DECIMALS. 75 



CASK III. 

To reduce Integers or Decimals to Equivalent Decimals (f Higher 

De/iominations, 

Divide by the number of parts in the next higher deno- 
mination ; continuing the operation to as many higher deno^ 
minations as may be necessary, the s^me as in Reduction 
Ascending of whole numbers. 

EXAMPLES. 

1. Reduce 1 dwt to the decimal of a pound troy* 



20 
12 



1 dwt 

0*05 oz 

0004-166 &c.lb^ Ans, 



2. Reduce 9fl? to the decimal of a pound: Ans. .0375/. 

3. Reduce 7 drams to the decimal of a pound avoird. 

Ans. -02734.375^.- 
' 4. Reduce '26^ to the decimal of a /. Ans. '0010333 &c. /. 

5. Reduce 2*15 lb to the decimal of a cwt. 

Ans. -019196 + cwt. 

6. Reduce 24 yards to the decimal of a. mile. 

^ Ans. -013636 &c. mile. 

7. Reduce *056 pole to the decimal of an acre. 

Ans. -00035 ac. 

8. Reduce 1'2 pint of wine to the decimal of a hhd. 

Ans. '00238 + hhd. 

9. Reduce 14 minutes to the decimal of a day* 

Ans. -009722 &c. da. 

10. Reduce *21 pint to thedecimalof a peck. 

Ans. -013125 pec. 

1 1 , Reduce 28" 1 2"' to the decimal of a minute. 

Note, When there are several numbers^ to be reduced all to the 
decimal of the highest : 

Set the given nuihbers directly under each otherj for divi- 
dends, proceeding orderly from the lowest denominatiori to. 
the highest. % 

Opposite to each dividend, on the left-hand, set such a 
number for a divisor as will bring it to the next higher name; 
drawing a perpendicular line between all the divisors and 
dividends. 

Begin at the uppermost, and perform all the divisions: 
qnly observing to set the quotient of each division^ as decimal 

parts. 



16 ARITHMETIC.^ 

parts, on the right-hand of the dividend next below It } so 
shall the last quotient be the decimal required. 

EXAMPLES. 

1. Reduce 17/ 9^rf to the decimal of a pomid* . 



4 
12 
20 



3 

9-75 
17*8125 
£ 0*890625 Ans. 



2. Reduce 19/ 17/ S^d to /. Ans. 19-86354fl66 &c. /. 

3. Reduce 15/ 6d to the decimal of a /. Ans. '775/. 

4. Reduce 74^ to t^e decimal of a shilling. Ans. '6S5s. 

5. Reduce 5 oz 1 2 dwts 1 6 gr to lb. Ans. *46944 &c. lb; 



iSKTrnffTTT 



RULE OF THREE in DECIMALS. 

PiEPARfe the terms, by reducing the vulgar fractions to 
decimals, and any compound numbers either to decimals of the 
higher denominations, or to integers of the lower, also the 
iirst and third terms to the same name : Then multiply and 
tlivide as in whole numbers. 

Note^ Any of the convenient Examples in the Rule of 
Three or Rule of Five in Integers, or Vulgar Fractions, may 
be taken as proper examples to the same rules in Decimals. 
— ^The following Examplei which is the first in Vulgar Frac- 
tions, is wrought out here, to show the method. 

Jf I of a yard of velvet cost |/, what will ^-^ yd cost ? 

yd / yd / s d 



1 = -375 . 


•375 : -4 :: -3125 : -333 &c. or 6 « 

•4 


T — * 


•375 ) -12500 ( •333333 &c. 

1250 20 
rf 1^1? . _ 




/ 6-66666 Sec: 
12 


« 


Ans. 6s Sd. d 7*99999 &c.=8rf. 




]>UOD£* 



DUODECIMALS. 77 



DUODECIMALS. 



1 • f 

DuobsciMALs, or Cross Multiplication, is a rule used 
by workmen and artificers^ in computing the contents of 
tneir works. 

Dimensions are usually taken in feet, inches, and quarters ^ 
any parts smaller than these being neglected as of no conse- 
quence. And the same in multiplying them together, or 
casting up the contents. The method is as follows. 

Set down the two dimensions to be multiplied together, 
one under the other, so ^t feet may stand under feety 
inches under inches, &c. ' 

Mukiptyeach term in the multiplicand, beginning at the 
lowest, by the feet in the multiplier, and 6et the result of 
each straight under its corresponding term, observing to 
carry 1 for every 12, from the inches to the feet. 

In like mai^ner, multiply all the multiplicand by the inches 
and parts of the multiplier, and set the result of each term 
one place removed to the right-hand of those in the multipli- 
cand; omitting, however, what is below parts of inches, only 
carrying to these the proper number of units from the lowest 
denomination. 

Or, instead of multiplying by the Inches, take such parts 
<£ the multiplicand as there are of a foot. 

Then add the two lines together, after the manner of 
Compound Addition, carrying 1 to the feet for 12 inches^ 
when these come to so many# 





EXAMPLES. 


J- Multiply 4 f 7 inc 
by 6 4 


2. 


Multiply Uf 9 inc. 
by 4 « 


27 6 




59 
7 4i 


Ans. 29 0\ 


Ans. ^6 41- 



t 

3. Multiply 4 feet 7 inches by 9 f 6 inc. Ans. 43 f. 6^ inc. 
y 4, Multiply 12 f 5 inc by 6 f 8 inc. Ans. 82 9J- 

5. Multiply 35 f 4| mc by 12 f 3 inc. An?. 433 4 J. 

6. Multiply 64 f 6 inc by 8 f 9J inc. Ans. b^o 8| 

INVOLUTION. 



1$ 



AIUTHMETIC: 



INVOLUTION. 



Involution is the raising of Powers from tny given 
number, as a root. ' ' ' ' 

A Power is a quantity produced bytnultiplying any given 
number, called the Root, a certain number of times conti- 
nually by itself. Thiis, •' * 

2 = g is the*0Qt, qr iBt ppwer of 2* 
2x2= 4 is the 2d power, or square of'il^ , 
2x2x2= 8 is, the 3d powerj or cubeiof 2. . 

2 X '2 X 2 X 2 == 16 is the 4th. power of 2, &ۥ 

/ 

jAnd in this tnanner may be caktflated the followitig Table 
of the first^nine powers of the fkaj 9 numbers. 



TABLE of th.^ jfirst Ninb Powers of Numbers. 



1 



1st 



4 



2c 



if 3d 



9. 



5 

6 

7 

s 
9 



16 



I 

8 



27 



04 



25 
36 



J25 



40 



2J6 



343 



64 512 

_! 

|8l|/2,9 



4th 



16 



81 



256 



625 



I2g6 



240] 



•5th j ^t1i'|'>ih 



■t 



' 1 



32 



64 



128 



243 



J024 



729 



4096 



2187 



16384 



3125 



15625 



7776 



16807 



409632768 



0561 



59019 



46656 



78125 . 



2799^6 



II 7649 823543 



I 



2621442097162 



531441 



8th 



256 



6561 



65536 



39062^ 



J 67961 6 



^•^'-••••^va 



5764591 



16777216 



4 782969 4304672 J 






4 * 



512 



.196.88 



262144 



19531 2 j: 



10077696 



40353607 



■S.t. 



1«4217 






»> 



387420489 



The 



r^ 



INVOLUTION. ^9 

The Index or Exponent of a Powerj is the number de- 
noting the height or degree of that power; and it is 1 more 
than the number of muUiplications used in producing the 
same. So 1 is the index or exponent of the 1st power or 
rooty 2 of the 2d power or square^ 3 of the third power or 
cube, 4 of the 4th power, and so on. 

Powers) that are to be raised, are usually denoted bj placing 
the index above the root er first power. 

So 2* S3 4 IS the 2d power of 2. 

.' ■ 2'= 8is the. 3d power of 2. ' 

2;* ac 16 is tb« 4th power of 2. 
540* is the 4th power of 540, &c. 

When two or more powers are multiplied together, their 
product is that power whose index is the sum of the expo* 
nents of the faaors or powers multiplied. Or the multipH-* 
cation of the ppmrs, answers to the addition of >tfaje indices. 
Thus, in the following powers of 2, 



1st 


2d 


3d 


4th 


5th 


6th 


7th 


8th 


9th 


10th 


£ 


4 


8 


16 


32 


64 


123 


256 


512 


1024 


fflp ^' 


«• 


2^ 


£♦ 


25 


2* 


2' 


2« 


go 


gto 



Here^ 4 X 4 ±=: 16, and 2 + 2 5= 4 its index; 
and 8 X 16 = 128, and 3 + 4 = 7 its index; 
ako 16 X 64 = 1024, and 4 + 6 = 10 its index. 



eTHfiR EXAMPLES. ' 

1. What IS the 2d power of 45 ? . Ans. -2025. 

2. What is the square of 4' 16 ? Ans*^ 17*3056. 
S. What is th« 3d power of 3*5 ? Ans. 42'875. 

4, What is the 5th power of '029? Ans. '00000002051 1 149. 

5, What is the square of-f- ? Ans. ^. 

6. What is the 3d power of 4? Ans. fj^. 

7. What is tjtie 4th ppw^r of 4: ? Ans. ^Vy* 



EVOLUTION. 



80 ARITHMETIC. 



EVOLUTIO^f. 

EvoLUTiONi or the reverse of Involution^ is the extracting 
or finding the roots of any given powers, 

■ , 

The root of any number, or power, is such a number^ 29 
being multiplied into itself a certain number of times, wul 
produce that power. . Thus, 2 is the square root or 2d root 
of 4, because 2' = 2 x 2=4; and 3 is the cube root of 3d 
root of 27, because 3' = 3 x 3 x 3 = 27. 

Any powei: of a gfiven number or root toay be found ex- 
actly, namely, by multiplying the number continually inta 
itself. But there are many numbers of which a proposed root 
can never be exactly found. Yet, by means of decimals, we 
may approximate or approach towards the root, to any de- 
gree of exactness. 

Those roots which only approximate, are called Surd 
roots; but those which can be found quite exact, are called 
Rational Roots. Thus, the square root of 3 is a surd root; 
but the square root of 4 is a rational root, being cfqual to 2 : 
also the cube root of 8 is rational^ being eqvial to 2 ; but the 
cube root of 9 is surd or irrational. 

Roots are sometimes denoted by writing the charstcter 4/ 
before the power, with the index of the root against it. 
Thus, the 3d root of 20 is expressed by ^0 5 and the square 
root or 2d root of it is v'20, the index 2 being always omit- 
ted, when oiily the square root is designed. 

When the power is expressed by several numbers, with the 
sign + or ~ between them, a line is drawn from the top of 
the sign over all the parts of it: thus the third root of 

45 -12 is ^45 - 12, or thus ^(45-12), inclosing the 
numbers in p^entheses. 

But all roots are now often designed like poweti, with 
fractional indices : thus, the square root of 8 is 8^, th6 cube 
root of 25 is 25"^, and the 4th root q{ 45 -^ 1« is 45- 1,8 J*, 
•r (45 -.18)^. 

TO 



SQUARE ROOT. M 



TO EXTRACT THE SQJIARE EOOT. 

* DfViDB the given number into periods of two figuret 
each, by setting k point over the place of units^ mother over 
the place of hundredsi and so on, over every second figure^ 
both to the left hand in integers, and to the right in de^ 

cimals. 

Find the greatest square in the first period on the left-hand, 
and set its root on the right-hand of the given number, after 
the manner of a quotient figure in Division. . 



■»•»• 



* The reason for separating the figures of the dividend into 
periods or portions of two places each^ is^ thdt the square of any 
single figure never consists of more thui two places $ the square of . 
a number of two figures, of not more than four places, and so i^ 
So that there will be as many figures in the root as the given nUin- 
ber contains periods so divided or parted off. 

And the reason of the several steps in the operation appears 
from the algebraic form of the square of any number of terms, 
whether two or three or more. Tlius, 

(a + b)^ = a» + 2ab + 6* =: a* + (2a + b) b, the squareof two 
tffirms ; where it appears that a is the first term of the root, and b 
the second term ) also a the first divisor, and the new divisor is 
2a 4- ^> or denize the first term increased by the s^copd. And 
hence the manner of extractipp is thvs : 

1st divisor a) a^ ^ 2ab -f 6* {a + b the root* 



a* 



2d divisor 2a -f & | 2a6 + b* 

b\2ab + bf 

Again, for a root of three parts, a, b, c, thus : 

(a + 6 + c)* =; a* + 2a6 + 6® + 2tfc + 2bc + e« = 

a^ + (2a + 6) 6 -f (2a + 26 + c) c, the 
squaie of three terms, where a is the first term of the root, b the 
second, and c the third term ; also a the first divisor, 2a -f ^ the 
second, and 2a -^ 2b + c the third, each consisting of the double 
fif the root increased by the next term of the same. And the mode 
of extraction is thus : 
1st divisor a) a* + 2a6 ^ 6^ + 2ac + 2bc + c'^ {a + b +c the root. 



a?- 



2d divisor 2a + b 

b 



2ab + i* 
2a6 + A» 



N 



3d divisor 2a + 26 + c I 2flc + 26c + c* 

c I 2flc + 26c + c* 

Vol. I. G Subtract 



sr -Aum^/^tic. 



»% 

• ^ 



Subtract the square dius found from the said period, and 
to the remaindet annex tNi^>^two fTg^ures of the ntext following 
pQriodffbrfidivi4^. ' 

* Doiri>le the rootabove aaeadoned fisr a divisor ; and.ftid^ 
how oiften k is contained in the said dividend^ exclusive of 
its right-hand figure ; and set that quotient figure botl)^ in 
the quotient and divisor. 

Multiply the whole augmented divisor by this last quotient 
figure, and subtract the p]t)duct from the said dividend, 
bnnging down to it the next period of the given number, 
foraaew^Uvidend. . 

Rppeat tb^ same procesi o^r agaln^ viz. find another new 
dsmei^. by douUipg all the figures new hund in the root ( 
fixmi wludi, and the last dividend, find the next figure of 
the root as before ; and so on through all die periods, to the 
last. 

Note^ The best way of doubling the root, to form the new 
divisor's, is by adding the last figure always to the last divisor,^ 
as appears in the following examples. — Also, aftejr the figures 
belo^ng to the given number are all exhausted, the operas, 
tipn may be continued into decimals at pleasure, by adding 
any number of periods of ciphers, two in each period, 

EXAMPLES. 

i. To find the square root of 29506624. 

• • • • 
29506624 ( 5433 the root. 
25 



104 
4 


450 
416 


1083 
8 


3466 
3249 


2 


[21724 
21724 



Note, When the rooi is to he extracted to many places of figures ^ 
the work may be considerably shortenedy thus'/ 

* • . ' * 

Having proceeded in the extraction after the common me- 
thod> till there be found half the required number of figures 

•'in 



S(^tTAR£ ROCyr. 



» 



in the robt^ or one figure more; tb^> Ibr ib(t fM, dii^e 
the last remainder by its eori^e^pohdihg divl^rrafter the ittali-' 
aer of the thu*d contraction in Division of Deciihdisi tluis, 

2. To find the root of 2 to nine placid of fi^M^ 

2 ( 1*4142195^ the tooti 
1 • • 



^ 100 
4j 96 



2ii 
1 



400 
3^1 



2824 

4 



11900 
11296 



28282 
2 



60400 
56564 



v.* 



28284 } 



3. What 

4. What 

5. What 

6. What 

7. What 
i. WlKit 
9. What 

10. What 

11. What 

12. What 



id the 
is the 
is (he 
is the 
is the 
is the 
is the 
is the 
is the 
is the 



square 
square 
square 
square 
square 
sqtiarje 
square 
square 
square 
square 



S8S6 ( 1856 
1008 
160 
19 

root of 2025 ? 
root of 17-3056 ? 
root of •000729 f 
root of S ? 
root of 5 ? . 
root of 6 ? 
root of 7 ? ^ 
root of 10 ? 
root of II ? 
root of 12? 



Am. 45. 

Ans. 4*16. 

Ans. *027. 
Ans. 1-732050. 
Ans. 2*236068. 
Ans. 2*449489. 
Ans. 2*645751. 
Ans. 3162277. 
Ans. 3*316624. 
Ans. 3*464201. 



RULES tOK TB£ s'qpARE ROaTS OF VUXOAR FRACTIONS 

AND MIXED NUMBERS. 

FitLsT prepare all vulgar fractions, by reducing theni to 
their least terins^ both for this and all other roots. Then 

1. Take tbe root of the numeratoi' and of the denominator 
for the respective terms of the root required. . Aiid fKb is the 
best way if the den6i£iiilator be a complete power: but if it 
be not, then 

2. Multiply the numeraitbr and denominator together; 
take tlie root of th^ product : this root being zidade the nume^ 
^ G 2 rater 



M 



ARITHMETia 



xatorto the denominator of the riven fraction, or mado tl^e 
denominator to the numerator of it> will form the fractio^al 
root required. 



That is, v^y 



f^a \/ah 



s^b b s/ab' 

And this rule will serve, whether the root be finite or infinite. 

3^ Or reduc(e the vulgar fraction to a decimal, and extracf 
its root, 

4. Mixed numbers may be either reduced to improper 
Actions, and extracted by the first or second rule» or the 
vulgar fraction may be reduced to a decimal, then joined tp 
the integer, ami the root of the whole extracted. 



SAMPLES. 



Ans. 



4» 

Ans. f . 



!• What is the root of |4 ? 

2. What i$ the root of^l 

S. What is the root of A ? Ans. 0-866025. 

4. What is the root of T^j. ? Ans. 0-645497^ 

5. What is the root of n| ? ^ns. 4-168333; 

By means of the square root also may readily be found the 
4th root, or the 8th root, or the l^th root, &C| that is, the 
root of any power whose index is some power of the numoer 
2 ; namely, by extracting so often the square root as is de* 
.noted by that power of 2; that is, two extractions fbr the 
4th root, three for the 8th root, and so on. 

So, to find the 4th root of the number 21035*8, extract 
the square root two times as follows : 



21035-8000 
I 



( 145-037237 (12-0431407 the 4th root, 
1 



24 

4 



-110 
96 



22 
2 



45 
44 



285 
5 



1435 
1425 



2404 I 10372 
4 9616 



■▼ .- 



29003 
3 



108000 24083 
87009 3 



75637 

72249 



20591(7237 3388 ( 140t 

687 980 

107 17 

Jlx, 2. Wh^t is th? 4th root of 97--^ ^ I 



n 



ROOT; §5 

TO EXTRACT THE CUBE ROOT. 

4 

I. By the Common Ruk^i 

. i. Having divided the given number into periods of three 
£gures each, (by setting a point over the place of units, and 
also over every third figure, from thence, to t&e left hand iri 
y^hole numbers, and to the right iirxtedmals), find the nearest 
less cube to the first period ^ set. its root in the quotient, and 
subtract the said cube ftoffl^the fifsrperiod f to the remainder 
bring down the second period, and call this the resolvend. 

2. To three times the square of the root, jiist found, add 
three times the root itself, setting this one place more to the 
right than the former, and calltliis siiiri the divisor/ Thctn 
divide the resolvend, wanting the last figure, by the divisort 
for the next figure of the root, which annex to the former ; 
calling this last figure #, and the part of the root before 
found let be called a. 

3. Add alltogether these three products, ns^ely, thrice 
^uare multiplied by ii^ thrice a multiplied by e s()uare, and 
^ cube, setting each of them one place more to the right than 
the former, and call the sum the subtrahend ; which must 
hot exceed the resolvend ^ bdt if it does, then lilake the last 
£gure e lessj and repeat the operation for finding the subtra- 
hend> till it be less than the resolvend. 

4. From the resolvend take the subtrahend, and to there- 
tnainder join the next period of the given number tdr s hew 
resolvend ; to which form a new divisor from the wlfole rOot 

'now found ; and froith ihence another figure of the root, as 
directed in Article 2, and so oii. 



^ 



♦ The reason, for pointing the givcaa number into periods of 
three figures eacb> is because the cube of one' figure never amounts 
io more than three pl^es. And, for a similar, reason, a given 
number is pointed into periods of four figures for the 4th root> tif 
£ve figures for the 5th root, and so on. 

And the reason for the-other parts of the rule depends on the 
algebraic formatioad of a ci^be : for, if the root consist of the two 
parts a -{- h, then its cube is as follows ; (a + 6)' = a' + 3a*6-f 
3aA* + b^ j where ei is the foot of the first part a' y the resolvend 
is 3a*6 + 3«i* -f J?, which is aUo.the same as the three parts of 
{he subtrahend ; also the divisor Is 3a* + 3tf, by which dividing 
the first two terms ©f the re^lvend 3o*^ + ao*, gives b for the 
le^kond part of the root 3 and so on. 

ixAMPLK. 



tft 



AWTHMET^, 



J^XAMPLC 



To extsaet tke cube root of 48328-544. 



S X S* sr 27 
S X ? 5B. .09 

Divis^ 279 



48228*544 ( 3$ '4 rpot. 
27 



■WP*<PV«^ 



21228 resoWend. 



^ly^'fnr f " ) ' '^' 



S X S* X 6 =c 162 
3x3 X 6* = 324 Vadd 
6» = 216 



3 X 36*. == 3888 
3 X 36 sc 108 



38988 



19656 subtrahend 



1572544 resolvend. 



5' X 36^ X 4 = 
3 X 36 X 'V = 

4^ = 



15552 
nSSSadd 
64 



1572544 subtrahcn*. 



0000000 remainder. 



Ex. 2/ Extraa the cube root erf" 57 1482' 19» 
Ex. 3. Extnct the cube root of 1628' 1582. 
£x« 4. Extract the cube root of 1332. 



* II. T* 49ctract tie Cube Root by a short ITay*. 

t. Bj trials, c»r by the table of roots at p. 90, &c, t^ke. 
the nearest rational cube to the given nunaber, whether it be 
greater or less ; and call it the assumed cube. 

2. Then 



f •. 



* The method usually giveb for extracting the cube root, is so 
exeedingly t^edioos> and difficult to be remembered, that various 
pth^r appro'xiiuiiting rules have heeh invented, viz. by Newton, 
llaphson, H^lley^ De Lagny, Simpson, Emerson, and several other 
mathematioiaDs ; but; ho one that I have yet seen, is so simple in 
^ts form, or seems to wiell adapted for general use, as ^hat above 
given. This rule is the same in effect as Dr. Bailey's, rational 

formula. 



hmtbm nai doaUe the assumsdrcube^ is to iht stm of the 
assumed cube and ctouble the given number, so isthe lOOt of 
the assumed cube, to the root required^ nearly^ Otf As the 
first sum Is to the difierence of the given and assumed cube^ 
so is the assumed root to the difference of the root» nearly. 

3. Again, by using, in like manner, the cube of the root 
last found as a new asstiiTietrcube, another root wiQ be ob- 
tained still nearer. - And $o on as £u: as we please ; uring d- 
ways the cube^ of the last found root, for the assumed cube. 



EXAMPLE. 

To £nd<the ctdse root of 31035*8. 

Here we soon find that the root lies between 20 and 30, 
and then between 27 and 98. Taking th^erefore 27, its cube 
is 19683, which is the assumed cube. Then 

. 2 2 



39366 4207 1-6' 

21035*8 19683 



»■ < ■■ 



As 60401-8 : .617S4-6 :: 27 : 27 '60471 

27 



f . k 



4322822 
1235092 



60401-8) 1667374-2 (27-6047 th« root neadf. 

4is;9338 

284 
42 



formoki) but m or e co m med t o usl y eapiessed ' , tnd-^ie first-intesti^ 
gation of it was given in my Traets, p. 49. ^ Tl^e algsbraic fom 
of it is this : 

As P H^ 2a : A «f 29 ; : r ; a. Or^ . . \ 

As B + aA : p fio; a : : r ; a^ v> r j ] 

^bere r is the given nittqber, .i^ Ite as^otattd^ieafrilt cobc r ^Ire- 
^be root of a^ and a the root of ? sought. 

Again, 



88 ARTTHMEnC. . 

Agam, for a secoad opdratlony the cube of tfaif rooit le 
91035*318645155623^ and the process bj the ktter method 
will be thus : 

21035-318(545, &c. ' 



• . 42070-637!29a 21035-8 

21035-8 . 21085-318645, &c. 

As 63106-43729 : diff. -481355 ;: 27*6047 : 

thediff. -0002 10560 



codseq. the r<>ot req. is 27-604910560. 

Ex. 2. To extract the cube root of •67. 
Ex. 3. To extract the cube root of *01. 

TO EXTRACT ANY ROOT WHATEVER*. 

Let p be the given power or number, n the index of t&e' 
power, A the assumed power, r its root, r the required root 
of p. Then say. 

As the sum of if + 1 times a and /i — 1 times p, 
is to the sum of » + 1 times ? and » — 1 times a^ 
so is the assumed root r, to the required root R. 

Or, as half the said sum of ii -f 1 times A, and »*- 1 times 
r, is to the difference between the given and assumed powers,- 
so is the assutned root r, to the difference bet'^een the true 
and assumed roots ; which difference, added or subtraaed^ as 
the case requires, gives the true root nearly, 
' ■ ■ ■ ■■ ' ■ ■ ■■ ■ ■ ■ ii 

Thatis, «+l. A + «— 1. P :/f+l'. P. +«— 1. a :: r :R. 



Or, «+1.4a+«— I. -y* : PcQ A :: r : Rco r. 

And the operation may be repeated as often as we please, 
by using always the last found root for the assumed root, and 
its nth power for the assumed power A. 



mmmimim'mmm^aKt^ttm 



* This is a very general approximating rule« of which that fbc 
the cube root is a particuk.r case^ and is the best adapted for 
practice^ andfor meinorjr, of any that I have yet seen. It was first 
disooveied in this form by myself^ and the investigation and use of 
it.wefe^ven at large ia my Tacts, p. 45, Dec. 



EXAMPLE. 



d£N£RAL tOOrtS. 



i& 



fXAMPLE. 

"to extract the 5th root of 21035*8. 

( 

Here it appears that the 5th root is between 7*3 and 7'4. 
^Taking 7*3, its 5th power is 20130'7 1593. Hence we have 
p = 21035-8, « = 5, r =7*3 and A = 20730-71593; then 

«+l. iA + «— U 4^ • ^ CO A :: r : rco r, that is, 
3x30730-11593+11x21035-8 : 305 084 :: 73 : 

3 2 7-3 



62192-14719 
42071-6 

1P4265-74779 



42071-6 915252 
2135588 



2227- 11 32 (-02 13605 = 11 cor 
7'3 = r, add. 



1. What 

2; What 

3. Wliat 

4. Wifiat 

5. What 

6. What 

7. What 

8. What 
9* What 

lOi What 

11. What 

12. What 
13. 'What 



OTHfiH iXAMPLES. 

s the 3d root of 2 ? 

sthe 3d root of 321 4? 

s the 4th root of 2? 

s the 4th root of 91 '4ft ? 

s the 5th root of 2 ? 

s the 6th root of 21035-8 ? 

s the 6 th root of 2? 

s the 7th root of 21035-8 ? 

s the 7th root of 2 ^ 

s the 8th root of 21035-8 ? 

s the 8th root of 2 ? 

s the 9th root of 2 1 0S5'*S ? 

sth^' 9th root of 2? 



7-321360 = R, true 


to the last figure^ 


Ans. 


1-259921. 


Ans. 


14-75758. 


Axis. 


1-189207. 


Ans« 3-1415999. 


A^s. 


I-148699. 


Ans. 


5-254037. 


Ans. 


1-122462. 


Ans. 


4-145392- 


Ans. 


1-104089. 


Ans. 


3-470323. 


Ans. 


1-090508. 


Ans. 


3-022239. 


Ans. 


l-080059.f 



•riT 



llie following is a Table of squares and cubes, as also the 
Square, roots aild cube roots, of ah numbers from 1 to 1000, 
JKrhich will be found very useful on many occasions, in nu- 
meral calculations, when xoots or powers are concerned. 



A TAtiS 



fO A TABLE OF SQUAWJSb CUMS^ and ROOTS. 



^. 



Number. 


Square. 


Cube. 


Square Roo 


t. « Cube Root. 


1 


I 


1 


1*0000000 


1000000 


3 


4 


8 


1-4142136 


1-259921 


3 


9 


. 27 


1-7320508 


1-442250 


4 


l6 


64 


2'C 


iJi • 1 ?i 


1*587401 


5 


25 


125 


^^ ^^ ^^ -^y -^r- ^^ ^^ -^^ 

2*2360680 


' 1*709976 


6 


36 


%\6 


2-4494897 


1-^17121 


7 


49 


343 


2-6457513 


1-912933- - 


s 


64 


. 512 


2*82842^1 


2K)octoob 


9. . 


81 


729 


. 3-ooooonf] 


2*Qfi0nB4 


10 


100 


1000 


3*1622777 


2*i8.4?l35 


11 


121 


1331 


3*3166248 


2*223986 


ji 


144 


1728 


3*444 r 016 


2*289428 


13 


169 


2i97 


3-6055513 


2-35i335 


14 


196 


2744 


37416574 


2*410142 


g^5 


2f5 


3375 


.3-8729833 

J , _ ^_. .^ _ 


2*466212 


250 


4096 

4913 


4-C 


i:i:i:i:t:i:« 


2 510842 


17 


289 


^ -^^ ■mm' -^^ ■^^ -m^ ■m^ '^^ 

4*1231056 


- '2*571282 


18 


324 


5833 


4-2426407 


2*620741 


19 


361 


6859 


4*3588989 


2-668402 


20 


4CX) 


8000 


4'4721360 


2-7144I8 


21 


441 


9261 


4-5825757 


2-758923 


22 


484 


10648 


4-6904158 


2*802039 


23 


529 


12167 


4*7958815 


2-843867 


24 


;?76 . 


13624 


4-8989795 


2-884499 


25' 


625 


156!15 


5-0000000 


2-924OJ 8 


26 . 


^7^ 


1757I5 


5-0990195 


2-962496 


27 


7'^9 


19683 


5-1961524 


3-000000* 


28 


784 


21952 


' 5*2915026 


3*036589 


29 


841 


24389 


5-3851648 


3-072317 


30 


900 


27000 


5-477^^6 


I 3-107232 


31 


961 


29791 


5*5077644 


3^141381 


32 


1024 . 


32768 


5*6568542 


> 3*174802 


33 ' 


IO89 


35937 


6*7445626 


3*207534 ' 


34 


1156 


39304 


5-8909519 


" 3-239612 


35 


1225 


42875 


5-9160798 


3*271066 


36 


1296 


, 46656 


6'OOOOOOCi 


1 . 3*3019i27 


^37 


1369 


50653 


6-082/625 


3^332222 


^8 


1444 


54872 


6-1644140 


► 3-361975 


^ 


1521 


59319 


^*:M49980 


3*391211 


40 


1600 


64000 


6-3245553 


3*419952 


41 


I68I 


68921 


6-4031242 


3*448217 


42 


1764 


74O88 


6-4807407 


3*476027 


w 4a 


1849 


79507 


6*5574a85 


3*503398 


44 


19^6 


85184 


66332496 


3*530348 


45 


2^ 


.911^ 


6*7082039 


3-556893 


46 


2116 


97336 


6*7823300 


3*563048 


47 


220^ 


103823 


6*8556546 


3*608826 


48 


2304 


110592 


6*9282032 


3*634241 


49 


2401 


117649 


7-0000000 


\ 3*659306 


50 


2500 


125000 


7*0710678 


3*684031 



• 



r. 



M 



ri ■ 



SQlTARtS, CUBES, AND ROOTS. 



91 



Tf umber. 



Square. 



51 
52 
53 
*54 
*55 
56 

57 
58 

,m 

6k 
Gl 
63 
64 
65 
66 

07 
68 

70 

71 
72 

73 
74 

75 
76 
77 

78 

79 
80 

81 
82 
83 
84 
85 
86 
S7 

88 
89 

90 
91 

'93 

94 

95 

9^ 

97 
98 

99 
100 



2601- 

2704. 

2809 

2916 

3025 

3rl36 

3249 
3364 
3481 
36Cp 
372i 
3844 
3969 
4096 
4225. 
^4356 
4489 
4624 

4791 

4960 

5041 

5184 

^329 

5476 

5625 

5776 

5929 

6084 

6241 

6400 

6561 

6724 

68^ 

7056 

7225 

7396 

7569 

77^ 
7921 

8100 
8281 
.8464 
8649 
8836 
9025 
9216 

9409 

9604 

9801 

10000 



Cube. 



132651 
140608 

148877 
157464 

166375 
175616 
195193 
195112 
205379 
216000 
22^81 
238328 
250047 
2S2144 
. 274625 
287496 
300763 
314432 
3285Q9 
343000 

357911 
373248 

389017 

405224 

421875 

438976 

456533 

474552 

493039 

512000. 

531441 

551368 

571787 
'592704 
614125 
636056 
658503 
68I472 

704969 
729000 

753571 
778688 
804357 
830584 
857375 
884736 
912673 

941192 
^970299 

1000000 



Square Root. * Cube Bool. . 



•• 



\ 



7'1414284 

7'21 11026 
. 7-2801099 

7-3484692 
7'4l6l985 
7-4833148 
7^40^44 

7'6l5773i " 

7*6811457 
77459667 

7\8iQ2497 

7-8740079 
' 7*9372539 

8-0000000 

8-0622577 
8*1240384 

8-1853529 
8-2462113 
8-3066239 
8-3666003 
8*4261498 
8*4852814 
8-5440037 
'^•6023253 
8*6602540 

S*7177979 ' 
8*7749644 

8-8317609 
8-888 1944 

8-9442719 
9-0000000 

9-0553851 

9*1104336 

9-1651514 

9-2195445 

9*2736185 

93273791 
9*3808315 

9-43398 11 

9*4868330 

9*5398920 

-9*5916630 

9*6436508 

916953597 

9-7467943 

9-7979590 
9-8488578 

9-8994949 
9-9498744 

10*0000000 



I 



3-708430 

3*732511 

3*7562^96 

3779763 

3-8O2953 

3-825862 

3*843501 

3*870877 

3*892996 

3*914867 

3-936497 
3-957892 

3*979057 

4-000000 
4-020726 
4*041240 
4-061548 
4-081656 
4-101566 
4- 12 1285 
4*140818 
4*l60l68 

4-179339 
4-198336 

4*217163 
4*235824 
4-254321 
4-272659 
4*290841 
4*308870 
4-326749 
4*344481 
4-362071 

4'379519 
4*396830 

4*414005 

4-431047 
4-447060 

4*464745 

4*481405 

4-497942 

4-514357 

4-530655 

4-546836 

4*562903 

4-578857 
4*594701 
4-610436 
4*626065 
4-641589 



^ 



-• I 



s. 



»2 



\ 



ARITHMETIC. 



N amber. 


Square. 
10201 


101 


102 


10404 


103 


10609 


104 


10816 


105 


11025 


106 


11236 


107 


11449 


108 


11664 


109 


11881 


110 


12100 


111 


12321 


112 


12544 


113 


12769 


114 


12996' 


115 


13225 


116 


13456 


117 


13699 


118 


13924 


• 119 


14J61 


120 


14400 


121 


14641 


122 


14884 


123 


15129 


. 124 


15376 


125 


15625 


126 


15876 


127 


16129 


128 


16384 


129 


16641 


130 


l6S0p 


131 


17161 


132 


17424 


133 


176S9 


134- 


17956 


135 


18225 


136 


18496 


137 


I8769 


138 . 

1 


19044 


139 


19321 


140 


196QO 


141 


I988I 


142 


20164 


143 


20449 


144 


' 20736 


145 


21025 


146 


21316 


i47 


2I6O9 


148 


21904 


ug 


22201 . 


150 


22500 



Cube. 



1030301 
.IO612O8 
1092727 
1 1 24864 
1157625 
1191016 
1225043 
1259712 
1295029 
1331000 
1367631 
1404928 
1442897 
1481544 
1520875 
I56O896 
1601613 
1643032 
1685159 
1 728000 
1771561 
1815848 
I86O867 
1 906624 
1953125 
2000376 
2048383 
2097152 
2146689 
2197000 
2248091 
2299968 
5K3 52637 
2406104 
2460375 
2515456 
2571353 
2628072 
2685619 
2744000 
2803221 
2863288 
2924207 
2985984 
3048625 
3112136 
3176523 
3241792 
3307949 
3375000 



Square Root. 



00498756 
0-p99i049 
0-1488916 
O-198O39O 
0-2469508 
0-2959301 
0-3440S04 
0-3923048 
0-4403065 
0*4880885 
0-5356538 
0-5830052 
0*6301458 
0-6770783 
0*7238050 
07703296 
0*8166538 
OS6278O5 
0-9087121 
0*9544512 

rooooooo 

10453610 
1*0905365 

1'1355287 
1*1803399 
1 224972a 
1 2694277 
1-3137085 

1*3578167 
1-4017543 
1-4455231 
1*4891253 
1*5325626 
1-57583^ 
1-6189500 
1-6619038 

1*7046999 
1 -7473444 

1-789^261 

1-8321596 

1-8743421 

1-9163753 

1-95 82607 

2-0000000 

2*0415946 

2*0830460 

2-1243557 

21655251 

2-2065556 

2-2474487 



Cube Root. 



4-657010 

4-672330 

4-687548 

4702669 

4717694 

4*732624 

4747459 

4762203 

4'77^56 

4'791420 

4*805896 

4*820284 i 

4-834588 

4*848808 

4-862944 

4-876999 

4*890973 
4*904668 
4*918685 
4*932424 
4*946088 

4-959675 
4-973190 
4*986631 
5*000000 
5-013298 
5-026526 
5-039684 
5-052774 

5*065797 

5*073753 
5-091643 

5-104469 
5*117230 
5-129928 
5*142563 

5-155137 
5^167649 
5*160101 
5*192494 
5-204828 
5*217103 
5-229321 
5*241482 
5-253588 

5*265637 
5-277632 
5-289572 
5-301459 

5-313293 



SQUARES, CUBES, akd ROOTS. 



9$ 



1 


Sfjoare. 


Cube. « 


Square Root. 


CubeKoot. 


151 


22801 


3442951 


12-2882057 


5325074 


152 


23104 


3511808 


12*3288260 


||-336803 
^•348481 


153 


23409 


zb%\^^^ 


1 2*3693 169 


154 


23716 


3652264 


12-4096736 


5360108 


155 


24025- 


3723875 


12-4498996 


5-371685 


156 


24336 


3796416 


12*4899960 


5*383213 


157 


24649 


386989a 


12-5:^99641 


5-394690 


158 


24964 


3944312 


12*5698051 


5-406120 


159 


25281 


4019679 


126095202 


'5-417501 


160 


25600 


4096OOQ 


12-6491106 


5*428835 


l6i 


25921 


4173281 


12*6885775 


5*440122 


l62 


26244 


4251528 , 


12 7?79221 


$-451362 


163 


26569 


4330747 


127671453 


5-462556 


164 


26896 


4410944 


12*8062485 


5-473703 


165 


27225 


4492125 


1 2-8452326 


5*484806 


166 


1155^ . 


4574296 


12-8840987 


5-^5865 


167 


27889 


4657463 


12-9228480 


$-506879 


168 


28224 


A'J^lQ^'i 


}2'9614814 


5-517848 


169 


28561 


4826809 


13-0000000 


5'52S775 


170 


28900 


4913000 


13-Q384048 


5*539658 


171 


29241 


5000211 


13-0766968 


5*550499 


T73 


29584 . 


5088448 


13-1148770 


5-561298 


29929 


5177717 


13-1529464 


5'572Q54 


174 


30276 


52^5024 , 


13-19(^060 1 


5-582770 


175 


30625 


5359375 


13-2287566 1 


j5 593445 


17? 


30976 


M5\77Q 


13*26(54992 


5-604079 


177 


31329 


5545233 


13-3041347 


5 6146/3 


175 


31684 


5^3971^'^ 


13-3416641 


5»625226 


179 


3204! 


5735339 


^13-3790882 


5-635741 


180 


32400 


5S320O0 


13-4164079 


5*646216 


181 


32761 


5929741 


13-4536240 


5-656652 


182 


33124 


6028568 


13-4907376 


5-66706I 


183 


33489 


6128487 


135277493 


5-677411 


184 

i 


33856 


6229504 


13-5646600 


5-687734 


185 


34225 


6331625 


13-6014705 


5-698OI9 


186 

> 


34596 


6434856 


13*6381617 


5*708267 


187 


3491^ 


6539203 


13*6747943 


5-/18479 


188 


35344 


6644672 


137113092 


5-728654 


189 


35721 


67512^ 


137477271 


5-738794 


190 


36l(jp 


6859000 


13-7840488 


5*748897 


191 


36481 


6967871 


13-8202750 


5-75S965 


192 


36864 


7077888 


13-8564065 


5768998 


193 


37249 


71 89057 


13-8924440 


5-77S996 


194 


37636 


7301384 


13-9283883 


5*788960 


195 


38025 


7414875" 


. 13-9642400 


5*798890 


J 90 


38416 


7520536 


^Kvi:i:«:i:i iii^ 


5-808786 
5-818648 


1 *^ 

197 


38809 


7645373 


14-0356688 


198 


39204 


7762392 


14-0712473 


5-828476 


190 


39601 


7880599 


141067360 


5*838272 


J ^^ 


4obqo 


8O()0Qqo 


14-1421356 


5-848035 



•« 



ARITHMETIC. 



Numb. 


Square. 


Ciibfe. 


Sqaare Root. 


Cube Root. 


201 


4O401 


8120601 


1417744^ 


5-857705 


202 


40804 


8242406 


14-2126704 


5-867464 


203 


41^ 


8365427 


14-2478068 


6-87? 130 


204 


41616 


S4Sg66^ 


14-2828560 


5-886765 


205 


42025 


8615125 


14-3178211 


5-890308 


206 


42436 


8*741816 


14-3527001 


5^905941 


2G7 


42849 


8869743, 


14-3874946 


5-915481 


208 


43264 


8998912 


14^222051 


-^•9^^4991 


209 


436S1 


9123329 


14-4568323 


5-934473 


210 


44100 


9261000 


14-4913767 


5-943911 


211 


44521 


9393931 


14-5258390 


5-953341 


212 


44944 


9528128 


14*5602198 


5962731 


213 


45369 ' 


9^^3597 


14-5945195 


5-972091 


214 


45796 


9800344 


14-6287388 


5-98U26 


215 


46225 


9938375 , 


14-6628783 


5-990727 


216 


46656 


10077696 


14-6969385 


6-000000 

9 


2J7 


47O89 


10218313 


14-7309199 


6-0Q9244 


2ia 


47524 


10360232 


14*7648231 


6-018463 


219^ 


47961 


10503459 


14-7986486 


6O27650 


220 


48400 


10648000 


14-8323970 


6-036811 


221 


48841 


10/93861 


14-8660687 


6-045943 


. 222 


49284 


10941048 


14-89960*4 


6055048 


223 


49729 


1 10895^ 


14*9331845 


6-064126 


' ' 224 


50176 


11239424 


14-9666295' 


6-073177 


225 


50625 


II39O625 


15-0000000 


6-082201 


226 


51076 


11543176 


150332964 


.6-09Hd9 


: 227 


51529 


11 697083 


15-0665192 


6-100170 


228 


51984 


11852352 


15-0996689 


6-109115 


. 229 


52441 


I2OO8989 


15-1327460 


6-118032 


230 


5290a 


12167000 


15- 1657509 


6-126925 


231 


53361 


12326391 


15-1986842 


6135792 


232 


53824 


12487168 


15-2315462 


6-144634 


233 


^4289 


12649337 


15-2643375 


6' 1 53449 


, 234 


54756 


1281290* 


15-2970585 


6162239 


235 


55225 


12977875 


15-3297097 


6-171005 


236 


65696 


13144256 


15-3622915 


6-1797^7 


237 


56169 


13312053 


15-3948043 


6-188463 


238 


56644, 


13481272 


15-4272486 


6-197154 


239 


57121 ^ 


13651919 


15-4596248 


6-205821 . 


240 


576OO 


13824000 


15-4919334 


6-214464 


241 


58081 


13997521 


^5-3241747 


6*223083 


242 


5S564 


14172488 


15-556349^ 


6-231678 


243 


59049 


14348907 


15*5884578 


6-240251 


244 


5g536 


14526784 


15-6204994 


6-248800. ' 


245 


60025 


14706125 


1^-6524758 


6-257324 


246 


60516 


14886936 


15-6843871 


6-205820 


247 


61009 


15069223 


15-7162336 


6-274304 


248 


61504 


15252992 


15-7480157 


6-282760 


249 


62001 


15438249 


15-7797338 


6-291194 


; 250 


62500 


15625000 


15*8113883 


6-299604 





SQUARES, CUtJES, and ROOTS: 95 


'^imlb. 


Square. 


Cube. 


Square Koot. 


Cube Root 


251 


63001 


15813251 


15-8429795 


6-307992 


232 ' 


63504 


16003008 


15-8745079 


6-316359 


253 


64009 


16194277 


15-9059737 


6-32470* 


254 


64516 


16367064 


15-9373773 


6-333025 


255 


65025 


16561375 


15*9687194 


6-341325 


256 


65536 


16777^16 


16-0000000 


6349602 


257 


66049 


16974593 


16*0312195 


6-357859 


258 


66564 


17173512 


16-0623784 


6-366095 


259 


67O8I 


17373979 


16-0934769 


6-3743 ID 


S<50 


67600 


17576000 


16-1245155 


6-382564 


261 


6812i 


17779581 


16-1554944 


6-390676 


262 


68644 


17984728 


16*1864141 


6-39S827 


263 


69169 


18191447 


16-2172747 


6-406958 


264 . 


69696 


18399744 


1 6-2480768 


6-415068 


265 


70225 


186096^5 


16-2788206 


6-423157 


266 


70756 


18821006 


16-3095064 


6-431226 


267 


71289 


190341 )3 


16-3401346 


6-439275 


266 


71824 


192468^2 


16-3707055 


6-447305 


26g 


72361 


19465 1O9; 


16-4012195 


6-455314 


270 


72900 


19683060 


16-4316767 


6*463304 


271 


73441 


19902511 


16-4620776 


6-471274 


272 


73984 


20123648 


16-4924225 


6-479224 


273 


74529 


20346417 


16-5227116 


6-487153 


274. 


75076 


20570824 


16-5529454 


6-495064 


275 


75625 


20796875 


16-5831240 


6-502956 


276 


76X76 


21024576 


16-6132477 . 


6-510829 


277 


76729 


21253933 


16-6433170 


6-51 8694 ' 


278 


77284 


21484952 


16-6733320 


6-526519 


279 


77841 


21717639 


16-7032931 


6-534335 


280 


78400 


21952000 


16-7332005 


6-542132 


28i 


78961 


22188041 


16-7630546 


6-549911 


282 


79^24 


22425768 


16-7928556 


6-557672 


263 


8OO89 


22665187 


l6-8226038 


6-565415 


284 


80^56 


22906304 


16-8522995 


& 573 139 


285 


81225 


23149125 


16-S8I943O 


6-580844 


286 


8 1796 


23393656 


16-9115345 


, 6-588531 


287 


82369 


23639P03 


16-9410743 


6-596202 


288 


62944 


23887872 


16-9705627 


6-603854 . 


289 


83521 


24137569 


17-0000000 


6-61 1488 


290 


84100 


24389000 


17-0293864 


6-619106 


291 


• 84681 


24642171 


. 17-0587221 


6-626705 


292 


85264 


24897088 


17-0880075 


6-634287 


293 


85849 


25153752 


17-1172428 


6-64 F851 


294 


'86436 


25412184 


17-1464282 


6-649399 


295 


87025 


25672375 


17-1755640 


6-656930 


296 


87616 


25934336 


17-2046505 • 


6-664*43 


297 


88209 


26196O73 


17-2336879 


6-671940 


298 


88804 


26463592 


17-2626765 


6-679419 


299 


89401 


2673O899 


17-2916165 


6-686882 


300 


90000 


27000000 


17-3205081 


6-694328 / 



p« 




ARITHMliTlC. 




1 

1 


Nuiob. 


Square. 


Cube. 


Square Root. 


Cube Root^ ] 




301 


90601 


27270901 


17-3493516 


6*701758 


, 


302 


91204 


27543606 


17-378 1472 


6*709172 




3W 


9I8O9 


27818127 


17-4068952 


6-716569 




304 


92416 


26O94464 


17-4355958 


6*723950 


« 


305 


93025 


28372625 


17-4642492 


6-731316 




soej 


93636 


28652616 


17-4928557 


6'7SS66& 


. 


307 


9^249 


28934443 


17-5214155 


^'7^5997 


• 


3^ 


94864 


292 181 12 


17-5499288 


6-753313 ' 




309 


95481 


29503629 


17-5783958 


6-760614 




310 


96100 


29791000 


17-6068169 


6*767899 




311 


96721 


. 3008023 1 


17-6351921 


6'775lt}S 




312 


97344 


303/1328 


17'66S5'/.17 


6-782422 


i 


3r3 


97969 


30664297 


17-6918060 


6'7eg66i 


M 


314 


QS5g6 


30959144 


17-7200451 


6-796884 


> 


315 


99'^2S 


31255875 


l7-748239a 


6*804091 




3)6 


99S6(i 


31554496 


17-7763888 


6-811284 




317 


100489 


3185^013 


17-8044938 


6-816461 




318 


J01124 


32157432 


^*S325545 


6-825624 




319 


^01761 


3^^461759 


17-8605711 


6-832771 




320 


102400 


3^768000 


17-8i88543(^ 


6-639903 




321 


103041 


33076161 


17-91<>4729 


6*847021 




. 322 


103684 


, 33386248. 


17-i;443584 


6-854124 




323 


101329 


33698267 


l/-i;722C08 


6-861211 




324 


ia\97<^ 


34012224 

» 


18-OUOOQOO 


6-868284 




.325 


105625 


34328125 


18-027/504 


6-875343 




326 


IO6276 


34645976 


180554/01 


6-882388 




327 


IO6929 


34fj65783 


l&-083i413 


0-889419 




328 


107584 


35287552 


18- 1107703 


6-896435 


■ 


329 


108241 


35611289 


lb- 13835/1 


6-903436 




330 


lOSgOO 


35937000 


18-1659021 


^-910423 




331 


IQ956I 


36264691 


16- 1934054 


6-917396 




332 


) 10224 


36594368 


I8r2/08072. 


6924355 


• 


333 


IIOS89 


36926037 


1 8*2482876 


6-931300 




334. 


111556 


37259704 


18-2756669 


6*938232 




335 


112225 


37595375 


18-3030052 


6-945 ug 




336 • 


112896 


37933656 


18-3303028 


6-952053 




337 


113569 


38272753 


18-35/5598. 


6-9^8943 




338 


114244 


38614472 


• 16-3847763 


6*965819 


■ 


339 


114921 


38958219 


18-41 19-)26 


6'97^6S2 




340 


U5600 


39304000 


18-4390889 


6-979532 




341 


116281 


3965 1821 


18*4601853 


6-986369 




342 


II6964 


4CKX)l688 


18-4932420 


0^993191 


• 


343 


117649 


46353607 


18-5?02592 


7-000000 




344 


118336 


40707584 


18r5472370 


7-006796 


■ 


345 


119025 


41063625 


I8-574I 756 


7-013579 


1 


346 


119715 


41421736 


18^10752 


7-020349 




347 


120409 


41 78 1923 


18-6279360 


7-027106 




348 


121104 


42144192 


18-6547581 


7-033850 


1 


349 


121801 


42508549 


18-6815417 


7-040581 


1 


350 


122500 


42875000 


18*7082869 


7-047208 


1 




,,v» 









SQUARES," CUBES, and ROOTS. 



97 



Cu^elt 



OQt. 



Numb. \ Square. | Culie. 



351 
352 
353 
354 
355 
356 
357 
358 
359 
360 
361 
362 
303 
364 
365 
366 
367 
368 

370 

371 

372 

373 
374 
375 
376 

377 
378 

379 
380 

381 

382/ 

383' 
384 
335 
386 

387 
38S 

389 
390 

391 
393 
393 
394 
395 
39S 

^97 . 

398 

399 

400 

■ ■ ■ m tA 



123201 
123i,04 
124609 
125316 
126025 
126736 
127449 
128164 
128881 
129600 
130321 
131044 
I3i;69 
132496 
133225 
133P56 
134689 
135424 

136161 
136900 
137641 
138384 
139I29 
135876 
140625 
141376 

1421^29 
142884 
1436^11 
144400 
145 161 
145924 
-.146^89 
147456 
148225 
148996 
149769 
150544 
151321 
152100 
152881 
153664 
154449 
15523*6 
156025 
156816 
157609 
158404 
1.59201 
160000 



Square Root. 



43243551 
43614208 

43986977 
44361 864 
44738875 
45118016 
45499293 
45832712' 

46268279 

46656000 

47045881 

47437928 

47832147 

46228544 . 

4862712^ 

4902789<5 
49430^3 
49836032 
50243409 
50(i5 >000 
5106481 1 
51478848 
51899117 
52313624 
52734375 

58 157376 
53582633 
540(0152 

54439939 
54872003 
5530634 1 

557429* 

56181887' 

56623 104 

57066625 

575 1 2456 

57960633 

58411072 

58863869 

593.1 9000 

59776471 

60236288 

6O69S457 
61162984 
61629875 
62099136 

^257077^ 
630i4792 

63521199 
64000000 

H 



87349940 

8761 6630 
87882942 

1^*8148877 
5-8414437 

8*8679623 

8«S944436 

8*9208879 
8*(H729.53 
8*9736660 
49*0000'X)J 
9^0262 j76 

9-0525589 
9-0787840 
9*1040732 
Q* 13 11265 
9*15724^1, 
9*1833261 

9*2093727 
9*2353841 
9-26l?60? 
9*2873015 

9-3132079 

9-339079^ 
9*3649167 
9-3907194 
9*4164878 
9*4422^^21 
9-4679^«3 

9-4935887 
9-5192213 
9-5448263 
9*5703858 

9-5959179 
9*62141(^ 

9*6468327 
9*6723156 

9'%77^^Q 
9*7230329 

9*74^4177 

9-7737199 
97989899 

9*8242276 
9*84943J2 

9-3740069 

9-89974S7 

9-92<18588 

9'9m^73 

9-9749844 

20*0000000 



7*0540O» 

7*06069« 
7067376 

7*074043 

7*060698 
7*087341 

7-093970 

7* 100588 

7- 107198 
7' 1 13786 
7-120367 

7*12693i 
7*133492 

7*140037 
7'iA6!i6g 

7*153090 

7159599 
7*l6609« 
7']7268i0 

M 79054 
7*185516 

71919W 
7' I984O5 

7*204832 
7*211247 

7*217632 

7*224045 
7-230»l27 
7*236797 
7*243156 
7-249504 
7*255b44 
/• 262107 
7*268482 
7274786 

7-281071 
7*287362 
7*293638 

7-299893 
7-306143 
7*312363 
7*318611 
7-324829 
7*33 I0i7 
7-337234 
7-343420 
7*349596 

7•a5^762 

7-36ISI17 
r'368(>6i 



V 



mm 



J 



98 



ARraiMETlC. 



Nuinb" 
401 


• Square, 
jocaoi 


Cube. . 
64^81201. 


Square Hoot. Cube Root. 




20-0249914 7*374198 




. 4K^2 


-l6l(x)4 


64964808 


20*G49t>377 7-380322 




; 403 ' 


1 $24091 


65450827 


20-0/48599 


7'3fa6l37 


1 


4M 


J6321(i 


6.939264 


20'0997512 


7-392542 




; 405 


16^025 


66430125 


20- 1240 116 


7*398636 




' 40«i 


164836 


66923416 


20-1494417 


7-404/20 


1 


j 407 


165649 


67419143 


201 7424 10 


7-410/94 




I 40iJ 


166464 


679U312 


20'\ijgoi)Qg 


7-410859 




i 409 


167281 


i- 684 17(^29 


20-2237484 


7-422914 




1 410 


468100 


- 68921000 


20-2484567 


7-4*28958 




! 4U 


468921 


6942653 1 


20-2731349 


7-404993 




4t2 


16&Z4,4 


6gg3452S 


20-297783 1 


7-44K)18 




* 413 

4 


. I70&l)g 


70^144997 


20-3224014 


7-447033 




! 414 


i;j3y6 


7^)957944 


yO-34(i98Ci9 


7-4;53039 




• 415. 


•a723i»5;. 


.71473375 


20-3715488 


7-459036 




4l(j 


17^^056 


71991296 


20:396078 1 


7-465022 




. 417 . 


17^«*89 


72511713 


20-4205779 


7'V(m9 




i 418 


1747/i4 


73034632 


20-4'W5OW3 


7'4f709(>^ 




' 410 


. }7^^^<il 


7356CO59 


20-4694895 


7*482924 




420 


. 176400 


74O8SOOO 


20-4939015 


7-488872 




: 421 


177241 


74618401 


20-5V82845 


7-494810 




422. 


178Q84 


75151448 


20-5426386 


7-500710 


■ 


423 


.178929 


75<)i()ijb7 . 


20-5669608 


7-506660 


1* 


424 


. \m)770. 


702'>5(yM 


20'59J 2603 


7-512571 




425 . 


. •,li)OGi25 


. 707(>5ty25 


20*6155281 


7-518473 




426 


18X476 . 


77'^i>^77^ 


20-6397674? . 


7-524365 




427 


. l?2y^ 


778544»3 


20-6639783 


7-330248 




: 128. » 


U3184, 


78402752 


20-688 1609 


7-536121 




4?^ 


IS4©'4l 


7895:)589 


, 20-71 23 W2 


7*541986 


h 


430 . 


J8I9G9 


7950;(X)0 


20-/304414 


7-547841 




43\ 


1 85/61 


.8U(J02991 


aO-7605395 


7-553688 




482 


. .186624, 


6U62 1 5ib 


•iO-7846(:97. 


7-559525 




^33 V 


-187489 


61 1 Q'17^7 


20*8086520 


7-565353 




434 * 


. IB 8-356 


B\7465iH 


20-8326667 


7-5/1173 


' 


435 ; 


-189225 


82S12S75 


:ilO'SM}6536 


7-576984 




tf^) 


I9O096 


82tiil856' 


20-6906130 


7-582786 




437s 


■i9<^^69 


8:i4l^3453 


20-9045450 


7'5S657y 


\ 


43S 


,101644 


84027672 


20-9:84495 


7'5ij4'M>3 




430, 


192721 


84(;04519 


20*9523268 


7-600138 




4*40, 


.. 193600 


85I8400O 


20-9761770 


7'605g05 




441 


if;4-J8i . 


85766121 


2100(XXK0 


7-611662 




44^ 


196364 


86350S88 


21 0337960 


7-617411 




44'Ji 

* 


l(;62t9 


t-6i3S307 


21-0475652 


7-623151 


1 


4^4 


nj7r^6, 


875283r84 


21-0713075 


7-628683 


' 


4-15 


1 i)S025 


88121125 


21-0950231 


7-034006 


I 
I 


440 . 


i9«yi6' ^ 


83716*536 


2l-Jlb7121 


7-0 1 (.321 


\ 


447 


I$9«09 . 


89314023 : 


21-1423745 


7-646027 


i 


448 


200704 


89915392 


21-16*'i010S 


7't)5i725 


^ 


44g 


20t601, 


<)()5 18849 


21 1896201 


7-657414 






202500 


91 1 25000 


21-2132034 


7-663094 








4 









I 



SQUARES, CUBES, and ROOTS. 



90 



Numb. 


Square, 


Cube. 


Square Root. Cube Root- 




451 


203401 


91733851 


21-2367606 7'6m76& 


• 


452. 


204304 . 


9234540s 


21-2602916 


7-6744^0 


■ 


453 


205209 


92i^5gt)77 


21/2837967 


7'680085 ^ 




454 


206116 


93576664 


& 1-307275 3 


7*685/32 




455 


20/025 


9^19^375 


21»33072C;0 


7'(^9^3}i • 




45^ 


2079:i6 . 


04818816 


21-3541565 


7*697002 




457 


20bb4^ 


95443993 


21-3775583 


7-/0L1624 




458 


209764 


9607l9Ja 


21 -4009346 


77oe23S 




459 


210681 


96702579 


21-424^2853 


7*713«44 




4(>0 


211600^ 


97336000 


21-4476105 


77x91*2 




4^1 


212521 


d7972J8l 


21-4709103 


7-/25032 




. 402 


213444 


9861 1 1 28 


i2 1*494 1853 


7730014 




463. 


214369 


99252847 


21-5174348 . 


7-7^61^7 




464 


215296 


99897344 


21-5406592 


7741753 




465 


216225 


103544625 


21-5638587 


7747310 


\ 


466 


217156 


101194696 4 


21*5870331 


7752860 




467 


2I8O89 


10184/563 


♦?l-6l(n828 


7-758402 




46S 


21902^ 


102503232 


21*6333077 


7763936 




m 


2i9'^6l 


103161709 


21*6564078 


776t)462 




470 


220900 


103823000 


21-6794834 


7774980 




471 


221841 


104497111 


217025344 


7 7^t90 




472 


222784 


105154048 


21-7255610 


7 795992 




473 


223729 


105323817 


21 -7485632 


7791487 




474 


224676 


;06496424 


21-7715411 


7*796974 




.475 


225023 


IO7I71875 


21-7944947 


7*602453 




476 


2265/6 


107850176 


21-8174242 


7-807925 


1 


^77 


227529 


108531333 . 


21-8403297 


7*813380 




47s 


228484 


10:^2153 52 


21-8632111 


7*818845 




479 


229441 


109902239 


21-88606^6 


7-824294 




480 


230 ICO 


110532000 


21-9O89023 


7'829/35 


• 


481 


231361 


. 111284641 


21-9317422 


7*835198 




482 


232324 . 


lilQ80l68 

* 


2 1 '9544984 


7-840594 ! 




483 


2332^9 


1126/8587 


21-9772610. 


7'846013 




.4S4. . 


234256 


U 3379304 


22-0000000 


7' 85 1424 




.4S5 . 


235225 


;i 14084125 


22*0227155 


7-856828 ' 


1 


486 


236196 


114791256 


22-0454077 


7-862224 




487 


237169 


115501303 


22-0680763 


7'S676i3 




488 


23S14A 


116214272 


220907220 


7-872994 




,489 


. 239121. 


116930169 


22* 1 1 33(444 


7-878363 




490 • 


240100 


1 1764.0000 


22-1359436 


7*883734 




:4i)i : 


241081 


3 18370771 


2.-I585198 


7*889094 


1 


1^92- . 


u 242064: 


. 119095488 


5?2- 1 3 10730 


7-8J4446 




. 493; 


2430*9 


11982315;^, : 


22*2036033 


7-8c;9791 ^ 




49* • 


244036 . 


J 205^3734 


i?2-226M0S 


7-905 1 29 




495 


24.'>025 


12128/375 


22-2485955 


7*910460 




496- . 


246016 


122023936 


22-2710575 


7-9^0734 




497 


247009 


122763473 


122*2934968 


7921100 




498 


24^004 


-123505992 


22-3 1 5/) 136 


7-9264O8 




.499>, 


249001 


124251499 


2^-33830/9 


7?9'5]7io 


■ 


soo 


2.0OCO3 


1 25C0 XXX) 


22-3606708 


7-937005 


^ 


'•m • -«w^p^». 


> 


ll 2 " 




- ._.. • J 


It 
* 



100 



ARITHMETIC. 



* Numb, i b<\\\sire. 



50I 
502 
503 

504 
505 

5Q7 
506 

srp 

510 

511 
512 

513 
514 
515 
Si\6 
517 

5m 

519 

520 

521 

^22 

523 

524 

525: 

526 ' 

527 

528 

52p 

5:^a 

531 

632 
' 535 
' 53^ 

535 

5?>7 
5S8 
539 
54U 
541 

*5 43 

544 
54 > 
5llt; 

518 
54 a 



4 



I 






251001 

25200^1 

2.W009 

254016 

255025 

256(30 

^57049 

258064 

2^908! 

200100 

261121 

262144 

263 1 6^ 

264196 

263225 

26eji25Q 

2()7289 

208324 
269361 
2;04(X) 
27)441 
2724 S4 
273529 
274576 
275625 

278784 
27984 1 
2^:0,00 
281(j6l 
283024 
2840S9 
285150 
2S6225 
28/296 
288369 
289444 
290521 
291<XX) 
29268 1 
293764 

*2(|4849 
■295936 
2y7025 
298II6 

299209 
30030f 
301401 
;i02500 



Cube.' 



11 ittii • 



25751501 

26506008 

27263527 

26024064 

28787625 

29554216 

30323843 

31(^512 

31b72229 

3265 10(X) 

.334328 51, 

342l77'i8 

350O5f;97 

35796744 

3659O875 

373te096 

38488413^ 

38(JS1832 

39708359 ' 
4O60S0CO 

41420/61 

42236643 

438/7824 
44703125 
4.' ^^ \ 576 
40363183 
47 J 97952 
48035889 
46877000 

4972129! 

5i)5j6S70'8 

51419437 
52273304 
53130375 
53990^).'!^ 
54854153 
5572O872 

574b*4000 
58340421 
5925W)088 

6(>l63(X)7 
6C9S9I84 
618/8625 
62771336 
63667323 
645(56502 

654601^9 
66375000 



Square Root. 



2i*3Wit)293 
22-40J35J5 
22-4276615 
22*449^)443 

22 4/22051 
22*4^)44438 
22-5l666a5 
22-5388553 
22-5610283 
22^83 1796 

22-pO53091 
22-6274 170 
226495033 
22*6715681 
22 -0936 11 4 
22-7156334 
227376340 
22759Q134 
227515715 
'22-b035085 
22-8254244 

22-84 73 ISA 

22-8691935 

22-891046^^ 

22-912878^ 

22-93 ^6899^ 

22-95^806 

22-9782500 

23 OCOOOQO 
23O2I72S9 
230434372 
23-065 1252 
23 0867928 
23- 1 084400 
23-13a':6yO 
23- 15 16738 
2M73i^05 
23-1948270- 
23 •2163735 
23-2379001 
23-2594067 ' 
23-2808935 
23-3023604 
23-S23807(} 
2:^-345 235 1 
23-3666429 
23'38:?0311 
23-4093998 
23-4307490 
23-4520788 



*. 



-ti^^»-*mi^t,-mftmim 



Cube Uooi 

75942293 
7*947373 
7'9^2«47 

7-958114 
7-963374 

7-9C8'r-7 

.7-97:>873 
7-97i^ii2 

7*984344 

7-9895f;9 
7*9t;4788 
8'OCCOOO 
6-005205 
8-010^03 
8'015595 

8*020779 

8*025957 

8-03U29 

8-036293 

8-041451 

8*046603 

8-05 1 748 

8-056886 

6-00^018 

806714a 

8-072262 

8-077374 

8-082480 

8*087579 

8O92O72' 

8-0()7758 

8-102838 

8-M7912 

8*11-2980 

8-118041 

8- 1 23096 



8M 28144- 
8*133 186 
8-138223 
8- 143253 
8*M8276 
8* 1 53293 
8*158304 
8- 163309 
8* 1 68308 
8- 1 73302 
8-176289 
8-|832<^ 
8-188244 
8-193212 



■mh 



SQUARES, CUBES, a^j> ROOTS. 101 



Numbr. 

.551 
552 
553 
554 
555 
556 
557 
558 
5 9 
500 
5b i 
5(52 
503 
564 
565 
566 
567 
508 

56g 

57P 

^572 
573 
574 

575 
576 

577 

57s 

5bO 
561 
582 
5^3 
584 
585 
580* 
587 
688 
589 
5(J0 

5^2 
5ij3 
^y4 
5i)5 

SyG 
• 5ys 

.00 ) , 



Square. 



303601 

304/04 

3058:k> 

3001)10 

308b-i5 

3 9 MO 

3102-19 

311364 

312-181 

3136CX) 

311721 

315844 

3\6j69 

3 1 80i)(i 

319225 

32035d 

321489 

322J24 

323761 

324poO 

3200.1 1 

3^^84 

328329 
329476 
330 J25 

33177O 

332929 
33-1084 

335241 
33(>400 
337561 
338724 

339889 
34 To 36 

342225 
34339O' 
S4456'9 
34^744 
316921 
348*00 
34'>28 1 
35(MC>4 
35V649 
35283 C) 
354025 
355216 
355409 
357604 
?588pi 

ii'iiiobo 



Cube. 



10728415 1 

lobigo'd'os 
10*91 123/7 

1704X^1464 
170,51875 

17I879'J10* 

172S08693 
17:^741112 

174676B79 
I756iCxx;0 

176558481 
17750-i328 

178453547 
i794J6i44 
180332125 
18l32M9'> 
18228^263 
183250432 
184220009 
i85l930J0 
186169411 
1 87149248 

18eri325l7 
1 89 119224 

1130109375 
191102976 

I92IOCOJ3 

193103552 

194104539 

1951i20JO 

190*122941 

197J37368 

193155297 
199 -70701 

2(i()20l625 
201 23005(3 
2{y22(j20J3 
203^j7472 
2043:i64oy 
305379O:xi 
206125071 
2074746%Q 

io^527^7 
20i)5&4584 
3lb:)4487d 
3 1 1 7O87 J6 
^12776173 
213847193 

214921799 
216000000 



Sqn«re Reot.^ Cube Uuoi. 



iZ., 



m^lmtm 



'23-4733892 
23-4946802 
i>3-5 159520 
23-5372046 
2D-558433a 
23-571:6322 

i!3 -0006 174 
230220^36 
'23-6431 b08 
23-6643191 
23-6S54386 
23-70J5392 
237276210 
237436842 
2376J7286 

23-790754.5 

23-8117618 
23-8327506 

23-8537209 
23-8746/28 
23-8956063 
23-9165215 
23-9374184 

£3-9582971 

23-9791576 

24-0000'JOO 
24-02)6243 
24^16306 
24-062>18:i 
24-0831892 
24-1039416 
21- 1*^46762 

24' 1453:^29 
24-y>>0t)19 
24-1807732 
24-2074369 
24-22S0829 
24-2487113 
24-i6932ii 
24-2899156 
24-3104916 
24-^310301 
24-35 i5Dl 3 
24-37:»U52 
24*39a6:^id 
24-4)31112 
24-4335834 
24-4540385 
24-47447fi5 



a, t 



8-198175 

H8-2^3131 
8-208082 

S-2IJO27 

8-2179U5 

8-222898 

8-22/825 

8-232746 

8-237001 

8-242570 

8-2-17474 

8-252371 

8-2572d3 

8-262149 

8-267029 

8-271903 

8-276772 

8-281635 

8-28649$ 

8-291344 

8-296190 

8-301030 

8-305865 

8-3lO.>94 

8-315517 
8-320335 

8-SI25I47 
8-329:'54 : 
8-334755 
8-33955 1 
8-34434 1 
8-349 12» 
8-353901 
8-358678 
6*363446 
8-368209 
8-372966 
S"J777^S 
8*382465 
8*3d7206 
8-391942 
8-396673 
8-4013 v)8 
8-f406U8- 
8-^10832 
8-(^ 15541 
8*420245 
8-424944 
8-429638 
I II-434327 



/ 



102 



ARITHMETIC. 



).\ Square. | 



•m.!^ 



Square 6oot. . Cube KcMDit. 



i\umb 



Cube. 



. 



(101 

603 

(3CH 
605 
606 
607 
60S 

009 
610 

611 

612 

613 

I 6u 

615 

6l6 

617 

618 

'619, 

620 

mi 
' 6^2 
. 623 

624 

Cii7 

(52^ 

629 

6:jo 
;63f 

i^ 

6?i 

635 

630 

.Qiit 

64! 
642 
•643 
jB44'* 
7545 
646- 



361201 
362404 
363609 
364316 
366015 
36/236 
368449 
36^664 
370831 
372100 
373321 
374544 

378325 
379^56 
3S0639 
381924 
383161 
384400 
385641 

386884 
3881 29 

389376' 
390625 
391 876 ' 
393129 ; 

391384 

393C"^U 

306900'' 

3()3|6l , 

399424' 

400^89 

401956 

403225 

404406 

405769 

: 407044 

408321 
. 409600 
410881 
412164 : 
41344Q . 
414736^ 
416025* . 
417^16*- 

4 J 9904 

42i2ai[ 



217081801 

21 8 167203 

219256227 
220348864 
221445125 
222543016 
223648543 
224755712 
225^66529 
22698 1 COO 
228'.)9913l 
229220928 

230346397 
231475544 
232608375 
233744896 
23458.5113 
236029032 
237 1 76(^59 
iJ3832S0 10 ' 
2394Q306J 
240641848 
241804367 
242970624 
244140625 
.245314376 
216191883 
'247073 1 .02 
2488581 89 
250047000 
251239591 
2524359(iS 

'? 53636! 37 
2546-10104 
2560478/5 
257559456 

.25847485^-." 
259694072 ' 
26091!^ J 19 
"262144000 
263374721 

'26460c528S 

26/5S47rq7 

' 26708^gir4 
2683361:25 
269586136 

'270fe-f0Qi3 

'272og7'T9'r 

.273359449 



24-5153013 
24-5356883 
21-5560583 
2i-5764115 
24-5967478 
24-61706731 
24.6373700 
24-65/6560 
24-6779254 
24-6981781 
24718^1142 
24-7386338 
247588368 
24-7790234 
247991935 
24-3193473 
24-8394847 

24 8596053 
24-8797106 

24-8997992 
24-9193716 
24-9399^78 

24-9599679 

24"97P9!)20 

25hob()boo 
25019992b 

25039968 1 

25-05'99282 
25*0798724 
25-0998008 
2.5-1197134 
25- 1396102 

25 15949 13 
25-1793566 
25-l<i9i063 
25-2190404 
25'23885b9 
25-2586619 
25-2784493 
25-2982213 
25-3179778 
25-3377189 
25-35/4447 
25-3771551 
25-3^K)8502 
25-416530! 

"25*4361947 
25-4558441 
25-4754784 
25:4950076 



- 



8-439009 
8-443687 
8-448360 
'8-453027 
8-457689 
8-462347 

8-466999 
B'47Wi7 

8-476289 

8-4S0926 
6-485557 
8-490184 
8-494806 
8-499423 
6-504034 
8-508641 
8-5 13243 
8-517840 
8-522432 
8'5270!8 
8-5J16CO 
8-53f>177 
8-540749 
8-545317 

8-549879 
8-5*4437 

8-558990 

8-563*537 
&-5660S0 
8'5726l8 
8-577152 
8*58 168O 
8-586204 
8-590723 
8-595238 

8*599747 
8-6()4252 
8 608752 
8-613248 
8-617738 
8-622224 
8-626706 
8-631183 
8-635655 
8-640122 
S-644585 
S-649Q43 
8653497 
6-657(^6 
8-662301 



SQUARES, CUB£6, AMD ROOTS. lOS 



: NuQib. 


Square. 


iCiibe, 


Square Ro^t,. 


Cub« ituol 


051 ^ 


423601 


275894451 


25-5147016 


8*666831 


()52 


425104 ^ 


277167BOS 


•25-5342()OX 


8*671266 


65^ 


426409 


27S4i5077 


25-a53il?47 


8-675697 


654 


427716. 


2797^626* 


25^7:^4137 . 


8-68W23 


655 


429025 


281011375 / 


25Jj9i9678 . 


8-684545 


656 


430336 


232300 H 6 


25-6124969/' 


«;6t^63 


657 


431649 


2835^33 ^ 


25-6310112. 


8^693976 


65S 


432:^64 


28489C»3I2 


»5-65i510;e 


8^697784 


659 


43t<i8l 


286191179 


^5'67(m^ 


8-702183 


660 


435600 


287^99000 . 


25-6^)54652 v 


8-7o6iS7 


661 


436921 


2dd80478b 


257099303 


s-7iO$«2 


662 


438244 


29J1 175211 


25-72oa!eto/ 


8-7Ui73 


.663 


439569 


29143i247 


25-7487864 


87l9r5.7 


664 


44O896 


292754444 


2^763195^^" 


8724111 


66 J 


442i25 


294C79;yi5 


25-7375939 


8-726518 


666 

m 


443556 


2954O8296 


25'80ai9753 


8-734891 


607 


444889 ' 


296740963. 


Z6'B2m4M'^ 


8-7372f;0 


ttoa 


446^2* 


2^j8Q77632 


25-8156960 


87446i4 


6og 


447561 


2994i.8309 


^ 2.5-86.«143 


874^9^4 


670 


4489CO 


3007(J3COO 


25-tB84))A82 


8-750340 


671 


450241 


3021 117n 


25-^036677* 


S754691 


^;2 


451564 


303464448 


25 922962^^^ 


8-754^0.i8 


673 


452929 


304821217 


25-9422435 : 


8763380 


674 


^54276 


306182024 


25-96150100 


8767719 


675 


455625 


307546875 


25-pg0762» 


8772053 


676 


456976 


3O89 15 776 


26-00 JO?O0 


8-7763.9'^ 


677 


458329 


3I0288733\ 


26019^2:57 


8780708 


673 


459684 


3 U 665752 


26*03 S4 331 


8785029 


(>79 


461041 


313046839 


20-0576284 


8-789346 


(580 


462400 


314432000 


26-0768096 


S7.c;36:59 


681 


463761 


315821241 , 

• 


26'0gr>ij767 


^'7979^i7 


082 


465124 


317214568 


26-115127 


8v^0i272 


6S3 


466489 


3Ifc?6n987 


25-1342687 


8-806572 


C'84 


467856 


320013534 


26-1533937 


8«8 108018 


6S5 


469225 


3il4l9l25 


26-1725047 


8-815159 


6q6 


470596 


32282S856 


26-1916017 


8-8 HM-^? 


6S7 


471969 


324242703 


26-21(K5S48 


8 823730 


688 


473344 


3 i56bo67a 


26:229754 1 


8-82^00j 


689 


474721 


327082769 , 


2J'248S095 


8*83 Z2S5 


690 


476100 


32850i)(J05 


a6- 26785 11 


8*836556 


091 


47748 1 


32^)93937 1 


26-2868789 


8-840822 


6ij2 


47S864 


331373888 


26-ji058929 


8-845085 


693 


480249 


332912557 


26-324.8932 


8*849344 


6(;4 


481636 


3:H25538> 


26-3438'797 


8-8583(^8 


O95 


483025 


33570237^ 


26-062 9 '>*27 


8-857849 


096. 


484446 - 


-337153336 


26-JJ8lkJH9 


8*86i695 


^97. 


48580) 


3^8J68872| 


26'hb07S76 


8*866337 


698/ 


: 457204 


. •34CXX)J?392 


26419)896 


b' 8/0^ 75 


^ 


.: 488601 


^156^099 


. 20>3tJ«fiO81 


8^87480,) 


70J 


490000'^ 


choooooo 


^6-iJ576131 


K'STiK)40 



10* 



AIUTHMETIC 



Numb. 


$qua». 


Cube. 


Square Root. Cube-Rodt. 


701 


49}iai . 


344472101 


26-4704O46 


8-8832116 


702 


492804 


34594SOO8 


26-4942826 


8*887438 , 


703 


494209 


347428927 


26*5141472 


,8-8ftl706 


7<H 


49^16 


848913664 


26*5329981 


8*895920 


705 


4970^ 


350402625 


2&5518361 


8*9001 30 


706 


493436 


351895816 


26*5706605 


8*904336 


707 


49.)849 


353393243 


26*5894716 


8-908538 


7O8 


50l2(S4 


354B94912 


26-6082694 


8-9l27|5* 
8-9i<»fl 


7<» 


M20SI 


S5640082^ 


26-6270539 


7iQ 


A04100 


35791 lOOQ 


26-6458252 


8-921121 


7U 


50S52I 


S59425431 


26-6645833 


8-925307 


7U 


50r)944 


360944128 


26-6833281 


8-929490 


' 7«3 


5033(!^ ^ 


862467097 


267020598 


8-Q33668 


7W 


3q979<> 


3*63994^4 


267207784 


8-937843 


.7*5 


511225 


365525675 


26-73©4S39 


8-942014 


7i« 


512^56 


367O61696 


26-7581763 


8-9 t6l 80 


717 


514089 


36bd0l8i3 


'iQ'77^bli7 


8-^50343 


7iS 


515524 


370146232 


267955210 


8-954502 


' 7W 


5IQ96I 


37i^94»« 


26-8141754 


8-958658 


•720 


5 J 8400 


373248000 


26-83^ 157' 


8-962809 


T^l 


5h9841 


3748C536i 


26*6514432 


QV^957 


722 


532284 


370367048 


Z6-8700577 


8-97110O 


723 


52^729 


377933067 


2d-«8S6593 


*»975240 


724 


524i7t> 


379^03424 


20-9072*81 


8-979^7^ 


725 


5^^(>25 
5j27076 


381078125 


2C)-92.56240 


8-933^08 


726 


382657 « 76 


26-f4438/2 


8-C;87<>37 


1%7 


5'2S5'H| 


3S4240583 


26'9629.75 


8-9917^2 


728 


52(}9ci4 


385828352 


26'9a 14751 


8'(;9588a 


729 


.531441 


367420189 


27-poooo :o 


9*000000 


730 


532^00 


3P9O 17000 ' 


27-0185122 


9»004113 


731 


534361. 


3ii0'il789l 


270370117 


9-00S222 . 


732 


535324 . 


3f;2223l63 


270554935 


9»0 12328 


733. 


53. 289 


393H32837 


27-0739727 


9-016430 


734 


53875<J 


3 J544d9D4 


27-09^43+4 


y020529 


735 


540225 


397<^'6.0375 


27' 1108834 


9-021.62^ 


7.'« 


541 696 


396688256 


271293199 


9-028714 


7'67 


543I()9 


400315553 


271477439 


9-032802 


738 


544f>14 


4019172/2 


27' 166 1^)54 


9*036385 


739 


546121 


403563419 


27-i 845544. 


9-040(:65 


740 


547600 


40522^1000 


27-20.94IO 


9*045041 


741 


5^19081 . 


46o8()'9021 


27-2213152 


9-0491 U 


742 


6bo:^<i^ 


4G85l84»r^ 


272396769 


9-053 1 83 


743 


552049 


4^0172407 


27-2580265 


9-05;2i8 


744 


553536 


4U&307S4 


27(2763634 


9*O0*J309 


745 


556025. 


. 41349^625 


27-}J946881 > 


9*065367 


746 


5A6516 • 


415160936 . 


27- 


) 130006 


9:069422 


7'^7 


^^W^^ 


416832723 


27- 


3313007 ^ 


$^073^72 


748 


5S9504 


418508992 
^30189749 


27&495887 


9^7519 


7-^0 


501091" , 


27-3678644 i 


9'081503 


75C> i 


t 562.W0 . 


4U1875O0O i 


-27-386127^ 


9-O856C0 



/ 



SQUARES, CUBES, and ROOTS. 



105 



Numb. 



7^1 

753 

75* 

7^7 
758 

769 

700 

701 
7d-i 
7tf3 
7W 
765 
766 

767 

76S 

769 
770 

771 
771 
77^ 
77^ 
775 
776 
777 
778 
7» 

780 
781 

7Q3 

784 
785 

786 
787 

788 
789 
790 

793 
.794 

795 

797 
798 
799 

800 



Square V' Ctibev 



3t>400X 
^03^04 
567009 
5685 \6 
570025 
57i539 
573049 

B745(k 
57Cf081 

V760O 
579121 
5S0044 
58'ilfi^ 
5SJ60i 
5833Ji5 
48^5(3 
588289 
589824 

5iniaoi 

59?900 
594441 

•95984 
597529 

^99^^ 

^100625' 

6m\76 
6o37'J9 

lk)5284 
600841 
608400 

09961 
611524 
613089 
614656 
616225 
617796 
619^69 
620944 
622521 
$24100 
6256S1 
627264 
628849 

. 63Q436 
632a25 
633616 
6352Q9 
636804 

^ 638r*01 
640000 



. 



