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Full text of "Concise mathematical operations; being a sequel to the author's class books:"

Norwich llniversitv Librai^, 

NortlAfleld, Vermont. 
Class NO. JT/O Boot^No.3fJ5^ 




University of California • Berkeley 



The Theodore P. Hill Collection 

of 
Early American Mathematics Books 



Digitized by the Internet Archive 

in 2008 with funding from 

IVIicrosoft Corporation 



http://www.archive.org/details/concisemathematiOOrobirich 



CONCISE 
MATHEMATICAL OPERATIONS; 



BEING A 



SEQUEL 



TO THE AUTHOR'S CLASS BOOKS 



WITH MUCH ADDITIONAL MATTER. 



A. WORK ESSENTIALLY PRACTICAL, DESIGNED TO GIVE THE LEARNER A PROPER APPRE* 

CIATION OF THE UTILITY OF MATHEMATICS ; EMBRACING THE GEMS OF 

SCIENCE FROM COMMON ARITHMETIC, THROUGH ALGEBRA, 

GEOMETRY, THE CALCULUS, AND ASTRONOMY. 



BY H. N. ROBINSON, A. M. 

FORMERLY PROFESSOR OF MATHEMATICS IN THE UNITED STATES NAVY J ATTTHOR OF 

ARITHMETIC, ALGEBRA, NATURAL PHILOSOPHY, GEOMETRY, 

SURVSYING, ASTRONOMY, ETC. ETC. ETC, 



CINCINNATI: 

JACOB ERNST, 112 MAIN" STREET. 
1854. 



Entered according to Act of Congress in the year 1854, bj 

H. N. ROBINSON, 

In the Clerk's Ofl&ce of the District Court of the United States 

for the Northern District of New York. 



PREFACE. 

This book is not designed to teach Mathematical Principles, but to apply 
and enforce them. It contains collections and groups of mathematical prob- 
lems which show the utility of science, and place its fruits in the foreground. 

Let no one expect to find any close connection between the different parts 
of this book, or even in any one part of it. System and connection is essen- 
tial in every theoretical work, but it would be as. absurd to look for it here 
as to look for a composition in a dictionary. 

That there is need for such a work as this, all would be convinced who 
could see but a tenth part of the letters that every accommodating mathema- 
tician is constantly receiving, requesting the solution of problems or the 
exposition of principles. 

Indeed, much important matter to be found in this volume, has been sug- 
gested and brought to the immediate notice of the author by letters received 
requiring his aid ; and to save the trouble of answering such letters in future 
was one inducement to publish this work. 

There is a great deal of perfectly barren mathematical knowledge in this 
country; particularly among those who have studied, not for science, but for 
a diploma. 

Not unfrequently do we meet persons who can demonstrate many, if not 
all the elementary problems in common Geometry, who, at the same time, 
cannot make the least application of them, and who seem to be unaware that 
they were ever intended for any practical use. 

Knowledge, so confined and abstract, is of doubtful utility, even as a 
mental discipline. Unless we take in a broad expanse, and unite both theory 
and practice, we perceive nothing of the beauties of the Mathematics. De- 
tached propositions and abstract mathematical principles, give us no better 
idea of true and living science, than detached words and abstract grammar 
would give us of poetry and rhetoric. Small acquirements in the Mathe- 
matics serve only to make us timid, cautious, and distrustful of our own 
powers — ^but a step or two further gives us life, confidence, and power. 

The efforts of the great mass, who attempt the study of the Mathematics, 
are very inefficient and feeble, because the motive is not sufficiently pointed 
and pressing. They study for the discipline of mind. 

]N'ow, we venture to assert, that those who study for any object so indirect 
and indefinite, can never be decidedly successful. And those who teach 
with no other view than giving discipline to the minds of their pupils, never 
more than half teach. The object, and the only object, should be to under- 
stand the subject studied, and if that understanding is attaiiied, the highest 
mental discipline that the subject can yield, will surely come with it. 

iii 



iv PREFACE. 

Let a person undertake the study of any science, Trigonometry for exam 
pie, with no other object than the discipline of the mind, and our word for it, 
the science will come to him with the utmost diflficulty ; and however long 
he may study, the spirit of the science will never find a lodgment with him. 
But let him be determined to understand it, for the purpose of being an 
architect, an engineer, or a navigator, and all is changed — beauties are now 
seen where none were discovered before, and the student is now sensible of 
possessing both knowledge and mental discipline. 

Let a person commence Astronomy, simply with a view to mental disci- 
pline, and when will he obtain a sound knowledge of that science ? We 
answer, never. But let him commence the study with a determination to 
understand it, and his efforts will be well directed, and science will come to 
him with ease, and with it will come a discipline of mind, the most pure 
and lasting that man can attain. 

There is another erroneous impression which serves, as far it goes, to ob- 
struct the progress of sound mathematical learning in this country. It is a 
vague, yet general idea, that Arithmetic, Algebra, Geometry, Trigonometry, 
and the Calculus, are distinct and separate sciences, and each is to be learned 
by itself and then carefully laid aside. The truth is, they are but diJOferent 
sections of the same science, and each one in turn may be used to illustrate 
the other ; and studied as a whole, under the direction of a philosophic 
t^^acher, the labor of acquisition would be very much reduced. 

Were we to say nothing in respect to our method of treating the square 
and cube roots in this volume, the mere arithmetician would undoubtedly 
depreciate it. He will perhaps still regard the method as unscientific, 
and call it a mere " cut and try" operation ; but when he finds the same thing 
in Geometry, and there finds lines which may represent all the different 
factors in any case, and sees the geometrical reason why the exact square root 
is always a little less than the half sum of two unequal factors, he must then 
admit that the cut and try method is not very unscientific after all. The 
truth is, in the hands of those who can take the geometrical view of it, and 
who can use it with judgment, this method is as scientific as any, and in 
many cases far more practical than the common rilles. 

The first principles of Geometry are, to a certain degree, abstract ; but the 
application of Geometry, as appears in this work, is far from being so ; and 
he must be a very practical mathematician who cannot find something here 
to amuse, to interest, or to instruct him. 

To the subject of finding sines and cosines, both'natural and logarithmic, 
for every minute of the quadrant, we call special attention — as strict attention 
to that subject in all its bearings, will so readily impress upon the mind of 
a learner, the importance of theoretical Geometry. 

To the practial application of Interpolation, we'also call attention. Some 
problems in Mensuration and Plane Trigonometry will be found very inter- 
esting to those who possess a taste for the Mathematics, and we have ex- 
tracted several different solutions from the works of others, to show how 



PREFACE. V 

diflferently diiFerent persons present the same thing. There are few mathe- 
matical students who could not be greatly benefitted bj a close perusal of 
Spherical Trigonometry and Astronomy as presented in this work. Any 
person who has the outlines of Astronomy and Elementary Mathematics can 
here have a view of all the details of a solar eclipse, in a comprehensible 
shape. 

There has been a great deal of unnecessary controversy about the Differen- 
tial and Integral Calculus, which we think can and ought to be wiped away. 
And we have here given a little foretaste of what we shall attempt if cir- 
cumstances prompt us to write a work on that subject. 

It is not for us to assume that we can make science clearer than others, 
but we have yet to see the works of an author who has made the least attempt 
to show the simple elementary nature of this science. They at once commence 
with the definition of constants and variables, and then direct what to do. 

We have yet to see the first book that expends a word in giving an idea of 
what the Calculus is, or what is the utility and object of the science, and we 
charge more than half the obscurity to this fact alone : hence we could not 
forbear being a little elementary when we came to that subject, and we leave 
it to those readers, who have fonnerly studied other works on this science, 
to say whether we have or can dispel any of the obscurity that has so long 
hovered around it. 

All sciences are obscure until they are applied. Even Arithmetic would 
be so in the abstract, and being alive to this fact we have extended the ap- 
plication of the Calculus to more subjects than we have hitherto observed in 
other works. For example, see the method of clearing lunar distances, and 
the use we made of the same principle in computing an eclipse. 

But neither in the Differential nor the Integral Calculus do we pretend to 
be any thing like full or perfect, even for a work of this kind. 

We have only thrown out a few practical remarks and problems, in our 
own unique manner, more to learn what is desired, and what can be appre- 
ciated, than for any thing else. 

"When we commenced, we did not intend to produce so large a volume; it 
grew on our hands; but we believe that this result will not be regretted 
by generous patrons. 



CONTENTS. 

PART I.— ARITHMETIC. 

SECTION I. 

Introduction, 13 14 

The Philosophy of Multiplication and Division, 14 18 

Canceling, 19 ^ 

Proportion, 23 ^29 

Cause and Eflfect, 26 ^29 

SECTION II. 

Exchange, -29 

Compound Fellowship, 30 32 

Problems in Mensuration and the Roots, 32 34 

SECTION III. 

Powers and Roots, 35 16 

Alligation Alternate, 46 48 

Position, 48— —50 



PART II.— ALGEBRA. 

SECTION I. 

Simple Equations, 51 59 

Problems Producing Simple Equations, 59 67 

Interpretation "of Negative Values, 67 68 

Finding and Correcting Errors, 69 71 

Pure Equations, 72 80 

Questions Producing Pure Equations, 80 84 

vii 



viii CONTENTS. 

SECTION II. 

Quadratic Equations, 84 89 

Special Equations in Quadratics, 89 ^96 

SECTION III. I 

Quadratic Equations containing more than one Unknown 

Quantity, 97 108 

UnTfTought Examples, 109 110 

SECTION IV. 

Problems producing Quadratic Equations containing more 

than one Unknown Quantity, 110 116 

Problems Selected from Various Sources, 116 123 

SECTION V. 

Problems in Proportion and Progression, 123 124 

Geometrical Progression and Harmonical Proportion, 125 129 

Proportion, 129 130 

Additional Problems, 130 138 

SECTION VI. 

Solution of Equations of the Higher Degrees, 138 161 

H'ewton's Method of Approximatic«i, 138 140 

Horner's Method, ^ 140 157 

l^ew and Concise Formula to find Approximate Roots in 

Quadratics, 143 144 

Cubic Equations, 148 152 

The Combination of Roots in the Fca-mationof Coefl5cients,.158 161 

Recurring Equations, 161 166 

SECTION VII. 

Indeterminate Analysis, 166 181 

Properties of Numbers, 166 170 

Indeterminate Problems, , . . , 171 181 



CONTENTS. k 

SECTION VIII. 

To determine the Number of Solutions that an E<iuation in 

the form AX-^BY=C, will admit of, 181 185 

SECTION IX. 

Diophantine Analysis, 185 192 

SECTION X. 

Double and Triple Equalities, 192 200 

Double Equalities, 193 195 

Triple Equalities, 195 200 

Application of the Diophantine Analysis, 201 ^202 



PART III.— GEOMETRY. 

SECTION I. 

Geometrical Theorems, 20.3 ^218 

Geometrical Constructions, 219 ^223 

Constructions for finding the Square Roots of Numbers,. . .221 ^223 

Geometrical Problems requiring the aid of Algebra, 224 ^241 

Numerical Problems, 240 ^241 

SECTION II. 

Trigonometry, • 243 ^268 

The Most Concise and Practical Method of finding the 

Circumference of a Circle, 242 246 

Interpolation : Its Utility in finding the Sines and Cosines 

of each Minute of the Quadrant, 246 248 

On finding Logarithmic Sines and Cosines, 248 251 

Solution of Trigonometrical Problems, 251 255 

Problems in Mensuration, 256 262 

Theorems on pages 219 and 220 of Robinson's Geometry, 262 267 

Additional Theorems, 267 ^268 



CONTENTS. 

SECTION III. 



Problems in Spherical Trigonometry and Astronomy, 269 ^272 

Problems from Page 215, Robinson's Geometry, 273 ^285 

Other Problems of like kind, 285 ^286 



PART IV.— PHYSICAL ASTRONOMY. 

kbpler's laws, 

Kepler's Laws, 287 ^289 

Propositions from Robinson's Astronomy, page 146, 289 291 

Increase of the Moon's Periodic Revolution, 292 293 

To find the Position of a Planet as seen from the Earth, 293 297 

SOLAR ECLIPSES. 

The Computation of the Eclipse of May 26th, 1854, for 

the Lat. of Burlington, Vt.,.\ 297 313 

The Elements, 297 398 

Tables for Correcting the^ Elements, 298 299 

To determine the time of the beginning of the Eclipse, 

and the Place at which it will be Central, 299 301 

Parallax in Altitude, 301 

Distance between the Centers, 302 305 

Point of first Contact, 305 

Greatest Obscuration, 305 309 

The End of the Eclipse, 309 313 

Summary, 313 

Correspondence, '. . . .313 314 



CONTENTS. xi 

THE CALCULUS. 

DIFFERENTIAL CALCULUS. 

Differential Calculus, .' 315 340 

Subject Defined, 315 

Logarithmic Differentials, 317 321 

Circulai- Functions, 322-^ — 330 

Differential Expressions for Trigonometrical Lines, 322 

Examples showing the Utility of the Calculus, 322 326 

Additional Examples in Circular Functions, 326 330 

Lunar Observations, 330 333 

Maxima and Minima, 334 34fr 

INTEGRAL CALCULUS. 

Introductory Remarks and Exercises, 340 342 

To Integrate two or more Expressions, 342 345 

Application of the Integral Calculus, 345 350 

To find the Value of a Semicircle to Radius Unity, 345 346 

Application of the Calculus to Surfaces, 346 348 

To Solids, 348 350 

Two Integrations from Poisson's Mecanique, 350 353 



MATHEMATICAL TABLES. 

TABLE I. 

Logarithms of Numbers from 1 to 10000, 1 20 

TABLE II. 

Logarithmic Sines and. Tangents, also Natural Sines, for 

<'very Degree and Minute of the Quadrant, . .21 65 



CONTENTS. 

TABLE III. 

Logarithms of Numbers from 1 to 1 10, including twelve places 

of Decimals, 66 67 

TABLE IV. 

Logarithms of the Prime N"umbers from 110 to 1129, including 

twelve places of Decimals, 67 69 

Formula for Computing the Logarithms of N'umbers beyond 

the limits of the Table, 69 

Auxiliary Logarithms, 70 

TABLE V. 

Dip of the Sea Horizon, 71 

TABLE VI. 

Dip of the Sea Horizon at different Distances from it, ^71 

TABLE VII. 

Mean Refraction of Celestial Objects, 71 



30536 



ROBINSON'S SEQUEL. 



PART FIRST. 

ARITHMETIC. 

SECTION I. 

We shah i)e very brief in this work on the subject of arithme- 
tic, only truching on such points as are generally neglected in 
the class r>om. 

Formerly all kinds of problems and puzzles were to be found 
in arithmetics ; but pure science, good taste, and the rapid ad- 
vancemert of the pupils, require that the works on arithmetic 
should bf concise and clear, and take no undue proportion of the 
student's ^ime and attention. 