423564/5 1 
425259008 

426957777 
428661064 

430868875 
432O8I216 
433798O93 
4355l9idl2 

437245479 
43897CiOOO 

44071108.1 
44245P72S 

444194947 
445943744 
447697 i 25 
449455096 
43121/663 
452984632 
454756609 
456533000 
45831401,1 
460099648 

4618699*7 
463684824 

465484375 

467288576 

469097433 

470910952 

472729*39 
474552000 

476*379541 
478211768 
480048i)b7 
43l890:iO'i 
48:17360255 
485587656 
487443403 
489303872 

49n69069 
493039000 
494913671 
496793088 

498677257 

$00566184. 

$02459«^75 

$0^3^336 

506261573 

50^16^592 

5IJDOV2399 
5 li>000000 



.- • 



Squa^ Root. | Cube Roou ^ 

9-069639 
9*093672 
9-097701 
9-101726 
9-105748 
9-10^766 
9'l.J37«l 
9*117793 
9*121801 
9.125895 
9-129806 

9ia38^ 

9-137/97 
9' 14.1 788 
9-145774 

9149757 
9-153737 
9-157743 
9-161686 
9-105656 
'9-169622 
y 173585 
9:177344 

9-181500 
9185452 
9-189401 

9- i 93347 
9-197^^9 

9*20i228 
9'i05l64 

9'20CK)96 

9*213(/25 

9-216950 

9-2Z0S72 

9-224791 

9-J2fel706 
9-432618 

9*237527 
9-240433 

9-2443ii5 
9-248234 
9-252130 
9-2o6J22 

9"25<;91 1 

9-2^3797 

9-207679 

9-27 155() 

9-275435 

9'279^0& 

0-283177 



27*404379'* 

274226184 

27*4408455 

27*4590604^ 

27*4772633 ' 

27*4954542 

27-5 1 36330 

275317998 

27-5499546 

27-5680975 

27-5862284 

27*6043475 

27'6224546 

27-6i05499 

27*6586334 

27-6767050 

27-6947648 

27712i*129 
2773O8492 

27-7488739 
27-7C68b68 

>277B4B880 

27-b028775 

27-8203555. 

27-S3i!i82i8 

27'S567766 

27^8747197 
27-8926514 

27-9105715 

27-9284801 

27-9463772 

27-96426/9 

27-9821372 

28-0000000 

280178515 

28-0356915 

28-053j'iO3 

28-07133/7 

28-0891438 

28-lo:i9386 

28-1247222 
28-1424946 
28- 160 J657 
28'17800?li 
28-1957444 
28-2m72Q 
28-2311 8 i4 
23-:j4 83938 

:48*:t66588i 

28-28427 1 2 



106 



arithmetk:. 



Nnmb. i Square.*- Cube. | Square Root. 



l 



•eoi 

602 
803 

eo4 
eo5 

800 

807 
60S 
8C9 

810 

811 
6i2 
813 
814 
$15 
BW 
817 
6)8 

Big 
8;30 

821 
822 
823 
$24 
825 
826 

627 
828 
839 
830 
8^1 
832 
833 
834 
835 
836 
837 

638 
839 

840 
841 
842 
843 

844 
845 
846 
847 
848 
849 

850 

'-■ 



641601 
(;43204 
64^809 

^464 16- 

648025 

6\()()36 

651249 

652864 

654481 

65&]0C> 

657721 

659344 

66O969 

662556 

664225 

665B56 

m4BQ 

669124 

670761 

672400 

674041 

675684 

677329 

P78976 

68062^5 

^82276 

6b3929 
68£«5S4 
687241 
688CjO0 

6go56i 
692224 
693689 

()i)5S56 
697225 
698896 

7oa569 

702244 

703921 
705(iOO 
70/281 
70896^4 
71O649 
712336 
714025 

7i74Cg 

719\04 
72OSO1 
■722500 



513922401 
515849^08 

5(77Sl<^'i7 
6J9718464 
52\6G0125 
523606616 
5255579-<3 
627514112 
529475129 
53I4410CO 

533411731 
535367328 

537^0^797 

539353144 

541343375 

543338496 

545336513 

547343432 

549353259 

551368000 

553387661 

555412248 

557441707 

559476224 

561515625 

56S55gg7Q 

565609283 
667tG3552 
5097227^9 
57 I787COO 
573856191 
' 57593036S 
576OO9537 
'580093704 
582182875 
584277056 
586376253 
5Se4fe0472 

590589719 
592704000 

594823321 

5960,47688 

599077107 
6(XJ 21 1,584 
60535 1 J 25 
^0549573^ 
60f 04 54 23 
609tOOI92 
611960049 
6I41250CO 



28-3019*34 
28-3100045 
26-^^7^546 
26-35489^8 
28-3/25219 
28*6i,01 39 1 
28-4077454 
28''4253408 
28-k4i9253 
•28-40O49B9 

26*4; 8061 7 
28-4956137 
28*;5U>549 
28*t$306852 
28')54 83048 

26-5657137 
28-5832119 
28-t0069:^3 
^8'6l8]760 
28-6356421 
28-P53Q976 
26-6705424 
«8-e879766 
28-^054002 
28-7228132 
28-7402157 
28-^5?6077 
C8-774&891 
28-7923001 
28-6037106 

28-82;o706 

28-8444102 

28-8617394 
28-8/90582 
28-8963666 
28-9136646 
28-<)i09523 
28-fi48i2(,»7 
28-|654(,67 
28-0827535 
Q9h0QCOO0 
290172363 
29-*344623 
29-*5 16?61 
29-4688837 
•V9-06()C'79l 
2.g- 1032644 
20- 12043 96- 
'29J370C46 
♦ S?9l5475(;5 



Cube Book! 

9-2^7044 

9'2gOL}07 
9^'.(|4767 

9-2i;b0\!3 

9-30i477 

9^306327 
9-310175 

9'3 14019 

9-317859 
9'32l697 

9-325532 
9*329363 

9*333191 

9-337016^ 
9-340838 
9-3446^7 
9-348473 
9'35!2285 
9-356095 
9*359901 
9{363704 
* 9*367505 
9»371302 
9*375096 
g-378887 
9-362675 
9*366400 
,9-390241 
9*3^«20 

y-.97796 

9-401 569 
9-405338 
9-409105 

9-412S69 
9-416630 

9-420387 
t;-424l4l 
9-427893 
(}'43l643 
9*435388 
9*439130 
9-44!i870 
9-4 4 6607 
9-4.C0341 
9^*540/ 1 

9';i57799 
9-461524 

9-J4 65247 
gH46896to^ 

0-472682 



SQUARES, CUBES, and ROOTS. lOT 



Namb. 

851 

852 

853 

854 

855 

856 

8S7 

858 

859 

860 

86l 

SdZ 

SO'3 

8C)4 

8b'5 

SQd I 

867 

868 
8(X) 

87b 

S7J 

872 
873 
874 
875 
87a 

877 
87s 

«79 

880 
881 
882 

883 

884, 

385 

835' 

887 

888 

889 

890 

891 

8ij2 

893 

894 

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896 

8fl7 
.8!«8 

.mo 



Square. 
7^^4201 

727609 

729316 
731025 
732736 
734449 

7361 61 

7:^7881 
739600 

7'<I32l 

7430 N 

744769 

746496 

748225 

749956 

751089 

753424 

755I6I 

756900 

75864 1 

760384 

762 129 

763376 

765()25 

767376 

769129 

770884 

77264 1 

774400 

776161 

777t;'24 

779689 

781456 
783225 
784996 

786769 
788544 
790321 
792100 
793881 
7956(:4 

?97449 
79^6 

801025 
80^816 
QO4606 
806404 
808201 
:^10QPO 



Cul?e. 



6l6i05O)l 
6184/0208 
620650-177 
622835864 
625026375 
627222016 

629422793 
631628712 

633839779 

636056000 
638277381 
640503928 

6*2735647 
4)44972544 
647214625 
64946I896 
651714363 
653972032 
656234909 
658503000 
6607763 1 1 
663054848 

665338617 
66762/624 
66992 I 875 
672221376 
674526133 
676836152 
679151439 
08 1472000 
Ci8i797S4l 
686»289<>8 

688465387 

69O6O7104 

693154125 

6i:5506456' 

69786410 J 

700227072 

702595369 

704969006 

707'M797l 
709732288 

712121957 

7if^5r6984 

7i6c)i7375 

7103231^6 

7217^4273 

72*150792 
72^572699 

7290000 JO 



Square Root. 



29-1719043 
29-18903f)0 
29*2061 637 
29-2232784 
2'^-2403830 

29'7'^74777 
292745623 
29^291^370 
29308701 8 
29-325/566 
29*342801 5 
293598365 
29*3768616 
29-39:^8769 
29' 4 108823 

29*4278779 
29*4148637 

29-4618397 
29-4788059 
29-4957624 
29-5127091 
29*5296461 
29-5465734 
29*5634910 
29*5803989 

29*5972972 
29-6141858 
29*6310648 
29-6479325 

29*6647939 
29-08,l6442 
2^-6984848 
29*7153 150 
'29-7321375 
29*7**8(?496 
29*7^57521 
29*7825452 
29-7993289 
29*8»U1030 
29'8328678 
29-8496231 
29-8663690 
29-8831056 
29-899«328 
29*9165506 
299332591 

29-949958:^ 
29-9366481 

29*933^287 
'30-0000000 



Cube Hoot. 






9-476395 
9*180106 
9-483 3 13 
9487518 

9-4'9rii9 
9*491918 
9-498614 

9-502307 
9-505998 
9*50C}()&5 
9*513369 
9*517051 
9*520/30 
9-524406 

9-528079 
9*53 1 749 
9-535417 
9-5.9081 
9-542743 
9-516402 
9-550058 
9*553712 
9-557363 
9-561010 
956^655 
9'56S297 

9-^719^7 
907^574 
9-^79208 
9-582839 
9*585468 
9-590093 
9593716 
9-597337 
9-600954 
9604569 
9-6081 81 

9-611791 
9-615397 
9-619001 
9-622t>03 
9-626201 

9-^X9797 
9-633390 
9-636981 
9-640569 
9-644154 
9-647736 
9-651316 
9-654893 



109 



ARITHMETIC. 



• 




* 






"Nucijir. 


Square. 


Cube. 


Square ilobt. 
300 J 66620 


Cube Root^ 


goi ' 


811801 


731432/01^ 


9-658468 


()C2 


813^04 


733870»08 


30-0333148 


^•662040 


903 


815409 


736314327 


;}0 0499584 


9-665C09 


904 


8I7216 


738763264 


3O'O605f>28 


9'^\7yy 


905 


8 19025 


741217625 


300832179' 


9-672740 


906 


620836 


743677416 


30-099^339 


9-676301 


907 


822049 


746142643 


30 1164407 


9-679860 


DOS. 


824464 


748013312 


301330383 


9-683416 


909 


826281 


751089429 


30-1496269 


9-686970 


910 


828100 


753571000 


30-1632063 


9-690521 


9ii 


S2992I 


756058031 


30-1827765 


9'6940t)9 


gn 


631744 


758550528 


301993377 


9-697615 


913 


S33569 


7610.8497 


30 2158899 


9-701158 


914 


835396 


763551944 ' 


35-2324329 


97^^8 


915 


83/225 


7660(X)875 


30-2489C69 


' 9-703236 


9ie> 


839056 


7^5Tb'2^ 


30-2654919 


9711772 


917 


840889 


771095213 


30-2820079 


9-7 1 5305 


918 


842724 


77362O032 


30-2985148 


9-718835 


y»9. 


8445^1 


;^76151559 


30-3150128 


9-722363 


920 


846400 


7786860C'0 


30-3315018 


9-'725888 


921 


84824 1 


731229,61 


30-34798 1 8 


9729410 


92i 


860084 


78377/448 


303644529 


9732930 


923 


85 1929 


. 7S6330467 


30-3809151 


9-736448 


924 


. 853/76 


786S89024 


30-3973^:^83 


973c9i3 


925 


855625 


791453125 


30-4138127 


9743475 


92t) 


85/476 


79402277s 


30-4302481 


9-746965 


' 927 


859329 


79^597983 


30-44^6747 


9-750493 


923 


861184 


, 799173752 


30-4630924 


9753998 


. ^29 


863041 


80I/650S9 


30-4795013 


9757500 


930 


864900 


804357000 


30'49590M 


9-761000 


931 


866/61. 


806954491 


30-5122926 


9704497 ; 


932 


- 868624 


809^75 j3 


30-5286750 


97S7992 


933 ^ 


870^89 


8 12166237' 


30-5450487 


9771484 


934' 


872:56 


814780504 


30-5614136 


9-774974 


935^" 


8/4225 


817400375 


30-5777(>97 


977846 J 


. 93^' - 


876oi>(i 


820025856 


30-5941171 


978i9-^6 


i)37 


^779^9 . 


822656953 


30-6104557 


9*7854 2.S 


938 


87y844 


825293*672 


30-6267857 


9788903 


. 939 . 


8S1721 


82793COI9 


30-6431069 


9792336 


• 940 


883 coo 


8305^4000 

1 


30-6594194 


9795361 


941 


8S5481 


83323/621 ; 


, 30-6757233 


979.^33 


• 942 


887304 


83^896888 


30-6920 1 85 


9-802303 


1 943 


8S9249 


838561807 


30-7083051 


' 9-^0027 1 


, 944 


S9II36 


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SQUARES, CUBES, and ROOTS. 109 



- 


Nurob. 


^ Square. 


. Cube. 


Square Root 


Cube KtM)t 
<r83392:i 




y5l 


9044m 


86C0S535 I 


30-S3S82S79 




. y«2 


90Q304 ■ 


802801408 


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953 


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(^*H0812 




(,54 


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958 


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959 


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30-96'7725 1 


9*861431 




960 


x^2 1 boo 


384736000 ^ 


' 30*9838668 


9*b6: 848 




9'n 


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\ 


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> 


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9 



110 ARITHMETIC. 



Of ratios, proportions, and PROGRESSIOKS- 

Numbers arc compared to each other in two difFerent 
>yay$: the one comparison considers the difference of the two 
numbers, and is named Arithmetical Relation ; and the dif-* 
ference sometimes the Arithmetical Ratio 2 the other consi- 
ders their quotient, which is called Geometrical Relation ^ 
and the quotient is the Geometricat Ratio. So, of these two 
numbers 6 and 3, the difference, or arithmetical ratio> is 
6 — S or 3, but the geometrical ratio is y or 2. 

There must be two numbers to form a comparison : the 
number which is compared, being placed first, is called the 
Antecedent ; and that to which it is compared} the Conse^ 
quenl. So, in the two numbers above, 6 is the antecedentf 
and 3 the consequent. 

If two or more couplets of numbers have equal ratios, or 
equal differences, the equality is named Proportion, and the 
terms- of the ratios Proportionals. So, the two couplets, 4*, 2 
and 8, 6,^ are arithmetical proportionals/ because 4 — 2 = 8 
-^ 6 ;= 2} and the two couplets 4, 2 and 6, 8, are geometri- 
cal proportionals, because 4 ^ 4 ^ ^> ^^^ same ratio. 

To denote numbers as being geometrically proportional a 
colon is set between the terms of och couplet, to denote' their 
ratio; and a double colon, or else a'mark of equality, betweeri 
the couplets or ratios. So, the four proportionals, 4, 2, 6, 3 
&re set thus, 4 : 2 : : 6 : 3, uhicli, means, that 4 is to 2 as 6 
is to 3 ; or thns 4 : 2 3= 6 : 3, or thus, 4 = yy both 
which mean, that the ratio, of 4 to 2, is equal to the ratio 
of 6 to 3. . 

Proportion is tUstinguished into Continued and Disconti-* 
nued. When ' the difference or ratio of the consequent of 
one couplet, and the antecedent of (he ndxt couplet, is not the 
' same as the common difference or ratio of the coufplets, the 
proportion is discontinued. 5o, 4, 2, ?, 6 are in discontinued 
arithmetical proportion^ because. 4 — 2=fc8 — 6 =2, where- 
as 8—2 = €; and'4, 2, 6, 3 are in. discontinued geometrical 
proportion, because 4 « |. a 2,«btit f =' 3, which is not 
the same. ' * 

But when tlie difference or ratio ,qf every two succeeding 
terms is the same quantity, the proportioh i§ &ai4 to be Conti- 
nued, and the num^jers thenwelyes make a seriets p/Cpntinued 

. '• \ . ;, /i?jc6pQrtionals, 



ARITHMETICAL PROieORTION. ill 

Proportionalsr oc a progressiox^t So 2, 4» 6, 3 form an arith* 
metical progression, because 4 — 2 = 6—4 = 8—6 = 2, all 
the same common difierencc ; and 2, 4, 8, i6 a geometrical 
progression, because $ = J = y^ r= 2, all the. same ratio. 

When the following terms of a pro^rression increase, oy 
exceed each other, it is called an Ascending Progression, ot 
Series ; but when the terms decrease, it is a descfjhding one. 

Ko, 0, 1, 2, 3, 4, &c. is ^ja ascending arithmetical progisession^ 
but 9, 7, 5, 3, 1, &c. is a descending arithmetical progression. 
Also 1 , 2, 4, 8, 1 6, &c. is an ascending geometrical prop-ession, 
and 16, 8, 4, 2j^ 1, $cc* is a de^^ding geometrical profession. 



,. ■.. i^ I i gStt 



ARITHMETICAL PROPORTJONj^iPROGRESSlON. 

In Arithmetical P/ogressionj^ the niimbers or terms hate all 
the same cbmmon difference. . Also, the first and last terms 
of a Progression, are called tlieTIxtremes j ai;id the? otter 
terms, lying between them, the Means. The mose useful 
part of arithmetical proportions, is tonttiined in the follow* 
ing theorems : , » • . ,. . '- 

TilUoREM 1. When four quahtities are in arithmetical 
proportion, the suaa^of the two ^tremes is equal to the dum 
of the two means. Thus, of the four % 4, d, 8, here 2 ^ 
8=4 + 6 = iOi 



* f 



Thkokesvt 2. In any continued arithmetical progression, 
the som of the two exti%mes is equal to the sum of any two / 
means that are equally distant from them, or tqual to double 
the middle term when there is an uneven number of terms* 
Thps, in the terms I, 3, 5, it is 1 + 5 = S + 3 ;= 6., 
And in the series 2, 4, 6, 8, 10, 12, 14, it is 2 + 14 = 4 
+ 12 = 6+10 = 8 + 8=16. 

TtoEoKiSM 3. Th^ difference between the extreme^ terms 
p£ an arithmetical progression, is equal to the common dif- 
ference of the series multiplied by one less than the nuipber 
of the terms. So, of the ten terms, 2, 4, 6, 8, 10, 12, 14, 
J 6, IS, 20, tbet <fom»dn differtoce is 2, and one less thaa 
tho number of terms 9 j then.tli^ifferenceof th^ extremes 
is 20-2 = 18, and 2 X 9 = IS also. , '. i. 

Consequently, 



112 AftltHiMETlC 

Consequently tW greatest ttrm is <^at t6 the least term 
^dded to the product of the eommon 4»ereiiee multiplied hf 
'1 less than the humber of terms. 

Theorem 4. The sum of all the terms, of any arithiue-' 
♦ical progression, is equal to tht sum of the two eiCtreines mul- 
tiplied by the* number of terms, and divided by 2; or the sum 
of the two iBxtrcmes multiplied by the number of the tcrms^ 
gives double the sum of all the termt in the series. ' 

This is made evident by setting the termi of the series in 
tin inverted order, under the same series in a direct order, and 
lidding the corresponding termi together in tliat order. Thus^ 
in the series I, 3, 5, 7, 9, n, l^t 15 1 
ditto inverted 15, 13, !!» 9> % 5, ^8, 1 ; 

the sums are 16+ 16 + 16 + 16+ 16 + 16 + J6 + 16, 
which must be double the sum of the single series, and is 
equal to the sum of the extremes repeated as often as are the 
number of the terms. * ■ 

From these theoreftis may reacfilf be fcund any one of 
these Qye parts ^ the two exlremos, the number of terms, th^ 
common difference, and the ium of all the ternis^ when any 
three of them are given ; is in the following problems : 

PROBLEM I. 

Xiivmihe ExtfrmeSf antlthe Number ofTemi$t to Jmdthe Suth 

of^dl the Terms ^ 

Add the extremes together, midtipty the sum by the num^ 
ber of terms, and divide by 2. . ' 

EXAJ^WS. 

1. The extremes being .3 and Wt and the nuic^wr of 
teriQs 9 \ required the sum of the terms ? 

19 ' ^ 



22 . 

Or, *— -; — X 9 = ~ X 9 = 1 1 )c 9 = 99, 



2 ) 198 ** 1 * 

' , ^ the same answti\ 

Ans. 99 

- 2. It Is reqiwred t© find the number of all the strokes a 
common clock strikes in ^ifie whole irtvohition of the ind^x, 
or in W hours? Ans. 78v 



1) 



ARITHIMETICAL PROGRESSION. US 

( 

lEx. S, How m^ny strokes do the clock* of Venice strike 
in the compass of the dzj, which go continually on from I 
to 24 o'clock ? . Ans. 300* 

4. What debt can be discharged in a^ ye^, by weekly 
payments in soithmetical progression, the first payment 
being 1/^ and the last or 52d payment 51 Si/ AnSj 135/ 4/. 

'^ 

PROBLEM It. 

Given the Extremes y and the Number <kf Terms ; to jind the 

Common Difference, 

SuBTjiACT the less extreme from the greater, and divide 
the remainder by I less than the number of terms; for th< 
i^ommoh difference. 

£XAMPL£2(. 

• ( 

1. The extremes bein^ 3 and 19, and the number of terms 
; required the tomimon difference ? 

19 
3 ^ 19- 3 16 , 

^ 9-1 8 



8) 16 
Ans. 2 



• / 



2. If the extremes be 10 and 10, and the number of terftii 
t\ \ what is the. common difierence, and the sum of the 
series ? Ans. the com. diff. is 3, and the sum is 840. 

3, A* certain debt can be discharged in one year, by weekly 
payments in arithmetical progression, the first payment being 
1 J, and the last 5/ 3/ i what is the cdmmcin difference of the 
terms ? Ans« 2i 

PROBLEM III. ' 

Xjiven one o£ the Extremes^ the Common Difference^ and the 
Number of Terms : to find the other Extreme .^ and the Sum of 
the Series. 

Multiply the common difference by 1 less thati the num- 
ber of teriiis, and the product will be the difference of the 
fextremes i Therefore add the produtt to the less extreme, to 
give the greater ; ot subtract it from the greater, to give the 
less extreme* 

Vol. I. I EXAMPLES. 



I 



114' AfttTHMETlC 

EXAMPLES. 

1. Giv^n the fcasf term 3, the connnoit difference ^, of an 
arithmetical series of 9 terms ^ to finxl the greatest term, and 
the sam of the series: 

8 



16 



19 the greatest tertii 
3 the least 



22 sum 
9 number of terms. 



2 ) 198 



99 the sum of the series. 



.2. If the greatest term be 70, the common difference if 
and the number of terms 21, what is the least term, ai\d the 
sum of ^he series ? 

Ans. The least term is 10, and the sum is 840. 

3. A debt can be discharged in a year, by paying I shilling 
the first week, 3 shillings the second, and so on, always 2 . 
shillings more every week ; what is the debt, and what will 
the last payment be ? 
Ans. The last payment will be 5/ 3/, and the debt is 135/ 45. 

PROBLEM IV. 

* 
To find an Arithmetical Mean Proportional between Two Givetr 

Terms, 

Add the two given extremes or term* together, and take 
half their sum for the arithmetical mean required. 

EXAMPLE. 

To find an arithmetical mean between the two numbers 4 

and 14. TT 

Here 

14 

4 



IT) 1» 
Ans. 9 the' mean required. 



;>v^ P160BLE5* 



ARITHMETICAL PROGRESSION. 115 

PROBLik V. 

to find Two Arithmetical Means between two Given Extremes. 

Subtract the less extreme from the greater, and divide 
the dijflference by % so will the quotient be the common dif- 
ference ; which being continually added to the less extreme^ 
or taken from the greater, gives the means. 

EXAMPLE. 

To find two arithmetical means between 2 and 8. 

Here 8 . 

2 . 



3)6 Then 24-2 = 4 the one mean; 
and 4 + 2 = 6 th« other mean; 



com. dif. 2 



PROBLEM Vl. 

t$ find any Number of Arithmetical Means between Tw Given, 

Terms or Extremes. 

Subtract the less extreme from the greater, and divide 
the difference by 1 more than the ntlmber of means required 
to be found, wblch will give- the common difference ; then 
this being added continually to the least term, or subtracted 
from the greatest, will give the mean terms required. 

EXAMPLE. 

To find five arithmeticarmeans between 2 and 14. 

Here 14 . 

2 



6)12 Then by adding this com. dif. continually, 

the means, are found 4, 6, 8, 10, 12. 



^dn. dif. 2 



See more of Arithmetical progfesslbn in. the Algebra. 

I 2 GFOMETRICAL 



116- ARITHMETIC. 



I 
/ 



GEOMETRICAL PROPORTION *^ PROGRESSION, 

In Geometrical Progression the numbers or terms have 
all the same multiplier or divisor. The mpst useful part of 
G^metrical Proportions is contained m the following^ 
theorems* 



Theorem 1. ,When four quantities are in geometrical 
proportion, the product of the two extremes is equal to the 
product of the two means. 

. Thus, in the four 2, 4, 3, 6, it is 2 x 6 = 3,x 4 = 12. 

And hence, if the product of the two means be divided by 
one of the extremes, the quotient will give the other extreme. 
So, of the above numbers, the prodirct of the. means 12 -f- 3 
= 6 the one extreme, and 12 -r- 6 =2 the other extreme \ 
and this is the foundation and reason of the practice m the 
Rule of Three. 



Theorem 2. In any continued geometrical progression, 
the product of the two extremes is equal to the product of 
any two means that are equally distant from them, or equal 
to the square of the mididle term when there is an uneven 
number of terms. 

■ 

Thus, in the terms 2, 4, 8, it is 2 x 8 =c 4 x 4 = 16. 

And in the series 2, 4, 8, 16, 32, €4, 128, 

it is 2 X 128 = 4 X 64 = 8 X 32 = 16 X 16 = 256* 



Theorem 3. The quotient of the extreme terms of si 
geometrical progression, is eqtial to the common ratio of the 
series raised to the power denoted by 1 less than thi number 
of the terms. Consequently the greatesit term is equal tof 
the least terra multiplied by the said quotient. 

So, of the ten terms 2, 4, 8, \^^ 32, 64, 128, 256, 512j 
1024, the common ratio is 2, and one less than the numbef 
of ternis is 9 ; then the quotient of the extremes is 10^4 4- 
2 = 512, and 2^ = 512 also. 



Theorem 



• x: V 



GEOMETRICAL PROGRESSION. 117 

TThborem 4. The suih of all the terms, of any geome- 
trical progression, is found by adding the greatest term to th^ 
difference of the extremes divided by 1 less than the ratio. 

So, the sum of 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 

1024 — 2 
(whoseratiois2),isl024+-— — -— = 1024 + 1022 = 2046, 

jL — 1 

The foregoing, and several other properties of geometrical 
proportion, are demonstrated more at large in the Algebraic 
part of this work. A few examples may here be added of 
the theorems, just delivered^ with some problems concerning 
mean proportionala. 

EXAMPLES. 

!• The least of ten terms, in geometrical progression^ 
being 1, and the ratio 2 ; what is the greatest term, and the 
sum of all the terms ? 

Ans. The greatest term is 512, and the simi 1023. 

2. What debt may be discharged in a year, or 12 months, 
by paying 1/ the first month,' 2/ the second, 4/ the third, and 
V so on, each succeeding payment being double the last^ and 
what will the last payment be ? 

Ans. The debt 4095/, and the last payment 2048/. 

PROBLEM I. 

t 

V - '. 

To find One Geemetrical Mean Proportional between any Tnv$ 

Numbers, 

9 

MuLTiPXT the two numbers together, and extract the 
square root of the product, which will give the mean propor- 
tional sought. ^ 



EXAMPLE* 

To find a geometrical mean between the two numbers 
S and 12.' 

12 
3 

36 (6 the mean. 



36 PROBLEM 



lia ARITHMETIC. 



PROBLEM II. 

To findr Two Geometrical ]Hean Proportionals iet^uneen any Ttu§ 

Numbers, 

Divide the greater number b7 the less, and extract tho. 
cube root of the quotient, which will give the common ratio 
of the terms. Then multiply the least given term by th<^ 
ratio for the first mean, and this mean again by the ratio foi' 
the second mean : or,- divide the^ greater of th^ two given 
terms by the ratio for the greater mean, and divide this agaiA 
by the ratiq for the less mean. 

E3CAMPLE. 

' To find two geometrical means between 3 and 24f. 

Here 3 ) 24« (8; its cub^ root 2 is the ratio. 
Then 3 x 2 =: 6, and 6 x 2 =s 12, the two means. 
Or 2* -7- 2 = 12, and 12 — 2 = 6, the same. 
That is, the two means between 3 apd 24, are 6 and 12. 



PROBLEM III. 

s 

To find any Number of Geornetrical Mea^s between Two Numbers^ 

Divide the greater number by the less, and exta'act such 
root of the quotient whose index is 1 more than the number 
of means required ; that is, the 2d root for one mean, the 3d 
iTpot for two means, the 4th root for three means, and so on ; 
and that root will be the common ratio of all the terms. 
Then, with the ratio, multiply continually from the first term!.* 
or divide continually from the last or greatest term^ 

Example. 

To find four geometrical means between 3 and 96. 

Here 3 ) 96 ( 32 ; the 5th root of which is 2, the ratio. 
Then 3x2=6,&6x2 = 12, &12x2 = 24, &24x2=48. 
pr96-^2=48,&48-r-2=24,&24-T-2=:12, &a2-7-2=6. 
' That is, 6,12, 24, 4?8, are the four means between 3 and 96. 

Of 



MUSICAL PROPORTION. |l» 



Of MlTSICiVL PROPORTION. 



There is also a third kind of proportion, called Musical^ 
which being but of little or no common use^ a very short ac- 
count of it may here suffice. 

Musical Proportion is when, of three numbers, the first 
has the same proportion to the third, as the difference between 
the first and second, has to the diderence between the second 
and third. 

As in these three, 6, 8, 12 ; 

where 6 : 12 :: 8 — 6 : 12 - 8, 
that is 6 : 12 :: 2: 4. 

When four numbers are in musical proportion ; then the 
first has the same ratio to the fourth, as the difference be- 
tween the first and second has to the difference between the 
third and fourth* 

As in these, 6, 8, 12, 18; 

where 6 : 18 :: H -^^ i 18 - 12> 
that is 6 : 18 :: 2 : 6. 

- When numbers are in musical progression, their recipro- 
cals are in/ arithmetical progression ; and the converse, that 
is, when numbers are in arithmetical progression, their reci^ 
f rocals are in musical progression. 

So in these musicals 6, 8, 12, their reciprocals 4i h A» 
are in arithmetical progressioo \ for 4 + A = A = T > 
^i T + T^T'-Ta ^^^^ i^> ^^^ sum of the extremes is 
equal to double the mean, which is the property of arithme«- 
ticals. 

The method of finding out numbers in musical propor- 
tion is best expressed by letters iji Algebra. 



FELLOWSHIP, OR PAJITNERSHIP. 

Fellowship is a rule, by which any sum or quantity 
may be divided into any number of parts, which shall be in 
any given proportion to one another. * 

By this rul^ ar^ adjusted the gains or loss or charges of 

partnei;5 



t 



12« ARrrHMETia 

psurtners in company ; or the effects of bankrupts* or 
legacies in case of a deficiency of assets or effects ; or the 
shares of prizes ; or the numbers of men to form certain de- 
tachments; or the division of waste lands among* a number 
of proprietors. 

Fellowship is either Single or Double. It is Single, when 
the shares or portions are to be proportional each to one sin- 
gle given number only 5 as when the stocks of partners are 
all employed for the same time : And Double; when each 
portion is to be propoirtional to two or more numbers ; as 
when the stocks of partners are employed for different times. 



SINGLE FELLOWSHIP. 

GENERAL RULE. 



Add together the numbers that denote the proportion of 
the shares. Then say, 

As the sum of the said proportional numbers. 
Is to the whole sum to be parted or divided, 
So is each several proportional number. 
To the corresponding share or part. 

Or, as the whole stock, is to the whole gain or loss. 
So is each man's particular stock, 
To his particular share of the gain or loss. 

To PROVE THE Work. Add all the shares or parts to^ 
gether, and the sum will be equal to the whole number to 
be shared, when the work is right. 

EXAMPLES. 

1 . To divide the number 240 into three such parts, a$ 
shall be in proportion to each other as the three numbers l^ 
2 and 3. 

Here 1 + 2 + 3 = 6, the sum of the numbers. 

Then, as 6 : 240 : : 1 : 40 the 1st part, 
and as 6 : 240 : : 2 : 80 the 2d part, 
' also as 6 : 240 :: 3 : 120 the 3d part. 



Sum of all 240, the proof. 



Ex. 2« 



\. 



SINGLE FELLOWSHfP. 121 

Ex.'2. Three persons, a,b, c, freighted a ship with 540 tuns 
of wine ; of which, a loaded 1 10 tuns, B 97, and c the rest : 
in a storm the seamen were obliged to throw overboard 85 
tuns I how much must each person sustain of the loss i 

Here 110+ 97 = 207 tuns, loaded by A and b j^ 
theref." 340 — 207 = 133 tuns, loaded by c. 

110 

110 : 27i tuns = a's loss j 
97 : 24^ tuns =: b's loss ; . 

133 : 33^ tuns =: ds loss ; 



Hence, as 340 : 85 

or as 4 : 1 

and as 4 : 1 

ajiso as 4 : 1 



Sum 85 tuns, th^ proof. 



3. Two merchants, c and d, made a stock of 120/; of 
which c contributed 75/, and d the rest : by trading they 
gained 30/ ; what must each have of it ?. 

Ans. c 18/ 15/, and d 11/5/. 

4. Three merchants, e, f, g, make a stock of 700/, of 
which E contributed 123/, F 358/, and G the rest : by trading 
they gain lg5/ 10/ ; what must each have of it ? 

Ans. E must have 22/ Is Od 2^^q. 
F - - - 64 S 8 Off. 
G - - - 39 5 3 l-/y. 

5. A General imposing a contribution * of 700/ on four 
villages, to be paid in proportion to the number of inhabitants 
contained in each j the 1st containing 250, the 2d 350, the 
3d 400, and the 4th 500 persons ; what part must each vil- 
lage pay ? Ans. the 1st to pay 116/13/ 4rf. 

the 2d^ - - 163 6 8 
the 3d - - 186 13 4 
the 4th - - 233 6 8 

6. A piece of ground, consisting of 37 ac 2ro i4<ps, is 
to be divided among three persons, L, M, and n, in propor- 
tion to their estates : now if l's estate be worth 500/ a year, 
lid's 320/, and n's 75/ ; what quantity of land must each one 
have ? ^Ans. l must have 20 ac 3 ro 39 {4^ ps.. 

M - - •- IS 1 30,*^. 
,N . - - 3 2344I. 

7. A person is indebted to o 57/15/, to p 108/ 3/ 8i, 
to Q 22/ 10 J, and to r 73/; but at his decease, his effects 



. * Contribution is a tax paid by provinces, towDs^ villages, 
^c. to excuse tbem from being plundered. It is pffid an provir 
sio)^$ or in money^ and sometimes in both. 

arc 



123 ARITHMETIC 

are found to. be^ worth no more that! 170/ 14s ; how must it 

be divided among his creditors ? 

Ans, o must have 37/ 15j- Bd2^^^^, 
P ... 70 15 2 ^T^^. 
Q^ - . - 14 8 4. OAVtV 
R - - - 47 U 11 2t^V 

Ex. 8. A ship, worth 900/, being entirely lost, ofwhich -g-be-r 
longed to s, \ to T, and the rest to y ; what loss will each 
sustain, supposing 540/ of her were Insured ? 

Ans. s will lose 457, r 90/, and v 225/, 

9. Four persons, w, x, y, and z, spent among them 25/, 
and agree" that w shall pay i of it, x 4> y |-, and z 4- ; that 
is, their shares are to be in proportion as -j-, ^, \y and ^ : 
what are their shares ? Ans. w must pay 9/ 8</ S^f^, 

• X - '.6 5 3ff. 
T - - 4 10 If^. 
Z - - 3 10 S-yV- 

10. A detachment, consisting of 5 companies, being sent 
into a garrison, in which the duty required 76 men a day \ 
what number of men must be furnished by each company, in 
proportion to their strength ; the 1st consisting of. 5 4 men, 
the 2d of 51 men, the 3d of 48 men, the 4th of 39, and the 
5th of 36 men ? * 

Ans. The 1st must furnish 18, the 2d 17, the 3d 16, the 
4th 13, and the 5th 12 men*. 



BOUBLE FELLOWSHIP.. 



Double Fellowship, as has been said, is concerned in 
cases in which th r stocks of partners are employed or conti- 
nued for different times. 



* Questions of this nature frequentiy occurring in military 
>9ervice« General Haviland^ an officer of great merit, contrived an 
ingenious instrument, for more expeditiously resolving them; 
which is distinguished by the name of the inventor, being called ^ 
]^avilaod. ^ 

RVLE, 



DOUBLE FELLOWSHIP. 123 

• I 

Rule*.-— Multiply each person s stock by the time of its 
continuance ; then divide the quantity, as in Single Fellow- 
ship, into shares, in proportion to these products^ by sayings 

As the total sum of all the said products, 

Is to the whole gain or loss, or quantity to be parted. 

So is each particular product, 

T^p the correspondent share of the gain or loss. 



EXAMPLES. 

I. A had in company 50/ for 4 months, and b had 60/ for 
5 months ; at the end of which time they find 24/ gained ; 
bow must it be divided between them ? 

Here 50 60 
4 5 

200 + 300 =: 500 



Then, as 5C0 : 24 : : 200 : 91 zz 9/ 12j = a's share, 
and as 500 : 24 :: 300 : 14^ = 14 8 = B's share. 

2. c and D hold a piece of ground in common, for which 
they are to pay 54/. c put in 23 horses for 27 days, and d 
21 horses for 39 days ; hpw much ought each man to pay of 
the rent f Ans. c must pay 23/ 5s 9rf. 

p must pay 30 14 3 

3. Three persons, e, f, g, hold a pasture in common, 
for which they are to pay 30/ per annum ; into which e put 
7 oxen for 3 months, f put 9 oxen for 5 months, and G put 
in 4 oxen for 12 months ; how much must each person pay 
of the rent ? Ans. e must pay 5/ lOi 6d l-r^g^. 

F . . 11 16*10 O-j^. 
G - - 12 12 7 2t%. 

4. A ship's company take a prize of 1 000/, which they 
agree to divide among them according to their pay and the 
time they have been on board : now the officers and midship- 
men have been on board 6 months, and the sailors 3 months ; 



s 

* The proof of this rule is as follows : When the times arc 
equal/the shares of the gain or loss ar^ evidently as the stocky, as 
JD Single Fellowship ; and when the stocks are equal, the shares 
are as the times; therefore, when neither are equal, the shares must 
be as their products. 

the 



124 ARITHMETIC. 

the officers have 40j a month, the midshipmen SOSf and the 
sailors 22/ a month ; moreover there are 4 officers, 12 mid* 
thipmen, and 110 sailors : what will each man's share be ? 

Ans. each officer must have 2S/ 2s 5d O-^^^. 
each midshipman - 17 6 9 3i7^« 
each sieaman - - 6 7 2 Oj^, 

Ek. 5. H, with a capital of 1000/, began trade the first of 
January, andj meeting with success in business, took in i as a 
partner, with a capital of 1500/, on the fii^t of March fol- 
lowing. I'hree months after that they admit K as a third 
partner, who brought into stock 2800/. After trading toge- 
ther till the end o^^ the year, they find there has been gained 
1776/ 10/ ; how must this be divided among the partners ? 

Ans. H must have 457/ 9s A^d 

J - - - 571 16 sy 

K - - - 747 3 14* 

6. X, Y, and z made a joint-stock for 12 months; x at 
first put in 20/, and 4 months after 20/ more ; t put in at 
first SO/, at the end of 3 months he put in 20/ more, and 2 
months afiier he put in 40/ more ; z put in at first 60/, and 
5 months after he put in 10/ more, 1 month after which he 
took out 30/; during the 12 months they gained 50/'^ how 
much of it must each have ? > 

Ans. X must have 10/18/ 6d S^q. 

Y - - 22 8 1 0^. 

z - - 16 13 4 0. 



SIMPLE INTEREST. 



Interest is the premium or sum allowed for the loan, or 
forbearance of money. The money lept, or forkorn, is called 
the Principal. And the sum of the principal and its interest,^ 
added together, is called the Amount. Interest is allowed 
at so much per cent, per annum ; which premium per cent, 
per annum, or interest of 100/ for a year, is called the rate of 
interest ; — So, 

Whe» 



SIMPLE INTEREST. 



IfS 



When interest is at 3 per cent, the rate is 3; 

- - - 4 per cent. - - 4; 

- - - 5 per cent. - - 5; 

- - - 6 per cent. - - 6; 

But, by lawj interest ought not to be taken higher than at 
" the rate of 5 per cent. 

Interest is of two sorts ; Simple and Compound. 

Simple Interest is that which is allowed for the principal 
lent or forborn only, for the whole time of forbearance. 
As the interest of any sum, for any time, is directly propor- 
tional to' the principal sum, and also to the time of continu- 
ance; hence arises the following general rule of calcula- 
tion. 

As 100/ is to the rate of interest, so is any given principal 
to its interest for one year. And again. 

As 1 year is to any given time, so is the interest for a year, 
just found, to the interest of the given sum for that time. 

Otherwise. Take the interest of 1 pound for a year, 
which multiply by the given principal, and this product again 
by the time of loan or forbearance, in years and parts, for the 
interest of the proposed sum for that time. 

NoUf When there are certain parts of years in the time, 
as quarters, or months, or days : they may be worked for, 
either by taking the aliquot or like parts of the interest of a 
year, or by the Rule of Three, in the usual way. Also, to 
divide by 100, is done by only pointing off two figures for 
decimals. 

EXAMPLES. 

1. To find the interest of 230/ 10/, for 1 year, at the rate 
«f 4 per cent, per annum. 

Here, As 100 : 4 ; : 230/ lOi : 9/ 4/ 4|rf. 

4 



100) 9,22 
20 




Ans. 9/ 4/ 4| J. 



3-20 



Ex.2, 



126 ARITHMETIC. 

Ex. 2. To find the interest of 54^/ 1 5sy for 3 years, at 5 per 
cent, per annum. 

As 100 : 5 :: 547-75 : 

Or 20 : 1 :: 547-75 : 27-3875 interest for 1 year. 

3 

/ 32-1625 ditto for 3 years. 
20 

J 3-2500 
12 



\ 



d 3 00 Ans. 82/ Ss Zd. 



3. To find the interest of 200 guineas, for 4 years 7 months 
»nd 25 days, at 4Y*pcr cent, per annum. 

ds / ds 

210/ As 365 : 945 :: 25 : / 

44 or 73 : 9*45 :: 5 : -6472 
5 



840 



105 73 ) 47-25 ( •64721 



345 



9-45 intei-est for 1 yr. 530 
4 19 



37-80 ditto 4v years. 
6 mo =5 4 4-725 ditto 6 month, 
1 mo = 1^ -7875 ditto 1 month. 
-6472 ditto 25 days* 

/ 43-9597 
20 



s 19-1940 
12 



d 2-3280 

4* Ans. 45/ ids 2^; 

q 1-3120 

4. To find the interest of 450/, for a year, at 5 per cent, 
per annum. Ans. 22/ lOj. 

5. To find the interest of 715/ i2s.6df for a year, at 4j. 
per cent, per annum. Ans» 32/ 4/ O^d. 

6. To find the interest of 720/, for 3 years, at 5 per cent, 
per annum. Ans. 108/. 

7. To find the interest of 355/ 15j for 4 years, at 4 per 
cent, per annum. Ans. 561183 ^d. 

, - Ex. 8. 



. ' COMPOUND INTEREST. 121 

Ex. 8. To find the interest of 32/ 5s Sd, for 7 years, at 4|. 

per cent, per annum. Ans. 94 \2s \d. 

9. To find the interest of 170/, for I4 year, at 5 per cent. 

per annum. Ans. 12/1 5j-. 

-^ 10. To find the insurance on 205/ 15j, for -J- of a year, af 

4 per cent, per annum. Ans. 2/ 1/ \\d, 

^ 1 1 . To find the interest of 319/ 6//, for 5^- years, at 3| per 

cent, per annuin. . Ans. 68/ I5s 9\d. 

12* To find the insurance on 1 07/, fof 117 days, at 4| per 

cent, per annum. Ans. 1/ 1 2s Id. 

-— IS* To find the interest of 17/ 5/, for 117 days, at ^ per 

cent, per annum. , Ans. bs Sd.' 

' r4. To find the insurance on 712/ 61, for 8 months, at 

74 per cent, per annum. Ans. 35/ t2s S^d. 

Ni^^m The Rules for Simple Interest, serve also to calcu- 
late Insurances, or the Purchase of Stocks, or any thing else 
that is rated at so much per cent. 

See abo more on the subject of Interest, with the algebraical 
expression and investigation of the rules, at the ^nd of the 
Algebra, next following. 1 



COMPOUND INTEREST. 

Compound I^nterest, called also Interest uppn Interest, 
Is that which arises from the principal and interest, taken 
together, tis it becomes due, at the end of each, stated time 
of payment. Though it be not lawful to lend mOhey at 
Compound Interest, yet in purchasing annuities, pensions, 
or leases in reversion, it is usual to allow Compound Interest 
to the purchaser for his ready money. 

Rules. — 1. Find the amount of the given principal, for the 
time of the first payment, by Simple Interest. Then con- 
sider this amount as a new principal for the second payment, 
whose amount calculate as before. And so on through all 
the payments to the last, always accounting the last amount 
as a new principal for the next payment. The reason of 
which is evident from the definition of Compound Interest* 
Or elsCi 

2. Find the amount of 1 ^ound for the time of the first 
payment, and raise or involve it to the power whose index 
is denoted by the number of paynients. Then that power 
multiplied by the gi^cen principal, will produce: the whole 

amount. 



\ 

/ 



128 ARltHMETIC. , 

amount. From which the said principal' being ^ubti^actdil/ 
leaves the Compound Interest of the same. As is evident 
from the first Rule. 

EXAMPLES. 

I. To find the amount of 720/, for 4 years, at 5 per cent, 
per annum. 

Here 5 is the 20th part bf^.lOO, aiid the inter^t of 1/fora 
year is ^V or '05, and its amount 1*05. Therefore^ 

1 . By the ist Rule. 2. By the 2d Rule. 
I s d r05 amount of l/« 

20)720 1st yr*s princip. 1-05 

36 1st yr's interest. 

1 • 1025 2d pow^rof it*' 

2Q) 736 2d yr's pr^ncip. 1'1025 

37 16 2d yr's interest. 

1 -2 1 5506 25 4th pow. of it, 

20) 793 16 3dyr*sprincip. 720 

39 13 94: 3d yt's interest. 

/ 875-1645 

20) 833 9 9^ 4th yr's princip. 20 

41 13 5|- 4th yr's interest , 

^ ^ / 3 -2900 

£ 875 3 3|. the whole amo^ 12 

■ or ans. required. ■ 

^3-4800 



2. To find the amount of 50/, in 5 years, at 5 per centj 
per annum, compound interest. Ans. 63/ 16/ 3^rf. 

3. To find the amount of 50/ in 5 years, or 10 half-, 
years, at 5 per cent, per annum, compound interest, the in- 
terest payable half-yearly. Ans. 64/ Os Id* 

4. To find the amount of 50/, in 5 years, or 20 quarters, 
at 5 per cent, per annum, compound interest, the interest 

^ payable quarterly. ' Ans. 64/ 2/ 0-|^. 

5. To find the compound interest of 370/ forbom for 6 
years, at 4 per cent, per annum. Ans. 98/ 3/ 4^. 

6. To find the compound interest of 410/ forborn for 24 
years, at 4^ pei* cent, per annum, the interest payable half- 
yearly. Ans. 48/ 4j 1 \\d* 

7. To find the amount, at compound interest, of 217/, 
forbom for 2^ years, at ^5 per oent. per annum, the interest 
payable quarterly. Ans. 242/ i 3j 44rf. 

Nate. See the Rules for Compound Interest algebraically 
investigated, at the end of the Algebra. 

ALLIGATION* 



ALLIGATION* ■ , 129 



ALLIGATION. 



1 \ 



Alligation teaches how to compound or mix together 
several simples of different qualities, so that the composition 
may be of" some intermediate quality, or rate. It ia com- 
monly distinguished into two cases, Alligation Medial, and 
Alligation Alternate*. 



ALLIGATION MEDIAL. 



Alligation Medial is the method of find ing the rate 
or quality of the composition, from havii^g. the quantities 
and rates of qualities of the several simples given* And it 
is thus performed : ' 

* Multiply the quantity of each ingredient by 'its rate or 
quality; then add. all the produclts together, and add also all 

r • ■('... 

• ■ ' -- -- — ■■ - -. f • j ■•■ ' I I ■ ■ ■ , ■ - ■ ,- J- - ;-| -jq 1 |^_iJi«lM^M 

i ' ' - * " ' 

* Demonstration* The Rhle is thus riroyed by Algebra. 

Let flj bg c be the quantities of the ingredients, 

arid wi, «, p their rates, or qualities> or prices 3 

then amy bn, cp iare their severar values, 

atid am + bn -f cp the sum of their values, 

also a -f 6 -f- c is the sum of the quantities, 

and if r denote the rate of the whole Compositioh^ . 

then a + 6 + c X r will be the value of the whole, 

conseq.a -{■ b + c X r zz am -^ bn -^ cp, 

and fit am + 4« + cp^a -\' i H-c, which is the Rule. 

• > ■ 

Note, If an ounce or any other quantity of pure gpld be reduced 
into I?4 equal, parts, these parts are called Caracts; but gold is often 
mixed with sonie base metal^ which is called the Alloy , and the 
mixture. is said to be of so many caracts fine, according to the. 
proportion of pure gold contained in it ; thus, if 22^ caracts of 
pare golfd, and 2 of alloy be mixed together, it is said to be 22 
caracts fine. 

If any,one of the simples be of litde or no value with respect to 
the rest, its rat6 is supposed to be nothing ; as water mixed with 
wine, and alloy with gold and silver. 

VolL K ^ the 



ru 



the quantities together into another sum ; then divide the 
former sum by the latter, that is, the sum of the product* 
by the sum of the qutotities, and: the quotient will be the 
rate or quality of the composition required*^ 



EXAMPLES* 

I. If three sorts of gunpowder be imxed togetbeis ^rz* 
50lb 9t I2d z pound, 441b at 9d, and 261b at Sd a pound ^ 
how much a pound is the composition worth i 

Here 50, 44, 26 are the quantities, 
and 12, 9, 8 the rates or qualities ; 
then 50 x 12 = 600 

44,X 9 = S96 

26 X 8=208 

120 ) 1204 ( lOrJv = 10^ 

Ans. The rate or price is 10^^ the poimd. 

^•2, A comporitioa bein|^ made of 5lb of tea at 7s per Ib^ 

9lb.at 8r 6^ per Ib^ and. 14ilbat 5/ lOi/per lb; whacis»^ 

♦lb of it worth ? Ans. 6s lO^rfr 

5» Mixe4-4-gsfioa»-of-wfiie-at-4j' ICtf-pcr gatf, with"7 gal- 

k)ns at 5s Sd per, gall, and 9|- gallons at 5s Si/pjer^gall^ 

what is a gallon of this composition worth ? * Ans. 5s 4^. 

4. A meahnan would mix 3 bushels of- flour Jat ^i 5d^ 
per bushel, 4 bushels at 5$ 6d per bushel, and 5 bushels at 
4j8^per bu&el; what is the: worth of. a bushel of this 
mixture ? • > Ans.- 4r iid^ 

5. A farmer nvixes 10 bushels of wkeat. at .5s the bushd, 
Vith 18 bushels of rye at 3x the bushel, and 20 bushels of 
barley at 2s per l>ushel:' how much is a bushtel of the mixture 
worth ? Ans; di^ 

6. Having melted tog^b^r. 7 *oz of gold of 22caracts fine, 
1240Z of 21 caracts fine, aild 17 oz of 19 caracts^fine : I 
Would know the fineness of tbfe composition ? 

Ansl 20ff caraets fine. 
1, Of what fineness Is that con^osit ion, which is made- by 
tiiixing,3lb of silyer of 9 oz fine, ,with 5lh 8 oz pf -lOoaSr. 
fine, fUid> lib lOoz of aUay.. Ansi 7|j;0Z» fine.' 



ALUGATIOK 



[ 131 ] 



ALLIGATION ALTERNATE.. 

: AttiOATiaN Alternate is the method of finding what 
quantity of any number of simples, whose rates -are given, 
will compose a mixture of a given rate. So that it is the re- 
verse of Alligation Medial, and may be proved by it. 

RULE I*. 

1. Set the rates of the simples in ^ column under each 
Other.-^2. Connect, or link with a continued liae, the rate 
of' each simple) which is less than that of the compound^ with 
one» or any number, of those that are greater than the com-' 
ppund ;* and. each greater rate with one or any number of the 
le»rftr-^* Write the difference between the mixture rate^and^ 
that of each of the simples^ opposite the rate with which they 
al-e Unked.-*-4. Then if only one difference atand ;^ainst 
any rate, it will be the quantity belonging to that rate ; but 
if there be seV.eral, their suni will be the. quantity. , 

- TUfe' examples may be, proved by the rule for AlHgation 
MediaL 



f Demonst, By connecting the less rate to the gteikter, atid 
placing the difference between them and the rate alternately, tho 
quantities resulting are such, that there is precisely as much 
gtified by 6ne quantity as is lost by the other, and therefore the 
gain' and loss upon the whole is equal, and is exactly the proposed 
rate : and the same will be true of any other two simples managed 
according to the Rale^ 

In like manner, whatever the number of siibples may be, and 
with how ttiany soever every one is linked, since it is always a 
less with a greater than the ^roean price, there will be an equal 
balance of loss and gain between every two, and consequently an 
equal balance on the whole. - a. e* o. « 

.It is obvious, from this Rale» that questions of this sort admit of 
a gneat variety of answers \ for, having found one answer, we may 
£nd as many more as we please,, by only multiplying or dividing 
each of the quaptities found, by 2, or 3, or 4, ^c: the reason of 
which is evident : for^ if two quantities, of two simples, toake a 
balance of loss and gain, with respect to the mean price, so must 
also the double or treble, the 4 or |- part, or any other ratio of these 
quantities, and so on, 0(2 in^mYt/m. 

These kinds of questions are called by algebraists indeterminate 
or unlimited problems 3 and by an analytical process, theorems may 
be raised that will give all the possible answers. 

' K2 EXAMPLES. 



X 



133 ARITHMETIC 



EXAMPLES. 



1. A merchant would mix wines at 16/, at 18j, and at 
' 22s per gallon, so as that the mixture may be worth 20/ the 

gallon : what quantity of each must be taken i 

/^16"^ 2 at 16/ 

Here20< 18x j2 at 18/ ^ 

i^22j/ 4 + 2 =: 6 at 22/. 
\Ans. 2 gallons at 16/, 2 gallons at 18/, and 5 at 22x. 

2. How much wine at 6s per gallon, and at 4s per gallon,^ 
must be mixed together, that the composition may be worth 
Ss per gallon ? Ans. 1 qt, or 1 gall, &c. 

3. .How much sugar at 4rf, at 6rf, and zt \ld per lb, mUst 
be mixed together, so that the composition formed by them" 
may be worth Id per lb ? 

Ans. 1 lb, or 1 stone, or 1 cwt, or any other equal quantity 
of each sort. 

4. How much corn at 2x 6rf, 3/ 8rf, 4j, and 4/ 8 J per 
bushel, must be mixed together, that the compound may be 
worth ifif lOd per bushel ? 

Ans. 2 at 2s 6d. 2 at 3/ 8 J, 3 at 4/, and 3 at 4/ Sd. 

'I 

5. A goldsmith has gold of 16, of 18, of 23, and of 24 
caracts fine : how much must he take of each, to make it ,21 
caracts fine ? Ans. 3 of 1 6, 2 of 1 8, 3 of 23, and 5 of 24.' 

6. It is required to mix brandy at 12/, wine at 10/, cyder 
at 1/, and water at per gallon together, so that the mixture 
may be worth 8/ per gallon ? 

Ans. 8 gals of brandy, 7 of wine, 2 of cyder, and 4 of water. 

' ' * . . . « 

RULE II. ^ 

* •)• 

When the whole composition is limited to a certain 
iquantity : Find an answer as before by linking ; th^en say, as 
the sum of the quantities, or differences thus determined, is 
to the given quantity ; so is each ingre4ient, found by link- 
ing, to the required quantity of each* 

examples. 

^ . How much gold of 1 5, 17, 18, and 22 caracts fine, must 
be mixed together, to form a composition of 40 oz of 20 ca^- 
racts fiuie ? . 

Here 




ALLIGATION ALTERNATE. 133 

- 2 
- - 2 

- 2 

5 + 3 + 2 = 10 

16 

Then, as 16 : 40 :: 2 : 5 
and 16 : 40 :: 10 : 25 
Ans. 5 oz of 15, of 17, and "of 18 caracts fine, and 25 oz of 
22 caracts fine*. 

Ex. 2. A vintner has wine at 4/, at 5/, at 5s 6i, and at 6s 
a gallon; and he would make a mixture of 18 gallons, so 
that it might be afforded at 5s 4rf per gallon ; how much of 
each sort must he take ? 

Ans. 3 gal. at 4 J, 3 at 5j, 6 at 5s 6t/, and 6 at 6/. 



* A great number of qtiestions might be here given relating to 
the specific gravities of metals, &c. but one of the most curious 
may here suffice. 

Hiero, king of Syracuse, gave orders for a crown lobe made 
entirely of pure gold ; but suspecting the workman had debased 
it by mixing it with silver or copper, he recommended the dis- 
povery of the fraud to the famous Archimedes, and desired to know 
the exact quantity of alloy in the crown. 

Archimedes, in order to detect the imposition, procured two 
other masses, the one of pure gold, the other of silver or copper, 
and each of the same weight with the former j and by putting each 
Separately into a vessel full of water, the quantity of water expelled 
by them determined their specific gravities 5 from which, andtheit 
given weights, the exact quantities of gold and alloy in the crowa 
may be determined. 

Suppose the weight of each crown to be lOlb, and that the 
water expelled by the copper or silver was 92lb, by the gold 52lb, 
and by the compound crown 64lb ; what will be the quantities of 
gold and alloy in th€ crown r 

The rates of the simples are 92 and 5*2, and of the compound 
04 5 therefore • 

fii I 9'^"^ ^^ of copper 
"■* I 52-^ 28 of gold 
And the sum of these is 12 + 28 =: 40, which should have been 
but 105 therefore by the Rule, ^ 

40 : 10 :: 12 : 3 lb of copper 1 , 

: 7lbofgold /t^e answer. 



40 : 10 : ; 28 



RULE 



IS* ARITHMETia 



HOLE III*, 

When one of the ingredients is limited to a certain quaii« 
tity ; Take the difference between each price, and the mean 
rate as before ; then say, As the difference of that simple, 
whose quantity is given, is to the rest of the differences se-^ 
verally, so is the quantity given, to the several quantities 
required. 



EXAMPLES* 

1. How much wine at 5/, at 5s 6df and 6s the gallon, 
must be mixed with 3 gallons at 4s per gallon, so tbat thp 
multure may be worth Ss 4d per gallon ? 

+ 2 = 10 



Here 64 




+ ^ = 10 

+ .4 = 20 

+ 4 = 20 

3 
3 : 6 
3 : 6 
Ans. 3 gallons at 5sy 6 at bs 6^, and 6 at 6x« 

ft. A grocer would mix teas at 12j, 10/, and 6x per lb, 
with $!01b at 4/ per lb • how much of each sort must he tako 
to make the composition worth %s per lb ? 

Ans. 201b at 4/, lOlb at ^x, lOlb at lOi, and 201b at 12/. 

3. How much gold of 15, of l7, and of 22 caracts fine, 
must be mixed with 5 oz of 18 caracts fine, so that the com- 
position may be 20 caracts fine ? 

Ans. 5 oz. of 15 caracts fine, 5 oz of 17, and 25 of 22« 



•♦*" 



« In the very same manner questions may be wrought when sc* 
veral of the ingredients are limited to certain quant ities, by finding 
first for one limits and then for another. The two last Rules can 
peed no demonstration^ as they evideody result from the firstj the 
1^904 of wl^icli h^ been already explained* 



fOSITl[0H« 



SINGLE POSITION. 1« 



POSITION- 



Position is a method of performing certain questibnSf 
which cannot be resolved by the common direct rules. It h 
sometimes called False Position, or False Supposition} because 
it makes a supposition of false numbers^ to work with the 
same as if they were the true ones, and by. tl^ir means dis- 
covers the true numbers sought. It is sometimes ako called 
Trial-and«£rror, because it proceeds by trials of false num- 
bers, and thence finds out the true ones by a comparisoa of 
the /rrvr/.'— Position is either Single or Double. 



SINGLE POSITION. 



Single Position is that by which a questmi k resotved 
by means of one supposition only. Questions which have 
their result proportional to their suppositionSj|^ belong to 
jingle Position : such as those which require the multiplica- 
tion or division of the number sought by any proposed num* 
ber ; or when it is to be increased or dimiotthed bj ifiself^ 
«r any parts of itself, a cevtain proposed nutnbe» ot times* 
The rule i& as follows : 

Takb or assume any nwnbeF fiur that whick is required, 
sad perform, the some operatttons with it, as aM described or 
performed in the question* Thensay, As the resiite cS the 
said operation, is to the position, or number assumed^ so is 
the result in the question, to a fourth term, which will be 
the number sought*. 



^1^ 



* The reason of this Rule is evident^ becanse it is auppoaad tbajt 
the results are proportional to the sapposUions. 

Thus, na I a i I nz I 2g 

a z 

or — : a : : — : z. 
n n 

a Q z z 

or — ± — &c : fl ; : — ± — &c s z, 
n tn u m 

and so on. 

EXAMP LES« 



1S« 



ARITHMETIC. 



EXAMPLES. 



1. A person after spending y and ^ of his money, has yet 
remaining 60/; what had he at first ? 

Suppose he had at first 120/. Proof. 



Now ^ of 120 is 40 
i of it is 30 



•§.ofl4«4is 4S 
i of 144 is 36 



their sum is 70 
rhich taken from 1 20 



their sum 84 
taken from 144 



leaves 50 
Then, 50 : . J20 : ; 60 



leaves 60 as 
144, the Answer. per question. 



2. What number is that, which being multipied by 7, and 
the product divide4 by 6^ the quotient may be 21 ? Ans. 18, 

S. What number is that, which being increased by i^ f, 
^nd ^ of itself, the sum shall be 15 i Ans. 36. 

4. A general, after sending X)ut a foraging i and -J- of his 
men, had yej; remaining 1000 : what number had he in com- 
mand ? Ans. 6000. 

5. A gentleman distributed 52 p€n9e among a number of 
ppor people, consisting of men, women, and children ; to 
each man he gave 6^, to each woman 4^, • and to each child 
Sd : moreover there were twice as many women as men, and 
thiice as ipany children as women. How many were there 
of each i . Ans. 2 men, 4 women,' and 12 children. 

6. One being asked his 'age, said, if ^ of the years 1 hav^ 
lived, be multiplied by 7, and -f- of them be added to the 
product, the sum will be 219. What was his age ? 

Ais. 45 years. 



|>OUBl.E 



I 137 ] 



DOUBLE POSITION, 

Double Position is the method of resolving certain 
questions by means of two suppositions of false numbers. 

To the Double R ule of Position belong such questions as 
have'their results not proportional to' their positions : such are 
those, in which the numbers souglit, or their parts, or* their 
multiples, are increased or diminished by some given absolute 
number, which is no known part of the number sought* 

RULE I*. 

Take or assume any two convenient numbers, and proceed 
with each of them separately, according to the conditions of 
the question, as in Single Position ; and find how much earJi 
result is different from* the result mentioned in the question, 
calling thes« differences the errors ^ noting also whether the 
results are too great or too little. 

* Demonstr,^ The RuKs is founded on tliis supposition, namely, 
thai the first errqr is to the second, as 'the difference between the 
true and first supposed number^ is to the diffbr^ce between the true 
and second supposed number 5 when that is no( the case, the exact 
answer to the question cannot be found by this Rule.— That thot 
Rule is true, according to that supposition^ may be thus proved. 

I^t a and b be the two suppositions, and a and b their results, 
produced by similar operation \ also r and s their errors, or the 
differences between the results a and b from the true result n ; 
and let x depote the number sought, answering to the true result 
If of the question. 

Then is n — a = r, and n — b = *. And, according to the 
supposition on which the Rule is founded, r : s :: x— a: x—bs% 
hence, by multiplying extremes and means, rx — r6 = ^«— *a j 

4hen, by transposition, rx — «x = rb — sa j and, by div«iio|i^ 

flf g(i • -* 

X — r iz the number sought, which is the rule when the^ 

r — « . 

results ^re both too little;. 

If the results be both too great, so that a and b are both greater 
than N 5 then n — • a z: — r, and n — bzz — *, orr and s are both 
negative ; hence — r ; — $ : nx — a : x ^b, but — r : —4 : : + r 
: -f *, therefore c ; s : : ^p— a ; x -^"b; and the rest will be ex- 
actly as in the former case. ^ 

But if one result a only be too little, and the other b too great, 
or one error r positive, and the other s negative, then the theoreiu 

becomes x = — ^ , wbioh la the Rule in this case, or when 

the errors are unl&e, 

Theo 



13S ARITHMETia 

Then multiply each of the said errors by the contrary sup- 
position, namely, tlie first positiQn by the second .error, jmd 
the second position by the first error. Then, 

If the errors are aJike, divide the difference pf th^ products 
by the difierence of the errors, and the quotient will be tl^e 
answer. 

But if the errors are unlike, divide the siun of tl^e produces 
by jhe sum of th^ errors, for the answer, 

Noicy The errors are said to be alike, when tb^y are either 
both too great or both too little^ ^md unlike, ^\^^ we is too 
great and the other too Uttle. 

EXAMPLES. 

1. What number is th^t, which being multiplied by 6^ 
%!t^ product increased by 18, and the sum divided by 9, the 
quotient shall ^e 20 ? 

Suppose the two numbers 18 and 30* Then, 



First Position. 

18 Suppose 
6 mult* 


iSecopd Position. 
30. 
6 


Rroof. 

27 


108 
1^ 


« 1 

add 


180 
18 




162 
18 


9) 126 


div. 


9) 198 




9} 1^ 


14 
20 


results 
tri^e res. 


22 
20 




20, • 


OA pos. SO 


errors unlike -^ 2 
mujt. ' 18 


1st 


pos. 


Igr- ( 2 180. 
rors ( 6 36 




« 






surn. 8 ) 216 


sum of products 






27 

< 


Answer 

* . # 


sought. 

RULE 11. 







Find, by trial, two numbers, as near the true number as 
convenient, and work with them as in the question ; mark- 
ing the errors which arise from each of them. 

Multiply the difierence of the two numbers assumed,, or 
found by trial, by one of the errors, and divide th^. product by 
the difference of the errors, wlien they are alik;e, but by their 
sum when they are unlike. 

AAi the quotient, last found, to the number belonging to 
the said error, when that number is too little, but; subtracjt 

It 



»• 



DOUBLE PpSmON. . 18? 

It when too greats and the result will give the true quantitf 

jBOUght *# 

examples; 

4 

1. So, the foregoing example, worked by this 2d rulet 
will be as follows : 



t: I. 



30 positions 18; their dif. 12 

^ 2 errors + 6 ; least error 2 

« . 9 

sum of errors 8 ) 24 ( 3 subtr. 
from the position 30 

leaves the answer ^7 

Ex. 2. A son asking |iis father how old he was, received 
this answer: Your age is no^ one-third of mine; but 4 
years ago, your age was only one-fourth of mine.. What then 
are their two ages ? , Aas. 15 and 4^,. 

3, A workman was hii^ed for 20 days, at 3/ per day, for 
every day he worked ; but with this condition, that for 
every day he played, he should forfeit* 1^. Now it so hap- 
pened, that upon t^e whole he had 2/ 4x to receive. How 
piany of the days did he work f Ans. 16, 

4« A and B began to play together with equal sums of 
money : a first won 20 guineas, but afterwards lost back ^ 
of what he then had; after which, B had 4 times as much as 
A. What sum did each begin with ? Ans. lOO guineas. 

5. Two persons^ A and B^ have both the same income^ 
A saves 4^ of his ; but b, by spending 50/ per annum more 
than A, at the end of 4 years finds himself 100/ in debt* 
What does each receive and spend per annum ? 

Ans. They recei^ 125/ per annmn; also A spends 1004 
and B spends 150/ per annum. 



■••^ 



1 

* for since, by the supposition, rtnix — a : x^b^ there* 
fore by divisioni r— ^ : t ; ; *— « ; «— ft, which is the ad Rule, 



"" PRACTICAI, 



iLd 



liO » * AfHTHMETIC. 



PRACTICAL QUESTIONS in ARITHMETIC. 

Quest. 1. The swiftest velocity of a cannon-ball, is 
about 2000 feet in a second of time. Then in what time, 
at that rate, would such a ball be in moving from the earth 
to the sun, admitting the distance to be 100 millions of 
miles, and tlie year to contain S65 days 6 hours ? 

Ans. Sy^^s^ years. 
Quest. 2. What is the. ratio of the velocity of light to 
that of a cannon-ball, which issues from the gun with a ve- 
locity of 150O feet per second; light passing from the sun to 
the earth in 14 minutes ? Ans. the ratio of 782222|- to I • 

Quest. 3. The slow or parade-step being 70 paces per 
minute, at 28 inches each pace, it is required to determine 
at what rate per hour that movement is ? Ans. Ixfl- miles. 

Quest. 4. The quick-time or stcp^ in marching, being 
55 paces per second, or 120 Y^r minute, at 28 inches each ; 
then at what rate per hour does a troop march on a route, 
and how Jong will they be in arriving at a garrison 20 miles 
distant, allowing a halt of one hour by the way to refresh? 

A r the rate is 3-rV miles an hour. 
^^^' 1 and the time 7| hr, or 7 h 17| min. 

Quest. 5. A wall was to be built 700 yards long in 29 
days. Now, after 12 men had been employed on it for 11 
days, it was found that they had completed only 220 yards of 
the wall. It is required then to determine how many men 
must be added to the former, that the whole number of them 
may just finish the wall in the time proposed, at the same 
rate of working. Ans. i men to be added. 

Quest. €. To determine how far 500 millions of gui- 
neas will reach, when laid down in. a straight line touching 
one another ; supposing each guinea to bean inch in diameter, 
as it is very nearly. Ans. 7891 miles, 728 yds,, 2 ft, 8 in. 

^ Quest. 7. Two persons, a and B, being on opposite 
sides of a wood, which is 536 yards about, they begin to go 
round it, both the same way, at the same instant of time ; a 
goes at the rate of 1 1 yards per minute, and B 34- yards in 
S minutes j the question is, how many times will the wood 
be gone round before the quicker overtake the slower ? J^ 

Ans. 1 7 times. 
Quest, 



. PRACTICAL QUESXIONS. Ut 

Quest. 8. a can do a piece of work alone in 12 days^ 
and B alone in 14"^ 'in what time wiil they btrth together per-f 
form a like quantity of work ? Ans. 6^ days* , 

. QupsT. 9. A person who was possessed of a ^-.share of a 
copper mhie, sold -J- of his interest, in it for \ 800/ v what was 
the reputed value of the whole at the same rate? Ans. 4000/* 

QifEsT. 10. A person after spending 20/ more than \ or 
his yearly income, had then remaining' 30/ more than the . 
half of it ; what was his jncome ? Ans. 200/. 

Quest. 11. The hour and mjnute hand of a clock are 
exactly together at 12 o'clock; when^are they next together? 

Ans. at I7V hr> or 1 hr, 5^^ rain-r 

Quest. 1 2., If a gentleman whose annual income is 1300/y. 
spend 20 guineas a week ; whether will he save or run in 
debt, and how much in the year ? i 'I Ans. save 403/* 

' Quest. 13, A person bought •! SlQ.or^nge^ at 2 a penny, 
and 180 more at 3 a penny; after ,>viiich, ^selling them out 
again at 5 for 2 pence, whether did he gain or lose by the 
bargain? '' Ans. he lost 6 pence. . 

Qubst. 14. If a qujintity of provislopis s^ves ,1500 men 
12 weeks, at thje^.|*ate of ?0' ounces a;4ay for each man ;> how 
many men will the .same provisions maintain for 20 weeks, at 
the rate of 8 ounces a day for each'm^n ? Ans. 225Q ttitxu 

Quest. 15. In the latitude*of London, the distance round 
the earth,'measured on. the parallel q£ latitude^ is about 15550 
miles; now as the earth turns round in.23|iours 56 nijfij^t;es, 
at what rate per hour is the city of J^p;i<J(xp carried by.jthis 
motion from west to east J A,ns< ^4p|r||' tpiles anl?i>u^» 

Quest. 16. A father left his son a fdrtune, J- of which he 
ran through in 8 months': f of the remamder lasted him 12 
months longer ; after which he had bare 820/ left. What 
sundr did the father bequeath his son?' > Ans/ 1913/6/ 8//. 

Quest. 17. If 1000 men, besieged in a town, with pro- 
visions for 5 weeks, allowing each man 16 ounces a day, be 
reinforced with 500 men more ; and supposing that they can- 
not be relieved till th^ er|d of 8 weeks, how many ounces a 
day must each man have, that the provision may last that 
time ? \ Ans. 6|. ounces. • 

Quest. 18. A younger brother received 8400/, which 
was just |- of his elder brother's fortune : What was the 
father worth at his death ? , Ans. 19200/. 

' - Quest. 



U2 ARlTHMfetiC 

. Qvi,sr. 19. . A person, looking on his j^stchj v^s. a^k^d 
Vlis^t >yas the time of^th^ day, who ansv^ere^i.It. U.bp(Wjeea 
5 and 6 ;. but a more particular answer being required, he 
uad that the hoitr and minute hands were then exaqtly toge- 
ther : What was the time ? Ans. 27-1^ >^i°' P^^ ^* 

.^ Quest. 20. If 20 men tin perform a piece of work in 
lb days, how many men will accomplish another ti^rice as - 

large in one-fifth of the time? Ans. 300* 

... . ..... 

Quest* 21, A father devised y^^ of his estate tp one of his 
sons, and -^ of the residue to another, and the surplus to his 
relict for life, 'the children's legacies were found to be 
5i4/ 6s 8d different: Then what money did he leave the 
widow the use of? ' Ans. 1270/ 1/ 9|4^. 

Quest., 22* A person, making his will, gave tp^ ouj? child . 
i% of his estate, and the rest to another. When these legacies 
caoie to be paid the one turned out 1 200/ more than the 
other : What did the testator die worth ? Ans. ^OOO/* 

.Quest. 23» Two persons, a and B, travel between 
£ondon and Lincoln, distant 100 miles, a from London, 
and B from Lincoln, at the same instant. ' After 7 hoUrs they 
meet on the road, when it appeared that a had rode \i miles 
an hour more than b. At what rate per hour, then did 
each of the travellers ride ? Ans. a 7|4V^nd b 6^4 miles. 

Quest. 24. Two. petrsons, A and b, travel betwGeti Lon- 
don and Exeter, a leaves Exeter at 8 o'clock in the morrj* 
ing'^ and walks at t}ie r^te of S miles an hour,-wtthout inter* 
mission; and b sets out from London at 4 o'clock the same 
evening, and walks for Exeter at the ritte of 4 miles an hopr 
cpnsfantly. , Now,.. supposing the distance betwejen the two 
cities to be 130 miles, whereaSouts on the road will they 
meet ? ; Ans. 691 miles from Exeter. 

• QuesTi|,25. One hiindred eggs -being placed on the 
ground, in ^ straight line, at the distance of a yard from each 
other : How far will a .perspn travel who shall bring them 
ope by one to a basket, which, is placed at one, yard from the. 
first egg ? Ans. lOlOO yards, or 5 miles and 1300 yds. 

Quest. 26. The ' clocks, of • Italy gd on to 24 hours: 
Then how many strokes do they strike in one complete re- 
volution of the index? Ans. 300. 

Quest. 27. One Sessa, an Indiaiijij having ipyented the 
game of chess, shewed it to his prince, who was so delighted 

with 



wftK iti tfekf he promised^ litnf aby iWstr'd Ire shbuld' isle ; on' 
wHith Sessa- requested that he might be allofwed one griiri'of^ 
wheat for the first square on the cjiess boards 2 for the secoxid^ 
4 for the third, and so on, d9ubiing contihualiy, to 64; the 
whold number of squares. Now, supposing a pint to contain 
7(180 of theje grains, and one quarter or 8 busheU to be worth 
^Is 6dy it is required to compute the value of* all the corji ? 

Ans. 6450468216285/ lis 3d 3j4J4^* 

. Quest. 26* A' person increased his estate annually by 
100/ more than the j- part of it ; and zt the end of 4 year^ 
found'that'his estate amounted to 10342/ 3/ 9rf. What had 
h6' at' first ? Ans: 46oO/. 

Quest. 29.' Paid 1012/ lOx for a principal of 750/,'takeii' 

irf^7 years before : at what rate per ceiit. p^r aftnum did I' 

piy interest ? Ansf. 5 pet cent* 

■ • '> . - . '• 

Quest. 30. Divide 1 000/ among A, b,' c j so as to give 

A 120 more, and b 95 less than o 

Ans. A 445^ b 2S0, c ^25- 

QuEST. 3U A person l)eing asked the hour of the day, 
said, the time past noon is equal tooths of the time till 
midnight. What was the time r Ans. 20 min. past ^- 

QuEsT. 82. Suppose that I have^ -/y of a ship worth 
1^00/; what part of her hiave I left after selling | of ^ of 
ftiy share, and what is it worth? Ans. -^^j worth 185A 

Quest. 33. Fart ItOO acres of land among A, b, q ; so 
fliat B may have 100 morfc thkn A, and c 64 more than b. 

Ans. A 312, B 412, c 476. 

Q0EST. 34i What number is that, from which if there 
be takien j- of |, ■ and 'to the remainder be 'added -^j of -j^, 
thfc^^utn will be 1 ? Ans. 9||- 

QCEsT; S3; There is a number which, if multiplied by |^ 
of 4 of li, will produce ,1 ; what is the square of that 
numbejr? ' ' • , ^ Ans. l-,2^. 

Quest. 36. What leugth must be cut off a board, S^- 
inclies broad, to contain a square fobt, cr as much as 12 
incBkes in length and 12 in breadth ? Ans. 1614 i-^ches. 

J Quest. 37. What sum of money will amount to 1 3S/ 2s 
IBd, in 15 months, at 5 per cent, per annum simple interest i 

Ans. 130/. 

Quest. 3'8. A father divided his fortune among his three 
sons, A, B, c, giving a 4 as often as b 3, and g 5 as often as 

B 6 J 



144 ARITHMETIC. 

B 6 ; what neas the whole legacy, supposing ^*s share wsfc* 
4000/. Ans. 9500/. 

Quest. 39. A young hare starts 40 yards before a grey- 
houndy and is not perceived by him till she has been up 40 
seconds ; she scuds away at the rate of 10 miles an hour, and 
the dog, on view, makes after her at the rate of 1 8 : hovr 
long will the' course hold, and^what ground wiU bcTrun ov^, 
tounting from the outsetting of the dog? 

Ans. 60^5y sec. and 530 yards run. 




UE^T. 40. Two young gentlemen, without private forr 
tiffle, obtain commissions at the same time, and at the age of 
ISy One thoughtlessly spends 10/ a year more than his pay ; 
butjfi shocked at the idea of not paying his debts, gives his 
creditor a bond for the money, at the end of every year, and 
also »^sures his life for the amount; each bond costs him. 30 
shillings, besides the lawful interest of 5 per cent, and to in- 
sure liis life costs hhn 6 per cent. 

The other, having a proper pride, is determined never to 
run in debt ; and, that he may assist a friend in need, perse- 
veres, in saving 10/ every year, for*» which he obtains an 
interest, of 5 per cent, which interest is every year added to 
his savings, and laid out, so as to ansfwerthe effect of com- 
pound mterest. 

Suppose these two officers to meet at the age 'of 50, when 
each recieives from Government 400/ per annum ; that the 
one, Seeding his past errors, is resolved in future to spend no 
more thto. he actually has, after paying the interest for what 
he owes,«^d the insurance oh his life. 

The oth^lk having now something before hand, means in 
future, to spen4 his* full income, without increasiog his stock. 

It is desirable to know how much each has to spend per 
annum, and wliat money the latter has by him to assist the 
distressed, or leaVM^ those who deserve it ? 

Ans. The reformed officer has to spend 66l 19/ lf*5389i 
per annum. 
The prudent officer has to spend 437/ 1 2^ 1 l^-4379i 

per annum. 
And the latter has saved, to dispose of,7 52/ 1 9/ 9' 1 896^. 



END OF THE ARITHMETIC. 



[ 14i' 1 



OF LOGARITHMS^ 



L 



rOG ARITHMS are ihade to facilitate troublesonie caIcH^ 
lations in numbers. This they do^ because they perform 
multiplication by only addition, and division by only subtrac- 
tidn, and raising of powers by multiplying the logarithm by 
the Index of the pbwer j and extracting of roots by dividing 
the logarithm of the number by the index of the root. For, 
lagarithms are numbers sd contri?ed| and adapted to othei^ 
numbers, that the sums and difierences of the former shall 
correspond to, and show, tlie products and <|uoti^nts of th^ 
fatter, &c. 

Or, niore generally, logarithms are the numerical ^tpp^ 
nents of ratios ; Or they are a series of numbers in arith^ 

metical 



Mhata 



* The invention of Logdritbchs is due to Lord Napier, Baron of 
Merckistton, in Scotland, and is properly considered as one of the 
inost useful inventions of modern times. A tabl^ of tbe<ie numben 
was first published by the inventor at Edinburgh, iii the year 
16 1 4, in a treatise entitled Canon Mirificum liOgnrtthmorum ; which 
was eagerly received by all the learned thfotighoat Europe. Mr. 
Henry Briggs, then professor of geometry at Qresham College^ 
soon after the discovery, went to visit the noble nventor ; aft»r 
which, they jointly undertook the arduous task of computing new, 
tables on this subject, and redacing them to a more cdnvenieht 
form than that which was at first thought of. But Lord Napier 
dying soon after, the whole burden fell upon Mr. Briggs, who, 
with prodigious labour and great skill, made an entire Canon, 9.0 
cording to the new form, for all numbers from i to 'iOOncs and 
from 9 XX,0\o 10 100, to 14 places of figures, and published it at 
London in the year 1(J24, in a treatise euiitled Arithmetica Loga<> 
jtlthmica, with directions for supplying the intermediate parts. 

Vol.1. L ' This 



U6 LOGARITHMS. 

metical progression, answering to another series of numbers 
in geometrical progression. 

Th /^* ^» ^* ^* ** ^* ^' Indices, or logarithms* 
tl> 2, 4, 8, 16, 32, 64, Geometric progression* 

^ rO, 1, 2, 3, 4, 5, 6, Indices, or logarithms* 
ll, 3, 9, 27, 81, 243, 729, Geometric progression. 

{0, 1, 2, 3, 4, 5, Indices, or logs. 