Severo problems do not teach science — but science will subdue 
all severe problems, and we would use problems only as a means 
of elucidating science. Algebraic problems, and problems in 
geometry and mensuration, should never appear in arithmetic, 
but old custom will not yet tolerate their expulsion. 

We shall pay particular attention to the metaphysique of the 
science. 

Numbers only of the same kind can be added together or subtracted 
from each other. 

Numbers are either abstract or concrete. Abstract numbers 
are unapplied and are mere numerals. Concrete numbers bring 
to the mind the particular number of things to which they refer. 

Arithmetic proper, comprises the system of notation, and the 
operations to be performed with abstract numbers only — vnihout 
any reference to their application whatever. 

The application of arithmetic includes all kinds of numerical 

13 



U ROBINSON'S SEQUEL. 

computations, and they are therefore endless in variety and char- 
acter. 

In the application of arithmetic, there are two distinct opera- 
tions, the logical one and the mechanical one; the thinking and the 
doing. 

The undisciplined direct their attention more to the doing than 
to the thinking, when it should be the reverse; and nearly all the 
efforts of a good teacher are directed to make his pupils reason 
correctly. 

If a person fails in an arithmetical problem, the failure is always 
in the logic, for false logic directs to false operations, and true logic 
points out true operations. 

Abstract arithmetic we shall not touch, except when necessary 
to illustrate a point before us. 

With these introductory remarks we commence with the follow- 
ing principles : 

1. Multiplication is the i^epetition of one number as many times 
as there are units in another. 

This is general, whether the numbers be large or small, whole 
or fractional. The mles in whole numbers and in fractions, apply 
to the mechanical operations only, and not to the one fundamental 
principle. 

2. When the multiplicand and multiplier are both abstract num- 
bers, the product is abstract, or a mere numeral without a name. 

3. iVb two things can be multiplied together. 

A multiplicand may have a name, as dollars, yards, men, <fec. ; 
then the multiplier must be a mere numeral, and the product will 
have the same nam>e as the multipHcand. 

4. Division is finding how many times one number can he sub- 
tracted from another of the same hind. 

There are other definitions to be found in books, which do very 
well in the main, but this is the only truly logical definition I can 
find. Division should never be considered in the light of sepa- 
rating a number into parts, for this is not true in all cases, and 
confusion often arises in fractions by this view of the subject. 



ARITHMETIC. 16 

5. Division corresponds to multiplication conversely, when we take 
the product for a dividend, the multiplicand for a divisor, and the 
quotient for a multiplier, 

6. In multiplication it is indifferent which of the two factors is 
called the multiplicand, the other must be an abstract multiplier. The 
name of the product (when known) is an infallible index to show 
which of the two factors is really the multiplicand. 

To illustrate principles three and six, we give the following 

EXAMPLES. 

1 . What will 763 pounds of pork come to at 8 cents per pound? 
At first view, this example seems to conflict with principle three, 

for, says the pupil, we multiply the pounds by the price per pound; 
but it is not so. 

Two pounds would cost twice as many cents as one pound, and 
763 pounds would cost 763 times as many cents as one pound ; 
therefore 763 is the abstract multiplier in the operation, and 8 
cents is the true multiplicand, and the product will be cents, as 
required. 

In the act of multiplying, it is indifferent how the numbers are 
written. 

2. Reduce 6£ 135. ^d. to pence. 

Here 20 and 5, as abstract numbers, must be multiplied together 
and 13 added, making 113 shillings, — but which of the two fac- 
tors 20 or 5, is the multiplicand? 

Here nine-tenths of those who teach arithmetic would call the 
6£ the multiplicand and 20 the multiplier ; but this is not so. A 
multiplicand suffers no change of name by being multiplied, and 
as the name of the product is unquestionably shillings, 20 shil- 
lings is the multiplicand, and 5, as an abstract number, is the 
multiplier, there being 5 times as many shillings in 5£ as in 1 £. 

By the same logic, to reduce 113 shillings to pence, 12, the 
number of pence in a shiUing, is the true multiplicand, which 
must be repeated 113 times, and then the product must of course 
be pence as required. 

Dollars can be divided by dollars, and by nothing else. Yards 
can be divided by yards, and by nothing else, and so on, for any 
other thing tha* might be mentioned. 



16 ROBINSON'S SEQUEL. 

This fact has not been sufficiently attended to ; indeed, it has 
scarcely been recognized by many teachers. 

It is true we can divide a number of dollars, yards, <fec. into 
equal parts, but we do so indirectly, in point of logic, while the 
mechanical operation is direct. 

That dollars can only be divided by dollars arises from the fact 

that division is but a short process of finding how many times one 

< quantity can be subtracted from another, and we can subtract only 

dollars from dollars, therefore we can divide dollars only by dollars. 

Example. — Divide 842 equally among 6 men. 

Now we cannot divide 842 by 6 men nor by 6 ; but if we give 
each man a dollar, that will require 86, and ^Q can be subtracted 
from 842 seven times. Hence we can give each man a dollar 
seven times, or we can give him 87 at one time. After the ope- 
ration is performed, we may call the 7, seven dollars, then the 6 
will be a mere number, and thus, indirectly, we may divide 842 
by 6. Practically, however, all such operations are performed 
abstractly, as 42, 6, and 7, taken as mere numbers, and then 
mere logic decides upon the names. For another example. 
Divide 11£ 7s, 8c?. into 4 equal parts. 

Lay out a £ in four different places — this will require 4j£. 
Now we are to consider how many times 4£ can be subtracted 
from 11 £, which is 2 times and 3£ over, which reduced to shil- 
lings and 7 added, makes 67 shillings ; this divided by 4 shillings 
gives 16, and 3 shillings over, which reduced to pence and 8 added, 
makes 44 pence, which for the same reason, divided by 4 pence 
. gives 11. The operation stands thus : 
' 4)11£ 75. 8c?.(2 



3 
20 

4)67(16 
64 

3 
12 



4)44(11 
44 



ARITHMETIC. 17 

The first divisor is 4£, the second 4s., and the third 4d.; but as 
the divisor and the quotient may change names, we may say 2£ 
\6s. lid is the original sum divided into four equal parts. 
The following example will further illustrate this philosophy : 
Divide 42 1£ 14s. M. among 3 men, 5 women, and 7 boys, giving 
each woman three times as much as a hoy, and each man double the 
sum to a woman. Required the shares for each, 
I will commence by giving one boy 1 £, 
One woman 3£. 
One man 6£. 
But there are 7 boys ; to give each 1 £, would require 7 j^ 
5 women each 3£ would require 1 5£, 
3 men each 6£, would require 18£ 

40£. 

Thus going once round giving each boy 1 £ and each man and 
woman their due proportion, would require 40 j£. We are now to 
'Consider how many such rounds it would take to consume 42 1£. 
In other words, we must divide 42 1£ by 40£, When each boy 
can no longer take a pound, we must in like manner go the rounds 
with shillings, then with pence, (fee. The operation is thus: 

40£)421£ 14s. Sd.(lO 
400 

21' 
20 



40)434(10 
400 



34 
12 



40)416(10| 
416 
Here then it is clear that each boy must have \£ ten different 
times, or which is the same thing in effect, 10£ at one time. Hence, 
simply changing the names of the divisor and quotient, 
Each boy is to have 10£ 10s. lOfc?. 

Each woman 3 times as much, or 31£ 12s. I^xl. 
And each man 63£ >5*> 2f'i, 

2 " 



18 ROBINSON'S SEQUEL. 

We will give but one more example to show that the divisor 
and dividend must be of the same name. 

The moon describes an arc in the heavens of 197° 38' 45" in 15 
days; how great an arc will it describe in 1 day? 

The fifteenth part of 197° 38' 45" is obviously the sum required, 
but how will the number 15 measure 197 degrees? If we drop 
the name of degrees and say 197, then we can divide it by 15, 
and this is the usual way — during the operation all names are prac- 
tically destroyed — and after the operation is over, the proper name 
is given according to the logic or philosophy involved in the ques- 
tion, and it is in this logic or philosophy where the unthinking fail, 
if they fail at all. 

We may also solive this problem by conceiving the moon to 
move 15° in one day, and then dividing 197° 38' 45" by 15°, we 
shall obtain an abstract number, each unit of which corresponds 
to a day. Then changing the names between the divisor and the 
quotient, 16° will become an abstract number, and the quotient 
will be degrees, ^(Scc. as required. 

15°) 197° 38' 45"(13 
15 



47 
46 



60 



J5)158(JO 
150 



8 

60 



16)525(36 
45 



76 

76 
Our first divisor was 15°, second 15', third 15", but by making 
these abstract numbers, the quotient will become 13° 10' 35", the 
answer. 



CANCELING. 19 



€ANCEL,ING. 



Within a few years the subject of canceling has been brought 
to the special notice of teachers and others, and like every other 
improvement, it has been opposed by some, and looked upon with 
distrust and indifference by others. But still, it being a real and 
substantial improvement, it is working its way ; and even at this 
day intelligent pupils are astonished that teachers should op- 
pose it as they sometimes do. Indeed, that teacher who would in 
any degree discountenance cancellation should be dismissed at 
once from the class room. 

Some few educationists had private reasons of a pecuniary 
nature for opposing cancellation; but the chief opposition arose 
from the disinclination of persons to break into old habits. 

Cancellation does not change the process of reasoning on a 
problem, iDut it requires a more general perception at a glance, 
and more rapidity of thought, than the old methods; hence the 
naturally dull did and do yet oppose it as a matter of course. 

The architect makes the design of a proposed building on paper, 
represents it inside and out, estimates the cost, suggests changes 
and improvements, and has it all in his mind before a stick of timber 
is prepared, or any serious labor commenced. It is economy to do so. 
An arithmetician should do the same ; he should be able to repre- 
sent what he proposes to do, on paper, look at it and consider it 
fully before he commences real labor. It is economy to do so, for 
then he may see counter operations that will cancel or abridge 
each other. Desirable as all this is, it is rarely thought of; — 
no sooner is an operation decided upon than the operator hastens 
to perform it, without thinking further until that is done. He then 
decides upon another step and performs it, — then another, and so 
on through the problem. 

Now as a general rule we would have each step of an operation 
distinctly indicated before it be performed, and then examined as 
a whole, the same as an engineer would examine a map, or an 
architect the plan of a building. We give the following exam- 
ples to exercise this faculty : 



20 ROBINSON'S SEQUEL. 

1. A merchaiU bought 526 barrels of flour at $4.50 per barrel^ 
and paid in cloth at $2.25 per yard. Bow many yards did it require? 

Ans. 1052. 

The following is the common method of thought and operation. 
We must find what the flour will amount to, and as soon as that 
thought is defined, the operator commences the multiplication. 
When that is done, then comes the thought about the cloth, and 
it is decided to divide the amount by $2.26 for the required re- 
sult, and the operation stands thus : 

526 
460 



26300 
2104 



225)236700(1052 

225 



1170 
1126 



450 
450 

Now we would not change the direction of the thought in the 
least, but we would have it continued to the end, and each opera- 
tion indicated as we go along.* 

The map of the whole operation stands thus: 
526-45 
225 ~ 

Here is a fraction, the numerator consisting of two factors, the 
denominator of one factor which is contained twice in 460. 
Hence twice 526 is the required result, and the mechanical ope- 
ration is just nothing at all. 

♦When two numbers are to be multiplied together, we write them with a point 
between, thus 4.6 indicates 4 multiplied by 6. If this is to be divided by 

any other number, say .3, we would write -1-. When two numbers are to 

be added, we write (+) plus between them; when one is to be subtracted, 
we write ( — ) minuB before that one. 



CANCELING. n 

2. How much will 540 yards of cloth cost at 3s. 4d., in dollars 
at 6 shillings each? 

The map of the operation is thus: — —- ?-. 

This reduces to 90 'SI-. Multiply one factor by 3, and divide 
the other by 3, (which will not change the value of the product), 
then 30-10=300, the result. 

N. B. In this work we do not pretend to explain prime and composite num- 
bers, what numbers will cancel with each other, and what will not. These 
things must be learned elsewhere. 

3. At \9.\ cents per pound what must he paid for four boxes of 
sugar, each containing 136 pounds? 

Map of the operation, = =63 dollars. 

4. What will one hogshead, or 63 gallons of wiTie, cost at Q\ 
cerUs a gill? Ans. ^126. 

Map of the operation, =126. 

The multiplication indicated in the pumerator reduces the 63 
gallons to gills, and as 6|- cents is one-sixteenth of a dollar, we 
divide by 16, which cancels the product of the fours in the nu- 
merator and leaves 63 to be doubled for the result. 

5. At 1^' cents a gill, how many gallons of cider can be bought 
for $24? Ans. 50. 

24* 100 
Map of part of the operation =:the number of gills, 

or 24-200 ,, .,, * ' ' 
=the gills. 

HT r .1 1 1 .• 24-200 200 ^^ . 

Map of the whole operation, = =50 Ans. 

^ ^ 3-4-2-4 4 

6. Jf a man travel 39 miles 20 rods in a dag, how many days 
leill be required to traverse 25000 miles? Ans. 640. 

As 320 rods make a mile, the following is the m? 
320-25 000 
320-39-1-20 



22 ROBINSON'S SEQUEL. 

Here numerator and denominator can be divided by 20, which 

, ,, ,. , 16-25000 

reduces the operation to . 

^ 16-39+1 

Because no further reduction can be made, this last indicated 
operation must be performed in full. 

In all cases, whether reduction can be made or not, we would 
insist on having the operations first indicated; and in practice, 
nine-tenths of the operations can be reduced. There is now and 
then one that cannot be reduced. Even when the plan of an 
arithmetical operation is laid down, judgment should be used in 
drawing out the final result, as the following example will illus- 
trate: 

Required the value of the following expression : 

4900\2 



/43y /144y /4900\ 
\80/ \ 95 / \ 24 / 



This occurs on page 156 of Robinson's University Algebra, 
and we have seen it literally carried out as indicated, in several of 
the best schools in the country; no reductions being made until 
after the numbers were squared ; thus making a long and tedious 
process. 

The proper way is to take the square root of the expression ; 

then we shall have l?.iil.l?00^ 
80 95 24 
Reducing does not change the value of the expression ; the 
first obvious reduction, is to divide the numerator and denominator 

43 6 490 
by 10 and 24 ; then the expression will stand thus, — •■ . . 

A still further reduction ffives • — • — =166, nearly. 

^ 2 19 1 ^ 

Now the square of 166.3, is the value of the required expres- 
sion. 

We square, because the square root was taken in the first step. 
We may do this, because we have no where changed the value 
of the expession, except in taking the root. 



PROPORTION. 23 



PROPORTIOIV. 

This manner of expressing an operation is most efficacious and 
practical in proportion. 