1, 10, 100, 1000, 10000, 100000, Geom. progres. 



Or 



Where it is evident, that the same indices serve equally 
for any geometric series ; and consequently there may be an 



This Canon was again published in Holland by Adrian Vlacq,. 
in the year l6U8, together with the Logarithms of all the numbers 
which Mr. £riggs had omitted -, but he contracted them down to 
10 places of decimals. Mr. Briggs also computed the Logarithms 
of the sines^ tangents, and secants, to every degree^ and centesm, 
or 100th part of a degree^ of the whole quadrant ; and annexed 
them to the natural sines, tangents, and secants, which he had 
before computed, to fifteen places of figures. These Tables, 
with their construction and use^ were first published in the year 
1633, after Mr. Briggs's death, by Mr. Henry Gellibrand, under 
the title of Trigonometria Britannica. 

Benjamin Ursinus also gave a Table of Napier's Logs* and of 
tines, to every 10 seconds. And Chr. Wolf, in his Mathematical 
Lexicon, sajs that one Van Loser had computed them to every 
single second, but his untimely death prevented their publica- 
tion. Many other authors have treated on this subject ;' but as 
their numbers are frequently inaccurate and incommodiously dis« 
posed, they are now generally neglected. The Tables in most 
repute at present, are those of Gardiner in 4to, first published in 
the year 1742 3 and my own Tables in 8vo, first printed in the 
year i78^> where the Logarithms cf all numbers may be easily 
found from 1 to lOCXXXXX) -, and those of the sines^ tangents, and 
secants, to any degree of accuracy required; 

Also, Mr. Michael Taylor's Tables iu large 4to, containing the 
common logarithms, and the logarithmic sines and tangents to 
cVery second of the quadrant. And, in France, the new book of 
logarithms by Calletj the 2d edition of which, in 1795, has the 
tables still farther extended, and are printed with what are called 
stereotypes, the types in each page being soldered together into a 
solid ^ass or block. 

Dodson's Antilogaritbroic Canon is likewise a very elaborate 
work, and used for finding the numbers answer'uig to any givea 
logarithm. ' 

endless 






tOGARITHMS. 147 

endless T^rkty of systems of Ic^arithms^ to the samecosi- 
mon numbersy by onlyt chaoaging the second term^ 2, 3/or 
IG, &c.^ of the geometrical series of whole numbers; and 
by interpolation the whoie system of numbers may be made 
to enter: the geometric series, and receive^thieir proportkmal 
logarithms, whether integers or decimals. 

It is alao apparent, from the nature of these ^ries,, that if 
any two indices be added together, their sum. will be the 
index of that number which is equal to the product of the 
two terms, in the geometric progression, to which those in- 
dices bdpng^ Thus, the indices 2 and S> beifig* zsMfii. to- 
gether, make 5*; and the numbers 4t and .8, on the termts 
corresponding to those indiees) being multiplied togetherj 
tnake 32, which is the number answering to. i^e itidex 5. 
" ■ ' • * 

In like manner, if any one index. b6 subtracted from 
another, the difierence willbe the index of that number 
which is equal to the quotient of the twa tenns to which 
those indices belong. Thus, the index 6> minus the index 
4, is = 2' ; and the terms corresponding to those indices are 
64 and 16, whose quotient is = 4, which is the number 
answering to the index 2* 

For the same reason, if the logarithm of any number 
be multiplied by the index of its power, the product will 
be equal to t^^e logarithm of that powef. Thus, the 
index or logarithm of 4, in the above series, i» 2; and 
. if this number be nmltiplied by 3j the product will be 
=s 6 ; which is the logarithm of 64^ or'Ae' third pDwer 
of 4.- •' . / -■ ' . .' - 



.>' 



And, if the Icgaritbm of any number bexliyided by the 
index of its root,.' the qtiOtient. yrill b&. eqiial to' the logarithm 
of that root. Thus, the iskdex . or logarithm of:.64 .is..6; 
and if this number be divided by 2, the quotient will b<e 
= 3; whiihris the logaritHm of 8, or the sqi^arerroot 
of 64. 

' - * « . < 

The logarithms most convenient for jpriactice, are such as 
are adapted" to a geometric series iricreasmg in a tenfold pro- 
portion, as in the last of the above forms *; and are those 
which are to be found, at present, in most of the common 
tables on this subject. The distinguishing mark of this 
system of logarithms is, that the index or logarithm of 10 
is 1 5 that of 100 is 2 5 that of 1000 is 3 5 &c* And, in 

L t, decimals. 



\ . 



148 LOGARITHMB. 

^•cimils, the logarithm of *1 b — 1 ; that of '01 i$^9i that 
of OOl is— 3} Slc* The log. of 1 being in every lystem. 
Whence it fellows, that the logarithm of any numbtr be- 
tween 1 and 10, must be and some fractioittl partem s^ 
that of a number between 10 and lOQ, wiU be 1 and sopie 
fractional parts \ and so on, fer any other number whatever. 
^d since the integral part of a logarithm, usually called the 
Index, or Characteristic, b always thus readily feund, it is 
commonly omitted in the tables ; being left to be suppiied 
by the operator himself, as occasion requires. 

Another Definition of Logaritiuns is, that the logarithm of 
any number is the index of that power of some other nmn^ 
her, which is eaual to the nven number. So, if there be 
N zs #*, then i» is the log. of M ; where n may be either pcK 
sitive or negative, or nothing, and the root r any number 
whatever, according to the different systems of k^sirithms. 
When nis TzOf then N is ss 1, whatever the value of r is-^ 
which shows, that the k)^. of 1 b always 0, in every systepi 
of logarithms. When jiis ::= 1. thenN b =? r/ so thatthe 
radix r b always that number whose log. b 1, in every 
system* When the raduL ris s 2*7l8281828<b59 &<:, the 
indices n are the hyperbolic or Napier's log* of the i^umtMurs - 
N; so that n is always the hyp. log. of the number N or 
(2-718 &c.)^ 

« But when die ra£x r is s 10, thaa the index n becomes 
the common or Briggs's log. of the number N: so that Ihe 
common log. of any number 10" or N, b n the index of 
thsit power of lO which is equal to the said number. Thus 
100, being the second power of 10, will have 2 for its loga^ 
rithm i and 1000, being the third power of 10, will have 9 
for its logarithm: hence also, if 50 be s: iO'-^^'^^ then 
b 1*69897 Ae cofdmon log. of 50. And, in^genentl, the 
following docuple series of terms, 

viz. 10*, lO', 10*, 10% 10*, 10-^, 10-*, 10"», lOr-*, 
or 10000, 1000, 100, 10, 1, -1, -01, -OOl, -OOOl, 
have 4, 8, 2, 1, 0, —1, —2, — S, —4, 

for their logarithms, respectively. And from this ^alf.of 
numbers and logarithms, the same properties itosily follow; ^ 
as above mentioned. 



I'ROBLSM* 






LOGARITHMS. H9 



PROBLEM. 

To compute the Logarithm to any of the Natural Numbers 

1, 2, 3, 4, 5, isfc. 

RULE 1*. 

Take the geometric series, 1, 10, IGO, 1000, 10000, &c. 
And apiply to it the arithmetic series, 0^ 1, 2, 5, 4, &c. as 
logarithms.— Find a geometric mean between 1 and 10, or 
between 10 and 100, or any other two adjacent terms of the 
series, between which the number proposed lies.-^Ii^ like 
manner^ between the mean, thus found, and the nearest ex- 
treme, find another geometrical mean ; and so on, tilli you 
arrive within the proposed limit of the number whose loga^ 
rithm is sought. — ¥ind also as many arithmetical means, in 
the same order as you found the geometrical pnes, and these 
will be the logarithms answering to the said geometrical 
means. 

EXAMPLE. 

Let it be required to find the logarithm of 9. 
Here the proposed number lies between 1 and 10. 
First, then, the log , of 10 il 1, and the log. of 1 is ; 

theref. ^_+0 -f- 2 = -J- ae -5 is the arithmetical mean, 

and v/ 10 X 1 = V 10 — 3*1622777 the geom. mean j 
hence the log. of 3-1622777 is '5. 

Secondly, tl ie log, o f 10 is 1, and the log. of 8-1622777 is 'B\ 

theref. 1 -f '5 ^ 2 = '75 is the arithmetical mean, 

and -v/10 X 3-1622777 = 5*6234132 is the geom. mean J 
hence the log. of 5*6234132 is '75. 

Thirdly, the log, of 10 is 1, and the log. of 5-62341 32 is -75 -, 

theref. 1 + '75 -r- 2 = -875 is the arithmetical mean^ 

and v^ 1 X 5-6235 1 32 =» 7 -4989422 the geom. mean ; 
hence the log. of 7-4989422 is -875. 

Fourthly, tfa elog. of lO ils 1 , and the log. of 7-49*9422 is '875 i 

theref. 1 -f -875 4- 2 =s -9375 is the arithmetical mean, 

and V iO X 7*4989422 » 8*6596431 the geonr. mean| 
hence the log. of 8*6596431 is *9S75. 



» *^M*4M4 



* The reader who wishes to inform himself more particularly 
coBceming the history, Bature, and constraction of Logarithms, 
may oonsoll the introdnction to my Math^malical Tables, lately 
published, where he will §nd his curiosity amply gratiled. 

Fifthly, 



150 LOGARITHMS. 

Fifthly, the l og.oflOi sl, and the log. of S*659643 1 is -9375; 

theref. 1 +'9375 ^-g = '96875 is the arithmetical mean, 

and v'lOx 8-65964'31 = 9*3057204 the geom. mean 5 
hence the log. of 9*3057204 is -96875. 

Sixthly, the log. of 86596431 is -9375, and the, log. of 
9- 30 57204 is '96875 ; 

theref. '9375 + '96875 -^ 2 = '95 3125 isthearith.mean, 

and -/8'6596431 x 9-3057204 = 8-9768713 the geo- 
metric mean ; 
hence the log. of 8-9768713 is '953125. 

And proceeding in this manner, after 25 extractions, it 
tri?l Ije fbimd^hat the logarithm of 8-9999998 is -9542425; 
which may be taBen for the logarithm of 9, as it differs so 
little from it, that it is sufficiently exact for all practical pur- 
poses. And in this manner were the logarithms of almost all 
the prime numbers at first computed. 



RULE II*. ' 

Let h be the number whose logarithm is required to be 
found ; and a the number next less than ^, so that h'^azz.Xy 
the logarithm of a being known \ and let s denote the sum 
of the two numbers a ^ h. Then 

, 1. Divide the constant decimal -8685889638 &c, by /, 
and reserve the quotient : divide the reserved quotient by 
the square of /, and reserve this quotient : divide this last 
quotient also by the square of /, and again reserve the 
quotient : and thus proceed, continually dividing the last 
quotient by the square of /, as long as division can be 
made. 

2. Then write these quotients orderly under one another, 
the first uppermost, and divide them re-pectively by the odd 
numbers, 1, 3, 5, 7, 9, 8cc, as long as division can be made; 
that is, divide the first reserved quotient by 1, the second by 
S, the third by 5, the fourth by 7, and so on. 

3. Add all these last quotients ^together, and the sum will 
be the logarithm of h -=-. a; thereJFore to this logarithm add 
also the given logarithm of the said next less number a, so 
will the last sum be the logarithm of the number h proposed* 



f For the demonstration of thi^ rule, s^ npy Mathematics 

Tables, p. 109, &c. 

/ ' ' That 



LOGARtTHMS, 



J51 



. That IS, 

/; 111 

Log. oi b IS log. a'\--jx {1+ "57+ ■57 + '77^ + ^^'^ 

where n denotes the constant given decimal '8685889638 &c. 



EXAMPLES. 



Ex. 1 . Let it be required to find the log. of the number 2, 
Here the given number b is 2, and the next less number a 
is 1, whose log. is 0; also the sum 2 -|- 1 = 3 zz /, and its 
square ,s* = 9. Then the operation will be as follows : 



3) 


•868588964 


1 ) 


•289529654 ( 


[ -28952965* 


9) 


•289529G54 


3 ) 


32169962 ( 


; 10723321 


9 ) 


32169962 


5 ) 


3574440 ( 


[ 714888 


9 ) 


3574440 


7) 


397160 ( 


: 56737 


9) 


397160 


9) 


44129 1 


; 4903 


9 ) 


44129 


11 ). 


4903 ( 


[ 446 


9 ) 


4903 


13 ) 


545 '( 


; 42 


9) 


545 


15 ) 


61 ( 


4 


9) 


61 




1 


» 


" 








log. of T 


- -301029995 








add log. 1 
log. of 2 


- 000000000 




- -301029995 



Ex. 2. To compute the logarithm of the number 3. 

Here 3 = 3, the next less number ^ = 2, and the sum 
« + ^ sr 5 = J, whose square s^ is 25^ to divide by which, 
always multiply by '04. Then the operation is as follows: 



5 
25 
25 
25 
25 
'^5 



•868588964 

173717793 

6948712 

277948 

11118 

445 

18 



1 
3 
.5 

"7 

9 

11 



173717793 

6948712 

271948 

11118 

445 

. 18 

log. of 4 - 
log. of 2 add 



•173717793 

2316237 

55590 

1588 

50 

2; 

'176091^60 
3Q1029995 



log. of 3 sought -477121255 

Then, because the sum of the logarithms of numbers, 
gives the logarithm of their product ; and the difference of 
^he logarithms, gives the logarithm of the quotient of the 

numbers ; 



15? 



LOGARITHMS. 



pumbers ; from the above two logarithms, and the logarithm 
of 10, which is 1, we may raise a great many logarithms', as 
in the following examples ; 



EXAMPLE S. 
Because 2x2 = 4, therefore 
to log, 2 - •30102!:'995| 
add log. 2- '301029995 J 

t^mmrn^^ I. ■ ■ ■ 

sum is log. 4 -6020599911 



FXAMPLE 4. 

Because 2 x 3 = 6, therefore 
to log. 2 - -301029995 
add log. 3. -477121255 

sum is log. 6, '778151250 



EXAMPt^ 5. 

Because 2^ = 8, therefore 
tnulu by 3 



301029995^ 



EXAMPLE 6. 

Becayse 3* = 9, therefore 
log. 3 - -4771212541^ 
muk. by 2 2 

gives log. 9 954242 09 



EXAMPLE 7. 

Because V* = 5, therefore 
from log. 1 1 OOOOr^OOOO 
take log. 2 -3010299951 

leaves log. 5 -69897000411 



EXAMPLE ^r 
Because 3x4 = 12, thereferer 
to log. 3 - -477121255 
add log. 4 -602055(991 



gives log. 8 -903089987 gives log. 12 1-0791 S124« 



■Wi W 



TT 



And thus, computing, by this general mle, the logarithms 
tb the other prim^ numbers, 7, U, 13, 17| 19, 23, &c, and 
then using composition and division, we may easily find as 
many logarithms as we please, or may speedily examine any 
logarithm in the taUe *• 



f^^^i^^mr^ 



TT^ 



>^»» 



* Th^te are, besides these, many other ingenious methods, 
which later writers have discovered for ^ding the logarithms of 
numbers, in a much tasiet way than by the ori^nal inventor ; bat> 
as ^bey cpoodt ^ u|iderstood w|thpci| a knowledge of some of the 
higher branches of t|ie mathematics^ it is thought proper to omit 
thefn» ana to r^fer the reader to those works which are written 
expressly on the subject. It would likewise much exceed the 
limits of this cbfnp^pdium, to point out all the peculiar artifices 
that are made use of for constructing an entire table of these num- 
bers } but any information of this kind^ which the learner may 
"ivtsh to obtain^ may be found in my Tables, before mentioned. 



Descrtftiw, 



LOOARlTHliMS. US 

Vescripiim and Use rfthe Tablr j/^Logae^thms. 

HavinC explained the manner of forming a table of the 
logarithms of numbers, greater than unity; tne next thing to 
be done is^ to show how the logarithms of fractional quan- 
tities may be found. In order to this^ it may be observed, ' 
that as in the former case a geometric series is supposed to 
increase towards the lefty from unity, so in the latter case 
it is supposed to decrease towards the right hand, still be- 
ginning with unit ; as exhibited in the general description, 
page 148, where the indices being made negative, still show 
the logarithms to which they belong. Whence it appear^^ 
that as -h 1 is the log. of 10, so — I is the log. of Vrr or '1 > 
and as + 2 is the log. of 100, so -- 2 is the log. of ^w or 
^01 : and so on* 

Hence it appears in general, th^t all numbers which con« 
sist of the same figures, whether they be integral, or frac- 
tional, or mixed, will have the decimal parts of their loga* 
rithms the same, but diSering only in the index^ which will 
be more or Jess, and positive or negative, according to the 
place of the first figure of the number. 

Thus, the logarithm of 2651 being 3-423410, the log. rf 
tVj or i^y or twyj ^c, part of it 5 wil^ be as'fbllows : 



Numbers. 


Logarithms. 


2 6 5 1 


3 -4 2 3 4 1 


2 6 5-i 


2-423410 


2 6-51 


1-423410 


2*6 5 1 


0-423410 


•2651 


-1-423410 


•0 2 6 5 J 


-2-423410 


2 6$! 


-3 -423410 



Hence it abo appears, diat the ixKlex of any logarithm, i% 
ftlways less by 1 than the number of integer figures which 
the natural number consists of; or it is equal to ihe distance 
of the first figure firom the f^e of iinit% ox first place of in- 
tegers, whether on the kft', cnr on the right, of it : and this 
index is constantly to b^ placed on the left-hand side of the 
decimal part of the logarithm^ 

When there are integers in the given number, the index 
IS always afiirmative ; but when there are no integers, the 
index is negative,' and b to be marked by a short line drawn 
l^efore it, or else above it. Thus, 

A number having 1, 2, 3, 4, 5, &c, integer places, 

l;|ie index of it& log. is 0, 1, % 3, 4| &c. or 1 less than those 
* places. 
^ And 



15* LQGAmTHMS. 

And a decimal fraction having its first figure in the 

1st, 2d, Sd, '4th, &c, place of the decimals, has always 
— 1, —2, —3, —4, &c, for the index of its logarithm, 
, It may also be observed, that though the indices of frac- 
tional quantities are negative, yet the decimal parts of their 
logarithms are always affirmative. And the negative mark( — ) 
may be set either before the index or over it. 

1. TO nND, IN THE TABLE, THE LOGARITHM TO AKY 

NUMBER*. 

1. If the given Number be less than 100, or consist of 
only two figures \ its log. is immediately found by inspection 
in the first page of the table, which contains all numbers 
from 1 to 100, with their logs, and the index immediately 
annexed in the next column. 

So the log. oi5 is 0*698970. The log. of 23 is V361728. 
The log. of 50 is 1 -698970, And so on. 

2. y^ the Number be more than 100 but less than 10000; 
that is, consisting of either three or four figures; the 
decimal part, of the logarithm is found by inspection in the 
other pages of the table, standing against the given number, in 
this manner; viz. the first three figures of the given number in 
the first column of the page, and the fourth figure one of those 
along the top line of it ; then in the angle of meeting are the 
last four figures of the logarithm, and the first two figures of the 
same at the beginning of the same line in the second column 
of the page : to which is to be prefixed the proper index, 
which is always 1 less than the number of integer figures. 

So the logarithm of 251 is 2*399674, that is, the decimal 
'399674 found in the table, with the index 2 prefixed, be- 
cause the given number contains three integers. And the 
log. of 34*09 is 1-532627, that is^ the decimal -532627 
found in the table, with the index 1 prefixed, because the 
given number contains two integers. . 

3. But if the given Number contain more than four figures^; 
take out the logarithm of the first four figures by inspection 
in the table, as before, as also the next greater logarithm^ 
subtracting the one logarithm from the other, as also their 
corresponding numbers the one from the other. Then say» 

As the difference between the two numbers. 
Is to the difference of their logarithms, 
So is the remaining part of the given number. 
To the proportional part of the logarithm. 

I I .1 1 ' I ■■■■■'■ JL I 

* See ,tfce table pf Logarithms^ after the Geometry, at; th^ endi 
of this volume. 

* . Whigh 



LOGARITHMS. 



155 



'Which part being addejd to th« less logarichm^ before taken 
out, gives the whole logarithm sought very nearly. 



EXAMPLE. 



To find the logarithm of the number 34t*0926. 
The log. of 340900, as before, is 532627. 
And log. of 341000 * - is 532754. 

Thediffs.^are 100 and . 127 



Then, as 100 : 127 : : 26 : 33, the proportional part. 
. This added to •- - - 332627, the first log. 
Gives, witii the index, 1 •532(360 for the log. of 34'0926. 

4. If the number consist both of integers and fractions, or 
is entirely fractional ; find the decimal part of the logarithm 
the same as if all its figures were integral ; then this, having 
prefixed to it the proper index, will give the logarithm re- 
quired. 

5. And if the given number be a proper vulgar fraction : 
subtract the logarithm of the denominator from the loga-, 
rithm of the numerator, and the remainder will be the loga- 
rithm sought ; which, being that of a decimal fraction, must 
always have a negative index. 

6. But if it be a mixed number ; reduce it to an improper 
fraction, and find the difference of the logarithms of the 
numerator and denominator, in the same manner as before. 



EXAMPLES. 



l.Tofindthe.log. off J, 
Log. of 37 - 1-568202 
Log. of 94 - 1*973 123 

Dif. logl of f|. - 1-595074 

Where the index 1 isnegative. 



2. To find the log. of 17||. 
First, n4iJ=\V- Then, 
Log, of 405 
Log. of 23 

Dif. log. of 17i^ 



2-607455 
1-361728 

1-245727 



II, TO FIND THE NATURAL NUMBER TO ANT GIVEN 

LOGARITHM. 

This is to be found in the tables by the reverse method 
to the former, namely, by searching for the proposed loga- 
riit]im among those in the table, and taking out the corre- 
sponding number by inspection, in which the proper number 
of integers are to be pointed off^, viz. 1 more than the 
index. For, in finding the number answering to any given 
logarithm, the index always shows how far the first figure 

must 



156 logarithms/ 

icnst be removed from the place of units, viz. to the Ifcfi; 
hand, or" integers, v^hcn tlxe index is affirmative; but to the-' 
right hand, or decimals, when it is negative. 



EXAMPLES* 

So, the number to the log. P532888 » 34*1K 
And the number of the log. 1*5S2882 is *34i 1. 

But if the logarithm cannot be exactly found in the table ^ 
take out the next greater and the next less, subtracting the 
one of these logarithms froiii the dther, as also their natural 
numbers the one from the other, and the less logarithm from 
the logarithm proposed. Then say> 

As the difference of the first or tabular logarithms^ 
Is to the difference of their natural numbers. 
So is the differ, of the given log. and the least tabular log. 
To their corresponding numeral difference. 
Which being annexed to the least natural number above 
taken, gives the natural number sought, corresponding to 
the proposed logarithm^ 



EXAMPLfi* 

So, to find the natural number answermg to the gives 

logarithm 1-532708. 
Here the next greater and next less tabular logarithms, 
with their corresponding numbers, are as below : 

Next greater 532754 its num. 34iOOO; given log. 532108 
Next less ,532627 its num. 340900; next less 532627 

Differences 127 — 100 — 81 



Then, as 127 : 100 :: 81 : 64 nearly, the numeral difiTer. 

Therefore 34*0964 is the number sought, marking oflTtwo 
integers, because the index of the given logarithm is 1. 

Had the index been negative, thus 1 -532708, its ccmtc- 
sponding number would hxvt been *S4Q064, whoUy d«- 
cimaL 



MWLTIPLI- 



£ 157 } 



MULTIPLICATION, bt LOGARITHMS* 



Ta^e out the logarithms of tlie j&ctorfi from the table^ 
then add them together, and their sum will be the logarithm 
of the product required. Then, bj^ means of the table, 
take out the natural number, answering to the sum, for the 
product sought. 

Ob^i^ving to add what is to be carried from the ijecimal 
part of the logarithm to the affirmative index or indices, or 
else subtract it from the negative. 

* Also* adding the indices together yrhen they are of the 
samfs kind, bom affirmative or both negative ; but subtract^ 
ing the less, from the greater, when uie one is affirmative 
gnd the other negative, >nd prefixing the sign of the greater 
to the remainder. 



EXAMPLES* 



. 1. To multiply 23-14 by 
5-062. . 
Numbers. Logs. 
23.14 - 1'364.363 
5-062 - 0-704322 



Product 1 17*1347 2*068685 



2. To multiply ^-531 92^ 

by 3-457291. 
Numbers. Logs* 
2-581926 - 0-41 1^44 
3-457291 - 0-538736 



Prod. 8-92648 - 0-9506W 



X To mult. 3-902 and 597-16 

^4 '0314728 all together. 

Numbers. Logs. 

3-902 - p-591287 

- 597-16 - 2-776091 

•03 14728- 2-497935 



>ro4. 7S-3333 - 1-865318 

9 

Here the — 2 cancels the 2, 
and the 1 to carry from the 
di^cimals is set down. 



4.To mult.3*586, and2-1046, 
and 0-8372, and 029.4-ali 
together. 

Numbers. Logs. 
3-586 . 0'55461Q 
2-1046 - 0-323170 
0-8372 -1-922829 
00294 — 2-468347 



Prod. 0'1857618--l-268956 



Here the 2 to carry cancels 
the— 2, and there remains thir 
I — 1 to set dawn . 



DITISIOK 



r 158 ] 



DIVISION BY LOGARrrHMS. 

RULE. 

• * « 

From the logarithm of the dividend subtract the loga- 
rithm of the divisor, and the number answering to the re- 
mainder will be the quotient required. 

Observing to change the sign of the index of the divisor, 
from affirmative to negative^ or from negative to affirmative^ 
then take the sum of the indices if they be of the same name, 
cr their difference when of different signs, with the sign of 
the greater, for the index to the logarithm of the quotient. 

And also, when 1 is borrowed, in the left-hand place of 
the decimal part of the logarithm, add it to the index of 
the divisor when that index is affirmative, but subtract it 
when negative ; then let the sign of the index arising from 
hence be changed, and worked with as before. 



EXAMPLES. 



I. Tt> divide 24163 by 4567. 

Numbers. Logs. 
Dividend 24163 - 4-383151 
Divisor 4567 - 3-659631 



Quot. 5*29075 0-723520 



3. Divide -06314 by -007241. 

Numbers. . Logs. 
Divid. -06314 -2-800.'^05 
Divisor -007241 —3-859799 



2.Tpdivide37-149by523-7e. 

Numbers. Logs. 
Dividend 37- 1 49 - 1-569947 
Divisor. 523-76 - 2-719132 



Quot. 8-71979 0-940506 
Herie 1 carried from the 



Quot. -0709275 -2-8508 15 



4.Todivide-7438byl2-9476. ' 

Numbers. Logs. 
Divid. -7438 -1-871456 ' 
Divisor 12-9476 1-112189 



Quot. -057447 - 2-759267 
Here the 1 taken from •the 



decimals to th6 —3, makes it — 1, makes it become —2, to 

becopie— 2,whichtakenfroi^ set down, 

the other — 2, . leaves re- ' 
maming. 



,' Note. As to the Kule-of-Three, or Rule of Proportion, 
it is performed by adding fhe logarithms of the 2d and Sd 
tcims, and subtracting that of the first term from their sum. 



INVOLUTION 



( 159 ] 



mVOLUTION BT LOGARITHMS. 

RtTLE. ' • . • '" 

Ta«e out the logarithm of the given number from the 
table. Multiply the log. thus found, by the index of the 
power proposed. Find the number answering to the pro- 
duct, and it will be the power required. 

Ifdie* In multiplying a logarithm with a negative index, 
by an affirmative number, the product will be negative. 
But what is to be carried from the decimal part of the loga^ 
rithtn,.will always be affirmative. And therefore their dif- 
ference will be the index of the product, and is always to be 
made of the same kind with the greater. ^ 



EXAMPLES. 



1. To square the number 
.2-5791. 
Numb. I'Og. 

Root 2'5791 - - 0-411468 
The index - - 2 



Power 6-65174 0-822936 



2. To find the cube of 
3-07146. 
Numb. Log. 

Root 3-07146 - - 0*487345 
The index - - S 



Power 28-9758 1 -462035 



8. To raise -09163 to the 4th 
power. 
Numb. Log. 

Root -09163 —2-962038 
The index - - 4 



Pow. -000070494 - 5 -8^8 152 



Here 4 times the negative 
index being — 8,and 3 to carry, 
the difference — 5 is thp index 
of the product. 



4. To raise 1 -0045 to the 
365th power. 
Numb. Log. 

Root 1-0045 - - 0-001950 
The index - - 365 



9750 
11700 
5850 



Power 5-14932 0*711750 



EVOLUTIOK 



[ 160 i 






'ZVOLXmON IT LOCillUtllMSl. 

Take the log: of the fiv^ pmmber out of the tabled. 
Divide the log. thus round by the index of the root. Thent 
dur number antveriag to the quotientj iTillbf the root. 

Hoii, When the index of the logsurithmj to be divided, W 
negative, and does not exactly contain the divisor, without 
some remainder, increase the index by such a number as will 
make it exactly divisible by the index, carrying the units bor-* 
rowed, as so many teni> to the left-hand place of the decimalf 
and then divide as in whole numbers* 



Ex. 1 .To find the square root 
of 365. 
Numb. Log. 

Power 365 2)2*562293 
Root 19-10496 1-28114.64 



"Ex. 3. To find the 10th root 

of 2. 

Numb. Log. 

Power 2 - 10 ) 0*301030 

Root 1-071173 0-030103 



Ex. 5. To find V' -093. 
Numb. Log. 

Power -093 t ) - 2-9684'»3 
Root -304^959 ~ 1-4842414 

Here the divisor 2 is con- 
tained exactly once in the ne 
gative index —2, and there- 
Fore the index of the quotient 
is —I. 



Exi 2. To find the 8d foot of 
12345. . 
Numb. Log* 

Power 12345 3)4091491 
Rpot 23-1116 1-363830J. 



Ex. 4. To find the 365th root 
of 1045. 
Numb. Log* 

Powerl'045 365)0^019116 
Root 1-000121 0-0000524 



Ex. 6. To find the ^-00048, 

Numb. Log. 

Power '00048 3)^4'681241 
RcfQt -0782973 - 2-893747 

Here thedirisor 3, not beipjBT exact- 
ly contained lA — ^, it is augmented 
by 2, to malw up 6, in which the di-* 
visor it contained just 2 times; then 
the 2, thus bon;owed, being ci^rriedt^ 
the decimal figure (>, makes 2^f which 
divided by d, ^ives 8, &c. 



Ex. 7. To find 3-1416 x 82 x fj. 
Ex. 8. To find -02916 x 751-3 X -^ 
Ex. 9. As 7241 : 3*58 :: 20-46 : ? 
Ex. 10. As v'724 : v^4| : : 6-927 : ? 



ALGEBRA. 



[ i«i J 



-■•;• • 



A L G E B R A; 



DEFINITIONS AND NOTATION. 

1. xjLLCxEBRA i^ the scieoce of computing by sy^nhplju 
It Is sometimes also called Analysis ; and is a general kind 
of arithlnetic, or universal way of computation. 

2. In this science, quantiti^^ of all kindaar^ re^es^ented by 
the letters of the alphabet. Ahd tbeo(}^rations' to be per* 
^ibrmed with them, as addition or sjbibtractiDQ^ &C9 aiie ile* 

noted by certain simple character^ instead; of being ^i^e^s^d 
hy words at^ength. ._..... 

3. In algebraical questions, s6mt quantities^ are kh^w^ oit' 
given, viz. those whose values are known: and others-un- 
known, or are to be found otit, vils. thos^ whos^ values Ve 
not known. The fonrter of these are represented by the 
leading letters of the alphabet, a, b^ r, dy &c \ and the lattef » 
or unknoi^ quantities, by the final lettei^, z> j^ at, u^ &<f.y 

4>. The fiiaracters lis^sd to denote the opiorations^ av^ 
chiefly the JFoUowing : ' 1. 

+ signifies addition, and is named //wi** " 

-^ signifies subtraction, and is named minus. , 

X or . signifies multiplication, and is named into^ . j 

-i- signifies division, and is named by* 

•/ signifies the square root ; ^ the cube root s ^ thi| 
4th root, &c i and ^ the /7th root. 

: : : : signifies proportion. • . 

ss signifies equality, and is named 9^iud iff. 

And so on for other operations. 

Thus a^b denotes that . the number tepresented by.^ is 
to be added to that represented by tfk 

a — b denotes, that the number represented by j is to; be 
subtracted from that repiiresented by ii. 

aKn b denotes the difference pf a ahd ^, when it, id not 
known which is the greater. - 

Vot. I. M n^, oli* 



162 ALGEBRA. 

aBf or a X if or a.i^ expresses the product^ by multipli- 
cation^ of the numbers represented by a and i. 

# -r i^, or-T-, denotes, that the number represented by # 

is to be divided by that which is expressed by i. 

a : b :: e : dy signifies that tf is in the same proportion to t, 
as r is to d» 

X =^ a -^ b -i^ CIS 7kn equation, expressing that x is equal 
to the diiSerence of a and by added to the quantity r. 

^a, or M^, denotes the square root of « j ^a, or ^i^, the 

cube root of ^ ; and ^^a^ or a^ the cube root of the square of « ; 

zho ^y or sT^f is the Mth root oia\ and ^a* or tf« is the 

iith power of the mth root of a, or it is a to the ~ power. 

d'^ denote* the square of as n^ the cube of ^ ; o^ the fotuth 
p0wer oi a.: and if^ the Mth power of a, 

d + ^ >^ r, or (rt 4- i^) <•, denoted the product of the compound 
^uc^tit]^ n^h multiplied^ by the ^i^le quantity r. U»ng 

the bar , or the parenthesis ( ) as a vi^culum^ to codtieGt 

•Several aimjple quantities into one compound. 

/ «.+ i-ra — ^*or-7 — y, expressed like a fraction, means 

flftft quotient oi<jL\ it.divided by a^b. 

i/a^-^^cd^ rov {fib^ + 4ri)^9 is the square root of the com* 

I. ,1. I 

poixn4 quantity /l^4«r^. And^v^^ + ^^f or r («^ + c£ff 
denotes the product'of c into the square r<>ot of the coxnpocUid 
quantity ^3 + r^. .,. • 



fl H- ^ — r , or (^j + ^ -^ f)', denotes the cube, or third 
power, of the compound quantity a -f ^ — r. 

3tf denotes that the quantity a IS' to be taken 3 times, and 
4 (o 4- ^) i^ 4* timed ^i + ^ And these numbers, 3 or 4y 
showing how often the quantities are td be taken, or multi- 
plied, are called Co-efficients. • . 

Also \x denotes that x is multiplcd by ^ ; thus f x * or 

- 5. Like Quantities, are those which consist of the same 
letters, and powers. As a and 3a ; ch* 2db and ^ab ; or 
3flf*Ar and -r bs^bc^ 

6. Unlike Quantities, are those which consist of different 
letters, or dffierent powers. As a and ^^ or 2tf and d^\ or 
%§ff' and %abc. • 

- 1. Simple 



DEFINITIONS' ANiJ NOTATlOiJ. iss 



it* 



Simple Quantities, are those which consist of On^ term 
onjy. As Say or Saby or Saic^ 

8. CompQund Quantities, are those which consist of two qj* 
more terms. A& « 4- ^, or 2<i — Sc, or a + 2b -^ Sf. 

9. And when, the compound quantity consists of tw^ 
terms, it is called a Binomial, as a + ^} when of three term^ ^ 
k is a Trinomial, as tf + 2^ ~ 3^r ; when of foiir terms^ a 
Quadrinomial, as 2a '^ $b -^ /^ -^ 4d \ and so - on. Also, a 
Multinomial or Polynomial, consists of many tefmsi 

10. A Residual Quantity, is a binomial having one of the 
terms negative. As a -^ ^^. 

1 1 . Positive or Affirmative Qu^titiesj are those which are 
to be added, oi' have the sign +. As a or + ^j, or ^J : for 
when a quantity is found without a sign, it is understood tq 
be positive, or have the sign +- prefixed. 

12. Negative Quantities, are those which ^ are to be sub** 
tracted. As — ii, or -'2tf^, or -3ai% 

13. Like Signs, are either all positive ( + ), (Jr all nega- 
tive (-). 

14. Unlike Signs, are when some are positive ( *f ), and 
©thers negative ( — ). 

1 5. The Co-efficient of any quantity, as shown above, is 
the number prefixed to it. As 3, in the quantity Sab. 

16. The Power of a quantity (a]y is its square {a^)f or 
cube («^), or biquadrate (^'*), &c; callfed also, the 2d power, 
or 3d power, or ^th power, &c. 

17. The Index or Exponent, is the number which denotes 
the power or root of a quantity. So 2 is the exponent of 
the square or second power /»* ; and 3 is the index of the 

cube or 3d power ; and ^ is the index of the square root, a^ 

or v^^ 5 and \ is the index of the cube root, d^y or ^a, 

18. A Rational Quantity, is that which Has no radical 
sign (-y/) or index annexed to it. Ag tf, or Sab. 

19. An Irrational Quantity, or Surd, is that which has 
not an exact root, of is expressed by means of the radical 

sign v/. As v^ 2, or v^j, or^a^y or ai^. 

20. The Reciprocal of any quantity, is that quantity in- 
verted, or unity divided by it. So, the reciprocal of tf, or 

a , I f. a , i 

— ,is — y and the reciprocal of "T" is — :• 

M2 21. The 



i64 ALGEBRA. 

21 • The letters by which any simple quantity is expressed^ 
may be ranged according to any order at pleasure. So the 
product. of a and b,, may be either expressed by ah^ or ta ; 
and the product of a, i, and r» by either abcf or acb^ or hacy 
or bcof or cab^ or cba ; as it matters not which quantities are 
placed or multiplied first. But it will be sometimes found 
convenient in long operations, to place the several letters 
According to* their order in the alphabet, as abc^ which order 
adso occurs most easily or naturally ta the mind. 

22. Likewbe, the several members, or terms, of which a 
compound quantity is composed, may be disposed in any 
order at pleasure, without altering the value of the signifi- 
cation of the whole. Thus, Sa — 2ab + Aabc may also be 
\\Titten 3fl -f ^abc — 2aby or ^abc -f 3 j — 2ab^ or — 2ab + 3« 
+ ^akcy &C5 for all these represent the same thing, namely, 
' the quantity which remains, when the quantity or term 2ab 
is subtracted from the sum of the terms or quantities Stf and 
4ahc. But it is most usual and natural, to begin with a po* 
sitive term, and with the first letters of the alphabet. 

a 

SOME EXAMPLES FOR PRACTICE, 

In finding the numeral values of various expressions^ or 

combinations, of quantities. 

Supposing /7 = 6, and 4=5, and c = 4,. and rf = 1, anct 
/ = 0. Then 

1. Will a' + Zab-c^ =^ 56 + 90 - 16 = UQ. 

2. And 2^3 -3^*3 -^ c^ = 432 -540 + 64 =—44. 



3. And a^ X a-\- b-2obc = 3« x 11-240 =156. 

a^ 216 

4. And r- + ^ =z — r- + 16 = 12 + 16 = 2S.. 

a -i- 3c 18" 

5. Arid v" 2^7+7^ or 2ac + r]^ = y/64i = 8. 

6. And x/c + ^ • \ =5? .+ — = 7. 

^ 2ac ' + e- 8 / 

„ ., 1 ^'--/^'-^^ 36-1 _35_- 



8. And s^h"- -ac^ V2ac + ^ = 1 + 8 = ^. 

9. Andx/b^-ac + ^?^cT^ = ^25-24^ + S = 3. 
10. And fl*^ + r - ^/ = 1^5. 

n. And 9/7^ - 10^* + r = 24. 



\ 



12. And 



ADDITION. fSS 



fl'= 



tt. And — X rf =3 45. 
c 

13. And-^X -7= ISi- 

c a ^ 

. ,flr + ^ if— 5 

14. And 7- = 11^ 

c a 

15. And 1-^=46. 



16. And — X ^ = 9: 4^ ::r 



17. And*-^x J— ^ = i- . ^ 

18. And^i+^— r— rf = 8, 

19. Andtf+^ — ^ — </ = 6»' 

20. Andfl*r x J' = 144. 

21. Andacd- d:=:2S* 

22. And tfV + y'e + rf ^^^ ^-^ 

23. And -1 X — ^ = 18J-. 

. 24. Andv^/i^- +^^"-i/tf* - ^* = 4-4936249. 
25. And 3^^ + ^a^ - ^' == 292-497942. 
56. And 4fl* — 3^1 ^a'--^ab = 72. 



ADDITION. 



Addition,^ in Algebra, is the connecting the qnaptities 
together by their proper signs, and incorporating or uniting 
into one term or sum, such as are similar, and can b^ united. 
As Sa + 2b — 2a = a + 2b j the sum. 

• The rule of addition in algebra, may be divided into three 
cases : one, when the quantities are like, and their signs like 
also ; a second, when the quantities are lite, but their signs 
unlike; and the third, when the quantities are unlike. 
Which are performed as follows*. 

CASE 



* The reasons on which these operations are fou tided, will rea- 
dily app^ar^ by a little reflection on the nature of the quantities to 



16« ALOEBRA. 

CASE I. 

TVben tie Quantities are Liiey and have Like Signs. . 

« 

Add the co- efficients together, and set down the sum j 
after which set the common letter or letters of the like 
quantities, and prefix the common sign + or — • 



be added, or collected together. For, with regard to the first ex- 
ample> where the quantities are 3a and 5a, whatever a represents 
in the one term, it will represent the same thing in the other^ so 
that 3 tiroes any thiqg and 5 times the same thing, collected 
together, must needs make 8 times that thing. As if a denote a 
shilling \ then \\a is 3 shiirmgj5> and ba is 6 shillings, and their sum 
8 shillings. In like manner, — 2a6 and — 7«^> or —2 times any 
thing, and —7 times the same thing, make —9 times that thing. 

As to the second case, in which the quantities are like, but the 
signs unlike ; the reason of its operation will easily appear, by 
reflecting, that addition means only the uniting of quantities to- 
g€Jther by means of the arithmetical operations denoted by their 
signs -(- and — , or of addition and subtraction ; which being of 
contrary or opposite natures, the one co-efficient must be sub- 
tracted from the other, to obtain the incorporated or united mass. 

As to the third case> where the quantities are unlike, it is plain 
that such quantities cannot be united into one, or otherwise added, 
than by means of their signs : thus, for example, if t^ be supposed 
to represent a crown, and h z shilling ; then the sum of a and b 
can be neither 2a nor 26, that is neither 2 crowns nor 2 shillings, 
but only 1 crown plus 1 shilling, that is a -f 6* 

in this rule, the word addition is not very properly used ; being 
much too limited to express the operation here performed. The 
business of this operation is to incorporate into one mass, or alge- 
braic expression, different algebraic quantities, as far as an actual 
incorporation or union is possible; and to retain the algebraic 
marks for doing it, in cases where the former is not possible. 
When we have several quantities, some affirmative and some ne- 
gative j and the relation of these quantities can in the whole or in 
part be discovered ; such incoi'por^^tion of two or more quantities 
into one, is plainly effected by the foregoing rules. 

It may seem a paradox, that what is called addition in algebra^ 
should sometimes mean addition, and sometimes subtraction. But 
the paradox wholly arises from the scantiness of the name giyen to 
the algebraic process; from employing an old term in a new and 
more enlarged sense. Instead of addition, call it incorporation, 
or union, or striking a balance, or any name to which a more ex- 
tensive idea may be annexed, than that which is usuj^Hy implied 
by the word addition ; and the paradox vanishes. 

Thus, 



f 



Thus, 2a added to 5a, mak;ps 8a. 

And —2ab added to — 7a^, makes — 9fl*. 

And 5a + 7^ added to la + Si, makes I2a + 10* 



M7 



OTHER EXAMPLES IfOR PRACTICE. 



3tf 

Sa 

I2a 

a 

2a 

Sla 



\5z 



- Six 

- lix 



hxy 

2hcf 
5hxy 
kxy 
Shxy 
&xy 

11 bxy 



2z 


Sjt* + Sxy 


2ax — 4; 


22 


x" + xy 


4ax — J? 


4z 


2x' 4- 4.Afy 


«Af — Sy 


z 


5x^ + 2afj? 


Sax — 5^ 


5^ 


4;v* ^ Sxy 


7a* — 2y 



15if* + 15*y l^ax — 15; 



Sxy 
X^xy 
22xy 
11 xy 
lixy 

ixy 



12/ 
7/ 
2/ 
4/ 

/ 
3/ 



4/1 •^ 4# 
Ai - 5* 
^a — * 
3a - 2* 
2tf - 7* 



30 - ISacf - Sxy 
23 — lOxY - 44ry 
14 - 14*1 - 7xy 
10 - 16;rt - 5xy 
16 - 20Ar^ - xy 



Sxy — 3jp + 4ai 
Sxy -* 4* + 3a* 
3iA[y •«" 5ac + 5a* 
xy — 2x + ab 
ifxy -^ ;if + lot 



i m» % > " > " ■ 



-^U^ 



i|ii . i m 



CASS 



\$$, ALGEBRA. 

CASl^ II. 

r 

tf^ifu tie Qjuantitiei ari Lite, but have Unlike Signs : 

Add all the affirmative co-efficients into one sum, and all 
the negative ones into another, when there ^e several of i 
kind. 'Then subtract the less sum, or the less co-efficient, 
from the greater, and to the remainder prefix the sign of the 
greater, and subjoin the common quantity or letters. 

So + 5a and — Sa, united, make + 2a. 
And — 5a and + Sa, united, make ^ 2a. 

m 

OTHEH EXAMPLES FOR PRACTICE. 

r- 5a + Sax^ 

-\- 4a + irax* 

+ 6fl — Sax^^ 

— 3a — eax": 

-fa + 5ax^ 



+ 


8*3 + 3^ 


— 


5x^ + 4y 


— 


16*' + 5y 


+ 


3x^ - 1y 


+ 


2x^ — 2y 



+ 3a — 2fl** - 8*3 -f i(5y 



— 


3i»* 


+ 


sby 


+ 4tf* + f 


— 


5a^ 


+ 


9by 


- 4ab + 12 


' / 


lO/i* 


— 


loby 


+ lab ~ 14. 


+ 


lOfl* 


— 


19by 


+ ab + $ 


+ 


Ua^ 


— 


2by 


- Sab - 10 



■ < 



I 



— Zax'^ + 10^ ax + 3;^ + ^^x^ 
+ ^M^ — 3x/ax -^ y — ,5tf*^ 

I r 

+ 5ax^ + 4x/ax + 4)^ + 2ax'^ 

- 6fl*^ — 12 V ax — 2y + 6^*^ 



CASE 



— / 



ADDITION. 1G9 



CASE III. 

When the Quantities are Unlike. 

Having collected together all the like quantitlesj,^ as ia 
the two foregoing cases, set down those that are unlike> one 
aft^r another, with their proper signs. 



%xy 
2ax 

Sxy 
6ax 


SXAMPLES. 

— 40;'*+ 3xy 

+ 4;r*— 2xy 
-3^7+ 4x* 

4.r^— 8a:' 


4^x— 130+30:^ 
5x* +3/i-r+9jr' 

i 
Ixy -4jr^ + 90 


'-2xy + Sax 


lax + 8 j:* + 7^^ 


9r*/ 

-7x*)» 
+ Saxy 
-4jr*j? 

• 


14tfjr— 2.r* 
5ax-^3xy 
8/ -4aj' 
3jr* + 26 


9+lQV'/jjr-5; 
2jr+ 7v/jrj> + 5j^ 
5y+ 3^i7X*-4; 
10— 4-v/Ar+4; 


• 






4x*y 
— 6;r/ 
'+3/jr 

-Ix'y 


4-v/x — , Sy 
2v^a:y+14r 
Sx + 2j^ 
-9 + 3^jrjr 


3/1* + 9 + a-^— 4 
2/j -8+2/1*— 3j: 
4a:f-2iJ*+18 —7 
-12+ tf-3.r*-2y 



Add /f + i and 3/i — 5A together. 
Add Stf— 8a: and 3/i— 4a: together. 
Add 6a:-5* + /i + 8to — 5/i -4a:+4i-3. 
Add /I +: 2*-3r- 10 to 3i— 4/i + 5c + lOand 5*-r. 
Add fl + i and a— h together. 

Add 3/1 + i- 10 to c^d-^a and —4^ + 2^-3^—7. 
Add 3/1* + b'^-c to^*-3tf' + fc-4. 
Add ^' + ^V-*' tP aA* -/i^r + A*. 

Add 9tf— 8* + 10Ar-6//-7r + 50 to 2ar-3/i— 5r + 4A + 
W-10. 

SVBIE ACTION/ 



170 ALGEBRA. 



SUBTRACTION. 

Set down in one line the first quantities from which the 
subtraction is to be made ^ and underneath them place all the 
other quantities comi>osing the subtrahend : ranging the like 
quantities under each other, as in AdditicMi. 

Then change all the signs ( + and — ) of the lower line, 
or conceive them to be changed ; after which, collect all the 
terms together as in the cases of Addition*. 

\ 

EXAMPLES* 

From 7tf*-3* 9x* - 4y + 8 SJcyS + 6jr— y 

Take 3a*- 8* 6x^ + 5;^ - 4 My-1 ^ 6jr-4y 

I r m. ^ i ' -i..i. .Ill 

Rem. 4a* + 5^ Sjt*— 19^+12 44:j^ + 4+ 12^ + Sy 



From 5xy—\ 6 4y*— Sj;— 4 — 20— 6j:-;5xjr 

Take -2ji'j^4- 6 2/ + 2j?+4 3xy-9j; + S^'2ay 

Rem. 7jry— 12 2/— 5;^-8 - 28 + St "Sa^y-^^ay 



-*~«— 



From 8jr*3f-f6 5^xy'{-2a:^xy lx^+2^x—l8 + Si 
Take -2x*j;+2 7 4/ xy -{- 3 - 2xy 9^-12 +5^+j: 

Rem. 



I 



* This rule is founded on the consideration, that addition and 
subtraction are opposite to each other in their nature and operation^ 
^s are thje signs -h and — , by which they are expressed and repre- 
sented. So that, since to unite a negative quantity with a positive 
one of the same iiind^ has the effect of diminishing it> or subduct- 
ing an equal positive one from it, therefore to subtract a positive 
(which is the opposite of uniting or adding) is to add the equal 
negative qu'antity. In like manner, to subtract a negative quan- 
tity, is the same in effect as to add or unite an equal positive one. 
So that, by changing the sign of a quantity from -f to — , of 
from — to +, changes its nature from a subduottve quantity to an 
additive one j and any quantity is ip effect subtracted^ by barely 
changing its sign, 

5xy 



MULTIPLICATION. 171 



Sxy - 30 7^3 _2 {a + h) Sjt/ + QOax/{xy +^ 10) 
Ixy - 5Q 2;r=-4 {a + h) 4xy -j- V2a*y\xy + 10) 



From /J 4- ^> take «— ^. 

From 4ii + 4^, take 3 + <»• ' 

From 4tf — 4^, take 3a + 55. 

From 8^1—1 2jr, take 4/ar — Sjt. 

From 2jr— 4a~2^ + 5, take 8-5* + a + 6t. 

From Sa-^b + c-d- 10, take c ^^a-d, ' 

From 3a+h+ c-d- 10, takei- 10 4--3zi. 

From 2^l5 + Ir^—^c + be -by take ^a^'-c + b\ 

From tf3 + 35V + ab^-ahc, take b" + (^b^-abc. 

From l2x + 6a'-U 4<40, take 45 - 3<i + 4ar + 6i- 10. 

From 2j:— 3fl + 45 4-6^-50, take 9tf+^ +65-6r-4a 

From 6«-45- 12t: + 12jr, take 2x-8^ + 45-5^, 



MULTIPLICATION. 



This consists of several xases, according as the fattors ar? 

simple or compound quantities. 

CASE i^ When both the Factors are Simple Quantities : 

First multiply the co-eiEcients of the two terms together, 
then to the product annex all the letters in those termsj 
which will give the whole product required. 

Ngte ^. Like signs, in the factors, produce + > -suid unlike 
jsigns — , in th^, products, 

I 

{:XAMPLES, 



* That this rule for the signs is true, may be thus shown. 

1. When + a is to be multiplied by + c; the meaning is, that 
-{- <2 is to be taken as many times as ther<^.are units in c ; and since 
the sum of any number of positive terms is positive^ it follows that 
-j- a X -}- c ip^kes -f- ac, 

% When 



2 

lOa 
2b 


ALGEBRA. 

• 

EXAMPLES. 

- Sa la 
+ 2^ -^c 


-ex 

i 1 1 


20ab 


-Cab 


-2Sac 


-i'2iax 


4ac 
-Sab 


Od'x 
4x 


-2x^y 


— ifXy 

- xy 


-\2abc 


36tf*jr* 


-6xy 


+4xy 


9 

-Sax 
4x 


1 

— ax 

-6c 


+Sxy 
-4 


— Sxyz 

— Sax 












CASE II. 


' f 


♦ 



When one of the Factors is a Compound Quantity ; 

Multiply every term of the multiplicand, or compound 
quantity, separately, by the multiplier, as in the former 
case; placing the products one after another, with the 
proper signs; and the result will be the whole product 
required. 



2. When two quantities are to be multiplied together, the re- 
sult will be exactly the same, in whatever order they are placed $ 
for a times c is the same as c times a, and therefore, wjiep — a if 
to be multiplied by + c, or + c by — a : this is the same thing 
as taking - a as many times as there are units in + c ; and as the 
sum of any number of negative terms is negative, it foljows that 

— a X -|- c, or + '< X — c make or produce — ac. 

3. When — « is to be multiplied by — c; here — a is to b« 
subtracted as often as there are units in c: but subtracting nega- 
tives is the same thing ^s adding affirmatives, by the demonstration 
of the rule for subtraction 5 consequently the product is c times a, 
•r + ac 

Otherwise. S^nce a —- a zr 0, therefore {a — a) X — c is also 
=: 0, because multiplied by any quantity, is still but 5 and 
since the first term of the product, or a X — c is 3: — ac, by the 
second case ; therefore the last term of the product, or — a x — c, 
niust be -{- ac, to make the sum iz 0, or — ac -^ ac zzQ', that is, 

— a X — c =: + ac 

EXAMPLES. 



MULTIPLICATION. ng 



EXAMPLES* 

Sa-Zc Sac- 4b 2a^-Sc+5 

2a 3a be 



10a*- 


•6ac 


• 

12j7- 
4a 


2ac 


i 


4xy 


X 





9fl V - 1 2ab 2a^bc - Uc" + 5bc 



25c -lb 4x-b + 3ab 

— 2a iab 






CASE III. 

When both the Factors are Compound Quantities; 

Multiply every term of the multiplier by eVery term of 
the mulriplicand, separately ; setting down the products one 
after or under another, with their proper signs ; and add the 
several lines of products all together for the whole product 
required. 

« + * Sx-i-Qy 2x^'\-xy--2f 

« + ^ 4x — by Sjt— 3y 



a^-k-ab I2x^+Sxy 6x^ -h Sx^y- 6x/ 

+fl4 + ** -IBxy—lOy"- - 6^*y - 3x/ + 6/ 

m^-\-2ab-\-b^ ISj;'*— 7xy-10f 6x^-3x^y-'9xy''^ey^ 



m-^-b x^-j^y a^-k-ab-^b 

— b ^^-{-y ' a -^b 



^ :-_ 

-ab-b"^ +yx'' +/ - a^b - ab"" - b^ 

^ ♦ - ^» ^*'-f2;!^*+/ a^ * * -b^ 

Notf> 



~i 



174 ALGEBRA. 

Note. In the multiplication of compound quantities, it is 
the best way to set them down in order, according to the 
powers and the letters of the alphabet. And in multiplying 
them^ begin at the left-hand side» and multiply from the left 
hand towards the ri^ht, in the manner that we write, which 
is contrary to the way of multiplying numbers. But in 
setting down the several products, as they arisei in the second 
and following lines, range them under the like terms in the 
lines above^ when there are such like quantkiee ; which is 
the easiest way for adding them up together. 

In many cases, the multiplication of compound quantities 
is only to be performed by setting them down one after 
another, each within or under a vinculum, with a sign of 
multiplication between them. As {a ^ b) x {a — b) x Sabf 

era -{■ b . a — i . Zab. 



exa;mples for practice. 

1. Muhiply lOac by 2a. Am. 20d'c. 

2. Multiply Sa^-2b by Zb. . Ans. 9a^b~6b\ 
3'. Muhiply Sa + 2i.by 3a- 2b. Ans. 9a^-4l^. 

4. Multiply jr* - ary + / by jr 4- ;>. Ans. a;^ + f. 

5. Mukiply a^ -^ i^b + ab"- -^ P by a-^b. Ans. a*-K 

6. Multiply a^ + ab + b''hYa''-ab + b\ 

7. Muhiply 3j:^-'2^y + 5 byx^ + 2a:y'^6. 

8. Multiply Sa^-2ax + 5jr* by Sd'-Ux-l:^. ' 

9. Multiply Sx^ + 2jry + 3/ by 2:c^-Sxy + 3/. 
10. Multiply d' + ab +b^bYa-2b. 



DIVISION. 

4 
\ 

Division in Algebra, like that in numbers, is the con^rse 
of multiplication j and it is performed like that of numberp 
also, by beginning at the Teft-hand side, and dividing all the 
p^rts of the dividend by the divisor, when they can be so 
divided ; or else by setting them down like a fraction, the 
dividend over the divisor, arid then abbreviating the fraction 
as much as can be done. This will naturally divide into 
the following particular cases. 

CISE 



DIVISION. 



175 



CASE I. 

When th^ Diwsor and Dividmd are both Simple QuantitUs; 

Set the terms both down as in division of numbers, either 
the 4ivisor before the dividend, or below it, li're the deno- 
minator of a fraction. Then abbreviate these terms as 
much as can be done, by cancelling or striking out all the 
letters that are common. :to them, both, and also dividing 
the one co-efficient by the other, or abbreviating them after 
the manner of a fraction, by dividing them by their common 
measure. ^ . 

Note.' Like signs in the two 'factors make + in the quo- 
tient ; and unlike signs make — ; the same as in multipli- 
cation *^ 



EXAMPLES. 






1. To divide 6^^ by 3 J. 






Here Sab -f- 3tf, or Za ) Gab ( ox 


6ab 

Sa " 


2b. 


t 
2. Also c-^c ^ — "^ 1 ; and abj: 

c 


-T- b.vy : 


atx a 
^ bxy - ji • 


3. Divide I6x*by 8^. 




Ans. 2jr. 


4. Divide 12j*j:* by - 'id'x. 




Ans. — 4*r. 


5. Divide — \5ay^ by ^ay. 




Ans. — 5j. 


6. Divide ■- ISax^y by — Sdfjrz. . 




Ans. --- • 

• 



* Because the divisor multiplied by the quotient, must produce 
the dividend. Therefore, 

1 . When both the terms are + , the quotient must be + ; be- 
cause + in the divisor X -f in the quotient, produces + in the 
dividend. 

2. When the terms are both — , the quotient is also -f ; be« 
cause ^ in the divisor X + in the quotient, produces — in the 
dividend. 

3. When one term is + and the other — , the quotient mast be 
—J because + in the divisor x — in the quotient produces — 
ki the dividend, or — in the divisor X + in the q^uotient gives -* 
Ih the dividend. 

So that the rule is general; viz. that like signs give -{-, and 
unlike signs give — > in the quotient. 

C4SE 



/» 



176 ALGEBRA, 



CASE II. 

When the Dividend is a Compound Quantiify and the Divisor 

Simple one : 

Divide every term of the dividend by the divisor^ a$ in 
the former case. 

EXAMPLES. 

ab + b'' a + b 

1. {ab + *^) -^2*,or --^ = -|- ^ia + ib. 

lOab + I5aa: ^, . „ 

2. (lOab + I5ar) -r 5a, or j^ = 2* + 3.r. 

30flfz-48z • . _ 

3. (30^2 -48«) -r- z, or * = 30^-48. 

4. Divide 6^?^- Stfo: +ahj 2a. 

5. Divide 3^*- 15 + 6.r + Qa by 3r. 

6. Divide 6fl^r + 12fl^J7-9fl'3 by 3tf*. 

7. Divide. 10^*x— 15-r*— 25jr by 5^. 

8. Divide lot^bc- iBacr" + $ad^ by - Sac. 

9. Divide 15^ + 3ay- 18/ by 2U. 
10. Divide -ZOd-b"- + 60^^' by-6a*.^ 



CASE III. , 

When the Divisor and Dividend are both Compound Quantities: 

1 . Set them down as in common division of hurabers^ 
the divisor before the dividend, ttrkh a small curvecJ line 
between them, and ranging the terms according to the 
powers of some one of the letters in both, the higher 
powers before th6 lower. ^ 

2. Divide the first term of the dividend by the first term 
of the divisor, as in the first case, and set the result in the 
cjuotient. 

3. Multiply the whole divisor by the term thus found, and 
' subtract the result from the dividend. 

4. To this remainder bring, down as tnany terms of the 
dividend as are requisite ' for the next operation, dividing as . 
before ; and so »n to the end, as ih common arithmetic. 

Not~e4 



F 



DIVISION. m 

Noie. If the divisor be not exactly contained in the divi- 
'dend^ the quantity which remains after the operation is 
finishddy may be placed over the divisor, like a vulgar frac- 
tion, and set down at the end of the quotient, as in common 
arithmetic* 



EXAMPLES. 



tf*- ab 












a^-4? 



^—2) a'— 6fl* + 12a-8 (fl*-4tf + 4 



r 





+ 




-8 
■8 


az 

+ 
+ 


- 




4<i- 

4«- 




il 


«) a^ + «» (, 




+ »' 






- 




N 


«3 




P 


- 


• '.•^' 



Tot. I. N «+*) 



lis ALGEfeRA. 

2^* 



a + x) a^-'ix* ( tf'-a'x + ff4P'-jr» 



*+* 



»* 


+v* 


• 


- 3«* 






— «**- 








«'«» 

/!»«». 






1 


-*»>- 
-«.'- 


8** 




— 


■2** 



EXAMPLES FOR PRACTltE. 

!. Divide u* + 4j;r + 4;ir* by a + 2;r. Ans. a + 2x. 

2. Divide tf^-3fl*« + 3iia*-z' hy a— z. 

Ans. II*— 2/jz + »*• 

5. Divide 1 by 1 + ii. Ans. 1 -tf + tf*-a' + &c, 

4. Divide 12;j«*-ld2 by 3*-6. 

Ans. 4w» + 84r* + 16jr + 32* 

A. Divide «5-5fl** + I'Oa^*^ - .lOay + 5/j**- ** by n*- 
24f* + F: Ans.. «' - 3^** + 3^** - ^. 

6. Divide 4823-96tf«*— 6*0^2 + ISOtf' by 2a;— 3^. 

7. Divide ^*-3**P»* + SS^x^^x^ by ^3-3*'x + 3**'^-«^ 
8« Divide fl^— o;^ by ^ — x 

9. Divide a' + 5a^x + Sax" + *' by a + x. 

10. Divide a* + 4fl*J*- 32** by « + 2*. 

11. Divide '24fl*-** by 3a- 2b. 



tSS! 



i^» i^i i M S -< 



ALGEBRAIC FRACTIONS. 

Algebraic Fractions have the same names and rules 
of operation, as numeral fractions in common aritjj^netic } as 
..appears in the following Rules and Cases. 

CASE 



JfRACIIONS. 17SI 

CfSB Z. 

To Reduce a Mixed Quantity to ah Improper Fraction. 

Multiply the integer by the denominator of the fraction^ 
and to the product add the huinerator> dr cozmect it wit^ 
its proper sign, .+ of — ) tli^n the denominator beipg set 
under this sum> will give the improper friction required* 

1* Reduce S4>.atid d — - t6 improper fractions. 

3x5+4 15 +4 : 19 , ^ 
First, 5f ^ j-^ = — ^ = -J, the Answeir^ 

. ■ * ax x^b a^—i . . 

And. tf — -= — — r^— 2= the Answer. 

^ X X X 

2* Reduce a + — and a -^ t6 improper fractions; 

. o a 



fl* axb+d* at-^d" 



First, 4+-7- = — *-7 == 



t 



the Amwer. 



And, a — ^ — -^— = '^^ = the Answer. 

a . a a 

. 3. Reduce 5^ Jo an improper fraction^ Ans. y * 

4. Redu(^e 1 to an improper fi^ction^ Ans. * ' 

fiC X 

6i Reduce 2i» — — to an improper frractioh^ 

4^ " ■ * 

6. ite4uce 12 4 ^t to an impfopiir fraction* 

7. Reduce * H — '- ^ to sin improper fraction* 

c 

8« Reduce 4 ^2x ^ to an improper fraction' ' 

tASE It. 

To Redwte an Improper Praction to a Jf%Ie wr Mixed Quantity! 

DitiiJi; the numerator! by thcf denpniinator, for the in* 
te^al part ; and set the remainder, if any, oter the deno^^ 
xmnator, for the fractional part j the two joined together will 
be the mi^ed gji^antity required. ' 



1$0 ALGEBRA. 



EXAMPLES. 



. 1. To reduce — and — -r^ to mixed quantities. 

3 p 

First, y 5= 16 -r 3 = 54, the Answer required. 
And, f*±^= ^M^ -5- 4 = « + J. Answer. 

"2. To reduce !flzif! and 5fl±if! to mixed quantities. 

First, ^^^^^ = 2ar - 3^ ^ ^ = 2« - — . Answer. 
c c 

And, ^^"^^"^ = i^HK*^ -T-fl+or = 3a: + — . Ans. 

3. Reduce — and -r- — — to mixed quantities. 

3jr* 
Ans. 64> and 2* . 

4. Reduce — - and -I--— to whole or mixed quan- 

2a a-b 

tities. 

3jr*^- 3/ • , 2j^^ - 2y3 - , . . 

5. Reduce ^, and to whole or mixed 

quantities. 

6. Reduce ■ — to a mixed quantity. 

7. Reduce ^ to a mixed quantity. 

3a3 + 2a*- 2a— 4? 



CASE III. 

To Reduce Fractions to a Common Denominator* 

Multiply every numerator^ separately, by all the deno^ 
minators except its own, for the new numerators ; and all the 
^ denominators together, for the common denominator. 

When the denominators have a common divisor, it will be 
^better, instead of multiplying by the whole denominators, to 
multiply only by those parts which arise from dividing by the 
common divisor. And observing also the several rules and 
directions as in Fractions in the Arithmetic* 

EXAMPLES. 



/ 



FRACTIONS. 181 



EXAMPLES. 



a - i 



1. Reduce — and — to a common denominator. 

X z 

Here — and — = £- and — , by multiplying the terms of 
X % xz x% 

the first fraction by z, and the terms of the 2d by x* 

/I V b 

2. Reduce , — , and — to a common denominator. 

X b c 

Here — , ---, and — = ^ — , — > and , by multiplying the 

X h c hex hex hex 

terms of the 1st fraction by hc^ of the 2d by cx^ and of the 

3d by hx. 

I* 

3. Reduce — and — to a common denominator* 

X 2c 

Ans. and — • 

2ex 2eK 

4. Reduce — anJ^^^^^I — to a common denominator. 

h 2c 

ifOe - , SaH-2i* 

*Ans. -rr—, and — -r . 

2bc' 2bc. 

' 5. Reduce — and — , and 4«f. to a common denominator* 

3* 2? 

Ans, and and 



%cx Qex Qex 

6. Reduce — and — and 2b + -7-. to fractions having a 

6 4 b * 

common denominator. Ans. — r— and ^ and , 

. 243 243 243 

7. Reduce — and and — III- to ^ commori deno- 

3 4 fl+3 

minator., 

8. Reduce — and — and — - to a common denominator. 

4a* 3a 2a 



CA8B 



n2 ALGEBRA, 



CASE IV. 

To find the Qriofist Common Meastfti rf the Tkrms jf # 

practian* 

Divide the greater term by the less, and thje last divisor 
by the last remainder, and so on till nothing remains ; then 
the divisor last used will be the common measure required 9 
just the same as in common numbers. 

But note, that it is proper to range the quantities according 
to the dimensions of some letters, as is shown in division. 
And note also, that all'the Itoers or figures which are com- 
mon to each term of the divisors, must be thrown out of 
them, or must divide theip, before they are used in the 
operation. 

£3(AMPLBS. 

J. To find the greatest common measure of -^ rr. 

«* + **) ac" + i^ 
6r m+b) ac" + 6^{c^ 



Therefore the greater Cbmmon measure isa + i. 
2. To^nd the greatest common mea^r^e of ^ 






or a+ i )d' + 2aL+f(a+L 



ai + ^ 
Therefore <f + ^ is the greatest common divisor. 

9« To find the greatest cozjunon devisor of-; — rrr, 

ao+29 



Ans* «— 2« 
4. Ta 



FRACTIONS. 183 

4. To find the greatest common divisor of — ^ ^ ,^ « 

Ans. fl* - f. 

5, Fiadtne greatest com. measure of ^ .t\ ^ •;r^ — t ^ % z ' 



CASE T. i^ 

I 

To Reduce a FractUn to its Lowest Terms* 

Find the greatest common measure^ as in the last pro- 
blem. Then divide both the terms of the fraction by tjie 
common nxeasure thus founds and it will reduce it to its lowest 
terms at once^ as was required. OriKvide the terms by any 
quantity which it may appear will divide them both, as in 
arithmetical fractions. 

\ 

EXAMPLES. 

ai + b* . 

1. Reduce ■ . , ^ *o i^s lowest terms. 
or + or 

ab + i^) ac" + he* 
dra + b ) 0c* + be'{c' 
0^ + ^V* 



Here ab + i^U divided by the common factor b» 

Therefore a + i^s the greatest common measure, and 

ab + b^ b 
hence a + b ) , , . = t*** the. fraction required- 

2. To reduce ; .■ ^, — r-rr to its least terms. 

c^+2bc + b 

tf* + 2fc + *" ) r' - b^c ( c 

- 2*^V2*V) c' + 2bc + b^ 

orc + b) i^ + 2bc+b^(c + i 
ji'+ be 



be + b' 
bc + b'' 



Thtjrefore 



lU ALGEBRA. 

Therefore ^ 4- ^ is the greatest common measure) and 
hence r + *) , ^ = , , is the fraction required* 

S. Reduce -r — rr^ to Its lowest terms. Ans. ^ , , ^ ■ 

fl* — J* 1 

4. Reduce — — rr to its lowest terms. Ans. ^ . ^ ^ 

fl* — ** 

5. Reduce --r — ^ ,, , ^ ^ — n to its lowest terms. ^ 

7. Reduce , . ^ . . ... to its lowest terms. 
a* + 2ai + b^ 



CASE vi. 



T^ tf^i/ Fractional Quantities together. 

If the fractions have a common denominator^ add 73\ the 
numerators together ; then under their sum set the common 
denominator, and it is done. 

If they have not a common denominator, reduce them to 
one> and then add them as before. 



EXAMPLES. 



T ^ J ^ 

1. Let ~- and — be given, to find their sum. 
3 * 

-T a A 4tf 3tf la 

Here -5- + t-"^ i^ ^ T^ ^ 75 '* ^'^^ ^^°^ required 



3 4 12 12 12 

.> 2i.nrl — 4'r\ 4\r\A 4-ViA«« ».«««« 



a b , c 
% Given -7-, — ^,anQ~7, to find their sum. 
PC a 



the sum required* 

3. Lit 



,-*\i\ 



FRACTIONS. IBS 

Sx* 2ax 

* 3- Let a r-and i H — ^ be added together. 

b c , ° 

^x^ , Sax * 3rjr* , , 2a5j: ^ 
Here^-— +* + — =^-^^+*+->^j- 

2abx~^3cx* - * 

= a + i H r , the sum required. 

^. *., 4!X ^ 2x , . 20*j:+6flx 

4. Add -r- and -rr together. Ans. — ■ ^ , ^. 

5. Add ---J — and — together. Ans. ^a. 

^ Ajj 2^—3 - 5tf , . . 9a — 6 

6. Add and -e together, Ans. "^ 

4 8^ , 8 

7. Add 2a -J •— to 4a H ^ — ^. Ans, 6a + 



,**na 



5 '4 20 

So" a+b 

8. Add 6a, and -— r- and —-7- together. 

5a 6a 3a + 2 

9. Add — , and --- and — - — together. 

3a a 

10. Add 2a, and — and 5 + — together. 

, , . , , ^ , Sa - 5a 

11. Add 8a + — and 2a— — together. 

4 o 



CASE VH. 

* To Subtract one Fractional Quantity from another . 

Reduce the fractions to a common denominator, as in 
addition, if they have not, a* common denohiinator. 

Subtract the numerators from each othler, and under their 
difference set the common denominator, and the work is 
done. 



* In the addition of mixed quantities^ it is best to bring the 
fractional parts only^o a common denominator, and to annex theiv 
sum to the sum of the integers, with the proper sign. And the 
same lule n^ay be observed for mixed quantities in subtraction 
also. 

(XAMPUS. 



^ 



1S6 ALGEBRA. 



EXAMPLES. 

1 . To find the difference of *f and ~. - 

4 7 

Hcrc^-l?=ii?^l^=,^is the dMTerence remiircd- . 

4 7 28 28 28 

£• To find the difference of and 



Here 



•4r 3^ 

4c Sb I2bc \2bc 



6tf?— 3ii— 124awr+ lefc . , ■»./*• • » 

— i ' IS the difference required. 

\2bc ^ 

3. Required the difference, of — f and -^. 

9 7 

4. Required the difference of 6« and — . 

4 

5. Required the difference of — a^d — • 

4 3 

6. Subtract?* from ?l±f. 

c b 

7.Take!l±.^fromll±i. 
9 5 

U. Take 2a^t^l^ from 4^i + £f. 



CASE Till. . 

Jb Multiply Fractional Qjianiities together* 

Multiply the numerators together for a new numerator, 
and the denominators for a new denominator*. 



* 1. When the numerator of one fraction, and the denominator 
of thf»-other^ can be divided byisome qqaptttyj which is common 
to both, the quotients may be used inAtfsad of them. 

a. When a fraction is to be n|ulti(^ie4 by an integer^ the pro- 
duct is found either by i^ultiplying the uum^i^tort or dividing th& 
denominator by it; and if the integer be the s^e with the deno- 
minator^ the numerator may be taken for the product. 

EXAMPLES. 



FRACnONB; 



i*r 



EXAMPLES. 



I. Required to find the product of --^ and — . 



Here -.— — r- = -r- 
8x5 40 



— the product required. 
2d' 



2. Required the'product of — , ^, and — . 

f^iliiiiif = i!f! = !i the i*oduct reqal^ ■ 
3x4x1 «4 , J4r *^ 

3. Required the product o£ — and • 

^ ^ b 2a -^c 

Here i — - — i = ! the product requum. 

4. Required the product of — and — . 

5. Required the product of— and — - 

4 3flf 

6. To multiply -^, and — ^i and ---- together. 



3r 



7. Required the product of 2fl+ — and — , 

2c b 

8. Required the product of — «-- — »- and -^ — , 

Sbc a -{rb 

9. Required the product of 8^, and ^ T , and ^ *" * 

10. Multiply «+4 --^byar-iL + _?l. 
*^' 'Za 4<i* 24? 4j?* 

CASE IX. 

2it Divide one Fractional Quantity by another. 

Divide the numerators by each other, and the denomina- 
tors by each other, if they will exactly divide. But^ if not^ 
then invert the terms of the divispr, and multiply by it 
exaetly is in muhi{£catiQ«i^. 

*'^^ — — ^^ — ^^^ — ^^^^ — ,^ — .._ ^ _ _, ^ ^^^^^ — ^_.,.^^^,^ ^^^.^ — . — ^__ — ^^.^ — .^ 

^ 1. if tin £raedoot'to be divided have a csramon deomninator^ 
take Ibe mmefto r t£ tbe dividend for a new nuineratDO and 
the numeratQjT of the Avisos ibr the new denomMiAtor. 

2. Wheii 



Ito ALGEBRA. 



EXAMPLES. 



1- Required to divide — by-^. 

Sa 5c 
2. Required to divide rr by t-v 

„ 3# 5r Sa 4d \2ad Sad , 

• 3a* 2a , 

4. To divide ^ . .» by - — r-^. 

3tf^ tf+i _ 8a*x(a+ ^) _ 3tf 

is the quotient r.equiredl 

5. To divide --r by t*. 

4 ^ 12 

6. To divide — by 3x. 

7. To divide — - — by — . 

9 ^ o 

' 8. To divide *- — - — : by -^. 

2^:— 1' ' 3 , 

4x Sa 
9. To divide -r by -t- 

5 ^ 5h 

2a "b Sac 

10. To divide —-J- by .-TT. 

11 n- -^ 5a^^5b^ , 6a*+5a* 

11. Divide ^ . ^ I 77 by r. 

2a* - 4ai -h 24* ^ 4a - 4* 



i2. When a fraction is to i>e divided by any quantity, it is tho 
Wne thing whether the numerator be divided by it, or the deno- 
minator multiplied by it. 

3. When the two numeratprs, or the two denominators, can be 
divided by some common quantity, Jet that be done, and thequo- 
tients used instead of the fractions first proposed. 

INVOLUTION. 



[ 18i> ] 

INVOLUTION. 

Involution is the raising of powers from any proposed 
root ; such as finding the square, cube, biquadrate, &c, of 
any given quantity. The method is as follows : 

* Multiply the root or given quantity by itself, as many 
tipes as there are units in the index less one, and the last pro- 
duct will be the power required.— Or, in literals, multiply 
the index of the root by the index of the power, and the 
result will be the power, the same as before. 

Note. When the sign of the root is +> all the powersof 
it will be 4- ; but when the sign is — , all the even powers 
will be .+f and all the odd powers — ; as is evident from 
multiplication. 

exampCes. 



«, the root 
tf* = square 
tf' z= cube 
tf * =r 4th power 
41* = 5th power 
&c. 



— 2a, the root 
-f- 4^* = square 
'— 8/j^ = cube 

+ 16^1* == 4th power 

J- 32a* = 5th power 



2ax^ 



, the root 



+ 



+ 



3A 
4a^j?* 

=: cube 

27*^ 

1 6a*x* 

■ I I «. 

81** 



= 4th power. 



a*, the root 
a^ = square 
a^ = cube 
if = 4th power 
**°= 5th power 
&c. 



- Zab\ the root 
+ 9a*i* = square 

- 27a'** = cube 

+ Sia'^b* = 4th power. 

- 243a**'° = 5th power. 



, the root 



2i 



^ = square 



« 



W 



=: cube 



16* 



- = biquadrate 



* Any powerof the product of two or more quantities, is equal 
to the same power of each of the factors, multiplied together. 

And an^ power of a fraction, is equal to the same power of the^p 
numerator, divided by the like [ibwer of the denominator. 

Also, powers or roots of the same quantity, are multiplied by 
one another, by adding their exponents ; or divided, by sobtract- 
ing their exponents. 

Hius, «^ X a* =««+*=; ff^. And a^^a« or --=: a = a. 

a* 



L 



190 AIX;^£BRA. 

j:-'a = root iji+tfnrool 



;r* + tfjr 


X 4- tf 


+ aj^ +2fl*x + fl' 


«' + 3tfT* + 3i^x + a' 






jf*— 2ar + a* square 
s -^ a 

jc^— 2aj;* +<!** 

jr5-3Ar*+ Mx-a^ 
Ae cubes^ or third powers, olx^a and x + ^. 

EXAMPLES. FOR PRACTICE. 

1. Required. the cube or 3d power of So'.' 

2. Reqjuired the 4th powi^.of 2i7'j. 

3. Re<|uired the 3d powej- of — 4iiV. 

4. To find the biquadrat^ of — --^. 

5. Retfuired the 5th power of a — 2x. 

6. To find.the 6th power of 2d^. 

Sir Isaac ICiTewtom^s Rdle for raising a Binomial to any 

Power whatever *. 

1. To find the Terms without the Co^efficiertts. The index of 
the first, or leading quantity, begins with the index of the 
given power, and in the succeeding terms decreases conti- 
nually by I, in every term to the last; and in the 2d or 
following quantity, the indices of the terms are O, 1, 2, 3, 4, 
&c, increasing always by 1. That is, the first teno will con-» 
tain only the 1st part of the root with the same index, or of 

— 1 -— — ' ' — ■■ I , ji ■ — ■ 

* Tbb ruW expressed in general tennsi is as follows t 

M 2 3 

^ ' 2 2 3 

Nate* The sum of the co-e£ident»^ in every power, is equd to 
the number 2, wl^n raised to that power. Thus 1-4- 1 =: 2 in 
the first power ; I + 2 + 1 = 4 == 2^ in the square ; i 4. 3 4- 3 
j+ 1 =: 8 =: 2^ in the cube, or third power ; and so on* ' 

the 



ItJVOLtfTIiDN. , -l$l 

I 

fllel^ame height as the inteticled 'power : and the kst teHn of 
the^ series will contain only the 2d part of the given teot, 
when raised aho to the same height'of the intended- p6wer: 
but an the other or intermediate terms will contain the ]pro^ 
ducts of some powers of both the members of the -root, in 
such sort, that the powers or indices of the 1st or leading 
•lembet will always decrease by 1» while those of the 2d 
member always increase by 1. 

2; To find the Co^iffictents. The first co-eifficient is always 
1, and the second' is the same as the index of the intended 

' jpew^er ; to find the 3d co-efficient, multiply that tA die ^d 
term by the index of the leadkig letter in the sameterm^ and 
divide the product by 2 ; and so on, that is, multiply the co- 
efficient of the term last found by the index of the leading 
quantity in that terhi, and divide ^he product by the number 
of terms to that place, and it will give the co-efficient of the 
term next following; which rule will find all the co-efficients^ 
one after another. 

, Nate. The whole number of terms Will be 1 more than the 
index of the given power i and when both tel^ms of the root 
are +, all the terms of the power will be + ; but if the '^e-r 
cond term be — , all the odd terms wilt /be' +,- afed all the 
even terms — , which causes the terms to be + and — alter-* 
nately. Also the sum of the two indices, in each term, is 
always the same number, viz. the index of the required 
power: and, counting from the middle of the series, both 
ways, or towards the right and left, the indices of the two 
terms are the same figures at equal distances, but mutually 
changed places. Moreover, the co-efiicients are the saihe 
numbers at equal distances from the middle of the series, 
towards the right and left ; so by whatever numbers the 
increase to the middle, by the same in the reverse order they 
decrease to the end. 

' tiAMPLES. 

\\ Let d-^^x be involved to the 5th power. ' 

The terms without the co-efficients» by the ist rule, 
#iU be 

a\ it%, c^x^y c^x^y #jr*^ jt^ 
and the co-efficients, by the 2d irule, will be 
. , ^5x4 10X3...1P„x2 ^>^\ 
l.^> 2 ' "^^^ 4^5^ 
or, 1,5, 10, 10, 5', \s 

Therefore the Ath power altogether is 
#5 + Ba\v + 10#»^* + lO^^jr^ + Sax^ + x\ 



But 



y 



192 ALGEBRA. 

But it is best to «et down both the cOhefficiehts and the 
powers of the letters at once, in ope line, without the inter- 
mediate lines in the above example^ as in the example here 
below. ^ 

2, Let fl — X be involved to the 6th power. 
The terms with the co-efl5cients will be 
a^-^ea^x + 15tf*x*- 20^x3 + ISd'j^-Gax^ + jt*. • 

S. Required the 4th power of ^ — jr. 

Ans. fl* - ^x + e(^x^ - ^ap^ + «♦. 

' And thus any other powers may be; set down ^t once^ in 
the same manner ; which is the be$t way. 



EVOLUTION. 



Evolution is the reverse of Involution^ being the method" 
of finding the square rootj cube root> &c>. of any given 
quantity, whether simple or compound. 

CASE I. To find the Roots of Simple QuMntities. 

Extract the root of the co-efficient, for the numeral 
part \ and divide the index of the letter or letters, by the 
mdex of the power, and it will give the root of the literal 
part ; then annex this to the former, for the whole root 
sought*. 



■■iiiUi 



* Any even root of an affirmative quantity^ may be either 4- 
or — : thus the square root of + c^ is either -^ a,ov — a ; be« 
cause + a X + a = + a*, and —a X — a r= + a^ also. 

But an odd root of any quantity will have the same sign as the 
quarittty itself: thus the cube root of -f a* is -f a, and the cube 
root of— a^is — ajfbr + ax +ax+a=:+«^, and —a X 
— fl X -^ azu -- a^. 

Any even root of a negative quantity is impossible; for neither 
+ a x-^a, nor —a X — a can produce —a*. 

Any root of a product, is equal to the like root of each of the 
fBictors multiplied together. And for the root of a fraction, take 
^he root of the numerator, and the root of the denominator. 

EXAMPLES. 



EVOLUTION. 19.S 

EXAMPLES. 

1. The square root of 4fl*, is 2/i, 

3* 

2. The cube root of 8«^ is 2a^ or 2a. 

3. Ihe square root of "^^> ^'' V^"5^j js ^\/5. 

4. ' The cube root of ' j , xs — r — ^2^, 

5. To find the square root of 2^***. Ans. fli*v^- 

6. To find thfe cube root pf — 64«^^ Ans. -4a**. 

7. To find^the square root of-^-r-, Ans. 9,abA/^* 

8. To find the 4th root of^U^b^. Ans. 3aby/b. 

9. To find the 5th root of - 32a*^. Ans. —^ab^b. 

CASE II. 

To find the Square Root of a Compound Quantity* 

This is performed like as in numbersi, thus : 

i . Range the quantities according to the dimensions of 

one pf the letters^ and set the root of the first term in the 

quotient. ^ , 

2. Subtract the square of the root, thus founds from the 
first term, and bring down the next two terms to tike re- 
mainder for a dividend) and take double the root for a 
divisor. 

3. Divide the dividend by the divisor, and annex the re- 
sult both to the quotient and to the divisor. 

4. Multiply the divisor, thus increased, by the term last 
set in the quotient, and subtract the pnx^uct from the 
dividend. 

And so on, always tlie same, as in common arithmetic. 

EXAMPLES. 

1 • Extract the square root of j*- ^Q}h + e^j*i* -- 4o3* + *♦. 
a'^-^^a^ + ^a^b^-^d^ + ** ( fl*-2fli + i* the root. 

2<l* - fitf* ) - 4tf ^^ + 6<i**** \ 






Vot. I. O 2. Find 



1 



194 ALGEBRA. 

< ^ 

2. Find the root of a^ + 4^a^b + lOaH"^ + l2aP + *♦. 









3. To find the square root of fl*'+ 4^' + 6«' + 4* + 1. 

Ans. fl* + 2« + l. 

4. Extract the square root of a^ — 2a^ + 2£^—a + ^. " 

.Ans. a:*— jr +4. 

5. It is required to find the square root of a* --at. 

CASE III. 

To find the Roots of any Powers in Genera/, 

This is also done like the same roots in numbers, thus t 
Find the root of the first term, and set it in the quotient. 
— Subtract its power from that term, and bring down the 
second term for a dividend. — Involve the root, last found, to 
the next lower power, and multiply it by the index of the 
given power, for a divisor. — Divide the dividend by the di- 
visor, and set the quotient as the next term of the root. — 
Involve now the whole root to the power to be extracted ; 
then subtract the power thus arising £rom the given power, 
and divide the first term of the remainder by the divisor first 
found; and so on till the whole is finished^. 

EXAMPLES. 



* As this method, in high powers^ ipay be thought too labo- 
rious, it will not be improper to observe, that the roots of com- 
pound quantities may sometimes be easily discovered, thus': 

Extract the roots of some of the most simple terms, and connect 
them together by the sign + or — , as may be judged most suit- 
able for the purpose. — Involve the componnd root, thus found, la- 
the proper power -, then, if this be the same with the given quan- 
tity, it is the root required. — But if it be found to differ oiily in 
some of the signs, change them fcom + to — , or from — to +, 
till its power agrees with the given one throughout 

Thus, 



EVOLUTION. 195 



EXAMPLES. 

i. To find the square root of a*--2a^t+Sa'k'^2aP + i\ 

— — ■ I . I. ■■ • !■ I 

2. Find the cube root of a^-^a^ + 21a^ - 44a' + 63^*- 

- 54fl + 27. 

f«-6^|i + 21a*-44tf'+ 63fl*— 54fl + 27 ( d'-2a + $. 






3tf^)-6a5 



ii* - 6^5> i2fl* - 8fl' = (^~2j)3 

a**-6^* + 21fl*-44fl5+63a*-54a +27 = {a^^2a-3y. 

3. To find the square root of «* - 2ab + 2ax + ^* - 
^x + j^. . Ans. a—b + j:. 

4. Find the cube root of a^ - Sn* + 9fl^-13/i' + ISa*- 
12a +,8. Ans. fl*-tf + 2. 

5. Find the 4th root of 81a* - 216a'* + 216a*** - 96a*» 
+ 16*^ Ans. 3a -2*. 

6. Find the 5th root of a^ - 10a* + 40a'- 80a* + 80/i 
— 32. . Ans. a— 2. 

!• Required the square root of 1 — :r*. 
8. Required the cube root of l—x'. 



i«i«i 



Thusy in the 5th example^ the root 3a— 2*, Is the differenpe of 
the roots of the first and last terms ; aad in the 3d example, the 
root a—* + x> is the sum of the roots of the ist^ 4th^ aod 6th 
terms. The same may also be observed of the 6th example, where 
the root is found from the first and last terms. 

O 2 SURDS. 






196 ALGEBRA. 

SURDS. 

SuKDs are such qutntkics m ha^« no exact rbot; 4nd are 
usually expre^ed by fractional indices, or by means of the 

radical sign a/. Thus, 3^, or ^/S, denotes the square root 

of S ; and 2^ or i/2S or V*, the cube root of the s^are of 
2 ; where the numerator shows the power to trhich the 
qutotity is tb be raised, and the denoniinator its root,. 

pHoblem I. 
Ta Reduce a Rational Quantity to the Form of a Surd. 

Raise the given quartity to the power denoted by the. 
index of the surd ; then over or before this nevr qiljiatity set 
the radical sign, and it will be of the form required. 

EXAMPLES. 

J • To reduce 4 to the form of the sqyare root. 
First, 4* = 4 x 4:=16; then Vl6 is the answer. 

2. To reduce Sa^ to the form of the cube root* 
First, 3^' X^d" X Za"- = {^a^f == 27a% 

then ^'Itf or (27/1^)"^ is the answer: 

3. Reduce 6 to the form of the cube root. 

Ans. (216)7or^l0. 

4. Reduce \ab to the form of the square root. 

Ans. /fiV. 

5. Reduce 2 to the form of the 4th root. Aiw. (U)^. 

I 

6. Reduce a'^ to the form of the 5th root. 

7. Reduce a-^-x to the forhi of the square root. 
S. Reduce a—x to the form of the cube root, 

PROBLEM II. 

' * I. " 

To Reduce^ Quantities to a Common Index. 

1. Reduce the indices of the gi^eh quantities to a com-i- 
mon denominator, and involve each of mem to the power 
denoted by its numerator ; then 1 set ov^ the common ^^ 
npminator will form the common index. Or, 

2. If 



I 



t 



SURDS. 197 

2. If tht common index be given, divide the indices 
of the quantities by the given index, and the quotients 
will be the new indices for those quantities. Then over ' 
the $sud quantities, with tbeir new indices, set the given 
ind^x, and they will make the equivalent quantities SQughiji 

EXAMPLES. 
I 1 ' 

1. Reduce 3^ and 5'^ to a common ind^x. 
Here 4 and I = ^^ and -1%. 

Therefore 3^ and 5tI=(3s)tV ^nd {5^)^=^'^S^ and '^S* 

= '°/243 and ^^25. 

1 

2. Reduce (^ and AT to the same common index -J* 

Here, 4.4-4 3=-|-x-^=3f the 1st indes^ 
and 1^4 = 1- X 4^ = f the 2d index. 

Therefore (tf®)^ and (H^)^, or x^cf and ^S^ are the quan- 
tities. 

3. Reduce 4^ and 5^ to the common index ^. 

Ansl 2567)Tand2S^. 

II 

4. Reduce d^ and x^ to the common index 4- 

Ans. (fl*)^and(.r^)"«'. 

- 5. ' Reduce a^ and .r^ to the same radical sign. 

Ans. iy/fl* and a/x^* 

6. Reduce (a + x)'^ and (a— j?*)^ to a common index. 

I I 

7, Reduce {a + ^)^ and (a— ip to a commoti mdex. 

PROBLEM III. , 

T& Reduce Surds to more Simpte Terms. 

Find put the gre^te^t power contained in, or to divide the 
given surd 5 take its root, and set it before the quotient pr 
the remaining quantities, with the proper radical sign b^ 
tween them. 

EXAMPLES* 

1. To reduce -v/82 to simj^er terms. 

Here v^32=: ^16 x2=^\6 x V2 =^4xV2=:4^2. 

2. To reduce ^320 to simpler terms. 

^320 = 3/64 X 5 =4/^4 X 4/5 = 4 X 4/5 = 4^/5. 

, S. Reduc 



19S ALGEBRA. 

3. Reduce -v/ 75 to its simplest terms. ^ Ans. 5a/S. 

4. Reduce %/f| to simpler terms. Ans. -rrV^^^* 
5/ Reduce ^189 to its simplest terms. Ans. 3'/7. 

6. Reduce V^^V to its simplest terms. Ans. :|-(/10, 

7. Reduce ^ISe^b to its simplest terms. Ans. Sax^Sb. 

Nate, There are other cases of reducing algebraic siu>ds 
to simpler forms, that are practised on several occasions ; one 
instance of which, on account of its simplicity and usefulness, 
may be here noticed, viz. in fractional forms having com* 
pound surds in the denominator! multiply both numerator 
and denominator by the same terms of the denominator, but 
having one sign changed, fr9m + to — or from — to +> 
which will reduce the fraction to a rational denominator. 
xi r^ A ^2Q + ^/12. • ^/5rf^3 

Ex. To reduce ^^^^3 , multiply it by z^s^%' *°^ 

. ^ . 16 + 2v^l5 , ' ^, .^3v/15-4^5 

It becomes — ^ — -; =8+^^15. Also, if — ,, ^ . — 7--; 

, . , . , s/lS-x/S ... 65 - Yv'^S 

multiply It by ,, ^ ' — TT, and it becomes — = 

'^ ^ ^ -v/15 — ^5 15 — 5 

65-35x/3 _ IS — 7^/3 
10 ■* 2 ' 

PROBLEM IV. 

To add Surd Quantities together. 

» 

i. Bring all fractions to a common denominator, and 
reduce the quantities to their simplest terms, as in the last 
problem. — 2. Reduce also such quantities as have unlike 
indices to other equivalent ones, having a common index.— 
S. Then, if the surd part be the same in them all, annex it 
to the sum of the rational parts, with the sign of multiplica- 
tion, and it will give the total sum required. 

But if the surd part be not the same in all the quantities, 
they can only be added by the signs + and — . 

EXAMPLES. 

1. Required toadd v'l^ and v'32 together. 



First, -1/^8= -/9x 2=3-v/2; and '•32=v^I6x 2=4-v/2: 
Then,.3-v/2 + 4-v/2 = (3 + 4) ^2 = 1^.2 =- sum required. 

2. It is required to add ^375, and VI 92 together. 

First, V375=V125X3=:5V3; and ^102= V64 x 3 =4^3 : 
Then, 5^3 + 4V3 = (5 + 4) V3 = 9?/3 = sum required. 

3. Kequired 



' 



.; SURDS. isy 

S. Required the sum of V27 and V48. Ans". 7 V^3. 

4. Required the sum. of V50 and V72. Ans. 11 \^ 2. 

5. Required the sum of ./I and -/-^ 

Ans. 4 V-iV o** A^l^- 

6. Required the sum of ^56 and ^189. Ans. 5^/7, 

7. Required the sum of V|- and^^ ' Ans* ^l^. 

8. Required the sum of 3 Va^b and 5Vl 6a^i» 

PROBLEM V. 

To find the Difference of &urd Quantities, 

Prepare the quantities the same way as in the last rule; 
then subtract the rational parts, and to the remainder annex 
the common surd, for the difference of the surds required. 

But if the quantities have no common surd, they can only 
be subtracted by means of the sign — . , 

EXAMPLES. 

I 

1. To find the difference between V320 and V 80. 

First, v^320= v'64 x 5—8^/5', and-/80=: ViGx 5=:4^^5, 
• Then S^ 5 — 4 -• 5 = 4 ^Z 5 the difference sought. 

2. To find the difierente between ^128 and ^54. 

First, 3/128=3/64 x 2=4^2 ; and 4/54= ^27 x 2=3J/2. 
Then 4^ — 3^ =v^2, the difference required. 

3. Required the difference of ^15 and v^48. Ans. ^3. 

4. Required the difference of 3/256 and ^32. Ans. 2?/4. 

5. Required the difference of \/^ and ^^. Ans. ^V6. 

6. Required the difference of ^/^ and 3/y . An*. -f^l$. 

7. Find the difference of v'24tf^^* and ^/54ab\ 

' * • Ans. {a-2by^{Sb''-2ab)^ 6a. 

' PROBLEM VI. 

To Multiply Surd Quantities together • 

Reduce the surds to the same index, if necessary ; next 
multiply the rational quantities together, and the surds toge- 
ther ; tnen annex the one product to the other for the whole 
product required ; which may be reduced to more simple 
terms if pecessary. 

EXAMPLES. 



200 ALGEBk A. 



EXAMPLES, 

!• Rei^ulred to find the product of 4^^/12 and Sv^2. 

Here, 4x3xv'12xV2 = A2v/12x2=12v'24=12^4x6 
= 12 X 2 X \/6 = 24 v' 6. the product required, 

2. Required to mukiply ^^i by |^|. 

Here ^x^>?/^x|/| = ^x^^=^\xl/ii=^^xix Vl8 = 
^^18, the product required. 

3. Required the product of 3^2 s^d 2V'8. Ans. 24. 

4. Required the product of |^ and |V12. Ans. ^^6. 

5. To find the product of j^s/i ^^^ Av^t« Ans. -^^ 15. 

6. Required the product of 2^14 and S^4. Ans. 12^7. 

ft 4 

7. Required the product of 2a^ »nd a^. Ans* 2a\ 

8. Required the product of {a + 3)^ and {« + *) ♦. 

9. Required the product of 01.^ + \/* and 2x— v^*. 

10. Required t^ product of (a + 2v/i)^, and (« - 2-v/*)' 

11 J, Required the product of 2x' and Sx"*. 

i J. 

12. Required the product of 4;? and 25^** 

/ 

PROBLEM mi. 



t 



y 



To Divide one Surd Qmntity by another. 

Reduce the surds to the same index, if necessary ; then 
take the quotient of the rational quantities, and annex it to 
the quotient of the surds, and it will give the whole quotient 
required 5 which may be reduced to more simple terms if 
requisite;. 

EXAMPLES. 

1. Requh^d to divide 6^/96 by 3 v' 8. 

Here 6 -r 3 .V(96 -i- 8)ifc2v^l2 =: 2v/(4xa) :?? 2x2v/3 
= 4v/3, the quotient required. 

a Required to divide 12^280 by 3^5. 

Here 12 -f- 3 = 4, and 280 -r 5 = 56 = 8 x 7 = 2^7 ; 
Therefore 4 x 2 x4/7 = 8^7, is the quotient i^equired. • 

3. Let 



SURD8. rSW 

S. Let 4i/50)be divided by 2V'5. Ans. .2^ 10- 

4. Let 6V100 be divided by 3^5. Ans. 2^20. 

5. Let I-v^tW be divided by ^Vf. Ans. -^^y/B. 

6. Let ^;^-j% be divided by |^|. Ans. -^l^SO. 

7. Let 4 v'/i, or -Jo^, be divided by j^^. Ans. |^^. 

8. Let ^^ be divided hja^. 

9. To divide Sa^ by 4/i *. "* 



PROBLEM VIIX. 



To Involve or Raise Surd Quantities to any Power* 

■ > 

Raise both the rationd part and the surd patrt. Or mul-» 
Uply the index of the quantity by the index of the power to 
which it is to be raised, and to the result annex the power 
of the rational parts^ which will give the power required* 



EXAMPLES. 

1. Required to £nd the square of |ii\ 

Fin^t, (|-)*=ix 1=^2^, and {a^Y=za^^^^a^:^a. 
Therefore (-|^j^)* = iV^i ^s the square required. 

2. Required to find the square of \a'^. 
Fkst, -i X 1 = ^, and {a^f = a^'-a\/a\ 

z 

Therefore {ia^Y ^ ^fi^l/a ia the square required. 

I 
* 3. Required to find the cube of |.v/6 or f x e''. 

First, {\y =z f X I X f z= ,V» and (fi^f =r 6* = 6^6 j 
Theref. (|--v/6)3 = ^s^ x 6^/6 = ^-^^s/^ytbQ cube required. 

4. Required the square of ,2 V2. An«k 4^4. 

I • 
5.. Required the cube of 3^, or -/S. Ans. Sy'S. 

6« Required the 3d power of y V 3* . Ans, | v^ 3. 

7* Required to find the 4th power of iy/2. Ans. 4» 

8. Required 



2M ALGEBRA. 

S. Required to find the mth power of tf". 
9. Required to find the square of 2 + VS. 

PROBLEM IX. 

To Evolve or Extract the Roots of Surd Quantities*. 

Extract both the rational part and the surd part. Or 
divide the index of the given quantity by the index of the 
root to be extracted ; then to the result annex the root of 
the rational part ^ which will give the root required. 

EXAMPLES. 

/ 

1. Required to find the. square root of IGv'S. 

First, v'lS = 4, and (6^^= 6^"^^ = 6^ ; ' 

II ' ^ 

theref. (16 V^6)^ =: ^.6\zz 4jJ/6, is the sq. root required. 

2. Required to find the cube root of -rr ^^* 
Fu-st, i/-,V = h and (^S)^ = 3^"="^ r: 3^ j 

theref. (^ V3)'''=j. . 3^ = ^5^3, is the cube root required. 

3. Required the square root of 6^ Ans. 6\/6^ 

4. Required the cube root of ^tf^i. Ans. ial/6, 

5. Required the 4th root of I6a\ Ans. 2Va. 

X. 

6. Required to find the mth root of x" • 

7. Required the square root of a* ~ 6a Vi + 9^. 



* The square root of a binom ial or r esidual surd, a -^ b, 
m-^h, may be foand thus : Take \/a« — ^ =z cj 



or 



. 7 .flf + C fl—C 

then ^/a +6 = •— r— + v^-tt-; 

— ^ — r a -f-c a — c 

and v^o- 6 =z •/-— ^T"' 



Thus, the square root of 4 + 2v^3 = 1 -|- ^3 ^ 

and the square root of 6— 2\/5 := ^5 — 1 . . 

But for the cube, or any higher root, qo general rule is 



known. 



inhntte 



SURDS. 20S 

INFINITE SERIES. 

An Infinite Series is formed either from division, dividing 
by a compound divisor, or by extracting^ the root of a coih- 
pound surd quantity ; and is such as, being continued, would 
run on infinitely, in the manner of a continued decimal 
fraction. 

But, by obtaining a few of the first terms, the law of the 
progression will be manifest; so that the series may thence be 
continued, without actually performing the whole operation. 

PROBLEM I^ 

To Reduce Fr^actional Quantities into Infinite Series by Division. 

Divide the numerator by the denominator, as in commcm 
division; then the operation, continued as far as maybe , 
thought necessary, will give the infinite series required. 

EXAMPLES. 

2ah 

1 . To change r into an infinite series. 

a -^ a 

. . 2i* 2*^ 2** 
M+b)2iA..(2b h-T r + &<^- 

2ab + 2b^ 



-2i* 

- 2**-— 
a 




2*» 

a ■ 

2i} 2b* 




2i* 


23' 


* 


,3 .&c. 



2. Let 



a04 ALGEBRA. 

t 

2. Let- be changed into an infinite series. 

1 — a 

I - tf ) I . (l+a + tf*+^ + a^ + &c. 

1 - a 



a 



a* 



«*-a» 



a' 



S. Expand -^---^ into an infinite series* 

Ans. — X (1 - — h-i r + &c.) 

4. Expand — ^. — 7 into an infinite series. 

a <r Or 

1 - ^ . 

5. Expand -r—r — into an infinite series. 

Ans. 1 — 2.r + 2j:*— 2;ir' + 2x*, &c j 

«* . . . 

€. Expand - — \ — rrr into an infinite series. 

.2* 34* 44' 

Ans. 1 ' 1 — r r> &c. 

fl ' a* a'* 

1 
7. Expand •; — ; — - = 4> into an infinite series. 
1 4- 1 

PROBLEM II* 

To Reduce a Compound Surd into an Infinite Series. 

Extract the root as in common arithmetic ; then the 
Operation, continued as far as may be thought necessary, will 
give the series required. But this method is chiefly of use 
in extracting the square root, the operation being too tedious 
for the higher powers. 

EXAMPLES. 



INFINITE SERIES. fiOi 

EXAMPLES. 

i. Extract theitoot of rf^— ;r* in an infinite $erte. 



2a Sa^ I6a^ 128a7 



a" 



'--Ta^-^ 






or* a:* . ;i"* 



2^---:^)- 






4a* 8a* 64a* 






a 4a3 ' 8«* 64a' 



6 



*• *» 



r*+i 



8iJ* • 16a' 



&c. 



640^ 



2. Expand \/ 1 + 1 = \/^, into an ininite series, 

Abs. 1 + i - i + rt - x-It &c- 

3. Expand V'l — i i^^^o ^^ infinite series. 

Ans. l-i-l-iV— i4t*^- 

4. Expand ^/a^ + x into an infinite series. 



- ' - *---»- 



5. Expand Va^--2bx — or* to an infinite series. 

PROBLEM III. 

3i Epitract any Root of a Binmud : or to Riduce a ^ BimmicJ 

Surd into an Infinite Series* 

Thi jj trill bp done by substituting the particular letters of 
the binomial, with their proper signs, in the following 
general theorem or formula, viz. 

(p + PQ^) n =p n + - ^^^—— B(^+ —^ C(^+ &C. 

and 



206 ALGEBRA. 

and it will give the root required : observing that p dehotes 

m 

the first term, (^the second term divided by the first, T the 
index of the power or root ; and A> b, c» d, 8cc, denote the 
several foregoing terms with theii: proper sighs. 



EXAMPLES. 

1.. To extract the sq. root of a^+ ^> in an infinite series. 
Here p = tf*> Q = -«f and — = — : therefore 

m m r 

F n = (a*) Q = (a^)^ zz a ss At the 1st term of the series. 
— AQ = i X a X — = — 5= B, the 2d term. 

m^n 1-2 4* 4» 4* , , 

— - — B(i X — r- X :r" X "1 = — rr-, = ^j the 3d term 
2« ^ 4 2fl a* 2.4a^ 

m-2« 1—4 4* 4* 34* - ,' 

-3^c^ ;- — X - — 3 X ^= 5;^:^ = D the.4th. 

> 4* 3.4* 
Hencea+g-^-53j^3 + 5;j;^ - ^^^^^' ' 

4* 4* 4^ 54» 

1 

2. To find the value of ;^ r-, oritsequal (a— a;)"*kin 

(a— x)* 

an infinite series*. 



* Note. To facilitate the application 'of the rule to fiactional 
examples, it is prgper to observe, that any surd may be taken fi*om 
tb^ denominator of a fraction and placed in the numerator^ and 
rice versa, by only changing the sign of its index. Thus, 

-^ =1 X jr^ or only jr-« 5 and—— = 1 X (a +4)"^ of 
(« + 4)-« y and • ^ .^ = tt^a + jp)-« j and f! == x^ X «^ ; alse 



— 1 



(^ _ ' ^.y i = («Hx*)* X (o«-^) 'i &c. 

H«rt 



Here p 



INFINITE SERIES. »)7 

r 

=fl, o== = — ^~ •*"> and — = -— = — 2 ; there*. ♦ 



«», 1 

p » =: fa)"* = a"* r:— :; = a, the 1st term of the series* 
* ^ .a* 

— AQ=:— 2 X — X = , = 2a"'^ =x B, the 2d term. 

2« ^ ^ a^ ^ fl* 

w — 2« ^ Sot* -X 4ir' . -t , 



Hence at* + 2a"'a: + 3a"*jr* + 4flr*jr'. + &c, or 

1 . 2x Sx^ AiX^ . 5x^ «.,.'• J 

~7 H i- + —7- + — r- H s- ^c, IS the series required, 

;«* ' ^' ' a* ^ ^5 ^ ^^ . ^ 

3. To find the value of-^ — » in an infinite series. 

a — x 

x^ x^ s^ 
Ans; a + x -{ +-5+--r&c 

4. To expand V ^^^^^ or ^-^,^^^4 in a series. 

. 1 X* 3x* 5^** , 
Ans. ^r-T + "—7- — TTij aCC. 



a* 



5. To expand tt: in an infinite series* 

*^ (a — Of 

2J 3i* 4J^ 5J* , 

Ans. 1 +— + ^ + ^ + "^ *^- 
a a* a* a* 



e. To expand Va^-x^ ox (a* -jt*)"^ in a series 



* x^ x?" 5x^ 



X ^ w %^— « 

■*"*•''" 2a ~ 8a' T6a»~"l28a^ 



7. Find the value ofMa'-i') or (a'-A')^ in a series. 

Ans.a- — --^-g^&c. 

S. To find the value of V(a^ + -^0 or («*+-^0^^"^ series. 

x^ 2x^° 6jr** 
Ans.a+-^--^^,+ -j5^&c. 

9. To 



90% 



ALGEBRA. 



9. To find the square root of ' n in an infinite serief* 

I 

. b M x^ ^^ 

AnS. 1 --;;;- +-;;-£ '"^ ^^' 



10. Find the cuW root of 



a' 



e^ + b^ 



2a' 



in a series. 



63 3J6 

Ans. I — -— r + -r-x- 






&c. 



ARITHMETICAL PROPORTION. 



Arithmetical Proportion is the relaticxi between 

two numbers with respect to their difference. 

Four quantities are in Arithmetical Proporti(Mi, when the 
difference between the first and second is equal to the dif- 
ference between the third and fourth. Thus, 4, 6, 7, 9, 
and Uy a -{- di b^ b -{- d, are in arithmetical proportion. 

Arithmetical Progression is when a series of quantities 
have all the same common difference, or when they either 
increase or decrease by the same common difference. Thus> 
2, 4, 6, 8, 10, 12, &c, are in arithmetical progression, hav- 
ing the common difference 2; and a, a -^ d, a + 2dy fl + 3</, 
a + 4^/5 a + 5dy &c, are series in arithmetical progression^ 
the eommon difference being d. 

The mpst useful part of arithmetical proportion is con- 
tained in the following theorems : 

« 

1. When four quantities are in Arithmetical Proportion, 
the sum of the two extreme^ is equal to the sum of the two 
means. Thus, in the arithmetical 4, 6, 7, 9, the sum 4 + 
9 = 6 + 7=:l3: and in the arithmetical fl, a+rf, A, p-k-d^ 
the sum a-^-b-^-d-zia + b+d. 

2. In any continued . arithmetical progression, the sum of 
the two extremes is equal to the sum of any two terms at an 
equal distance from them. 

tAus, 



ARITHMETICAL PllOPORTION. *p^ 

Thus, ifAe ^^ries bfe 1, 3,5, 7, 9, 1 r, &c. 
Th^a 1 + 11 = 3 + 9 = 5 + 7 = 12. . ■ 

S. Tlie fast t^rm of an^ increasing arithmetical series, is 
equal to th« first term increased by the product of the 
common 'difference multiplied by the ntimber of tenn^ lessr 
one ; but in.a.4^rea^g>;Serie9r t,^^ is^t tQrmhiii<e^ual to. the 
first term lessened by i^inesaid ^pduct^ 

/ Thus, the 20th tewti of the^ s«fies 1 , »V 5is7, 9j &c, is sss' 
1 + 2 (20-- 1 J «= r 4- 2 i^ 19 = 1 -^^W^zz'SQ. •- 

And the «th term of tf, a^d, a^^Qd, a-r^Sd, «-'4i/,'&c, 

4. The sum of all th^ fvrffisimHty series in arithmetical 
progression, is equal to half the sum of the two extreines 
multiplied by the number ojTterm^. ^ - ^ ,.. t. 

Thus, the sum.of 1, 3, 5,%4>i &ci coi^iinved to the lOfli 

. (1 + 19) X 10 '20x10 : • 
term, is =5^ -^ =: — r- — a: lo x lO =s 100, 

Andthisvmo£ntermso{a,a+d^a + 2d, a+^d^toa^-md, " 

her- ...: 



\ 



1^ The first^tern)^,^ jaAHnKreasing arithmetti:al seriei^is 1, 
the comman, difference 2^ and thts namber of teritos'421 } re« 
<)uired the si^^pf ^^ series ?. , . ~ r i ^^ :: I 

First, 1 +2 x: 20^ 1 +40'== 41, isltHe lifet^term. 

14:41' ' '^^" '-' ■ I ^ ^ ->- ^ 
Thenf ^ - ^ fe'«0 ==-21 x 20 =i: 420, {heWm required. 

•. .' *' ' i -- « L -w: j: 4 . .if/j-* .'■t ! ^'^ff /^ 
2k The first term- of a decreosing^arithnietical so^es i# 1 99^ * 
like common dffiTcnrence'l^j and tht dumbe'r of tettxik 37 ; re-- 
quired th* ^m 6f the serieaf ? • :i '^j ' -^ 

; JFirst, 1 9d - 3 . e^ = 1 dp -- ISiS h J, iS'lhe last term,' . ,;, , 

^ Then ^ ^ X 67 = lOQ ?^ ^7 ;=,,6700, .th,e sum, ric^r 
quired.'-' ^^ ' ■ ■'■"'' ' ^ -••'^- • '• '-' '- •^^;- 
^3. Tp find the^sjim pf JOO^terins ©fjtjbe. ns^fllil^nUi^i^bm 

Y>VL- 1. , P . ^« He<]vured 



4. * Required the ^um pf 9^ terma of Ike ^d omilbers 

1, 3, 5, 7, 9, &c. ._ . . . -. • ; An* 9811. 

5. T2^ ^st term of a decres^ng.oH^ui^tic^i^rit^ls lO, 
the oamojQn. diff$i!9nce f, and the Dtin^r of te^m^Sl ) 
i^eqaked^.supapf, the serves? . r. Ana. 140. 

6. <)biejhitiidl«d' stoned beiR^ 'plai!e4«ii the ^6uhd, ins 
straight line, at the dktanc^ of ^' yitrdft frtittt each bther ; 
how far . viU zi^spri ^ tra?el» . whp \Ah9li bring tben\ ooe by 
one to a iyA9kst%_y^ck is- pi^ed 2 y^ds ftom^ihe first 
stone f ^ . Ans. 11 miles ^d 840 yards* 



*v .'!..' /I *•• ! o? 'r. 



APPXICATJON OF ARITHMETICAjL PROGRESSION 

' ' TO- MBLfTA^T 'AfTAIl^S. ^' ; 

■ .1 X "i • ■• ■■ - • • ■ 

OCTESTION I. 

A Triangular Battalion f, consisting of thirty ranks, 
in which tiie first rank is forifled of vone man biify, the 

second 



* The sum of aoy miAiber in) df IMUH: 4f the arithmetical 
aeries of odd number \,Z,5f 7, 9, &C| is equal to the square (n^) 
pf that number. That is, 

If 1 1^ '3, ^^ ^i 9» he, be (he nnoiiMn/than ^!U ^ 

. ^^i %\t^A\ 6\ Uc, be the suiiB 0£ Bpa, d^ dse, tormi; 

Thus, + 1 =: 1 or I2,the*tfm^i tfertn, • 

I .lb ^3 f?i 4or2^^the8u*o68t^ms . - 

4 + 52= o or 3^, the sum of 3 term9, 

, 9 + vj s: igor4>the-fium.of4 t6rm4>.icfr * . 

For, by the 3d theorem, 1+2 («— 1) = l'+ 2«— 2 = 2ii— 1 
i8thQli(5tterm^wfaenitfa9e''Ot|mberof'jlbrms ig.n; ti» tBia hist t^ui 
2»— \/Mi t^ flrat tpfro. h give^2n ^ sum of thexearfarciBes, or 
n half the sum of the extremes ; then, < by the 4tb thi^oretpi nXfi 
=r 71/^ is the sum of , all. the term?. Hence it appears in general> 
that half the suioa df the extremes;, is dlways iht satne as the Hub-*. 
ber of the tei^s n ^ ,^Dd that the sum of all the tcmk,'k the ^ame 
as ihi^squai^ of the same ndiriber, n^. 

See more on Arithmetical Proportion in the ArithoqtetiCj^. 
p. 111. 

'y.^'iti^ti^lM bittiiron, is tp be understood, a body of Iropps. 
Wg^ in the form of 9 triangle^ in which the Jl^ks exdedd ^ch 



ARITHMETICAL t^ROGRESSION. 21 1 

rtcond of S, the third of 5 5 and so on : What is the 
strei^[di of such a triangular battalion ? 

'^ Ans#eri 900 iiten. 



* J 



<^J£ST20K II, 

A detachment having 12 snccesshedays til march, with 
orders to advance the first docf only i ieagae$j the second S^, 
and so on, increasing I4 league each day's inarch: What is 
the length of the whole marth^ and what is the last day's 
inarch? 

Ansjver, the last day's march is I8j l^^igues, and 12S 
leagues is the length of the whole march* 



qjJESTION III. 

A brigade of sappers*^, having carried on 1$ yards of 
3ap the first night, the second only 13 yards, and so on, 
decreasing 2 yards every night, till at last Uiey carried on in ' 
one night only 3 yards : What is the number of nights thef 
Were employed ; and what is the whole length of tne sap ? 

Answer, they were employed 7 nights, and thq length of 
the whole sap was 63 yards. 



n . ' » i 



other by an equal number of men : if the first ranl^ consist of one 
man only^ and the difference between the ranks be also 1, tbea 
its form is that of an^equilateral triangle -, and when the dljfbrenc^ 
between tlw ranks is more than i, its form may then be an 
isosceles or scalene triangle. The practice of forming troops fa 
this order, which is now laid aside^ was formerly held in greatet 
esteem than forming them in a tfolid square^ a^ admitting of a 
greater fronts especially when the troops were to make simplj a 
stand on ail sides. 

^ A brigade of sappers, consists generally of 8 men» divided 
equally into two parties. While one of these parties is advancing 
the sap, the other is furnishing the gabions* fascines* and other 
necessary implements : and wnen the first party is tired, t^e 
second takes its place, and so on, till each man in turn has heed 
at the head of the sap. A sap is a snudl ditch, between 3 and -^ 
feet in breadth and depth -, and is distinguished froin the trench 
by its breadth only, the trench having between 10 and 15 feet, 
breadth. As an encouragement to sappers, the pay for all the 
work carried on by the whole brigade,, is given to the survivors* 

F 2 QJJESTfQN 



212 ALGEBRA. 

QUESTION IV, 

A number •£ gabions ^ being given to he placed in she 
ranks, one above the other, in such a manner as that each 
rank exceeding one another equally, the first may consist of 
4 gabions, and the last of 9 : What is the number of gabions 
in the six ranks; and what is the difference between each 
rank f 

Answer, the difference between the ranks will be 1» and 
the number of gabions in the six ranks will be 39. 
' * « • 

QUESTION V. * 

Two detachments, distant from each other 37 leagues, and 
both 'designing to eccupy an advantageous post equi-distant 
from each other^s camp^ set out at different times ; the first 
detachment increasing e^^ery day's march 1 league and a 
^half, and the second detachment increasing each day's march 
2 leagues : both the detachments arrive at the same time ; 
the first after 5 days' march, and the second after 4* diiys' 
faiarch ; What is ^he number of leagues marched by each 
detachment each day ? 

The progression -^, 2^xy, 3-,^, 5-^, 6^y answers the con* 
ditions of the first detachment : and the progression 14> 3|<» 
^Ti ^h smswers the conditions of the second detachment. 

QUESTION VI. 

A^eseitery in>his flight, travelling at the rate of 8 leagues 
a day ; and a detachment of dragoons being sent after him^ 
with orders tamiarch the first day only 2 leagues, the second 
5 leagues, the third 8 leagues, and so on : What is the 
number of days necessary for the detachment to overtake the 
deserter^ and what will be the number of leagues marched 
before He is overtaken ? 

Answer, 5 days are necessary to overtake him \ and coij- 
^quently 40 leagues will be the extent of the march. 



-*-•■ 



* Gabions are baskets^ open at both ends, made of ozicr twigs, 
smd of a cylindrical form : those made use of at the trenches are 
2 feet wide, and about 3 feet high $ whicb^ being filled with earthy 
serve as a shelter from the enemy's fire: and those made use of 
to construct batteries, are generally higher and broader. There 
is another sort of gabipn, made use of to raise a low parapet : its 
height is from 1 to 2 ftpt, and 1 foot wide at top, but somewhat 
les^ at bottom, to give room for placing the muzzle of a firelock 
between them : these gabions sen^e instead of sand bags. A sand 
bag is generally made to contain about a cubical foot of earth. 

• QUESTION 



PILING OP BALLS. 



(UTESnOH TII. 



A convoy • tSstant 95 leagues, having orders to join its 
camp, and to march at the rate of 5 leagues per day; its 
escort departing at the same timci with orders to march the 
first day only hidf a league, and the last day 94 leagues; 
and both the escort and convoy arriving at the same time : , 
At what distance is the escort m>m the convoy at the end of 
«ach march ? 



OF COMPUTING SHOT OR SHELLS IN A FIHISHED PILE. 

'Stiot and Shells are generally piled in three different 
forms, called triangular, square, or oblong piles, according 
as their base is either a triangle, a square, or a rectangle. 
Fig. 1. C G Fig. 2, 



ABCD, fig. 1 , is a triangular pile, 
EFGH, £g. 2, is a square pile. 

E A Fig. 



ABCDEP, iig. 3, is an oblong pile. 



* fi^ convoy is generally meant a snp^y of ammunition or 
provisions, conveyed to a town or army. The body of men that 
{Hard this supply^ is cslled Mcort. 

A triangular 



314 ALGEBRA. 

A triangulir pile is formed by the continual Ikying of 
triangular horizontal co.urse^ of shot one above another, i% 
such a manner, as that the sides of these courses, called rows, 
decrease by unity from the bottom row to the top row^* 
which ends always in 1 shot. 

A square pile is £Drm^d by the continual laying of square 
horizontal courses of shot one above another, in such a man- 
ner, as that f he sides of these courses decrease by unity from 
the bottom to the top row, which ends also in 1 shot. 

In the triangular and the square piles, th& sides or faces 
being equilateral triangles, the shot contained in those faces 
form an arithmetical progression^ having for first tem^unity, 
and for lait term and number of terms, the shot contained 
in the bottom row ; for the number of horizontal rows, or 
the number counted on one of the angles from the bottom to 
the top, is always equal to those counted on one side in the 
^ttom : the sides or faces in either the triangular or square 
piles, are called arithmetical triangles ; and the nuoaibers 
contained in these^ are called triangular numbers: abc, fig. 1, 
£FG, fig. 2, are arithmetical triangles. 

The oblong pile may be -conceived as formed from the 
square pile abcd ; to one side or face of which, as ai), a 
number of arithmetical^triangles equal to the face have been 
added : and the number of arithmetical triangles added to the 
square pile, by means of which the oblong pile is formed, 
is always one less than the shot in the top row ; or, which 
is the same, equal to the difference between the bottoooTrow 
of the greater ^ide and that of the lesser. 



QIJESTIOH Till. 

To find the shot in the triangular pile abcd, fig. ], the 
b6ttdm row ab consisting of 8 shot. 

SOLUTION. 

The proposed pile consisting of S horizontal courses, eack 
of which forms an equilateral triangle ^ that is, the shot 
contained in these being in an arithmetical progression, of 
which the first and last term, as also the number of terms, 
•are known ; it follows, that the sum or these particular 
^ourses, or of the 8 progressions, will be the shpt contained 
in the proposed pile i then . 

The 



PILING OS BALLS. 



tlS 



Hie sW of the fitst or lowet > 
^ trianj^oiaf coarse vttlbe 5 




8+ 1 X 4 ss'^e 


the second - 

' ( 1 
* < 


7+a X3^=g|f 


the itivd \ ' m .' ^ * «. 


6 + 1x3 &: 2t 

§ 


the fourth - 


5 ^- I X 2i s= 15 


the fiita •- •* • - 


♦ + I xS = 19 


the sixth - - - 


3 + 1 X li =s $ 


the seventh * - 


2 + 1x1 = 3 


the eighth - - 


1 + ix i* I 


X 


Total - I20sl 



J 



in the pile propo$e(4 

qt^ESTioN IX. 

To find the shot of the square pile xfgHj fig. 2, the bot^ 
torn row £F consisting of 8 shot. 

SOLUTION. 

The bottom row containing 8 shot> and the second only 7 ; 
that isj the rows forming the progression, S, 1, 6, 5, 4, 5,2, 1 p 
in which each of the terms being the square root of the shot 
contained in each separate square course employed ill formin^r 
the square pile j it follows, that the sum of the squares of 
these roots will be the shot required : and the sum of the 
squares, divided by 9» 7, €, 6, 4> S^ 2, 1 , being 204, expresses 
the shot ia the proposed pile. 

QJTESTION X. 

To find the shot of the oblong pile abc^def, fig. 3 ; in 
which >F «s 16, sind xc =b 7. 

SOLUTION. 

The ^oblong pile proposed, consisting of the square pile 
ABCD, whose bottom, row is 7 shot ; besides 9 arithmetical 
triangles or progressions, in which the first and last term^ ?s 
also the number of term$, are known ^ it follows^ that, 
if to the contents of the square pile - 14Q 

we add the sum of the dth progression - 252 , 

their total gives the contents required - S92 shot* 

REMARK I. 

The shot in the triangulin: and the square piles, as also 
the dbiot in each, horizontal course, may at once be ascer<^ 

tained 



v^ 



ALGEBRA. 



tauned by the followmg table : the Optical c<diimn a, coiu* 
taint the shot in the bottom row, from 1 tx> 30. izicliisiTe ; 
the column b contains the triangular numbers, cr number 
of eslch course ; the column c contains the sum of the 
triangular numbers, that is, the shot eontaihed in a trian^ 
gular pile, commonly called pyramidal numbers ; the column 
B contains the square of the numbers of the column a, that 
is, the shot contained in each square horizontal course ; and 
the column £ contains' the sum of these squares or shot in a 
square pile.* 



c 


B 


A 


D 


E 


Pyramidal 


Triangular 


Natural 


Square of 

thj* vtAtuml 


Sumofthese 

square 

numbers. 


numbers. 


numbers. 


numbers. 


LUC llV«*iil|flil 

Qumbers. 


1 


I 


1 


. 1 


I- 


4 


3 


2 


4 


5 


10 


6 


S 


9 


14 


20 


^ 10 


4 


16 


30 


35 


15 


5 


25 


55 


56 


21 


6 


36 


91 


34 


28 


7 


49 


140 


120 


36 


8 


64 


204 


165 


45 


9 


81 


285 


220 


55 


10 


100 


385 


28(J 


66 


11 


121 


506 


♦ 364 


78 


12 


144 


650 


455 


91 


13 


169 


8I9 


5d0 


105 


14 


19^ 


1015 


680 


120 


15 


225 


1240 


816 


136 


16 


^56 


1496 


969 


153 


17 


289 


I78d 


1140 


171 


18 


324 


2109 


1330 


190 


19 


,361 


2470 


1540 


i 210 


20 


400 2870 t 



^Thus, the bottom row in a triangular pile, consisting of 
9 shot, the contents will be 165 ; and when of 9 in the square 
pile, 285.— ^In the same manner, the contents either of a 
square or triangular pile being given, the shot in the bottom 
roynfc may be easily ascertained. 

The contents of any oblong pile by the preceding table 
may be also with little trouble ascertained, the less side not 
exceeding 20 shot, nor the difference between the less and 
the greater side 20. Thus^ to find the shot ia an oblong pile, 

... the 



PILING OF BALLS. <11 

the less side bein^ 15, and the ^eaten95, we are first to 
imd the contents of the square pile, by means of which the 
oblong pile may be conceived to be formed $ that is, we tare 
to find the contents of a square pile, whose bottom ro# is 
15 shot ; which being 1240, we are, secondly, to add these 
1240 to the product 2400 of the triangular number 120» 
answering to 15, the number expressing the bottom row of 
the arithmetical triangle, multiplied by 20, the number of 
those triangles; and their sum, being 3640, expresses the 
number of shot in the proposed oblong pile. 



REMARK II. 

\ 

The following algebraical expressions, deduced from the 
investigations ot the sufns of the powers of numbers in 
arithmetical progression, which are seen upon many gunners* 
callipers *, serve to compute with ease and expedition the shot 
or sheUs in any pile. • 

That serving to compute any triangular ) ^^^ + 2x /i -f 1 x^ 
pile, is represented by I 6 

\ 

That serving to compute any square ) ^ -f 1 x 2fg 4- 1 x ft 
pile, is represented by ) 6 



In each of these, the letter n represents the number in the 
bottom row : hence, in a triangular pile, the number in the 

bottom row being 30 ; then this pile will be 30 + 2 x SO + I 
X V ^ 49$0 shot or shells. In a square pile, the number 
in the bottom row being also 30; then this pile will be 

30 + 1 X 60 + I X ?/ = 9455 shot. or shells. 
That serving to compute any obiong pile, is represented by 

2/1+1 +3«ix« + 1 X « . ,1-1. 1 J 

■ ' ■ • — i in which the letter n denotes 



* Callipers are large compasses, with bowed shanks, serving to 
take the diameters of convex and concave bodies. The gunners' 
callipers concisx of two thin rules or plates, which are moveable 
quite round ^ joint, by the plates folding one over the other : the 
length of each rule or plate is t> inches, the breadth about 1 
inch It is usual to represent, on the plates, a variety of scales, 
tables, proportionH, 6kC, such as are esteemed useful to be kuovq 
by persons employed about aitillery ; but; except the nfMsuring 
of the caliber of shot and cannon^ and the measuring of saliant and 
re-entering at^gles, none of the articles, with which the callipers 
are usually filled^ are essentia^ to that instrument. 

the 



fit ALGEBRA. 

the number of coimcSy aBcl the letter m the number of shat^ 
less ODe> in the top row : hence, in an oblong pile the num- 
ber of courses being 30» and the top row 31 ; thb pile wiU 

be 60 + 1+90 X 30+1 x V" = 23405 shot or shells. 



GEOMETRICAL PROPORTION, 

Geometrical Proportion contemplates the relation of 
quantities con^dered as to what part or what multiple one 
is of another, or how often one contains, or is contained 
in, another.— Of two quantities compared together, the first 
is called the Antecedent, and the second the Consequent. 
Their ratio is the quotient which arises from dividing the 
one by the other. 

Four Quantities are proporticmal, when the two couplets 
have equal ratios, or when the first is the same part or mul- 
tiple of the second, as the third is of the fourth. Thus, 
3, 6, 4, 8, and a, ar^ b, br, are geometrical proportionals. 

or bt 
For ^ = |. = 2, and — = — = r. And they are stated 

thus, 3 : 6 :: 4 : 8, &c. 

Direct Proportion is when the same relation subsists be- 
tween the first term and the second, as between the third and 
the fourth : As in the terms above. But Reciprocal, or 
Inverse Proportion, is when one quantity increases in the 
same proportion as another diminishes : As in these, S, 6, 8, 
4 ; and these, a, ar^ br^ b^ 

The Quantities are in geometrical progression, or con- 
tinuous proportion, when every two terms have always the 
same ratio, or when the first has the same ratio to the 
second as the second to the third, and the third to the 
fourth, §cc. Thus, 2, 4, 8, 16, 32, 64, &c, and a, ar^ at^, 
ar^, ar^y ar^, &c, are series in geometrical progression. 

, The most useful part of geometrical proportion is con- 
tained rtr the following theorems ; which are similar to those 
in Arithmetical Proportion, using multiplication for addi- 
tion, &c. 

. 1, When 



r 



GEOMETRICAL PROPORTION. 819 

1. W^jen four qujuitities are in geometrical j>roportion, 
the product of the two extremes i$ equal to the product of 
the two means. As in these, 3, 6, 4, 8, where 3x8=6 
X 4f=i24:i aad in tht^se, a» ar% b^ br^ where ax kr^ar x 

2. When four quantities are in geometrical proportion, 

the product of the means divided by either of the extremes 

gives the other extreme. Thus, if 3 : 6 : : 4 : 8^ then 

6x4 6x4 

■ ■ =: 8, and- ■ = 3} also if tf : ar :i b : br, then 

= ir, or — r— = a* And this is the foundation of the 

a or 

Rule of Three. 

9* In any continued geometrical progression, the product 
of the two extremes, sekI that of any other two terms, 
^quaUy ^stanl: fr(»n them, are equal to each other, or equal 
to the square of the middle term when there is an odd 
number of them. So, in the series 1, 2, 4, 8, 1(5, 32, 64, &c, 
it is 1 x 64 = 2 X 32 = 4" X 16 = 8 x 8 = 64. 

4. In any continued geometrical series, fhe last term is 
f qual to the first multiplied by such a power bf the ratio as 
is denoted by 1 less than the number of terms. 'Thus, in the 
series, 3, 6, 12, 24, 48, 96, 8cc, it is 3 x 2^ = 96. 

5« The sum of any series in geometrical progression, is 

{bund by multiplying the last term by the ratio, and dividin|r 

the difference of this product and the first term by the difc 

ference between I and the ratio. Thus, the sum of 3, 6, 

192 X 2 — 3 

12, 24, 48, 96, 192, is — -^ = 384- 3 == 381. And 

2— i 

the suro of n terms of the series j, or, ar^, ar^^ ar^, &c, to 

, . ar^"^ X r— tf af—a r" — 1 

I jg rz ~ = -a. 

' r—l r- 1 r — 1 



6. When four quantities, «, tfr, ft, Ar, or 2, 6, 4, 12, are 
proportional ; then any of the following forms of those quan- 
tities are also proportional, viz. 

1. Directly a : ar :: b : Ar ; or 2 : 6 : : 4:12. 

2. Inversely, ar :a : : ir : A j ar 6 : 2 : : 12 : 4. 

3. Alternately, j : 6 : : «r : ir; or 2 : 4 : : 6:12. 

4. Coxn^ 



9ib AtCEBIlA. 

4. Comp6\md€dlyfO:a+ar::b:b+kp\ or2:S::4:I6. 

5. Dividediy, a i ur^a iihi br~-b\ or 2 : 4 :: 4 : 8. 

6. Mixed, ar^g znr^a :: br^bibr^b\ or 8 : 4 :: 16 : S. 

7. Multiplication, aciarc: : fc : brc ; or 2.3 : 6.3 : : 4 : 12* 

8. Division, — :—:;>: ir; or 1 : 3 :: 4 : 12. 

c c 

9. The numbers a, b, c, d^ are in harmonical proportion, 

when A I d II a^nh I czo d\ or when their reciprocals 

1111 
•^% "T* ""■> ^> are in arithmetical proportion* 



EXAMPLES. 

1. Given the first term of a geometrical series 1, the ratio 
2> and the number of terms 12 ; to find the sum of the series? 

First, 1 X 2" = 1 X 2048, is the last ternu 

i„ 2048x2 — 1 4096 — 1 , , . 

Then — ss ■ == 4095> the sum required. 

2. Given the first term of a geometric series -f-, the rati© 
I, and the number of terms 8 ; to find the sum of the series? 

Krst, I X {\y = I X TTT = TTTf »s the last term. 
Then (f ^3^ X 4) -4- {\-i) = (i-yW) ^ 1 = fH X x 
=: f 44'> ^^ ^^^ required. 

3. Requited the sum of 12 terms of the series 1, 3, 9, 27, 
SI,&c. Ans. 265720. 

4. Required the sum of 12 terms of the series 1, -f, ^t 
Tjy Tr> &ۥ Ans. ttHtt* 

5. Required the sum of 100 terms of tha series I, 2, 4, 8, 
16, 32, &c. Ans. 1267650600236229401496703205375. 

3ee more of Geometrical Proportion in the Arithmetic* 



I 
I 



SIMPLE EQUATIONS. 

An Equaticm is the expression of two equal quantities, 
with the sign of equality (=•) placed between them. Thus, 
10^4 =x 6 is an equation, denoting the equality of the quan- 
tities J.0 — 4 and 6. 

Equations 



SIMPLE EQUATIONS. -221 

Equations are either simple or compound. A Simple 
Equation, is that which contains only one power of the un- 
known quantity, without including different powers. Thus, 
x^a Si b + c, or ax^ = 4, is a simple equation, containing 
only one power of the unknown quantity iZ*. But jt*— Sat 
:;^&^ is a compound one. 

Reduction of Equations, is the finding liie value of the 
unknown quantity. And this consists in disengaging that 
quantity, from the known ones ; or in ordering the eqtia^ 
tion so, that the unknown letter or quantity may stan<i 
alone on one side of the equation, or of the mark of equality, 
without a co^efficient ; and all the rest, or the known quan- 
tities, on the other si€le.«-4n general, the unknown quantity 
i$ disengaged from the known ones,> by performing always 
the reverse operations. So, if the known quantities are con- 
nected with it by + or addition, they must be subtracted ; if 
by minus (—), or subtraction, they must be added'; if by 
multiplication, we must divide by them ; if by division, we 
^ust multiply ; wheQ it is in any power, we must extract 
the root ; and when in any radical, we must raise it to the 
power. As in th^ foUdwiiiff particular rules ) which are 
tbundedon the general principle of performing equal operas 
tiQits on equal quantities; in which case i| is evident that 
the results must still be equal, whether by equal additions, 
or ^ubtraoliops, or multiplicatipns^ or divisions, or roots, or 
powers, 

FARTXCULAIt RULE I, 

« 

When known quantities are connected with the unknown 
by + or — } transpose them to the other side of the ecjua- 
tion, and change their signs. Which is only adding or sqb-' 
tracting the same quantities on both sides, in order to get 
all the unknown terms on one side of the equation, and ail 
the icnow|i pnes on (he other side *« 

- - Thusy 



' » Wii I " ■ ■ . ■! . ■'■■ * '■' ^- " ^1 ■■■»■>! 



♦ Here it is earnestly recommended that the pupil be aq-f 
customed^ at every line or stejj in the reduction of the equation $». 
to name tbo particular operation to be performed on the equation 
in the last line^ in order to produce the next form or state of the 
eqoation> in applying each of these rules, according as the particular 
^rm of th^ equation may require 5 applying them according to the. 

order 



2^2 - ALGEBRA. • 

Thus, i(x 4- 5=S ; then transposing 5 givte jr=r8 — 5==$. 

And, if a; — 3 -{-7=^9 then transposing the 3 and 7, gives 
j:=:9 + 3— 7 = 5. 

Alsb, if X ^ s -f b=s cd: then by transposing dr ^nd 5, 

itisx =: a — b + cd. 

In like manner, if 5jc — 6 = 4x + 1 0, then by transposing 
6 and 4jr, it is 5r-.4?a: = 10 + 6, or .r = 16, 

RUtE II. 

"When the unknown tferm is multiplifed by aiiy quantity ; 
idiride all the terms of the equation by it. 

Thus, if /7jr=^5 — 4a; then dividing by a, gives x ^i-^if. 

And, if 3x + 5 xs 20 ; then first tron^osing 5<fi^^ SJt: 
sc 15 } and then by dividing by 3, it is x = 5. 

In like manner, i{ax+3ab=4€^^f then by dividing by <j, it 

4r* 4:^^ 
is x+Sb = ; and then transposing 8^, gives x =- — 3A 



RULE III. 



When the unknown term is divided by any quantity ; we 
tnust then multiply all the terms of the equation by that di- 
visor ', which takes tt away. 

Thus, if— =i= 3 +2 : then mult, by 4, gives a: = 12 4 8 = 20*. 

4 

And, if — = 3i + 2^ - rf; 

a • ' ■ 

then by mult, a, it gives x = Sab + 2ac — ad. 

5 

Then by trani^posing 3, it is ^x = 10. 
^d multiplying by 5, it is 3x = 50. 
Lastly dividing by 3 gives j; = 16y. 



order in which tbey are here placed -, and beginning every line 
with the words Then by, as in the following specimen^ (^ Bt- 
amjples > which two words will always bring to his recoUectioo^ 
that He is to pronounce what particular operation he is to perform 
dn the last \\{ie, in order to give the next -, allotting atways a 
single line for each operation^ and ranging the equations neatly 
just under each otlier^ in the several lines^ as they are successively 
produced. 

Rule 



SIMPLE l£QUATIONS. S&S 



RULE lY. 

Whek the unkn(^#n; ^tlamtiiy is incloded in any root or 
surd : transposes tiie rest; of the terms, if thtsre be any, bjr 
Rule 1 ; then raise e^ch side to such a power as is deao^ 
by the index of the surdf viz* square each side when it is 
the square root ; cube each side when it is the cube ro9t; &c. 
which clears that radical. 

Thus, if V.r— 3 = 4f ; then transposing 3j gives Vt=s7 j 
And squaring both sides giires -r = 49. 

And, if V2J+To K « : . 
Then by squaring, it becomes 2x + 10 = 64 j 
And by transposing 10, it is 2x =: 54/^ 
Lastly, dividing by 2, gives x zz 27. 



Also, if3/3.r+4 + 3 =6: . v 

Then by transposing 3, it is ^3^ + 4 ^ 5;.'" 
And by cubing, it is 3;i' + 4 = ^7 ; * ^ 

Also, by transposing 4, it is Sjt ^=: 23 ; 
Lastly, dividing by 3, gives x ss 7.J, 



R«L£ V. 



I 
I 
% 



r 

Whe j? that side of the equation which contains the un- 
known quantity is a complete power, or can easily be reduced 
to one, by rule 1, 2, or 3 : then extract the rbbt of the said 
power on both sides of the equation ; that is, extract the 
square root when it is a square power, or th^ cube root 
when it is a cube, &c. 

Thus, if JT* + Sj? + 16 = 36, or {x + 4)' = 36 : 
Then by extracting the roots, it is r + 4 = 6 ; 
And by transposing 4, it is x ;= 6 — 4 = 2. 

Andif 3^*-19 = 2l +3^. 
Then, by transposing 19, it is 3j:^,r= 75 ; 
And dividing by 3, gives .r* =s 25 ; 
And extracting the root, gives jr = 5. 

' Also, if fr*— 6=s 24. 
Then transposing 6, gives f x* = 30 ; 
And multiplying by 4, gives 3:r* = 120; 
Then dividiag by 3, gives x* = 40 $ 
Lastly, extracting the root, gives x = V40 = 6*324555« 



RULE 



.f2iy ALGEBRA. 



EULE VU 

:. When there is any analogy or proportion, it. is to be 
changed into an equation, by multiplying the two extreaae 
tdrois together ) and the two means together, and xmaking 
the one. product equal to the other. 

Thus, if 2x : 9 : : 3 : 5. 
Then, mult, the extremes and means, gives lOx =: 27 v 
And (dividing by 10, gives ^r- =: 2-^, 

And i{ ^x : a :: 5i :2c. 

Then mult, extremes and means gives icx = S^t ; 

And multiplying by 2, gives Sex si \Oab ; 

\Oab 
Lastly, dividing by 5r, gives a: =-- — • 

Also, if 10— X : fjr : : 3 : 1. 
Then mult, extremes and means, gives 10-<-r r= 2r ; 
And transposing ir," gives 10 = 30*^ 
Lastly, dividing by 3, gives 3 j. = ;c, 

nvm VII. 

When the same quantity is found on both sides of ait 
equ?.tion, with the same sign, either plus or minus, it may be 
left out of both : and when every term in an equation is 
either multiplied or divided by the same quantity, it may b^ 
strucl? ou^ of them all. 

Thus, i[3x + 2az^2a + b: \ , 
Then, by taking away 2^, it is 3x =: k^ 
And, dividing by 3, it is :r = ^6. 

Also if there be 4ax + 6ah = lac. 
Then striking out or dividing by fl, gives 4j: + 6i =; 7r, 
Then, by transposing 6^, it becomes ^x r: 7^—6^; 
And then dividing by 4? giv^s ;r = ^r— J^. 

Again.if|x-J=V>-.J. . ; , . 

Then, taking away the ^^ it becomes -|.r =: */ 5 
And taking away the 3's, it is 2jr =: 10 ; 
Lastly, dividing by 2 gives ^ n 5, 



MISCELLANEOUS KXAMpLESt 



i\ 



1. Given 7;r- 18 = 4«r + 6 ; to find the value of \r.^ 
First, transposing 1 8 and 5j; gives 3;v 2: 24 ) 
Then dividing by 3, gives jr =r 8. 

1. . ^' Qlvesx 



SIMPLE EQUATIONS. . 2t& 

I 
% GivenSO— 4a:-12 s=s^— Ipjr; to find i*. 

Firtt transposing 20 and 12 and IOjT, gives 6x » 84 ; 

Then dividing by 6, gives j: = 14. 

^. Let 4tfjr- 5^ = 3<& + 2<? be given ; to find x. 

First, by trans. 5* and 3dXi.it is ^ax-^Sdx = 5* + 2r; 

5ft4-2r 
Then dividing by 4fl— 3</, pves a: = ■ ^> • 

4* Let 5;r*— 12x c= 9 J? + 2jr* be given; to find *. 
. Fii^, by dividing by x, it is 5j: — 12 =: 9 + 2jr ; 
Then transposing 12 and 2x, gives ^jt rt 21 ;- 
Xastly, dividing by 3, gives j; =» 7. 

5. Giveii 9fljr'— ISabx^ ^ 6ax^ + I2ax^ \ to find x. 
First, dividing by SoJi^, gives Sx— 5^ = 2jr + 4; 
Then transposing Bb zad 2Xf gives x :=:, 5b -h 4. 

* 

6. Let -r --+-j- = 2be given, to find X. 

I 3 4 5 

First, multiplying by S, gives x^-^x + ^x =s 6 ; 
Then multiplying by 4, gives x -{- Y^ == 24. 
Also njultiplying by 5, gves J7r = 120 ; 
Lastly, dividing Dy 17, gites x = 7^. 

^ ^. ^ .r— 5 jr ; i*— 10 ^ " 

7. Given — ; — + — = 12 — : to find x. 

3 ^ ^ .. S- '■ ^ . . .^^., .^ . 

First, mult, by S, gives x—S -f 4x =,*36 — jr +'lO ; 
Thep transposing 5 and x, gives 2jr -f* l-^ = 51 ; 
And multiplying by 2, gives 7jr =: 1 02 ; 
Lastly, dividing by 7, gives jrt=:14|, , 

Sx ' ' 
^ S^ Let V'^T" + '^ ~ ^^> ^ givai ; to find x. 

First, transposing 7frgiyes ^^x = 3 ; 

Then squaring the equation, g|lves -j^ tsz>9\ 
'Then dividing by 3, gives ^ as= 3 ; 
1/astlyi multipiying by 4, gives a: = 1 2. 

. 9.'' Let* 2^ + 2 Vi?47r* =^J^=L, be^gitreti; tb firid x. 

Fiht, mult, by ^tf*+ jr*, gives 2xVa* + a:* + 2^* + 2^"- 

Then transp. 2a* and2jr%gives2;rv'a* + jr*=3«*— 2x'5 
xoVoL-I. Q Th'^ 



\ i*r 



226 ALGEBRA. 



i • I. -ii'V .H'ft. 



Then by squiring, it is 4a:*x^+ x*^3tf* - ar* 5 

That is, ^a^x^ + 4x* = 9tf * - 1 2/i*a;* + 4jr* j 

By taking 4x* from both sides, it is 4tfV=9ii*— 12tfV j 




Laftly extracting the root, gives v ^^ 



CXIMPLBS FCfR Pfticmd^ ' 

1. Given ^»— 5 + 16 = 21 ; to find n. Ato. at = 5* 

2. Given 9* — 15 £rT + 6j to find x. Ans. x = 4|.. 

3. Given 8-3i'+ 12=30— 5x+4; to find ;r. Ansjr=7; 

4. Given x + -pr— i* =185 to find x. Ans, or r: 12. 

5. Given S*'+4;» +2=5* - 4 j to find at. Ans. * = 4; 

6. Givgn 4fljr + ^a— 2 cr ^jr— *x j to find x. 

Ans. X =: 



9a+3^ 

7. Giv€4i \x^\x + T^ = T> ^o fi»d r. Ans. ^ ^/if*J 4ji 

8. Given v^4+x = 4 — v^jr; to find jr. Ans. xzz2\. 

X* 

9. Given 4ta + x ^ -r — -— ; to deter, x* Ans. ^= — 2a. 

4a+x 

V 



10. Given V4ji* + jr*x=;J/4** + jr*j to find x. 

Ans. X tSL'V- 



2^ 



^MMM 



4a 



11. Given V;c + V^2a + x zr- ^ ; to find x. 

Ans. 4? irfiPir 

12. Given r-~*" +t — ^ =2 2* ; to find x. 

l+2x 1 — ; 3* 

Ans. X fSr^V— -r—- 



13, Given j + x = Va* + x VW + x*| to find k. 

Ans. j: ts a^ 

a 



w 



SIMPLE'EQUATIONS. «27 

OV REDUCING DOUfiLB, TRIPLE, .ftc. £<^A'fiIONs/ CON- 
TAINING TWO, THREE, OR MORS UNX3K0WN jqUAN- 
TITIES. 

i»ROBL$M I. ; 

7i Exterminate Two Unknown Sluantities; Or^ to Reduce 
the Two Simple ^puLtiom tontetining fkentfto a Single 
4m. , 

RULE L 

Find the T?lue pf one of the pn^nown letters, in terms 
-W ^e other qiantjti^^ in eskfix 4af the (equations, by the 
l&ethods already explained. Then put those two values 
equal to each other tor a new equation, with only oi&e yn- 
known quantity in it, iiriiose value is to be found .as .before. 

N&te. It is ev ident that we must first begin to find the 
values of that letter which are easiest to be found in the two 
proposed equati^is* 

EXAMPLES* 

1. Given [ll t%Zu]^ '^ ^^ * ^^^- 

17 — Sv 
In the lstequat.traf»p.SyanddAv.by2,gives;rs%i-m--^; 

14 4- 22/ 
Jbi the 2d transp. 2v and div. by 5, gives * =s r^J 

Putting these two values equal, gives "^^ ^» 

Then mult, by 5 and 2, gives 28 + 4y =z SS-- 15yj 
Transposing 28 and 15y, gives 19y = 57; 
And dividing by 19, gives ^ =ai S. 
And hence, ffp = 4. 

Or, to do the same by finding two valuc^ of ^, thus: 
In the 1st equat. tr. 2k and div. by ^, gives^ = 

s 

5:^— 14 

In the 2d tr. 2yand 14, anddiv. by 2, ja^vesj^ ;=■ 

art 

«,,.,, 1 1 . 5«— 14 17-2* 

Jrutting these two values equal, gives -t— -rr= . ■' ■ 

Mult, by 2^ and by 3, gives lBx^.^2. =s. ^^ -t'4^| 

" Q2 Transp. 



2U ALGEBRA^ ^ 

Tramp. 42 and 4r, gives^ 19x cr J6 ; 
Dividing by 19, gives x ;= 4. 
Hence j^ = 3, as before. 

' d. Given J|^j;|j;=;^> to find* and;. 

Ans. * = « + *, and j^ = ^—44- 

St Given 3* + ^ = 22, and Sy+«f = 18 ; to find x and >. 

Ans. X s 6} and j^ :^ 4. 

4. Given {jj + ?{= ^, { ; to findirand,. 

Ans. jr = 6, and v = 3. 

'^. 2* 3v 22 3*p 2ir 67 - . 

,5. Given — -l-'f = — , and— +-^«rr; to find /? 

and y. Ans. x = 3, and ; = 4. 

6. Given * + 2j^ sz /, and i^*- 4/ = i/* ; to find x and jr. 

Ans. X = — ^, and jr =— jj— 

7. Given ;» — 2v n rf, and xiyiiaib; to find x and y. 

ad , W 
Ans. X = r, and f = -r 



RULE U. 

Find the value of one of the unknown letters, in ooly one 
of the equations, as in the former rule ; and substitute this 
value instead of that unknown quantity in the other equation, 
and there will arise a new equation*^ with only one unknown 

quantity, whose value is to be found as before. 

" . ... * 

Nate. It is evident that it i5,l?est to begin first with that 
letter whose value is easiest found in the given equations. 

EXAMPLES* ' , 

1. Given ^l^ 1 1 Zu]> to find ' and j. 

This will admit, of fom* ways of solution; thus : First, 

17— 3v 
in the 1st eq. trans. 3y and div. by 2, gives x =s — ^~^* 

. 85/-^ I5y 
This val. subs, for x in the 2d, gives — ^ — 2y = 1 4 j 

Mttlt. by 2, this becomes 8S - liy-- 4y =: 28 j 

Transpv 



SIMPLl EQUATIONS. 22§ 

Transp. 15y and 4y and 28, gives 57 c= 19y ; , 

And dividing by 19> gives 3 szy. 

Then^=: — s~ = 4. 

2dly, in the 2d trans! 2y and div. by S^ gives ^ = — ^-A 

5 

This subst. for :r in the 1st, gives 2! + 3j^;s 17 j 

Mult, by 5, gives 28 + 4y + 15y = 85 j 
Transpos. 28, gives 19y =: 57 ; 
And dividing by 19, gives y =: 3. 

Then x = ' ' = 4, as before. 



3dly, in the 1st trans. 2r and div. by 3, gives j>= — ^^^^5 

3 

This subst* for ^ in the 2d, gives 5:c — = 14 . 

Multiplying by 3 gives 15*1;' -^ 34 + 4jr s= 42 $ 

Transposing 34, gives . 1 9x = 76 ; 

And dividing by 19, gives x = 4. 

„ 17-2;r ^ 

Men^e^ 1=: — - — <*ss 3, as before. 

4thly,in the 2d tr. 2^ and 14 and div. by 2, gives y=: ^"^ \ 

This substituted in the 1st, fives 2x + = IT; 

Multiplying by 2, gives i 9x — 42 =; 34 j 

Transposing' 4<2, gives 19x == 76 ; 

And dividing by 1 9, gives x r: 4, 

5x— 14 
Hence y = — r — =: 3, as beforfc. 



2. Given 2;r + 3y =: 29, and 3x - 2jy — 11 ; to find * 
andy. . Ans. jt = 7, and j^ == 5. 

, . 3. Given {J 1 J f ^2} ' '" ^'^ -^ '"'^> 

Aris. a? = 8, and v = 6. 

4. Given 



299 AIGBBRA* 

4. GiTCi! {^'J^jl I '^1} ; to fftid T^iy. 

' Am^ jr =: 6, and y = 4. 

5. Given -r- + 3^ = 21, and ~- + S>r = 29 ; to find ;r 
and jf^ Ans. *r ss 9, and j^.= 6. 

«. Given 10 - ^ -i + 4, and ^ + ~ - 2 = 
— I 1 ; to find X and y. An$. x zz 8, and v s: 6, 

7. Given x : y : : 4 : 5^ and ap'— / = 37 j to find x and jr. 

Ans. tT = 4j and y = 3. 



I.ULE III. 

I/ET the given equations be so multiplied, or divided^ ScCt^ 
, and by such numbers or quantities, as will make the terms 
which contain one of the unknown quantities the same ia 
both et[uations ; if they afe not the sarte when £rst pro- 
posed. 

Then by adding or siibtracting the equations, according 
as the signs may require, there will remain a new equa- 
tion, with only one unknown quantity, as before. That isj^ 
add the two equations when the signs are unlike, but sub- 
tract them when the signs are alike, to cancel that common 
term. 

Note. To make two unequal terms become equal, as abovcn 
multiply each term by the co-efficient of the other. 



EXAMPLES. 

^^^^" {S + S^ = le} ' ^^ ^^ * ^^^ J'- 

Here we msly either make the two first terms, containing n^ 
equal, or the two £d terms, containing j^, equal. To make 
the two first terms equal, we must multiply tji€ 1st equation 
by 2, and the 2d by 5; but to mlike the two 2d terms equal, 
we must multiply the i st equation by 5, ancl the 2d by 3 5 
as follows* 

1. B7 



SIMPLE EQUATIONS. • 231. 

i 

1. Bj making the two first terms equal : 

Mult, the 1st equ. by 2, gives lOor— Gy ss 18 ^ 

And mult, the 2d by 5, gives 10.r + 25j> = 80 y 

Subtn the upper froi^ the under^ gives Sly a 62| 

' And dividing by 51| gives ys£ 2* 

d + 3f 
Hfince> from the Ist given equ* .r s: ■ / ■ ■ f ' ==. 9i» 

• £• By making the twQ 24 terms equal: 

Mult, the 1st equat. by 5, gives 2Sx — 1 5jf =5 45 ;j 

And mi|lt. the 2d by 3, gives 6ur -f 15; =;; 4S ; 

Adding these two, gives Six s 93 ; 

And dividing by 31,^ gives x =s S. 

^ 5x — 9 

^ Hence^ from the 1st equt^y = — ''^— s: 2* 

MISCELLANEOUS fiXAMPLSS* 

;r-hS v-f6 

1. Given — - + 6t =^ 21, and ^-~ + 5x ;a 2S 5 to 

» s 

find X and jiv Ans. x ^ 4, and ;^ =9 S* 

2. Given — --^ + 10 = 13, an4 -^ J - + 5 = 12 ; 
to find X and y. Ans*> str 5, and jr s: 3. 

3. Given— j-^+— =slO,and ^ + ^-^^9 

to find X and y. Ans. ^ = 8, and y =: 4* 

4. Given Sx + 4y = 38,and4x— Sy = 9; tofindxandy. 

Ans. x :;;: 6, and y sb 5« 

FROBLSM I|. 

To ExterffunaU Three or More Unknown Quantities i Or, to 
Reduce tie Single Equations, containing them, to a Singly 
one. 

nxsiA. 

This may be done by any of the thret methods i^ the Ia|t 
problem: viz. 

1. After the manner of the first rule in the la$t problemi 
find the value oip one of the unknoven letters in each of th« 
given equations : next put Vrro of these values equal to each 
other, and then one of these and a third value equal, and so 
on for all the values of it ; which gives a ntwsct of equations, 

with 



i3« * ALGEBRA. 

i 
I 

t 

with which the same process is to be repeated, and so on 
till there is only one equation, to be reduced bj the rules for 
a single equation. 

2. Or, as in the 2d rule of the same problem, find the 
value of one of the unknown quantifies in one of the equa-» 
tions only; then substitute this value instead of it in the 
other equations ; which gives a new set of equations to be 
resolved ^ before, by repeating the operation. 

3. Ots as in the 3d rule, reduce the equations, by multi- 
plying or dividing themi so as to make som^ of the terms to 
agree: then,. by adding or subtracting them, as the signs 
may require, one of the letters may be exterminated, &c, as 
before. 



EXAMPLES. 

1. Given -I ;ir + 2j/ + 32 = 16 >• ; to find 4f,^, and a» 
(Ar+3j/ + 4z = 2l) 

1. By the 1st method : 
Transp. the terms containing^ and z in each ^ua.. gives 

• - » =s 9 — J/ — SB, 

4?= 21 — 3y — . 42} 
Then putting the 1st and 2d values equal, and the 2d and 3d 

values equal, give 

9 — y - as = 16 — 2j/ - 3«, 
16 _ 2y - 32S = 21 - 8j^ - 4z ; 

In the 1 St trans. 9, 25, and 2yj gives j^ = 7 — 2z ; 
In the 2d trans. 16, 3z, and 3y, gives j/'= 5 — - a;; 

Putting these two equal, gives 5—2= 7 — 2zj 
Trans. 5 and 3z, gives 2 = 2.. 

Hence 3^ = 5-2 = 3, and x = 9— j^— 2 =; 4. 

Jdly. By the 2d method : 

From the 1st equa. x = 9—^ — 2; 
This value of «: substit. in the 2d and 3d, gives 

9 -f y + 22 == 16, 
.U + 2y+.32 = 21; 
In the 1st trans. 9 and 2z, gives 3/ =; 7 — 23^; 
This substit. i|i the last, gjves 23 — 2 == 21 j. 
Trans, z and 21, gives 2 = 2. 
Pence again J/ =; 7 — 2z == 3, and x = 9— 3^-2 = 4. 

3dly. By 



SIMPLE EQUATIONS. 3SS 

Sdly. By the 3d method : subtracting the 1st tqu. from 
the 2d, and the 2d from the 3d, gives 

J/ + 25f = 7, 

y + z=: 5i 

Subtr. the latter from the former, gives z = 2, 

Hence y = 5 — z = 3, and x =^ 9—7/—Z =i 4-. 



C x+ 1/+ z^lSl 
\. Given < x + 3j/ + 25r = 38 >.; 

L ^+j;y + i2 = ioy 




to find X, j^, and s, 

« 

Ans. a? = 4, ^ = 6, 5r = 8. 

= 271 

^. Given ^ x + yj/ + |z = 20 >• ; to find jr,3^, and s. 

z = 163 
Ans. X = 1, j^ = 20, at = 60. 

4, Given jt — j/ = 2, jr — ;a =: 3, and y — a = 1 j to 
find a;*, ^, and z, Ans. r = 7 ; j/= 5 ; 2; = 4. 

2jr + 3y + 4z = 34T 

5. Given ^Sx + iy + 5z = 46^ i to find x, y, ajid z. 
4^ + 5y+ 6z =583 



A COLLECTION OF QUESTIONS PRODUCING SIMPLE 
^ EQUATIONS. 

Quest. 1. To find two numbers, such, that their sum 
shall be 10, and thair difference 6. 

Let JT denote the greater number, and 1/ the less *. 
Then, by the 1st condition x + j/ = 10, 
And by the 2d - - a; — 3/ = 6, 
Transp. y in each, gives jr = 10 — ^, 

and X = 6 + y J 

Put these two- values equal, gives 6+^ = 10-*j/; 

Transpjos. 6 and — t/^ gives - 2^ == 4 j 

Dividing by 2, gives - - ^ = 2. 

And hence - - - - x =:^ 6 +j/ = 8. 



* In all these solutions^ as many unknown letters are always 
used as there are unknown numbers to be founds purposely the 
better to exercise the modes of reducing the equations : avoiding 
the short ways of notation^ which> though g'^ving a shorter solu- 
tion, are for that reason less usefiil to the pupil, as affording less 
exercise io practising the several rules in reducing equations. 

Quest. 2. 



SS4 ALGJEBRA. 

Quest. 2. Divide lOQ/. simong A» u, c, se tlut a may 
have 20/. more than b, and b 10/. more thaa c. 

Let A' = a's share, j^ = b's, and z = c's. 
Then :t' + j^ + » = 100, 
X =cj/ +20, 
j^ =5 z 4- 10. 
In the 1st substit. y + 20 for x^ gives 2y + 2 + 20 = 100; 
In this substituting 2 + 10 for j<, ^ves Ss; + 40 :s 100 ; 
By transposing 40, gives • Sf; ^ 60 ; 

And dividing by S, gives - - z = 20. 

Htocej^ = 2 + 10 = 30, and x = j/ + 20 = 50. 

« 

Quest. 3. A prize of 500/. is to be divided between two 
persons, so^^as their shares may be in proportion as 7 to 8; 
required the share of each. 

Put X and y for the two shares ; then by the question, 

7 : 8 : : jr :y, or muk. the extremes 
and the means, ly = Sx, 

and a:H-j/ = 500; 
Transposing J/, gives x = 500 — j^ ; 
This substituted in the 1st, gives 7j/ = 4000 — St/^ 
By transposing 8y, it is 15j^ = 4000 ; 
By dividing by 15, it gives y = 266^-1 
And hence x = 500— j/ =' 233|-. 

Quest. 4. Whatnumber is that whose 4th part^exceeds 
its 5th part by 10 ? 

Let X denote the numba* sought. 
Then by the question ^x — |jr = 10 ; 
- By mult, by./4, it becomes x — ^:r = 40 j 

By mult, by 5, it gives x = 200, the number sought. 

Quest. 5. What fraction is that, to the numerator of 
which if 1 be added, the value will be 4 } but if 1 be add^ 
to the denominator, its value will be j- ? 

X 

Let --- denote the fraction. 
^ 3/ 

Then by the quest. = », and —^ — = '. 

The 1st mult, by 2 andy, gives 2^ + 2 = v ; 
The 2d mult, by 3 and J^ + 1, is ,3^ =:^ + J ; 
The upper taken from the under leaves .r— 2 = 1 j 
By transpos. 2, it gives x z=z s. , 

And hence J/ = 2^ + 2 = 8 ; and the fraction is |. 

Quest. «. 



\ 



SIMPI^'EQUATIONS. 285 

Quest. 6. A hbourer engaged to serve for SO days on 
these conditions : that for every day he worked^ he was to 
receive 20d, but for wu^ry day he played, or was absent, he 
was to forfeit \()d* Now at the end of the time he had to 
receive just 20 shillings, or 240 pence. It is required to 
find how many days he worked^ and how many lie was 
idle ? 

Let j: be the days worked, and^ the days idled«^ 
Then 20x is the pence earned^ and iOy th^ forfeits ; 
Hence, by the question - x +j/ ^ 80, 

and 20x — IC^ = 240; 
The 1st. mult, by 10, gives lOx + lOj^ = 300 j 
• These two added give - 30jr = 540 ; • 

This div. by 30, gives - ;r = 18, the days worked j 
Hence . - j^=30— a:=sl2, the days idled. 

Quest. 7. Out of a cask of wine, which had leaked :^way |, 
80 gallons were drawn ; and then, being gaged, it jappeared 
to be half full; how much did it hold? 

Let it be supposed to have held x gallons. 

Then it would have leaked -^x gallons, 

Conseq. there had been taken away ^x + 30 gallons. 

Hence ix=zix + 30 by the question. 

Then mult, by 4, gives 2x = a: -+- 120; - 

And transposing x, gives x 2= 120 the contents. 

Quest. 8. To divide 20 into twd such parts, that 3 times 
the one part added to 5 times the other may make 76. 

Let X and^ denote the two part^. 
Then by the question - - :r + 3^ = 20, 

and Sx + 5t/ = 76. 
Mult, the 1st by 3, gives - 3-r + 8j^ =s 60; 
Subtr. the latter from the former, gives 2y = 16 ; 
And dividing by 2, gives - - ^ = 3. 

Hence^ from the 1st, - x s= 20 — j/ = 12. , 

QUBST. 9. A market woman bought in a certain number 
pf eggs at 2 a penny, and as many more at 3 a penny, and 
sold them all out again at the rate of 5 for two-pence, and 
by so doing, contrary to expectation, found she lost 3^.; 
what number of eggs had she ? 

Let X ss number of eggs of each sort. 
Then will ^x = cost of the first sorty 
And ^ = cost of the s^coi^sott ^ 

But 



285 ALGEBRAr 

Bat 5 r 2 : : 2r (the whole number of eggs) : ^ ; 
Hence ^x = price of both sorts, at 5 for 2 pence ;• 
Then by the question ^r + jx— 4ir = 3 ; 
Mult, by 2, gives - *• + f^— 1-^ = 6 ; 
And mult, by 3, gives 5x — V-^ = 18; 
Also malt, by 5, gives x = 90, the number of eggs ol 
each sort. 