We shall make no attempt to elucidate the principles of propor- 
tion, our attention for the present being entirely on numerical op- 
erations. 

EXAMPLES. 

1. If 9,ciut. ^qr. ^Ub. of stigar, oo&t ^£ Is. 80?., what wUl 
Shcwt. Iqr* cost? 

cwt. qr. lb. cwt. qr. £, s. d. 

Statement. 2 3 21 : 35 i : : 6 1 8 
This example is taken from an old but popular book, in which 
the solution covers about two pages. The sugar is reduced to 
pounds, and the money, to pence. The result of the proportion 
is then obtained in pence, which being reduced, gives 73£. 
We do it thus : Reduce the sugar to qrs. Then the proportion is 

11| : 141 : : 6£ U, Sd. 
Multiplying the two first terms of this proportion by 4, which 
does not change the proportion, then we have 
47 : 141-4 : : 6£U. M. 
or 1 : 3-4 : : 6£ U. M. 

Therefore 12 times the third term is the result, 73£. 

2. If 3cwt. of sugar cost 9j2 Is., what will 4cwt. Sqr. 26lb. cost 
nt the same r<ate? Ans. 1 5£ 25. 3c?. 

We give the following solution just as it appears in a very pop- 
ular book : 



4.7 



cvi. 


cwt. 


qr. 


lb. 


£s.d. 


3 


: 4 


3 


26 


:: 9 2 


4 


4 






20 


' 12 


19 






182 


7 
84 


7 
133 






12 


2184 


4 


4 








336Z§. 


: bbm 


- 


: 2184 



»We take tire old scale of 28 pounds to the quarter. 



2L4 ROBINSON.'S SEQUEL. 

336/6. : 558/6. : : 2184 
558 



17472. 
10920 
10920 


3627: 




336)1218672( 
1008 




2106 
2016 


12)3627 


20)302 3rf. 
15£ 2s. 


907 
672 



2352 
2352 

If the question had called for the cost of 2 pounds more of 
sugar, it would have called for the price of ^wt 
Then the proportion would have been 

3:5:: 9.1£. £ s. d. 

Whence the cost of bcwi. would be 15 3 4 



For the cost of 2 pounds we have 
3-112 : 2 :: 182 shillings. 
or 168 : J : : 182 
or 84 : 1 : r 91 : lyV shillings. 



2lb. cost 1 1 



15 2 3 An^. 



3. If \h\ hashdt of dautr co&t Si 56^, how many bushels can b& 
bought for S95|? • A^is. 9y\y^. 

Statement, 156.26 : 95.75" : r 15.625. 

When a statement is properly made, drop all names and ope- 
rate as abstract numbers; then the proper name can be given to 
the result by the rules of logie-, or rather, the true name comes 
as a matter of course. Those who operate by rule and without 
thought and close observation., would make very tedious work of 
this. 

rru c.-L .• • *v, 95.75(15.625) 

The map of the operation is thus: — ^v ^ — ^. 

(156.25) 



PROPORTION. 26 

The factor in the denominator is 10 times one of those in the 
numerator, therefore the operation reduces to 

^^'^1=9.515 Ans. 
10 

4. 1/240 bushels of wheat can be purchased at the rate of $22^ 

far 18 bushels, and sold at the rate of ^33^- for 22 1 bushels, what 

would be the profit? Ans. $60. 

240 •291 
18 : 240 : : 22\ : cost=_ — ^1 

18 

221 : 240 : : 33| :• sale=?^??i 

' 22i 

Cost =240^=^^^=20. 15=300 dollars. 
18-2 3-2 

Sale =^:l^lll=?:^^=3. 120=360 dollars. 
221 2 

A complete proportion consists of four terms; and in problems, 
tho unknown answer is generally one of them ; and were it not for 
old prejudices, it would be conducive to perspicuity to represent 
the unknown term by a symbol, say x. Then a problem stated, 
would no longer consist of three terms, but of four. 

At first a young learner will not comprehend a symbol nor an 
equation, and his confusion arises from, the "very simplicity of the 
thing. 

Notwithstanding the aversion of learners to the use of symbols^ 
the aversion must be avercome before they can enter the first por- 
tals of science ; and a little firmness on the part of the teacher willi 
remove every difficulty in a very short time. 

When a proportion is complete, the ratio between the first coupr- 
let is the same as the ratio between the second couplet. Thus, 

3 : 6 : : 8 : 16. 

Here the proportion is true, because 6 divided by 3 gives the same 
quotient as \Q divided by S. 

Such a trial will test any proportion. 

Suppose in this proportion that 16 is not known, and represented 
hj XI then it becomes 3 : 6 : : 8 : a; . 

Whence 5 = ^. Or, x = ^t 

3 8 a 



26 ROBINSON'S SEQUEL. 

That is, when the three first terms of a proportion are given, 
the fourth is found by multiplying the second and third terms 
together, and dividing by the first. 

In any proportion the product of the extremes is equal to tJie 
product of the means; and from this principle any one of the 
terms of a proportion can be found, provided the other three are 
given. 

A term may consist of two or more factors, and one of those 
factors unknown : in such cases, the unknown factor may always 
be found from an equation formed by the product of the extremes 
<md means. 

Thus 3 : 6 : : 2a; : 16. Whence 6-2-a;=3-16. 

r. 3-16 . 

Or x= =4. 

6-2 

The foregoing is designed to prepare the way for such problems 
as are usually found under compound proportion, which we shall 
call 

CAUSE AND EFFECT. 

After several years reflection, we have come to the conclusion 
that the only clear and scientific method of presenting compound 
proportion is that of cause and effect. 

It is an axiom in philosophy that equal causes produce equal 
effects; a double cause a double effect, &c. In short, effects are 
proportional to tJieir causes. 

Now causes and effects that admit of computation, that is, in- 
volve the idea of quantity, may be represented by numbers, which 
numbers have the same relation to each other as the things they 
represent. 

EXAMPLES. 

l.Ifl m£n in 12 days dig a ditch QO feet- long, Zfeet wide, and 
^ feet deep, in how many days can 21 men dig a ditch 80 feet long, 
^feet wide, and 8 feet deep? 

Here 7 men in 12 days perform 84 days work; the force or 

cause of removing 60* 8*6 cubic feet of earth, which is the effect. 

In how many days (we say x days) can 21 men remove 80* 3* 8 

cubic feet of earth? Hence we have this proportion : 

Cause. Effect. Cause. Effect. 

7-12 : 60'8-6 : : 21a: : 80'3-8. 



PROPORTION. 27 

Here is a case where a factor in one of the terms is unknown, 
and that factor is the answer to the question. 

A proportion is equally true when the same factors are rejected 

from corresponding terms. This is hut another form of canceling. 

In this proportion, we observe the factor 7 in each cause, and the 

factor 8 in each effect. Expunging- these, the proportion becomes 

12 : 60-6 : : ?,x : 80-3 

Similarly 2 : 6 : : x : Q Whence x=2^ days, Ans. 

2. If ^ men huild a wall in 12 days, how long would it require 
20 men to huild it? Ans. 3 3 days. 

Questions of this kind are usually placed under the rule of three 
inverse; they do in fact belong to compound proportion, or rather, 
to cause and effect ; but the effect being the same in the supposi- 
tion, and in the demand, (that is the building of one wall,) it 
may be omitted and only three quantities used. 

The following statement banishes all confusion : 
Cause. Effect. Cause. Effect, 
6-12 : 1 : : 20a; : 1 
As effect=effect, therefore cause=cause, that is, 20a;=6*12. 

3. i)^ 4 men in 2\ days, worlcing 8^ hours a day, mow 6f acres 
of grass, how many acres (ans. x acres,) will 15 men mow in 3| 
days hy working 9 hours a day? Ans. 40]-- acres. 

Here the unit of cause is one hour's work for a man. 
C. E. C. E. 

^ 4-2i-8i : 6f : : 15-3|-9 : x 

Multiply the 1st and 3d terms by 4, then 

4-2i-33 : 6f : : 15-15-9 : x 
Because 2|- is contained in 15 six times ; and the 1st and 3d 



terms contain the factor 3 


: therefore 


4-11 : 


6| : : 6-15-3 


^ Or 2-11 : 


3A- : : 6- 15-3 


Or 6-11 : 


10 : : 6-15-3 


Or 11 : 


10 : : 15-3 : 



X 
X 
X 

a;=VV=40j^/4fW. 
The reader will observe that we give but specimen examples ; 
one of a kind : the preceding one was given on account of the 
fractional factors. 



28 . ROBINSON'S SEQUEL. 

4. What is the interest of $240 /or 3^ years, at 6 per cent J 
This question simply demands the effect of loaning S240 for 
3^ years, in case $100 in one year yields $6. 

Cause. Effect. Cause. Effect. 
100-1 : 6 : : 240-31 : x. 

Whence x= ~. 

100 

This equation shows the common rule for computing interest. 
That is : 

Multiply the principal by the rate per cent.; that product by the 
time, and divide by 100. 

Now let us take this same example and reduce the time to 
months, then the proportion will stand thus : 
100-12 : 6 : : 240-42 : x. 
Cast out the factor 6 from the first couplet, then 
100-2 : 1 : : 240-42 : x. 

Divide the 1st and 3d terms by 2, then we shall have 

940-21 
100 : 1 I : 240-21 : x a;=fll_— • 

100 

This equation shows a special rule to compute interest at 6 per 
cent, which is, 

Multiply the principal by half the number of months and divide 
by 100. 

6. WTiat is the interest of $1248, /or 16 days, 30 days taken for 
a month, and 12 months in a year ? 

Cause. Effect. Cause. Effect. 
100. : 6 : : 1248 : x 
Days 360. 16. 

Divide the first couplet by 6, then 

1248-16 
100-60 : 1 : : 1248-16 : x ^=~loO^O 

This equation shows a special rule to compute interest for days 
at six per cent, which is thus. 

Multiply the principal by the number of days, divide by 60, and 
that quotient by 100. 



PROPORTION. S9 

6. The interest on $98, at 8 per cent., was $25.48 : what was 
the time? Arts. 3 years 3 months. 

Cause. Effect. Cause. Effect. • 

100-1 : 8 : : 98-a; : 25.48 

Whence a:=?^ =3 y. 3 m. 
8-98 ^ 

This equation shows the following rule to find the time in 
interest problems when the other elements are given: 

Rule. Multiply the interest by \00 and divide by the product of 
the principal and rate. 

These general rules refer only to forms. It is not intended that 
they should be literally followed. In the last equation 8 and 
98, the principal and rate, are multiplied inform as they stand, 
and the fraction can be canceled down. 

The great detriment to improvement has been, that both teacher 
and taught, have clung close to the letter of the rules. 



SECTION II. 

We shall touch on but few points in this section ; and only such 
as will bear on conciseness of operations. 

We give but one example in Exchange and per centage — it is 
the following : 

A merchant bought sugar in New York at 6 pence a pound, New 
York currency ; and while on his hand the wastage was estimated at 
5 per cent.; and interest on first cost at 2 per cent.; how many cents 
shall he ash per pound to gain 25 per cent. Ans. ^-^^j. 

To reduce pence, IS'ew York currency, to cents, we must mul- 
tiply by If ; to increase any quantity 5 per cent, we must multiply 
by 1^1, and so on for any other per cent.; hence the index of the 
operation is as follows : 

6 25 105 102 125 

T ' 24 ' 100 ' 100 * Too 

This will cancel to a considerable extent. This form is a gene- 
ral rtile for all problems of the kind. A loss in any problem, 
of 3 per cent, for example, is brought in by the factor jVa , and so 
on for any other estimated loss, expressed as per centage. 



30 ROBINSON'S SEQUEL. 



COMPOUND FEI.I.OWSIIIP. 

Under this head gains and losses must be proportioned by the 
products of capital and time. We give a iew peculiar examples. 

1 . Two men commenced partnership for a year; one put in $ 1 ,0C0 
at the commencement, and four months afterwards the other put in 
his capital: at the close of the year they divided their gains equally. 
What capital did the second put in ? Ans. $1,600. 

For a mere arithmetical student, who had never been tauofht 
the use of symbols, this would be a very puzzling problem. We 
are therefore opposed to taking up the time of students with diffi- 
cult problems, except so far as may be necessary to show them 
the necessity and- advantages of symbols and true science. This 
is a very good example to illustrate the utility of symbols : 

Let X = the required capital. It was in trade 8 months", and 
as their gains were equally divided, therefore the products of 
capital and time of each must be equal ; that is, 

8a;=12-1000 or, a:=??22=1500. 

2 

2. A, B and C had a capital stock of $5762. A's money was 
in trade 5 months, B's 7 months, and C's 9 months. They gained 
$780, which was divided in the proportion of 4, 5 and 3. Now B 
received $2087 and absconded. What did each gain and put in, 
and did A and C gain or lose by B's misconduct, and how much? 

., , 780-4 780 p 780-5 ^ 780-3 780 

^'s share= = jd = C= = 

12 3 12 12 4 

As gains are divided in proportion to capital multiplied by the 
time it is in trade ; conversely then, capital must be in propor- 
tion to the respective gains, divided by the respective times. 

Their proportional gains are 4, 5, 3, which divided by the times 
5, 7, and 9, give j, f, and } for their proportional shares of the 
capital. 

But these numbers being fractional are inconvenient. We will 
multiply each by 5-7-3 or 106, which gives the proportional 
numbers 84, 76, 36, the sum of which (194) may be taken as 



COMPOUND FELLOWSHIP, ?l 

the number of shares composing the capital, §5762; and ^'3 
capital is 84 such shares, i?'s 75, and C's 35. That is, 

^'s capital =!6^^. B-s=''J^t C's ^^I^l^- 
^ 194-1 194 194 

A and C gain by B, §465.57-f.. 

S. In a certain factory were employed, men, woTnen, and boys. 
The boys received 3 cents per hour, the women 4, and the men 6 ; 
the boys worked 8 hours a day, the women 9, and the men 12; the 
boys received $5 as often as the women ^10, and for every $10 
paid to the women 824 were paid to the men : how many men, women 
and boys were there, the whole number being 59 ? 

Ans. 24 men, 20 women and 15 boys. 
Boys. Women. Men. 
Sums per hour, 3 cts. 4 cts. 6 cts. 

No. of hours, 8 9 12 

Sums paid to one of each class, 24 36 72 

Proportional sums paid to one of each, 2 3 6 

The sums paid to all of each class divided by the sums paid 
one of each class will give the proportional number in each class. 

. 5 . 10 24 
That is ^ • -^ ' -Q are the proportional numbers of persons res- 
pectively. Multiply by 6 to clear of fractions, for fractional 
numbers cannot apply to persons. Then the proportional numbers 
will be 15, 20, 24, and as these numbers make 59 they are the 
numbers in fact. 

4. A, B, arid C, are employed to do a piece of work for $26.45: 
A and B together are supposed to do | of it, A and G ~^, and B 
and C If, and are paid proportionally to that supposition : what is 
each man's share? Ans. A $11.50, J3 $5.75, C$9.20. 

This problem is algebraic, and the operation is algebraic 
whether the symbols be used or not, and this is true of many 
other problems found in Arithmetics. 

Here A works with JB and with C, and we must discover what 
he is supposed to do, working alone. It is done thus : 
A+B= f. (1) 

-4+C = A (2) 

£+C=ii. (3) 



S2 ROBINSON'S SEQUEL. 

By addition, 2( J+^+(7)=:iA_[.i^_|_t i=^||. 

Dividing by 2, ^-|-i?+(7=|f. (4) This equation 
shows that the anaount each one was supposed to do was over 
estimated. 

Equation (3) taken from (4) gives A=z^% — H=^H' 

- (2) from (4) - B=.U-H--^- 

- (1) from (4) - (7=H-M=^V 

For A's portion of the money we have the following proportion: 
f 4 : i^ : : 26.45 : A's part. 

Or, 23 : 10 : : 26.45 : _^^1:^= $11.50. 

23 

6. A person after doing *} of a piece of work in 30 days, calls in 
an assistant, and together they complete it in 6 days : in what tlni€ 
could the assistant alone do the whole work^ Aiis. 2 If days. 

If the person could do | in 30 days, he could do ^ in 10 days, 
and in one day he could do jV ^^ the whole work. Therefore, it 
would require 50 days for him to do the whole work alone. 
Again, §■ of the work being done | remained to be done ; on this 
the first person worked 6 days and did /„ ^^ it- Then ( | — /o ) 
or ^^ remained for the assistant to do in 6 days ; hence he must 

do aVo ill ^^^ day, or in one day. Therefore, to do th-e 

Sly 

whole he w^uld require 21f days, the answer required. 



PROBI.CM§ IN MENSURATION AND THE ROOTS. 

Mensuration and the Roots belong to Geometry and Algebra, 
but custom requires that some practical problems under these 
principles, should appear in every Arithmetic. We select such 
examples as will illustrate numerical brevities. 

1 . How many feet in a board 22 inches wide at one end, 8 i^cht^ 
wide at the other, and 14 feet long ? Ans. 17^ feet. 

Index to the operaticm, — 



MENSURATION AND THE ROOTS. 3S 

2. A man bought a farm 198 rods long, 150 rods wide, at 832 
per acre ; what did the farm come to? Ans. $5940. 

T J * *!, r 198-150-32 ,^„ „^ 
Index to the operation, =198-30. 

3. If the forward wheels of a coach are four feet in diameter, and 
the hind wheels 5 feet, how many more times will the former revolve 
than the latter in going a mile, estimating the diameter of a circle to 
the circumference, as 7. to 22.? 

Circumference of the fore wheels, = ; hind wheels, ' 

7 7 

There are 5280 feet in a mile ; this, divided by the circumfer- 
ence of each wheel will give the number of revolutions of each 
wheel. 

The fore wheels revolve times. 

22-4 

5280*7 
The hind wheels revolve times. 

22-6 



-n-ff 5280-7/1 

Dmerence 



22 
22-20 22 



^'(H) 



^, , . 5280-7 264-7 ,„ ^ ^. . 
That IS = ._=12-7=84 Ans. 



4. The bin of a granary is 10 feet long, 5 feet wide, and 4 feet 
high; allowing the cubical contents of a dry gallon to contain 26 8| 
cubic inches, how many bushels will it contain? Ans. \Q\^^-^. 

10-12-6-12-4-12 50-12-5-12-6 



Index to the operation. 



1341 



5. A man wishes to make a dstern 8 feet in diameter to contain 
60 barrels, at 32 gallons each and 231 cubic inches to a gallon: what 
must be the depth of the cistern? Ans. 61|- inches. 

The diameter of the cistern is 96 inches ; 
Its area is 96 -96 -(0.7854.) 

60-32-231 I 



The index to the operation is 



96 -96 -(0.7854) 
3 



34 ROBINSON'S SEQUEL. 

6. What will it cost to build a wall 240 feet long, 6 feet ?dgh, 
and 3 feet thick, at $2>.^5 per 1000 bricks, each brick being 9 inches 
long, 4 inches wide, and 2 inches thick? Ans. $336.96. 

Index 240'12'6'12'3'12-(3.25) 
1000-9-4-2 . 

7. The bung diameter of a cask is 38 inches, the head diameters 
inside the staves 28 inches, and the length 45 inches: how many 
wine gallons will it contain? Ans. 167.89-|-. 

N. B. The cask is conceived to be two equal /rwi-^wms of cones 
joined by their greater diameters. {^See Geometry.) 

Index to solution (5^^+28^+28-38),7854-45 

3'231 
Observe that the decimal 0.7854 is divisible by 231 : quotient 
.0034. Therefore we may have the following rule to find the num- 
ber of wine gallons in a cask : 

Rule. To the square of the head diameter add the square of 
the bung diameter, and the product of the two diameters : multiply 
that sum by ^ of the length of the cask and by the decimal .0034. 

8. A man bought a grindstone which was 48 inches in diameter 
and 5 inches in thickness, for $10. When he had ground down 3 
inches of its radius, a neighbor proposed to purchase it from him at 
the same proportional price, in case he would deduct 4 inches each 
way from the center ^ allowed to be the limit to which it could be used. 
What should the purchaser pay? Ans. $7.58-)-. 

Statement (43''— 8"^). 7854 : (42''— 8^). 7854 : : 10 : Ans. 
Or (48^'— .8'') : (4^^ —8'') : : 10 : Ans. 
Or 66-40 : 50-34 : : 10 : Ans. 

oca" 

Or 56-2 : 6«17 : : 10 : Ans.=^^. 

112 

9. If a mxin 6 feet in height travel round the earth, how muxk 
further must his head travel than his feet? 

Ans. 37 ^-Q feet nearly. 
Let D= the diameter of the eai*th in feet ; then rti>= the cir- 
cumference in feet. (D-\-12)= the diameter, and rti>+13rt=s 
the circumference traveled by the man's head. 
The difference =.1^(3.1416)=^««. 



POWERS AND ROOm 3ft 

« 

SECTION UIs 

PO^VERS ANO KOOTS. 

The common methods of operation, as taught under this head, 
Rre in general the besk One object in this work is to show some 
peculiarities which will in some instances abridge labor, awaken 
investig'ation, and inspire originality of thought. 

We give the following delinitions : 

1 . Any number multiplied into itself is called the square of that 
number. Or we may say (he product of two equcd factors pro- 
duces a square^ Either factor is called the root. 

2. The product of threie equal factors is a cube or third power, — ■ 
of four equal factors, a fourth power, and so on. One of the 
equal factors is a root in all cases. 

3. A square number multiplied by a square number, will pro- 
duce a square numberv 

N. B. This is obvious in Algebi-a for a^ multiplied by b^ pro- 
duces a^b^ , obviously a square, whatever numbers may be repre* 
sented by a and b. 

4. A square number divided by a square number will give a 
square number, either whole or fractional. 

5. A cube number multiplied by a cube number will give a 
cube number. 

6. If a root is a composite number, its power (square or cube 
as the case may be) can be separated in square or cube factors: 
but if the ix)ot is a prime number, the power cannot be so sepa^ 
rated. 

We will soon show the practical utility of these principles. 
While operating in powers and iHDots w-e should have the fol- 



lowing table before us : 








Numbers, j 1 j 2 | 3 | 4 


1 5 1 6 


1 7 i 8 1 9 


1 10 1 


Sq. or 2d poAver, | 1 | 4 ! 9 jl6 


1 25 1 36 


1 49 1 64 1 81 


! 100 1 


Cube or 3d power,! 1 ! ^ !27 j64 


il25 |216 


|343 |512 |729 


iiooo i 



Powers being obtained from roots by simple multiplication, 
there is no room for much artifice. 



56 ROBINSON'S SEQUEL. 

Sometimes the application of the following properties of num- 
bers will be useful : 

The square of the difference of two numbers is equal to the sum 
qf the squares, less twice the product of the two numbers. 
Algebraically, (a—hy=a'-\-b^—2ab. 
The square of the sum of two numbers is equal to the sum of the 
squares added to twice the product of the number. 

Algebraically (a+5)2 =a2 +6^ +2aJ. 

EXAMPLES. 

1. What is the square of 79? Ans. 624L 

(79)2 = (80—1)2 =6401—160=0241. 

2. What is the square of 83? Ans. 6889, 

(83)2 ^(80+3)2 =64094-480=6889. 

3. What is the square of 97? Ans. 9409, 

(97)2=(100— 3)2 = 10009— 600=9409. 

4. What is the square of 971? Ans. 942841, 

(971)2 =(970+1)2 =940901-f 1940=942841. 

5. What is the square of 29? Ans. 841. 

(29)2 = (30— 1)2=901— 60=841. 

These formulas are useful when one or all of the integers are 
large. 

We shall now turn our attention to the extraction of square 
root. We suppose the reader understands the common method, 
which as a general operation is the best. 

To call out thought, however, we will require the square root 
of 9409, on the supposition that we know nothing of the common 
rule, and only know thai two equal factors of the numbers 9409 are 
required. 

The first thought is, that if we divide any number by any factor ^ 
the quotient will be another factor . 

Take 100 for one factor. Divide 9409 by 100, and the other 
factor is 94, omitting the decimal ; but these factors are not equal. 
The factors sought then, or rather one of them, is more than 94, 
and less than 100. Hence it must be near the half sum of these 
two numbers ; that is, near 97. 

By trial we find 97 correct. 



POWERS AND ROOTS. 37 

N.. B. The half sum of two unequal factors, is always a little 
greater than one of the equal factors, because the sum of two une- 
qual factors which form a product, is always greater than the sum 
of two equal factors. 

EXAMPLES. 

1. Find the square root o/" 841 ; that is, we demand two equal 
f Victors, which, multiplied together, will produce 841. 

Assume any factor : say 25. 

25)841(33, (plus a fraction, which we omit) is 
the corresponding factor. But these factors are not equal, and 
the equal factors must be near their half sum ; that is, near 29. 
By trial, 29 is found to be the number exactly. 

2. Find the square root, or two equal factors of the number 
444889.. 

Divide by 6. 6 )444889 ^ 

74148 

Here the two factors scre-veri/ uneqzcal, but we can bring them 
to a proximate equality, by conceiving one multiplied by 100, and 
the other divided by 100. The factors will then be 600 and 741, 
nearly. The half sum of these is 670, which must be near one 
©f the equal factors sought. 

Now divide. 670)444889(664 

4020 

4288 
4020 



2689 
2680 



These factors being so nearly equal, and there being a slight 
remainder, the half sum of the two (667) may be relied upon as 
tlie true -root. 



38 ROBINSON'S SEQUEL. 

3. Find the square root of 3. Am. 1.7320508. 

The only two factors in whole numbers are 1 and 3 ;* these are 
so unequal that their half sum, 2, will be entirely too large. 
Hence I will assume one factor to be 1.7. 



1.7)3. (1.7647 
1.7 

130 
119 



110 



1.7647 
1.7 

2)3.46T7"" 



JQ2 1.7323 root nearly. 

80 

68 

120 
Making another trial with the assumed factor, 1.732, we find 
the result as stated in the answer. 

4. Find the square root o/ 181. Ans. 13.45362-f.. 

If we allow ourselves to have some knowledge of square num- 
bers, we can find a factor near in value to one of the equal factors 
sought. Thus the square of 12 is 144, and of 13, 169 ; therefore 
one of the equal factors of 181 is more than 13. Assume it 13.5, 
the other factor is, then, 13.4074 ; the mean of these is 13.4537. 

Taking this as the assumed factor, wo approximate still nearer 
to the root by a like operation ; and thus we can approximate to 
any degree of accuracy required. 

By admitting that every figure in a root demands two places in its 
second power, we can come near the root at the first assumption. 
For example : 

5. Find the square root c/ 617796. Ans. 786. 
Separate the power into periods, as in the common operation ; 

the superior period is 61 ; the square root of this is near 8, and 
being three periods the root is near 800. Assume 780, then di- 
vide by it,, thus, 

* In our geometrical problems we shall give a scientific and satisfactory 
method of reducing unequal to the equivalent equal factors. 



POWERS AND ROOTS. 39 

780)617796(792 
5460 

7179 
7020 



1596 

1560 



36 
The half sum of 780 and 792, is 786 ; the answer. 

By the last example we perceive that the square root of the 
product of two factors which are nearly equal, is very nearly equal 
to the half sum of the two factors. It is a little less. In the 
last example there was a small remainder, which was rejected ; 
had there been no remainder, 786 would have been too great for 
the root. 

The square root of the producf of two square factors is equal to the 
product of the square root of those factors. That is, the square 
root of a^6^ is the square root of a^ into the square root of b^; 
in short it is aX^- 

To apply this principle I adduce the following examples : 

1. A section of government land is a square of 640 acres. What 
is the length, in rods, of one of its sides ? Ans. 320. 

This problem requires the square root of the product of the two 
factors, 640 and 160. 

The product of two factors is not affected by multiplying one and 
dividing the other by the same number. 

Now multiply the factor 160 by 10, and divide the other by 10, 
then 1600 "64 will be the equivalent factors ; both square factors ; 
their roots are 40 and 8. Hence, the value sought is 40* 8=320. 

Again. Take the original factors, 640, 160. Divide 640 by 
2, and multiply 160 by 2, which gives 320, 320. 

As the factors are now equal, one of them is the root sought. 

2. A man has 50 ^ acres of land in a. square form; what is the 
length of one of its sides ? Ans. 90 rods. 



Index. V50|-160 = 7ifi- 160 = ^405-20=^8100=90. 



40 ROBINSON'S SEQUEL. 

3. Find the square root of the product of the two factors^ 1 8 
aw«?32. 

Equivalent factors, 9 and 64 roots 3-8=24. 
" Or, 36 and 16 roots 6-4=24. 

Again, — it — =25, which is too great for one of the equal 

factors by 1, because the factors are so unequal. 

In working square root, it is important that the teacher should 
be able to show to his intelligent pupils, that the square on the 
hypotenuse of a right angled triangle is equal to the sum of the 
squares on the other two sides, notwithstanding they have never 
been students in geometry. 

To give an ocular demonstration of this important truth, we 
present the following figure ; 




The line PQ separates two equal squares. The triangle a is 
the right angled triangle in question, its right angle at P, x and 
y are its two sides, and the side opposite the right angle P is 
called the hypotenuse. In each square are four equal right 
angled triangles. Let them be taken away from each square, and 
in one square the square ^will be left, and in the other square 
the two squares A and B will be left. 