Quest. 10. . Two persons, a and b, engage at play. 
Before they begin, a has SO guineas, and b has GO. After 
a certain number of games won and lost between them, a 
rises with three times as. many guineas as B. Query, how 
many gttineas did A win of B ? 

Let X denote the number of guineas A won. ' 
Then a rises with 80 + x. 
And B rises with 60— x ; • 

Theref. by the quest. 80 + x = 1 80 - Zx\ 
Transp. 80 and 3 a;, gives 4x = 100 ; 
Anc^ dividing by 4, gives x =: 25, the guineas won. 



QUESTIONS FOR PRACTICE. 

1. To determine two numbers such, that their difference 
may be 4, and the difference of their squares 64. 

Ans. 6 and 10. 

2. To find two numbers with these conditions, viz. that 
half the first with a 3d part of the second may make 9, 
and that a 4th part of the first with a 5th part of the se- . 
cond may make 5. Ans. 8 and 15« 

. 3. To divide the number 20 into two such parts, that a 
3d of the one part added to a fifth of the other, may 
make 6. Ans. 15 and 5. 

4. To' find three numbers such, that the sum of the 1st 
and 2d shall be 7, the sum of the 1st and 3d 8, and the 
sum of the 2d and 3d 9. Ans, ^^ 4, 5. 

5. A father, dying, bequeathed his fortune, which was 
2800/. to his son and daughter, in this manner ; that for 
every half crown the son might have, the daughter was ta 
have a shilling. What then were their two shares ? 

Ans. The son 2000/. s^nd the daughter 800/. 

6. Hiree persons. A, B, c, make a joint contribution, 
which in the whole amounte to 400A : of which sum b con- 

tributes 



SIMPLE EQUATIONS. ?37 

tribiiites twice as much as a and 20/. more \ and c as much 
as A and b together. What sum did each contjribute ? 

Ans. A 60A B UO/. and c 200/1 

7. A person paid a bill of lOOA with half guineas and 
crowns, vsing in all 202 pieces ; ..how many pieces were 
there of each sort ? 

Ans.. 180 half guineas, and 22 crowns. 

8. Says A to B, if you give me 10 guineas of your money, 
I shall then have twice as much as you will have left : but 
says B to A, give me lO of your guineas, and then I shall 
liave 5 times as many as you. How many had each ? 

Ans.- A 22, B 26: 

9. A person goes to a tavern with a certain quantity of 
money in his pocke't, where he spends 2 shillings; he then 
borrows as much money as he had left, and going to another 
tavern9 he there spends 2. shillings also; then borrowing 
< again as much money 'as was left, he went to a third tayerQ^ 
whi^e likewise h^ speixt 2 shillings; and thus repeating. the. 

« same at a &urth tavern, he then had nothing remaining. 
What sum had he at first ? Ans. 3/. 9 J. 

10. A man with his wife and child dine together at an 
inn. The landlord charged 1 shilling for the child; and 
for the woman he charged as much as for the child and i^% 
much as for the man ; and for the man he charged as much 
as for the woman and child together.. How much was that 
for each ? Ans. The woman 20d. and the man 32rf. 

4 

11. A cask, which held 60 gallons, was filled with a 
mixture of brandy,. wine, and cyder, in this manner, viz* 
the cyder was 6 gallons more than the brandy, and the 
wine was as much as the cyder and \ of the brandy. How 
much was there of each ? 

'^S.' Ans. Brnndy 15, cyder 21, wine 24- 

12. A general, disposing his army into a square form, 
finds that he has 284 nien more than a perfect square ; but 
increasing the side by 1 man, he then wants 25 men to be 
a complete square. Then how many m^n had he under his 
command ? Ans. 24000. 

13. What number is that, to which if 3, 5, and 8, be 
severally added, tl^e three sums shall be in geometrical pro- 
gression? Ans. 1. 

13. The stock of three traders amounted to 860/. the 
shares of the first and second exceeded that of the third 



SS8 ALGEBRA. 

hf 240 ; and the sum of tbe 24 and Sd extdeded the fint 
by 260, What was the share of each ? ^ 

Ans. The 1st 200, the 2d 300, the 3d 260. 



15*. What two ntimbers are those, whicht being in* the 
ratio of S to 4, their product is equal to 12 times their swni 

Ans. 2 i and 29. 

16.' A certain company at a tavern, when they came to 
settle their reckoning, found that had there been 4 more in 
company, they might have paid a shilling a-piece less than 
they did ; but that if there had beea 3 fewer in company. 
tliey most have paid a shilling a-piece more than they did. 
What then Was the number of persons in company, what 
each paid, and what was the whole reckoning ? 

Ans. 24 persons, each paid 7i. and the whole 
reckoning 6 guineas. 

17* A jockey has two horses ; and also two saddles, the 
one valued at 18/. the other at 3/. Now when he sets the 
better saddle on the 1st horse, and the wor#e on theM, it 
makes the first horse worth double tbe 2d : but when he 
places the better saddle on the 2d horse, and the worse on 
the first, it makes the 2d horse worth three times the 1st,. 
What then were the values of the two horses t 

Ans. The Ist (i/., and the 2d 9/. 

IS. What two numbers are as 2 to 3, to each of which 
if 6 be added, the sums will be as 4 to 5 ? 

Ans. 6 and 9» 

19fc What are those two numbers, of which the greater 
is to the less as their sum is to 20, and as their difference is 
to 10? Ans. 15 and 45. 

'20. What two number*^ are those , whose difference, sum, 
and product, are to each other, as the three numbers 2, 
3, 5 ? Ans. 2 and lO. 

21. To find three numbers in arithmetical progression, 
of which the first is to the third as 5 to 9, and the sum of 
all three is 63. Ans. 1 ^, 21, 27. 

22. It is required to divide the number 24 into two such 
parts, that the quotient of the greater part divided by the 
less, may be to the quotient of the less part divided by the 
greater, as 4 to 1. Ans. 16 and 8« 

23. A gentleman being asked the age of his two sons, 
answer<»d, that if to the sum of their ages 18 be added, ^ 
the result will be double the age of the elder ; but if 6 be ' 

taken 



^o 



QUADRATIC EQUATIONS. Wl^ 

ttffeen fitfm the difference of ttueir age«, the remainder will 
be edual to the age of the yotmger. What then were their 
nfges? ^ Ans. SCand 12. 

24. To find four numbers such, that the sum of the 1 st, 
2d, and Sd shall be 13 ; the sum of the 1st, 2d, and 4th, 
15 ; the sum. of the 1st, 3d, and 4th, 18 ; and lastly th^ 
sum of the 2d, 3d, and 4th, 20. Ans. 2, 4, 7, 9. 

25. To divide 48 into 4 such parts, that the first increased 
liy 3, the second diminished by 3, the third multiplied by 3, 
and the 4th divided by 3, may be all equal to each other. 

Ans. 6, 12, 3, 27. 



QUADRATIC EQUATIONS. 

Quadratic Equations are either simple oh compound. 

A simple quadraticequatton, is that which involves' the 
jliqtare of the unknovOvi iquantityonly.. Asax^ = h. And 
the ^ktion of such quadratics has been aheady given 'in 
simple equattoms. 

A compound qtladr^atic equation, is that^which contains 
the square of the unknown quantity in one term, and the ^ 
ifilfst power in' another term. As ax^ -^^ hx ^c. 

All compound quadratic equations, after being properly 
r^dticed, fall under the thr^e following forms, to which 
they miliist ' always be reduced by preparing them for sola- 
tion. 

1. x^ 'Y oxsz b 

2. a^ -- ax 1=: b 

3. jr* - oT = - A 

The general method of solving quadratic equations, is by 
what is caUed completing the square, which is ^s follows : 

1. Reduce the proposed equation to a proper simple form, 
as usual, such as the forms above ; namely, by transposing 
all the terms which <C6ntain the unknown quantity to one 
side of the elation, and the known terms to the other; 
placing the square term first, and the single power second ; 
dividing the equation by the co*efficient of the square or 
first term, if it has one, and changing the signs of all the 
terms, when that terhi happens to do negative, as that 
term must always be made positive before the solution. 
Then the proper solution is by completing the square as 
folio W8| viz. 

2. Complete 



xYr 



S40 ALGEBRA. 

2. Complete the unknown side to a squarei in this man- 
ner»' viz. Take half the co-efficient of the second term, and 
square it } which square add to both sides of the equation, 
then that side which contains the unknown quantity will be 
a complete square. 

3. Then extract the square root on both sides of the 
equation *, and the value of the unknown quantity will b^ 

determined. 



* ^l^l^he square root of any quantity may be either + or — , 
therefore all quadratic equations admit of two solutions. Thus, 
the square root of + n« is either + nor-.ii5 for+nx+it 
and — n X — n are each equal to + n*. But the square root of 

— n*, or \/— n% is imaginaiy or impossible, as neither + n nor 

— n, when squared, gives — n\ 

So, in the first form, x^+ax:nb, w here x - f ia Is found =: 

^b + i^*, the root may be either + ^FTT^, or — i/hTj^S 
since either of them being multiplied by itself produces b -f ^\ 
And this ambiguity is expressed by writing the uncertain or double 

Vign ± before •/^ + |^' ; thus x =z ± •^ + ^a" — ^a. • 



In this form, where x r: ± \^b + ^a^ — itf» the first value of 

X, viz. X zr + \/b + Ja« —^a, is always affirmative; for since 
^* 4- ft is greater than ^*, the greater square must neces- 
sarily have the greater root ; therefore \/ft + ^a* will always 
be greater than •Jo*, or its equal ia-, and consequendy + 

^b + .^* — ^ will always be affinnative. 



The second value, viz. x iz — ^b -f- i«* — i^ will always be 
negative, because it; is composed of two negative terms. There- 
fore when x' + ax — b, we shall have x zz + ^/b + Ja' — ^a 

for the affirmative value of x, and x r: — ^ft -f -Ja^ — ^a for 
the negative value of x. % * . 



In the second form, where x = ± v^6 4. ^^ -f ^a the first 

value, viz. x = + v^ft i- |a^ + |fl is always affirmative, since it 
is composed of two affirmative terms. But the second value, viz. 

JP = -^ -v/^ + -Ja^ + io, will always be negative j for since 

ft + Jos is greater than Ja^ therefore ^/b + ^a* wi ll be gr eater 

than \/ia% or its equal ia 5 and consequently -^ v^ft + ia' + ia 
is always a negative quantity. » 

Therefore, 



QUADRATIC EQtJA^tONS- Sit 

determined, making the' root of the known side either + or 
— , which will giv6 two roots of the Equation, or two values 
of the unknown quantity* 

'Notey 1. The root of the first side of tKe equation, is 
always equal to the rpdt of the first term, with half the 
co-efficient -of the second tei^m joined to it, with its sign, 
whether '+ or — • ' ' * 

2. All equations, in, which there are two terms including 
the unknown qijiantjt jr, and which have the index of the one 
just double that^ of the other, are resolved like quadratics, 
by completing the square, as above. 



1" 



Thus, x^ + ^-^ =* K or ^^ + ^^ ™ *> o** ^ + ox^ :2:^ 
are the same as quadraticsi and the value ,of the unki^qwo 
quantity may be determined accordingly* ./ 



t 



■ ^ i < ■ M <*■ III I ' 1*1 I II I II ■ I III II ■ ' I I II I U ' 1 • 



Therefore, when a^ — ax.zzb, we shall have x zf - f . y^^ 4- ^ 
+ la for the affirmative value of a:; and j? =* — y6 -f ia^+^a 
for the negative vake of a? j i^ that' in both the first and second 
forms> the unknown quanti^ has alwi^sbtro values^ one of which 
is positive^ and the otb^r negative. 

But, in the third form, where ar =z ± v'ifl* — ^ + 1«* both^tho 
values of X will be positiv e/ when Ja^ is greater tli^n 0, FoxthfS 
first vjflue, viz. ar' =: -f ^/^a^ - b + -Ja will then be affirmative, 
being composed of two affirmaiiveteirmd; !«> 



The secQnd value, viz* .^r 5= .;--t;. Vi^^ — 6 4- . J^ y a^jna* 
tive also 5 *for since ^o^ ,5 jrreatei; than Ja* — 6^, therefore, Vi** ot 

^a is greater than \/Ja^ -r 6^ aqd consequently -r \/Ja*— 6.^ fa 
will always be an affirmative quantity. So that, when afi ^ ax 

= ^— 5, we shall have a: =: 4- \/|o* — ^ + ia, and,alsp x = — 

v^Jo* — 6 + f aj for the valuesDf X, botti positive^ * ' 

But in thifc third forpa, if b be greater than ^a?, the solution of 
th6 proposed -question wilt be impossible/ For ^ince the sqUdre of 
any quantity (whether jthat quantity be affirmative or negative) 
is always amrinative, the square 'robt^of a negdfcive quaotily ik im- 
possible, and cannot be assigned. But when b is greater than 
Ja^ thenirt*'— b is a negative quantity j add therefore its root 

V'Jo* — ^ is impossible, or imaginary; consequently, in that case, 

j: n ^a ± ^^a^ — b, or thfe two toots or values of Jf, are\>otli 
impossible, 01 imaginary qu^tities; .1 

Voii. L R ' EXAMPLES. 



^^ 



34f ALGEBRA. 



EXAMPLES. 



1. Given jt* + 4j: = 60 •, to find r. 

First, by completing the square, or^ '\'4fX + 4 ^ i4\ 

Then, by extracting the root, jr + 2 =z ± 8 j 

Then, transpose 2, gives x = 6 or — 10, the tw« roots. 

2. Given x*— 6x + 10 = 65 ; to find 4% 

First, trans. 10, gives j;^— 6r = 55 ; 
Then by complet. the sq. it is x^—'Sx + 9 =: 64 j 
And by extr. the root, gives x-^S =: ±: S'f 
Then trans. 3, gives j* = II or — 5. 

5. Given 2-r* + ar-30 as 60; to find x. ^ 

- First by transpos. 20, it is 2jr* + Bar rr 90;' 

Then div. by 2, gives .r* + ^'*'= 4-5*; 

And by compL the sq. it is jt* + 4jr -f- 4 = 49; 
"-Then extr. th e roo t , it is ar +-^ =^ ± 7}' " ' 
«. Ahd iransp. 2, gives Jr = 5 or — 9. 

4.' Given So:'- - Ix V '^* = ^4-^ to find .r. \ . . ' . 

. First diy. by 3, giv^s ,ar*— jt + 9f= 21-; 
. Tbeja transpos. ^3, giye« jt*— itr ni'— I:} i ' : • -J . i 

And compl. the sq. gives ar^-^x + ^ =s ^; t 

Then extr.- the root gives or — 4^ = dt -Jr* 
^ And transp. 4^, gives x = J or ^. 

J5. Given i^*— ^ +.30i =,52* ; tp find jt. 

First by transpos. 30 j., it is -Jjr*— ^jr as 22^; 
Then mult, by 2 gives jr*— l^r = 44^ ; 
And by compl. the sq! it is x*— |!rH- ^ = 44$; 
.' Then extr. the root, gives \r—^ = -t 6^; 
And transp. -J-, gives o^ = 7 or — 6-y. 

6. Given ijr*— ^jr = r ; to find x. 

he 
First by div. by a, it is 1:* x » •^; 

' Then compl. the sq. rives x* x + -:~x = 1 r-» 

And extrac. the root, gives X --=. ± ^-^j^-; 

r,^, i . ,4ac + i* A 

Then transp. —, gives x = ± ^-_^_+ — 

7.. Given x* — 2<wr* =: h\ to find r. 

First by compl. the sq. gives x^ -* 2a4r* + a* i= a* +*s ^ 



QUADRATIC EQUATIONS. 24^ 

And extract, the root, gives x*- ^a'=i ±, ^/et + h\ 
Then transpos. j, gives a;*=±v'^* + * + *» 



iri tm ■*■ 



And extract, the root, gives ^ = ± v'* i V^.^ + ^* 

And thus, by always tising similar words at each lihe, the 
pupil will resolve the following examples. 



EXAMPLES. FOR PRACTICE. 



) I.. 



1. Given x*— 6x— 7 == 33 j to find ^r, i^wJf^ 10. 

2. .Given x*— 5jr— 10 = 44; to find x. Ans. ar = 8. 

3. Given 5jr* + 4x — 90 = 114 ; to find x, Ans. x = 6- 

4. Given -J^x*— ^x + 2 » 9; fo find x. . A6s. > = 4. 

5. Given 3x*— 2x* = 40 ; to find x. Ans. x == 2. 

6. Given -fr— ^J-V^*^ = ^il to find x,. Ans. x = 9. 

7. Given ix* + |x = | ; to find x. Ans. x = '727766. 

S. Given 05^ + 4x^ =» 12 i to find .r. 

Ans. .r =^ = 1-259921. 

9. Given x* + 4x = tf* + ^5 to find x. ' 

. Ans. jr = y a*^^- 2. 

•<^lSTIpNS PRODUCING qUADRATiC EqUATIOMS. 

1. To find two numbeps v^hose difference is 2, and 

product 80. 

Let X and y denote the two required numbers *. * 
Then the first condition gives x—yzz 2, f 
And the second gives xy = 80. 
Then trsnsp. y in the 1st gives x » y + 2 ; 
This value of x substitut. in the 2d, i8^*4-2y 5= 80; 
Then comp. the square gives j^ + 2y + l=8i; 
And extrac. the root gives ^ +1=9^ 
And transpos. 1 gives y = 8; 
« And therefore x ssj/ 4-2 = 10. 



* These questions^ like those in simple equations^ are also 
solved by using a^ many unknown letters, as are the numbers 
required, for the better exercise io reducing equations ; not aim« 
ing at the shortest modes of solution, which would not' afford so 
much useful practice. 

R 2 2. To 



244f ALGEBRA. 

2. Tq' divide the number 14- into two such parls> that' their 

product may be 48. 

■ > • 

Xet X and 7/ denote the two numbers. 
. T^ien i^e Ist condition gives x + t/ « 14, 
. j,Apd the 2d gives xy ==48. 

Then transp. ^ m the 1st gives * = 14 -3/ ; 
This value subst. for x in the 2d, is 143/-^ = 48 ; 
Changirig all the signs, to make the square positivejj 

' ^vesy-14y=-48; 

- Then compl. the square gives /— 14j; + 49 = 1 > 
And extrac. the root gives 5^— 7 = i 1; 
Then tnnspos. 7> gives > .= aor «> the two parts. 

» , . ■ ' ' 

S. Given the sum of two nunibers = 9, and the sum of 
their' sqiiareV = 45 ; to find those niimbers. 

Let jr and y denote the two numbers, 

* Then by* the 1st condition x + y = 9. 

. A"nd'bythe-2d:r*+/^45. ' ' 

Then transpos. y ixK tho. 1 st.gives xz^S-^y^ 
Tl^% value '.sjbst. in jJ>e^2d gives 81 - 18;; + 2/ = .45 ^ 
Then transpos. .8J, give? 2/- ISy = — 36 •, 
And dwiding by 2 gives / — 9y ±= — 1 8 -,' 



8 Z — 9 . 



,: The^ con>pL the sq. gives y^-9y -^ V 
And extrac. the root gives ;;— | = ± i ; 
Then transpos. ^ gives j? == 6 or 3, the two numjpers., 

4. What two numbers arc those, whose sum^ pro4uct,' and 
difference of their squares, are all e^ual to each other ? 

' Let .«• ^ndj/ denote the two numbers. 

Then the ist^md 2d expression give x + y sz xy^ 

And the 1 st and 3d give at + j? = *"*—/• 

Then the la^ eqoa. div. by x + y^ gives 1 ^ x^-y ^ 

'Axi^ -transpos. y^ gives y -^ \ i^ x\ 

This vaL substit. in the 1st gives. 2;; + 1.3±:y +31^ 

And transpos. 2^, gives 1 3= /— y 5 

Then complet. the sq. gives i| isc y* —)••{- ^f- ; 

And extracting the root giVea iV 5 = y — 4 j 

And transposing ^ gives 4>v/5 + t == y > 

And therefore \r = y -f- 1 = ^^ 5 + 4. 

And if. thc^e. expressions be turned intp numbers, by ex- 
tracting the root of 5, .&c, they give ar = 2*6I8a + , 
andy= 1-6180 +• 

.- ' * ' - 

5. Theye are four numbers in arithmetical progression, of 

which 



QUADRATIC "E^^UATIONS. fe4S 

wMch the product of the two extremes is 22f,'and that of 
the means 40; what are the numbers? . ^ 

Let X = the less extreme, 
and J/ ^ the common diflferencc^' 
- Then x, .r+y, jr+2y, jr+Sy, will be thefourntJmbcrs. 

Hence by the Irt c<»dition i* + diry « Sa, 

And by the 2d x^ -{- 3xj/ + 2y =? 40,< 

Then subtracting the first from the 2d ^ives 2y* =5 18 j 

And dividing by 2 giv^y* — 9j 

And extracting the root gives y = 3. . 

Then substit. 3 for y in* tie 1st, gives 4:* .+ 9t = 22; 

And completing the square gives :r* + 9.r + y =s '|»j 

Then extracting the root gives > + 4 =*: V,> 

And ilransposing f give^ x =» 2 the least number. 

Hence the four numbers «re 2, 5, 8, 11. 

6. To find S numbers in geometrical progression^ whose 
fjMnn shall be 7, and the sum of their squares 21, . 

Let XyT/j and z denote the three numbers sought. 
Then by the 1st condition 0% ±=j/% 

And by the 2d x + 1/ + ;ar =7, 

And by the 3d jr* +y + «* = 21. 

Transposing y in the 2d giy?es x + z =7 —3/$ 

Sq. this equa. gives x"^ + 2xz +;8^ = 49 — 14j^+j/^} 

Substi. 2y* for 2jr2f, gives x'^+ 2y*+ ;2^= 49 — 14y -|-y* ; 

Siibtr. j/* from each side, leaves ^* + y* + ^= 4?9 — 1 4y ; 

Putting the two values of x' + v* + 2* 7 ^ i ^ n ia.. 
.0 • 1 V •^ ' ' > 21=49-' 14V; 

equal to each other, gives 3 "^ ' 

Then transposing 21 and 14y, gives 14y = 28; 

And dividing by 14, gives y = 2. 

Then substit. 2 fory in the 1st equa. gives xz =; 4, 

And in the 4th, it gives ^4-^ = 5; 

Transposing z in the last, gives x ^ S^z%' 

This substit, in the next above, gives 5lf—z^ :^ 4; : 

Changing all the signs,.gives «*— 5a == --4^ . 

Then completing the square, gives «* — 5a; + V ^ ii 

And extracting the root gives 3 — 4- = i'4» 

Then transposing 4, gives z and x = 4 and 1} the twa 

, . other numbers; 

So that the three numbers are }, 2, 4» ' 

QUESTIONS FOR PRACTICE. - 

1. Wha,t number is that which added to its square makes 
♦2? Ans. 6. 

a- Ta 



S46 ALGEBRA. 

2, To find two numbers such» that the less may be to the 
greater as the greater is to 12j and that the sum of their 
squares may be 45. Ans. S and ۥ 

S. What two numbers are those, whose difference i$ S^ 
and the difference of their cubes 98 ? Ans. S and 5* 

4. What two numbers are those whose sum is 6, and the 
sum of their cubes 72 ? Ans. 2 and 4. 

5. What two numbers are those, whose product is 20^ 
and the difference of their cubes 61 ? Ans. 4 and 5. 

6. To divide the number 1 1 into two suqh parts, that the 
product of their squares may be 784. Ans. 4 and 7. 

7. To divide the number 5 into two such parts, that the 
sum of their alternate quotients may be 44, that is of the 
two quotients of each part divided by the other. 

Ans. 1 and 4. 

8. To divide 12 into two such parts, that their product 
may be equal to 8 times their difference. Ans. 4 and 8. 

9. To divide the number 10 into two such parts, that the 
square of 4 times the less partf may be 112 more than the 
^uare of 2 times the greater. Ans. 4 and 6. 

10. To find two numbers such, that the sum of their 
squares may be 89, and their sum multiplied by the greater 
may prbduce 104. Ans. 5 and 8. 

11. What number is that, which being divided by the 
product of its two digits, the quotient is 5^ ; biit when 9 is 
subtracted from it, there remains a numberil|Naving the same 
digits inverted ? Ans. 32. 

12. To divide 20 into three parts such, that the continual 
product of all three may be 270, and that the difference of 
the first and second may be 2 less than the difference of the 
second and third. Ans. 5, 6, 9. 

13. To find three numbers in arithmetical Sprogression, 
such that the sum of their squares may be 56^ and the sum 
arising by adding together 3 times the first and 2 times the 
second and 3 times me third, may amount to 28. ^^ 

Ans.^'2, 4, 6. 

14. To divide the number IS into three such parts, that 
their squares may have equal differences, andj that the sum 
of those squares may be 75. * Ans. 1, 5, 7, 

15. To find three numbers having equal differences, so 
that their sum may be 12, and the sum of their fourth powers 
962. Ans. 3, 4, 5. 

16. To 



CUBIC, &c. EQUATIONS. 2« 

16. To find three numbers having equal differences, and 
such that the square of the least added to the product of the 
two greater may make 28, but the square of the greatest 
added to the product of the two less may make 44. 

' Ans. 2, 4, 6. 

17. Three merchants, A, b, c, on comparing their gains 
find, that among them all they have gained 1444/.; and that 
9^s gain added to the square root of a's made 920/. ; but if 
added to the square root of c^s it made 912. What were 
their several gains ? Ans. A 400, B 900, c 144. 

"18. To find three numbers in arithmetical progression, so 
that the sum of their squares shall be 93 ; also if the first be 
multiplied by 3, the second by 4, and the third by 5, the 
sum of the products may be 66» Ans 2, 5, 8. 

10. To find fpur numbers such, that the first may be to the 
second as the third to the fourth ; and that the first may be 
to the fourth as 1 to 5 ; also the Second to the third as 5 to 
9 ; and the sum of the second and fourth may be 20. , 

Ans. 3, 5, 9, 15. 

20. To find two numbers such, that their product added 
^o their sum may make 47, and their sum taken firom the 
sum of their squares may leave 62. Ans. 5 and 7. 



RESOLUTION OF CUBIC AND HIGHER 

EQUATIONS. 

A Cubic Equation, or Equation of the 5d degree or 
power, is one that contains the third power of the unknown 
quantity. As a?^— (ur* -j^ bx =:c, 

A Biquadratic J or Double Quadratic, is an equation that 
contains the 4th power of the unknown quantity : 

As x^ — ax^ -f bx^—cx = d* 

An Equation of the 5th Power or Degree, is one that 
contains the 5th power of the unknown quantity : 

As pe^-^ax* + bx^-^cxP' + dx =^ e. 

And so on, for all other higher powers. Where it is to 
be noted, however, that all the powers, or terms, in the 
equation, are supposed to be freed from surds or fractional 
exponents* 

There are many particular and prolix rules usually given 
(pr the solution of some of the above-mentioned powers 

or 



(tiS ALGEBRA. . 

:or equations* • But they may be all readily solved by the 
following easy rule of Double Position, sometimes called 
Trial-and-Error. 



RULE. 

« 

1. Find, by trial, two numbers, as near the trtie roofas 
you can, aiid substitute them separately in the given equation, 
instead of the imknown quantity 5 and find how much the 
terms collected together, according to their signs + or — , 
differ from the absolute known term of the equation, mark- 
ing whether these errors are in excess or defect. 

2. Multiply the difference of the two numbers, found or 
taken by trial, by either of the errors, and divide the pro- 
duct by the difference of the errors,. when they are alike^ 
but by their sum when they are unlike. Or say. As the 
difference or sum of the errors, is to the difference of the 
two numbers, so is either error to the correction of its sup- 
posed number. 

3. Add the quotient, last found, to the number belonging 
to that error, when its supposed number is too little, but 
subtract it when too great, and the result will give the true 
root nearly. 

4. Take this root and the nearest of the two former, or 
any other that may be found nearer \ and, by proceeding in 
like manner as above, a root will be had still nearer than 
before. And so on to any degree of exactness required. 

Note ] . It is best to employ always two assumed numbers 
that shall differ from each other only by unity in the last 
jBgure on the right hand j because then the difference, or 
multiplier, is only I. It is also best to, use always the least 
error in the above operation. 

Note 2. It will be convenient also to begin with a single 
figure at first, trying several single figures till there be found 
the two nearest the -truth, the one too little, and the other 
too great ; and in working with them, find only one morQ 
figure. Then substitute this corrected result in the equation^ 
for the uftknown letter, and if the result prove too little, 
substitute also the number next greater for the second sup- 
position ; but contrarywise, if the former prove too great, 
then take the next less number for the" second supposition ; 
and in working with the second pair of errors, continue the 
quotient only so far as to have the corrected number to four 
places of figures. Then repeat the same process again with 
^us last corrected number, and the next greater or less, *s 

th^ 



CUBIC, &c. EQUATIONS, 



249 



the case may require^ carrying the third corrected number 
to eight figures; because each new operation commonly 
doubles the number of true figures. And thus proceed to 
any extent that may be wanted. 



Examples. 



Ex. 1. To find the root of the^ubic equation x^ +**+ 
9c = IDO, or the value of * in it. 



Here it is soon found that 
x lies between 4 and 5. As- 
sume therefore these two num- 
bers, and the operation will be 
3IS fellows : 

1st Sup. 2d Sup. 

4 - X - 5 

16 - ^* . 25 



64 



X' 



84 - sums - 
100 but should be 



125 

155 
100 



Again, suppose 4*2 and 4'3, 
and repeat the work as foU 
lows: 



1st Sup. 2d Sup. 

4'2 - jr - 4*3 

17-64 ^ js* ^' 18-49 

74-088 - x' - 79-507 



95-928 
100 



sums 



102-297 
100 



— 16 - errors - +55 



the sum of which is 71. 
Then as 71 : 1,:: 16 : -2. 
Hence »r = 4'2 nearly. 



— 4-072 errors + 2-297 



the sum of which is 6*369. 
As 6-369 fl :: 2-297 : 0*036 
This taken fi-om - 4* 300 



leaves x nearly = 4*264 



Again> suppose 4-264, and 4-265, and work as follows: ' 
4-264 ^ X - 4-265 

18*181696 - ;^* - . 18*190225 



77-526753 

99-972448 
100 



X 

sums 



77-5S1310 

100-036535 
100 



-0-027552 - errors - +0O36533 
the sum of which is -064P87. 
Then as -064087 : -001 : : -027552 : 0-0004299 
To this adding - 4-264 

gives X very nearly = 4-2644299 



The 



250 



ALGEBRA. 



The work of the example above might have been much 
shortened, by the use of the Table of Powers in the" Arith- 
metici which would have given two or three figures by in- 
q>ection. But the example has been worked out so particu- 
larly as it is, the better to show the method. 

Ex. 2. To find the root of the equation x^ — 15jr* + 63x 
ss 50, or the value of x in it. 
Here it soon appears that x is very little above 1. 



Suppose therefore 1*0 and 1-1 > 
and work as follows : 



1-0 - 



1-1 



63*0 - 
-15 
1 *- 


6^x - 
-15jr* 
x^ - 

sums - 

errors - 
sum •f the 
:1 ::1:'03 
100 


69-3 
-18-15 
1-331 


49 - 
50 


52'4S1 
50 


-1 - 
3-481 
As 3*481 ; 


+ 2-451 

errors. 

correct. 


Hence x'= 


1*03 


nearly. 



Again, suppose the two num* 
bers 1-03 and 1-02, &c, ad 
follows : 
103 - AT - I '02 



64-89 - 63Jf 64*26 
- 15-9135 — 15;v*- 15-6060 
1-092727 x^ 1-061208 



50-069227 sums 49*715208 
50 50 



+ -069227 errors — •284793 
•284792 



As -35401 9 : -01 : : -069227 : 

-0019555 
This taken from 1 '03 



leaves 4: nearly = 1*02804 



Note 3. Every equation has as many roots as it contains 
dimensions, or as there are units in the index of its highest 
power. That is, a simple equation has only one value of 
the' root ; but a quadratic equation has two values or roots, 
a cubic equation has three roots, a biquadratic equation ha$ 
four roots, and so on. 

And when one of the roots of an equation has been found 
by approximation, as above, the rest may be found as follows. 
Take, for a dividend, the given equation, with the known 
term transposed, with its sign changed, to the unknown side 
of the equation ; and, for a divisor, take x minus the root 
just found. Divide the said dividend by the divisor, and 
the quotient will be the equation depressed a degree lower 
th^n the given one. 

Find 



CUBIC, &c. EQUATIONS. 251 

Find a root of this new equation by approximation, as 
before, or otherwisei and it will be a second root of tlye 
original equation. Then, hj means of this root, depress 
the second equation one degree lower, and' from thence 
£nd a third root, and so on, till the equation be reduced to 
a quadratic ; then the two roots of tjiis being found, by the 
method of completing the square, they will make up the 
remainder of the roots. Thus, in the foregoing equation, 
having found one root to be 1 "02804, connect it by minus 
with r for a divisor, and the equation for a dividend, &c, as 
follows : 

X - J -02804) x^ -• ISx" + eSx - 50 ( x* — 13-97l96r + 

48-63627 = 0. 

Then the two roots of this quadratic equation, or ' 

X* - 13'97196jr se - 48*63627, by completing the square, 
are 6'57653 and 7-39543, which are also the other two 
roots of the given cubic equation. So that all the three 
roots of that equation, viz. x^— 15x^ + 6Sx = 50, 

d fi« 576*58 J^^d the sum of all the roots is found to b^ 

j»T«ofi'.iofl5, beine equal to the co-efficient of the 

and 7-39543 V ^ 1 ^r 1 • 1 • t_ 1 i- 
>2d term of tlje equation, which the sum of 

15-00000 V^^ roots always ought to be, when thejr 
/ are right. 



sum 



Note 4. It is also a particular advantage of the foregoing 
rule, that it is not necessary to prepare the equation, as for 
other rules, by reducing it to the usual final form and slate 
of equations. Because the rule may be applied at once to an 
unreduced equation, though it be ever so much embarrassed 
by surd and compound quantities. As in the following 
example : 

Ex. 3. Let it be required to find the root x of the equation 

Vl*4**-(*' + 20)'- + -v/i96A?"--(*' + 24)* =z 114, or the 
value of X in it. 

By a few trials, it is soon found that the value of x is but 
little above 7. Suppose therefore first that at is = 7, and 
then X = B. 



First, 



t$2 ALG£BRA. 

Firsts when .r s= 7j Second, when x ±z B. 

I 47-906 - ^144x*— (a:* + 20)» - 46'476 

65-384 - v^ 196^* -(a:* +24)2 . 69*283 

■ > ■ ■ ■ ■ I 'I >' II 

1 1 3-290 - the sums of these - 1 1 5*759 

114*000 - the true number - IH'OOO 



—0-710 - the two errors - +1*759 
+ 1-759 



■ V 



As 2-469 : 1 :: 0*710 : 0*2 nearly 

70 



Therefore t = 7*2 nearly 

Suppose again jt = 7-2, and then, because it turns out toa 
great, suppose jc also = 7*1, &c, as follows : 

Supp. jr' = 7*2 Supp. ;r =: 7'I 

47*990 - *Vl44:r*-~(^* + 20)* ^. 47-973 
66*402 - ^/196jr*~(a:* + 24)* - . 65*904 

114*392 - the sums of these - 113*877 
1 1 4*000 - the true number - 1 1 4*000 



+0*392 - the two errors - -0*123 

0*123 



■ I ' 



As -515 : -1 :: '123 : -024 the correction, 

7*100 add 



Therefore X = 7*124 nearly the root required. 

Note B, The same rule also, among other more difficult 
forms of equations, succeeds very well in what are called 
exponential ones, or those which have an unknown quan* 
tity in the. exponent of the power ; as in the following 
example : 

• Ex. 4. To find the value of x in the exponential equation 
x"" = 100. 

For more easily resolving such kinds of equations, it is 
convenient to take the logarithms of them, and then com-, 
pute the terms by means of a table of logarithms. Thus, 
the logarithins of the two sides of the present equation are 

ff X log, 



CUBIC, &c. EQUATIONS. 



235 



X X iog. of iT = 3 the log. of 100. Then, by a few trials, 
it is soon perceived that the value of x is somewhere be- 
tween the two numbers S and 4, and indeed nearly in the 
middle between them, but rather nearer the latter than the 
former. Taking therefore first x = S'5, and then n 2*6, 
and working with the logarithms, the operation will be as 
follows : 



First Supp. X =« 3*5. 
Log. of 3-5 =: 0-54.4068 
then 3*5 x log. 2^'S^ 1*904238 
the true number 2-OQOObO 



error, too little, — •095762 

002689 



Second Supp. x = 3'6. 
Log. of 3-6 = 0-556303 
then 3-6xlog.3-6=2-00&689 
the true number 2-000000 



error, too great, +.002689 



•098451 sum of the errors. Then, 



As '098451: '1 ; : '002689 : 0-00273 the correction 

taken from 3-60000 



leaves - 3-59727 = x neatljr. 

On trial, this is found to be a very small matter too little. 
Take therefore again, x = 3-59727, and next = 3*59728, 
and repeat the operation as follows : 



* Firtt, Supp. i" 2= 3-59727. 
Log. of 3-^9727 is 0*555973 
3-59727 X log. 

- of 3-59727 = 1-9999854 
the true number.2-0000000 



error^ too little, -0-0000146 

- 0-0000047 



Second, Supp. or = 3-59728. 
Log. of 3-59728 is 0*555974 
3*59738 X log. 

of 3-59728 = 1-9999953 
the true number 20000000 



error, too little, - 0-0000047 



0-0000099 diff. of the errors. Then, 

As -0000099 : -00001 : : -0TO0047 : 0*00000474747 the cor. 

added to - 3-59728000000 



gives nearly the value oi.^ = .3-59728474747 

Ex. 5. To find the value of x in the equation jt^ + lO^r* 
+ 5x = 260. Ans. x = 4-1 179857. 

Ex. 6. To find the value of x in the equation ;r' — 2;r = 50. 

Ans. 3-8648«i. ^^^ 

Ex. 7. 



H -. 



U4 ALGEBRA. 

Ex. ?• To find the value of x in the equation jr' + 2x* 
— 2Sx = TO. Ans. x = S'lS^S?. 

Ex. 8. To find the value of x in the equation x^ - IIa:^ 
+ 54* = 350. Ans. x = 14"95407. 

Ex. 9. To find the value of x in the equation x^— 3x* 
— 75x = 10000. Ans. x = 10-2609. 

Ex. 10. To find the value of x in the equation 2x*- 16x' 
+ 40x*— SOx = - 1. Ans. x = 1-284724. 

Ex. 1 1 . To find the value of x in the equation x^ + 2Af* 
+ 3*3 + 4x* + 5;t = 54S21. Ans. x = 8-414455. 

Ex. 12. To find the value of x in the equation x* =-- 
1234567^9. Ans. x = 8*6400268. 

Ex. 13- Given 2x*-7x5 -f llx^-Sx = 11, to find x. 

Ex. 14. To find the value of x in the equation 

{3x''- 2v/ Jr + 1 )^ - (t* - 4Xv^x + 3s/x)^ = 56. 

Ans. X = 18-3i0877. 

2a resolvf Cubic Equations by Cariarfs Rule. 

•Though the foregoing general method, by the applica^ 
tion of Double Position, b^ the readiest way, in real practice^ 
of finding the roots in numbers of cubic equations, as well 
as of all the higher equations universally, we may here add 
fhe particular method commonly called Cardan's Rule, for 
resolving cubic equations, in case any person should choose 
occasionally to employ that method. 

* 

The form that a cubic equation must necessarily have, to 
be resolved by this rule, is this, viz. z^ :|e js =» 5, that is, 
wanting the second term, or the term of the 2d power z\ 
There/ore, after any cubic equation has been reduced down 
to its final usual form, x^ + px^ 4- ^x = r, freed from the 
coefficient of its first term, it will then be necessary to take 
away the 2d term px^ ; which is to be done in this manner : 
Take \py or \ of the coefficient of the second term, and 
annex it, with the contrary sign, to another tmknown letter 
s, thus z—-jpf then substitute this for x, the unknown 
letter in the original equation x' +^* + ^x = r, and there 
will result this reduced equation z^ ^ a» zz i, of the form 
proper for applying the following, or Cardan's rule. Or 
take r = far, and d = 4^, by which the reduced equation 
takes this form, ;&' )|c ^cz^ 2d. 

Then 



CUBIC, &c. EQUATIONS. 255 

Then substitute the values of ^r and d in this 



form, z =l^d+ V(^* + ^0 +{/^- V(^*+f^), 

or * = V^+V(^T^-|^^7^^, 

and the value of the root z, of the reduced equation s^ 4b 
az = i, wjll be obtained. Lastly, take x = «— -j^* which 
will give the value of x, the required root of the original 
equation x^ + px^ + qx czr^ first proposed. 

One root of this equation being thus obtained, then de- 
pressing the original equation one degree lower, after the 
manner described p. 250 and 251, the other two roots of 
that equation will be obtained by means of the resulting 
quadratic equation. 

N^e, When the coefficient a, or c, is negative, and c* is 
greater than d''^ this is called the irreducible case, because 
then the solution cannot be generally obtained by this rule. 

Ex. To find the roots of the equation x^^Sx^ + lOxsrS. 

First, to take away the 2d term, its coeflicient being ^ 6y 
its 3d part is — 2 ; put therefore ;r = « + 2 ; then 

4r' = z3^6x*+ 12a -f 8 
- 6x^ = - ez^ - 24.Z - 24 
-J^lOx = + lOz + 20 

theref. the sum z^ ♦ — 2z + 4=8 

or z' * — 22 = 4 
Here then «= — iS, ^ = 4, ^== — |, rf = 2. 



Tiieref. 4/£/4--v/K+^) = e/^i5+ v/(4-,V)= ^2 + -/ i^o=: 
i/2 + V^V^ = 1-57735 



and 3 /i-^(^H-^ ^) = >^^2-^/(4-:^V)=^2 - ^/'^^ = 

4/2 — V>/3 = 0-42265 
then the sum of these two is the value of z = 2. 
Hence :r zz z + 2 = 4, one root of x in the eq. r'— 6x* + 

lOo: = 8. 

To find the two other roots, perform the division, &c, as 
in p.. 251, thus : 

x-4) J^— 6jr*+ 10jr-8(jr"-2;r + 2=B0 

-2x^+ IOt 
-2ar*+ Sjt 



2ar— 8 
2ar— 8 



Hence 



tS6 ALGEBRA. 

* 

Hence x^—2it = — 3, or a?f — 2ir +1 r: — 1, and r — 1 

= ± \/-'l I r c=: 1 + v' —1 or = .1 - v' - *» the two 
other sought. 

Ex.2. Tofindtherootsof jt'— 94:*.+ 28Jr = 30. 

Ans. X :i= 3, or = 3 +v/ — 1, or =3 — -v/ — I. 

Ex. 3. To find the roots of x'^r-T-f* + 14r =t 20. 

Ans. x = 3, or = i + ^ — 3, or =* 1 — iy/' — 5. 



OF SIMPLE: INTEREST. 

As the interest of any sum*, ft)r any-time, is directly, pro- 
portional to the priocijxil sum^ and. to the time ; there^re 
• the interest of i pound) for 1 yoar> beine multiplied by any 
given principal sum', and by the time of its foH^earance^ in 
y^ars and part$> will give it^ interest for that time. That isj 
if there be put : 

# 

r = the rate of interest of 1 poimdper annum, 

p = any principal sum lent, 

/ = the time it is lent for, and 

a = the amount or sum of principal and interest; then 
IS prt = the interest of the sum p; for th^time /, and conseq. 
p +prt or p X {I +rt) = a, the'amount for that time. 

From this expression, other theorems can easily be de- 
duced, fo}r finding arty -of the qtiaiui£jes abpVe mentioned: ^ 
which theorems, collected together,. will be as below i 

1st, a "=: p + prfy the amount, 

^^' ^ =" 1 4- rt *^^^ principal, 

^ - a—p , 

3d, r = — -y the rate, 
pt 

4th, / = , the time. 

pr 
» 

For Example. Let it be required to find, in what time 
any principal sum will double itself, at* any rate of simple 
interest. 

In this case, we must use the first theorem, a = /) + prty 
in which the amount a must be made = 2p, or double the 
principal, that* i», /) + prt ;= 2/?, ov-prt = /;, or r/ = 1 ; 

and hence / «= — . 

r 

Here, 



COlaPOUND INTEREST, $Sl 

Here^ r being thfe iaterest of 1/^ for 1 yearj it follows, 
that the doubling at simple interest, is equal to the quotient 
t)f any sum divided by its interest for 1 yfear. So, if the 
Ihate o^ interest be 5 per cent, then 100 -^ 5 =i^2d, is^the 
time of doubling at that rate. 

Or the 4th theorem give^ at 6nc6 

a—p ^p — p 2—11 

^ = -— ^=± -r~ = — 1—==— i the same as before* 

pr pr TV 



k! 



Compound iNXEREST? 

•■ . » 

Besides the quiantities concerned io Simple ]^tfrest| 
hamely^ 

p ^,the principal sum, * 

r =r the ratie or interest of 1/. for 1 year, 

a = the whole aittoiim of the principal and interest, 

i = the time, 

th^re is another qUatltity employed in Compound^ Interest, 
viz. the ratio of the x^te of interesti which is the amount 
pf 1/. for 1 time of payment, and which here let be denoted 
by R, viz. 

R. ax i 4- ^j the amount of lU for 1 time. 

Then the particular, amounts for the several tiities rpay 
he thus coiiipflted, viz; As I/« is to its amount for any time, 
isb is any proposed priiicipal sum> to its amount for die same 
time \ diat is, as 

1/. : R : : p : pR, the Ist yearns amount, 
12. : R : : pR : j^R*, the 2d year's amount, 
1/. t R : ; pR* t j^R', the 3d year's amount, 

and so on. . , . 

Therefore, in general, pBl^ =i a is the amount for the 
t year,, or t time of piiyiiaent. Whence the following general 
theorems are deduced : , 

Ifit, a = /RV the amount^ 

a 
2d, ^ = n^t ^^ principal, 

3d, R = .J/—, the ratio, 

^ _ log. of <7 — log. of^ , 

: Vol. I. S From 



1>5« 



ALGEBRA. 



,v. 



From which, anyone 6( th^'qu&mmes imify be foElndr 
Wh^n the rcsft are giteh^ 

A3 to the whole interest, it is found.by barely subtracting 
the principal p from the amount a. 

Example\ Suppose it be required to find, jn how many 
years any principal, sum will double itself, at any proposed 
rate of cote[^Qttn4 interest. 

In this case the 4th theorem must be employed, making 
41 = 2/ ; and then it is 

log. fl— log./ log. 2^— log. p _ log. 2 

"" log. Rk "" log. R. " log. R' 

So, if the rate of interest be 5 per cent, per annum; then 

lR BB 1- + *05 rr 1*05 j and hence . ^ . 

log. 2 -301030 , ^ ^^^^ , . ' 
/ = .:; — ^-— : = ■ ^ ,^^ = 14-2067 nearly ; 
log.J-05 -021189 ^ ■ 

that is, any sum doubles' itself in 14^- years nearly^ at the 

rate of 5 per cent, per anmiixi compound interest* . 

Hence, and from the like question iti Simple Interest^ 
' above given, are deduced the times in Which any sum 
doubles itiself, at several rates of interest, both simple and 
compound; Viz. ' • . 



ii^mt^^i 



At' 

2 

24 
3 

3i 
4 

5 
6 
7 
8 

9. 
10 



''AtSimp.Int 



> 



per cent, per annum 
interest, 1/. 6r any 

other sum, will 
double itself in the 

following years. 



m 



50 

40 

33| 

28f 

25 

22| 

20 

16|. 

14| 

12^ 

IH 

10 






At Comprint. 



•*»—«• 



ill 35-0028 . 
28-0701 
23-4498 
20-14S3 
17-6730 . 
15-7473 fr 
14-2067 § 
11-8957* 
!0'2448 
'9-0065 
8-04S^ 
7-2725 



.1 



The 



f 



COMPOUND INTEREST- 



2SB 



The following Table wili very mttdi facilitate caloilitipiis 
of compound interest on any sum, for any number of yeacSf 
at various rat«s of imef est- 

The Amounts of 1/. in any Number of Tears. 



Yrs. 


3 


34 


.4 


■■4,i 


. 1 '^i 


> 


1*0300 


1*0350 


10400 


1^0450 


1*0500 


1-06DO 


2 


1-0509 10712 


IO8I6 


10920 


11025 


1-1236 


3 


10927 


1-1087 


]-124g 


1-J412 


1*1576 


11910 


4 


V1255 


1-1475 


1-1699 


1-1925 


1-2155 


1*2625 


5 


1.1593 


1-1877 


1-2167 


1-2462 


l-!?763 


1-3392 


6 


11941 


1*2293 


1-2653 


1*3023 


1*3401 


1-4165 


7 


1*2299 


1-3723 


1*3159 


1-3609 


1*4071 


1*5036 ^ 


8 


1*2668 


1-3168 


1-3686 


1*4221 


1*4775 


15939 


9 


1-3048 


1 -3629 


1*4233 


1*4661 


1*5513 1 


i*669il 


10 


1-3439 


1*4106 


1*4802 


1-5530 


1-6289 


17909 


n 


1-3842 


1-4600 


j-5«9^ 


xm^ 


17103 


1-8933. 


12 


1-4258 


1*5111 


I'60lO 


1-6959 


17959 


2-0l2» 


13 


1*4685 


J -5649, 


1-6651 


l'77'^2 


1*«856 


2-13'^- 


14 


1-5126 


l*6l87* 


1*7317 


1-8519 


1-9799 


2-2609 


15 


1-5580 


1-Q753 


1-8O09 


1*9355 


2-0789 


2-3966 


16 


1-6047 


1-7340 


1*8730 


2*0224 


2*1829 


2-5404 


17 


1-6528 


17947 


1-9^79 


2-1134 


22920 


2-6928 


18 


1-7024 


1-8575 


20258 


2*2035 


2-4066 


2-8543 


19 


1-7535 


1-9225 


2- 1068 


2*30/0 


2-5270 3-0256 


20 


1-8061 


i'989B 


21911 


2-4U7 


2-6533 3-2071 



\ 



The use of this Tafeje, which contains all the poivers, :E^j 
to the 20th power, or the anjounts of 1/, is cniefly tp c;^T- 
culate the interest, or the aoiount of any principal sum, for 
any time, not more than 20 years. 

For example, let it be required to find, to how much 92$I* 
will ai^punt in 15 years, at the rate of 5 per cent per fttmuisi 
^oropQund interest. 

In the table, on the line 15, and in the column 5 per cent. 

is the amount of 1/, viz. - - 2*0789 

this multiplied by the principsil - 523 ^ 



gives the amioimt 

or 
and therefoi*e the interest is 



10S7-2647 
1087/. 5/. '$IJ. 
564/. 5s. 3^, 



Noie 1. When the rate of interest i» to be determined tt^ 
any other time than a fear ; its suppose to 4- a ye^r, 4^^^a 
year, &c ; the rules are still th« Mfiie \ '• but theii^ s^mHH 

S 2 express 



^260 ALGEBRA. 

« 

express, that time» and r must be taken the amount (or that 
.time also. 

Note 2. When the compound interest, or amount, of any 
sum, is required for the parts of a year ; it may be deter- 
mined in the following manner : 

. 1//, For any time which is some aliquot part of a year :— 
Find the amount of 1/. for 1 year, as , before } then that 
root of it which is denoted by the aliquot part, will be the 
amount of 1/. This amount being multiplied by the prin- 
cipal sum, will produce the amount of the given sum as 
required. 

2</, When the time is not an aliquot part of a year : — 
Reduce the time into days, and take the 365th root of the 
amount of 1/. for 1 year, which will give the amount of the 
same for 1 day. Then raise this amount to that power 
whose index is equal to the number of days, and it will be 
the amount for that time. Which amount being multiplied 
by the principal sum, will produce the amount of that sum 
as before.— And in these calculations, the operation by loga- 
rithms will be very useful. ' 



OF ANNUITIES- 



ANNUITY IS a term used for ""any periodical income^ 
arising from money lent, or from houses, lands, salaries, 
pensions, &c. payable from time to time, but mostly by 
annual payments. 

"" Annuities are divided into those that are in Possession, 

and those in Reversion : 'the former meaning such as have 

_ commenced ; and the latter such as will not liegin till some 

particular, event has happened, or till after some certain 

time has elapsed. 

When an annuity is forborn for some years, or the pay*- 
moits not made for that time, the annuity is said to be in 
Arrears. 

An annuity may also be for a certain number of years ;. 
or it may be without any limit, and then it is called a Per- 
petuity. 

The Amount of an annuity, forborn for any number of 
•years, is the sum arising from the addition of all the annul- 
.f i^s for that number of years j together with the interest due 
Upon eAchjafter it becomes due. . 
^i. : 4j The 



ANNUITIES.' 261 

The Present Worth or Value of an annuity, is the price or 
sum which ought to be given for it, supposing it to be bought 
off^ or paid all at once. 

Let a = the annuity, pension, or yearly rent ; 

n = the number of years forborn, or lent for ; 
ft = the amount of 1/. for 1 year i 
m = the amount of the annuity ; 
V = its value, or its present worth. 
Now, 1 being the present value of the sum R, by propor- 
tion the present value of any other sum a, is thus found : 

a • 

as R : 1 : : tf ; — • the present value of a due 1 year hence. 

In like manner — is the present value of u due 2 years 

a a A u o 

hence ; for r : 1 : : — : ~y- So also — ^ -^, — j, &c, will 

be the present values of a, due at the end of 3, 4, 5, &:c, 
years respectively. Consequently the sum of all diese, or 

— +-I+ A+r: + &c = {— + -5+-i + ri&<^-)x«» 

R R* R' k* ^R R* R^ R* ' 

continued to n terms, will be the present value of all the n 
years' annuities. And the value of the perpetuity, is the 
sum of the series to infinity. 

But this series, it is evident, is a geometrical progression, 

.1 . ' . 

liaving — both for its first term and common rsrtio, and the 

R ~ .•» 

number of its terms n ; therefore the sum v of all the terms^ 
or the present value of all the annual payments, will be 
Jl \^ l^ 

R R R" R" — 1 tf 

« = — -y— x«,or=j^3^x- 

1 

R -^ . ■■ 

When the annuity is a perpetuity; n being infinite, R** 

. 1 
is also infinite, and therefore the quantity — becomes = 0, 

therefore " x -^ also = ; consequently the exprei- 

R — 1 R 

sion becomes barely v = — jr ; that is; any annuity divided 

by the interest of 1/. for } year, gives the value of .the per- 
petuity. So, if the rate of interest be 5 per cent. 

Then lOO/i -r- 5 = 20a is the value of the perpetuity at 
S per cent : Also IQOa -r- 4 = 25a is the value of the per- 
petuity 



r- 



26^ ALGEBHA. 

• • • • 

petuifj' at 4 f)6r teni : Atid 100^ -r 3 £= SS-^ h tlie valut of 

the perpetuity ^t 5 pcp cent t ^nd so on. 

Again, because the amount of 1/. in n years, fe H", its 

increase in that time will be R*^— I ; but its interest for one 

single ytdr, or the annuity answering to that increase, is 

R — 1 ; therefore as R — 1 is to r" — 1, so is a to iw ; that 

r"— 1 

is, m =: r X a. Hencc^ the several cases relating to 

R — 1 

Annuities in Arrear, will be resolved by the foUowipg 
equations : 

m = X a ■= vR" i 

R — 1 

. • R" i~ 1 a m 

R - i . R - 1 ^ 

|t» - 1 R" - 1 

WR — W -f tf 
■ log. 

log. >yf — log. v _^ ^i 

"" log. R ^ log. R * 

log. ^ »- log. V 
. X.pg. R =2 — -^ ' 



■•r " 

# 

n 



1 1 fl 

^R.P V^ R - 1 

'.Ih this last theorem, r denotes the present value 6f an 
annuity in reversion, after p years, or not commencing till 
after the first j& years, being found by taking the difference 

between the two values 7 x — r and x --;;:, for 

R — 1 R" R — I RP 

n jeTLi^ and p years. 

But the amount and present value of any annuity for any 
number of years, up to 21, will be most readily found by the 
two following tables. 



I - 



TABLE 



AumimES, 



2$$ 



TABLE I. 



The Amoiint of an Anhiiity of 1/. at Compound Interest. 



im 



atSperc. 



M «^-^ 



"^ 



1 

2 

r 3: 
4 

5 
6 

7 

8 

9 

10 

11 

12 
J3 
14 
15 
16 
■.17- 

a8 
19 

20 
21 



3f per cJ 4 per c. 



.10000 
^3'OaOO 



1*0000 

2T03i0 

.^fOpQgrl 3-iQQ2. 



4-1836 

5-^091 

d-4684 

7*6625 

d-8923 

10-1591 

1 1 -4^9 

12-8078 

14-1920 
15-6178 
1 70863 
18-5989 
20-1569 
>2l'5?6l6 
33-4144 

26-8704 
28-6765 



4-2149 
5-3625 
6-5502 
7.7794 

9-0517 
10-3685 

11*7314 
13-1420 
14-6020 
16-1130 
17-6770 
19-2957 
20-9710 
227050 

24-^^7 
^-3572 

28-2797 



1-0000 
2-0400 

a-i2i6 

4-2^465 
5-4163 
6-6330 

7*89.83 
9-2142 
10-5828 
12*0061 
13*4864 
15:0258 
16-6268 
18-2919 

20-3236 
21-8245 
S3-^75 
25*6454 

«7*6712 
297781 



4^ per c. 



30-2695131-9692 



1-0000 
2*0,450 
3-1370 

4-278:? 
5-4707 

6-7169 

8*0192 

9*3800 
10*8021 
12-2892 
13-8412 
15-4640 

171599 
1 8-9321 

207841 

22-7193 
24-7417 
26*8J51 
290636 
31'3714 
'33-7831 



5 per c. 6 per c. 



1-0000 

20500 
3-1525 
4-3101 
5-5256 
6*8019 
8-1420 
9*5491 
11-0266 

i«*5779 
14-2068 

15*9171 
17-7130 
19-5986 
21-5786 
23-6575 

25-8404] 
28-1324 
30-5390 
33*0660 

135*7193 



1*0000 
20600 
6-1836 
4-3746 

5-6371, 
6-9753 
8-3938 j 

9*8975 
11-4913 
13-1808 
U»97i6 
16-6699 

18*8821 
21-0151 
23-2760 
25-6725 
28-2129 

30-9057 
33-76001 
S6'7856l i 

39-9927 1 



tab;l^ 11. Th^ Pr^s^t ^Vs^lue of an Annuity of 1/. 



Yrs. 



1 
2 
3 
4 
5 
6 

7 

8 

9 
10 

11 

12 

\3 
14 
15 
16 

17 

18 

19 
20 

21 



at9p^c< 



0-9709 
l-9i35 

2-8286 
3-7171 

4-5797 

5-4172 

6-2303 

70197 
7*7861 
8-5302 
9-2520 
9-9540 
10-6350 
11-2961 

11 -9379 
12-5611 
131601 
13-7535 
14-3238 
14-8775 
15*4150 



3t per^.[4 per c 



09662 

1*8997 
2-8016 

3*6731 
4-5151 
5-3286 
6-1145 
6-8740 

7:0077 
8-3166 

9*01 16 
9-6633 

iG-3027 

:i 0-9205 

11-5174 

12*0941 

12-6513 
13*1897 
13-7098 
14-2124 
14*6980 



0-9615 

1-8861 

2*775 1 

3-6299 
4-4518 

5-2421 

6-0020 

6-7327 
7-4353 
8-110J9 
8-7605 
9-3851 

9*9857 

10-563 1 
11-1184 
11-6523 

12- 1657 
li-6593 
131339 
13-5903 
140292 



4t per c. 



0-956& 

1*8727 
2-7490 
3-5875 
4-3900 

5-1579 
5-89i7 
6-5959 

7-268S 

7-9127 

8*5289 
9- 1186 
9*6829 
10-2228 
10-7396 
1 1 -2340 
11-7072 
12-1600 
12-5933 
130079 
13-4047 



5 perc. 



ii< >i 



0*9524 
1-8594 
2-7233 
3-5460 

4-3295 

5-0757 
5->864 

6-4632 
71078 

7*7217 
8*3054 

8-B633 

9*3936 

9*8986 

10*3797 
10-8378 
11-2741 
11-6896 
12*0853 
12-4622 
•12*8212 

li ■ 



per e. 



0-9434 
1*8334 
2-673^ 

3-4651. 

4-2124< 
4*9173 
5-5824 
6-2098 

6*8017 
7*3601 

7*8869 

8-3838 
8-8527 
9-29501 

9*7^23 

10*1059 
10-4773 
10-8276 
11*1581 

11*4699 
117641 



Ta 



sH ALGSBRA. 

n find thf Ammnt cf any annuity forbore a eertain numbir tf 

years* 

TiiKE out the amount of 1/. from the £r$t table^ for the 
proposed rate and time ; then multiply it by the given 
annuity; and the prodyct will be the amount , for the same 
number of years, -and rate of int^r$st^-^And the converse to 
find the rate or time, 

£xaf^. To £nd how much an annuity of 50/. will amount 
to in 20 years, at Si per cent, compound interest. 

On the line of 20 years, and in the column of 3, f^r cent, 
$tands 28*2797^ which is the amount of an annuity of l/« for 
the 20 years. Then 28-2797 x 5Q, givw 1413-985/. a:; 
■ 1413/. 19/. 8d. for the answer required. 

* To find the Present Value of anyannwtt^foranj number of 
j^^tfrx.— Proceed here by the 2d table, in the same manner as 
iabove for the Ist tabl^, and the present; vorth require will 
be found. 

Exam, \. To find the present value of an annuity of SOU 
which is to continue 20 years, at 3^ per cent.-f-^y the table, 
the present value of 1/. for the given >ate and time, is 
14-2124 ; therefore 14-2124 x 50=710*62/. or 710/. 12/. 4rf. 
IS the present value required^ * . 

Exam. 2. To find the present value of an annuity of 20A 
to commence 10 ye^rs hence, and then to continue for l\ 
jezef longer, or to terminate 2 1 years hence, at 4 per cent. 
mterest. — In such cases as this, we hate to find the difierence 
between* the present values of two equal annuities, for the 
two given time? y which therefore will be done by $ubtract-i 
ing the tabular value of the one pieriod from thatdf the^ 
Other^ and then multiplying by the given annuity. Thus, 

tabular value for 21 years 14*0292 

ditto for - 10 years 8 1105^ 

.■I . ■ 1 

the difference 5'918i 

multiplied by 20 



H. ' I' " 



■ I 



gives - '1J8466/. 

or - - 1 1 8/. 7/. 34*/. the zns9ftr\ 



ENP OP THE ALGEBRA. 






•a 



sess 



i^MM*^ 



55SS55 



!te«9MBavs 



GEOMETRY. 



• - DEFINITIONS. 

1. jhL point is that which has position, 
h%it no magnitude, nor dimensions ^ neither 
length, breadths nor thickness. 

2. A Line is length, without breadth or 
thickness. 

3. A Surface or Superficies, is an extension 
or a figure of two dimensions, length and 
bri^adth ', but without thickness. 

4. A Body or 6oli4, is a figure of three di- 
mensions, namely, lepgth, breadth, and depth, 
or thickness. 

5. Lines are either Sight, or Curved, or 
JVIixed of these two. 

6. A Right Line, or Straight Line, lies all 
in the same direction, between its extremities ^ 
^d is the shortest distance between two points. 

When a Line is mentioned simply, it means 
t Right Line* 

7. A Curve continually changes its direction 
between its extreme points. 

8. Lines are either Parallel, Oblique, Per- 
pendicular, or Tangential. 

9. Parallel Lines are alw&ys at the same per- 
pendicular distance; and they nevermeet, though 
ever so far produced. 

10. Oblique lines change their distance, and 
would, meet, if produced on the side of the 
least distance* 

IL One line is Peipendicular to another, 
when it inclines not more on the one side 

than 




t:v 





— * 



266 GEOMETRY. 



•— / -.^«-. 






than the ©ther, or when the angles on both 
sides of it are equaL 

12. A line or circfe is Tangehtial, or a 
Tangent to a circle, or other curve, when it 
touches it, without cutting, when both are pro- 
duced. 

13. An Angle Js the inclination or opening 
of two lines, having difierent directions, and 
meeting in a point* 

14. Angles are Right or Oblique, Acute, or 
Obtuse, 

15. A Right Angle is that which js made by 
cuae iine. perpendicular to another. Or when 
the angles on each side are equal to One an* 
Qther^ they are right angles. 

; 16. An Oblique Angle is that which is 
made by two oblique lines ; and is either le$s 
or greater than a nght angle. 

l7. An Acute Angle is less than a right 
angle. 

1. IS. An Obtuse Angle is greater than a right 
angle. /■ v ' ' 

i9. Superficies are either Plane or Curved. 

20. A Plane Superficies, or a Plane, is that with which 
a right line may, every way, coincide. Or, if the line touch 
the plane in two points, it will touch it in every point. But, 
if Bot, it is curved. 

21. Plane Figures are bounded either by right lines or 
curves. 

22. Plane figures that are bounded by right lines have 
names according to the number of their sides, or of their 
angles ; for they have as miny sides as angles ; the least 
number being three. 

23. A figure of three sides and angles is called a Triangle. 
And it receives particular denominations from the relations 
of its sides and angles. 

24. An Equilateral Triangle is that whos^ 
three sides are all .equal. 

25.. An Isosceles Xnangle is that which has? 
two sides equal. 

26. A 




DEFINITIONS. 



26t 



^ 




zx 



[ 




* 26. A Scalene Triangle is that whose three 
sides are all unequal: ^ . ' 

27. A Right-angled Triangle is that which 
has one right-angle. > 

.28. Other triangles are Oblique- angled, and 
are either Obtuse or Acute. ^ 

29. An Obtuse-angled Triangle has one ob- 
tuse an<rle. 

SO. Alt Acute-Angled Triangle has all its 
diree angles acute- 

8K A figure of Four sides and angles is 
called a Quadrangle, or a Quadrilateral- 

'32. AParallelogram is a quadrilateral which 
&as both its pairs cf opposite sides parallel. 
And it takes the following particular name$» 
viz. Rectangle, Square, Rhombus, Rhomboid. 

33. A Rectangle is a parallelogram having 
a right angle. 

34*. A Square is an equilateral rectangle; 
having its length and breadth equal. 

35. A Rhomboid is an oblique-angled pa- 
rallelogram. 

S6. A Rhombus is an equilateral rhomboid; 
having all its sides equal, but its angles ob- 
lique. 

37. A Trapezium is a quadrilateral which 

hath not its opposite sides parallel. 

'* . . ' 

. 38. A Trapezoid has only.one pair of oppcH 
site sides paralleK 

39, A Diagonal is a iitae joining any two 
opposite angles of a quadriiateraL 

40^ Plane figures that have more than four sides are, in 
general, called Polygons : and they receive other particular 
names, according to the number of their sid,es or angles. 
Thus, 

41. A Pentagon is a polygon of five sides; a Hexagon, of 
six sidfcs;*a Heptagon, seven; an Octagon, eight; a No- 
fiagon, nine ^ a Decagon, ten ; an Undecagon, eleven ; and a 
Dodecagon, twelve sides» 

42. A 



C7 



i6i 



GEOMETRY. 



42. A Regular Poljgon has all its skies and all its angk§ 
equol. — If they are not both equal, the polygon is Irregular. 

43. An Equilateral Triangle is also a Regular Figure of 
three sides, and the Square is one of four j the former being 
also called a Trigon, and the latter a Tetragon. 

44. Any figure is equilateral, when all its sides are equal : 
and it is equiangular when all its angles are equal. When 
both these are equal, it is a regular figure. 

45. A Circle is a plane figure bounded by 
a curve line, called the Circumference, which 
55 every where equidistant from a certain point 
within, called its Centre. 

The circumference itself is often called a 
circle, and also the Periphery. 

46. The Radius of a circle it a line drawn 
from the centre to the circumference. 

47. The Diameter of a circle is a line 
drawn through the centre, and terminating at 
the circumference on both sides. 



4S. An Arc of a circle is any part of the 
circumference. 



5 I 




49. A Chord is a right line joining the 
extremities of an arc. 



50. A Segment is any part of a circle 
bounded by an arc and its chord. 

51. A Semicircle is half the circle^ or a 
segmeilt cut off by a diameter. 

The half circumference is sometimes called 
the Semicircle. 

52. A Sector is any part of a circle which 
IS bounded by an arc, and two radii drawn to 
i$s extremities. 

53. A Quadrant, or Quarter of a circle, is 
a sector having a quarter of the circumference 
for its arp-, and its two radii are perpendicular 
to each other. A quarter of the circumference 
» sometimes called a Quadrant. 







•j^ 



5*. Th« 



JJEPlMrtlONS. 



26!l 



/k 



D K 

V-. 




54. The Height or Altitude of a figure is 
ft perpendicular let fall from an angle, or its 
vertex, to the opposite side, called the base. 

65, In a right-angled triangle, the side op- 
posite the right angle is* called tlie Hypothe- 
nuse 'y and the other two sides arecafted the 
X^egs, and sometimes the Base and Perpends- . 
cukr. 

56. When an angle is denoted by three 
letters, of which one standi at the angular 
point, and' the other two on the two sides, 
that which stands at the angular point is read 
in the middle. 

57. The cirCunrference of every circle is 
supposed to be divided into 360 equal parts, 
called Degrees : and each degree into QO Mi- 
nutes, each minute into 60 Seconds, and so on. 
Hence a semicircle contains 1 80 degrees, 2XidL 
a quadrant 90 degrees. 

58. The Measure pf an angle, is an arc of 
any circle contained between the two lines 
which form that angle , the angi,ilar point being 
the centre; and it is estimated by the number 
of degrees contained in that arc. 

59. Lines, or chords, are said to be Equi- 
distant from the centre of a circle, when per- 
pendiculars drawn to them from the centre 
are equal. . 

60. And the right line on whichthe Greater ^ 
Perpendicular falls, is said to be farther from 
the centre. . 

6 k An* Angle In a segment is that which 
is contained by two lines, drawn from any 
point in the arc of the segment, to the two 
extremities of that arc, > 

62. An Angle On a segment, or an arc, is that which is 
contained by two lines, drawn from any point in the opposite 
or supplemental part of the circumference, to the extremi- 
ties of the arc, and containing the arc between them. 

63. An angle at the circumference, is that 
whose angular point is any where in the cir- 
cumference. And an angle at the centre, is 
that whose angular point is at the centre, 

1.^ \ . 6% A 






270 



GEOMETRY. 




64. A right»Uned figure is Inscribed in a 
circle^ or the circle Circumscribes it, when 
all the angular points of the figure are in the 
circumference of the circle. 

65. A right-lined figure Circumscribes a 
circle, or the circle is Inscribed in it, when all 
the sides of the figure touch the circumference 
of the circle. 

66. One right-lined figure is Inscribed in 
another, or the latter Circumscribes the former, 
when all the angular points of the former 
are placed in the sjuies of the latter. 

67. A Secant is a line that cuts a circle, 
lying partly within, and partly without it. 



68. Two triangles, or other right-lined figures, are said tp 
be mutually equilateral, when all the sides of the one are 
equal to the cotresponding sides of the other, each to each r 
and they are said to be mutually equiangular, when the 
angles of the one are respectively equal to those of the other. 

69. Identical figures, are such as are both mutually equi- 
lateral and equiangular ; or that have all the sides and all the 
angles of the one, respectively equal to all the sides and all the 
angles of the other, each to each ^ so that if the one figure 
were applied to,, or laid upon the other, all the sides of the one 
would exactly fall upon and cover all the sides of the other; 
the two becoming as it were but one and the same figure. 

• 70. Similar figures, are those that have all the angles of 
the one equal to all the angles of the other, each to each, ind 
the sides about the equal angles proportional; 

74. The Perimeter of a figure, is the sum of all its sides 
taken together. 

72. A Proposition, is something which is either proposed 
to be done, or to be demonstrated, and is either a problem or 
a theorem. 

4 

73- A Problem is something proposed to be done. 

74. A Theorem , is something proposed to be demonstrated. 

75. A Lemma, is something wliich is premised, or demon- 
strated, in order to render what follows more easy. 

76. A CoroUary, is a consequent truth, gained immedi- 
ately fi-om some preceditig truth, or demonstration. 

* 77. A Scholium, is a remark or observation* made upon 
something going before it; ' * '^ *' 

' # -^ AXIOMS. 



[ 211 ] 



, « 



AXIOMS. 

» • . ■ ' • 

i. Thincs which are equal to the sam^ thing are equal 
to each other. 

5. When equals are added to equals^ the wholes are equal. 

, 3. When equals are taken from equals, the remains arc 
equal. " * , \ 

4. When equals are added to unequals, the wholes are 
unequal. 

-5. When equals are taken from unequal% the remains are 
: unequal. ' -y 

6. Things which are double of the same tbiag) or eqml 
things, are equal to each other. 

, 7. Things which are halves, of the samd thingi tsfe eqUaL 

• ' . , 

^. Every whole is equal to all its parts taken together. ' 

. 9. Things which coincide, or fill the same space, are iden- 
tical, or inutually equal in all their parts. 

10. All right angles are equal to one another.. 

1 1» Angles that have equal measures, or arcs, are equaL 



THEOKEM I. 

• 

If two Triangles have Two Sides and the Included Angle 
in the one, equal to Two Sides and the Included Angle 
in the other, the Triangles will be Identical, or equal in all 
respects. 

In the two triangles aec, def, if 
the side ac be equal to the side df, 
and the side Bc equal to the side ef, 
and the angle c equal tq ^he angle f \ 
then will the two triangles be identi- 
cal, or equal in all wsp^ts, ■ ^ , ; 

.. for xioiiceive the triatijjfe. £BC it^ be -applied to, of ^e«d 
On^ the t]4angle VB^yitk iMin a ttiAnii^ that tho^poiiit fc may 




S72 



GEOMETRY. 



- \ 




coincide with the point f^ and the side ac with th< $ide Jiff 
which is equal to it* 

Then, since the angle F k equal to the angle c (by hyp.)^ 
the side bc will fall on the side £f. Also, because AC is 
equal to df, and BC equal to £F (by hyp.)> the point A will 
coincide with the point d, and the point B with the point £ ; 
consequently the side ab will coincide witfi the side db. 
Therefore the two triangles are identical^ and have all their 
other corresponding parts equal (ax. 9), namely, the side ab 
equal to the side D£, the angle A to the angle t>, and th^ 
angle B to the angle e. <^ £. d, 

THEOREM tU 

When Twb Triangles have Two Angles and the included 
Side in the one, equ^ to Two Angles and the included Side 
in the other, the Triangles are Identical, or have their other 
sides and angle equal. ^ 

Let the two triangles aBc, t)EF| 
have the angle A equal to the angle 
D, the angle B equal to the angle £, 
and the side ab equal to the side D£; 
then these two triangles will be idetl^ 
tical. 

« 

For, cenceive the triangle abc to be placed on the triangle 
PEF, in such manner that the side ab may fell exactly on the 
equal side de. Then, since the angle a is equal to the angle 
D (by hyp,)> the side ac must fall on the side df ; and, in 
like manner, because the angle B is equal to the. angle £, the 
side BC must fall on the side ef. Thus the three sides of the 
triangle abc will be exactly placed on the thr^e sides of the 
triangle def: consequently the two triangles are identical 
(ax. 9), having the other two sides ac, Bfc, equal to. the two 
DF, EFy and the remaining angle c equal to the remaining 
angle f. <^ e, d. 

theorem III. 

In an Isosceles triangle, the Angles at the Base are equal. 
Or, if a Triangle have Two Sides equal, their Opposite 
Angles will also be equal. 

U the triangle abc have the side AC equal 
to the side bc; then will the angle d be 
equal to the angle a. 

For, conceive the angle c to be bisected, 
or divided into two «qua^>aru, by the line 
CD, making the angle A(ii> equidi.to the 
angfe*BCD. Then, 




THEOREMS. 



273 




Then, the two triangles acd, bcd, have two sides and 
the comained angle of the ,one, equal to two sides and the 
contained angle of the other, viz. the side ac equal to bCj 
the angle acd equal to bcd, and the side cd common; 
therefore these two triangles are identical, or equal in all 
respects (th, 1); and consequently the angle a equal to the 
angle b. q.. e. d. 

CotqL 1. Hence the line which bisects the verticle angte 
of an isosceles triangle, bisects the base, and is also perpendi- 
cular to it. 

Corol. 2. Hence too it appears, that every equilateral tri- 
angle, is also equiangular, or has all its angles equal. 

THEOREM IV. 

When a Triangle has Two of its Angles equal, the Sides 

Opposite to them are also equaL 

If the triangle abc, have the angle a 
equal to the angle B, it will also have the side 
AC equal to the sideBc. 

For, conceive the side a B to be bisected 
in the point d, making ad equal to db; 
and join dc, dividing the whole triangle into 
the two triangles acd, bcd. Also conceive 
the triangle acd to be tiu-ned over upon the 
triangle bcd, so that ad may fall on bd. 

Then, because the line ad is equal to the line db (by k0.), 
the point A coincides with the point B,.and the point d with 
the point d. Also, because the angle a is equal to the angle 
B (by hyp.), the line ac will fall on the line bc, and the ex- 
tremity c of the side ac will coincide with the extremity c 
of the side bc, because dc is common to both ; consequently 
the side ac is equal to BC. Q- E. D. 

CoroL Hence every equiangular triangle is also equila* 
' teral* 

THEOREM v. ^ 

When Two Triangles have all the Three Sides in the one, 
equal to all the Three glides in the other, the Triangles are 
Identical, or have also their Three Angles equal, each to each. 

Let the two triangles abc, abd, 

haVe their three sides respectively 

. equal, viz. the side ab equal to ab, 

AC to AD, and BC to bd j then shall 

the two triangles be identical, or have 

^ their angles equal, vijs. those afigles 

Vol. I. T thuc 



(ijrt>t4^l^t^ 




» * 



i74 



GEOMETkY. 




that are opposite to the equal sides ; 
namely, the angle bac to the angle 
BAD, the angle abc to the angle abd, 
and the angle c to the angle d« 

For, conceive the two triangles to 
be joined ^ together by their bngest 
equal sides, and draw the line cd. 

Then, in the triangle acd, because the side AC is equal 
to AD (by hyp.), the angle acd i^ equal to the angle adc 
(th. 3). In like manner, in the triangle bcd, the angle 
BCD is equal to the angle bdc, because the side Be is equal 
to BD. Hence then, tne angle acd being equal to the angle 
ADC, and the angle bcd to the angle bdc, by equal addi« 
tions the sum of the two angles acd, bcd, is equal to the 
sum of the two adc, bdc, (ax. 2), that is, the whole angle 
ACB equal to the whole angle adb. 

Since then, the two sides Ac, CB, are equal to the two 
sides AD, DB, each to each, (by hyp.), and their contained 
angles acb, adb, also equal, the two triangles abc, abd, 
are identical (th. 1), and have the other angles equal, viz. 
the angle bac to the angle bad, and the angle ab& to the 
angle abd. q. £• d. 

THEOREM VI. 

When one Line meets another, the Angles which it 
makes on the Same Bide of the other, are together equal to 
Two Right Angles. 

Let the line ab meet the line cd : then 
will the two angles abc, abd, taken to- 
gether, be equal to two right angles. 

For, first, when the two angles abc, 
ABD, are equal to each other, they are both 
of them right angles (def. 15). 

But when the angles are unequal, suppose BE drawH per- 
pendicular to CD. Then, since the two angles ebc, ebd, 
are right angles (def. 15), and the angle ebd is equal to the 
two angles eba, abd, together (ax. 8), the three angles, EBC, 
EBA, and abd, are equal to two right angles. 

But the two angles ebc, eba, are together equal to the 
angle abc (ax. 8). Consequently the two angles abc, abd» 
are also equal to two right angles, q. £. D* 

CoroL 1. Hence also, conversely, if the two angles ABC, 
abd, on both sides of the line ab, make up together twa 
right angles^ then cb and bd form one contixraed right 
Ijne CD. 

Corol. 





THEOREMS. ?75 

CoroL 2. Henccy all* the injgles which can be made» at 
any point B, by any number of lines, on the same side of 
the right line cd, are^ when taken all together^ equal to twQ 
right anglesi. \ 

Corol. S. And, as all the angles that can be made on th« 
other side of the line CD are also equal to two right angles ; 
therefore all the angles that c^m be made quite round a point 
s, by any number of lines, are equal to four right angles. 

CoroL 4. Hence also the whole circumfer- 
ence of a circle, being the sum of the mea- 
sures, of all the angles that can be made about 
the centre f (def. 57), is the measure of four 
right angles. Consequently, a semicircle, or « 

180 degrees, is the measure of two right 
angles ; and a quadrant, or 90 degrees, the measure of ene 
right angle. 

THEOREM VII. 

When two Lines Intersect each other, the Opposite Angles 

are equal. 
Let the two lines ab, cd, intersect in 
the point E; then will the angle aec be 
equal to the angle bed, and the angle 
ABD. equal to the angle ceb. 

For, since the line CE meets the line 
AB, the two angles aec, beg, taken 
together, are equal to two right angles (th. ^). 

In like manner, the line be, meeting the Jdne CD, majces 
the two angles bec, bed, equal to two right angles. , \ 

Therefore the sua\ of the two angles aec, bec> is equal 
to the sum of the two bec, bed (ax. 1). < 

And if the angle bec, which is common, be taken away 
'from both these, the remaining angle aec will be equal to 
the remaining angle bed (ax. 3). . 

And in like manner it may be shown, that the angle aed 
is equal to the opposite angle bec. 

THEOREM VIII. 

When One Side of a Triangle is produced, the Outward 
Angle is Greater than either of the two Inward Opposite 
. Angles. 

T2 Let 




276 



GEOMETRY. 




Let ABC be a triangle, having the 
side AB produced to d ; then will the 
outward angle cbd be greater than 
either of the inward opposite angles a 
•r c. 

For, conceive the side bc to be bi- 
sected in the point £, and draw the line 
ACT, producing it till bf be equal to A£i 
and join bf. 

Then, since the two triangles A EC, bef, have the side 
AE =s the side ef, and the side ce = the side be (by suppos.) 
and the included or opposite angles at s also equal (th. 7), 
therefore those two triangles are equal in all^ respects 
(th. I), and have the angle c = the corresponding angle 
EBF. But tlie angle cbd is greater than the angle ebfj 
consequently the said outward angle cbd is also greater than 
the angle c 

In like manner, if cb be produced to G, and AB be bi- 
sected, it may be shown that the outward angle ABG, or its 
equal CBD, is greater than the other angle a. 




J>B 



THEOREM IX. 

The Greater Side, of every Triangle, is opposite to 
the Greater Angle ; and the Greater Angle opposite to the 
Greater Side. 

Let ABC be a triangle, havbig the side 
AB greater than the side ac ; then will 
the angle acb, opposite the greater side 
AB, be greater than thp single B, opposite 
the less side ac. 

For, on the greater side ab, take the 
part AD equal to the less side ac, and join cD. Then, since 
BCD is a triangle, the outward angle adc is greater than, 
the inward opposite angle b (th. 8). But the angle acd 
is equal to the said outward angle adc, because ad is equal 
to AC (th. 3). Consequently the angle acd also is greater 
than the angle B. And since the angle acd is only a part 
of i\cB, much more must the whole angle ACB be greater 
than the angle b. q. e. d. 

Again, conversely, if the angle c be greater than the angle 
B, then will the side ab, opposite the former, be greater than 
the side, ac, opposite the latter. 