Now, from each of the two equal squares on each side of PQy 
we took equal sums — which must leave equal sums. That is, 
A+B=H. 

When we operate on a right angled triangle, we may divide 
the two given sides by the same number, if we can do so without 



POWERS AND ROOT^S. 41 

a remainder on either side, and then operate with the quotients 
as we would with the original numbers. But in conclusion we 
must multiply the result by the number which we divided bj. 
This is working on a similar reduced triangle. 

j:xa.mples. 

1. The two sides of a right angled triangle are 312 and 416; 
what is the hypotenuse ? Ans. 520. 

Divide by 5 2)312. 416 

Divide by 2) 6 8 

3 4 

Square 3 and 4, add those squares which make 25 ; the square 
root of 25 is. 5. 

Multiply 5- 2 -52=520. Ans. 

2. A hawk, ^perched on a tree 77 feet high, was brought down by 
a sportsman 1 4 rods distant on a level with the base ; what distance in 
yards did he shoot? Ans. 81.15-1-yards. 

14 rods reduced to feet is 14- 161=7 -.33. 
Now without reduction we shall be obliged to square 77 and 
231. But we may operate thus, 

7)7-33 77 
11) 33 n 



32-1-12 = 10. ^10=3.1622+. 

Ans. in feet, = 77 (3.1622). Ans. in yards, 77 (1.054). 



CUBE ROOT. 



The object of the Cube Root is to find three equal factors, ex- 
actly or approximately, Avhose product will give any required 
sum. The reason of its being called cube is because the three 
factors may be correctly represented by the length, breadth, 
and height of a geometrical cube. The product of three unequal 



42 ROBINSON'S SEQUEL. 

factors may be represented by a geometrical solid of unequal 
length, breadth, and height, called a parallelopipedon. 

While operating for cube root it is convenient to have the cube 
numbers before us. 

Roots, 123456789 10 
Cubes, 1 8 27 64 125 216 343 512 729 1000 

We see by these cubes that one figure in a root may Kave 
three places in its corresponding power. 

Therefore separate the power into periods of three figures each, 
beginning at the units ; the number of periods will show the num- 
ber of figures in the root. 

Now as we are to have nothing to do with the common methods 
of extracting cube root, all we are permitted to know is the divi- 
sion of the power into periods, and the fact that three 'equal factors 
of the power are required. 

EXAMPLES. 

1. Extract the cube root of 84604519 ; or in other woi'ds,find three 
equal factors whose product will produce this number. Ans. 439. 

Here are three periods 84'604'519, which show that there 
must be three figures in the root. The superior period is 84, and 
84 referred to the line of cubes, its place would be between 64 
and 125, whose roots are 4 and 5. Hence the root sought for is 
greater than 400 and less than 500 ; I should judge it not far 
from 440. Therefore I assume 440 as one of the factors of the 
number. 

440)84604519(192283 
440 

4060 
3960 



1004 
880 
1245 
880 
"3661 
3520 
"1319 
1320 



CUBE ROOT. 43 

Now if one factor is 440, the product of the other two is 192283, 
very nearly ; (not exactly, for the last figure 3 is too large by a 
very small fraction.) 

We will now operate for two equal factors of the number 
192283, and if our first factor is near an equal factor, that same 
factor is near an equal factor in this number ; therefore try it thus, 
440)192283(437 
1760 



1628 
1320 



3083 
3080 

Here we have three factors, 440, 440, 437, whose product will 
give the number 84604519, within 3 units. These factors are 
not all equal, and of course are not the factors required ; but they 
are so nearly equal that one-third of their sum will be one of the 
equal factors required. That is, 

440 

440 

437 
3)1317 

439 Ans. 

2. Find the cube root, or three equal factors of the number 
32461759. Ans. 319. 

By the aid of the periods we perceive that the factors must be 
greater than 300, but nearer 300 than 400. 

Assume then 312 to be near one of the equal factors sought for. 
Divide by 312 twice, or once by the square of 312. That is, 
97344)32461759(333.3 
292032 



325855 
2 92032 
"338239 
292032 



46207 
It is now obvious that the product of the three factors, 312, 
312, and 333.3, will produce very nearly the given power; but 
these factors are not all equal, and equal factors are required ; 



44 . ROBINSON'S SEQUEL. 

but they are so nearly equal that } of their sum, 319.1, can be 
relied upon as extremely near the root required. A factor, or root, 
determined in this manner from unequal factors, will always he a 
little in excels of the true value required. Hence, in this case we 
will omit the one-tenth and take 319 as nearer the root sought, 
and on trial find it to be the root exactly. 

We will now give one of the most difficult examples. 

3. Find the approximate cube root of 16. Ans. 2.519842. 

The factors of 16 are 2- 2- 4; the sum of these is 8, which 
divided by 3 gives 2.66 for the first approximation to equal factors, 
but as these factors are so unequal, 2.66 must be in excess. 
Therefore we assume 2.5 to be near one of the equal factors re- 
quired. To find the other two factors, divide twice by 2.5, or 
once by 6.25. 

Thus, 6.25)16.00(2.56 

12 50 



3500 
3125_^ 
' / ~375a 

3750 

Here we have three factors, 2.5, 2.5, 2.56, whose product will 
give 16 exactly ; they are not all equal however, but being nearly 
so \ of their sura, 2.52, is very nearly equal to the root sought : 
{it must he a very little in excess). 

Now if we repeat the operation with 2.52 as an assumed factor 
and find two other corresponding factors, ^ of the sum of the three 
will be the root to a high degree of approximation. 

4. Find an approximate cube root of QQ. Ans. 4.041240. 

By the cube numbers we find that 4 must be near one of the 
equal factors, therefore divide by the square of 4. 
16)66(4.125 
64 
20 
16 
40 
32 



CUBE ROOT. 46 

Hence the root sought must be a very little less than ^ of the 
sum of 4, 4, 4.125 ; that is, a very little less than 4.0416. 

For a nearer approximation take 4.041 as one of the factors of 
66, and find the other two, (fee. 

5. Jf^ind an approximate cube root of 21. Ans. 2.758923. 

That is, find three factors, as near equal as possible, whose pro- 
duct will be 21, or very nearly 21. 

We know that 27 has three factors, each equal to 3 ; therefore 
the equal factors of 21 must be each less than 3, and as we can- 
not expect to find the equal factors at the first trial, we will 
assume 2.7 and 2.8 to be two of the factors, their product is 7.56 ; 
hence, the third corresponding factor is found by the following 
division : 

7.56)21.00(2.777 
15 12 

5880 ' ' 

5292 



5880 
5292 



5880 
5292 
588 
Here we have three factors nearly equal, whose product is very 
near 21 ; one-third of their sum is 2.759, which must be a little 
greater than the root required. We will therefore assume 2.75 
as one of the equal factors sought, and find the other two corres- 
ponding factors, and one-third of their sum will be an approxi- 
mate cube root of 21. 

It is not necessary to give more examples. 

When it is necessary to multiply several numbers together and 
extract the cube root of their product, we may often evade or 
abridge the operation by resolving the numbers into cube factors. 

EXAMPLES. 

1 . What is the side of a cubical mound, equal to one 288 feei 
long, 216 feei broad, and 48 feet high? Ans. 144. 



46 ROBINSON'S SEQUEL. 

288=2»12»12 
216=6- 6- 6 
48=4-12 



Product, 288»216«48=123>63«8 

Whence, y288- 216 -48=12*6*2=144. Am, 

S. Required t^ie cube root of the product of 448 by 392 in a brief 
manner. 

N. B. Divide by the cube number 8 ; then it will appear that 
448=:8'8-7 
and 392=s8-7»7 



Product, 448-392=^83-73 

Whence, 3^448*392^8- 7=66 Ans. 

3. Find the cube root of the pn)duct of the two fuctors 192 and 
1025 in as brief a manner as possible, Ans. 60. 

The three last examples are rare cases ; nevertheless they serve 
to awaken thought, afid for this purpose they were introduced. 



JlI.I.I«ATIOM AI>TE»]VATE. 

No arithmetical rule is more difficult to be comprehended by 
young pupils than this. 

The operations are generally very trifling, but the rationale is 
rarely discovered. 

For this reason we shall be a little unique in our exposition of 
the principle — we shall resort to an experiment in philosophy. 

It is clear to the comprehension of almost every one, that two 
bodies balanced on a fulcrum, the heavier body must be nearer 
the fulcrum than the lighter body. 

Thus two bodies bal- 



anced on the fulcrum F, 
2 pounds at the distance 
of 6, will balance 6 pounds at the distance of 2. 




ALLIGATION ALTERNATE. 47 

Or when we have the distances, we can take those distances, 
or their proportion for corresponding weights if we alternate them. 
That is, the long distance must go on the opposite side of the 
fulcrum, and there become weight, and so of the other distance, 
and there will be a balance. 

We shall apply this principle in the following example. 

1. A grocer has two kinds of sugar , one at 9 cts., the other at 16 
cts. per pound ; he wishes to make a mixture worth 1 1 cts. per pound : 
what proportion of the two kinds shall he take ? 

Here two quantities are to be balanced on tho, fulcrum 11. 
The difference between 9 and 11 is 2 ; place C 95 

the 2 opposite 16. The difference between 11 and 11 ^ 
16 is 5 ; place this opposite 9. (^16 2 

The result is, that 5 pounds at 9 cts. = 45 cts. 
and 2 pounds at 16 cts. = 32 cts. 
Makes 7 pounds worth 77 cts., which is 11 cte. 
per pound as required. 

We may now expand the problem and add another kind of 
sugar, worth 10 cts. per pound. 

Then make a mixture, worth 11 cts., with sugars worth 9, 10, 
and 16 cts. per pound. 



9\ 5 
16^ 2+1 



Link each price below 1 1 to the one above. 
Make a balance between 9 and 16 as before, 
then between 10 and 16, and all will be bal- 
anced as required. The result is. 6 pounds at 
9, 5 at 10, and 3 pounds at 16 cts. That is, 13 pounds of this 
mixture is worth 143 cts., which is 11 cts. per pound as required. 

On the same principle, any number of ingredients may be re- 
duced to any given mean price or quality. 

We give but one example. 

Mix 6 bushels of oats, worth 20 cts. per bushel, with 8 bushels of 
oats worth 25 cts. per bushel, with rye at 70 cts. per bushel, and 
wheat at 80 cts., and sell the mixture at 75 cts. per bushel ; what 
proportion of rye and wheat will there be in the mixture ? 

Ans. Rye 14 bushels, wheat 160' bushels. 

6 bushels at 20 cts. will cost 120 cts. and 8 at 25 cts. will cost 



48 ROBINSON'S SEQUEL. 

200 cts.; whence the 14 bushels of oats will cost 320 cts., or 22IJ 
cents per bushel. 



76 



22f 5 
80^ 6+621 



6 bushels of oats. 
6 bushels of rye. 



51} bushels of wheat. 
Here we have a true mixture worth 76 cts. per bushel, but the 
mixture contains only 6 bushels of oats : it must contain 14, there- 
fore multiply each of these quantities by y. Then 6* Y = 14. 
57}- V = 160. 

Alligation is of little or no practical utility, yet it serves as well 
as any other arithmetical operation to discipline the mind. 



POSITION. 

SINGLE POSITION DOUBLE POSITION. 

Before Algebra became a popular study many algebraic prob- 
lems appeared in common Arithmetics, and were solved by special 
rules, which were drawn from the results of algebraic investiga- 
tions. But at the present day all such problems in Arithmetic 
are improper ; as much so as to travel 500 rdiles in a pri- 
vate carriage by the side of a railroad track. 

Problems in Single Position produce equations reduceable to 
this form: a:f—m. (1) 

Problems in Double Position produce equations in this form : 
ax-\-bz=m (2) 

Not knowing the value of x in equation (1) we assume some 
known number, x, which may not be the true one, and if it is not, 
the result will not be the given number m ; let it be m\ Then we 
shall have : ax'=m' (3) 

Divide equation (1) by (3), then 
X —-.m ^ 
X m' 
Converting this into a proportion, we have 
m : m '. '. x' '. V, 



POSITION. 40 

The result of this proportion, put into words, is the rule of 
Single Position given in all the old Arithmetics. 

Rule. Assume a number and find the result of the supposition ; 
then say : As the result of the supposition is to the given result, so is 
the supposed number to the true number. 

We give but a single example. 

A and B have the same income, ^contracts an annual debt 
amounting to | of it : B lives on | of his income, and at the end 
of 10 years lends to A money enough to pay off his debts and has 
$160 to spare : what is the income of each ? Ans. $280. 

For the sake of convenience we will take some number divisible 
by 6 and 7 ; therefore take 35 for the supposed income of each. 

Then ^'s debt in one year is $5, in 10 years $60. 

B saves ^, or $7, in one year, in 10 years $70. 

B lends A 60 and has $20 left as the result of the supposition. 
Then, 20 : 160 : : 35 : 280. Ans. 

Now let us suppose the income to be 1, or unity. Then ^*s 
debts in 10 years amount to y . 

B saves in 10 years -^j", or 2. 

B pays A's debts ; he then has (2 — y>)=^. 

Whence, 4=160, or4=40, or 1=280. Ans. 

This manner of working by fractions some teachers call Arith- 
metic, but it is Algebra in disguise. 

Let X be the income, in place of 1, and the identity will be 
obvious. 

To show the arithmetical rule for Double Position we take the 
equation 

ax-\-b=m. (1) 

1st. Suppose a; to be represented by the assumed number x', 
and m-\-e' the result of this supposition, e' being the excess, or 
error. Then, 

ax'+b^^m+e'. (2) 

Again, assume another number, say x" and e" the second error. 
ax"+i=m-{-e". (3) 

Subtract (1) from (2) and 

a (x' — x)=e'. (4) 

(1) from (3) alx'''-<c)=e\ (6) 

4 



60 ROBINSON'S SEQUEL. 

Divide (4) by (5) and we have, 

x' — X e' 



X — X e 
Whence, e"x — e"x^e'x" — e'x. 

ex" — e"x' 



And a;=- 



This last equation, put in words, is the rule given in all the 
Arithmetics of a former day. It is substantially this : 

Make two distinct suppositions and note the results. Take the 
diflference between the given result and the result of each suppo- 
sition, which difference call error. Then, 

Multiply the first error hy the last supposition, and the last error 
hy the first supposition. Divide the difference* of these products hy 
the difference of the errors, and the quotient will he the number 
required. 

EXAMPLE. 

A has 820; B ha^ as many as A and half as many as C; and 
C has as many as A and B both. How many dollars had ea^h? 

Ans. A 820; B 860; C 880. 
Suppose C had 60. 
Then B had 30+20=50. 

A had 20. But 20+50 is not 60, the error 
therefore is 10. 

1st sup. 60. Error 10. 660 



2d sup. m. Error 7. 420 

3)240 

Ans. 80 



Again, suppose Chad QQ. 
Then B had 33+20=53. 
A had 20. 
But 20+53 is not 66. Error 7. 
By Algebra, 

Let 2a:= C's. Then a;+20=^'s. 20=^'5. 

Then ^x=x-\-^0. ic=40. 2x=i^0.Ans. 
By comparing the last operation with the operation of the arith- 
metical rule, and then applying it, we perceive the folly of retain- 
ing the old rules for mere arithmetical purposes. 

The rule of Double Position, however, is of importance in 
solving exponential equations in Algebra. 

* The diflference is Algebraic dz ; hence some Arithmetics give two cases 
to the rule — Qne when the errors are alike, the other when unlike. 



PART SECOND 

AI.O£BRA. 

SECTION I, 

In Algebra, as in Arithmetic, we shall only touch here and 
there on such points as might come up in the school-room, and 
present some difficulties. Hence this work will seem to want 
connection. 

When we indicate the solution of a problem in Robinson's 
Algebra, University edition, we shall refer to it by Article and 
number of the problem, and not write the problem in full. 

When the problem is to be found only in some other book, or 
is original here, it will be written out in full. 

ROBINSON'S ALGEBRA, 

SECTION II. 

CHAPTER I. 

EQUATIONS. 

None of the questions in this chapter require the aid of a key, 
«intil we come to the 15th, page 65. 

fl5.) (^J_::±L-^Y=='1^1^-J^= his stock at the 
\ 3 /3 9 3 

'commencement of the third year, before his expenses are taken 

out. Hence, 

Reduced gives a-= 14800, Ans, ' 

(16.) Put a=99, .'r=time past. Then v. — :rc=time to come, 
and per question, 

-^ _= ...,.,. ,a's=54. 

3 6 

(17.) Let ^= the whole composition. 
Then per question, 



H ROBINSON'S SEQUEL. 

_+10=nitre. 

- — 4|^= sulphur. 

— 4- — — 2= charcoal. 
21^7 

By addition, f^-j-^-j-^f +31+12 =ar.. 

Multiply by 6, and drop 5x from both sides, and we have 
l^+21+^=a;. Or, 4a:+21 -7+60=7^:. . . .rr=69. 

(18.) Put a= 183; a:=what the first received; then a — ^ar= 
2d received. 

Per question, ^^Sa—3x ^^^^^ 

^ 7 10 

(19.) Put a=68, x= the greater part, and a — ar= the less. 
84— <r=3(40— a+a;) a:=42, 

(20.) The distance from ^ to ^ put =2x. 
The distance from C to D *' =3a*. . 

Then, 3 times the distance from B to C must be 

-+ — or the distance is, -+-. 
2^ 2 6*2 

Hence the whole distance is, 5a:+-+-=34.. 

6 2 

(21.) Letar=fhe flock. 

The first party left him ?f — 6. The second left -—3—10=2 
^ "^ 3 3 • 

(23.) Observe that for every vessel he broke he lost 12 cents: 
S cents fee and 9 cents forfeiture. 

300— .I2a;=240 x=5. 

(24.) Had he not been idle he would have been entitled to o^ 
cents. But he was idle x days at a loss of (i+c) cents. 

Hence, ab — (b'^c)x=sd, x= . 



ALGEBRJL. 63 

{"25.) Put 6a;= less part. Then a — 5«= the greater part. 

3 
Per question, a — 7a; = 20a; — _ (a — 6a;) 

7a—49x== 1 40a;— 3a+16a; 
or, 204a;=10a=10-204 
or, a;=10 
Therefore, 5a;=60= the less part 

(26.) Let 8a;=the price of the horse. 
Then a — 8a;=: the chaise. «3=34i. 

Per question, 2a — 16a; — 3a;=24a; — -(a — 8a;) 

5 
or, 2a = 43a; — -(a — 8a;) 

14a=301a;— 6a+40a; 
19a=341a; or, a;=19 

8a;=152. Ans, 

(29.) Let 5x= his money. 

After he first lost and won 45., he had 4a;-|-4. 
He again lost and won, and then had 3a;-|-3-{-3. 
I of this must equal 20, or, 3a;-l-6=24. x=6 

5a;=30. Ans. 

(30.) Let 3a;= the income. 
Then 2a;= the family support. 

a;— ?^ =^=70. Hence, . - . . 3a'=70 • 9. 
3 3 

31, 32, and 33 require no explanation. 
(36.) Last year the rent was x dollars. 

This year it is x4-^= 1 890. 
^ ^100 

(36.) Is the (35) in g-eneral terms. 



(37.) Let 7a;= equal the income. 

5 



7a; 
Then x= A's annual debt. — =what B saves. 



7a; 

—— <c=16 or a;=40. 7a;=280. Ans. 

5 



54 ROBINSOlSr^S SEQUEL. 

In general terms, 

6 2 4 . 

(38.) f^4-^4.?^=a 

(39.) Let 10aj= the income. 

Then 2a;-|-100= the sinn spent. 
Sar-f- 35= " sum left. 
7^+135= 1 Oa: the whole, 
or, 45=ar 450. Ans. 

(40.) Let 2lx= the income. 

Then 3x-\-a=: the sum spent. 

7x-\-b = the sum left. 
10ar-f-«+^=21a:= the whole. ^^ (a-f-^) 



11 



(41.) 2a;4.4 : 3a:-f4 : : 5 i 7. 

(42.) Let x^ — 7= the number. 

Then, per conditions, x — \=.Jx^ — 7 
a;2_2ir-f-l=:a:2— 7 
or, 2:= 4 and x^ — 7=^. 

(43.) ^'s rate of travel is \ miles per hour. 
J5's rate of travel is f miles per hour. 
A is in advance when B sets out, ^f mdks. 
Let ar= the hours after B starts. 

Then^ — = — + — Reduced gives a;=42. 

3 5 5 



CHAPTER II. 
EQUATIONS- IN TWO UNKNOWN QUANTITIES. 

(6.) Add the two equations together, representing {x-^y) by 4, 

and we have is+^5=50 or 5=5 •3. 

But ir+9y=21-3 

Subtract x-\- y= 5-3 

8y=16-3 or, y=e. 



ALGEBRA. ti 

(7.) B^ adding the two equations we have 

65=50 
or, a:-|-y=10 
but 4a-+y=34 
Hence 3x =24 or, ar=8. 

(9) and (10) are resolved same as (6) and (7). 

(11.) From the first equation we have 

y=2:r— 80. 
Transpose — 8 in the second equation and we have 

6 ^3 4 ^ 
Multiply by 60 and we have 

1 2a;+l 22/+20a:=30y— 1 52r+35 • 60 
or 47a;=18y4-35-60 
Substituting the value of 1 By, we have 
47a;=36.r— .18-80+35-60 
or lla;=— 240-6+350-6=110-6 
Hence, x=60. 

(14.) Bringing unknown terms to the first members of the 
equations and we have 



x=4. 





^_^— — 1 




4 


_2 


_3 
















X y 




y 


X 


2 


By addition. -=- or 










J 


x 2 










(15.) Putc 


f=50. 










Then, 


a:+3a : 


y— a : 




3 


: 2 


And 


X — a : 


y+^a : 


: 


5 


: 9 




2a:+6a 


=3y— - 3a 






(1) 




^x — 9a 


=53^+10a 






(2) 



Multiply (1) by 5, and (2) by 3 ; then, 

10:P+30a=15y— 15a (3) 

27a:— 27a= 1 5y-[-30a (4) 

Subtract (3) from (4) and 

17a: — 57a=45a 
17a;=102a 

ar=6a=6 • 50^300. 



M ROBINSON'S SEQUEL. 

(16.) Divide the numerator of the second member of the first 
equation by its denominator, and we have 

Hence, Ux-\-Uy=m (1) 

Multiply the second equation by (3y — 4) and we shall have 

9^_12.=(i51=?^^LPj^^^^ 10. 

or, no^n.=(l^ti}M(^tz^- 

4y— 1 
440y_48a^— 1 1 0+1 2a:= 453y— 48a:y— 604+64a;. 
0=13y+52a:— 494 
or, 4a;+y=38 (2) 

Add (1) and (2), and we have 

15(x+y)=165 
or, a;+y=ll (3) 
(3) from (2) gives 3a;=27 x=9. 

(17.) Multiply the 1st equation by 14 and we have 
42a;— 7y=49 
Add — <c+7y=33 



41a; =82 or, 


x=2. 


(18.) ^^2^^i6 




a;-^=-3 
6 




Subtract ?y-|-?y=19 
3 5 




10y+9y=19-16 y-. 


= 16. 


(19.) Divide 2d by the 1st, and a;— y=2. 
But a:-|-y=8. 




(20.) Multiply the first equation by (X'\-y), and the second 
by 9, and we have 

4(a;+y)2=9(a:2— y«) 9(x^-^y^)=9'36. 
4{x+yy=9'36. 
Hence, x-^y=9. 
Divide the first equation by this last, and we have a?— y=4. 



ALGEBRA. 



(21.) 



_4y 



64y3 

' 27 



27 ^ 
372/3=37-27 y=3. 

(22) and (23) require no remarks. 

(24.) The first equation gives 

a;+24y= 91 (1) 

Add 40a:-l-y=763 (^) 

Multiply (1) by 40, and subtract equation (2), and 

969y=2877 or y=3, 

(26.) From 1st equation take the 2d, and we have 

2ia;+5y=60. 
Divide by 2^ and we have x-\-2f/=24: 
But 1^+2^=19 

^x =5 or, ...a;=10, 

(26.) Add the two equations, and 

Y(^+y)+K^+y)+2o=^+y 

or, -+-+20=s .5=120. 

By 2d equation, 1(120) — 5=y=35. Ans. 



(7.) Given 



CHAPTER III. 
EQUATION'S OF THREE. OR MORE UNKNOWN QUANTITIES. 

Sy=u-\-x-\-z 
Az:=u-\-x-\-y 
u^=x — 14 
Subtract 2d from the 1st, and 

2a; — ^y=y — x or 

Subtract 2d from the 3d, and 

42 — 3y=y — z or 

Add 3d and 4th and 



to find u, X, y and z. 



3x=4y (1) 



5z=4y (2) 
4z=2x-{-2/—'14 (3) 



■^' 



58 



ROBINSON'S SEQUEL. 



Multiply (3) by 5 and (2) by 4 and 

202=10a:+ 5y— 70 

202= \6y 

0=10a;— lly— 70 
' <» 30x— 33y=210 

From equation ( 1 ) 30a: — 40y= 

7y=210 

y= 30 



(X 



(8.) 



+1+^=62 
'3^4 



I 3^4^5 ^ 

4^5^6 



b 



To avoid numerical multiplication, and really to understand 



Igehra as applied here, observe that 


62+38=100 


Put a=60; then 62=a+12=a+5. 




Clearing of fractions, we have 




QxJ^ 4y+ 32=12a+126 


(1) 


20a?+l 5y+ 1 22=60a— 1 66 


(2) 


\5,x-\-ny-\-\0z=ma—QQb 


(3) 


Multiply (1) by 4, and subtract (2). 




Then, 4a;+y=636— 12a 


(4) 


Subtract (3) from (2), and 




5ar+3y+22= 456 
3 





(5) 



15a;+9y+63=1356 
Subtract \2x-{-^y\-Qz= M b-\-Ma 

~3ar+ y ^TTT6— 24a 
Subtract (5) from (4), and we have 

a:=(12a— 486)^12(a— 46)=12-2. Ans. 
That is, a:=24 or 26. 

Now, equa,tion (4) gives us 

864-y=63J— 6a 

y=(55—a)6=6- 12=60. 

(9.) By adding the three equations and reducing, we have 
4a;+3y+22=3a (1) 



ALGEBRA. 50 

By adding the 2d and 3d, reducing and doubling, we have 
l0x-{-4y+2z=4a (2) 

Subtracting (1) from (2), and we have 

6x+y=a (3) 

Adding the 1st and 3d, and reducing, we have 

x-\-2y=a (4) 

From (3) and (4) we readily find x and y. 

'2x +y —2z =40 (1) 

4y —X +32 =35 (2) 

(10.) < Su +t =13 (3) 

y J^u +i=15 (4) 

Sx—y-\-St—-u=49 ( 5) 

It is easier to eliminate t than any other letter. 

Subtract (3) from (4), and we have 

y—2u=2 (6) ^ 

Three times (3) taken from (5), gives 

.Sx—y—10u=10 (7) 

Add (6) and (7) and divide the sum by 3, and 

x—4u=4 (8) 

Double (6), and subtract it from (8), and we have 

x=2y (9) 

Eliminate z from equationi^ (1) and (2), and we have 
4a;+lly=190 

But4x=Sy. Then 1%= 190, or y=10. 



PROBLEMS PRODUCING SIMPLE EQUATIONS, INVOLVING TWO 
OR MORE UNKNOWN QUANTITIES. 

(1.) Let X, y, and z represent the numbers. 

xy=600 (1) 
xz=SOO (2) 
yz=200 (3) 
Multiply (1) and (2), and divide the product by equation 3, 
and we have x^ =900 a:=30. 



60 ROBINSON'S SEQUEL. 

(2.) Let X, y, and z represent the numbers. Then per question 

y+fZ:!= 70 (2) 

o 

^+y+^=190 (3) 
Double (1) and subtract (3), gives a;=50. 

This problem calls the pupil's judgment into exercise. He does 
not know in the first place which is greater, x or z', hence he 
must try both suppositions, and the one that verifies equation (2) 
is right. 

(3.) Let a;, y, and z represent the shares, and put a=120 

^ — T(y+^)=« 

y—({x+z)=a 

Clearing of fractions, we have 

Ix — 4y — 42= 7a (1) 

_-3;y_(_8y— 32=8a (2) 

— 2a;— 2y+92=9a (3) 

Double (1), and to the product add (2), and we have 
liar— ll2=22a 

a;— z= 2a (4) 

Double (3), and to the product add (1), and we have 
— lla;+222=lla 

—x-\- 2z— a (6) 

Add (4) and (5), and we have g=3a=360. 

Another solution. 
Assume, x-\'y-\-z=s. 

By the question {/-^(/+^)=- 
as before, i y— f(^+^)=« 

Clearing of fractions, and we have 

7a;_4y— 42=7a (1) 

8y— 3a;— 32= 8a (2) 

90— 2a:— 2y=9a (3) 



ALGEBRA. » 

To ix-\-4y-\-4z=::4s 

Add (1) 7ar — 4y — 42= 7a 

And ll^ ^4s+7a (4) 

To 3x-{-Sy-{-Sz=3s 

Add (2) Sy—3x—3z=Sa 

And Tly ^35-{-8a (6) 

To 2a:+2y+22=25 

Add (3) 90— 2a;— 2y=9a 

And ~Uz ^28-\-9a (6) 

Add (4), (6) and (6), observing that {x-^-y-\-z)=s, and 
ll5=95+24a 
Whence, «=12a 

This value of*, put in (4), gives 

llx—4Sa-^7a=55a 
or a?=:5a=600. Ans. 