For, if ab be ^ not greater than AC, it must be either 
equal to it, or less than i^. But it cannot be equal, for 

then 




THEOREMS. 277 

• 

thtn the angle c would be eqyal to the "angle B (th. 3), 
which it is not, by the supposition. Neither can it be less, 
for then the angle c would be less than the angle B, by the 
former part of this ; which is also Contrary to the supposi- 
tion. The side ab, then, being neither equal to AC, nor less 
than it, must necessarily be greater, q^. fi. D. 

« THEOREM X. v 

The Sum of any Two Sides of a Triangle is Greater than 

the Third Side. 

Let ABC be a triangle ; t^en will the » p 

^um of any two of its sides be greater than ...••* i 

the third side, as for instance, AC + cb 
greater than ab. 

For, produce AC till CD be equal to 
cb, or AD equal to the sum of the two 
AC + cb; and join bd: — ^Then, because 
CD is equal to cB (by constr.), the angle D is equal to the 
angle cbd (th. 3). But the angle abd is greater than the 
angle cbd, consequently it must also be greater than the 
angle d. And, since the greater side of any triangle is op- 
posite to the greater angle (th. 9), the side ad (of the tri- 
angle abd) is greater than the side ab. But ad is equal 
to AC and cd, or ac and cb, taken together (by constr.) ; 
therefore ac + cb is also greater than ab. <^ e. d. 

Cors/. The shortest distance between two points, is a singly 
right line drawn from the one point to the other. 

THEOREM XI. 

The Difference of any Two Sides of a Triangle, is Less 

than the Third Side. 

Let ABC be a triangle ; then will the 
difference of any two sides, as ab— ac, 
be less than the. third side bc. 

For, produce the less side ac to d, 
till AD be equal to the greater side ab, 
so that CD may be the difference of the 
two sides AB — AC ; and join bd. 
Then, because ad is equal to ab (by constr.), the opposite 
angles d and abd are equal (th. 3). But the angle cbd 
is less than the angle abd, and consequently also less than 
the equal angle d. And since the greater side of any triangle 

is 




rts 



GEOMETRY. 



IS opposite to the greater angle (th. 9), the side co (of ihoj 
triangle bcd) is less than the side bc. q. £• D. 




THEOHEM XII. 

When a Line Intersects two Parallel Lines, it m^es the 
Alternate Angles Equal to each other. 

Let the line ef cut the two parallel 
lines AB, CD $ then will the angle aef be 
equal to the alternate angle efd. 

For if they are not equal, one of them 
must be greater than the oth^ ; kft it be 
£FD for instance which is the greater, if 
poissible ; and conceive the line fb to be 
drawn ; cutting off the part or angle efb equal to the angle 
JLEF ; and meeting the line AB in the point B. 

Then, since the outward angle aef, of the triangle bef, 
is greater than the inward opposite angle efb (th, 8)^^ and 
tunce these two angles also are equal (by the constr.) it 
follows, that those angles are both equal and unequal at the 
same time : which is impossible- Therefore the angle efd 
is not unequal to the alternate angle aef, that is, they are 
equal to each other. <^ £. d. 

CoroL Right lines which are perpendicular to one, of two 
parallel lines, are also perpendicular to the other. 




T.HEOB.EM XIII. 

When a Line, Cutting Two other Lines, makes the Al- 
ternate Angles Equal to each other, those two Lines are Pa^ 
rallel. 

Let the line ef, cutting the two lines 
as, CD, make the alternate angles aef, 
PFE, equal to each other; then will ab 
be parallel to cd.. 

For if they be not parallel, let some 
other line, as fg, be parallel to ab* 
Then, because of these parallels, the 
angle aef is equal to the alternate angle efg (th. J 2). But 
the angle aef is equal to the angle efd (by hyp.) There- 
fore the angle efd is equal to the angle efg (ax. 1) ;.that 
is, a part is equal to the whole, which is impossible. There-^ 
fore no line but cd can be parallel to AB. q. e. d, 

CoroL Those lines which are perpendicular to the same 
line, are parallel to each other. 

theorem 



^■V i 



THEOREMS. 



m 




THEOREM XIV. 

When a Line cuts two Parallel Lines, the Outwar4 
Angle is Equal to the Inward Opposite one, on the Same 
Side ; and the two Inward Angles, on the Same Side, equal 
to two Right Angles. 

Let the line £F cut the two parallel 
lines AB, CD; then will the outward angle 
EGB be- equal to the inward opposite 
angle ghd, oh the same side of the line 
£v; and the two inward angles bgh, 
GHp, taken together, will be equal to 
two right ^gles. 

For, since the two lines ab, cd, are 
parallel, the angle agh is e<^ual to the alternate angle ghd, 
(th. 12). But the angle agh is equal to the opposite an?l^ 
£GB (th. 7). Therefore the angle ZGB is also equal to the 
angle ghd (ax. l). (^ e. d. 

Again, because the two adjacent angles EGB, bgh, are 
together equal to two right angles (th. 6) ; of which the 
single EGB has been shown to be equal to the angle ghd; 
therefore the two angles bgh, ghd, taken together, are 
^so e<|val to two right angles. v 

Coroi. 1. And, conyers^y, if one line meeting two other 
lines, make the angles on the same side of it equal, those 
two lines are parallels* 

CoroL 2. If a line, cutting two other lines, make the sum 
of the two inward angles, on the same side, less than two 
right angles, those two lines will not be parallel^ but will 
meet each other when produced. 



G 



H 



D 



THEOREM XV, 

Tho3|L LInfs which are Parallel to the Ssune Lines tfC 

Parallel to each oth^r* 

Let the Lines ab^ cd, be each of 
them parallel to the line ef ; then shall A. '* 
the lines ab, cp, be parallel to each q^ 
other, _ 

For, let the line gi be perpendicular ^^ 
to EF. Then will this line be also per- 
pendicular to both the lines ab, cp (corol. th. 12), andcon- 
i^(|uentl^ the two lines aB| cd, are parallels (cor<^- th. 13), 

Qi E. d; 

THEOREM 



—J 



feftO GEOMETRY. 



THEOREM XVI. 




When one Side of a Triangle is produced, the Outward 
Angle is equal to both the Inward Opposite Angles taken 
together. 

Let the side AB, of the triangle 
ABC, be produced to d; then will the 
outward angle CBDbe equal to the sum 
of the two inward opposite angles a 
and c. 

For, conceive be to be drawn p;v- 
rallel to the side ac of tlie triangle. 
Then bc, meeting the two parallels AC, ^E, makes the 
alternate angles c ^d cbe equal (th. 12). And ao, 
cutting the same two parallels ac, bp, makes the inward 
and outward angles on. the same side, A and ebd, equal to 
each other (th. 14). Therefore, by ecfual additions, the 
sum of the two angles a and c, is equal to the sum of the 
two cbe and ebd, that i$| to the wl^ol^ angle CBD (by 
ax- 2). <^£. D, 

THEOREM XVII, 

In any Triangle, the sum of all the Three Angles is tqual 

to Two Right Angles. 

Let ABC be any plane triangle ; then 
the sum of the three angles a + b + c 
i^ equal to two right angles. 

For, let the ride ab Be produced to d. / ^ ^^ 

Then the outward angle cbd is equal J> !> 

to the sum of the two inward opposite 
angles a + c (th. 16), To each of these equals add the 
inward angle b, thep will the sum of the three inward 
angles a + b + c be equal to the sum of the two adjacent 
angles abc+cbd (ax. 2). But the sum of these two last 
adjacent angles is equal to two right angles (th. 6). There- 
fore also the sum of the three angles ofthe triangle a+b + c 
is equal to twp right angles (ax, 1). q. E. d. • 

Corgi. 1. If two angles in one triangle, be equal to two 
angles in another triangle, the third angles will also be'equal 
(ax. 3), and the two triangles equiangular, 

Corol. 2. If one angle in one triangle, biB equal to on^ 
angle in another, the sums of the remaining angles will also 
be equal (ax. 3), 

Cord. 





THEOREMS. «si 

V 

\ 

Coroh 3. If one angle of a triangle be right, the sum of 
the other two will also be equal to a right angle, and each 
of them singly will be acute, or less than a right angle. 

CoroL 4. The two least angles of every triangle are acutet 
or each less than a right angle. 

THEOREM XVIII. 

In any Quadrangle, the sum of all the Four Inward Angles, 

is equal to Four- Right Angles. 

Let ABCD be a quadrangle; then the 
sum of the four inward angles, a + B + 
c + D is equal to four right angles. 

Let the diagonal ac be drawn, dividing 
tl>e quadrangle into two triangle, aec, adc. 
Then, because the sum of the three angles 
of each of these triangles is equal to two 
right angles (th. 17} ; it follows, that the sum of all the 
angles of both triangles, which make up the four angles of 
the quadrangle, must be equal io four right angles (ax. 2). 

q. E. D. 

Coroh L Hence, if three of the angles be right ones, the 
fourth will also be a right angle. 

Cor^l. 2. And, if the sum of two of the four angles be 
equal to two right angles, the sum of the remaining two will 
also be equal to two right angles. 

« 

THEOREM XIX. 

In any figure whatever, the Sum of all the Inward Angles, 
taken together, is equal to Twice as many Right Angles, 
wanting four, as the Figure has Sides, 

Let .ABCD^ be any figure ; then the 
sum of all its inward angles, a + b + 
c + D + E, is equal to twice as many 
right angles, wanting four, as the figure 
has sides. 




For, from any.point p, within it, draw 
lines PA, PB, PC, &c, to all the angles, 
dividing the polygon into as many tri- 
angles as it has sides. Now the sum of the three angles of 
each of these triangles, is equal to two right angled (th. 17) ; 
^erefbre the sum of the angles of all the triangles is equal 
to twice as many right angles as the figure has sides. . But 
the sum pf all the angles about the point P, which are so 

^ ' ' many 




282 GEOMETRY. 

many of the angles of the triangles, but no part of the in- 
ward ai}«{k^s of the polygon, is equal to four right angles 
(corol. 3, :h. ^), arid niUbt be deducted out of the former 
Sum. II<'nc( ii follows that the sum of all the inward angles 
c ^' T.or^rori alone, a + b+c + d + e, is equal to twice 
as ..^y iight angles as the figiu*e has sides, wanting the 
sai«i iour right angles, q. £. D* 

THEOREM XX. 

"When every Side of any Figure is prq^uced out, the 
Sum of all the Outward Angles thereby made, is equal to 
Four Right Angles. 

Let A, B, c, &c, be the outward 
singles of any polygon, made by pro- 
ducing all the sides ; then will the sum 
A + B + c + D + Ej of all those outward 
angles, be equal to four right angles. 

For every one of thes^ outward angles, 
together with its adjacent inward angle, 
make up two right angles, as A+a equal 
to two. right angles, being the two angles 
made by one line meeting another (th* 6). And tbere 
being as many outward, or inward angles, as the figure has 
sides ; therefore the sum of all the inward and outward 
angles, is equal to twice as many right angles as the figure 
has sides. But the sum of all the inward angles, with four 
right angles, is equal to twice as many right angles as the 
figure has sides (th. 1 9). Therefore the sum of all the in<* 
ward and all the outward angles, is equal to the sum of all 
the inward angles and four right angles (by ax. 1). From 
each of these take away all the inward angles, and there 
remains all the outward angles equal to four right angles 
(by ax. 3). 

THEOREM XXI; 

A Perpendicular is the Shortest Line that can be drawn 
from a Given Point to an Indefinite Line. And, of any 
other Lines drawn from the same Point, those that are Nearest 
the Perpendicular, are Less than those More Remote. 

If AB, AC, AD^ &c, be line^ drawn from 
the given point a, to the indefinite line de^ 
of which AB is perpendicular. Then shall 
the perpendicular AB be less than ac, and AC 
less than ad, &c. ji c Jft II 

foFf ib^ ^gle B bemg a rigbt one^ the 

angle 







THEOREMS. sas 



angle c is acute (by cor. 3, th. 17), and therefore less than 
the angle B. But the less angle of a triangle is subtended 
by the less side (th. 9). Therefore the side AB is less than 
the side ac. 

Again, the angle ACB being acute, as before, the adja- 
cent angle aCd will be obtuse (by th. 6) ; consequently the 
angle d is acute (corol. 3, th, 17), and therefore is less than 
the angle c. And «ince the less side is opposite to the lesi 
angle, therefore the side ac is less than the side Ap. 

t q. E. !>• 

CoroL A perpendicular is the least distance of a givea 
point from a line. 

THEOREM XXII. 

The Opposite Sides and Angles of any Parallelogram ai^ 
equal to each other 5 and the Diagonal divides it into two 
Equal Triangles. 

Let ABCD be a parallelogram, of which 
the diagonal is bd ; then will its opposite 
sides and angles be equal to each other, 
and the diagonal bd will divide it into two 
equal parts, or triangles. 

For, since the sides ab and dc are pa- 
jrallel, as also the sides ad and Bc (defin. 
32), and the line bd, meets them ; therefore the alternate 
angles are equal (th. 12), namely, the angle abd to the angle 
CDB, and the angle adb to the angle cbd. Hence the two 
triangles, having two angles in the one equal to two angles 
in the other, have also their third angles equal (eor. 2, th. 17), 
namely, the angle A equal to the angle c, which are twp of 
the opposite angles of the parallelogram* 

Abp, if to <tbe equal angles abd, cdb, be added the 
equal angles cbd, adb, the wholes will be equal (ax. 2)^ 
hamdy, the whole angle abc to the whole adc, which 
are the other two opponte angles of the parallelogram. 

, (^ E. D, 

Again, since the two triangles are mutually equiangular ,i 
^nd have a side in each equal, viz. the common side bd 5 
therefore the two triangles are identical (th. 2), or equal in 
all respects, namely, the side ab equal to the opposite sidef 
DC, and AD equal to the opposite side BC, and the wiiole 
triangle abd equal to the whol^ triangle scd. iq. s. b. 

Carets 





284 GEOMETRY. ^ 

CoroL 1 . Hence, if one angle of a parallelogram be a right 
angle, all the other three will also be right angles, and the 
parallelogram a rectangle. 

CoroL 2. Hence also, the sum of any two adjacent angles 
of a parallelogram is equal to two right angles. 

THEOREM XXIII. 

Evert Quadrilateral, whose Opposite Sides are Equal, is a 
Parallelogram, or has its Opposite Sides F^llel. 

Let ABCD be a quadrangle, having the 
opposite sides equal, namely, the side ab 
equal to pc, and ad equal to bc ; then 
shall these equal sides be also parallel, and 
the figure a parallelogram. 

For, let the diagonal bd be drawn. 
Then, the triangles, abd, cbd, being 
mutually equilateral (by hyp.), they are 
also mutually equiangular (th. 5), or have their correspond- 
ing angles equal ; consequently the opposite side$ are paralUl 
(th. 13); viz. the side AB parallel to Dc, and AD parallel to 
BC, and the figure is a parallelogram, q. £. d. 

THEOREM XXIV. ' 

^HOSE Lines which join the Corresponding Extremes of 
two Equal and Parallel Lines, are 'themselves Equal and 
Parallel. 

Let AB, DC, be two equal and parallel lines ; then will 
, the lines ad, bc, which join their extremes, be also equal 
and parallel. [See the fig. above.] 

For, draw the diagonal bd. Then^ because ab and dc are 
parallel (by hyp.), the angle abd is equal to the alternate 
angle bdc (th. 12). Hence then, the two triangles having 
two sides and the contained angles, equal, viz. the side ab 
equal to the side DC, and the side BD common, and the con- 
tained angle abd equal to the contained angle bdc, they 
have the remaining sides and angles also respectively equal 
(th. I); consequently AD is equal to Bc, and also parallel 
to it (th. 12). (^ E. D. 

THEOREM XXV. 

Parallelograms, as also Triangles, standing on the 
Same Base, and bettyeen the Same Parallels, are equal to 
^ach other. ' . 

Let 



THEOREMS. 



UBS 




Let ABCD, ABEF, be two parallelo- 
jrams, and abc,- abf, two' triangles, 
staiiding on the same base ab, and be- 
tween the same parallels ab, de ; then 
will the parallelogram abcd be equal to 
the parallelogram abef, and the triangle 
ABC equal to the triangle abf. 

For, since the line de cuts the two ^ 

parallels af, be, and the two ad, bc, it makes the angle e 
equal to the angle aid, and the angle d equal to the angle 
BCE j(th. H) ; the two triangles, adf, bce, are therefore 
equiangular (cor. 1, th. 17); and haying the two correspond- 
ing sides, AD, BC, equal (th. 22), being opposite sides of a 
parallelogram, these two triangles are identical, or equal in 
all respects (th. ^). If each of these equal triangles then be 
taken from the whole space abed, there will remain the 
parallelogram abef ip the one case, equal to the parallel* 
ogram abcd in the other (by ax. 3). 

Also the triangles ABC, Abf, on the same base ab, and 
between the same parallels, are equal, being the halves of 
the said equal parallelograms (th. 22). (^ e. d. 

Corol. 1. Parallelogramsv or triangles, having the same 
base and altitude, ai*e equal. For the altitude is the same as 
the perpendicular or distance between the two parallels, which 
is every where equal, by the definition of parallels. 

Corol. 2., Parallelograms, or triangles, having equal bases 
^nd altitudes, are equal. For, if the one figure be applied 
with its base on the other, the bases will couicide or be the 
same> because they are equal : and so the two figures, having^ 
the same base and altitude, are equal. 

THEOREM XXVI. 

If 2^ Parallelogram and a Triangle stand on the Same 
Base, and between the Same Parallels, the Parallelogram 
will be Double the Triangle, or the Triangle Half the Pa- 
rallelogram. 

Let abcd be a parallelogram, and abe 
a triangle, on the same base ab, and between 
the same parallels ab, dej then will the pa- 
rallelogram ABCD be double the triangle 
ABE, or the triangle half the parallelogram .- 

For, draw the diagonal ac of the paral- 
lelogram, dividing it into two equal parts 
(th* 22). Then because the triangle?, abc, 

ABE, 




286 



GEOMETRY. 



ABE, on the same base, and between the same parallek, are 
equal (th. 25) ; and because the one triangle abc is half the 
parallelogram abcd (th. S2), the other equal triangle abb is 
also equal to half the same paralielogram abcd. a. £. d. 

Cor<^» 1 . A triangle is equal to half a parallelogram of the 
same base and altitude^ because the altitude is the perpendi- 
cular distance, between the parallels, which is eriery where 
equal, hj the definition of parallels. 

CoroL 2. If the base of a parallelogram be half that of a 
triangle, of the same altitude, or the basq of the triangle be 
double that of the parallelogram, the two figures will be 
equal to each other. 

THEOREM XXVII. 

Rectangles that are contained by Equal Lines, are Equal 

to each other. 




Let BD, FH, be two rectangles, having 
the sides ab, bc, equal to the sides ep, 
FG, each to each ; then will the rectangle 
BD be equal to the rectangle fh. 

For, draw the two diagonals ac, eg, 
dividing the two parallelograms each into 
two equal parts. Then the two triangles 
ABC, EFG, are equal to each other (th. 1), because they 
have the two ^ides ab, bc, and the contained angle B, 
equal to the two side's ef, fg, and the contained angle f 
(by hyp). But these equal triangles are the, halves of the 
respective rectangles. And because the halves, or the tri- 
angles, are equal, the wholes, or the rectangles db, if f» are 
also equal (by ax. 6). <^ E. D. . 

CoroL The squares on equal lines are also equal ; for 
every square is a species of rectangle. 

N ' THEOREM XXVJkl. 

The Complements of the Parallelograms, which are 
about the Diagonal of any Parallelogram, are equal to each 
other. ' 

Let AC be a parallelogram, at} a dia^ 
gonal, EiF parallel to ab or dc, and 
GiH parallel to ad or bc, making ai, 
ic complements . to the parallelograms 
EG, HF, which are about the diagonal 
DB : then will the complement Al be 
equal to the complement ic. 

For, 



I) Gr 


C 


eN '- 


/r 


>r 


A JI 


15 



THEOREMS. 



S81 



For, since the diago&al )>B bisects the thtee parallelograms 
Ac; EG, HF (th. 22); thcffcfore, ^ the whdte triangle dab 
being equal to the ^^hole triangle dcb, and the parts DEly 
IHB, respectively equal to the parts DGi, ifb^ the remaining 
parts Ai, ic, must akio be equal (by ax. S). q. E. d. 

THEOKEM XXIX. 

A TRAPEZOID) or Trapezium having two Sides Parallel, 
is equal to Half a Parallelogram, whose Base is the Sum of 
those two Sides, and its Altitude the Perpendicular Distance 
between them. 

Let ABCD be the trapezoid, having its 
two sides ab, dc, parallel; and in ab 
produced take be equal to dc, so that 
AE may be the sum of the two parallel 
sides; produce dc also,^and let ef, gc, 
BH, be all three parallel to ad. Then is 
AF a parallelogram of the same altitude with the trapezoid 
ABCD, having its base A E equal to the sum of the parallel sides 
of the trapezoid; and it is to be proved that the trapezoid 
ABCD is equal to half the parallelogram af* 

Now, since triangles, or parallelograms, of equal bases 
and altitude, are equal (corol. 2, th. 25), the parallelogram 
dg is equal to the parallelogram he, and the triangle CGB 
equal to the triangle chb ; consequently the line BC bisects, 
or equally divides, the parallelogram af, and abcd is the 
half of it. Q. E. D. 




(^ , H C 



THEOREM XXX. 

• • 

The Sum of all the Rectangles contained undar one 
Wh<^ Line, and the several Parts of another Line, any way 
divided, is Equal to the Rectangle contained undei* the Two 
Whole Lines. • 

Let AD be the one line, and AB the 
other, divided into the parts kE, ef, 
fb ; then will the rectangle contained by 
AD ahd AB, be equal to the sum of the 
rectangles of ad and ae, and ad and ef, 
and AD and fb : thus expressed, ad . ab 

*s AD . AE + AD . EF + AD . FB. 

For^ make the rectangle Ac of the two whole lines ad, 
•AB ; and draw eg, fh, perpendicular to AB, or parallel to 
AD, to which they are equal (th. 22). Then the whole 
rectangle AC is made up of all the other rectangles ag, 

^ BH, 



EjnB 



S8S 



GEOMETRY. 



S-ELC 



A — t i -j 



EH» Fcv But these rectangles afe contain- 
ed by ADand AB» eg and bf, fh and fb ; 
which are equal to the rectangles of ad 
and A^ AD and £Ff ad and FB9 because 
AD is equal to each of tlie two, bg^'fh. 
Therefore the rectangle ad • ab is equal to the sum of aU 
the ether rectangles aj> . ae, ad . bf, ad . FS. q^ e. d. 

CoroL If a right line be divided into any two parts ; the 
square on the whole line> is equal to both the rectangles of 
the whole line and each of the parts. 



I 



"n 



THEOREM XXXI. 

The Square of the Sum of two Lines, is greater than th6 
Sum of their Square s> by Twice the Rectangle of the said 
lines* Or, the Square of a whole Line, is equal to the 
Squares of its two Parts, together with Twice the Rectangle 
of those Parts. 

* Let the line ab be the sum of any two £ H p 

lines AC, CB : then will the square of ab 

be equal to the squares of AC, cB, together ^ 

with twice the rectangle ®f Ac . cb. That 

is, ab* = AC* + CB* + 2AC . CB. 

For, let ABDE be the square on the sum 
or whole line ab, and acfg the square 
on the part AC. Produce cf and gf to the other sides at H 
and I. 

From the lines en, gi, which are equal, being each 
equal to the sides of the square ab or bd (th. 22), take the 
parts CF, OF, which are also equal, being the sides of the 
square af, and there remains fh equal to fi, which are 
also equal to dh, di, being the opposite sides of a parallelo- 
gram. Hence the £gure hi is equilateral : and it has all 
its angles right ones (corol. 1, th. 22); it is therefore a 
sqtiare on the line fi, or the square of its equal CB. Also 
the figures ef, fb, are equal to two rectangles under AC , 
and CB, because gf is equal to ac, and fh or fi equal 
to CB. But the wjkole square ad is made up of the four 
figures, viz. the two squares af, fd, and the two equal rect- 
angles EF, FB. That is, the square of ab is equal to the 
squares of AC, cb, togeth^ with twice the rectangle of AC, 
CB. q. E. D. 

CoroL Hence, if a line be divided into two equal parts ; 
the square of the whole line, will be equal to four times the 
square of half the line, * . « 

THEOREM 



1?HEOR£M&N 



289 




THEOREM XXni. 

Thb Square of the Difference of tw^o lines, is less tLaii 
tke Sum of their Squares^ by Twice the Rectangle of the. 
said Lines. 

- Let Ac» Bc> be any two lines, and ab 
their diSerence : then will the square of ab 
be less than the squares of ac, bc, by 
twice the rectangle of ac and bc. Or, 

AB* s= AC* + BC* — 2AC . BC. 

For, let ABDE be the square on the di& 
ference ab, and acfg the square on the 
line AC. Produce £d to h ; also produce 
Db and Hc, and draw ki, making bi the sqiislr^ of the 
other line bc. 

Now it is visible that the square ad is less than the two 
squares af, bi, by the two f ectangles ef^ di; But gf is 
equal to the one line ac, and gE or fh is equal to the other 
line BC ; consequently the rectangle bf, contained under eg 
and GF, is equal to the rectangle of ac and bc. 

Again, fh being equal to ci or bc or dh, by adding the 
common part hc, the whole Ht will be equal to the whole 
FCy or equal to ac ^ and consequently the figure di is equal 
to the rectangle contained by Ac and bc 

Hence the two figures ef, di, are two rectangles of the 
two lines AC, bc ; and consequently the square of ab is 
less than the squares of ac, bc, by twice the rectangle 

AC . BC. Q^ E. D* 

THEOREM XX^llt. 

TttB Rectangle under the Siim and Difference of two 
Lines, is equal to the Difference of the Squares, of those 
Lines. 

Let ABj AC, be any two unequal lines ; E ly If 

then will tl>e difference of the squares of 
AB, AC, be equal to a rectangle undet 
their sum and difference. That is. 



& 



TJ 



b i> 



3 



A 



AB* — AC*?=AB + AC . AB -T AC. 

For, let ABDE be the square of ab, and 
ACFG the square of ac. Produce db 
till BH be equal to ac ; draw hi parallel 
to AB or ED, and produce Fc both ways 
to I and K. 

Then the difference of the ttliro squares AD, Af, is evi- 
VoL.L U dendy 



I 



290 CEOMETRY. 

dently the t^o rectangles ef, rb. But the rectangles XiPy^- 
Biy are equal, bei!ng contained under eq[ual lines ; lor bk and 
BH are each equal to AC, and ge is equal to cb, being each^ 
equal to the difference between ab and AC, or their equals 
AE and AG. Therefore the two bf, kb, are equal to the two 
KB, Bi, or to the whole kh ; and consequendy kh is equal 
to the difference of the squares ad, af. But kh is a rect- 
angle contained by DH, or the sum of ab and AC, and by KS^ 
or the difference of ab and AC, Therefore the difference of 
the squares of AB, AC, is equal to the rectangle under tfaeiff 
sum smd difference* q,. £. D. 

THfiORKM XXXIV. 

In any Right-angled Triangle, the Square of the Hypo* 
fhenuse, is equal to the Sum of the Squares of the other tw« 
Sides. 

Let ABC be a right<<uigled triangle, 
having the right angle c ; then will th« 
square of the hypothenuse ab, be equal 
to the sum of the squares of the other 
two sides AC, CB, Or ab* s= ac* 

+ BC*. 

For, on AB describe the square ae, 
and on ac, cb, the squares A6, bh; 
then draw CK parallel to ai> or be ; 
and join ai, bf, cd, cb. 

Now, because the line AC meets the two CG, cb, so as tq 
make two right angles, these two form one straight line gb 
(corol. 1, th. 6). And because the angle fac is equal to the 
angle dab, being each a right angle, or the angle of a square ; 
to each of these equals add the common angle bac, so will 
the whole angle or. sum fab, be equal to the whole angle or 
sum cad. But the line fa is equal to the line ac, and the 
line ab to the line ad, being sides of the same square ; so 
that the two sides fa, ab, and their included angle fab, are 
equal to the two sides ca, ad, and the contained angle cad, 
each to each ; therefore the whole triangle afb is equal to 
the whole triangle acd (th. 1). 

But the square ag is double the triangle afb, on the 
same base fa, and between the same parallels fa, gb 
(tt. 26); in like manner, the parallelogram *k is douNe 
the triangle ACD, on the same base ad, and between the 
same parallels ad, ck. And since the deubles of equal 
things, are equal (by ax. 6); therefore the square AG is equal 
t« the parallelogram *AK. 

In 




THEOREMa. 



29i 



In like manner, the other square Bk is proved equal to 
the. other parallelogram bk. Consequently the two squared 
AG and BH together, are equal to the two parallelograms ak 
and BK together, or to the whole square ae. That is, the 
sum of the two squares on the two less sides, is equal to th» 
square on the greatest side. q. e. d. 

CoroL 1 . Hence, the square of either of the two less sides. 
Is equal to the diflFerence of the squares of the hypothenuse 
and the other side. (ax. 3);. or, equal to the rectangle con- 
tained by the sum and diflFerence of the said hypothenuser 
9nd other side (th. 33). 

.CoroL 2. Hence also, if two right-aingled triangles have 
two sides of the one equal to two corresponding sides of 
the other; their third sidefs will alsd be equal, and th^ 
triangles identical; 




THEOREM XXXV. 

In any Triangle, the Difference of the Squares of the 
two slides, is Equal to the Diderence of the Squares of thc^ 
Begihems of the Base,- or of the two Lines, or Distances, 
include between th^ Extremes of the Base and the Perpen^*^ 
dicuiar. 

Let ABd b^ any triangle, having 
CD perpendicular to Ai ; then will 
the difference of the squares of ac,. 
BC, be equal to. the difference of 
the squares of AD, BD; that is^' 

Ac* — BC* = AD* — BD*. 

For^ since Ac* is equal to ad* + Of \ >t ; „.^^ 

and Bc* is equal to bD* + cD* f 
Theref. the difference between ac* and Bd*, 
is equd to the difference between ad* + cd* 

and BIT* + CD%' 
* or equal to the difference Between ad* and bd*, 
by taking away the common square cd* q. b. d«: 

Corol, The rectangle of the sum and difference of the 
two sides of any triangle, is equal to the rectangle of the 
^um and difference of the distances between the perpendK» 
culai: and the two extremes of the base, or equal to the 
rectangle- of the base and the difference or sum of the 
segments, according as ,the perpendicular falls within or 
' tirithout the triangle, 
i . F 2 - That 



r 



m GEOMETRY. 



That IS, AC + BC • AC -* BC =5 AD + BD . AD — BD 

————*< ■ -■ ■ 

Or, AC 4* BC • AC — BC = AB. AD — BD in the 2d figure. 
And AC -{- BC . AC — BC = AB. AD + BD iQ the Ist figure. 



THBOREM XXICVI. 

In any Obtuse-angled Triangle, the Square of the Side 
subtending the Obtuse Anglei is Greater than the Sum of 
the Squares of the other two Sides, by 'Twice the Rectangle 
of the Base and the Distance of the Perpendicular from the 
Obtuse Angle* \ 

Let ABC be a triangle, obtuse angled at B, and cd perpen- 
dicular tp AB ; then will the square of ac be greater than the 
squares of ab, bc, by twice the rectangle of ab, bD. That 
is, AC* = AB* + BG* + 2ab . BD. See the 1st fig. above» 
or below. * 

For, since the square df the whole line ad is equal to the 
squares of the parts ab, bd, with twice the rectangle of 
the same parts ab, bd (th. SI); if to each of these equsds 
there be added the square of cD, then the squares of ad, cd, 
will be equal to the squares of ABy BD^ CD, with twice the 
rectangle of ab, bd (by ax. 2). . 

But the squares of ad, cd, are equal to the sqnare of ac; 
and the squares of bd^ cd, equal to the square of BC (th. 34) ; 
therefore the square of ac is equal to the squares of ab, bc,. 
together with twice the reaangle of ab^ bd. (^ k. d. 



THEOREM JCXXVII. 

In any Triangle, the Square of the Side subtending an 
Acute Angle, is Less than the Sqiuares of the Base and the 
other Side, by Twice the Rectangle of the Base and the 
Distance of the Perpendicular from the Acute Angle. 

Let ABC be a triangle, having Q 'C 

the angle A acute, and cd perpen- y 

dicttlartoAB; thenwill the square y^ j 

©f BC, be less tjian the squares of jT I 

AB, AC, by twice the rectangle a W 
of AB, AD. That is, bc* = AB* 

+ AC* — 2AB . ad. 



J. *. 



fi.V, 



THEOREMS. «93 

For, in fig. 1, AC* is = bc» + ab* + 2ab . bd (th. 36). 
To each of these equals add the square of AB, ^ 
then is AB* + Ac* = bc* + 2ab* + 2ab . bd (ax. 2), 

or = bc* + 2ab . AD (th. 30). 

Q.£. D. 

Again, in fig. 2, AC* is = ad* + DC* (th. 34). 
And AB* = AD^ + db* + 2 ad . db (th. 31). 
Theret ab*+ ac* = bd* + dc* + 2ad* + 2 ad , db (ax. 2), 

^or = bc* 4- 2ad* + 2ad . db (th. 34), 
or = BC* + 2ab .ad (th. 30). Q^ E. D. 

THEOREM XXXVIII. 

In any Triangle, the Double of the Square of a Line 
drawn from the Vertex to the Middle of the Base, together 
with Double the Square of the Half Base, is Equ<il to the 
Sum of the Squares of the other Two Sides. 

Let ABC be a triangle, and cd the line c 

drawn from the vertex to the middle of 
the base ab, bisecting it into the two equal 
parts AD, DB ; then will the sum of the 

squares of AC, cb, be equal to twice the £ DlirB 

$um of the squares of CD, bd •, or ac^ -f ' ^ 

CB* = 2cD* -j- 2db*. 

For, let CE be perpendicular to the' base ab. Then, 
since (by th. 36) AC* exceeds the sum of the two squares 
ad* and CD^ (or bd* and cd*) by the double rectangle 
2aj> . D£ (or 2bd . D£) ; and since (by th. 37 )^ bc^ is less 
than the same .sum by the said double rectangle :; it is mani- 
fest that both AC* and bc* together, must be equal to that 
sjun twice taken ; the excess on the one part making up the 
ilefiect en the other, q. e. d. 

THEOREM XXXIX. 

In an Isosceles Triangle, the Square of a Line drawn 
from the Vertex to any Point in the Base, together with the 
Rectangle of the Segments of the Base, is equal tQ the 
Square of one of the Equal Sides of the Triangle. 

Let ABC be the isosceles triangle, and 
CD a line drawn from the vertex to any 
point D in the base : then will the square of 
AC^ be equ^ to the square of cb, together 
with the rectangle of ad and db. That iS| 
Ac* = cao* + AD • db. 




£§♦ 



GEOMETWt. 



For, let CE bisect the vertical ai^e ; then ^ill ft alsf 
bisect the base AB perpendicularly, making ae =r eb (cor. 1, 
th. 3). 

But, in the^triangle Acp, obtuse angled at d^ ^he squarp 
ac* is ^ CD* + AD* + 2ad . DE (th. 36), 

or 5= CD* + AD . AD 4- 2de (th. 30), 

or = CD* + AD . AE + DE, 

or = CD' + AD . BE + D^i 
or = CD* 4- AP . DB. 

Q. E. Df 





ITIEOllEM XL* 

In ahy Parallelogram, the two Diagonals Bisect each 
pther ; an^ the Sum of their Squares is equal to the Sum of 
the Squares of all the Four Sides of the Parallelogram* 

Let ABCD b^ a parallelogram, whos^ 
diagonals intersect each other in e : then 
will A^ be equal to ^c, and be to ed ; 
and the sum of the squares of ac, bd^ will 
be equal to the sum of the squares of ab, 
Bc, CD, da. That is, 

AE = jEc, and Bp == ED, 
and Aq* + BD* =;: AB* + BC* + cp* + DA*. 

For, the triangles a^^b, dec, are equiangular, because 
they have the opposite angles at e equal (th. T), and the two 
lines AC, bd, meeting 'the parallels ab, dc, make the 
angle bajb equal to the angle dce, and the angle ab]^ equal 
to the angle CD^, and th^ side ab equal to the side Dp 
(th. 22); therefore thes^ two triangles are identical, and 
have their corresponding sides equal (th. 2), viz. ae = EC, 
and BE = ED. :" ^ 

* 

' Again, sinde ac is bisected in e, the sum of the squares 
AD^ + DC* = 2a^^ + 2de* (th. 38). - 

In like manner, ab* + bc* == 2ae* + 2be* or 2de*. 
• Theref. fB-'-f bc*+cd*+pa* = 4?ap*+4de* (ax. 2). 

But, because the square of a whole line is equal to ^ 
^mes the square of half the. line (cor. tl^. 31), that 13, ac* =5 
jl^AE*, and bd* ±= 4de*. ' ' ' 

' Theref. ab* + bc? + CD* + da* s= ac* '+ Bf)« (ax. 1 ). 

' ^ ■■'■ ■ '"' ^'" ' . ■"-'■■■ ^ • (J. E.D. 



THEOREM 




THEOREMS. 391 



TpiBOUBM XU. 

If? a Line^ drawn through or from the Oetitre of a Circle, 
Bisect a Chord, it will be Perpendicular to it ; or, if it be 
Perpendicular to the Chord, it will Bisect both the Chord 
and the Arc of the Chor4. 

Let AB be any chord in a circle, and cd , 
a line drawn from the centre c to the 
chord. Then, if the chord be bisected in 
the point d, cd will be perpendicular to 

For, draw the two radii CA, cb. Then, 
the two triangles acd, bcd, having CA , 
equal to CB (def. 44), and cd common, also 
AD equal to db (by hyp.); they have ail the three sides 
of the oqe, equal to all the three sides of the other, and so* 
have their apglea also equal (th. 5). Hence then, the angle, 
ADC being equal to the angle bdc, these angles are right 
angles, and the line cd is perpendicular to ab (def. 1 1). 

Again, if CD be perpendicular to ab, then will the chord 
AB be bisected at the point d, or have ad equal to db ; and 
the arc A£b bisected in the point ^, or have ab equal eb. 

For, having drawn CA, cb, as before,. Then, in the tri- 
angle ABC, because the side cA is equal to the side cb, their 
opposite angles a and b are also equal (th. 3). Hence then, 
in the two triangles acd, bcd, the angle a is equal to the 
angle 9f and the angles at d are equal (def. 11); therefore 
their third angles are also equal (corol. I, th. 17). And 
having the side cd common, they have also the side ad 
equal to the side db (th. 2). 

Also, since the angle ace is equal to the angle bch, the 
arc AE, which measures the former (def. 57), is equal to the 
arc be, which measures the latter, since equal angles must 
have equal measures. 

CoroL Hence a line bisecting any chord at right angles, 
|Kisses through the centre of the circle. 

THEOREM XLIU 

If More than Twp Equal Lines can be drawn from any 
Point within a Ciycle to the 'Circumference, that Point wiH 
he the Centre. 



ms 



GEOMETRY. 




Let ABC be a circlei and d a point 
within it : then if any three lines^ ^9 
JOB, DC, drawn from the point d to the 
circumferenqe, be equal to e^h other^ 
the point D will be the centre. 

For, draw the chords ab, BCj which 
let be bbected in the points £, c, and 

join DE, DF. 

I Then, the two triangles, dais, dbi, 
have the side da equal to the side db by supposition, and 
the side ae equal to the side eb by hypothesis, also the side 
DE common : therefore these two triangles are identical, an4 
have the angles at e equal to each other (th. 5) ; conser 
quentiy D£ is perpendicular tp the middle of the chord ab 
(def. 11), and therefore passes through the centrie of the 
circle (corol. th. 41). 

In like manner, it may be showi) that df passes thrqugh 
the centre. Consequently the point d is the centre of the 
circle, and the three equal lines da, db, dc, are radii. 

ft; ?• ^? 




theorem xliii. 

> 

If two Circles touch one another Internally, the Centres of 
the Circles and the Point of Contact will be all in the 
Same Right Line. 

Let the two circles abc, ade, toucl^ 
one another internally in the point a ; then 
will the point A and the centres of those 
circles be all in the s^me ri&;ht line. 

For, let F be the centre of the circle 
ABC, through w^iich draw the diameter 
AFC. Then, if the centre of the other 
circle can be out of this line ac, let it be 
supposed in some other point as G ; through which draw the 
line FG cutting the two circles in b and d* 

Now, in the triangle afg, the surti of the two sides fg, 
GA, is greater than the third side af (th. 10), or greater than 
its equal radius fb. From each of these take away the 
common part fg, and the remainder ga will be greater 
than the remainder GB. But the point p being suppo^ 
the centre of the inner circle, its two radii, ga, gd, are 
equal to each other 5 consequently gd will also he greater 
than GB. But ade being the inner circle, cu is necessarily 

' less 



V » 



THEOREMS- 



3sn 



less tlian cb. So that cd is both greater and less than CB; 
whkh is absurd. Con^^uently the centre g cannot be ont 
of the line afc. a. £. d. 




THEOREM XLIV. 

J 

If two Circles Touch one another Externally, the Centres 
!of the Circles and the Point of Contact will be ^ in 
the Same Right Linew 

Let the two circles abc, ade, touch one 
another externally at the point a ; then will 
thje point of contact a and the centres of the 
two circles be all in the same right line. 

' For, let F be the centre of the circle abc, 
jhrough which draw the diameter afc, and 
produce it to the other circle at E. Then, if 
the ccntreof the other circle ade can be out 
.of the line fe, let it, if possible^ be supposed 
in some other point as G ; and draw the lines 
AG, FBDG, cutting the two circles in b and d. 

Then, in the triangle afg, the sum of the two sides af, 
AG, is greater than the third side fg (th. 10). But, f and c 
being the centres of the two circles, the two radii GA, CD^ 
are equal, as are also the two radii af, fb« Hence the s;um 
of ga, af, is equal to the sum of gd, bf ; and therefore 
this latter sum also, gd. bf, is greater than gf, which is 
absurd. Consequently me centre G cannot be out of the 
)ine ef. q. e. p. 

THEOREM XLV. 

Any Chords in a Circle, which are Equally Distant from 
the Centre, are Equal to each other ; or if they be Equal 

. to each other, they will be Equally Distant from the 
Centre. 

Let ab, cd, be any two chords at equal 
distances from this centra g; then will 
these two chords ab, cd^ be equal to each 
other. 

For, draw the two radii ga, go, and 
the two perpendiculars ge, gf, which are 
the equal distances A-om the centre G. 
Then, the two right-aneled triangles, GAE, Gcf, having 
the side ga equal the side gc^ ana the side gs equal the 

side 




49S 



GEOMETUT; 




sde GWf said the angle at x equal to ib0 
abgle at f, therefore the two trtanfi^es 
GAB, GCF, are identical (cor. 2, th. S4)« 
and have the line ae equal the line cf. 
But AB is the douUe of ae, and cd is the 
double of CF (th. 41)^ therefore a8 b 
equal to cd (by ax. 6). q. e. d. 

Again> if the chord ab ^ equal to the chord co ; then 
will their distances from the centre, G^t gf, also be equal to 
each other. 

ToVf since ab is equal cd by soppoeition, the half abi^ 
equal the half of. Also the radii ga, gc, being equal, as 
well as the right angles £ and f,. therefore the third sides 
are equal (cor. 2, tlu 34), or the distance GE equal the diis- 
tance gf. q. e. d» - 



2UEJI 




THEOREM XL VI. 

A Une Feq>endicular to the Extremity of a Radius, is a 

Tangent to the Circle. 

L'et the line adb be perpendicular to the 
radius CD of a circle ; then shall ab touch 
the circle in the point d only. 

For, from any other point £ in the line 
AB draw cfe to the centre, cutting the . 
circle in f. 

Then, because the angle D, of the triangle 
CDE, is a right angle, the angle at E is acute (th. 17, cor. 3), 
and consequently less than the angle d. But the greater 
side is always opposite to the greater angle (th. 9); there- 
fore the side ce is greater than the side cd, or greater than 
its equal cf. Hence the point e is without the circle ; aUd 
the same for every other point in the line ab. Consequently 
the whole line is without the circle, and 'meets it in the' 
point D only. 



.w- *'., . ~ 



THEORBiif 



THEOREMS. «d» 



THEOREM XLTn. - - 

I I 

When a Line is a Tangent to ia Circle, a Radius drawn to 
the Ppint of Contact is Perpendicular to the Taagent. 

Let the line ab touph the circumference of a circle at th^ 
point d; then will the radius cd be perpendicular to thi* 
tangent ab. [See the last figure.] 

' For, the line , ab being wholly without the circumference 
except at the point d, every other line, as ce drawn froni 
the centre c to the line ab, must pass out of the circle to 
arrive at this line. The line cd is therefore the shortest that 
can be drawn from the point c to the line AB, and conse- 
quently (th. 21) it is perpendicular to that line. 

CoroL Hence, conversely, a line drawn perpendicular to 
a tangent, at the point of eontact> passes through die centra 
of the circle. 



THEOREM XLVITI. 

The Angle formed by a Tangent and Chord is Measured bjr 

Half the Arc of that. Chord. 

Let ab be a tangent to a circle, and cd 
a chord drawn from the point of contact c \ 
then is the angle' bcd measured by half the 
arc CFD, and the angle acd measured by 
half the arc cgd. 

For, draw the radius ec to the point of 
contact, and the radius £F perpendicular to 
the chbrd at h. . ' 

Then, the raius ef, being perpendicular to the chord 
CD, bisects the arc cfd (th. 41)« Therefore cf is half the 
arc CFD. 

in the triangle ceh, the angle h being a right one, the 
sum of the two remainiiig angles £ and c is equal to a right 
angle (cofoL 3, th. 17), which is equal to the angle bce, 
because. the radius c£ is perpendicular to the tangent. Froia 
each of these equals take awaf the common part or angle c^ 
and there remains the angle £ equal' to the angle bcd. 
But the angle e is measured by the arc of (def. 57;, which 
Is the half of^CFD \ therefore the equal angle bcd must 
also have the same measure^ namely, half the arc cfd of 
^$he chord CD. 

Again, 




soo 



GEOMITRY. 




Again> the line gef, being perpendicular 
to tnc chord CD, bisects the «rc cg0 
(th» 41). Therefore co is half Ae arc 
CGD. Now, since the line ce^ meeting 
SG, makes the sum of the two angles at E^ 
equal to two right angles (th. 6)» and the 
fine XJX makes with ab the sum of the two 
angles at c equal to two right angles } if from these^ twa 
equal sums there be taken away the parts or angles ceH and 
mcH, which have been proved equal, there remains the angle 
CEG equal to the angle ach. But the former of these, 
C£G» being an angle at the centre, is measured by the arc 
CG (def. 57) ; consequently the equal angle acd must also 
bave the same measure CG> which is half the arc qgd of the 
chord CD. Of e« d. 

CorcL 1. The sum of two right angles U n^asured bjr 
lialf the circumference. For the two angles qcd, Acb» 
which make up two right andes, are measured by the arcs 
CF> cG, which make up halt the circumference^ fg being 
a diameter* 

CdroL 2. Hence also one right angle must have for it^ 
measure a quarter of the circunuerencej or 90 degrees. 



THEOREM XLIX. 



An Apgle at the Circumference of a Qrcle, is measured by 

Half the Arc that subtends it. 

Let bac be an angle at the circumference; 
It has for its measure, half the arc Bc. which 
subtends it. 

For, suppose the tangent »e passing 
tlirough the point of contact A. Then, the 
angle dag being measured by half the arc 
ABC, and the angle dab by half the arc ab 
(th. 48); it follows, by equal subtraction, that the difference^ 
or angle bag, must be measured by half the arc bc, which 
it stands upon. q. £• p. x. 




THEOREM 



t. 




THEOREMS, 901 



THEOREM t- 

AU Angles in the Same Segment of a Circle^ or Standing on 
' the Same Arc, are Equal to each othen 

Let c and d be two angles in the same 
segment acdb, or, which is the same thing, 
standing on the supplemental arc aeb ; then 
win the angle c be equal to the angle d. 

For each of these angles is measured by 
lialf the arc aeb ; and thus, having equal 
measures^ they are equal to each other (ax. 11). 

* 

THEOREM U. 

An Angle at the Centre of a Orcle is Double the Angle at 
the Circumference, when both stand on the Same An:. 

Let c be an angle at the centre c, and 
D an angle at the circumference, both stand- 
ing on the same arc or same chord ab: then 
will the angle c be double of the angle D| or 
the angle d equal to half the angle c. 

For, the angle at the centre c is measured 
by the whole arc aeb (def. 57), and the angle atthecircum* 
ference d is measured by half the same arc aeb (th. 49) % 
therefore the angle D is only half the angle c, or the angle 
c double the angle o. 

THEOREM LU. 

f An Angle in a Semicircle, i;^ a Right Angle. 

If ABC or adc be a Semicircle ; then 
any angle D in that semicircle, is a right 
angle. 

For, the angle n, at the circumference, 
is measured by half the arc abc (th. 49), 
that is, by a quadrant of the circumference. 
But a quadrant is the measure of a right 
angle (corol. 4, th. 6 ; or corol. 2, th. 48). Therefinre the 
angle d is a right angle* 

THEOREM 






aos GEOMETRT. 



THEO&SM LIII. 

The Angle formed bjr a Tangent to a Circle, and a Chord 
drawn fsom the Point of Contact, is Equal to the Anglt , 
in the Alternate Segment* 

If AB be a tangent, and AC a chord, dnd 
]> any angle in the alternate segment adc ; 
then will the angle D be equal to the angle 
BAG made by the tangent and chord of the 

lore AEC. 

For the angle d, at the circumference, 
is measured by half the arc aec (th. 49) ; 
and the angle bag, made by the tangent and chord, is also 
measured by the same half arc Asc (th. 48) ; therefore these 
two angles are equal (ax. 11)* 

THEOREM tlV. 

The Sum of any Two Opposite Angles of a Quadrangle 
Inscribed in a Circle, is Equal to .Two Right Angles. 

iiET ABCD be any quadtilaterat inscribed 
in a circle ; then shall the sum of the two 
opposite angles a and c, or b and i>, be 
equal to two right angles. 

For the angle A is meascrf ed by half the 
;rrc dcb, which it stands on, and the angle 
t by half the arc dab (th^ 49) ; therefore 
Ae sum of the two angles a and c is measured by half the 
turn of these two arcs, that is, by half the circumference. 
But half the circumference is the measure of two right 
angles (corol. 4, th. 6) ; therefore the sum of the two oppo- 
site angles a and c is equal to two right angles. In like 
inanner it is shown, that the sum of the other two opposite 
angles, d tnd b, is equal to two right anglesw <^ £• b. 

THEOREM LV. 

If any Side of a Quadrangle, Inscribed in a Circle, be 
Produced out, the Outward Angle will be Equal to the 
Inward Opposite Angle; 

If the side Ab, of the quadrilateral 
AECD, inscribed in a circle, be produced 
to £ ; the outward angle dae will be equal . 
to the inward opposite angle c* 

For, 




\ / 





•THEOREMS. .463 

r For, the sum of the two adjacent angles DAfi ancToAB is 
equal to two right angles (th, 6) ; and the sum of the two 
opposite angles c and "bkh is also equal to two right angles 
(th. 54) ; therefore the former sum, of the two angles dae 
and DAB, is equal to the latter sum, of the two c and dab 
(ax. l). From each of these equals taking away the com- 
mon angle dab, there remains the angle dae equal the 
angle c. q. £. p. 

TttEOREM LVI* 

Any Two Parallel Chords Intercept Equal ArcsSi 

LfiT the two chords ab, c&, be parallel : 
then will the arcs ac, bi>, be equal; or 
AC = bo. 

■ ^ 

For, draw the line bC« Then, becaiise 
the lines ab, cd, are paraUel, the alternate 
angles b and c are equal (th. 12). But the 
angle at the circumference B, is measured by half the ar« 
AC (th. 49) ; and the other equal angle at the circumference 
c is measured by half tiie arc bd : therefore the halves of thef 
arcs AC, bd, and consequently the arcs themselves^ are alsa 
equal. Q. £. D« 



*rME0REM Ltir. 

"When % Tangent and Chord are PaJr^llel to each odieri thejr 

Intercept Equal Arcs. 

Let the tangent abc be parallel to the 
chord DF* ; then are the arcs bd, bf, equal y 
that is, bd = BF. 

For, draw the chprd bd. Then, be- 
cause the lines ab, df, are parallel, the al- 
ternate angles D and b are equal (th. 12). 
But the angle b, formed by a tangent and chord, is measured 
■b^ half the arc BD (th. 48) ; and the other angle at the cir- 
cumference D is measured by half the arc bf (tn. 49); there- 
for^ the arcs bd, bf, are equali . (^ B. d. ' 




THEOHEM 



5M GEOMETRY. 



THEOREM LVIII. 




The Ancle formed, Within a Circle, by the Intersection of 
two Chords, is Measured by Half the Sum of the Two 
Intercepted Arcs* 

Let the two chords ab, cd, intersect at 
the point e: then the angle aec, or deb, is 
measured by half the sum of two arcs ac. 

For,, draw the chord af parallel to cd. 
Then, because the lines af, cd, are parallel, 
and AB cuts them> the angles on the same 
aide a andi\D£B are equal (th. 14). But the angle at the 
circumference A is measured by half the arc bf, or of the 
sum of FD and db (th. 49) ; therefore the angle £ is also 
measured by half the sum of fd and db* 

Ag2un, because the chords af, cd, are parallel, the arcs ac^ 
FD, are equal (th. 56) i therefore the sum of the two arcs ac, 
DB» is equal to the sum of the two fd, db ; and consequently 
the angle e, which is measured by half the latter ^nm, is also 
measured by half the former, q. £. d. 



THEOREM LIS. 

The Angle formed. Without a Circle, by two Secants, h 
Measured by Half the Difference of the Intercepted 
Arcs* 

Let the angle X be formed by two se-> 
cants EAB and ecd; this angle is measured 
by half the difference of the two ztcs 
AC, DB, intercepted by the two secants. 

Draw the chord af parallel to CD. Then, 
beca^ise the lines af, cd, are parallel, and 
AB cuts them, the angles on the same side a 
and BED are equal (th. 14). But the ^igle A, at the circum* 
ference, is measured by half the arc bf (th* 49), or of the 
difference of Df and DB : thenefore the equal angle £ if 
also meas.ured by half the difference of df, DB. 

Again, because the chords af, cd, are parallel, thf arcs 
AC> FD, are equal (th* 56) ; therefore the difib^nce of the 

two 




TttEOREMS. 



soi 



ttro arcs At, DB, IS equal to the. difference of the twopp, 
DB. Consequently the angle E, which is measured by half 
the latter difference;, is also measured by half the former. 

q. £. 01 



THEOUBM LX. 



Th^ Angle formed by Two Tangents, is Measured by Hilf 
the Difference of the two Intercepted Arcs. 

Let £B^ £b^ be two tangents to a circle 
at the points a, c; then the angle £ is 
measured by half the difference of the two 

arcs CFA, CQA* 

r 

For, draw the chord af parallel to ed; 
Then, because the linies af, ed, are pa- 
rallel, iand eb meets them, the angles on 
the same side a and £ are equal (th. 14'). 
But the angle A, formed by the chord af ahd tangent AB, 
is measured by half the arc AF (th. 48) j therefore the* equal 
angle E is also measured by half the same arc Af, or half the 
difference of the ards cfa and cf, or cga (th. 57). 




CoroL In like manner it is proved, that 
the angle E, formed by a tangent ec», 
and a secant eab, is measured by half 
the difference of the two intercepted arcis 
t A and CFB, ' 




J) F 



Theorem lxu 

When two Lines, meeting a Circle each in two Points, Cut 
one another, either Within it or Without it ; the Rect- 
angle of the Parts of the one, is Equal to the Rectangle of 
the Parts of the other ; th^ Parts of each being measured 
from the point of meeting to the two intersections with ^ 
the circumference. 



Vol. L 



X 



Let 



X)^ 



GEOMfeTRt. 




Let* the two lines ab, ^d, meet each 
other in E^ then the rectangle of ae, eb, 
will be equal to the rectangle of CE, EP. 

Qr^ A£ • EB = C£ . £0. 

FoTf through the point £ draw the dia- 
meter FG ; also, from the centre h draw 
ihe radius dh« and drai;^ hi perpendi- 
cuhr to CO. 

Then, since deh is a triangle, and the 
perp. HI bisects the chord cd (th. 41), the 
line CE is equal to the difference of the 
segments di, ei, the sum of them being 
DE. Also, because h is the centre of the 
circle,' and the radii dh, fh, gh, are all equal, the line Bc; 
is equal to the sum of the sides dh, he ; and ef is equal to 
their difference. 

But the rectangle of the sum and difference of the two 
tildes of a triangle, is equal to the rectangle of the sum and 
difference of the segments of the base (th. 35) ; therefore 
the rectangle of fe, eg, is equal to the rectangle of cE, ED. 
In like manner it is proved, that the same rectangle of fe, 
XG, is equal to the rectangle of ae, eb. Consequently the 
rectangle of ae, eb, is also equal to the rectangle of CE| £|> 
(ax. 1). (^ E. D. 

Corol. 1. When one of the lines in the 
second case, as de, by revolving about the 
point E, comes into the position of the tan- 
gent EC or ED, the two points c and D 
nmning into one^ then the. rectangle of cE, 
ED, becomes the square of ce, because cb 
and DE are then equal. Consequently the 
rectangle of the parts of the secant, ae . eb, 
is equal to the square of the tangent, €e\ 

Carol. 2. Hence both the tangents ec, ef, drawn from 
the saine point £, are equal ; since the square of each is equal 
to the same rectangle or quantity ae • eb* 




THEOREM LXU. 



ta Equiangular Triangles, the Rectangles of the Correspoiid- 
ing or Like Sides^ taken aljiernately, are Equal. 



Lfit 




THEOREMS: S07 

, L^T ABCf DEF, be two equiangular 
triangles, having the angle a = the 
angle d, the angle b •= the angle e, 
afld the angle c =^ the anglp f ; also 
the like sides ab, de, and ac^ df^ 
being those opposite the equal angles: 
then will the rectangle of ab, df, be 
equal to the rectangle of Ac, de. 

In BA produced take AG equal to df ; and thrdugi the 
three points b, c, g, conceive a circle bcgh to be described^ 
ftieeting CA produced at H, and join Gif . 

Then the angle G is equal to the angle C on the same arc 
BH, and the angle h equal to the angle b on the^same arc cg 
(th. 50) ; also the opposite angles at a are equal (th. 7) : 
therefore the trlanglfe agiI is equiangular to the triangle 
acb, and consequently to the triangle dfe alsoi But the 
' two like sides AG, df, are also equal by supposition; conse- 
quently the two triangles agh, dfE, are identical (th. 2), 
having the two sides AG, AH, equal to the two df, db, each 
to each. • 

But the rectangle ga . AB is equal to the rectangle 
HA . AC (th. 61): consequently the rectangle d^ * ab is 
equal the rectangle de . AC q. e. d. . 



THEOREM LXIII. 

iThe Rectangle of the^two Sides of any Triangle, is Ecjual to 
the ReCtangl^ of the Perpendicular on the third Side and 
the Diameter of the Circumscribing^ Circle. . 

Let CD be the perpendicular, and cU 
the diameter of the circle about the triangle 
ABC 5 then the, rectangle CA . cb is ±s the 
rectangle cd . cB. 

For, join BE : theii in the two triangles 
_ACD, ECB, the angled A and E are equal, 
standing on the same arclBC (th. 50) j also the right angle D 
is equal the angle b, which is also a right angle, being in 
a semicircle (th. 52) : therefore these two triangles have also 
their third angles equal, and are equiangular. Hence, ac, 
CE, and CD, cb, being like sides, subtending the equal angles,' 
the rectangle AC . cb, of the first and last of them, is equal to 
the rectangle ce . cd, of the other two (th. 62). 

X 2 THEOREM 




N 




io# GEOMETRY. 



tHEOKEM LXIV. 

The Scpare of a line bisecting any Angle of a Tnangfe^ 
together with the Rectangle of the two Segments of the 
opposite Side, is Equal to the Rectangle of dbe two other 
Sides including the bisected Angle. 

LsT CD bisect the angle, c of the triangle 
IBc J then the square cd* + the* rectangle 
AD . DB is 2=: the rectangle Ac . cB. 

For» let CD be produced to meet the cir- 
cumscribing circle at e, and join ae. 

Then the two triangles ace, bcd, are 
equiangular : for the angles at c are equal 
by suppositioDf and the angles B and E are equal, standing 
on the same arc AC (th. 50) v consequently the third angles 
at A and d are eqvtal (coro). I, th. 17): also ac, cd, and 
GE, CB, are like or corresponding sides,, being opposite to- 
equal angles : therefore the rectangle ac . Cb is = the 
rectangle cd • ge (th. 62). But the latter rectangle cef. ce 
is = cr^ + the rectangle cd . de (th. 30) ; therefore also 
the former rectangle AC • cb is also = cd* -|- cd. • db, or 
equal to CD*^ + AD . db,. since C2> . de is = ad . db (th. 61)« 

q. E« D. 
THBOREM LXV. 

The Rectangle of the two Diagonals of gny Quadrangle 
Inscribed in a Circle, is equal to the sum of the two Rect- 
angles of the Opposite Sicbss* 

Let abcp be any quadrilateral inscribed 
in a circle, and ag, bd, its two diagonals : 
then the rec^tangle AC . bd is = the rect- 
angle ab . DC + Jthe rectangle ad . bc- 

For, let CE be drawn, making the angle 
BCE equal to the angle dca. Then thetwa 
triangles acd, bce, are equiangular ; for the angles A and 
b are equal, standing on thesame.arc dc; and theangles* 
dca, bcb, are equal by supposition ; consequently the third 
angles adc, B£c,,are also equal t also, Ac, bc, and ad^ be> 
are like or corresponding sides, being opposite to the equal 
angles : therefore the rectangle AC . BE is =s^ the rectangle I 

A.D . bc (th.. 62)» 

Again, 




THEOREMS. 3<» 

Again, the two triangles abc, D£C| are eqniangidar : for 
tlie angles bag, bdc, -are equal, standingon the same arc bc; 
and the angle dce is equal to the angle fiCA, by adding the 
common tingle ace to the two «qua:l angles dca, bc£ ; there- 
fore the third angles £ and abc are also equal : but AC, Dc> 
and AB, DE, are the like sides : therefore the cectangle AC • 
"DE is = the rectar^le ab . dc (th. 62), 
-Hence, by equal additions, the sum of the rectangles 

AC • BE + AC . DE is = AD . SO -|- AB . DC. Bllt the 

•ibrmer «um of tb^ rectangles AC . be + ac . 0E is = the 
rectangle AC . bb (tb« 30): therefore the same rectangle 
AC . bd is equal tp the latter suro^ the rect. ad . Bc + the 
rect. AB . DC (ax. I). <^ e. n. 






OF RATIOS AND PROPORTIONS, 

DETlNiTIONS. 

' Def. 76. Ratio is the proportion or relation which one 
magnitude bears to another magnitude of the same kind, 
with respect to quantity. 

JVi^. The measure> or quantity, of a ratio, is conceived, 
by considering what part or parts the leading quantity, called 
the Antecedent, is of the other, called the Consequent ; or 
what part or parts the number expressing the quantity of the 
•former, is of the number denoting in like manner the latter. 
So, the ratio of a quantity expressed by the number 2, to' a 
like quantity expressed by the number 6, is denoted by -6 
divided by 2, or | qr 3 : the number 2 being 3 times con- 
tained in ^, or tlie third part of it. In like manner, the ratio 
of the quantity 3 to 6, is measured by ^ or 2 ; the ratio of 
.4 to 6 is ^ or 1|.; that of 6 to 4 is f or |; &c. 

77. Proportion is an equality of ratios. Thus, 

78. Three quantities are said to be Proportional, when the 
ratio of the first to the second is equal to the ratio of the 
second to the third. As of the three quantities A (2), b (4), 
c (8), where 4 = J. = 2, both the same ratio. 

19. Four quantities are said to be Proportional, when the 
ratio of the first to the sec6nd,*is the same as the ratio of the 
third to the fourth. As of the four, a (2), b (4), c (5), D (10), 
•where i- = V* = ^> ^^^ ^^® ^^"^^ ratio. 



«ie GEOMETRY. 

■ 

Noie. To denote that four quantities, a, b> c, d^ are pro* 

portional, thej are usually stated or placed thus, a : b : : c : Q; 

^nd read thus, a is to B as c is to d. But vhen three 

€|uantities are proportional, the middle one is repeated, and 

they are written thus, a : B : : fi : c. 

80. Of three proportional quantities, this middle one is 
said to he a Mean Proportional between the other two j ^d 
.the last, a Third Proportional to the first Jtnd second. 

81. Of four proportional quantities, the last is said to be 
a Fourth Proportional to the other three, taken in orfier. 

' ' ' ' . • . ' . 

82. Quantities are said to be Continually Proportional, or 

in Continued Proportion, when the ratio is the same bietween 
every two adjacent terms, viz. when the first is to the second, 
a^ the second to the third, as the thij-d to the fourth, as the 
fourth to the fifth, and so on, all in the same common ratio* 
As in the quantities 1, 2, 4, 8, 16, &c; where the com« 
mon ratio is equal to 2. - ' -' - 

83. Of any number of quantities, A, b, c, d, the ratio of 
the first A, to the last D, is said to be Compounded of the 
ratios of the firs^ to the second, of the second to the third, 
and so on to the last. . * 

, 8'k Inverse ratio is, when the antecedent is made the 
consequent, and the consequent the antecedent. — ^Thus, if 

1 : 2 : : 3 : 6 5 then inversely, 2 : 1 : : (5:3. 

> ■ - * • 

SS, Alterrjate proportion is, when antecedent is compared 
with antecedent, and '^consequent with consequent. — Asy if 
1 : 2 : : 3 : 65 then>.by alternation, or permutation, it will be 

1 :3 ;;2 ;6. ' *. 

» 

S(), Compounded ratio is, when the sura of the antecedent 
and consequent is conipared, either with the consequent, or 
with the antecedent.— Thus, if 1 : 2 ! : 3 : 6, then by compo- 
sition, 1 + 2 : 1 : : 3 + 6 : 3, and J + 2 : 2 ; : 3 + 6 : 6. 

87. Divided ratio, is when the'difference of the antecedent 
and consequent is compared, eikher with the antecedent or 
with the consequent. — Thus, if 1 : 2 : : 3 : 6, then, by division, 
2-^1 : 1 :: 6-3 : 3, and 2— 1 : 2:: 6-3 : 6. 

^ ■ I « I 

Note. . The term Divided, or Division, Here means sub- 
tracting, or parting; being used in the sense opposed to^com- 
pounding, or adding, in def. 86. 



TH£OR|SM 



THEORIJMS, a>I 



THEOREM LXVI. 



Equimultiples of any two Quantities have the same Ratio as 

the Quantities themselves,. 

Let a and b be any two quantities, and f»A, wb, any 
equimultiples of them*, m being any number whatever : then 
will mA and mB h^ve the same ratio as a and B;, or 
A : B : : mA : mB. 



ror — ^ = •— , the sa:me ratio. 



Corol. Hence, like parts of quantities have the same ratio 
as the wholes ; because the wholes are equimultiples of th« 
like p9rt5i or A and i ax^e like parts of i»a and tn^* 



THEORBM LXVJI. 

If Four Quantities, of the Same Kind; be Proportionals^ 
thf y will be in Proportion by Alternation or Permutation, 
or th^e Antecedents will have the Same Ratio as the Con^ 

' sequents. 

Let a : b : : f»A : mB ; then will a '" mWi^ x mB, 
For 5;: |», an^ — • ^ niy both the same ratio, 

THEOREM LXVnx. 

If Four Quantities be Proportional ; they ^ill be in Pro* 
portion by Inversion, or Inversely, 

Let a : b : : mA : ms ^ then will B : a : : /»B : mA^ 

mtmm A A 

For — = - — , both the same ratio. 
;;;b B 



theorem lxix. 

Jf Four Quantities be Proportional ; they will be , in Pro* 
portion by Composition and Division* 

Let a : b :: mA : mB'f 

Then will B ± A : A : : /wB ± iwA : mA, 

and B ± A : b : : /wb ± mA : i»b. 

_ mA A mB B 

ffor, _, , — = 2"-; — 9 and 



m^±mA, » ± a' mB±mA » ;t A 



318 GEOMETRY. 

Coroi. It appears from hence, that the Sum of the Greatest 
and Least of four proportional quantities, of the same kind, 
exceeds the Sum of the Two Means. For, since — - - 
A : A + B : : ffifA : niA + nm% where A is tiie lea^t, and 

mK + w» the greatest ; then f« + 1 • A + mB, the sum of 

the greatest and least, exceeds « + i . A + P the sum o^ 
the two means. 

THEOREM I-XX. 

If, of Four Proportional Quantities, thiire be taken mj 
Equimultiples whatever of the two Antecedents, and any 
Eqoimnltiples. whatever of the two Consequents ; the 
'quantities resulting will still be proportional. 

Let A : b : : ffiA : /«b ; also, let px antl pmi^ be any 
equimultiples of the two antecedents, and ^B and qm^ any 
equimultiples of the two - consequent* \ then will - r - - - 
^A : ^B : : pmA. : qm'B, ' 

for - — = ^^, both the jame ratio. 
pmk pA 

THEOREM LXXI. 

If there be Four Proportional Quantities, and the iwfl 
Consequents be ,eifher Augmented, or Diminished by 
Quantities that have the Same Ratio as the respective 
Antecedents ; the Results and the Antecedents will still 
be Proportionals. 

Let A : b : : mA : /tie, and tiA and nmA any two quan-: 
pities having the sai; e ratio as the two antecedents j then will 
A : B ± «A ; : fwA : mB ± nmA. 

wB ± nmk B ± «A ^ , , 

ror = , both the same ratio. 

mA K ^ 



THEOREM LXXIJ. 

If any Number of Quantities be. Proportional, then any 
one of the Antecedents will be to its Consequent, as the 
Sum of all the Antecedents is to the Sum of all the Con- 
sequents. 

Let a \ b :i tnA ', ntfi \\ nA '. wB, &c ; then will * - - ^ 
A : B : : A + wA 4" ^A : : B + ^B + //B, &c. 

_ 5 + WiB + «B B _ 

J or -— =c •— , the same ratio, 

A + IWA + /7A 4 

theorem 



THEOREMS. 4 It 



THEOKEU IXXIir. 

|f a Whole Magnitude be to ?i Whole, as a Part taken from 
tbe first, is to a Part taken from the other ; then th^ Re- 
mainder wijl be to the Remainder, as th^ whpl^ to the 
whole. 

JLet a : 3 : : — A : — B ; 

n . n 

tjpten will a:b::a -a:b B^ 

n n 

B-^B B 

For •■ ; ■ - ^ ■ ' ' = — , both the same ratia^ 

• • A-rr-^ A A ' 

» - 



THEOJIEM J.iXIV. 

If any Quantities be Proportional ; ^heir Squares, or Cubes^ 
or any Like Powers, or Roots, of them, will also be Pro- 
portional. 

Let a : b : : »a : ms ; then will a* : B° : : iw^a" : jw^b**. 
For --—-; = — -, both the same ratio. 

«j"A° a' ■ 



THEOREM LXXV. * 

^ there be tvo Sets of Proportionals 5 then the Products 01? 
Rectangles of the Corresponding Terms will also be Pro- 
portional. 

JjET a 2 b : : ota : fwBj 

and p : D : : «c : «D J 

^hen will AC : bd : : mnKC : mftBD^ 

rfmtBT> bd , - , ' 
or ' = — , both the same ratio. 

mtJAC AC 



THEaHEM LXXVI. 

Jtf Refer Quamtities be Proportional; the Rectangle or Product 
of the two Extremes, will be Equal to the Rectangle or' 
Product of the two Means. And the converse. 

X»^r A : 8 : : mA : mB ; 

then & A X nmszu X mAss foxB. as is evident. 

theor;^ 



«M 



GEOMETRY. 



THEOREM .LXXVIT. 

If Three Quantities be Continued Proportionals ; the Rect- 
angle or Product of the two Extremes, will be Equal to 
the Square of the Mean. And the conveise. 

Let a, mA, w*A be three proportionals, 
or A : mk : : wa : m^K ; 
' then b A X /»*A == «i'A% as is evident. 



THROREM LXXVIII. 

If any Number of Quantities be Continued Proportionals \ 
the Ratio of the First to the Third, will be duphcate or the 
Square of the Ratio of the First and Second; and the Ratia 
of the First and Fourth will be triplicate or the cube of 
that of the First and Second ; and so on. 

. Let a, mA, f»^A, im'a, &c, be proportionals $ 



. OTA 

then IS = iM , 



nP'A 



m' 



= iM ; but =; «*; and =5 



s 



&c 




TBEOREU LXXIX. 

Triangles, and also Parallelograms, having equal Altitude^ 

are to each other as their Bases. 

Let the two triangles aoc, def, have 
&e same altitude, or be between the same 
parallels ae, cf ; then is the surface of 
the triangle adc, to the surface of the 
triangle def, as the base ad is to the 
base DS. Or, ad : de : ; the triangle 
ADC, : the triangle def. 

For, let the base ad be to thp base de, as any one nupa- 
ber m (2), to any other number » (3) ; and divide the respec- 
tive bases into those parts, ab, « bd, dg, gh, he, all 
equal to one another ; and from the pcrints of division draw 
the lines bc, fg, f», to the vertices c and f. Then will 
these lines divide the triangles adc, def, into the. same 
number of parts as their bases, each- equal to the triangle 
ABC, because those triangular parts have equal bases and 
altitude (corol. 2, th. 25) ; namely, the triangle abc, equal 
to each of th^e triangles bdc, dfg, gfh, hfb« Sq that 
the triangle -adC| is 10 the triangliei qfE| aithe nuimber of 



THEOREMS. 



S15 



parts « (2) of the former, to the number n (3) of the latter, 
chat is, as the base ad to the bas^ djk (cl6f. 79). 

^ In like manner, the parallelogram adki is to the parallel- 
pgram defk, as the base ad is to the base de; each of 
these having the same r;itio as the number of their partSj 
m to «. Qi ;e. p. 







THEOREM LXXX, 

Triangles, and also Parallelograms, having Equal Bases, 

to each other as their Altitudes, 

> 

Let ABC, BEF, be two triangles 
having the equal bases ab, be, and 
whose altitudes are the perpendiculars 
CG, FPl \ then will the triangle ABC : 
the triangle bef : : cg : fk. 

' -For, lei BK be perpendicular to AB, 
and equal to €G; in which let there 
be taken Bx. s? fh ;- drawing Ak and AL« 

» 

Then, triangles of equal bases and heights being equal 
(corol. 2, th. 25), the triangle abk is = ABC, ind the tri* 
atigle ABL == B^F. But, considering now abk, abl, as two 
triangles on the bases bk, bl, and having the same altitude 
AB, these will be as their bases (th, 19), namely, the triangle 
ABK: the triangle abl : : bk : bl. 

But the triangle abk = abc, and the trialigle abl = bef« 

also BK = CG, and bl = fh. 
Theref. the triangle abc : triangle bef :: cg : fh. 

« 

And since parallelograms are the doubles of these triangles, 
having the same bases and altitudes, they will likewise have 
to each other the same ratio as their altitudes, q. £. D. 

Coral. Since, by this theorem, triangles and parallelo^ams, 
^hen their bases are equal, are to each other as thew alti- 
tudes 5 and by the foregoing one, when their altitudes are 
equal, they are to each other as their bases ; therefore uni- 
versally, when neither are equal, they are to each other in 
the compound ratio, or as the rectangle or product of their 
}>ases and altitudes. 



THX0S.EU 



31$ GEOMETRY. 



THEOEEU LXXXI. 

If Four Lines be Proportional ; the Rectangle of the Ex- 
tremes will be Equal to the Rectangle of the Means. 
And) conversely, if the Rectangle of the Extremes, of four 
Lines, be equal to the Rectangle of the Means, the Four 
Lines, taken alternately^ will be FfoportionaL 

Let the four hues a, b, c, d, be 

^oportionals, or A : b : : c : d; a,— 

then will the rectangle of a and d be i^ 
fequal to the rectangle of b and c j j^' 

or the rectangle A ^ d = 9 • c 







A 


1 f U 


*^ i 



For, let the four lines be placed 
with their four extremities meeting 
in a common point, forming at that 

point four right angles ; and draw lines parallel tb them to 
complete the rectangles p, q, r, where p is the rectangle 
of A and D, Q^the rectangle of b and .c> and % the rect* 
angle of b and d. 

Then the rectangles p and R| being betFe^n the same 
p3trallels> are to each other as their bases a and B (th. 7:9/; 
and the rectangles q and r, being between the same pa- 
rallels, are to each other as their bases c and D* 3ut the 
ratio of a to b^ is the same as the ratia of c to p, by hypo* 
thesis 'j therefore the ratio of p to r, is the same as the ratio 
«of <^ to R 5 and consequently the rectangles P and q are 
tqual. Q. B. D. 

Again, if the rectangle of a and d, be equal to the 
rectangle of b and c; these Imes will be proportional^ qr 
A : b : : c : o. 

For, the rectangles being placed the sjime as before : then, 
because parallelograms between the same parallels, are to one 
another as their bases, the rectangle P : g : : a : Bi anil 
Q^ : R :: c : D. But as p and q^are equal, by supposition, 
they have the same ratio to r, that is, the ratio of a to MP 
equal to the ratio of c to d, or a : b :: c : d, q^ e. d. 

Corol, I. When the two means, namely, the second and 
third terms, are equal, their rectangle becomes a square of 
the second term, which supplies the place of both the second 
and third. And hence it follows, that when three lines arQ 
proportionals, the rectangle of the two extremes is equal ta 






a ' 



THEOREMS. z\1 

the square of the mean ; and, conversely, if the rectangle of 
the extremes be equal to the sq^uare of the meaa, the three 
lines are proporti(»ials. 

CoroL 2. Since it appears, by the rules of proportion lit 
Arithmetic and Algebra, that when four quantities are pr«i- 
portionai, the prockict of the extremes is equal to the product 
of the two means ; and, by this theorem^the rectangle of the 
extremes is ^qual to the rectangle of the two mean^ » it fol- 
lows, that the area or ^pace of a rectangle is represented 03f 
expressed by the product of its length and breadth multiplied 
together. And, in general, a rectangle in geortietry is simi- 
lar to the product of the measures of its two dimensions of 
length and breadth, or base and height. Also, a square, it 
similar to, or represented by, the measure of its side multi- 
plied by itself. So that, what is shown of such products, is 
to be understood of the sqtiares and rectangles. 

CoroL 3. Since the same reasoning, as in this theorem, 
holds for any parallelograms whatever, as well as for the 
rectangles, the same property belongs to all kinds of paral- 
lelograms, having equal angles, and also to triangles, which 
are the halves of parallelograms ; namely, that if the sides 
about the equal, angles of parallelograms, or triangles, be 
reciprocally proportional, the parallelograms or triangles 
will be equal ; and, conversely, if the parallelograins or 
triangles be equal, their sides about the equal angles will be 
reciprocally proportional ' 

Corol. 4. Parallelograms, or triangles, having an angle in 
each equal, are in proportion to each other as the rectangles 
of the sides which are about these equal angles. 



THEOREM LXXXII. 

If a Line be drawn in a Triangle Parallel to one of its 
sides, it will cut the two other Sides Proportionally. 

Let de be parallel to the side bc of the 
triangle abc ; then will ad ;db : : ae : ec. 

For, draw be and CD. Then the tri- 
angles dbe, dce, are equal to each other, 
because they have the same base de, and 
are between the same parallels j^e, bc 
(th. 25). But the two triangles ade, bde, 
on the bases adi, i>h, ha?e the saotie alti- 

tudei 




) 



J 



SIS 



GEOMETkt- 




tude; and the two triangtes ade, cde, 
on the bases ae, ec, have also the! same 
altitude; and because triangles of the same 
altitude are to each other as their bases> 
therefore 

the triangle Ade : bdk : : ad : db, 

and triangle adr : cde : : ae : £c. 

But BDE is ts CDE } and equals must ha^e to equals the 
same ratio; therefore ad : db ;: ae : £c* q. e. d* 

CS^ol. Hence, also, the whole lines ab, Xc, are propor- 
tional to their corresponding proportional segments (corol< 



VIZ* AB : AC 

and AB : ac 



• * 






ad 
bd 



AE, 

C£« 




THEOREM hXXXllU 

A Line which Bisects any Angle of a Triangle, divides the 
opposite Side into Two Segments, which are Propottional 
to the two other Adjacent Sides. 

LtT the angle acb, of the triangle abc, ^ 

be bisected by the line cd, making the 
angle r equal to the angle / : then will the 
segment ad be to the segment db, as the 
,iide AC is to the side cb^ Or, - - - - 
AD : db : : AC : CB. 

For, let BE be parallel to cd, meeting 
AC produced at £• Then, because the line bc cuts the two 
|mrallels cd, be, it makes the angle cbe equal to the alter-*^ 
Hate angle / (th. 12), and thereifore also equal to the angle 
r, which is equal to / by the supposition. Again, because 
the line ae cuts the two parallels dc, be, it makes the 
angle E equal to the angle r on the same side of it (th. 14). 
Hence, in the triangle bce, the angles b and e, being each 
equal to the angle r, are equal to each other, and conse- 
quently their opposite sides cb, ce, are also equal (th, 3). 

But now, in the triangle abe, the line cd, being drawn 
par^lel to the side be, cuts the two other sides AB, A £, pro- 
portionally (th. 82), making ad to db, as is ac to cb or to 
Its equal CB. Q. £. D* 



THEORBM 



THEOREMS. 



519 



Theore:^ Lxxxir. . 

Equiangular Triangles are Similar^ or have their Like Sidct 

Proportional, 

Let ABC, DEF, be two equiangular tri- 
angles, having the angle a equal to che^ 
angle D, the angle B to the angle £, and 
consequently the angle c to the angle v, 
then will ab : ac : : D£ : d^. 

For, make dg = ab, and dh = ac, 
and join gh. Then the two triangles 
ABC, DGH, having the two sides ab, ac, 
equal to the two dg, dh, and the con- 
tained angles a and d also equal, are iden- 
tical, or equal in all respects (th. 1 ), namely, 
the angles b and c are equal to the angled G and if. Bat the 
angles b and c are equal to the angles £ and f by the hypo* 
thesis ; therefore also the angles g and h are equal to the' 
angles £ and f (ax. 1), and consequently the line gh is pa^ 
rallei to the side ef (cor. 1, th. 14). 

Hence then, in the triangle def, the line gh, being pa- 
rallel to the side ef, divides the two other sides propor- 
tionally, making dg : dh : : de : df (cor. th. 82). fiut 
DO and DH are equal to ab and ac ; therefore also - - - "^ 
JB : AC : : DC : df. q. e. p. 





THEOREM LXXXV. 

Triangles which have their Sides Proportional, are Equi* 

angular. 

In the two friangles abc, def, if 
AB : DE : : AC : df : : bc : ef ; the two 
triangles" will have their corresponding 
angles equal. 

For, if the triangle abc be not equian- 
gular with the triangle def, suppose some 
Other triangle, as deg, to be equiangular 
with ABC. But this is impossible : for if 
the two triangles abc, deg, were equi-* 
angular, their sides would be proportional 
(th. 84). So that, ab being to de as AC 
to dg, and ab to de as Bc to eg, it follows tliat OG and 
SG, being fourth proportionals to the same three quantitiest 



Cr F 




880 



Gfi(»fETRY* 



as well as the two pr» ef, the former D<f, £G, irdoici hi 
equal to the latter, df, ef. Thus then, the tiKro tmngk^ 
DBF, d£g, having their three sides equal, lifroilld be identkal 
(th« 5); which is absurd^ since their angles ate^ un^qual^ 



THEOREM LXXXVt- 



Triangles, which have an Angle in the on^ Equal to zh Angte 
in the other, and the Sides about these angles Proportionals 
Equiangular. 



Let ABC, DEF, be two triangles, having 
the angle a = the angle d, and the sides 
AB, AC, proportional to the sides D^, Dt: 
then will the triangle abc be equiangular 
with the triangle def» 

For, make dg = ab, and dh = AG| 
and join gh. 

Then, the two triangles abc, dgh, 
having two sides equal, and the contained 
angles a and d equal, are identical and 
equiangular (th. l), having the angles G 
and H equal to the angles B and c« But, since the side^^ 
DG, DH, are proportional to the .ides de, df» the line GH is 
parallel to ef (th. 82); hence the angles e and f are equal to 
the angles g and H (th. 14), and consequently to their equals 
m and c* q« s. d. 




THEOREM LXXXVII. 

In a Right- Angled Triangle, a Perpendicular from the Right 
Angle, is a Mean Proportional betweex^ the Segments of 
the Hypothenuse ; and each of the Sides^ about the Right 
Angle, is a Mean Proportional between the Hypothenuse 
and the adjacent segment. 

Let ABC be a right-angled triangle, and . 
CD a perpendicular from the right angle 
c to the hypothenuse ab; then will 

CD be a mean proportional between ad and db ;. 

AC a mean proportional between ab and ad ; 

Bc a mean proportional between ab and BD. 

.' Or^^AD : CD : : CD ; DB$ and ab : »c : : ac : bd | attd 
AB : AC : : AC : AD. 

For J 




THEOREMS* 



i^i 



For, the two triangles abc, adc, having the right angles 
at c and D equal, and the angle a common, have their third 
wangles equal, and are equiangular (cor. 1, th. 17). In like 
manner, the two triangle^ abc, bdc, having the right 
angles at C and d equal, and the angle b common, have 
their third angles equal, and are equiangular. 

Hence then, all the three triangles abc, apc, bdc, 
being equiangular, will have their like sides proportional 
(th. 84); 



VIZ. Ab : ci> * : CD 


: db; 


and A£ t AC :: AC 


: AD} 


and AB : BC : : BC 


: BD. 



^ <^E. D* 

CoroL Because the angle in a semicircle is a right angle 
(th. 52) ; it follows, Ihat if, fi-om khjr point c in the peri- 
phery of the semicircle, a perpendicular be drawn to the 
diaxtieter ab ; and the two chords ca, dB^ be drawn td 
the extremities of the diameter : then are ac, bc, cd, the 
mean proportionals as in this theorem, or (by th. 17)^ *• * ^ 

tD* ac AD . DB; AC* ss AB • AD; and BC^ = AB . dDk 




THEOREM LXXXVIII* 

jEquiangular or Similar Triangles, are to each other as the 

'Squareis of their Like Sides. 

Let abc, def, be two equi- 
angular triangles, ab and de 
being two lik6 sides : then will 
the triangle' abc Ije to the tri- 
angle DBF, as the square of AB 
is to the $quaxe of D£, or as 
AB* to de\ 

For, ht At and dn be the 
isquares on ab and dk; also draw their diaj^onals bk, eM, and 
the perpendicuhrs cg, fh, of the two triangles. 

Then, since equiangular triangles have their like sides 
proportional (th. 84), in the two equiangular triangles abc, 
def, the side ac : df : : ab : de ; and in the two acg, 
DFH, the side Ac : df : : cg : fh ; therefore, by equality 
CG :, FH : : ab : de, or cg : ab :: fh : DE. 

But because triangles on equal bases are to each . other as 
their altitudes, the triangles ABC, Abk, on the same base 
ab, are to eath other, as their altitudes cg, ak, or ab: 

Vol. L Y and 



Z22 



GEOMETRY. 



and the triangles def^ dem, on the same base PE| are as their 
altitudes fh> dm^ or d£ ; 

that isy triangle abc : triangle abk ; : CG : AB9 
and triangle def : triangle dem : : f h : ds. 

But it has been shown that cG : ab : : FH : i>£ } 
theref. of equality A abc : aabk : : abef : Adem,. 
or alternately, as a abc : adef : ; >^ abc t A dem. 

But the squares al, dn, being the double of the triangles 
ABKj D£M« have the same ratio with them \ 
therefore the A abc : adef : : square al : square dn* 

. , (i. E. i^ 




THEOREM LXXXIX. 

All Similar Figures are to each other, as the Squares of their 

Like Sides. 

Let abcd£| fghik, be 
any two similar figures, the 
like sides being ab, fg, and 
BC,'GH,and so on in the same 
order : then will the figure 
ABCDEbe tothefigurcFGHiK, 
as the square of.AB to the 
square of eg, or as ab* to fg*. 

For, draw £e, bd, gk, g^x dividing the ^^es into a» 
equal number of triangles, by Knes from two equal angles- 
b and G^ 

The tsffo figures being similar (by stippos.)i they are equi- 
angular, and have their like sides proportional (def. 67). 

Then, since the angle A is ^^ the angle f, and the side^ 
ab, A£, proportional to the sides fg, fk, the triangles 
ABE, fgk, are equiangular (th. 86). In like manner, the 
two triangles bcd, ghi^ having the angle c = the angle H^ 
, imd the sides «c, CD, proportional to the sides gh. Hi, are 
also ecfuiangdar. Also, if from the equal angles aed, fki> 
there be taken the equal angles aeb, fkg, there will remain 
the equals bed, GKI ; and if from the equal angles cde, 
HiK, be taken away the equals cde, hig, there will remain 
the equals bde, gik 5 so that the two triangles bde, gik, 
having two angles equal, are also equiangular. Hence each 
triangle of* the one figure, is equiangular with each corre* 
sponding triangle of the other. 

But equiangular triangles are similar, and are propottfofial 
to the squaj'es of their tike sides (tit. 88).* 

TWeforr 



THEO&tMS. 



t2% 



Therefore th^ A ab» : a fgk : t ab* :sg% 
and A BCD :' A ghi : : bc* : gh* 
and A bdb : A gik : : de^ : IK 



.> 



But ds the two polygons are similar^ their like sides are pro- 
portional, iand consequently their squares also proportional i 
so that all the ratios ab* to pg% ^d bc* to gh*, and de* to 
1K% are equal among themselves, and consequently the cor- 
responding triangles also, abe to fgK) and bcd to ghi, and 
jBDE to GIK, have all the same ratio, viz. that of ab'^ to fg* : 
and hence all the antecedents, or the figure abode, have to 
mil the con^quents, or the figure fghik, still the «ame ratio^ 
Viz. that of AB* to FG* (th. 72). q. e. d. ^ 



THEOREM XC« 





Similar Figures Inscribed in Circles, have their Like Side«, 
and also their Whole Perimeters, in the Same Ratio- as this- 
Diameters of the Circles in which they are Inscribed. 

iiET ABCDE, fghik, 

be two similar -figures, 
inscribed in the circles 
whose diametof's are al * 
and FM ', then will each 
side A6, BC, &c, of the 
t)ne figure be to the like 
dde gf, gh^ &c> of the 
"other figure, or the whole perimeter ab + bc + &c, of tlie 
one figure, to the whole perimeter fg + GH 4" &c, of the 
Other figure, as the dis^meter al to the diameter f^. 

For, draw the two corresponding diagonals AC, fh, as 
also the lines bl, gm. Then, since the polygons are similar, 
they are equiangular, and their like*5ides have the same ratio 
(def. 67) ; therefore the two triangles abc, fgh, have the 
angle b =st the angle G, and the sides ab, bc, proportional 
to the two sides fg, Ch, consequently these two triangles 
are equiangular (th. 86), and have the angle acb = fhg. 
But the angle acb = alb, standing on the same arc ab; ^ 
and the angle fhg = fmg, standing on the same arc fg; 
.therefore the angle alb = fmg (ax. l). And sjince the 
wangle abl. = fgm, being both right angles, because in a 
.semicircle ; therefore the two triangles abl, fgm, having 
two anglCj* ecjoal, are equiangular ; and cpnsequentiy their 





.if 4 GEOMETRr. 

like sides afe profjortional (th. 8*) ; hence AB : ><J : ; the 
diameter al : the diameter fm. 

In like manner, each side bc, cd, &c, has to each side 
CH,m, &c, the same ratio of al to fm; and consequently 
the sums of them are still in the same ratio j viz, AB + bc -f 
c©, &c : FG + CH + HI, &c : i the diam. AV : the diami. 

FM (th. 72). £. B. D. . 

t 

THEOREM XCi; 

Similar Figures Inscribed in Circles, are to each other as 
the Squares of the Diameters of those Circles. 

Let abcdb, fghik, 
be two similar figures, in-' 
scribed in the circles 
whose diameters are al 
and FM ; then the surface 
of* the polygon abcde 
will be to the surface of 
the polygon fghik, as AL* to fm*. 

For, the figures being similar, are to each other as- the 
/ squares of their like sides, ab* to FG* (th. 88). But, by 
the last theorem, this sides ab, fg, are as the diameters al, 
FM ; and therefore the squares of the sides ab^ to fg*, as the 
squares of the diameters al* to fm* (th. 74). Consequently 
the polygons abcde, fghik, are also to each other as the 
squares of the diameters al* to fm* (ax. l). <^ £• d. 



theorem xcii. 

The Circumferences of all Circles are to each other as their 

Diameters. 

Let d, di denote the diameters of two circles, and c, <:, 
their circumferences ; 

then will d :.i/ : : c : r, or D : c : : rf : r. ' 

For (by theor. 90), similar polygons inscribed in circles 
have their perimeters in the same ratio as the diameters of 
those circles. 

Now, as this property belongs to all polygons, whatever 
the number of the sides may be ; conceive the numbe^r of the 
sides to be indefinitely great, and the length of each inde- 
finitely smalU till they coincide with the chrcumference of 

the 



THEOREMS. 3-25 

t 

the circle, and be equjd to it, indefinitely near. . Then the 
perimeter of the j)olygon of an infinite number of sides, is 
the same thing as the circumference of the circle. Hence h: 
appears that the circumferences of the circles, being the same 
as the perimeters of such polygons, are to each other in the 
i^ame ratio as the diameters of the circles, q^ s. d. 



THEOREM XCllI. 

t 

The Areas or Spaces of Circles, are to each other as the 
Squares of their Diameters, or of their Radii. 

Let a, ay denote the areas or spaces of two circles, aad 
JD, dy their diameters; then a :^ : : d^ : i/^. 

Fot (by theorem 91) similar polygons inscribed in circles 
are to each other as the ^squares of the diameters of th^ 
circles. 

Hence^ conceiving the numher of the sides of the poly- 
gons to be increased more and more, or the length of the 
aides to become less and less, the polygon apprpaches nearer 
and nearer to the circle, till at length, by an infinite ap<> 
proach, they coincide, and hecome in effect equal ; and then 
it follows, that the spaces of the circles, which are the same 
as of the polygons, will be to each other as the squares of the 
diametersof the circles. (^ £. J>. 

• CorQh The spaces of circles are also to each other as the 
squares of the circumferences ; since the circumlFerences are 
in the same ratio as the diameters (by theorem 92). 



THEOREM XCIV. 

The Area of any Circle, is Equal to the Rectangle of Half 
its Circumference and Half its Diameter. 

Conceive a regular polygon to he ^^f^ 

'inscribed in the'ciTcle^; and radii drawn to //\ 

-all the angular points, dividing it into as j/ . 

xnaAy -equal triangles as the polygon has K . 

. ^idesji one of which is abc, of which the \x / 

jiltituid^ is the perpendicular CD from the -S^F^li 
centre to the base ab.' 

Then the triangle abc, being equal to 
-a rectan^e bif half the base and equal altitude (th. 26, cor. 2), 
is equal to the^ectangle of the half base ad aod the a^titudeoD ; 

£onse«> 



I 



S^6 



GEOMETRY, 




consequently the whole polygon, or all 
the triangles added together which com- 
pose it, is equal to the rectangle of the 
conunop altitude cd, and the halves of all 
the sides, or the half perimeter of the po- 
lygon. 

Now, conceive the number of sides of the polygon to ht 
indefinitely increased ; then will its perimeter coincide with 
the circumference of the circle, and consequently the alt^ 
tude CD will become equal to the radius, and the whole 
polygon equal to the circle. Consequently the space of the 
circle, or of the polygon in that state, is equal to the rect^ 
angle of the radius and half the circumference/ ^ £• n. 



9e 



OF PLANES AND SOLffiS, 



\ • 



DEFINITIONS. 

DjtF. S8. The Common Section of two Pl^es, is the 
line in which they meet, to cut each other* 

89. A Line is Perpendicular to a Plane, when it is per-* 
. pendicular to every line in that plane which meets it. 

90. One Plane. is Perpendicular to Another, when every 
Kne of the one, which is perpendicuJar to the line of their 
common section, is perpendicular to the other* 

91. The Inclination of one Plane to another, or the ^gle 
they form between them, is the angle contained by two 
lines, drawn from any point in the common section, and at 
right angles to th« same, one of these lines in each plane^ 

92. Parallel Planes, are such -as being produced ever so 
far both ways, will never meet, or which are evcf]^ where at 
an equal perpendicular distance* 

9&. A. Solid Angle, is that which is made by three or 
more pltoe irfglcs, meeting tac-h pthcr in the sam^ point. 

94f. Similar 



J 



DEFINITIONS- 



327 



.'94. SiirvilafS^Uds^ contained by plane figures, 3re such as 
liave all their so)id angles equal, each to each, and are bound- 
ed by the same number of similar planes, alike placed* 

^ 95. A Prisrn, is a solid whose ends are parallel, equal, and 
like plane figures^ and its sides, connecting those ends, are 
parallelograms. - • 

> • 

96. A Prism takes particular names according to the figure 
of its base or ends, whether triangular, square, rectangular, 
pentagonal, hexagonal, &c* 

?7. A Right or Upright Prism, is that which has the 
planei, of the jides perpendicular to the plane* of the endU 
or bi^e. 



t^ 



hi 




9a» A Porallelopiped, or Parallel opipedon, is 
.^ . '^m bounded by six parallelograms, every 
o. , lie two of which are equal, alike, and pa- 

■•'9. A Rectangular Parallelopipedon, is that whose bound- 
ing i lanes are all \rectangles, which are perpendicular to each 
©her. 

100. A Cube, is a square prism, being bounded 
by six equal square sidei or faces, and are perpen* 
dicalar to each other. 

I (;1 . A Cylinder is a round prism, havij^g cir- 
cles for its ends ; and is conceived to be formed 
by I he rotation of a right line about the circum- 
ferences of two equal ^nd parallel circles, always 
parallel to the axis. 

1 02. The Axis of a Cylinder, is the right line 
joinirrg the centres of the two parallel circles, about vhich 
the ligure is described. 

10^. A Pyramid, is a solid, whose base is any 
right-lined jiane figure, and jts sides triangles, 
having all their vertices meeting together in a 
point above the base, called the Vertex of the 
pyramid. 

104. A pyramid, like the prism, takes particular names 
'.from the figure of the base. ' ^ 

105. A Cone, is a round pyramid, having a cir- 
.cula^ base, and is qonceived to be generated by 

the rotation of a right line about the circum- 
.ference of a circle, one ^nd of which is fixed at 
. a point above the piw^e pf tjiat circle. 

' ' • -^ 106. The' 





528 GEOMETRY. 

J Of, The Axis of a cone, is the right line, joining th« 
vertex, or fixed point, and the centre of the circle about 
H^hich the figure is described. 

107. Similar Cones and Cylinders, are such as have their 
altitudes and the diameters of their bases proportional. 

108. A Sphere, is a solid bounded by one curve surface^ 
which is every where equally distant from a certain point 
within, called the Centre. It is conceived to be generated 
by tlie rotation of a semicircle about its diameter, which re- 
mains fixed. 

109. The Axis of a Sphete, is the right line about which 
th^ semicircle revolves; and the centre is the same as that of 
the revolving semicircle. 

110. The Diameter of a Sphere, is any right line passing 
through the centre, and terminated both ways by the surface. 

1 H. The Altitude of a Solid, is the perpendicul^ drawn 
from the vertex to the opposite side or base. 

THEOREM XCV, 

A Perpendicular is the Shortest Line which can be drawfi 

from any Point to a Plane. 

Let ab b^ perpendicular to the plane ju 

DE ; then any other line, as AC, drawn \ 

from the same point a to the plane, will \ 

be longer than the line ab. ^ ^"ul \ 

In the plancf draw the line Bc, j^oining <?^ 

the points B, o. ' ' 

- 'Then, because the line ab is perpendi- 
cular to the plane de, the apgle B is a right angle {def. 90), 
and coi^sequently greater than the angle c ; therefore the 
line AB> opposite to the less angle, is less than any other lint 
AC, opposifjB the greater angle (th. 21), q. E. D. 

THEOREM XCVI. 

A Ferpendic^lar Measures the Distance of any Point from 4 

PJane, 

.The distance of one point from another is measured by a 
right line joining them^ because this is the shortest line which 
can be drawn from one point to another. So, also^ the 
^stance from a point to a line, is measured by a perpendi- 
ipular^ l^ecause this line is the shortest which can be drawn 

fronat 




THEOREMS. 



S29 



from tlie point to the line. In like manner, tbc distance 
from a point to a plane, must be measured by a perpendicular 
jdrawn from that point to the plane, because this is the 
shortest line which can be drawn from the point to the 
j>tane» 



THEOREM XCVII. 



. The Common Section of Two Planes, is a Right Line, 

Let acbda, akbfa, be two planes 

cutting each other, and A, b, two points £, 

in which the two planes meet ; drawing . \ F-* 

the line ab, this line will be the common 
intersection of the two planes. 

For, because the right line ab touches 
the two planes jn the points a and b, it — ^ 

touches them in all other points (def. 20): 
this line is therefore common .to the two planes; That is, * 
^he cpmmon intersection of the two planes is a right line. 



^ 



^ 



u 



$ 



THEOREM XCVllh 
I 

\i a Line be Perpendicular to two other Lines, at their 
Common Point o!F Meeting ; it will be Perpendicular to 
the Plane of those Lines. 

Let the line ab make right angles with 
the lines AC, ad 5 then will it be per- 
pendicular to the plane cde which passes 
jthrough these lines. 

If the line ab were not perpendicular to 
the plane cde, another plane might pass 
through the point a, to which the line ab ^ 
would be perpendicular. But this is im- 
possible -, for, since the angles bag, bad, are right angles^ 
this other plane must pass through the points g, d.. Hence, 
this plane passing through the two points a, c, of the line 
AC, and through the two points a, d, of the line ad, it will 
"pass through both these two lines, and therefpre he the same 
jplane with the former, q. e. d. 




TH]£0R|(M 



sso 



GEOMETRY. 



THJEbUBM XCIX. 



If Two Lines be Perpendicular to the Same Plane, thejr wM 

be Parallel to each other. 



S 



> 



Let the two lines ab, cd, be both per- 
pendicuVar to the same plane ebdf ; then 

will Afi ha paralLed to cD. 

« 

For, join B, i>, by the line bd in the ' 
plane. Then, because the^ lines ab, cd, . 
are j^^erpendicular to the plane £F, they are 
both perpendicular to the line bd (def. 90) in that plane^ 
and cor sequent ly they are parallel to eacl^ other (corol. 
th. I'i). Q. E. D. 

CoroL It two lines be parallel, and if one of them be 
perpendicular to any plane, the other will also be perpendi- 
cular to the same plane. 






THEOREM C 

If Two Planes Cut each other at Right Angles, and a Line 
be drawn in one of the Planes Perpendicular to their 
. Common Intersection, it will be Perpendicular to the 
. other Plane. 

Let the two pknes acbd, abbf, cut 
each other at right angles; and the line^ 
CG be perpendicular to their common sec- 
tion ab ; then will cg be also perpendicular 
to the other plane aebf. 

For, draw eg perpiendicular to ab. 

Then, because the two lines gc, gEj are 

perpendicular to the common intersection 
^ ABy the angle cge is the angle of inclination of the two 

planes (def. 92). But since the two planes cut each other 
. perpendicularly, the angle of mclination cge is a right 

smgle. And since the line cq is perpendicular to the. two 

lines GA, ge> in the plane aebf, it is therefore perp^ndi* 

cular to that plane (th. 9^)* q. e. v. 




TKBORSM 



THEOREMS- 



SSI 



THEOI^EM CI. 

If one Plane Meet another Plane, it will make Angles 
with that other Plane, which are together equal to two 
Right Angles. - 

Let the plane acbd meet the plane asbf; these planes 
xnake with each other two angles whose sum is equal to two 
right angles. 

For, through any point g, in the common section ab^ 
draw CD, ef, perpendicular to ab. Then, the line co 
^akeswjth EFtwo angles together equal to two right angles. 
But thesie two angles are (by def. 92) the angles of inclina- 
tioii of the two planes. Therefore the two planes make 
jingles with each other, which are together equal to two 
right angles* 

CoroL In like manner, it may be demonstrated, that planes 
icfhich intersect, have their vertical or opposite angles equal; 
also, that parallel planes have their alternate angles equal ; 
(ind so on, as in parallel lines. 



THfcOUEM CII. 



y 





If Two Planes be Parallel to each other ; a Line which is 
Perpendicular to one of the Planes, will also be Perpendi-» 
Cular to the other. 

I 

tiET the' two planes CD, ef, be parallel, 
jind let the line ab be perpendicular to the 
plane cD ; then shall it also be perpendi- 
cular to the other plane ef. 

For, from arty point G, in the plane £F, 
draw GH perpendicufar to the plane cd, and 
4raw^AH, BG. 

Then, because ba, gh, are both perpendicular to the 
plane cd, the angles k and h are both right an'gles. Attd 
because the planes cd, j^f, are parallel, the perpendiculars 
3A, gh, are equal (def, 93). Hence it follovrs that the 
lines ;3G, AH, are parallel (drf. 9). And the line ab being 
perpendicular tp the line ah, is also perpendicular to the 
parallel line bg (cor. th. 12). 

In'likemanner it is ptoved, that the line ab is p«rpen- 
4i£Dlar to aU otl^r 4ines whic^^a be drawn from^ the point b 

in 



532 



GEOMETRY, 



in the plane ef* Therefore the line ab is perpendicular tm 
the whole plane £f (def. 90). q. b. d. 



THEOREM cm. 

If Two Lines be Parallel to a Third Line, though not in the 
same Plane with it ; they will 6e Parallel to each other* 

Let the lines ab, cd, be each of them ^ . 
parallel to the third line £F, though not in 
the same plane with it ; then will ab be pa* 
lallel to cj>. 

For, frem any point G in the line ef, let 
GH, Gi, be each perpendicular to ef, in the 
planes eb, ed, of the proposed parallels. 

Then, since the line ef is perpendicular 
to the two lines GH, gi, it is perpendicular 
to the plane ghi of those lines (th. 9S). And because SF 
is perpendicular to the plane <yHi, its parallel ab is also per- 
pendicular to that plane (cor» th. 99), For the same reason, 
the line cd is perpendicular to the same plane ghi. Hence, 
because the two lines ab, cd, are perpendicular to the sam(^ 
plane, these two lines are paiydlel (th. 99). q. e. d* 




theorem cit. 

If Two Lines, that meet each other, be Parallel to Twq 
other Lines that meet each other, though not in the sanje 
^Plane with them j the Angles contained by those Lin^s 
will be equal. 

Let the two lines ab, bc, be parallel to 
the two lines de, ef ; then will the ajigle 
ABC be equal to the angle def. 

For, make the lines ab, bc, de, ef, all 
lequal to each other, and join ac, df, ad« 
BE, cf. 

Then, the lines ad, be, joining the equal 
and parallel lines ab, de, are equal and 
parallel (th. 24). For the same reason, cf, be, are equal 
and parallel. Therefore adj cf, are equal and parallel 
(th. 15); and consequently also ac, df (th. 24). Hence, 
the two triangles abc, d35f, having aU^heir sides equ^l, 




THEOREMS- 



S3* 



^ach to <^acli> have their angles also equal, and consequently 
the angle ABC = the angle def. (^ £. d. 




THEOREM cr. 

The Sections made by a Plane cutting two other Parallel 
Planes> are also Parallel to each other. 

3LBT the two parallel planes ab, cd, be 
cut by the third plane efhg, in the lines fXSL=::^K 

£F, GH : these two sections ef, GH,.will 
be parallel. 

Suppose EG, FH, be drawn parallel to 
each other in the plane efhg ; also let 
EI, FK, be perpendicular to the plane cd ; 
and let IG,KH, be joined. 

Then eg, fh, being parallels, and ei, fk, being both 
perpendicular to the plane CD, are also parallel to each other 
(th. 99) ; conisequently the angle hfk is equal to the angle 
CEi (th. 104). But the angle fkh is also equal to the angle 
fcio, being both right angles; therefore the two triangles are 
equiangular (cor. 1 th. 17) ; and the sides fk, ei, being 
the equal distances between the parallel planes (def. 93), it 
follows that the sides fh, eg, are also equal (th. 2). But 
these two lines are parallel (by suppos.}, as well as equal; 
consequently the two lines e^, gh, joining those equal pa- 
Tallek, are also parallel (th. 24). <^ e. d. 



theorem cvr. 

If any Prism be cut by a Plane Parallel to its Base, the Sectioa 
will be Equal and Like to the Base. 

LsT AG be any prism, and il a plane ' 
parallel to the base AC ; then will the plane 
IL be equal and like to the base ac, or the 
two planes will have all their sides and all 
their angles equal. 

For, the two planes ac, il, being paral- 
lel, by hypothesis ; and two parallel planes, 
cut by a third plane, having parallel sections 
(th. 105); therefore IK is parallel to ab, and kl to BC, and 
I,M to CD, and im to ad. But ai and bk are parallels 
(by def. 95) ; consequently ak is a parallelogram ; and the 
opposite sides ab, ik, are equal (th. ^2). In like manner, 

it 




S34 



GEOMETRY. 





It is shown that kl is =s bc» attd IM = ci>y 

uid IM ^ AD» 'or the two planes Ac, il, are 

mutually equilateral. But these two planes, 

havingtheir corresponding sides parallel, have 

tlie angles contained by them also equal 

(th. 104), namely, the angle A = the angle t, 

the angle B =: the angle K, the angle c = the 

angle. L, and the angle d = the angle m. So 

that the two planes AC, tL, have all their corresponding 

sides and angles equal, or they arc equal and like. Qi £. d» 

s 

THEOREM CVII. 

If a -Cylinder be cut by a Plane Parallel to its Base> the 
Section will be a Circle, Equal fo the Base. 

Let af be a cylinder, and ghj any 
section parallel to the base abc; then will 
CHI be a circle, equal to ABC. 

For, let the planes ke, kf, pass through 
the axis of the cylinder mk, and meet the 
section GHi in the three points il, I, 4- -, 
and join the goints as in the figure. 

Then, since kl, ci, are parallel (by 
def» l0'2) ; and the plane Ki, meeting the 
two parallel planes ABC, GHi, makes the two sections KC, Lr, 
parallel (th. 105) j the figure klic is therefore a paraI-» 
lelogram, and consequently has the opposite sides li, kg, 
equal, where KC is a radius of the circular base. 

In like manner, it is shown that lh is equal to the radius 
KB ; and that any other lines, dr;iwn from the point L to 
the circumference of the section ghi, are all equal to radii 
of the base 5 consequently Giii is a circle, and equal to ABC. 

(^ £. D. 

THEORElvr CVIII. 

All prisms and Cylinders, of Equal Bases and Altitudes^ ave 

Equal to each other. . 

Let AC, DF, be two 
prisms, and a cylinder, 
en equal bases ab, de, 
and having equal alti-» 
tudes £c, FF ; then will 
^the solids AC, DP, be 
-cquaL 

FoT) let FQ, Rs, be 

any 



C 



J^---^ 



Q, 



IB 




i^C3s 




THEOREMS. ■ 355 

any two sections parallel to the bases, and e^idlstant from 
them. Then, by the last two theoi^ms, the section pq^is 
equal to the base ab, and the section ks equal to the base ■ 
DK. But the bases ab. DE, are equal, by the hypothesis; 
therefore the sections pa, rs, are equal also. In like manDer, 
it m»y be shown, that any other corresponding sections are 
equal to one another. 

Since then every section in the prism ac, is equal to its 
_ corresponding secfion in the prism or cylinder DF, the prisms 
and cylinder themselves, which are composed of an equal 
sumheror all these equal sections, must also be eiual H..E.D. 

Carol. Every prism, or cylinder, is equal to a rectangular 
parallelopipedon, of an equal base and altitude. 

TUSORBM cix. 

Rectangular Parallelopipedons, of Equal Altitudes, are to 
each other as their Bases. 
\XT AC, EG, be two rectan- 

rularparallelopipedons, having 
the equal altitudes ad, eh ; 
thea will the solid ac be to the 
solid EG, as the base ab is to 
the base ef. 

For, let the proportion of the 
base AB to the base ef, be that 
of anyone number m (3) to any 

Other number n (2). And conceive AB to be didded into m 
equal parts, or rectangles, ai, i.k, mb (by dividing an into 
that number of equal parts, and drawing II, K^f, parallel 
to Bn). And let ef be divided, in like manner, into n equal 
parts, or rectangles, eo, pf : all of these pans of both bases 
being mutually equal among themselves.. And through the 
lines of division let the plane sections lr, ms_, pv, pass 
parallel to aq, et. 

"nien, the parallelopipeJons ar, ls, mc, ev, pg, are all 
equatj having equal base; and altitudes. . Therefore the solid 
AC is to the solid eg, as the number of parts in the former, 
to the number of equal parts in the latter ; or as the number 
of parts in ab to the number of equal parts in ef, that is, as 
the base ab tOjbe base Et. q. e. d. 

CsrU. From this theorem, ajid .the corollary to the last, it 
appears^ that all prisms and cylinders of equal alfitudest »» 



336 



GEOMETRY. 



to each other as their bases ; every prism at^d cylinder beiagf 
equal to a rectangular parallelopipedon of an equal base and 
altitude. 



THEOREM ex. 



Rectangular Parallelopipedons, of Equal Bases^ are to each 

other as their Altitudes. 



L 



B 



^' 



Ct 



^. 4 



a 



C 



Let ab, cd, be two rectan- 
gular paraUelopipi^ons, stand- 
ing on the equal bases ae,cf; 
then will the solid ab be to the 
solid CD, as the altitude £ B is to 
the altitude fd. 

For> let AG be a rectangular 
parallelopipedon on the base ae^ 
and its altitude eg equal to the altitude fd of the solid CD. 

Then ag and en are equal, being prisms of equal ba^S 
and altitudes. But if hb, hg, be considered as bases, the 
solids ab, AGy of equal altitude ah, will be to each other 
as those bases hb, hg. But these bases hb, hg^ being 
parallelograms of equal altitude he, are to each other a^ 
their bases eb, eg ; therefore the two prisms ab, ag, are 
to each other as the lines eb, eg. But ag is equal to 
CD, and EG equal to fd; consequently the prisms aC|CD, 
are to e^ch other as their altitudes eb, fD^ that is^ - - ** 
'ab : CD :: eb : fd, q^ e. d. 

CoroU 1. From this theorem, and the corollary to theorem 
108, it appears, that all prisins and cylinders, of equal bases^ 
are to one another as their altitudes. 

CoroL 2. Because, by corollary 1, prisms and cylinders are 
as their altitudes, when their bases are equal. And, by the 
corollary to the last theorem, they are as their bases, 'when 
their altitudes are equal. Therefore, universally, when nei- 
ther tire equal, they are to one another as the product of their 
bases and altitudes. And hence also these products are the 
proper numeral meaisures of their quantities or magnitudes^ 

THEORBM CXI. 

Simijar Prisms and Cylinders are to each other, as the 
Cubes of their Altitudes^ or of any other Like Linear Di* 
mensions. 

Let abcd, efgh, be two similar prisms ; then will the 
prism CD be to the prism gh, as, ab^ to sif' or ad' to £h'. 

- ^ ' For 



o 



p 



Pot the solids M'e to e^ch other tis 
the product of their bases and alti- 
tudes (th. 110, cor. 2), that Ts, as 
AC . Alb to £G . £H. But the bases,' 
i>6ing similar planes, tire to each 
other as the squares* of their like -^^ [y'*' j; 
sides, that is, ac to eg as ab^ to 
£F^; therefore the solid cd is to 
the solid Gii, a^ ABf . ad to ^f' . EH. - 

But KD and fh, being . similar planes, have their like side^ 
proportional, that is, ab : ef : : ad : bh, - - - - - ^ 
qr AB* : £f* : : ad*: eh**: thereCore ab*. ad : ef*. sh : : ab^ :,ef% 
or : : ad' : eh' ; conseq. the solid cd : solid gu : ; ab' : 
15f' ;: AD* t bh'* <^ e. d. 




6i 



THEOREM exit 




In any Pyramid, a Section Parallel to the Base is similar to 
the Base; and these two planes are to each other as the 
Squares of their Distances from th<» Ydrtexi 

Let abcd be a pyramid, and e^o a sec'^ 
tion parallel to the base BCD, also Aiii a 
line perpendicular to the two planes at H and 
I : then will bd, ]sg, be two siniilar planes, 
and the plane BD will be to the plane bg, as 

AH* to AI*. 

For, join ch, ti. TheUj because a plane 
cutting two parallel planes, makes parallel 
sections (th. 105), therefore the plane ABC, 
meeting the two parallel planes bd, eg, mak^ the secticrns 
Bc, ef, parallel : In like manner^ the plane acd makes 
the sections CD, fG, paraUel^ Again, because two pair of 
parallel lines make equal angles (th. 104), the two ef, fg, 
which are parallel to BC, CD, make the angle bfg equal 
rthe angle bcd. And in like mannef it is shown^ that 
each angle in the plane eg is equal to each angle in the 
plane BD, and consequently those two planes are equian-^ 
gular. 

Again, the three lines ab, Ac, ADj making with the 
parallels bc, eF, and cb, fg, tqual angles (th. 14)j and 
the angles at a being common, the two triangles ABC, aef, 
&re equiangular, as also the two triangles ACD, afg, and 
have thcrerore their like sides proportional, namely, - - - 

VOL.L Z AC 



d58 



GEOMETRY. 



AC : AF 



BC 



BP : : CD : VG. And in 




like manner it may be shown, that aU the 
lines in the plane fg, are proportional to all 
the corresponding lines in the base bd. 
Hence these two planes, having their angles 
equal, and their sides proportional, are 
similar, by def. 68. Ip-- — ^ 

But, similar planes being to each other as the squares of 
their like sides, the plane bd : eg : : bc* : ef% or : : AC* : 
AF*, by what is shown above. Also, the two triangles 
AHC, AiF, having the angles H and i right'ones (th. 98), 
. and the angle A common, are equiangular, and have there- 
fore their like sides proportional, namely, ac : af : : ah : ai, 
or AC* : AF* :: ah* : ai*. Consequently the two planes 
BD, EG, which are as the former squares ac*, af*, will 
be also as the latter squares ah*, ai'^ that is, - - - - - 
BD : EG :: ah* : Ai*. Q. E. D. 

THEOREM CXIII. 

X 

t 

In a Cone, any Section Parallel to the Base is a Circle ; and 
this Section is to the Base, as the Squares of their Distances 
from the Vertex. . 

Let abcd be a cone, and ghi a section 
p^vallel to the base BCd; then will ghi 
be a circle, and bcd, ghi, will be to each 
other, as the squares of their distances 
from the vertex. 

For, draw alf perpendicular to the 
two paralld planes; and let the planes 
ACE, ADE, pass through the axis of the 
cone AK.E, meeting the section in the three 
points H, I, K. 

Then, since the section ghi is parallel to the base BCD, and 
the planes CK, dk, meet them, hk is parallel to ce, and 
IK to DB (th. 105). And because the triangles formed by 
these lines are equiangular, kh : EC : : ak : ae : : Ki : ed. 
But EC is equal to ed, being radii of the same circle ; there- 
fore KI is also equal to kh. And the same may be shown of 
any other lines drawn from the point K to the perimeter of 
the section Gj^i, which is therefore a circle (def. 4?4). 

Again, by similar triangles, al : af : : ak : 
:: KI : ED, hence al* : af* :: Ki* : ed*; but Ki* 
circle ghi : circle bcd (th. 03); therefore A L* : 
circle ghi : circle bcd. q. e. d. 




AE or 
ED* : : 
af"^ ;: 



THEOHEM 



THEOR^MS^ 



S30 





TkEOR^M ckiv; . ^ 

All Pjramidsy and Cones, of Equal Bases and Altitudes, art* 

Equal io one another^* 

Let abc^ def, 
ie any pyramids and 
cone, of ^qiial ba^es 
BC, £F, and equal , 
altitudes AG, dh: 
then will the pyra-, 
mids and cone abc 
and DEF, be equal* 

For, parallel to the 
bases and at equal distances AK, po, from the vertices, 
suppose the planes tK, lm, to be di^wn. ' ....... 

Then, by the two. preceding theorems, -------* 

DO* : DH* i: LM : sf, and 
AN* 2 AG* : : IK : BC. 
But since an*, ag*, are equal to d6% dh*, 
therefore iK : BC : : lm :.ef. But bc is equal to fcp, 
by hypothesis j therefore ik is also equal to lm. « 

In lik^ thaiinef it is shown, that any other sections, at 
equal distance from the vertex, are equal to each other. 

Since theii, every section in the cone, is 6qual to the cor- 
responding section iii the pyramids, and the heights are equal,' 
the solids abc^ d£f, /:omposed of all those sections, must be 
equal also; q. IS. jb. 




TilEOR^M CXV.' 

Every Pyramid is tlie l*hird Part of a Prism o^ the Same' 

Base and Altitude. 

Let abcd£f bef a prism, and ubsp a ^ 

pyramid, on the same triangular base d£^: 
then will the pyramid BDef be a third part 
tf the prism abcdef* 

For, in the plancfs of the thi-^e sidrfs df thcf 

?rism, draw the diagonals bf, bd, cd. 
i'hen the two planes bdf, bcd, divide the 
whole prism into the three pyramids bdef, ^abc, 'iJBCF^^ 
^hich are proved to be all equal to one another, as follows. 
Since the opposite end^ of the prism are equal to each other, 
ihe pyramid nfhbse base is abc and vertex D, is equal to the 

a 2 pyraxjrwd 




f 



Sit) 



CEOMETRT. 





pyramid whose base is def and vertex B 
(th. 114), being pyntmids of equal base . 
and altitude. 

But the latter pyramid, whose base is 
B£F and vertex b> is the same solid as the 
pyramid whose base is bef and vertex t>i 
and this is equal te the third pyramid 
whose base is bcf and vertex d, bei^ig py- 
.ramids of the same altitude and equal ba^es 
BEF, seF. 

Consequently all the three pyramids, which compose the 
prism, are equal to each other, and each pyramid is the 
third part of the prism, or the prism is triple of the pyra- 
mid, q. E. D. 

Hence also, merj pyramid, whatever its figure may be, is 
the third part of a prism of the same base and altitude ; 
since the base of the prism^, whatever be its figure, may^ 
divided into triangles, and the whole solid into trian^lar 
prisms and pyramids. 

CoroL Any cone is the third part oJF a cylinder, or of a 
prism, of equal base and altitude ; since it has been proved 
that a cylinder is equal to a prism, and a . cone equal to 
.a pyramid, 6f equal base and altitude. 

ScholU/m, Whatever has been demonstrated of the propor- 
tionality of prisms, or cylinders, holds equally true of pyra- 
mids, or cones ; the former being always triple the latter j* 
viz. that similar pyramids or cones are as the cubes of their 
like linear sides, or diameters, or altitudes, &c. And the 
-same for all similar solids whatever, viz. that they are in pro- 
portion to each other, as the cubes of their like linear dimen- 
sions, since they are composed of pyramids every way similar. 



THEOREM CXVI, 

If a Sphere T^e cut by a Plane, the Section will be a Grcle^ 

Let the sphere aerf be cut by the 
plane abb ; then will the Sisction adb 
be a circle. 

Draw the chord ab, or diameter of 
the seotion ; perpendicular to which, or 
to the section adjb, draw the axis of the 
sphere ecg.f, thrpugh the centre c, 
which will bisect the chord ab in the 
point G (th. 41). Also, join cA, CBj 




an^ 



THEOREWS. 



341 



9nd draw cd^ cd^ to any point d in tb^ pemmetm? of the 
section AOB. 

Thenj because CG is perpendicular to the plane adb, it 
is perpendicular both to ca and gd (def. 90). So that cga, 
COD are two right-angled triangles/ having the perpendicular 
CG common y and the two hypothenuses^cAy cd^ equal, being 
both radii of the sphere ; therefore the .third sides ga^ gd, 
are also equal (cor. 2» th. 34). |p like manner it is shown, 
that any other line, drawn from the centre G to the circum- 
ference of the secdcm adb, is equal to ga or gb ; conse- 
quently that section is a circle. 

CfTol* The section through the centre, is a circle having 
the same centre and diameter as the sphere, and is called a 
great circle of the sphere ; the oiher plane sections being 
Uttle Circles. 



THEOREM CXVII. 




Every Sphere is Two-Thirds of its Circumscribing Cylinder. 

. Let abcd be a cylinder, circum- 
scribing the sphere efghj then will 
the sphere efgh be two-thirds of the 
cylinder abcd. 

For, let the plane ac be a section of 
the sphere and cylinder through the 
centre i. Join ai, bi. Also, let fih 
be parallel to ad or bc, and eig and 
KL parallel to ab or DC, the base of 
the cylinder ; the latter line kl meeting Bi in M, and the 
circular section of the sphere in n. 

Then, if the whole plane hfbc be conceived to revolve 
about the line hp as an axis, the square fg will describe 
a cylinder ag, and the quadrant ifg will descrihe a hemi- 
sphere efg, and the triangle ifb will describe a cone iab. 
Also, in the rotation, the tliree lines or parts kl, kn, km, as 
radii, will describe corresponding circular sections of those 
solids, namely, kl a section of the cylinder, kn a section of 
the sphere, and km a section of the cone. 

Now, fb being equal to Fi or IG, and kl parallel to 
fb, then by similar triangles IK is equal to km (th. 82). Arid 
since, in the right-angled triangle ikn, in* is equal to ik* 
+ KN* (th» 34; J and because kl is equal to the radius ig 

or 



343 



GEOMETRY. 




or IN, and KM rr IK, therefore kl* is 
jequal to km* + kn*, or the square of 
the longest, radiusi of the sud circular 
sections, is equal to the sum of the 
squares of the two others. And ber 
cause circles are to each other as the . 
squares of their diameters, or of their 
raidii, therefore the circle described by 
KL is equal to both the circles de- 
scribed by KM and kn ; or the section of the cylinder, U 
equal to both the corresponding sections of the sphere and 
cone. And as this is always the case in every parallel posir 
tion of KL, it follows, that the cylinder eb, which is com- 
posed of all the former sections, is equal to the hemisphere 
SFG and cone iab, .which are composed of all the latter 
sections. 

But the cone iab is a third part of the cylinder eb 
(cor. 2, th. 115) 5 consequently the hemisphere efg is equ4 
to the remaining two-thirds ; or the whole sphere bfgh 
equal to two-thirds of the whole cylinder abcd. q^ e. d. 

CoroL 1. A cone) hemisphere, and cylinder of the same 
base and altitude, are to ieach other as the numbers 1, 2, 3. 

CoroL 2. All spheres are to each other as the cubes of their 
diameters ; all these being like parts of their circumscribing 
cylinders. 

Corol. 3. From the foregoing demonstration it also api^ 
pears, that the spherical zone or frustrum egnf, is equal 
to the difference between the cylinder eglo and the cone 
IMQ, all of the 3ame common height ik. And that the 
spherical segment pfn, is equal to tl^e difference between 
the cylinder ablo and the conic frustrum. A(^I9> all of tbo 
iwne common altitude FK, 



V * » 



♦ ■ * 






fROBLEMS. 



■-■:.x fy^V^'i—^' 



t s" ] 






PROBLEMS. 



PROBLEM !• 



Ar 



To Bisect a Line ab ; that Is, to divide it into two Equal 

Parts. 

From the two centres a and B, with 
any «qual radii, describe arcs of circles, in- 
tersecting each other in c and d; and 
draw the Une cd, which will bisect the 
given line ab in the point £. 

For, draw the radif ac,,. bc, ad, bd. 
Then, because all these four radii are equal, 
and the side cd common, the two triangles 
ACO, BCD, are mutually equilateral : consequently they are 
also mutually equiangular (th. 5), and have the angle ag^ 
equal to the angle bce. 

Hence, the two triangles ace, bce, having the two sides 
AC, cE, equal to the two sides Be, ce, and their contained 
angles equal, are identical (th. 1), and therefore have the 
^de A£ equal to eb« q^ b. d. 



w 



problem II. 




To Bisect an Angle bac. 

From the centre a, with any radius, de- 
scribe an arc, cutting off the equal lines 
AD, AB ; and from the two ceni^res d, e, 
with the same radius, describe arcs intersect- 
ing in F ; then draw af, which will biisect 
the angle a as required. 

For, join dp, ef. Then the two tri- 
angles ADF, AEF, having the two sides 
AD, DF, equal to the two AE, EF (being equal radii), an4 
the side af common, they are mutually equilateral ; conse* 
quently they are also mutually equiangular (th. 5), and have 
the angle baf equal to the angle caf. . 

Sciolium. In the same manner is an ^c of a circle b)^ 

PROBLEM 



3H 



GEOmTBLY. 




PROBLEM III> 

t 

At a Given P<Mnt c, in a Line ab, to Erect a Perpendicular^ 

From the given point c, with any radius^ 
cut off any equal parts cp, C3j of the given 
line; and, from the ty^ centres d and e, 
with anyone radius, describe arcs intersecting 
is F ; then join cf, which will be perpendi- 
cular as required. 

For, draw the two equs^ radii df, £F. Then die tw^ 
triangles cdf, cef, having the two sides cd, of, equal tp 
die two CB, EF> and cf common, are mutually equilajteral; 
consequently they are also mutually eqmangular (th. 5), and 
have the two adjacent angles at c^qual to each other; there^ 
fore the line cf is perpendicular to ab (def. 1 1). 

Otherwise. 

Wh^n the Given Point c is near the End of the lin^. 

From any point d, assumed above the 
line, as a centre, through the given point 
c describe a circle, cutting the given line 
at E ; and through e and the centre D, 
draw the diameter edf ; then join cF, 
which will be the pci^ndicular required. 

For the angle at c, being an angle in a semicircle, is a 
right angle, and therefore the line cf is a perpendicular 
(bydef. 15), 




PROBLEM ly. 

From a Given Point a, to let fall a Perpendicular on a 

' given Line Be. * 

From the given point A as a centre, with ^ ^ 

any convenient radius, describe an arc, cut- 
ting the giving line at the two points d and 
E ; and from the two centres n, E, with 
any radius^ describe two arcs, intersecting 
at F 3 then draw agf, which will be per- 
pendicylar to ec as required. 

For, draw the equal radii An, AE, and 
DP, £.F. Then the two triangles adf, aef, having; the two 
sides AD> df, equal to the two ae, ef, and af common^ are 

mutually 



%: 




PHOBIXMS. SU 

inBtually equilateral; consequently they are also mutuallf 
equiangular (th. 5), and have the angle dag eqnal the angle 
BAG. Hence then, the two triangles adg, aeg, haying 
the two sides AD, ACry equal to the two ae, AGt and their 
included angles equals are therefore equiangular (th. 1), and 
have the angles at G equal; consequently a g is perpendicular 
to BC (def. 11). " 

Otherwise. 

When the Given Point is nearly Opposite the end of the 

Line. 

From any point d, in the given line 
B€, as a centre, describe the arc of a 
circle through the given point A, cutting 3—. /''' J -JgC 
90 in B ; ai3 from the centre b, with the *• .. ^\ 

radius ea, describe another arc, cutting 
the former in f ; then draw agf, which 
will be perpendicular to bc as required. 

For, draw the equafl radii da, df, and ea, ef. Then the 
two triangles dae, dfe, will be mutually equilateral ; conse- 
quently they are also mutually equiangular (th. 5), and,have 
the angles at d equal. Hence, the two triangles dag, dfg, 
having the two sides da, dg, equal to the two df, dg, and 
the included angles at d equal, have also the angles at G 
equal (th. l); consequently those angles at G are right 
angles, and the line AG is perpendicubr to dg* 



PROBLEM v« 

At a Given Point a, in a Line ab, to make an Angle Eqpial 

to a Given Angle c. 

From the centres a and c, with any one 
radius, describe the arcs de, fg. Then, 
with radius de, and centre f, describe an 
arc, cutting fg in o. Through G draw 
the line AG, and it will form the angle re- 
quired. 

For, conceive the equal lines or i:adii, 
DE, FG, to be drawn. Then the two triangles cde, afg, 
being mutually equilateral) are mutually equiangular (th. S), 
and have the angle at a equal to the angle c« 

FROBLBM 




S46 GEOMETRT. 



PROBLEM VI. 





Through a Given Point a, to draw a Line ParaQel ta a 

Given Line Bc. 

From the given point a draw a line ad 
to any point in the given line bc. Then 
draw the line eaf making the angle at A 
equal to the angle at d (by prob. 5); so 
shall £P be parallel to bc as required. 

For, the angle d being equal to the alternate angle A) the 
lines BC> bf, are parallel, by th. 1 9. 

PROBLEM VII. 

To Divide a Line ab into any proposed Number of Equal 

Parts. 

Draw any otlier line ac, forming any 
angle with the given line ab ; on which 
set off as many of any equal parts, ad, de, 
£F, PC, a^ the line ab is to be divided into. • 
Join BC ; parallel to which draw the other 
lines FG, £H, Di : then these will divide 
AB in the manner as required. — ^For those parallel lines di-» 
vide both the sides ab, ac, proportionally^ by th. 82. 

PROBLEM VIII. 

To find a Third Proportional to Two given Lines ab/ac. 

Place the two given lines ab, ac, 

forming any angle at a j and in ab take a a 

also AD equal to ac. Join bc, and A C 

draw DE parallel to it ; so will AE be r 

the third proportional sought. ^^^^C\ 

For, because of the parallels bc, dk, ^'^^ — Tiri 
the two lines ab, ac, are cut propor- 
tionally (th. 82) ; so that ab : ac : : ad or Ac : AE ; there- 
fore A£ is the third proportional to ab, ac. 

m 

PROBLEM ly. . 

' To find a Fourth Proportional to three Lines ab, Ac, ad. 

Place two of the given lines ab, ac, making any 
angle at A; also place ad on ab. Join bc; and paralld 

to 



PROBLEMS. 



3« 



to It draw ds : $o shall ae be the fourth 
proportional as required. 

For, because of the parallels Bc, de, 
the two sides ab, ac, are cut propor- . 
tionally (th. 82) ; so that - - » - - 
AB : AC : : AD : AE. 




PROBLEM X. 

To find a Mean Proportional between Two Lines ab, bc 

Place ab, ^c, joined in one straight a rB 

line AC : on which, as a diameter, describe 15 — c 

the semicircle adc ; to meet which erect 

the perpendicular £p \ and it will be the 

mean proportional sought, between AB 

and bc (by cor. th. 87). ^ A^ oTlfe 



, .1) 




PROBLEM XI. 

To find the Ceptre of a Circle. 

Draw any chord ab ; and bisect it per- 
pendicularly with the line ep, which will be 
a diameter {th. 41, cor.). Therefore cd 
bisected in o^ will give the centre, as re- 
quired.. 

PROBLEM XII. 

To describe the Circumference of a Circle through Three 

Given Points a, b, c. 

From the middle point b draw chords 
BA, Bc, to the two other points, and bi- 
sect these chords perpendicularly by lines 
meeting in o, which will be the centre. 
Then from the centre o, at the distance 
of any one of the point3, as OA, describe 
^ circle, and it will pass through the two 
Other points b, c, as required. 

For, the two right-angled triangles oad, obd, having the 
sides ad, pB, equal (by constr.), and od common with 
the included right angles a^ d equal, have their third sido« 
OA, OB^ also equal (th. I ). And in like manner it is shown, 
that oc is equal to ob or OA. So that all the three oA, be, 
pc, being equil^ will be radii of the same circle. 

' PROBLEM 




MS 



GEOMETRY. 



1 





PROBLEM XIII. 

To draw a Tangent to a Circle, through a Given Point a» 

When the given point a is in the cir- 
cnmference of the ciicle : Join a and the 
centre o ; perpendicular to which draw 
BAc» and it will be the tangent, by th, 46. 

But when the given pdint a is out of 
the circle: Draw ao to the centre oj 
on which as a diameter describe a semi- 
circlei cutting the given circumference in 
D ; through which draw badc, which 
will be the tangent as required. 

For, join do> Then the angle ado» 
in a semicircle, is a right angle^ and con- 
sequently AD is perpendicular to the ra- 
dius ix>> or is a tangent to the circle (th. 46). 

PROBLEM XIV. 

On a Given Line b to describe a Segment of a Circle> to^ 

Contain a Given Angle a 

At the ends of the given line make 
angles dab, dba, each equal to the 
given angle c. Then draw ae, B£> 
perpendicular to ad, bd ; and with the 
centre £, and radius ea or eb, describe 
a circle ; so shall afb be the segment 
required, as any angle f made in it will 
be equal to the given angle c. 

For, the two lines ad, bd, being 
perpendicular to the radii ea, eb (by c<Mistr.), are tangents 
to the circle (th. 46) ; and the angle A or b, which is equal 
to the given angle c by construction, is equal to the angle f 
in the alternate segment apb (th. 53). 

problem XV. 

To Cut off a Segment 'from a Circle, that sihall Contain a 

Given Angle G. 

Draw any tangent ab to the given 
circle ; and a chord ad to make the 
angle Cab equal to the given angle c ; 
then dea will be the segment required, 
any angle E made in it being equal to 
the given angle c. 

For 





PROBLEMS. 



S4f 



For 4he angle A, made by the tangent and chord, which 
is equal to the given angle c by construction, is also equal 
to any angle E in the alternate segment (th. 53). ' 



P|LOBL£M 2;VI« 

To make an Equilateral Triangle on a Given Line A3. 

« • 

From the centres a and b, with the 
distance iLB| describe arcsy intersecting in c. 
JDraw AC, bc, and abc will be the equi- 
lateral triangle. 

For the equal radii ac, bc, are, each of 
them, equal to ab. 




PROBLEM XVII.' 

To make a Triarigle with Three Given Lines ab, Ac, bc 

With the centre a, and-distance ao, 
describe an arc. With the centre B» and 
distance B€, describe another su'c, cutting 
the former in c. Draw ac, bc,. and 
ABC will be the triangle required. 

For the radii, or sides of the triangle, 
Ac, BC, are equal to the given lines AC| 
^c, by construction. 




PROBLEM XV m. 

To make a Square on a Given Line ab« 

Raise ad, bc, each perpendicular and 
^qual to AB ; and join dc ; so shall abcd 
ibe the square sought. 

For all the three sides ab, ad, bc, are 
«qual) by the construction, and dc is equal ^ 
and parallel to ab (by th. 24); so that all the 
four sides are equa]^ and the opposite ones are parallel. 
Again, the angle A or B, of the parallelogram, being a right 
angle, the angles are all right ones (cor. 1, th. ^2), Hence, 
then, the ii^re, having all its sides equals and all its angles 
xightf is a square (def. 34). 



4 



J \ 



PROBLEM 



S50 



GEOMETRY. 





PROBLBM XIX. • 

To make a Rectangle, or a Parallelogram} of a Given Lengtfi 

and Breadth, ab, bc. 

Erect ad, bc» perpendicular to ab, and 
each equal to bc ; then join Dc, and it is 
done» 

The demonstration is the same as the last 
problem. 

And in the same manner is described any oblique paral- 
lelogram, only drawing ad and BC to make the given oh" 
lique angle with ab, instead of perpendicular to it* 

PROBLEM XX. 

To Inscribe a Circle in a Given Triangle ABC. 

Bisect any two angles a and b, with 
the two linesvA d, bd. From the inter- 
section D, which will be the centre of 
die circle, draw the perpendiculars de, 
DF, DG» and they will be the radii of the 
circle required. 

For, since the angle DAE is equal to 
the angle dag, and the angles at £, g, 
right, angles (by constr.), the two triangles ade, adG, are 
equiangular ; and, having also the side ad common^ they are 
identical, and have the sides de, dg, equal (th. 2). In like 
manner it is shown> that dp is equal to de or dg. 

Therefore, if with the centre D, and distance DE, a 
drcle be described, it will pass through all the three point* 
E, F, G, in which points also it will touch the three sides of 
the triangle (th. 46), because the radii de^ df, dg, are per- 
pendicular to them. ^ 

PROBLEM XXI. 

To Describe a Circle about a Given Tjiangle abc^ 

Bisect any two sides ieith two of the 
perpendiculars db, df, dg, and d will ho^ 
the c^itre. 

For, join da, db, dc. Theft the two 
right-angled triangles DA£,DBE,have the 
two sides DE, ea, equal to the two de, ^. ^ . ^ 

EB, and the included angles at e equal : 
those two triangles are therefore identical 




PROBLEMS. 



351 



(th. 1), ^nd have the side da equal to DB. In like manner 
it !s shown, that dc is also equal to da or db. So that all 
the thre^ da, db, dc, being equal, they are radii of a circle 
passing through A, B, and c. * . , 



PROBLEM XXII. 

To Inscribe an Equilateral Triangle in a Given Circle. • 

Through the^ centre c draw any dia- 
meter AB. From the point b as a centre, 
with the radius bc of the given circle, 
cLescribe an arc dce. Join ad, ae, de, 
and ad£ is the equilateral triangle sought. 

For, join db, do, eb, ec. Then dcb 
is an equilateral triangle, having each 
side equal to the radius of the given cir- 
cle. In like manner, bc£ is an equilateral triangle. But 
the angle ade is equal to the angle ABB pr cbe, standing 
on the same arc ae % also the angle aed is equal to the 
angle cbd, on the same arc ad ; hence the triangle dae has 
two of its angles, ade, aed, equal to the angles of an 
equilateral triangle, and therefore the third angle at a is 
also equal to the same ; so that triangle is equiangular, and 
therefore equilateral. 




problem XXIII. 

To Inscribe a Square in a Given Circle. 

Draw two diameters Ac, bd, crossing 
at right angles in the centre e. Then 
join the four extremities a, B, c, d, Vith 
right lines, and these will form the in- 
scribed square abcd* 

For the four right-angled triangles 
aeb, bec, ced, dea, are identical, be- 
cause they have the- sides ea, eb, ec, ed, 
all equal, being radii of the circle, and the 
four included angles at e all equal, be- 
ing right angles, by the construction. Therefore all their 
third sides ab, bc, cd, da, are equal to one another, and the 
figure ABCD is equilateral. Also, all its four angles, a, b, c, d, 
are right ones, being angles in a semicircle. Consequently, 
the figure is a square. 

problem 





nsi GEOMETRY. 

prob;.£M XXIV. 

Tp OescrU)e a Square about a GIvoi Circle; 

Draw ^Dro cliameters ac, BD|Cro$smg 
at right angles in the centre e. Then 
through their four extremities draw.FG, 
IH, parallel to AC^ and ft, gh, parallel 
to BDj and they will form the square 

FGHI. 

Forj the opposite sides of parallelo- 
grams being equals fg and IH are each 
equal to the diameter Ac^ and fi and gh each equal to the 
diameter bd ; so that the figure is equilateral. Again, be- 
cause the opposite angles of parallelograms are equal, all the 
four angles f^ g> h, i, are right angles, being equal to the 
opposite angles at £. So that the figure fghi, having its 
sides equal, and its angles right ones, is a squarej and its sides 
touch the circle at the four points a, b, c, d, being perpen- 
dicular to the radii drawn to those points. 

PROBLEM XXV. 

To Inscribe a Circle in a Given Square* 

Bisect the two sides ^o, fi, in the points a and s 
(last fig.)» Then through these two points draw ac parallel 
to FG or iH, and bd parallel to fi or CH. Then the point 
of intersection e will be the centrcj and the four lines^EA, 
£B, EC, £D, radii of the inscribed circle. 

For, because the four parallelograms ef, eg, eh, ei, have 
their opposite sides and angles equal, therefore all the four 
lines EA, £B, EC, ED, are equal, being each equal to half a 
side of the square. So that a circle described from the centfe 
X., with the distance £A, will pass through all the points 
A, 9, c, D, and will be inscribed in the square, or will touch 
its four sides in those points, because the angles there are 
right ones. 

PROBLEM XXVI. 

To Describe a Circle about a Given Square, 
(see fig. Prob. xxiii). 

Draw the diagonals /c, bd, and their intersection t 
will be the centre. 

For the diagonals of a square bisect each other (th. 40), 
making E4, eb, eg, ed, all equal, and consequently thes^ 
are radii of a circle passing through the four points a, b, c, d. 

PROBLKM 



* 

■i 



/- .. .» 




1^1 



1»RDBLEM XXYU; 



.!■ '} 



Td Cut a CiVen L»n« ja Ejttreme and Me^an Ratb» 



XjsT. AB be the eiven line to be divK 
in e^ictreine and mean ratio, that is, so 




divided 
so as 
tKat the whole line maybe to the greater . 
p^t,»a$ the greater part is to the less part. 

I>ra1?:BC perpendicular to. AB, and equal 
to .half AB* Join AC Vrapd with tentre c . 
and disftance Qit, describct the circle. sd| 
then with centre A arid distance ad, 4e* 
scribe the arc DB ; so shall ab be divided in 
£ in extreme and mean ratioj or so that 
AB : A£ :: AE : eb'. . 

For, produce Ac to thd circumferencef at ^. Then, ABfF 
being a secant, and ab a tangent, because b U a right angle : 
^ therefore the rectangie^AF.AD is^qual to ab* (cor. 1 th. Gl); 
consequently the means and extremes of these are proportional 
(th. 17), viz. AB : af or ad -f- rip : : ad : Ab. But a« 
is equal to ad by construction^ and ab = 2bc r: d^j 
therefore, ab : Ate' + ab :: a^ : ab^ 

mid by division^ ab : ▲£ : : ab ; sb; « 

t 

PROBLEM XXYllU 

To Inscribe an Isosceles Triangle in a Givefn Circle, that 
shall have each of the Angles at the Base Double the' 
Angle at the Vertex. ^ 

iDKAW any diameter ab of th^ givea - 
circle ; and divide thji radius CB, in thef 
point d, in extreme and mean ratio, by the 
last problem. From, the pomt b apply the 
chords bB, bf, each equal to the greatef 
part CD. Then join Afe, af, ef j and Aef 
will be the triangle requif ed* 

For, the chords Bg,' b1*, befing equals 
their arcs are equal j therefore the supplemental arcs and 
chords AE, AF^ are also equal 5 consequently the triangle ajsp 
is isosceles, arid haa the angle B equ^l to the angle F$ aiscX 
the angles at G are right angles. "V 

Draw cf and Dt. Then, »c : «d : : cd : Bd, or 
Be : Bf : : bp : fiiD'by constr. And ba : bf : : bf : bg 
^by th. S7)i But bC = ^^ba j therefore bg = ^bd = gd j 
therefore the two triangles cbf, cpFi are identical (th. 1),! 

Vox.. L A a and 




SI4 



GEOMETRY. 




and each equiangular to ibf and act (th. 87). ^ Tberefori 
their doubles, BFb, af£| are ^isosceles and eqtiiangfiitar> as 
well as the triangle bcf ; having the two sides bc» cf, equals 
and the angle b common w^h the triangle bfiX Buf cet 
is ^ DF or BF ; th^efore the angle c == the angle dfc 
(th. 4) ; consequently the angle BDi-, which is equal to the 
sum of these two equal angles (th. 16), is double of one of 
them c; or the equal angle b or cfb double the angle C. 
So that cbf is an isosceles triangle, having each of its two 
equal angles double of the third angle c. Consequently the 
triangle abf (which k has been shown is equiangular to the 
triangle cbf) has sdso each of its mg\e» at the base double 
the angle a at the vertex^ 

PROBLEM XXIX. 

To Inscribe a Regular Pentagon in a Given Crcle. 

Inscribe the isosceles triangle abc 
having each of the angles abc, acb, 
double the angle ]6ac (prob. 28). Then 
bisect the two arcs adb, aec, in the 
points D, E ; and draw the chords ad, d^, 
a£, £€, so ^ shall ADBCE be the inscribed 
equilateral pentagon required. 

For, because equal angles stand on equal arcs, and, double 
angles on double arcs, aho the angles arc, acb, being each 
double the angle bag, therefore the arcs adb, aec, subtending 
the two former angles, one each double the arcs Be subtending 
the lat;ter. And since the two former arcs are bisected in D 
and £, it follows that all the five arcs ad, db, bc, ce, ea, 
are equal to each other, and consequently the chords also 
which subtend them, or the five sides of the pentagon, are 
all equal. 

Note* In the construction, the points d and e are most 
easily found, by applying bd and C£ each equsri to bc^ 

problem' XXX. 

To Inscribe a Regular Hexagon in a Circle. 

Apply the radius Ao of the given circle 
3s a chord, ab, bo, cd, &c, quite round the 
circumference, and it will complete the re- 
gular hexagon abcdep. 

For, draw the r.uiii ao, bo, co, i5o, Eo, 
Fo, completing six equal triangles } of* 
which any one, as Abo, being equilateral 

(bf 




PROBLEMS^ iU 

(jby constr.) its i£ree angles ^ all equal (cor. 2, th. 3), and 
any one <»f them, as-^bs, is one-third of the whole, or of tw^ 
right iandes (th. IT), or bne-sixtii of four right angles. But 
the whole circumfeirence is the measure of fchir right angles 
(ipor. 4, th. 6). Therefore the arc ab is one-sixth of the 
tircumference of the circKj and consequently 4ts chord Ap 
one side of an isquilateral hexagon inscribed in the circle* 
And the sanie bf the other thor<k* 

Cpr'of. The side pf a regular hexagoh is equal to th6 radius 
bf the circumscribing circle, or to ihi chord of oii^si^h 
{tet df thie circumference: 

ipROBtfiM xtxi: - 
To descnbe a Regular Pentagon or Hexs^n about a Circle. 

In the given circle inscAbe i^ regular 
polygoh of the szm6 name br hiuhber 
of sid^s, ' as abcde, by one of the 
foregoing problems. Then to all its 
angular points draw tangents (by' 
prob. 13), and these will form the cir* 
cumscribing polygon required. 

For, all the chords, or sides 6t / . 

the insc^ribing figure, ab. Be, &c, beiiig equal, and all thl* 
radii Oa, ob, &c, being equal, all the vertical angles about the 
point 6 are equal. But the angles OBF, oaf^ oXd, obc,- 
made by the tangents and radii, are right singles; therefore 
b£F 4- OAF = tWd Hght iUigles, and oAg + 6qg = two 
right angles j consequently, also, Ab& + Af5 =£ two right 
angles, and Aob + agb :£: twd right angles (cor. 2, th. 18): 
Hence, then, the angles Aofi + Afs being =at aob + acb» 
of which aob is = aoe } consequently the remaining angles 
F and G are also equal; In the same manhclr it is shown^ 
that all the angles f, G| Hy I, k, are equal. 

Again, the tangents from the same point #e, f a^ ari equal> 
as alsb thie tstogents AG, gb (con 2) th. 61,) ; and the angles 
F and G of the isosceli^ triangles afe, ag^, are 6qual^ as 
weir as their opposite sides ae, ab j coiisequently those two 
triangles are identical (th. l)i and have thWr oth^r sidei 
£F, fa, Ag, gb, dll equal, and fg equal to the double of 
iny one of them: In like manner it is shown, that all the 
bther sides gh, hi^ ik, kF, are equal tb FcT, or double of th^ 
tangents gb, bh, &c. 

Hence, then, the circumscribed figure is bcith equilaterat 
and equiangular, which was tb be showa* 

A a 2 C,ro/. 




ceotfisfrsuH 



'Cor$l Tl\ei 



icirdc touches^theiBftddlnQfdie:^j€t 



To Iivscribe a Circle In a Regular Polygon* 

BisjEOT any two sides of the polygon 
by the perpendiculars go^ fp» and tmir 
intersection o. will be the centre of the 
inscribed circle^ and OG or of will b<r 
the radius. 

For the perpendicul^s tq the t^ngei^ts 
AF, AG, pass through the centre (cor. 
th« 47); and the insciibed circle touches . 
the middle points f, G| by the last corollary, .AlsOf the twm 
sides AG, AO, of the right-angled. triangle aoO« being equa) 
to the two sides AFf Ao,. of the right-angled triangle aof, the 
third sides of, og, will also be equal (cor. th. 45). Therefore 
the circle described witH the centre o and radius oa, will 
pa^ through f, and will touch the sides in the points g ami 
F. And the same for all the other sides of the figure^ 





PROBLEM XXXIII. 

To Describe a Circle about a Regular Polygon. 

BisBCT any two pf the angles, c and Dj 
\vith the lines co, do 5 then their inter- 
section o will be the centre of the cir- 
cumscribing circle } and QC, or OD, will 
be the radius. 

For, draw ob, oa, oe, &c, to the 
angular points of the giten polygon. 
Then the triangle ocb is isosceles, having the angles at c 
and D equal, being the halves of the equal angles of the 
polygon BCD, ODE ; therefore their opposite sides Co, do, 
are equal (th. 4)« But the two triangles ocd, ocb, having 
the two sides oc, cd, equal to the two oc, cb, and the in- 
cluded angles ocdj ocb, also ^qual, will be identical (th. l), 
and have their third sides bo, od, equal. In like manner it 
is shown, that all, the lines. o a, ob, oc, od, oe, ai'c equal. 
Consequently a circle described with the centre o and radius 
oa, will pass through all the other artgular points, B, c, D, 
&c, and will circumscribe the polygon. 

pkoblsm 



■FROSLEMS. 



857 




5 IJ » 



PROBLEM ZXXIV. 

To make a Square Equdi to the Sum ol two ^w I^ore Given 

^Squares^ 

Let ab and ac be the sides of two 
given squares. Draw two iadefihite 
lines AP, AQ9 at right singles 'to each 
other I in which place the sides ab^ ac, 
of the given square^ ; jom bc ; then a 
square described on bc, will be equal to 
the sum of the two squares described oh 
AB and AC (th. S4). 

in the ^ame-lnanaer, a sqiiare may beiisnadie jequai , to tbi^ 
sutli of the tfaoree ($rfiii>re giVen sqi^ares. -Foiyif 4^9 AQgAHh 
be taken as the sides of the given square^^i then,^ makio^ 
' A£ = Bc^ AD =: AD, and drawing de^ it is evident that the 
square on de will be equal to the sum of the three squares 
on AB, AC5 AD. Andjjo on for more squares. 

PkOBLlE^ XXXV. • 

To make a Square Equal to the Difference x^yf two Given 

Squares. 

Let ab and A'c, tikett in the sa'me 
straight line> be equ^l to the sides of th^ 
two given squares.-^Fh5iil th^ cfehtte A, 
with the distance ab, describe a circle ; and 
make cd perpendicular to ab, meeting the 
circumference in d : so shall a square described on cD be 
equal to ad^— ac% or ab*— ac% as requiri^ ^cpr, th* 34); 

pRbBLfini xxxf I. 

To make a Triangle Eqtod to ^ Given Qtiadn^de ABcb. 

Draw the diagonal ac, ana parallel 
to it i^, meting ba produced at £, and. 
join C£ ; then will the triangle C£B be 
equal to th^ given quadrilateral abcd. 

For, the two triangles ace, acd, her 
ing on the same base ac, and between 
the same parallels ac, D£, are equal (th^ 25) ^ therefore, 
if ABC be added to each^ it wjU znake bce equal to abc^ 
(ax. 2). 

PiU)BL£M 





S5» 



GEOMETRt. 



PROBLEM ZXXVir. 

To make a Triangle Eqoal l;o a Given Pentagon abcde* 

DiCAw DA and db, and also ef. 
eC) parallel to them, meeting ab pro- 
duced at F and g ; then drai^ df and 
T>G ; so shall the triangle dfg be equal 
to the given pentagon abcd'e. • 

For th^ triangle d^a = deAi and 
the triangle dgb ss dcb (th. 25)} 
therefore! by adding DAB'to the equals^ 
the tinms are equd (ax. 3), that is, dab «f DAF + DBG 
SB' DAB 4- p^E + DBC> or the triangle dfg as to the 
{>entagon ABODE* 




^ 




t-^ 



PHOBUEM XXXVIlt. 

f t 

To make a Rectangle Equal to a Given Triangle Ape. 

' Bisect the base ab In d; then raisf 
D£ aiid BF perpendicular to ab, and 
meeting cf parallel to ^b^ at B and f : 
so shall DF be the rectangle equal to the 
given trismgle abc (by cor. 2, th. 26). 

PROBLEM XXXIX. 

To make a Square Equal to a Given Rectangle abjqO* 

FB.0DUCE one side ab, till ^e be 
equal to the other side bc. On as as 
a diametei* describe a circki meeting 
BC produced at f : then will bf be the 
side bf the square bfgh, equal to the 
given rectangle bd, as required; as ap^ 
pears by cor. th. 87, and th. 77. 



vT ^.«'* .* 




ATTT 



*.; » 



APPLICATION 






i 350 ] 



•t * 



APPLICATION o» ALGEBRA 

TO 

- ' GEOMETRY. 

W H E N it is proposed to resolve a gtometrical problem 
algebraically, or by algebra, it is proper, in the first place^ 
to draw a figure that shatltepresent the several parts or con- 
'ditions of the problem^ and to suppose that figure to be the 
true one* Then, hairing considered attentively the nature of 
the problem, the figure is next to be prepared for a solution, 
if necessary, by producing or drawing such lines in it as ap« 
j)ear mcfitx^onducive to tbit end. This done, the usual sym« 
bols or letters, for known and.unknown quantities, are em* 
ployed to denote the several parts of the figure, both the 
knbwn and unknown parts, or as many of them as necessary; 
as also such unknown line or lines as may be easiest found, 
whether required ot not. Then proceed to the operation, 
by observing the relations that the several parts of the £gure 
have to each other \ from which^ and the proper theorems 
in the foregcMug elements of geometry, make out as many 
equations independent of each other, as thel« are unknown 
quantities employed in them : the resolution of which equa- 
tions, in the same manner as in arithmetical problems, will, 
^determine the unknown quantities, and resolve the problem 
proposed. » . 

As no general rule can be given for drawing the lines, and 
selecting the fittest . quantities to, substitute for, so as always 
to bring out the most simple conclusions, because different 
problems requure different mode^ of solution^ the best way to 
gain experience, is to try the solution of the same problem 
in different ways, and then apply that which succeeds best, 
to other cases of the sanaeicind, when they afterwards occur. 
The following particular directions, however, may be of 
some use. 

\4t% In preparing the figure, by drawing lines, let them bo 
either parallel or perpendicular to other lines in the Bgurey 
Or so as to form similar triangles. And if an angle be given; 
it will be proper to let the perpendicular be opposite to that 
anglei ftnd to fall from one end of a given line^ if possible. 

2rf, 



SCO APPLICATION or AtGEBRA 

2di In selecting the qu^mities proper to substitute fopj^ 
those are to be phosen, whether required or not, which li^ 
nearest the known or given parts of the figure, and by means 
of which this 9j^jt gdj^ceqt p^t$Aiy3^Iexp/jpss!Bd by addi- 
tion and subtraction only, without using surds. 

3^9 When two lines or quantities are alike related to other 
parts of the figure or problem, the best way i§, not to make 
Dse of either* pf them separately, but to substitute for their 
sum, or diflfer^nce, or rectangle, or the sum of their alternate 
quotients, or for some line or lines, in the figure, to ^hidi 
tKey have botji thp ^aipe t^hx\on*^. • \ t , 

. . 4/A, When thei ayea^ or the p^metpr, of a. figure, b p^^n^ 
pat such parts of it as have <^y .^ temotfi .rdation /to the 
parts required: it is sometime! .of pse to assume asuathies 
figure similar to the proponed om^f having. one side iiqualito 
linity, or some other known qnastity. Fpr, hepce th» qthtc 
parts of the figure may be &und, by tbe/kifow2\ proportions 
pj'the like sides, Or parts, ind h> an equ;iti6a hei«btainfid|? 
For {examples, takf tJb^ following .pcobleins, : j - '« 



Jn a RighUangkd Triangle^ having f^ven thf JB^sf (&)^ and iik 
.^ Sum 7f the Hyfoihenusc and Berpfndict^Iar (9) ; to, find ba^ 
these two Sides. ' . > , , 

Li^T. ABC represent the proposed triangle, 
right-angled at b. Put the base ab = 3 =:^ ^ 
and the sum AC + pc of tlie hypothehus.^ ' ^ 
ihd perpendicular = 9 == j; also, let 'jr iae- 
pote the hypothenuse AC, and jj; the perpen- 
dicular BC. 

Then by the question r-'^-f+j'—'Cj 
and by theorem 34-, - - " - - or* =i / -1- ^J 
By transpos. y in the 1st jcqu. gives ^ = / — j;, 
"JThis valup of X substi. in the 2d, 

-gives - - - - - ^* - '2/y -J* / =: f + (% 

Takingaway/onbothsideslcayes r — 2sy = ^% 

Bv tr^spo?, ^sy and i% gives - /* — i^ == 2/v, 

■ ' ^ J* - ^* 

And dividing hy2s, gives -r •. ■ •.'. ■ =^ jp ?s 4?=?:]9C» 

Hfnce x=^j — y==5;= Ap. 

' N. B. In thi§ solQt.}pns and the^fQlIowipg onfs^ ^^ iiota^; 
^opi§ miidc? b$ jising ^ m^l tmb^9V© l^Umi ^ ?ftd jr^ a% 

ther^ 




there are tHikpearo ;»cleff^f the triangle^ a^par^e letter far 
each; ippceft^igiKfi |,o ais^ng .only -pije ^\nknoMrn letter iicir 
ox\f «de, Jttid i^pfcee^ng ^e othfff mknown side in ticvms 
of that letter ai\d the^iyen sum or difference of tli^e sides; 
though this'l^at4:er, way Y'^ould render the sqlation shorter and 
SQDlier J because the fbrn^er way, gives occasion for more and 
better practice in reducing equations, which is the very end 
aQd reason for which these problems are given at all« 



t * JP.ROBLEM II- 

/« a Right-angled Triangle J having given the Hypothenuse (5) ; 
and the Sum of the Base and P&rpendicular (7) ; tofiftd both 
these two Sides. 

^ Let A3C represebtt the prp.posed triartgle, right-angled at 
p. Put the given hypQthen.usie Ac = 6 zr ^?, and the sum 
AB + BC of the base and perpendicular = 7 = / ; also let -r 
denote the base ab, and j> the perpendicular fiC. 

. Then by the question <- •- • - ;r rf- y-^ s 

< ^A {yy tl^^rem 34 . - .-^ t *• ^* +/ == «* 

,JJy Irajnspos. yin.tb^ 1st, gives • :c v?: s -r- y 

Byaabstitu. th^v^lu. for -r, gives, x* -r 2jy + 2j?* = tf* 

By transposing /% gives - ^ 2/ — ^J)? = a* — j* 

By dividipg.t)y"2, gives - - - f — sy zn ^a^ — ^ 

JBy .cpmpl^^iiig tjbte s<jLiare; gives / — j;y + ^^^ = 4^* — i^, 
By e|Etracjfeij3g the root, gives - y ^ is ^s/^d^ — ^s*- * 

J^^^ transposing -JsT, gives - - y = -Jj" ± V'i^^ — :J/ = 

4 and 3, the values of x and j». 



PROjBLEM III. 

In a Rectangle i having giveft the Diagonal {10\ and the Penmen 
terj or Sum of all the pour Sides (28) 5 toJin4 (^'^h of the Sides 
severally. 

Let abcd be the proposed rectangle; 
and put the diagonal AC = 10 = dy and 
half the perimeter AB + BC or Ap + 
PC = i 4 = a ; also put one side ab = .r, 
and the other side BC =? y. Hence, by -^ 
right-angled triangles, - - - - .r* + / = //* 
And by the question -- - - ^+^ = a 
Thk^n by transposing)} in the 2d, givers x :ss a — y 
Thisvaluesubstitutedinthe lst,£iye* a^ -r 2^ + 2/ =» ^ . 

Transposing 




362 



APPUCATION OP ALGEBRA 



Transposing tf*, gives - - - 2)>* — ^ay nt' -^ if 

And dividing by 2, gives - - j^ — jrjr =: 4rf* — 4** . 

By completing the square, it is / — «^ + ^ =s -^ * * ^a* 

And extracting the root, gives y — ■J^srv'i^— J^* 

And transposing ia, «vcs - y = i^ ± -/i^ "• i** = •^ 
or 65 the values oix and y. 



PROBLEM IV. 

Having given the Base and Perpen£eular of any Triangles ta 
find the Side of a Square Inscribed in the same* 

Let ABC represent the given ,triangle» 
and EFGH its^ inscribed square. Put the - 
base AB = ^/the perpendicular CD = ^ 
and the side of the square gf or gh =* 
Di == X i then will ci = CD — di =x 
« — or. ' 

Then, because the like lines in the 
Similar triangles abc, gfc, are propor* 
tional (by theor. 84, Geom.)> ab : cD : : GB : ci, that 
hf h I a ', \ X \ a ^ X* Hence» by multiplying extremes 
and means, ai^ix sz ax, and transposing ix, gives ai ^ax 

+ hx^ then dividing by a + i, gives x = — -j--j = or 

or GH the side of the inscribed square : which therefore is 
of the same magnitude, whatever the species or the angles of 
the triangles may be. 




PROBLEM V. 

In an Equilateral Triangle, hatting given the lengths of the three 
Perpendiculars, drawn from a certain Point within, on the 
three Sides ; to determine the Sides. 

Let ABC represent the equilateral tri- 
angle, and D£, DF, DG, the given per- 
pendiculars from the point d. Draw the 
lines DA, DB, DC, to the three angular 
points; and let fall the perpendicular cu 
on the base ab. Put thje ttu*ee given per- 
pendiculars DE = tf, DF = h, DG = c, 
and put jr =z AH or bh, half !th« side of 
the equilateral triangle. Then is ac or bc = 2 x, and by 

right -angled t riangles the perpendicular CH = v^AC*— ah* 

= V4.r* — X* SB v'S^r* = x^ a. 

Now, 




% TO GEOMETRY. M 

Now, since the area or space of a rectangle, is expressed 
by the product of the base and height (cor. 2, th. ^1 GeQlD.)!^ 
and that a triangle is equal to halfa rectangle of equal base 
and height (con 1, th. 26), Jt follows that, 

the whote triangle abc is =tAb x ch =ir x x-/S=n4r*^St 

the tfiangle^BD 5= ^ab x t)G = j: x r = «r, 
the triangle bcd =^ ^bc x de = o: x « = «x, 
the triangle acd = ^ac x dp = a:* x * = ^x. 

But the three last triangles make up, or are equal tm^ llie 
whole former, or great triangle; 

that is, ^v^3 :=: ax -^-hx ^ cx'^ hence, dividing byx^i^ 
X ^3 :=: a +i +^> and dividing by -/f, " 

X = -jjr — ^, half the side of the triangle sought. 

.. Also, 3ince the whole perpendicular ch is = x^% it it 
therefore =: ^ + J + ^. That is, the whole perpencBcofar 
CH, is just equal to the sum of all the three smaller perpen- 
diculars DE + DF + DO taken together, wherever the point 
D is situated. 

PROBLEM VI. 