(4.) Resolved in the book. 

Let X, y, u, and z rejpresent their ages, and 8 their sum. 

Then, «— 2=18 

5 — «=16 

^ — y^=\4 

8—x—\2 



By addition, 35 =60 s=20. 

(6.) Let x^=A*& shillings. > 

y=B*s 

z—C'a " 

After the first game they will have as here represented 
X — y — z=A 
2y =B 

2z =0 ' 

After the second game, 

2ar— 2y— 22=^ 
3y— .t— z—B 
4z =(7 



n ROBINSON'S SEQUEL. 



After the third game 




4x—4y—4z== 16 


(!) 


6y— 22;->22= 16 


(2) 


70— ar— y== 16 


(3) 


Sum, a:+y-{- 2=3*16 


(4) 



Add (3) and (4) and we have 

82=4-16 2=8. 

(6.) This problem is resolved in the book, by equation 7, 
Art. 53. 

(7.) Let X represent the better horse, and y the poorer 

a:+15= Uy+10) 

x+10=.my+15) 
Therefore, |(y+10)=||(y+15)+5 
Reduced gives .,, y=s50. 

(8.) Let x= the price of the sherry. 

y= brandy. 

Put a=78. 

2x-{- y=3a 
7x-\-2y=9'a-\-9 
'3x =3a-f9 

X = a-\S— 81s. Am, 

(9^,) Let x=:A's time. y=^B's time. 

Then, _=th-e part that B can do in one day, 

4 , 4_^^1 
xy 16 4- 



Hence , ys=48. 



36^3 
J 4 

(10.) 2ar _2 f+?=^ 

^+7 3 '^f 5 

Sx=y-\-7 5a:+10=6y. 

(11.) , 2y , 3ar 

^3 ^4 

(12.) Let .r= the greater, and (24 — x)= the less, 

X , 24 — X ..4.1 



24 — ^ X 



ALGEBRA. 63 

x^ : ^24— a:)2 : : 4 : 1 
By evolution, x : 24 — x : : 2 : 1 
(13.) Let a:= the number of persons. 
y= what each had to pay. 
Then, xy=: the amount of the bill. 

Put (a;+4) (y— 1)= the bill. 

Also-, (x—3) (y+l)= the bill. 

xy-\-47/ — X — 4=xy 
xy—Sy-{-x—S=:xy 

4y — X — 4=0 

— 3y-^x — 3=0 

By addition, y — 7=0 

(14.) 10x-\-y=4x-\-4y or, y=:2x» 

10x-{'y^27=10y+x 
' or, 9x +27= 9y 

X + 3= y=^2x, hence, «=3. 

(16.) Let x= the digits in the place of lOO's. 
y= *' in the place of lO's. 

z= " the units. 

W-^y-\'Z=\l Z=:z2x 

100ic-f-10y+2+297= lOO^+lOy-f-a? 

99a;+297= 99^ 
x-\-3=z=:2x Hence x=3. 

/ic \ T + a;— 40 X — 20 , a:— 10 . ^, 

(16.) Let , ., and _ — represent the parts. 

Then, f=12+f:r?2+5ld?=90 .=100. 

2 ^ 3 ^ 4 

(17.) Let X represent the part at 5 per cent, and (a — x) the 
part at 4 per cent. Then 

5x 1 4a — 4x , 

ioo"*" 100 
Hence x=:100b — 4a. 

(18.) To avoid high numerals, and of course a tedious opera* 
tion, Put a= 5000; then 2a= 10000, 3a=s 16000, 

—=1500, and A^^=800. 
■ 10 100 



64 ROBINSON'S SEQUEL. 

Put x=. A'& capital, and r — l=^'s rate. 
ar+2a= ^'s ** r=B'& rate. 

a;-j-3a= (7's " r+l==C"s rate. 
'' r X — X I 16a r x-\-2a r 



By conditions 



100 ' 100 100 

rx — X .3a_rx-{-3ar-\-x-\-Sa 

~ioo ~^io Too 

Reducing (1), gives a:=(16 — 2r)a 



(1) 

(2) 



Hence, 



32— 4r=27-~3r, 



or. 



a 
. . . .r=5. 



(19.) Put a=1000, X and y to represent the two parts, and r 
and t the rates expressed in decimals. 
' x-\-y=z\3a 
rar= ty 
te=360 
ry=490 
Divide (3) by (4), and we have 

/J\/a;\ 36 
W\y/ 49 



Then by conditions, 



(1) 

(2) 

(3) 
(4) 

(6) 



From (2) we have 



x_t 
y r 
t . 



Substitute the value of — in equation (5), and 



.36 
49 



or 



By returning to equation (1) we have _iL-|-y=13a. 

13y=13a*7 or y=7a. 

(20.) Let Xy y, and z, represent their respective ages. 
Then by conditions given, x — y=. z 
5y-{-2z — ^a;=147 

(21 .) Let X, y, and z, represent the respective property of each, 
and put 5= their sum. 



ALGEBRA. 66 

fX'\-3y-\-3z=i7a a=100. 

Conditions, }y-\-4x-\-4z=58a 

(z-^5x-\-5y=63a 

Add 2x to the 1st equation, 3y to the 2d, and 4z to the 3d, 

observing that x-\'y-\-z=s ; then we shall have 

3s=47a+2a; (1) 

45=58a+3y . (2) 

65=63a+42 (3) 

3s— 47a 

. or, x= 

2 

45 — 58a 

5s— 63a 
z=. 

4 

3*— 47a , 4s — 58a , bs — 63a 



By addition, s 



2 ' 3 ' 4 
Hence, 5=19a. 

This value of s, put in equation (1), gives a?=5a=500. 



(23.) Let X, y, and z represent the respective sums. 




.+^=<. 


(1) 




y+|=a 


(2) 




1 ^ 


. (3) 


ac+y=2a 
3y+2=3a 
42+ar=:4a 

From the 1st 


or, 4z-\-12y= 

or, 4z-f- x= 

—x-\-12y= 

24x-\-12y= 


= 12a 
: 4a 
: 8a 
:24a 



25x =16a 



(24.) This problem is resolved in the work, by the 13th exam- 
ple, page 80, (Art. 61.) 
5 



6 ROBINSON'S SEQUEL. f 

(25.) Let aj-sthe greater, and y the less. 

^x — y=0 or, ...2ir=3y. 

(26.) a:+Ky+^)=«=61 

2y+(^+2/+2)=3a 
324-(^+y+2) = 4a 
a;=2a— 5 (1) 

y=i(3a— 5) (2) 

2=H^a- s) (3) 

s= 2a— 5-|-i(3a— s)+^(a— s) 
6s=12a— 6«-|-9a — 35-f-2a— 25 

175=29a or .5=29-3=87. 

Ifow equations (1), (2), and (3), will readily give x, y and z. 

(27.) Let x=A'&, y JB's, and z=C's sheep. 
Then by the conditions, 

a:_|_8— 4=y-[-g— 8 
i(y+S)=x+z-S 
i(z+S)^x+y-S 
x+n= y+ z (1) 

y^24=2x-\-^z (2) 

2-|-32=3a;+3y (3) 

Add (1) and (3), and we have x-\-44=3x-\-^. 

Double (1), and subtract (2), and we have 

2x — y=2y — 2x or, 4x=3i/ 



y=8. 



(30.) This is a repetition of the 1 0th example, page 89, in- 
serted here by oversight. 



But 


2a;+4y=44 




4a;-f-8y=44-2 




ny=44-2 or 


(28.) 


ar+l_l X __1 




y 3 y+1 4 


(29.) 


«+2_6 X _1 




y 7 y+2 3 



ALGEBRA. ♦ ^ 

(31) Let a?i=:-4's money, and y=zB's. 

x-5=Uy+5} (1) 

a?4-5s=r:3y— 15 (2) 

Subtract (1) from (2), and we have 

1 =%— 1 5—1—- or ; . . V = 1 K 

2 2 

(32.) Let a:=s= the number of bushels of wheat flour. 

And y=as •<* '^ barley ^^ 

Then the cost of th« whol« will be expressed by 

10a'-)-4y 
The sale at 11 shillings v/ill be H^c-j-lly 

Now by the coiiditi<ms, 

lOx^iij : llx^llif : ; 100 : i43| 
Multiply the last two terms by 4, and 

iOx-^4:i/ : n.r+!ly : : 400 ^ 575 
t)ivide the two last terms by 25, and 

10a;+% : lla;4-lly : : 5'6 : 23 
^H^2y : llar-fllt/ : : S ^ 23 ^ 

ii5.r-f-46y=a:88a:+8% 
27a:=42y 
9;c=14y 
These co-efficients, 9 and 14, give th« lowest proportion la 
whole numbers. Th« proportion was oaly required. 

(33.) Let iOir-{ry represent the number. 

Kow the question gives us Q =si^v 
And Q'^^hy 

i(10^+y)=2;r-f i or .,y=3, 

INTEEPRETATIOJf OF NEGATIVE VALUES. 

(Art. 55.) 

(4.) Let X represent the years to elapse. 

Then 30+a:=3(154-.c) x^—1^. 

To make this equation true, the years required must be takea 
■mibtr<ic4ively^ 



ee ROBINSOK'S SEQUEL, 

(6.) Let x= tlie man's daily wages, and y= the son's, 
7a:-j-3i/=22 (1) 

6x-\- y=18 (2) 

'2x-\-4y=40 

3x-{- y=18 x=4, y= — 2. 

(6.) Let j-ss man's wages, y= wife's, and z= the son's, 

10rc+ 8y-f- 62=1030 cts, (1) 

nx+lOy-^ 42=1320 cts. (2) 

16a:4-102/-f'122:=]385 cts, (3) 

Subtract (2) from (3), and we have 

3a:+82=65 (4 

Multiply (1) by 5, and (2) by 4, and take their difference, 
and we have 2x-\-\4z^= — 130 

a-4- 72= —65 (6) 

3a:-|-2l2=— 3-65 

(4) l'?Hr ^^== ^ 

By subtraction, 13z= — 4 '65 

2==_-4'5= — ^20, 
As z comes out with a minus value, it shows that the son had 
no wages, but the reverse of it, he was on expense. 

(7.) _ 10ar-f-4y+32=1150 (1) 

9a;-f8y-f-62=1200 (2) 

7a:_f_6y-j-42= 900 (3) 

Double (I), and subtract (2), and 11:? = 1100 ar=100 ct*. 

(8.) ^+1_3 X _5 





V 


6 ♦ 




y+I 


7 








5a:+6=r3y 






(1) 






Ix 


= 5y+5 






(2) 


Add 




I2x 
3x 


=8y 
=2y 






(5) 


iract (2) 


from 


(1) and we 
—2x+5=- 


have 

-iy—b 






(4) 



ar=— 10 by adding (3) and (4). 

y=— 15 
The result coming out minus, shows that there is no such 
arithmetical fraction. Algebraically, however, '_|| will answer 
the conditions. 



ALGEBRA. 69 

FINDING AND CORRECTING ERRORS. 

This subject has been suggested to us by circumstances. 
Those who have not been as severely disciplined in this matter 
as ourselves, are too much inclined to assume that the error must 
be in the answer, when it is more likely to be in some portion of 
the data. In short, we believe the following exposition will be 
of use to many. We shall illustrate by examples. 

1 . A young man, who had jiist received a fortune, sptnt |- of it 
the first year, and | of the remainder the next year ; when he had 
$ 1 420 left. WJmt was his fortune ? ^W5. $ 11 360. 

Let x=- the income ; then if he spent | of it, f would be left, 
and the next year he spent |xi=ro- 

Then, ——^-^=1420. 

• 8 10 

The value of x in this equation is ^38933}. 

This result shows a great error, somewhere. If an error 
existed in the answer, it is probable it would be in one or two 
figures, at most. But every figure of our result differs from the 
given answer ; and besides, it comes out with a fraction, which is 
against probability. 

We will therefore assume, that the stated answer is correct, 
and that the error is in 1420, 

To test this, write 11360 for x, and a for 1420 in the equation. 

Then, a=('?— A^ 11360=^- 1 136=852. 

\8 10/ 4 

By this supposition, 1420 should be 852 ; but we can conceive 
of no mistake, either in the writer or the printer, that would be 
likely to change 1420 to 852, they are so entirely unlike in all 
respects. We therefore assume, that 1420, as well as the answer, 
is correct. 

Now, it only remains to find the error in one of the fractions, 
I or ^. To try | let it be represented by m. 

Then, (1 — m)= the portion he saved the first year ; | of this, 
or — - — = the portion he spent the second year. 



TO ROBINSON'S SFQUEL. 

Hence, ( \—m)x—^ilz:^}^= 1 420 

6( 1— «»)ar— 4( 1— m)a:= 1 420 • 5 
Or, (1— m)ar=1420'5 

Write the value of a; in this equation and we have 
(!—»») 11 360= 1420 '5 

1136. 
Whence, »i=f 

Here is the error ; | was written or printed, by mistake, in the 
example, when it should have been f . The error was in one 
figure, only, and this is generally the case. 

2. A company at & tavern, wli&n they came to pay thehr hill, found 
that had there been 4 more in company, they would have had a shil- 
ling a piece less to pay : hut had there been 8 less in company, they) 
must have paid a shillihg a piece less. How mawy were in company ,, 
and what did each have to pay ^ 

Ans. 24 persons. Each paid 7 shillings. 
Let x=: the number of persons. 

y= the number of shillings each had to pay. 
Then, .ry= the amount of the bill. 
By the 1st condition [x-\-A)[y — \)=Lxy (1) 

By the ad - (;3r-.8)(y+l)=.ry (2)^ 

Expanding, and omitting xy on both sides, we have 
43/ — X — 4=0 
— 8y4-.r— 8=0 
By addition, — 4y— 12=0 or, y— — 3. 

This value of y, substituted in one of the preceding equations^ 
gives 3-= — 16. 

That is, there were 16 less than no persons in company, and 
each paid 3 less than no shillings ; in short we have a complete 
absurdity in all respects. No change in numbers expressed in 
the answer will remedy the matter, and indeed with the present 
data, no other values can be assigned to a? and y. 

Now to find where the error is in the data, let m represent 4, 
and n stand in the place of 8, in equations (1) and (2), and in 
place of X write 24, and of y write 7. 



to find J 
X and y | 

[ y=2, or 1 



ALGEBRA. Tl 

Then (1) becomes (24+w)6=7-24 (3) 

And (2) becomes (24— w)8=7-24 (4) 

Then m=4. and »=3. 

Here then we find that 8 in the example was printed in the 
place of 3. This correction being made every thing corresponds. 