In a Right-angled Triangle, having given the Base (S)» 
and the Difference between th^ Hypothenuse and Perpendi- 
cular (1) 3 to £nd both these two Sides. 

PROBLEM VII, 

In a Right-angled Triangle, having given the Hypotbenme 
(5), and &e Difference between the Base and Perpendknhr 
(J } $ to determine both these two Sides. 

PROBLEM VXII. 

Having given the Area, or Measure of the Space, <if a 
Rectangle, inscribed in a given Triangle \ to iletermiiie the 

Sides of the Reetangle. 

... . . ^ - - , . 

^ PROBLEM IX. 

In a Triangle, having ^ven the Ratio of the two Sidfls^ 

together with both the Segments of the Base, made by a 
t^erpendicular from the Vertical Angle ; tp determine the 
Sides of the Triangle. 

PILOBLEM X. 

In a Triangle, having given the Base, the Sum of the 
other two SidtS| and the Length of a Lint drawn from the 

Vertical 



• I 



SM APPLICATIOH 69 ALGEBRA 

Vertical Angle to the Middle tkftheiBate; to find the odes 
of the Triangle* 

PR09JUEM XI. 

In -tt Triangle, having given the two Sides abotit At 
Vertical Angle, with the Line bisecting that Angle^ and tei^ 
ininating in iht Base i to find the Base* 

PROBIJSM XIZ. 

To determine a Right-angled Triangle; Imving given the 
Lengths of two Lines drawn from the acute angles^ to the 
Middle of the opposite Sides. 

PROBLEM Xllt. 

To determine a Right-angled Triangle; hiiving giretl the 
Perimeter^ »ad the Radius of its Inscribed Circle. 

PROBLEM XIV. 

T^o determine a Triangle ; having given the Base> the 
Perpendicular, and the Ratio of the two Sides. 

PROBLEM XV. 

To determine a Right-angled Triangle ; having given the 
H jpothenuse> and the Side of the biscribed Square. 

* • 

PROBLEM XVI. 

To determine the Radii of three Equal Circles, described 
in a giv^n Circle, to touch each other and also the Circum- 
ference of the given Circle. 

^ PROBt-EM XVlI. 

In a Right-angled Triangle, having given the Perimeter, or 
Sum of all the Sides, and the Perpendicular let fall from the 
Right Angle on the Hypothenuse ; to deterihine the Tri- 
angle, that is, its Sides. 

PROBLEM XVin.. 

To. determine a Right-^angled Triangle; having given the 
Hypothenuse, and the Difference of two Lines drawn from 
tlie two acute* angles to the Centre of the Inscribed Cii-cle. 

PROBLEli 



/" 



' / 



TO GEOMETRY. 365 






PROBLEM XIX. 



To determine a Triangle; having given the Base, the Per- 
pendicular, aikd the Difi^nce of tHe tgvvt) other Sides. 

PROBLEM XX. • 

To determine a Triangle; having given the Base, the Per- 
pendicular, and the Rectangle or Product of the two Sides. 

N PROBLEM :^xi. 

To determine a Triangle ; having given the Lengths of 
diree Lines drawn from the three Angles, to the Midcfie o£ 
the opposite Sides. 

PROBLEM XXII. 

In. a Triangle, having given all the.thsee Sides; to find 
the Radius or the Inscribed Circle. 

PROBLEM XXIII. 

To determine a Right-angled Triangle; having given the 
Side of the-Inscribed Square, and the Radius of the Inscribed 
Circle. 

PROBLEM XXIV. 

To determine a Triangle, and the Radius of the Inscribed 
Circle ; having given the Lengths of three Lines drawn 
from the three Angles, to tlie Centre of that Circle. 

PROBLEM XXV. 

To detefirSnc a Right-angled Triangle ; having given the 
Hypothenuse, and the Radius of the Inscribed Circle. 

PROBLEM XXVI- 

To determine a Triangle; having given the Base, the 
line bisecting the Vertical Angle, and the Diameter of the 
7 Circumscribing Circle. 



LOGARITHMS 



LOGARITHMS 



OF THE 



NUMBERS 



tROtt 



1 tQ 1000. 



aBBSB 



N. 


Log. 


N. 


Log. 


N. 


Log. 


N. 
76 


Log. 


1 




26 


1-414973 


51 


1-707570 


1-880814 


riT. ♦ »:rt:fi 


« 

2 


0-3O103O 


27 


1-431364 


52 


1-716003 


77 


1-886491 


S 


0*477121 


28 


1-447158 


53 


1-72*276 


78 


1-8^2095 


4 


0*602060 


29 


1-46239S 


54 


1-732394 


79 


1-897627 


5 


0-698970 


'so 

1 


1-477121 


56 


1-740363 


80 


1-903090 


€ 


0-778151 


Isi 


1-491362 


56 


1-748188 


81 


1-908485 


7 


0-845098 


32 


1-505150 


57 


1-755875 


82 


1-913814 


8- 


0-903090 


33 


1-518514 


58 


1-763428. 


83 


1-919078 


9 


0^54243 


34 


1-531479 


59 


1-770852 


84 


1-924279 


10 


1-000000 


35 


1-5440P8 


60 


1-778151 


85 


1-929419 


11 


1041393 


36 


1-556303 


«1 


1-785330 


86 


1-934498 


12 


1'079181 


37 


1-568202 


62 


1-792392 


87 


1-939519 


13 


Ml 3943 


i38 


1-579784 


63^ 


1-799341 


68 


1-944483 


14 


1-146128 


139 


1-591065 


64 


1 -806 1 80 


89 


1-949390 


15 


1-176091 


40 


1-602060 


65 


1-812913 


90 


1-954243 


16 


1-204120 


41 


1-612784 


66 


1-819544 


9iL 


1-959041 


17 


1 ^30449 


42 


1-623249 


67 


1-826075 


92 


1-963788 


IS 


1-255273 


43 


1-633468 


68 


1-832509 


93 


1-968483 


19 


1 -278754 


44 


1-643453 


69 


1*838849 


94 


1-973128 


20 


1-301030 


45 


1-653213 


70 


1-845098 


95 


1-977724 


21 


1-822219 


46 


1-662758 


71 


1-851258 


96 


1-982271 


22 


1-342423 


47 


1 -672098 


72 


1-857833 


97 


1-986772^ 


25 


1-361728 


48 


1-681241 


73 


1-863323 


98 


1-991226 


24 


1-380211 


49 


1-690196 


74 


1-869232 


99 


1-995635 


25 


1*397940 


50 


1 -698970 


75 


1-875061 


'100 


2-000000 



N« B. In the following table, in the last nine colunlns of eech page, where 
the first or leading figures change from 9's to O's, large dots are now in- 
troduced instead 9f the 0*s through the rest of the line, to catch the eye^ 
and to indicate that from thence the corresponding natural number ia 
the^rst column stand's in the next 'lower line, and its annexed first tiiv'C* 
figures of the Logarithm ia the second ^column; 












LOftARrrHMS. 








>67 


N.' 





1 


2 


3 


4 


5 


6 


7 


8 


9 


100 


DOOOOO 


0434 


0868 


1301 


1734 


2166 


2598 


3029 


3461 


31891 


101 


4^4321 


4751 


5181 


5609 


6038 


6466 


6894 


7321 


7748 


8174 


102 


8600 


9026 


9451 


9876 


•300 


•724 


1147 


1570 


1993 


2415 


103- 


012831^ 


3259 


3680 


4100 


4521 


4940 


5360 


5779 


6197 


6616 


104 


7033 


7451 


7868 


8284 


8700 


9116 


9532 


9947 


•361 


•775 


105 


021189 


1603 


2016 


2428 


2841 


3252 


3664 


4075 


4486 


4896 


106 


5306 


5715 


6125 


6533 


6942 


7350 


7757 


8164 


8571 


8d78 


107 


9384 


9789 


• 195 


•600 


1004 


1408 


1812 


2216 


2619 


3021 


lOS 


033424 


3826 


4227 


4628 


5029 


5430 


5830 


6230 


6;£29 


7028 


109 


7426 


7825 


8223 


8620 


9017 


9414 


9811 


•207 


•602 


•998 


no 


041393 


1787 


2182 


2576 


2969 


3362 


3755 


4148 


4540 


4932 


111 


5323 


5714 


6105 


6495 


6885 


7275 


7664 


8053 


8442 


8830 


112 


9218 


9606 


9993 


•380 


•766 


1153 


1538 


1924 


2309 


2694 


113 


053078 


3463 


3846 


4230 


4613 


4996 


5378 


5760 


6142 


6524 


114 


6905 


7286 


7666 


8046 


8426 


8?<05 


9185 


9563 


9942 


•320 


115 


060698 


1075 


1452 


1829 


2206 


2582 


2958 


3333 


3709 


4083 


116 


4458 


4832 


5206 


5580 


5953 


6326 


6699 


7071 


7443 


7815 


117 


8186 


8557 


8928 


9298 


9668 


••38 


•407 


•776 


1145 


1514 


118 


071882 


2250 


2617 


2985 


3352 


3718 


4085 


4451 


4816 


5182 


119 


.5547 


5912 


6276 


6640 


7004 


7368 


7731 


8094 


8457 


8819 


120 


9181 


9543 


9904 


•266 


•626 


•987* 


1347 


1707 


2p67 


2426 


121 


082785 


3144 


3503 


3861 


4219 


4576 


4934 


5291 


5647 


6004 


122 


6360 


6716 


7071 


7426 


7781 


8136 


8490 


8845 


9198 


9552 


123 


9905 


•258 


•611 


•963 


1315 


1667 


2018 


2370 


2721 


3071 


124 


093422 


3772 


4122 


4471 


4820 


5169 


5518 


5866 


6215 


6562 


,125 


6910 


7257 


7604 


,7951 


8298 


8644 


8990 


9335 


9681 


•026 


126 


100371 


0715 


1059 


1403 


1747 


2091 


2434 


2777 


3119 


3462 


127 


3804 


4146 


4487 


4828 


5169 


5510 


5851 


6191 


6531 


6871 


128 


7210 


7549 


7888 


8227 


8565 


8903 


9241 


9579 


9916 


•253 


129 


110590 


0926 


1263 


1599 


1934 


2270 


2605- 


2940 


3275 


3609 


130 


3943 


4277 


4611 


4944 


5^78 


5611 


5943 


6276 


6608 


6940 


131 


7271 


7603 


7934 


8265 


8595 


8926 


9256 


9586 


9915 


•245 


132 


120574 


0903 


1231 


1560 


1888 


2216 


2544 


2871 


3198 


3525 


133 


3852 


4178 


4504 


4830 


5156 


5481 


5806 


6131 


6456 


6781 


134 


7M)5 


7429 


7753 


8076 


8399 


8722 


9045 


9368 


9690 


••12 


135 


130334 


0655 


0977 


1298 


1619 


1939 


2260 


2580 


2900 


3219 


136 


3539 


3858 


4177 


4496 


4814 


5133 


5451 


5769 


6086 


6403 


137 


6721 


7037 


7354 


7671 


7987 


8303 


8618 


8934 


9249 


9564 . 


138 


9879 


• 194 


•508 


•822 


1136 


1450 


1763 


2076 


2389 


270^ 


139 


143015 


3327 


3639 


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LOGARITHBte. 










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1943 


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160 


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4049 


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5109 


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5638 


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6166 


6430 


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7221 


165 


7484 


7747 


8010 


8273 


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9060 


9323 


9585 


9846 


166 


220108 


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1675 


1936 


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4015 


4274 


4533 


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168 


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6600 


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7115 


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7630 


169 


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9170 


9426 


9682 


9938 


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0704 


0960 


1215 


1470 


1724 


1979 


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2488 


2743 


171 


2996 


3250 


3504 


3757 


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4264 


4517 


4770 


5023 


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172 


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173 


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182 


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3412 


3628 


3844 


4059 


4275 


4491 


4706 


4921 


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202 


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203 


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7710 


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9630 


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1118 


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1542 


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1966 


2177 


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2600 


2812 


3023 


3234 


3445 


3656 


206 


3867 


4078 


4289 


4499 


4710 


4920 


5130 


5340 


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5760 


207 


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6390 


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7436 


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0769 


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211 


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3930 


4082 


4235 


•4387 


4540 


4692 


285 


4845 


4997 


5150 


5302 


5454 


5606 


5758 


5910 


6062 


6214 


286 


6366 


6518 


6670 


6821 


6973 


7125 


7276 


7428 


7579 


7781 


287 


■7882 


8033 


8184 


8356 


8487 


8638 


8789 


8940 


9091 


9242 


288 


9392 


9543 


9694 


984.5 


9995 


.146 


.296 


.447 


.597 


.748 


289 


460898 


1048 


1198 


1348 


1499 


1649 


1799 


1948 


2098 


2248 


290 


2398 


2548 


2697 


2847 


2997 


3146 


3296 


3445 


3594 


3744 


291 


3*893 


4042 


4191 


4340 


4490 


4639 


4788 


4936 


5085 


5234 


292 


5383 


5532 


5680 


5829 


5977 


6126 


6274 


6423 


6571 


6719 


293 


6868 


7016 


7164 


7312 


.7460 


7608 


7756 


7904 


8052 


8200 


294 


8347 


8495 


8643 


8790 


8988 


9085 


9233 


9380 


9527 


9675 ' 


295 


9822 


9969 


. 116 


.263 


.410 


.557 


.704 


.851 


.998 


^1145 


296 


471292 


1438 


1585 


1732^ 


187S 


2025 


2171 


2318 


2464 


2610 


297 


2756 


2903 


3049 


3195 


3341 


3487 


3633 


8779 


3925 


4071 


298 


4216 


4362 


4508 


4653 4799 


4944 


5090 


5235 


5381 


5526 


299 


sell 


5816 


5962 


6107 6252 


6397 


6542 


6687 


6832 


6976 











OF NUMBERS. 








371 


x^ • 


■■■■"' 0' 


1 


2 


3 


4 


5 


6 


7 


8 
8278 


9 : 


300 


477121 


7266 


7411 


7555 


7700 


7844 


7989 


8133 


8422 ; 


;^01 


%8566 


8711 


8855 


8999 


9143 


9287 


9431 


9575 


9719 


9863 


302 


480007 


0151 


0294 


04^8 


0582 


0^25 


0869 


1012 


M56 


1299 


303 


. 1443 


1586 


1729 


1872 


2016 


2159 


2302 


2445 


2586 


2731 


304 


2874 


3016 


3159 


3302 


344i> 


3587 


3730 


3872 


4015 


4157 


305 


4300 


4442 


4585 


4727 


4865 


5011 


5153 


5295 


5437 


5579 


306 


5721 


5863 


6005 


6147 


62S9 


6430 


6572 


6714 


6855 


6997 ' 


307 


7138 


7280 


7421 


7563 


7704 


7845 


7986 


8127 


8269 


8410 


308 


8551 


8692 


8833 


8974 


91U 


9255 


9396 


9537 


9677 


9818 


30i^ 


9958 


..99 


.239 


,380 


.520 


.661 


.801' 


.941 


1081 


1222 


310 


491362 


1502 


1642 


1782 


1922 


2062 


2201 


2341 


2481 


2621 


311 


2760 


2900 


3040 


3179 


3319 


3458 


3597 


3737 


387^ 


4015 


312 


4155 


4294 


4433 


4^572 


4711 


4850 


4989 


5128 


5267 


5406 


313 


5544 


5683 


5822 


5960 


6099 


6238 


6376 


6515 


6653 


6791 


314 


6950 


7068 


7206 


7344 


7483 


7621 


7759 


7897 


8035 


8173 


315 


8311 


8448 


8586 


8724 


8862 


8999 


9137 


9275 


9412 


9550 


316 


9687 


9824 


9962 


..99 


.236 


.374 


.511 


• 648 


.785 


. 922 


317 


501059 


1196 


1333 


1470 


1607 


1744 


1880 


2017 


2154 


2291 


318 


2427 


2564 


2700 


2837 


2973 


3109 


3246 


3382 


3518 


3655 


319 


3791 


3927 


4063 


4199 


4335 


4471 


4607 


4743 


4878 


5014 


320 


5150 


5286 


5421 


5557 


5693 


5828 


5964 


6099 


6234 


6370 


321 


6505 


6640 


6776 


6911 


7046 


7181 


7316 


7451 


7586 


7721 


«22 


7856 


7991 


8126 


8260 


8395 


8530 


8664 


8799 


8934 


9068 


323 


9203 


9337 


9471 


9606 


9740 


9874 


. • . ^ 


. 143 


.277 


.411 


324 


510545 


0679 


08J3 


0947 


1081 


1215 


1349 


1482 


1616 


1750 


•325 


1883 


2017 


2151 


2284 


2418 


2551 


2684 


2818 


2951 


3084 


•326 


3218 


3351 


3484 


3617 


3750 


3883 


4016 


4149 


4282 


4415 


327 


4548 


4681 


4813 


4946 


5079 


5211 


5344 


5476 


5609 


5741 


328 


5874 


6006 


6139 


6271 


6403 


6535 


6668 


6800 


^932 


7064 


329 


.7196 


7328 


7460 


7592 


7724 


7855 


7987 


8119 


3251 


8382 


a30 


8514 


8646 


8777 


8909 


9040 


9171 


9303 


9434 


9566 


9697 


331 


9828 


9959 


..90 


.221 


.353 


.484 


.615 


.745 


.876 


1007 


332 


521138 


1269 


1400 


1530 


1661 


1792 


1922 


2053 


2183 


2314 


:I33 


2444 


25175 


2705 


2835 


2966 


3096 


3226 


3356 


3486 


3616 


334 


3746 


3876 


4006 


4136 


4266 


4396 




4526. 


4^56 


4785 


4915 


335 


5045 


5174 


5304 


5434 


5563 


5693 


5822 


5951 


6081 


6210 


S36 


6339 


6469 


6598 


6727 


6856 


6985 


7114 


7'243 


7372 


7501 


337 


7630 


7759 


7888 


8016 


8145 


8274 


8402 


853 L 


8660 


8788 


338 


8917 


9045 


9174 


9302 


9430 


9559 


9687 


a815 


9943 


..72 


339 


530200 


0328 


0456 


0584 


0712 


0840 


0968 


1096 


1223 


1351 


340 


1479 


1607 


1734 


1862 


1990 


2117 


2245 


2372 


2500 


2627 


341 


2754 


2882 


3009 


SI 36 


3^64 


3391 


3518 


3645 


3772 


3899 


342 


4026 


4153 


4280 


4407 


4534 


4661 


4787 


4914 


5041 


5167 


343 


5294 


5421 


5547 


5674 


^800 


5927 


6053 


6180 


6306 


6432 


344 


6558 


6685 


6811 


6937 


7063 


7189 


7315 


'7441 


7567 


7693 


345 


7819 


7945 


8071 


8197 


8322 


8448 


8574 


8699 


8825 


8951 


346 


9076 


9202 


9327 


9452 


9578 


9703 


9829 


9954 


..79 


.204 


347 


540329 


0455 


0580 


0705 


0830 


0955 


1080 


1205 


1330 


1454 


348 


1579 


1704 


1829 


1953 


2078 


2203 


2327 


2452 


2576 


2701 


349 


2825 


2950 


3074 


3199 


3323 


3447 


3571 


3696 


3820 


3944 



372 




350 





544068 


351 


5307 


352 


6543 


353 


7775 


354 


9003 


355 


550228 


356 


1450 


357 


2668 


358 


3883 


359 


5094 


360 


6303 


361 


7507 


362 


8709 


363 


9907 


364 


561101 


365 


2293 


366 


3481 


367 


4666 


368 


5848 


369 


7026 


370 


8202 


371 


9374 


372 


570543 


373 


1709 


374 


.2872 


375 


4031 


376 


5188 


377 


6341 


378. 


7492 


379 


8639 


380 


9784 


381 


580925 


382 


2063 


383 


3199 


384 


4331 


385 


5461 


386 


6587 


387 


7711 


388 


8832 


389 


9950 


390 


591065 


391 


2177 


?92 


'3286 


393 


4393 


294 


5496 


395 


6597 


396 


7695 


397 


8191 


398 


9883 


999 


600973 



LOGARTTHMS 



4192 
5431 
6666 
7898 
9126 
0351 
1572 
2790 
4004 
5^15 
6423 
7627 

8829 
..26 
1221 
2412 
3600 
4784 
5966 
7144 
8319 
9491 
0660 
1825 
2988 
4147 
5303 
6457 
7607 
8754 
9898 
1039 
2177 
3312 
4444 
5574 
6700 
7823 
8944 
..61 
U76 
2288 
3397 
4503 
5606 
6707 
7805 
8900 
9992 
1082 



4316 

5555 

6789 

8021 

9249 

0473 

1694 

2911 

4126 

5336 

6544 

7748 

8948 

• 146 

1340 

2531 

3718 

4903 

6084 

7262 

8436 

9608 

0776 

1942 

3104 

4263 

5419 

6572 

7722 

8868 

.. 12 

1153 

2291 

3426 

4557 

5686 

6812 

7935 

9056 

.173 

1287 

2399 

3508 

4614 

5717 

6817 

7914 

9009 

. 101 

1191 



4440 4564 



5678 
6913 
8144 
9371 



0595 0717 



1816 
3033 
4247 
5457 
6664 
7868 
9068 
.265 
1459 
2650 
3837 
5021 
6202 
7379 
8554 
9725 
0893 
2058 
3220 
4379 
5534 
6687 
7836 
8983 
• 126 
1267 
2404 
3539 
4670 
5799 
6925 
8047 
9167 
.284 
1399 
2510 
3618 



8024 
9119 
.210 

1299 



5802 
7036 
8267 
9494 



1938 
3155 
4368 
5578 
6785 
7988 
9188 
.385 
1578 
2769 
3955 
5139 
6320 
7497 
8671 
9842 
1010 
2174 
3336 
4494 
5650 
6802 
7951 
9097 
.241 
1381 
2518 
3652 
4783 
5912 
7037 
8160 
9279 
.396 
1510 
2621 
3729 



4724 4834 
5827 5937 
6927 7037 



8134 
9228 
.319 
1408 



4688 
5925 
7159 
8389 
9616 
0840 
2060 
3276 
4489 
5699 
6905 
8108 
9308 
.504 
1698 
2887 
4074 
5257 
6437 
7614 
8788 
9959 
1126 
2291 
3452 
4610 
5765 
6917 
8066 
9212 
.355 
1495 
2631 
3765 
4896 
6024 
7149 
8272 
9391 
.507 
1621 
2732 
3840 
4945 
6047 
7146 
8243 
9337 
.428 
1517 



4812 
6049 
7282 
8512 
9739 
0962 
2181 
3398 
4610 
5820 
7026 
8228 
9428 
.624 
1817 
3006 
4192 
5376 
6555 
7732 
8905 
.076 
1243 
2407 
3368 
4726 
5880 
7032 
8181 
9326 
.469 
1608 
2745 
3879 
5009 
6137 
7262 
8384 
9503 
.619 
1T32 
2843 
3950 
5055 
6157 
7256 
8353 
9446 
.537 
1625 



r 



4936 
6172 
7405 
8635 
9861 
1084 
2303 
3519 
4731 
5940 
7146 
8349 
9548 
.743 
1936 
3125 
4311 
5494 
f6673 
7849 
9023 
.193 
1S59 
2523 
3684 
4841 
5996 
7147 
8295 
9441 
.583 
1722 
2858 
3992 
5122 
6250 
7374 
8496 
9615 
.780 
1843 
2954 
4061 
5165 
6267 
7366 
8462 
9556 
.646 
1734 



8 



5060 
6296 
7529 
8758 
9984 
1206 
2425 
3640 
4852 
6061 
7267 
8469 
9667 
• 863 
2055 
3244 
4429 
5612 
6791 
7967 
9140 
.309 
1476 
2639 
9800 
4957 
6111 
7262 
8410 
9555 
.697 
1836 
2072 
4105 
5235 
6362 
7486 
8608 
9726 
.842 
1955 
3064 
4171 
5276 
6377 
7476 
8572 
9665 
.755 
1843 



5183 

6419 

7652 

88S1 

. 106 

1328 

2547 

3762 

4973 

6182 

7387 

8589 

9787 

.982 

2174 

3362 

4548 

5730 

6909 

8084 

9257 

.426 

1592 

2755 

3915 

5072 

6226 

7377 

8525 

9669 

.811 

1950 

30S5 

4218 

5348 

6475 

7599 

8720 

9838 

.953 

2066 

3175 

4282 

5386 

6487 

7586 

8681 

9774 

.864 

1951 









J 


OF NUMBERS. 


^ 






373 


'N. 


. (y 


1 


2 


3 


4 


5 


6 


'j « 


9 


400 


602060 


2169 


2277 


2386 


2494 


2603 


2711 


2819 


2928 


3036 


401 


. 3144 


3253 


3361 


3469 


3573 


3686 


3794 


3902 


4010 


4118 


402 


4226 


4334 


4442 


4550 


4658 


4766 


4874 


4982 


5089 


51:97, 


403 


5305 


5413 


5521 


5628 


5736 


5844 


5951 


6059 


6166 


6274 


404 


6381 


•6489 


6596 


6704 


6811 


6919 


7026 


7133 


7241 


7348 


405 


7455 


7562 


7669 


7777 


7884 


7991 


'8098 


8205 


8312 


8419 


406 


8526 


8633 


8740 


8847 


8954 


9061 


9167 


9274 


9381 


9488 


407 


9594 


9701 


9808 


9914 


. .21 


.128 


.234 


.341 


.447 


.554 


408 


610660 


0767 


0873 


0979 


1086 


1192 


1298 


1405 


1511 


1617 


409 


1723 


1829 


1936 


2042 


2148 


2254 


2360 


2466 


2572 


2678 


410 


2784 


2890 


2996. 


3102 


3207 


3313 


3419 


3525 


3630 


3736 


411 


3842 


3947 


4053 


4159 


4264 


4370 


4475 


4581 


4686 


4792 


412 


4897 


5003 


5108 


5213 


5319 


5424 


5529 


5634 


5740 


5845 


413 


5950 


6055 


6160 


6265 


6370 


6476 


6581 


6686 


6790 


6895 


414 


7000 


7105 


7210 


7315 


7420 


7525 


7629 


7734 


7889 


T943 


415 


8048 


8153 


8257 


8362 


8466 


8571 


8676 


8780 


8884 


8989 


416 


9093 


9198 


9302 


9406 


9511 


9615 


9719 


9824 


9928 


..32 


417 


620136 


0240 


0344 


0448 


0552 


0656 


0760 


0864 


0968 


1072 


418 


1176 


1280 


1384 


1488 


1592 


1695 


1799' 


1903 


2007 


2110 


419 


2214 


2318 


2421 


2525 


2628 


2732 


2835 


29^9 


3042 


3146 


420 


3249 


3353 


3456 


3559 


3663 


3766 


3869 


3973 


4076 


4179 


421 


4282 


4385 


4488 


4591 


4695 


4798 


4901 


5004 


5107 


52 lO 


422 


5312 


5415 


5518 


5621 


5724 


58S7 


bm\i 


6032 


6135 


623S 


423 


6340 


6443 


6546 


6648 


6751 


6853 


'6956 


7058 


7161 


7263 


424 


7366 


•7468 


7571 


7673 


7775 


7878 


7980 


8082 


8185 


8287 


425 


8389 


8491 


8593 


8695 


8797 


8900 


9002 


9104 


^206 


9308 


426 


9410 


9512 


9613 


9715 


9817 


9919 


..21 


.123 


.0S4 


• 326 


427 


630428 


0530 


0631 


0733 


0835 


0936 


10S8 


1139 


1241 


1342 


428 


1444 


1545 


1647 


1748 


1849 


1951 


2052 


^153 


2255 


2356 


429 


2457 


2559 


2660 


2761 


2862 


2963 


3064 


3165 


3266 


3367 


430 


3468 


3569 


3670 


3771 


3872 


3973 


4^74 


4175 


4276 


4376 


431 


4477 


4578 


.4679 


4779 


4880 


4981 


5081 


5182 


62S3 


5383 


432 


5484 


5584 


5685 


5785 


5^86 


5986 


6087 


6187 


6287 


6388 


433 


64S8 


6588 


6688 


6789 


6889 


6989 


7089 


7189 


7290 


7390 


434 


7490 


7590 


7690 


7790 


7890 


7990 


8090 


S190 


8290 


8389 


435 


8489 


8589 


8689 


8789 


8888 


8988 


908i8 


9188 


9287 


9387 


436 


9486 


9586 


9686 


9785 


'^^^S 


9984 


..84 


. 183 


.283 


.3*2 


437 


64048 L 


0581 


0680 


0779 


0879 


0978 


1077 


1177 


1276 


1375 


438 


, 1474 


1573 


1672 


1771 


1871 


1970 


2069 


2168 


2267 


2366 


439 


2465 


2563 


2662 


2761 


2860 


2959 


3058 


3166 


3255 


3354 


440 


3453 


3551 


3650 


3749 


3847 


3946 


4044 


4143 


4242 


4340. 


441 


4439 


4537 


4636 


4734 


4832 


4931 


'5029 


5127 


5226 


5324 


442 


5422 


5521 


5619 


5717 


5815 


5913 


6011 


6110 


6208 


6306 


4'43 


6404 


6502 


6600 


6698 


6796 


6894 


6992 


7089 


7187 


^2&\ 


444 


7383 


7481 


7579 


7676 


7774 


7872 


7969 


8067 


8165 


S262 


445 


8360 


8458 


8555 


8653 


8750 


8848 


894S 


9043 


9140 


'9237 


446 


9335 


9432 


9530 


9627 


9724 


98^1 


991,9 


..1=6 


. 113 


.21,0. 


447 


650308 


04O5 


0502 


0599 


0696 


0793 


0890 


0987 


1084 


ilfi 


448 


1278 


1375 


1472 


1569 


1666 


17-62 


1^59 


1956 


3053 


2150 


,449 


2246 


2343 


2440^25361 


2633 


2730 


2826 


2923 


3019 


3116 



374 



LOGARITHMS 



450 
451 
452 
453 
454 
455 
456 
457 
458 
459 
460 
461 
462 
463 
464 
465 
466 
467 
468 
469 
470 
471 
472 
473 
474 
475 
476 
477 
478 
479 
480 
481 
482 
483 
484 
485 
486 
487 
488 
489 
490 
491 
492 
493 
494 
495 
496. 
497 
149^ 
499 







653213 
4177 
5138 
6098 
7056 
8011 
8965 
9916 

660865 
1813 
2758 
3701 
4642 
5581 
6518 
7453 
8386 
9317 

670246 
1173 
2098 
3021 
3942 
4861 
5778 
6694 
7607 
8518 
9428 

680336 
1241 
2145 
3047 
3947 
4845 
5742 
6636 
7529 
«420 
9309 

690196 
1081 
1965 
2847 
3727 
4605 
5482 
6356 
7229 
8101 



3309 
4273 
5235 
6194 
7152 
8107 
9060 
..U 
0960 
1907 
2852 
3795 
4736 
5675 
6612 
7546 
8479 
9410 
0339 
1265 
2190 
3113 
4034 
4953 
5870 
6785 
7698 
8609 
9519 
0426 
1332 
2235 
3137 
4037 
4935 
5831 
6726 
7618 
8509 
9398 
0285 
1170 
2053 
2935 
3815 
4693 
5569 
6444 
7317 
8188 



2 



3405 

4369 

5331 

6290 

7247 

8202 

9155 

. 106 

1055 

2002 

2947 

3889 

4830 

5769 

6705 

7640 

8572 

9503 

0431 

1358 

2283 

3205 

4126 

5045 

5962 

6876 

7789 

8700 

9610 

0517 

1422 

2326 

3227 

4127 

5025 

5921 

6815 

7707 

8598 

9486 

0373 

1258 

2142 

3023 

3903 

4781 

5657 

6531 

7404 

8275 



3502 
4465 
5427 
6386 
7343 
8298 
9250 
• 201 
1150 
2096 
3041 
3983 
4924 
5862 
6799 
7733 
8665 
9596 
0524 
1451 
2375 
3297 
4218 
5137 
6053 
6968 
7881 
8791 
9700 
0607 
1513 
2416 
3317 
4217 
5114 
6010 
6904 
7796 
8687 
9575 
0462 
1347 
2230 
3111 
3991 
4868 
5744 
6618 
7491 
8362 



3598 

4562 

5526 

6482 

7438 

8393 

9346 

.296 

1245 

2191 

3135 

4078 

5018 

5956 

6S92\ 

7826 

9759 

9689 

0617 

1543 

2467 

3390 

4310 

5228 

6145 

7059 

7972 

8882 

9791 

0698 

1603 

2506 

3407 

4307 

5204 

6100, 

6994 

7886 

8776 

9664 

0550 

1435 

2318 

3199 

4078 

4956 

5832 

6706 

7578 

8449 



3695 

4658 

5619 

6577 

7534 

8488 

9441 

.391 

1339 

2286 

3230 

4172 

5112 

6050 

6986 

7920 

8852 

9782 

0710 

1636 

2560 

3482 

4402 

5320 

6236 

7151 

8063 

8973 

9882 

0789 

1693 

2596 

3497 

4396 

5294 

6189 

7083 

7975 

8865 

9753 

0639 

1524 

2406 

3287 

4166 

5044 

5919 

6793 

7665 

8535 



3791 
4754 
5715 
6673 
7629 
8584 
9536 
.486 
1434 
2380 
3324 
4266 
5206 
6143 
7079 
8013 
8945 
9875 
0802 
1728 
2652 
tS574 
4494 
5412 
6328 
7242 
8154 
9064 
9973 
0879 
1784 
2686 
3587 
4486 
5383 
6279 
7172 
8064 
8953 
9841 
0728 
1612 
2494 
3375 
4254 
5131 
6007 
6880 
7752 
8622 



38^8 
4850 
5810 
6769 
7725 
8679 
9631 
.531 
1529 
2475 
3418 
4360 
5299 
6237 
7173 
8106 
9038 
9967 
0895 
1821 
2744 
3666 
4586 
5503 
6419 
7333 
8245 
9155 
..63 
0970 
1874 
2777 
3677 
4576 
5473 
6368 
7261 

8153 
9042 
9930 
0816 
1700 
2583 
3463 
4342 
5219 
6094 
696S 
7839 
8709 



Tl 



3984 
4946 
5906 
6864 
7820 
8774 
9726 
.676 
1623 
2569 
3512 
4454 
5393 
6331 
7266 
8199 
9131 
-.60 
0988 
1913 
2836 
3758 
4677 
5595 
6511 
7424 
8336 
9246 
. 154 
1060 
1964 
2867 
3767 
4666 
5563 
6458 
7351 
8242 
9131 
. .19 
0905 
1789 
2671 
3551 
4430 
5307 
6182 
7055 
7926 



4080 

5042 

6002 

6960 

7916 

8870 

9821 

.771 

1718 

2663 

3607 

4548 

5487 

6424 

7360 

8293 

9224 

. 153 

1080 

2005 

2929 

3850 

4769 

5687 

6602 

7516 

8427 

9337 

.245 

1151 

2055 

2957 

3857 

4756 

5652 

6547 

7440 

8331 

9220 

. 107 

0993 

1877 

2759 

3639 

4517 

5394 

6269 

7142 

8014 



879618883 



OP NUMBERS. 



915 



N. 





500 


698970 


501 


9838 


502 


700704 


503 


1568 


504 


2431 


505 


3291 


506 


4151 


507 


5008 


508 


5864 


509 


6718 


510 


7570 


511 


8421 


5152 


9270 


51S 


710117 


514 


0963 


515 


1807 


516 


2650 


517 


. 3491 


518 


4330 


519 


5167 


520 


6003 


521 


6838 


5^2 


7671 


525 


8502 


524 


9331 


525 


720159 


526 


0986 


527 


1811 


528 


2634 


529 


3456 


530 


4276 


531 


5095 


532 


5912 


533 


6727 


534 


7541 


5^5 


8354 


536 


9165 


537 


9974 


538 


730782 


539 


1589 


540 


2394 


541 


3197 


542 


3999 


543 


4800 


544 


5599 


545 


6397 


546 


7193 


547 


7987 


548 


8781 


549 


9572 



9037 
9924 
0790 
1654 
2517 
3377 
4236 
5094 
5949 
6803 
7655 
8506 
9355 
0202 
1048 
1892 
2734 
3575 
4414 
5251 
6087 
6921 
7734 
8585 
9414 
0242 
1068 
1893 
2716 
3538 
4358 
5176 
5993 
68Q9 
7623 
8435 
9246 
..55 
0863 
1669 
2474 
3278 
4079 
4880 
5679 
6476 
7272 
8067 
8860 
9651 



9144 
..11 
0877 
1741 
2603 
3463 
4322 
5179 
6035 
6888 
7740 
8591 
9440 
0287 
1132 
1976 
2818 
3659 
4497 
5335 
6170 
7004 
7837 

sees 

9497 
0325 
1151 
1975 
2798 
3620 
4440 
5258 
6075 
6390 
7704 
8516 
9327 
. 136 
0944 
1750 
2555 
3358 
4160 
4960 
5759 
6556 
7352 
8146 
8939 
9731 



9231 9317 
98 . 184 
0963 1050 
1827 1913 
2689 2775 
3549 3635 
4408 
5265 
6120 
6974 
7826 
8676 
9524 
0371 
1217 
2060 
2902 
3742 
4581 
5418 
6254 
7088 
7920 
87511-^834 



9580 
0407 
1233 
2058 
2881 
3702 
4522 
5340 
6156 
6972 
7785 
8597 
9408 
.217 
1024 
1830 
2635 
3433 
4240 
5040 
5838* 
6635 
7431 
8225 
9018 
9810 



4 I 5 



9404 
.271 
1136 
1999 
2861 
3721 
4494 1 4579 



5330 
6206 
7059 
7911 
8761 
9609 
0456 
1301 
2144 
2986 
3826 
4665 
5502 
6337 
7171 
8003 



9663 
0490 
1316 
2140 
2963 
3784 
4604 
5422 
6238 
7053 
7866 
8678 
9489 
.298 
1105 
1911 
2715 
3518 
4320 
5120 
5918 
6715 
7511 
8305 
9097 
9889 



6 



5436 
6291 

7144 
7996 
8846 
9694 
0540 
1385 
2229 
3070 
3910 
4749 
5586 
6421 
7254 
8086 
8917 
9745 
0573 
1398 
2222 
3045 
3866 
4685 
5503 
6320 
7134 
7948 
8759 
9570 
.378 
1186 
1991 
12796 
3598 
4400 
5200 
5998 
6795 
7590 
8384 
9177 
9968 



9491 
.338 
1222 
2086 
2947 
3807 
4665 
3522 
6376 
7229 
8081 
8931 
9779 
0625 
1470 
2313 
3154 
3994 
4833 
5669 
6504 
7338 
8169 
9000 
9828 
0655 
1481 
2305 
3127 
3948 
4767 
5585 
6401 
7216 
8029 
8841 
9651 
.459 
1266 
2072 
2876 
3679 
4480 
5279 
6078 
6874 
7670 
8463 
9256 
..47 



9578 
.444 
1309 
2172 
3033 
3393 
4751 
5607 
6462 
7313 
8166 
9015 
9863 
0710 
1554 
2397 
3238 
4078 
4916 
5753 
6588 
^421 
8253 
9083 
9911 
0738 
1563 
2387 
3209 
4030 
4849 
5667 
6483 
7j397 
8110 
8922 
^732 
•540 
1347 
2152 
2956 
3759 
4560 
5359 
6157 
6954 
7749 
8543 
9335 
. 126 



9664 
.•531 
1395 
2258 
3119 
3979 
4837 
5693 
6547 
7400 
8251 
9^00 
9948 
0794 
1639 
2481 
3326 
4162 
5000 
5836 
6671 
7504 
8336 

9165 
9994 
0821 
1646 
2469 
3291 
4112 
4931 
5748 
6564 
7379 
8191 
9003 
9813 
• 621 
1428 
2233 
3037 
3839 
4640 
5439 
6237 
7034 
7829 
8622 
9414 
.205 



19751 
.617 
1482 
2344 
3205 
4065 
4922 
^778 
6632 
7485 
8336 
9185 
..33 
0879 
1723 
2566 
3407 
4246 
5084 
5920 
6754 
7587 
8419 
9248 
..77 
0903 
1728 
2552 
3374 
4194 
5013 
5830 
6646 
7460 
8273 
9084 
9893 
.702 
1508 
2313 
3117 
3919 
4720 
5519 
6317 
7113 
7908 
8701^ 
9493 
.284 



37« 








LOGARITHMS 










JN. 
550 





1 


2 


3 


4 


5 


6 


7 


8 


9 


740363 


0442 


0521 


0560 


0678 


0757 


0836 


0915 


0994 


1073 


551 


1152 


lli30 


1309 


1388 


1467 


1546 


1624 


1703 


1782 


1860 


552 


1939 


2018 


2096 


2175 


2254 


2332 


2411 


2489 


2568 


2646 


553 


2725 


2304 


2S82 


2961 


3039 


31*18 


3196 


3275 


3353 


3431 


554 


3510 


35S8 


3667 


3745 


3823 


3902 


3980 


4058 


4136 


4215 


655 


4293 


4371 


4449 


4528 


4606 


4684 


4762 


4840 


4919 


4997 


556 


5075 


5153 


5231 


5309 


5387 


5465 


5543 


5621 


5699 


5777 


557 


^ 5855 


5933 


6011 


6089 


6167 


6245 


6323 


6401 


6.479 


6556 


558 


6634 


6712 


6790 


6868 


6945 


7023 


7101 


7179 


7256 


7334 


559 


7412 


7489 


7567 


7645 


7722 


7800 


7878 


7955 


8033 


8110 


560 


8188 


8266 


8343 


8421 


8498 


8576 


8653 


8731 


8808 


8885 


561 


8.963 


9040 


9118 


9195 


9272 


9350 


9427 


9504 


9582 


9659 


562 


9736 


9814 


9891 


9968 


..45 


. 123 


.200 


.277 


.354 


.431 


56'3 


750508 


0586 


0663 


0740 


0817 


0894 


0971 


1048 


1125 


1202 


564 


1279 


1356 


1433 


1510 


1587 


1664 


1741 


1818 


1895 


1972 


565 


2048 


2125 


2202 


2279 


2356 


2433 


2509 


2586 


2663 


2740 


566 


28(6 


2893 


2970 


3047 


3123 


S200 


3277 


3353 


3430 


3506 


567 


3583 


3660 


3736 


3813 


3889 


3966 


4042 


4119 


4195 


4272 


568 


4348 


4425 


4501 


4578 


4654 


4730 


4807 


4883 


4960 


5036 


569 


5112 


5189 


5265 


5341 


5417 


5494 


5570 


5646 


5722 


5799 


570 


5875 


5951 


6027 


6103 


6180 


6256 


6332 


6408 


6484 


6560 


571 


6636 


67J2 


6788 


6864 


6940 


7016 


7092 


716a 


7244 


7320 


572 


7396 


7472 


7548 


7624 


7700 


7775 


7851 


7J927 


8003 


8079 


573 


8155 


8230 


8306 


8382 


845« 


8533 


8609 


8685 


8761 


8836 


574 


8912 


898^ 


9063 


9139 


9214 


9290 


9366 


9441 


9517 


9592 


575 


9668 


9743 


9819 


9894 


9970 


..45 


. 121 


. 196 


.272 


.347 


576 


760422 


0498 


0573 


0649 


0724 


0799 


0875 


0950 


1025 


1101 


577 


1176 


1251 


1326 


14p2 


1477 


1552 


1627 


1702 


1778 


1853 


578 


1928 


2003 


2078 


2153 


2228 


2303 


2378 


2453 


2529 


2604 


579 


2679 


2754 


2829 


2904 


2978 


3053 


3128 


3203 


3278 


3353 


580 


3428 


3503 


3578 


3653 


S727 


3802 


3877 


3952 


4027 


4101 


581 


4176 


4251 


4326 


4400 


4475 


45 5Q 


4624 


4699 


4774 


4848 


582 


4923 


4998 


5072 


5147 


5221 


5296 


5370 


5445 


5520 


5594 


583 


5669 


5743 


5818 


5892 


5966 


6041 


6115 


6190 


6264 


6338 


584 


6413 


6487 


6562 


6&36 


6710 


6785 


6859 


6933 


7007 


7082 


585 


7156 


723Q 


7304 


7379 


7453 


7527 


7601 


7675 


7749 


7823 


586 


7898 


7972 


8046 


8120 


8194 


8268 


8342 


8416 


8490 


8564. 


587 


8668 


8712 


8786 


8860 


8934 


9003 


9082 


9156 


9230 


9303 


588 


9377 


9451 


9525 


9599 


9673 


9746 


9820 


9894 


9968 


..42 


589 


770115 


0189 


0263 


0336 


0410 


0484 


0557 


0631 


0705 


0778 


590 


0852 


0926 


0999 


1073 


1146 


1220 


1293 


1367 


1440 


1514 


591 


1587 


1661 


1734 


1808 


1881 


1955 


2028 


2102 


2175 


2248 


992 


2322 


2395 


^468 


2542 


2615 


2688 


2762 


^835 


2908 


2981 


593 


3a55 


3128 


3201 


3274 


3348 


3421 


3494 


3567 


3640 


3713 


594 


3786 


3860 


3933 


4006 


4079 


4152 


4225 


4298 


4371 




595 


4517 


4590 


4663 


47S6 


4809 


4882 


4955. 


5028 


5100 


5173 


596 


5246 


5319 


5392 


5465 


5533 


561^1 


M»S 


5756 


5829 


5902 


597 


5974 


6047 


6120 


6193 


6265 


6398 


64ii 

7l»T 


6483 


6556 


6629 


598 


6701 


6774 


6846 


6919 


6991^ 


7064 


7209 


7282 


7354 


599 


7427 7499 


7572 


7644 


7717 


7789 


7862 


7934 


8006 


8079 



'▼, 











OF NUMBftftS. 








3l^ 


600 


O 


1 2 


3 


4 


5 


6 7 


8 


9 


778151 


8224 


8296 


8368 


8441 


8613 


85851 


8658 


8lf30 


8802 


601 


8874 


8947 


9019 


9091 


9163 


9236 


9308 


9380 


9452 


9524 


602 


^596 


9669 


9741 


9813 


9885 


9957 


..29 


.101 


.173 


. 245 


603 


7S0317 


0389 


0461 


0533 


0605 


0677 


0749 


0821 


0893 


0965 ' 


^4 


1037 


1109 


1181 


I25;i 


1324 


1396 


1468 


1540 


f6l2 


16^4 


605 


17S5 


1827 


1899 


1971 


2042 


2114 


2186 


3258 


2329 


2401 


.606 


2473 


2544 


2616 


2688 


2759 


2831 


2902 


2974 


3046 


3117 


607 


3(89 


3260 


3332 


3403 


3475 


3546 


3618 


3689 


3761 


3832 


60d 


3904 


3975 


4046 


4118 


4189 


4261 


4332 


4403 


4475 


4546 


609 


4617 


4689 


47G0 


4831 


4902 


4974 


5045 


^116 


518*7 


5259 


610 


5330 


.5401 


5472 


5543 


5615 


sese 


5757 


5828 


5899 


5970 ' 


611 


6041 


6112 


6183 


6^54 


6325 


6396 


6467 


6538 


6609 


6680 


612 


6751 


6S22 


6893 


6964 


7035 


7106 


nil 


7248 


7319 


7390 


613 


7460 


7531 


7602 


7673 


7744 


7815 


7885 


7956 


8027 


9098 


614 


8168 


8269 


8310 


8381 


8451 


8522 


8593 


8663 


8734 


8804 


615 


8875 


8946 


9016 


9087 


9157 


9228 


9299 


9369 


9440 


9510 


616 


9581 


9651 


9722 


9792 


9863 


9933 


...4 


..74 


.144 


.215 


617 


790285 


0356 


0426 


0496 


0567 


0637 


0707 


0778 


0848 


0918 


618 


0988 


1059 


1129 


1199 


1269 


1340 


1410 


1480 


1550 


1620 


619 


1691 


1761 


1831 


1901 


1971 


2041 


2111 


2181 


2i25« 


2322 


620 


2892 


2462 


2532 


2602 


2672 


2742 


2812. 


2862 


2952 


3022 


621 


3092 


3162 


3231 


3301 


3371 


3441 


3511 


3581 


3651 


3721 


622 


3790 


3860 


3930 


4000 


4070 


4139 


4209 


4279 


4349 


4418 


623 


4488 


4558 


4627 


4697 


4767 


4836 


4906 


4976 


5Q45 


5115 


624 


5185 


5254 


5324 


5393 


5463 


5532 


5602 


5672 


5741 


5811 


625 


5880 


5^49 


6019 


6088 


6158 


6227 


6297 


6366 


6436 


6S05 


626 


6574 


6644 


6713 


6782 


6852 


6921 


6990 


7060 


7129 


7198 


627 


7268 


7337 


7406 


7475 


7545 


7614 


7683 


7752 


7821 


7890 


628 


7960 


8029 


8098 


8167 


8236 


8305 


8374 


8443 


8513 


8582 


629 


8651 


8720 


8789 


8858 


8927 


8996 


9065 


9134 


9203 


9272 


630 


9341 


9409 


9478 


9547 


961'6 


9685 


9754 


9823 


9892 


9961 


631 


800029 


0098 


0167 


0236 


0305 


0373 


0442 


0511 


0580 


0648 


632 


0717 


0786 


0854 


0923 


0992 


1061 


1129 


1198 


1266 


1335 


633 


1404 


1472 


1541 


1609 


1678 


1747 


1815 


1884 


1.952 


2021 


634 


2089 


2158 


2226 


2295 


2363 


2432 


2500 


2568 


2637 


2703 


635 


2774 


2842 


2910 


2979 


3047 


3116 


3184 


3252 


3321 


3389 . 


636 


3457 


3525 


3594 


5662 


3730 


3798 


3867 


3935 


4003 


4071 


637 


4139 


4208 


4276 


4344 


4412 


4480 


4548 


4616 


4685 


4753 


6S8 


4821 


4889 


4957 


5025 


5093 


5161 


5229 


5297 


5365 


5433 


639 


5501 


5569 


5637 


5705 


5773 


584 L 


5908 


5976 


6044 


61121 


640 


6180 


6248 


6316 


6384 


6451 


6519. 


6587 


66155 


6723 


6790 


641 


6858 


6926 


6994 


7061 


7129 


7197 


7264 


7332 


7400 


7*67 


642 


7535 


7603 


7670 


7733 


7806 


7873 


7941 


8008 


8076 


8143 , 


643 


8211 


8279 


8346 


8414 


S481 


8549 


8616 


8684 


8751 


8818 


644 


8386 


8953 


9021 


9083 


9156 


9223 


9290 


9358 


9425 


9492 


645 


9560 


9627 


9694 


9762 


9829 


9896 


9964 


..31 


..98 


. 165 


646 


810233 


0300 


0367 


0434 


0501 


0569 


0636 


0703 


0770 


0337 


647 


0904 


0971 


1039 


1106 


1173 


1240 


1307 


1374 


1441 


1508 


645 


1575 


1642 


1709 


1776 


1843 


1910 


1977 


2044 


2111 


2178 


.649 


2245 2312 


2379 


2445 


2512 


2579 


2646 27lfi 


2780 


2847 . 



Voi. 1. 



Cc 



378 



LOGARITHMS 



£i _°- -i 



650 812913 



651 
652 
653 

654 

655 1 

656 

657 

658 

659 

660 

661 

662 

66S 

66^ 

665 

666 

661 

668 

669 

670 

671 

672 

673 

674 

675 

676 

677 
678 
679 
680 

|681 
682 
683 
684 
685 
686 
687 
688 
689 
690 
691 1 
692 

1693 
694 
695 
696 
697 
698 
69» 



8581 
4243 



29^0 
3648 
4314 



4913 4980 



5578 
6241 
6904 
7565 
8226 
8885 
' 9544 
820201 
0858 
15141 
2168 
2822 
3474 
4126 
4776 
5426 
6075 
6723 
7369 
8015 
8660 
9304 
9947 
830589 
1230 
1870 
2^09 
3147 
3784 
4421 
5056 
5691 
6324 
6957 
7588 
8219 
8849 
9478 
840106 
0733 
135^ 
1985 
2609 
3233 
S855 



5644 
6308 



3047 
3714 
4381 
5046 
5711 
6374 



3 



6970 7036 



44771 



7631 

8292 

8951 

9610 

0267 

0924 

1579 

2233 

2887 

3539 

4191 

4841 

5491 

6140 

6787 

7434 

8080 

8724 

9368 

• .11 

0653 

1294 

1934 

2573 

3211 

3848 

4484 

5120 

5754 

6387 

7020 

7652 

8282 

8912 

9541 

0169 

0796 

1422 

2047 

2672 

3295 

3918 



4539 



7698 
8358 
9017 
9676 
0333 
0989 
1645 
2299 
2952 
3605 
4256 
4906 
5556 
6204 
6852 
7499 
8144 
8789 
9432 
• ..75 
0717 
1358 
1998 
2637 
3275 
3912 
4548 
5183 
5817 
6451 
7083 
7715 
8345 
8975 
9604 
0232 
0859 
1485 
2110 
27S4 
3357 
3980 
4601 



3114 

3781 

4447 

5113 

5777 

6440 

7102 

7764 

8424 

9083 

9741 

0399 

1055 

1710 

2364 

3018 

3670 

4321 

4971 

5621 

6269 

6916 

7563 

8209 

8853 

9497 

.139 

0781 

1422 

2062 

2700 

3338 

3975 

4611 

5247 

5881 

6514 

7146 

7778 

8408 

9038 

9667 

0294 

0921 

1547 

2172 

2796 

3420 

4042 

4664 



3181 

3848 

4514 

5179 

5843 

6506 

7169 

7830 

8490 

9149 

9807 

0464 

1120 

1775 



3247 

3914 

4581 

5246 

5910 

6573 

7835 

7896 

8556 

9215 

9873 

0530 

1186 

1S41 



2430 2495 
3083 3148 



3735 
4386 
5036 
5686 
6334 
6981 
7628 
8273 
8918 



3800 
4451 
5101 
5751 



3314 

3981 

4647 

5312 

5976 

6639 

7301 

7962 

8622 

9281 

9939 

0595 

1251 

1906 

2560 

3213 

S8&5 

4516 

5166 

5815^ 



8 



6399 6464 



7046 
7692 
8338 
8982 



9561 9625 



.204 
10845 
1486 
2126 
2764 
3402 
4039 
4675 
5310 
5944 
6577 
7210 
7841 
8471 
9101 
9729 
0357 
0984 
1610 
2235 
285U. 
3482 
4104 
4726 



.268 

0909 

1550 

2189 

,2828 

3466 

4103 

4739 

5373 

6007 

6641 

7273 

7904 

8534 

9164 

9792 

0420 

1046 

1672 

2297 

29;21 

3544 

4166 

4788 



7111 

7757 

8402 

9046 

9690 

.332 

0073 

1614 

2253 

2892 

3530 

4166 

4802 

5437 

6071 

6704 

7336 

7967 

8597 

9227 

9855 

0482 

1109 

1735 

2360 

2983 



3381 

4048 

4714 

5378 

6042 

6705 

7367 

9028 

8688 

9346 

0661 
1317 
1972 
2626 
3279 
3930 
14581 
5231 
5880 
6528 
7115 
7821 
8467 
9111 
9754 
.396 
1037 
1678 
23171 
2956 
3593 
4230 
4866 
5500 
6134 
6767 
7399 
80S0 
8660 
9269 
9918 
0545 
1172 
1797 
2422 
3046 



4229 
4850 



3606 3669 



4291 
4912 



3448 
4114 
4780 
5445 
6109 

mil 

7433 

8094 

8754 

9412 

..70 

0727 

1382 

2037 

2691 

3344 

8996 

4646 

5296 

5945 

6593 

7240 

7886 

8531 

9175 

9818 

.460 

1102 

1742 

25S1 

3020 

3657 

4294 

4929 

5564 

6197 

6830 

7462 

8093 

8723 

9352 

9981 

0608 

1234 

1860 

2484 

3108 

3731 

4353 

4974 



3514 
4181 

4847 

5311 

6175 

6838 

7499 

8160 

8820 

9478 

. 136 

0792 

1448 

2103 

2756 

3409 

4061 

4711 

5361 

€010 

6658 

7S05 

7951 

8595 

9239 

9S8S 

.525 

1166 

1806 

2445 

3083 

3721 

4357 

491^3 

5627 

6^1 

6894 

7525 

8156 

8786 

9415 

..43 

0671 

1297 

1922 

2547 

3170 

3793 

4415 

5036 











OF NUMBERS. ' 




1 




1 


N. 
700 





1 


2 


3 
528* 


4 


5 


6 
5470 


7 


8 
5594 


9 ' 




845098 


5160 


5222 


5346 


5408 


5532 


5656 


1 

■ 


701 


5718 


5780 


5842 


5904 


5966 


6028 


6090 


6151 


6213 


6275 


1 


702 


6337 


6399 


6461 


6523 


6585 


6646 


6708 


6770 


6832 


6^94 


( 


703 


6955 


7017 


noi9 


7141 


7202 


7264 


7326 


7388 


7449 


7511 




704 


7573 


7634 


7696 


7758 


78*19 


7881 


7943 


8004 


8066 


8128 




705 


8r89 


8251 


8312 


8374 


8435 


8497 


8559 


8620, 


8682 


8743 




708 


8805 


8866' 


8928 


8989 


9051 


9112 


9174 


9235 


9297 


9358 




707 


9419 


9481 


9542 


9604 


9665 


9726 


9788 


9849 


9911 


^972 




708 


850033 


0095 


0156 


0217 


0279 


0340 


0401 


0462 


0524 


0585 




709 


0646 


0707 


0769 


0830 


0891 


0952 


1014 


1075 


1136 


1197 




710 


1258 


1320 


1381 


1442 


1503 


1564 


1625 


1686 


1747 


1809 




.711 


1870 


1931 


1992 


2053 


2114 


2175 


2236 


2297 


2358 


2419 




712 


2480 


25*41 


2602 


2663 


2724 


2785 


2846 


2907 


2968 


3029 




713 


3090 


3150 


32 U 


3272 


3333 


3394 


3455 


3516 


3577 


2637^ 




714 


3698 


3759 


3820 


388-1 


S941 


4002 


4063 


4124 


4185 


4245^ 




715 


4306 


4367 


4428 


4488 


4549 


4610 


4670 


4731 


4792 


4852 




716 


4913 


4974 


5034 


5095 


5156 


5216 


5277 


5337 


5398 


5459 




717 


5519 


5580 


5640 


5701 


5761 


5822 


5882 


5943 


6003 


6064 




718 


6124 


6185 


6245 


6306 


6^66 


6427 


6487 


6548 


6608 


6668 




719 


6729 


6789 


6850 


6910 


6970 


7031 


7091 


7152 


7212 


7272 




720 


7332 


7393 


7453 


7513 


7574 


7634 


7694 


7755 


lSi5 


7875 




721 


7935 


7995 


8056 


8116 


8176 


8236 


8297 


8357 


8417 


8477 




722 


8537 


8597 


8657 


8718 


8778 


8838 


8898 


8958 


9018 


907i 


« 


723 


9138 


9198 


9258 


9318 


9379 


9439 


9499 


9559 


9619 


9679 




724 


9739 


9709 


9859 


9918 


9978 


..38 


.,98 


. 153 


.218 


• 278 




725 


860338 


0398 


0458 


0518 


0578 


0637 


0697 


0757 


08 n 


0877 




726 


0937 


0996 


1056 


1116 


1176 


1236 


1295 


1355 


141e5^ 


1475 




727 


1534 


1594 


1654 


1714 


1773 


1833 


1893 


1952 


2012 


2072 




728 


2131 


2ldl 


2251 


2310 


2370 


2430 


2489 


2549 


2608 


2668 




729 


2728 


2787 


2847 


2906 


2966 


3025 


3085 


3144 


3204 


3263 




730 


3323 


3382 


3442 


3501 


3561 


3620 


3680 


3739 


3799 


3858 


' 


731 


3917 


3977 


4036 


4096 


4155 


4214 


4274 


4333 


4392 


4452 




732 


4511 


4570 


4630 


4*689 


4748 


4808 


4867 


4926 


4985 


5045 


- 


733 


5104 


5163 


5222 


5282 


5341 


5400 


5459 


5519 


5578 


5637 




734 


5696 


5755 


5814 


5874 


5933 


5992 


6051 


6110 


6169 


6228 




735 


6287 


6346 


6405 


6465 


6524 


6583 


6642 


6701 


6760 


6819' 




736 


6878 


6937 


6996 


7055 


7114 


7173 


7232 


7291 


7350 


7409 




737 


7467 


7526 


75.85 


7644 


7703 


7762 


7821 


7880 


7939 


7998 




738 


80S6 


8115 


8174 


8233 


8292 


8350 


8409 


8468 


8527 


8586 




739 


8644 


8103 


8762 


8821 


8879 


8938 


8997 


9056 


9114 


9173 




740 


9232 


9290 


9349 


9408 


9466 


9525 


9584 


.9642 


9701 


9760 




741 


9818 


9877 


9935 


9994 


..53 


.111 


. 170 


.228 


.287 


.SU 




742 


870404 


0462 


0521 


0579 


0638 


0696 


0755 


0813 


0872 


093Q 




743 


0989 


1047 


ll06 


1164 


1223 


1281 


1339 


1398 


1456 


1515. 


1 


'H4 


1573 


1631 


1690 


1748 


1806 


1865 


1923 


1981 


2040 


2098 




745 


2156 


2215 


2273 


2331 


2389 


2448 


2506 


2564 


2622 


2681 




746 

^ 


2739 


2797 


2855 


2913 


2972 


3030 


3088 


3146 


3204 


3262., 




i^r 


3321 


3379 


3437 


3495 


3553 


3611 


3669 


3727 


3785 


3844. 




748 


3902' 


3960 


4018 4076 


4134 


4192 


4250 


4308 


4366 


4424 


f 


749 


- im 


4540. 4598 1 4656 


4714 


4772 


4830 


4888 


4945 

— r-J 


5003 





^) 



580 



LOGARITHMS 



N. 
750 





1 


2 


3 


4 


5 
5351 


6 


7 


8 


9 


^75061 


5119 


FirT 


5235 


5293 


5409 


5466 


5524 


5582 


751 


5640 


5698 


5756 


5813 


5871 


>5929 


5987 


6045 


6102 


6160 


752 


6218 


6276 


6333 


6391 


6449 


6507 


6564 


6622 


6680 


6737 


753 


67i^5 


6853 


6910 


6968 


7026 


7083 


7141 


7199 


7256 


7S14 


754 


7371 


7429 


7487 


7544 


7602 


7659 


7717 


7774 


7832 


7889 


755 


' 7947 


8004. 


8062 


8119 


8177 


8234 


8292 


8349 


8407 


8464 


756 


8522 


8579 


8637 


8694 


8752 


8S09 


8866 


8924 


8981 


9039 


757 


9096 


9153 


9211 


9268 


9325 


9383 


9440 


9497 


9555 


9612 


758 


9669 


9726 


9784 


9841 


9898 


9956 


.. 13 


..70 


.127 


. 185 


759 


880242 


0299- 


0356 


0413 


0471 


0528 


0585 


0642 


0699 


0756 


760 


08 1 4 


0871 


0928 


0985 


1042 


1099 


1156 


1213 


1271 


1328 


761 


1385 


1442 


1499 


1556 


1613 


1670 


1727 


17^4 


1841 


1898 


762 


1955 


2012 


2069 


2126 


2183 


2240 


2297 


2354 


2411 


2468 


763 


2525 


2581 


2638 


2695 


2752 


2809 


2866 


2923 


2980 


3037 


764 


3093 


3150 


3207 


'3264 


3321 


3377 


'3434 


3491 


3548 


3605 


765 


3661 


3718 


3775 


3832 


3888 


3945 


4002. 


4059 


4115 


4172 


766 


422^ 


4285 


4342 


4399 


4455 


4512 


4569 


4625 


4682 


4739. 


767 


4795 


4852 


4909 


4965 


5022 


5078 


5135 


5192 


5248 


5305 


76S 


5361 


5418 


5474 


5,531 


5587 


5644 


5700 


5757 


5813 


5870 


769 


5926 


5983 


6939 


6096 


6152 


6209 


6265 


6321 


6378 


6434 
699S ' 


770 


6491 


6547 


6604 


6660 


6716 


6773 


6,829 


6835 


6942 


771 


7054 


7111 


7167 


7223 


7280 


7336 


7392 


7449 


7505 


1561 


772 


7617 


7674 


7730 


7786 


7842 


7898 


7955 


8011 


8067 


8123 


773 


8179 


8236 


8292 


8343 


8404 


8460 


8516 


ft573 


8£29 


8685 


774 


8741 


8797 


8853 


8909 


8965 


9021 


9077 


9134 


9190 


9246 


775 


9302 


9358 


9414 


9470 


9526 


9582 


9638 


9694 


9750 


9806 


•776 


9862 


9918 


9974 


..30 


..86 


.141 


. 197 


.253 


.309 


.365 


177 


890421 


0477 


0533 


0589 


0645 


0700 


0756 


0812 


0868 


0924 


778 


0980 


1035 


1091 


1147 


1203 


1259 


1314 


1370 


1426 


1482 


779 


1537 


1593 


1649 


1705 


1760 


1816 


1872 


1928 


1983 


2039 


780 


2095 


2150 


2206 


2262 


2317 


2373 


2429 


2484 


2540 


2595 


78J 


2651 


2707 


2762 


2818 


2873 


2929 


2985 


3040 


3096 


3151 


782 


32Q7 


32^2 


3318 


3373 


3429 


3484 


3540 


3595 


3651 


3706 


783 


3762 


3817 


3873 


3928 


3984 


4039 


4094 


4150 


4205 


4261 


784 


4316 


4371 


4427 


4482 


4538 


4593 


4648 


4704 


4759 


4814 


785 


4870 


4925 


4980 


5036 


5091 


5146 


5201 


5257 


5312 


53^7 


7«6 


5423 


5478 


5533 


5588 


5644 


5699 


5754 


5809 


5B6^ 


5980 


787 


5975 


6030 


6085 


6140 


6195 


6251 


6306 


6361 


6416 


6471 


7?8 


65i2^Q 


6581 


6636 


6692 


6747 


6S02 


6857 


6912 


6967 


7022 


78^ 


7077 


7132 


7187 


7242 


7297 


7352 


7407 


7462 


7517 


7572 


790 


7627 


7682 


7737 


7792 


7847 


7902 


7957 


8012 


8067 


8122 


791 


. 8176 


823L 


8286 


8341 


8396 


8451 


8505 


8561 


8615 


8670 


792 


8725 


8780 


8835 


8890 


8944 


8999 


9054 


9109 


9164 


9218 


793 


9273 


9328 


9383 


9437 


9492 


9547 


9602 


9656 


9711 


9766 


794 


9821 


9875 


9930 


9985 


,.39 


..04 


.149 


,203 


.25B 


.312 


795 


9Q0367 


0422 


04718 


0531 


0586 


Q640 


0695 


0749 


0804 


0$^ 


796 


09J3 


0968 


1022 


1077 


1131 


1186 


1240 


1295 


1,349 


1404 


797 


145S 


1513 


1567 


1622 


1676 


1731 


1785 


1,840 


4394 


1948 


fSS 


2003 


2057 

< 


2112 


2166 


2221 


227:5 


2329 


2384 


243? 


2432 


799 

*- It il 


2547 


2601 


2655 2710 


2764 28 IB '2873 •2927' 29^1 

■ ■"">■■ ". — . — : '«■■■ ' j'l 


303^ 











OF NUMBERS. 




. 




3«I 


1— 





1 


2 


3 
3253 


4 


5 


6 


7 


# ' 


9 


800 


903090 


3 J 44 


3199 


3S07 


336? 


3416 


3470 


3524 


3578 


801 


3633 


3687 


3741 


8795 


3849 


3904 


3958 


4012 


4066 


4120 


802 


4174 


4229 


4283 


4337 


4391 


4445 


4499 


4553 


4607 


4661 


803 


4716 


4770 


4824 


4878 


4932 


4986 


5040 


5094 


5148 


5202 


804 


5256 


5310 


5364 


5418 


5472 


5526 


5580 


5634 


5688 


5742 


805 


5796 


5850 


5904 


5958 


6012 


6066 


6119 


6173 


6227 


6281 


806 


6335 


6389 


6443 


6497 


6551 


6604 


6658 


6712 


6766 


6820 


807 


6874 


6927 


6981 


7035 


7089 


7143 


7196 


7250 


7304 


7358 


808 


7411 


7465 


7519 


7573 


7626 


7680 


7734 


7787 


7841 


7895 


809 


7949 


8002 


8056 


8110 


8613 


8217 


8270 


8324 


8378 


8431 


810 


8485 


8539 


8592 


8646 


8699 


8753 


8807 


8860 


8914 


8967 


811 


9021 


9074 


9128 


9181 


9235 


9289 


9342 


9396 


9449 


9503 


812 


9556 


9610 


9663 


9716 


9770 


9823 


9877 


9930 


9984 


..37 


813 


910091 


0144 


0197 


0251 


0304 


0358 


0411 


0464 


0518 


0571 


814 


0624 


0678 


0731 


0784 


0838 


0891 


0944 


0998 


1051 


1164 


815 


1158 


1211 


1264 


1317 


1371 


1424 


1477 


1530 


1584 


1637 


816 


1690 


1743 


1797 


1^50 


1903 


1956 


2009 


,2063 


2116 


2169 


817 


2222 


2275 


232:3 


2381 


2435 


2488 


2541 


2594 


2647 


2700 


818 


27;53 


2806 


2859 


2913 


2966 


3019 


3072 


3125 


3178 


3?31 


819 


3284 


3337 


3390 


3443 


3496 


3549 


3602 


3655 


3708 


8761 


820 


3814 


3867 


S920 


3973 


4026 


4079 


4132 


4184 


4237 


4290 


821 


4343 


4396 


4449 


4502 


4555 


4608 


4660 


4713 


4766 


4819 


822 


4872 


4D25 


4977 


5030 


5083 


5136 


5189 


5241 


5294 


5347 


523 


5400 


5453 


5.005 


5558 


5611 


5664 


5716 


5769 


5822 


5875 


824 


5927 


5980 


6033 


6085 


6138 


6191 


6243 


6296 


6349 


6401 


85?5 


6454 


6507 


6559 


6612 


6664 


6717 


6770 


6822 


6875 


6927 


^ 826 


6980 


7P33 


7085 


7138 


7190 


7i?43 


7295 


7348 


7400 


7453 


827 


7506 


7558 


7611 


7663 


7716 


7768 


7820 


7873 


7925 


7978 


< 828 


8030 


8083 


8135 


8188 


8240 


8293 


8345 


8397 


8450 


8502 


829 


S55$ 


8607 


8659 


8712 


8764 


8816 


8869 


8921 


8973 


9026 


'830 


9078 


9180 


9183 


9235 


9287 


9340 


9392 


9444 


9496 


9549 


831 


9601 


9653 


9706 


9758 


9810 


9862 


9914 


9967 


..19 


..71 


832 


920123 


0176 


0228 


0280 


0332 


0384 


0436 


0489 


0541 


0593 


833 


0645 


0697 


0749 


0801 


0853 


0906 


0958 


1010 


1062 


1114 


834 


1166 


1218 


1270 


1322 


1374 


1426 


1478 


1530 


1582 


1634 


835 


1686 


1738 


1790 


1842 


1894 


1946 


1998 


2050 


2102 


21H 


836 
837 


2206 


225§ 


2310 


2362 


2414 


2466 


2518 


2570 


2622 


2674 


2725 


2777 


2829 


2881 


2933 


2985 


3037 


3089 


3140 


3192 


838 


3244 


3296 


3348 


3399 


3451 


3503 


3555 


3607 


3658 


3710 


859 


3762 


3814 


3865 


3917 


3969 


4021 


4072 


4124 


4176 


4228 


840 


4279 


4331 


4383 


4434 


4486 


4538 


4589 


464) 


4693 


4744 


841 


4796 


4848 


4899 


4951 


5003 


5054 


5106 


5157 


5209 


5261 


842 


53 12 


5364 


5415 


5467 


5518 


5570 


5621 


5673 


5725 


.5776 


843 


5828 


5879 


5931 


5982 


6034 


6085 


6137 


6188 


6240 


6^91 


844 


6342 


6394 


6445 


6497 


6548 


6600 


6651 


6702 


6754 


6805 


845 


6857 


6908 


6959 


7011 


7062 


7114 


7165 


7216 


7268 


7319 


Us 


7370 


7422 


7473 


7524 


7576 


7627 


7678 


7730 


7781 


7832 


847 


7883 


79^5 


7986 


8037 


8088 


8140 


8191 


8242 


8293 


8345 


848 


8396 


8447 


8498 


85^9 


8601 


8652 


8703 


8754 


«805 


8857 


ill 


8908 


8959 


9010 9061 


9112 


9163 


9215 


9266 


^317 


9368 



382 




• 




LOGARITHMS 




' 




» 


N. 
850 





1 


2- 
9521 


3 
9572 


4 


5 


6 


7 


8 


9 


929419 


9470 


9623 


9674 


9725 


9776 


9827 


9879 


851 


9930 


9881 


..82 


..83 


.134 


. 185 


.236 


.287 


.338 


.389 


852 


930440 


0491 


0542 


0592 


0643 


0694 


0745 


0796 


0847 


0898 


853 


0949 


1000 


1051 


1102 


1153 


1204 


1254 


1305 


1356 


1407 


854 


1458 


1509 


1560 


1610 


1661 


1712 


1763 


1814 


1865 


1915 


855 


1966 


2017 


2068 


211^ 


2169 


2220 


2271 


2322 


2,372 


2423 


856 


2471. 


2524 


2575 


2626 


2677 


2727 


2778 


2829 


2879 


2930 


857* 


2981 


3031 


3082 


3133 


3183 


3234 


3285 


3335 


3386 


3437 


858 


S487 


3538 


3589 


3639 


3690 


3740 


3791 


3841 


3892 


3943 


859 


3993 


4044 


4094 


4145 


4195 


4246 


4296 


4347 


4394 


4448 


860 


4498 


4549 


4599 


4650 


4700 


4751 


4801 


485^ 


4902 


4953 


861 


5CK)3 


5054 


5104 


5154 


5205 


5255 


5306 

• 


5356 


5406 


,5457 


862 


5507 


5558 


5608 


5658 


5709 


5759 


5809 


5860 


5910 


5960 


863 


6011 


6061 


6111 


6162 


6212 


6262 


6313 


6363 


6413 


6463 


864 


6514 


6564 


6614 


e^^Qo 


6715 


6765 


6815 


6863 


6916 


Q^^i^ 


865 


7016 


7066 


7117 


7167 


7217 


7267 


7317 


7367 


7418 


7468 


866 


7518 


7568 


7618 


7668 


7718 


7769 


7819 


7869 


7919 


7969 


867 


8019 


8069 


8119 


8169 


8219 


8269 


8320 


8370 


8420 


8470 


868 


8520 


8570 


8620 


8670 


8720 


8770 


8820 


8870 


8919 


8970 


869 


9020 


9070 


9120 


9170 


9220 


9270 


9320 


9369 


9419 


9469 


870 


9519 


9569 


9619 


9669 


9719 


9769 


9819 


9869 


9918 


9968 


871 


940018 


0068 


0118 


0168 


0218 


0267 


0317 


0367 


0417 


0467 


872 


0516 


0566 


0616 


0666 


0716 


0765 


0815 


0865 


0915 


0964 


873 


1014 


1064 


1114 


1163 


1213 


1263 


1313^ 


1362 


1412 


1462 


874 


1511 


1561 


1611 


1660 


1710 


1760 


1809 


1859 


1909 


1958 


875 


2008 


2058 


2107 


2157 


2207 


2256 


2306 


2355 


2405 


2455 


876 


2504 


2554 


2603 


2653 


27Q2 


2752 


2801 


28.51 


2901 


2950, 


877^ 


3000 


3049 


3090 


3148 


3198 


3247 


3297 


3346 


3396 


S445 


878 


3*95 


3544 


8593 


3643 


3692 


3742 


3791 


3841 


3890 


8939 


879 


3989 


4038 


4088 


4137 


4186 


4286 


4285 


4335 


4384 


4433 


880 


4483 


4592 


4581 


4631 


4680 


4729 


4779 


4828 


4877 


4927 


881 


4976 


5025 


5074 


5124 


5173 


5222 


5272 


5321 


5370 


5419 


882 


5469 


5518 


5567 


5616 


b^^h 


5715 


5764 


5813 


b%%2 


5912 


883 


5961 


6010 


6059 


6108 


6157 


6207 


6256 


6305 


6354 


6403 


884 


6452 


6501 


6551 


6600 


6649 


6698 


6747 


6796 


684^ 


6894 


885 


6943 


6992 


7041 


7090 


7140 


7189 


7238 


7287 


7336 


7385 


886 


7434 


7483 


7532 


7581 


7680 


7679 


7728 


7777 


7826 


7875 


887 


7924 


7973 


8022 


8070 


8119 


8168 


8217 


8266 


8315 


8364 


888 


8413 


8462 


8511 


8560 


8609 


8657 


8706 


8755 


8804 


8853 


889 


8902 


8951 


8999 


9048 


9097 


9146 


9195 


9244 


9292 


9341 


890 


9390 


9439 


9488 


9536 


9585 


9634 


9683 


9731 


9780 


9829 


891- 


' 9878 


9926 


9975 


..24 


..73 


.121 


.170 


.219 


.2^7 


.316 


892 


950365 


0414 


0462 


0511 


0560 


0608 


0657 


0706 


Q754 


0803 


893 


0851 


0900 


0949 


0997 


1046 


1095 


1143 


1192 


1240 


1289 


894 


1338 


1386 


1435 


1483 


1532 


1580 


1629 


1677 


1726 


1775 


895 


1823 


1872 


1920 


1969 


2017 


2066 


2114 


2163 


2211 


2260 


896 


2308 


2356 


2405 


2453 


2502 


2550 


2599 


2$47 


2696 


2744 


897 


2792 


2641 


2889 


2938 


2986 


3Q34 


3083 


3131 


3180 


3228 


898 


3276 


3325 


3373 


3421 


3470 


3518 


3566 


3615 


3663 


8711 


899 


3760 3808 

1 . 


3856 


3905 


3953 


4001 


4049 


4093 


4146 


4194 



OF iJUMBERS. 



'^ 



Sdi 



N. 1 





1 


£ 


3 


4 


5 


6 


7 


8 

4628 


9 


900 < 


J54243 - 


*291 ' 


4.339 - 


4387 


4435 


4484 


4^532 


4580 


4677 


901 


472.5 - 


4773 - 


4^821 


4869 


4918 


4966 


5ai4 


5062 


5110 


5158 


902 


51207 


5255. 


5«03 


5351 


5399 


54^1.7 


5495 


5543 


5592 


5640 


903 


5688 


5736 


5784 


5832 


5880 


5928 


5976 


6024 


6072 


6120 • 


904 


6168 


6216 


6265 ' 


6313 


6361 


6409 


6457 


6505 


6553 


6601 » 


905 


66k9 


6691 


6745 


6793 


6840 


6888 


6936 


6984 


7032 


7080 ' 


906 


7l28 


7176 


7224 


7272 73201 


7368 


7416 


7464 


7512 


7559 


907 


7607 


7655 


7703 


7751 


7799 7847 


7894 


7942 


7990 


8038 


908 


8086 


8134 


8181 


8229 


8277 8325 


8373 


8421 


8468 


8516 


909 


•8564 


8612 


8659 


8707 


8755 


8803 


8850 


8898 


8946 


89&4 


910 


9041 


9089^ 9137 


9185 


9232 


9280 9328 


9375 


942S 


9471 


911 


9518 


9566 9614 


9661 


9709 


9757 


9804. 9852 


9900 


994*1 


912 


9995 


..h 


..90 


. 138 


. 185 


.2.33 


.280 


.328 


.376 


.42S( 


913 


96047 1 


0518 


0566 


0613 


0661 


0709 


0756 


0804 


0851 


0899 


914 


0946 


0994 


1041 


1089 


1136 1184 


1231 


1279 


1326 1374 1 


915 


1421 


1469 


1516 


1563 


1611 1658 


1706 


1753 


1801 


1848 


916 


1895 


1943 


1990 


2038 


2085 


2132 


2180 


2227 


2275 


231i2 


917 


2369 


2417 


2464 


2511 


2559 


2606 


2653 


2701 


2748 


2795 


918 


2843 


2890 


2937 


298^ 


3032 


3079 3126 


3174 


3221 


3268 


919 


3316 


3363 


3410 


3457 


3504 


3552 


3599 


3646 


3693 


3741 


920 


3788 


3835 


3882 


3929 


3977 


4024 


4071 


4118 


4165 


4212 


92 r 


4260 


4307 


4354 


4401 


4448 


4495 


4542 


4590 


4637 


4684 


922 


4731 


4778 


4825 


4872 


4919 


4966 


5013 


5061 


5108 


5155 


923 


5202 


5249 


5296 


5343 


5390 


5437 


5484 


5531 


5578 


5625 


924 


.5672 


5719 


5766 


5813 


^860 


5907 


5954 


6001 


6048 


6095 


925 


6142 


6189 


6236 


6283 


6329 


6376 


6423 


6470 


6517 


6564 


926 


6611 


6658 


6705 


6752 


6799 


6845 


6892 


6939 


6986 


7033 


927 


7080 


7127 


7173 


7220 


7267 


7314 


7361 


7408 


7454 


7501 


928 


7548 


7595 


7642 


7688 


7735 


7782 


7829 


7875 


7922 


7969 


929 


8016 


8062 


8109 


8156 


8203 


8249 


8296 


8343 


8390 


84^6 


930 


8483 


8530 


8576 


8623 


8670 


8716 


8763 


8810 


8856 


8903 


931 


8950 


8996 


9043 


9090 


9136 


9183 


9229 


9276 


9323 


9369 


932 


9416 9463 


9509 


9556 


9602 


9649 


9695 


9742 


9789 


9835 


933 


9882 
970347 


9928 


9975 


. .21 


..68 


. 114 


.161 


.207 


.254 


.300 


934 


0393 


0440 


0486 


0533 


0579 


0626 


0672 


0719 


0765 


935 


0812 


0858 


0904 


0951 


0997 


1044 


1090 


1137 


IJ83 


1229 


936 


1276 


1322 


1369 


1415 


1461 


1508 


1554 


1601 


1647 


1693 


937 


1740 


1786 


1832 


1879 


1925 


1971 


2018 


2064 


2110 


2157 


ns 


2203 


2249 


2295 


2342 


2388 


2434 


2481 


2527 


2573 


2619 


939 


2666 


2712 


2758 


2804 


2851 


2897 


2943 


2989 


3035 


:i082 


940 


3128 


3174 


3220 


3266 


3313 


3359 


3405 


3451 


3497 


3543 


941 


3590 


3636 


3682 


3728 


3774 


3820 


3866 


3913 


3959 


4005 


942 


4051 


4097 


4143 


4189 


4235 


4281 


432(7 


4374 


4420 


4466 


943 


45U 


4558 


4604 


4650 


4696 


4742 


4788 


4834 


4880 


4926 


944 


4972 


5018 


5064 


5110 


5156 


5202 


5248 


5294 


5340 


5386 


945 


5432 


5478 


5524 


557b 


5616 


5662 


5707 


5753 


5799 


5845 


946 


5891 


5937 


5983 


6029 


6075 


6121 


6167 


6212 


6258 


6304 


947 


6350 


6396 


6442 


6488 


6533 


6579 


6625 


6671 


6717 


6763 


948 


6803 


6854 


6900 


6946 


6992 


7037 


7083 


7129 


7175 


7220 


949 


7266 


7312 7358 


7403 


7449 


7495 


7541 


7586 


7632 


7678