In an Algebra recently published, I find the following example 
given for solution. It contains an error — find that en*or. 
3. Given 

( Ans. ar=6, or 3. 

'\-:i L_!l_= - +rk fin /I 

a;=y3 4-2. 

This example is given under equations of the second degree ; 
but when we attempt a solution, the resulting equation will pro- 
duce an equation of a higher degree. 

As the answers are given in small commensurate numbers, it 
is probably highly probable that they are correct, or rather should 
not be changed. 

Because the second equation verifies with both answers, we 
must regard that as correct. 

It is also very improbable that an error should exist in regard 
to the radical sign of square root, but in regard to the exponent 
|, an error there is very possible. On this supposition we will 
substitute the values of x and y, in the first equation, and write 
m for the exponent ; then we shall have, 

2 , V6+2_ 17 
(6+2)""^ ^ ^^-^^ 

Multiplying by ^/B and we shall have 

^+-?=1! or, lVi+i6=17 

8" ^2 4 8"^ 

Whence, 878=8^=8" That is, m=\ 

From this we learn that the printer inverted the terms of the 

fractional exponent. This being corrected, all is harmonious. 
We shall give other examples in finding errors as we naturally 

come in contact with them. 



7» ROBINSON'S SEQUEL. 

^ PURE EQUATIONS. 

We omit all the examples in Robinson's Algebra to the 17th, 
page 136. From thence we touch all that can require any notice 
in a work like this. 

(17.) Divide the numerator by its denominator, in each mem- 
ber, and we have 

i--_^L_=i^ 1^ 



j6x-^2 4J6X+6 

Drop 1 and change signs, and clear of fractions, and 

iej6x-{-24=Uj6x+30 . Hence x=6. 

(18.) Cube both members, and 

Hence? 64-{-x^ — 8a;= 16+8a;+a;* or a;=3. 



(19.) By clearing of fractions, 6-|-ar-|-^a;^-j-6a:=15 
By reduction, ^a;^+6a;=10 — x Square, <fec. 

(20.) I z I Z 3 J^ 



N *+ V* — ^/* — V*=2 



\lx-\- Jx 

Multiply by \x-{- Jxy and we have 

x-^Jx—Jx^—x=^Jx 

2X'\-2jx—2j~x^—x=3jx 

^^Jx=2jx^^ 

4x^ — 4xJx-\'X=^4x^ — 4x af=x|. 

(21.) Resolved in the work. 

(22.) Resolved the same as 17. 

^_ _ 2& _^_ Ih 



[Ob= 
( U. 136. ) 



Hence, 6jaa;-\-10b=7 JaX'{'7b or x= 



ALGEBRA. 73 

(23.) Square both members and we have 



Jx^-{-12=2+x Square, &c. x=2. 

(24.) Multiply numerator and denominator by the numerator, 
and we have 

(V5±l±V_4fZ=9. 
1 



Take square root and transpose sj^x, and^4a;-|-l=3 — J4x. 
Square, 4x-\-l=9—6j4x-\-4x. Hence, . . . .x=^. 

(25.) Square root gives a — x=Jb, or x=a — Jb. 
(26.) Clearing of fractions and we have 

4— 4a;2=3 x^ = l-. a;=±i. 

(27.) Take the square root of both members, and 

.2*1 . 

=- or x=i5. 

x—1 2 

(28.) Resolved the same as the 21st. 

(29.) Clearing of fractions we have 

Jx'—9x-{-x—9=S6 

tjx^ — 9x = 4 5 — x 
a;2_9ar=452— 90a;+.^2 

8l2;=45-45 

9-9a;=5-9-5-9 a;=5-6=25. 

(30.) Resolved the same as (17) and (22.) 
Dividing numerators by denominators, we have 

3-^=3-i^- 
Va;-f2 V^+40 ^ .. j. 

Drop 3 from both sides, change signs, and divide by 5. and 
clear of fractions, and then 

2^+80=21 J^-|-42. Hence x=4. 

(31.) Multiply numerator and denominator of the first member 

Then, _(>/HV^z::^=— 



14 ROBINSON'S SEQUEL. 

Multiply by a and take the square root, and 

— an 

Jx+Jx—a= 

Jx — a 

tjx' — ax-^-x — a=an 

Jx^ — ax=^[n-\-\)a — x 

Drop x^ , and divide by a, and 

— x=^(n-\-\Ya — ^nx — 2a? 

l+2» 

(32.) Resolved in the work. 
(Art. 90.) 

(4.) Observe that 180=9-20 189=9-21. Put a=9. 
a:2y-|-a:y2_20a •^ja.j.ya—.gla 

Multiply the first equation by 3, and add it to the second, 
and a;3+3a;2y4-32:2/2_|_y3_8i«=a3 
cube root, a:-|-y=a=:9 

The rest of the operation is obvious. 

(6.) Divide the first equation by (x-\-y) and 
x^—xy-\-y^=xy 
x^ — 2a;3/-|-y^ =:0 or x — y=^^. 
Hence a;=2 y=2. 

(6.) x-\-y : a; : : 7 : 5 xy-\-y^^\tQ, 

5x-{-5y=7x 

5y=2x or ir=fy. 
Put this value of x in the second equation, and 
^y^+y^ = 126 
7y2 = 126-2 
ya = 18.2=36 y=d=6. 

(7.) From the first equation we have 

5x — 5y=s4y 5x=9y 



ALGEBRA. 7i 

181y2 = l81 -25 or y»=25. 

(8.) From the proportion we have 

5jj/=3jx or ^5y=9x. 

The rest of the operation is obvious. 

(9.) Extract square root and 

ia:+i=3 or a;=7|. 

(10.) From the first proportion 

x-\-y=3x — 3y or 4y=2x 

Hence 8y^=a;=*. 

8y3_y3_56 y=*=8 y=2. 

(11), (12) and (13) resolved in the work. 

(14.) f^::J^_6 or «-H=6 

From the second, .xy=5. 

(15.) Divide the first equation by (^-\-y), and 
x^ — xy-\-y^=2x7/ 
x^—<^yJ^y^=xyz=\Q (1) 

Add 4x y =64 

a;2+2a^+y2= 80=16-5 

Square root, x-\'y=4j5 

Square root of (1) a; — y=4 

2x =4^5+4. 

(19.) Double the 2d equation, and add and subtract it from 
the 1st, then 

a;^ -\^2xy-\-y' =a+2 J 

x' — ^y-\-y^ =a — 25 

x+y=:Ja-\-2b 



76 ROBINSON'S SEQUEL. 

(Art. 92.) 

(6.) Add the two equations and extract square root, and we 

have x^-\-y ^ = ±4 ( 1 ) 

Separate the first member of the first equation into factors, and 

we have x^(x^+]/^) = \2 (2) 

Divide (2) by (1) and x^=z±zS x=9, 

(6.) Is of the same form and resolved the same as (5.) 

(7.) Add the two equations, and extract square root, and we 

3 3 

have X* -{-y * = Ja-\-b 

3 / JJ 3. \ 

But a;*,U*+yV=a 



rr=- «' or 


r ( "' 


(u+6y 


\(a+b^. 


(8.) Resolved in (Art. 90,) of this Key. 




(9.) Square the first equation, and 




x+2x^y^+y=^5 


(1) 



Difference, ^.x^y"" =12 (2) 

Subtract (2) from (1) and 

By evolution, x^ — y^ =±1 

But, x^ +y^ = 5 

2ic2 =6 or 4. 



The following are not in Rol)inson's Algebra. They are mostly 

from Bland's Problems. We shall number them in order. 

18 
(1.) Given x^-\-'3x — 7=a;+2+— to find the values of x. 



Ans. +3, —3, —2. 



ALGEBRA. 77 

Reducing x'-\-2x=9+— 

X 

Factoring x'(x-{-2)=9(x-\-2) 

This equation will be verified by putting x-\-2=0 

Then will a;=— 2 

Again dividing both members by (ir-j-2) and x^=9 

Whence x=zt:3. 

As a general thing we shall not give all the roots to equations. 
Imaginary roots we shall not pretend to give, except in rare 
cases, or unless we have an ulterior object in view. 



(2.) Given ^^^-7^+^^-?-==^^ to find a;. 

^ ' axe 

Let P=z^a-\-x thenP^=a-f-^ ^^^ the equation 

become* — ~h — =-^^ 

ax c 

Or Px-\-Pa __Jx 

ax c 

That ia P(a:+a)=^-ar^ 

c 

Whence P^:=z-'x^ 

c 

By taking the cube root we shall have 

By squaring and replacing the value of P^ we shall have 

Let the known co-efficient ( ^ V be represented by m. 

Then a-\-x^=mx or, a;= ^ 

m — 1. 

(3.) Giren (gjf I^+ (''+») •'=? to find x. 



n ROBINSOlS^S SEQUEL. 

This 5s the same as example 2, in case «=2, therefore we may 
jump to the conclusion at once. 

Thus, a+x=^(^y^x. 

(4. ) Given i«^' +y ' = i^+y)^!/) ^ find the values of x and y. 

Dividing the first equation by («+y) and the second by ay 
we obtain 

«^— ^+2/^— a;y (1) 

^+y=4 (2) 

Transposing ccy in (1) from the second member to the first 
gives x^ — ^y-^y^ =0 

Whence x — y^^O or, x^ssy 

These values substituted in (2) give ics=2 ys=2 
In this example we divided one of the equations by {x^y)^ 
therefore (x-^-y) must contain a root of that equation. (See theory 
of equations.) That is, ar-j-yz^O, or, xss: — y, and — y substitu- 
ted for X in either equation will verify it 

(5 ) Given i^^'+^/M'^yC^+S^) =^8) to find the values of 
^ '^ ix^J^y^— Sx^—Sy^ =^12} X and y. 

Ans. xssi or 2. 
y=2 or 4. 
Multiply the first equation by 3 and to the product add thti 
second ; then 

x^'^Sxy{x+y)'^y^tsti2l6 (1) 

Cube root a:+y=6 (2) 

By squaring and transposing (2) becomes 

x^-^y^z=^36—2xy (3) 

By the aid of (2) and (3) we perceive that the first equation 
is equivalent to 

36— 2a:y+62y=68 or xy=Q (4) 
From (2) and (4) we find x aiid^. 



(6.) Given 



ALGEBRA. 
1 



S y ~Q X 7 



79 



To find the values 
' of X and y. 



Put 
And 



shall have 
And 



7(x—yy_7(x—yy^l 
4 y 4 X 9 

Ans. a:s=| y=i-. 

P^(x+y)^ F^^(x-^y) 

Q=(x—y)^ Q'''==(x—y) 
Divide the first equation by f , and the second by I, then we 
P , P^64 
~y x^ 63 

y X 63 
Equation (1) reduced, becomes 

(^+y)P^64 
xy 63 

F*_64 
xy 63 

xy 63 
Bivide (3) by (4) and we shall have 

JL_=16 

Whence F^2Q 

And P3^8$3 

That is a;+y=8a: — 8y or 9y='7x 

Ix^ 

From this last, xy=s * which being substituted in both (3) 

y 



That is 
Also (2) becomes 



(1) 

(8) 

(3) 
(*) 



ftnd (4) gives 
Whence 



9P' 



Ix^ 



.64 
'63 



or, 



9P'»^64 

2 



i>»==|, Thatis(*+y)^=^ (6)- 



Substituting ~ for y in equation (6) and we have 
y 

3 






O) 




ROBINSON'S SEQUEL. 




Cubing 


(•+i)"=i;- 




Tkatis 


••('+D'4:- 




Or, 


2232 83 
= — X 

92 93 


4=:-x or a;= 



Ans 



The following solutions refer to problems in Robinson's Alge- 
bra, Chapter V., Art. 93. They number from (5) to (13). 

QUESTIONS PRODUCING PURE EQUATIONS. 
(5.) Let ic-|-2/= the greater number. 

And X — y= the less. 
Difference 2y=4 Sum =2x 

2x{4xy)=1600 Hence ic=10. 

(6.) Let x-\-y= the greater number. 
X — 1/= the less. 
2y : x—ij : : 4 : 3 or ic=|y- 

( a;2_y2)( a;_y)=504 

(Vy^— y")(|y— y)=504 

(Vy')(iy) =604 Hence y=4. 

(7.) Let 8a:= the length of the field, and 5x= its breadth. 

Then = — = the acres. 

1^0 4 

Ix^ X 8a:= the whole cost. 

^6x= the rods around the field, 

13X26a:= the whole cost. 

Hence 2a;3 = 13 -262; or a;=13. 8ar=104. An9, 

(8.) Let 5x= the length of the stack. 

4x= the breadth. 

Then, —= the height. 





ALGEBRA. 




5x*ix'-^'4x= the cost in cents. 
2 


Also, 


5x'Ax'9,^A= the cost in cents. 


Hence, 


5x'4x*ll'4x=5x'4.X'2M 




7x'2x=224 or 



81 



a;=4. 



(9.) Put a;*— 7= the number. 



Then, x-^Jx^-\-9=9 



Jx''-{-9=9—x or a?=4. 



(10.) Let X represent ^'s eggs ; then 100 — ar= ^'s eggs. 

18 At ' o Tt, . 

■■As price. _=^'s price. 



100— re X 

Hence, _i^=?(100— ;r) 
100— a; x^ ^ 

9a;2=4(100— a:)2 
3a;=2(100— <?;) ir=40. 

(11.) Let x-\-y= the greater number, 
X — y= the less. 
2x=6 or x=3 
(x-\-y)^=x^-\-Sx^y+3xy^+y^ 
(a:— y) ^ =zx^ — Sx^y-}-3xy^ — y » 
2x^ +6a:y2=72 
Divide by 2a; and a;^+3y^ = 12. Hence, y=l. 

(12.) Let x= one number. 

Then, a'x= the other. , 

a 

(13.) Let x^ and y^ represent the numbers. *♦ 

Then a;^ +2/^ = 100 ^' 

« +y = 14. 



82 ROBINSON'S SEQUEL. 

The following are additional problems, and for the sake of dis- 
tinction we shall mark them (a), (6), <fec. 

(a) Two men, A and B, lay out some money on specvlatwn. A 
disposes of his bargain for $11, and gains as much per cent, as 
B lays out ; Ws gain is $36, and it appears that A gains four timss 
a« mtich PER CENT, a* B. Required the capital of each ? 

Ans. A's capital $5, B's $120. 
Let x=:B's gain per cent. 

Then 4x=A*s gain per cent ; also what B lays out. 
Per question, 100 : ar : : 4a; : 36 

4a:2=36-100 or x=30. 
Whence, 4x or 120 is A's gain per cent. 

Therefore, 220 : 100 : : 11 : ^'s capital =1^^= 5. 

^ 22 

(b) A vintner draws a certain quantity of wine out of a full cask 
which holds 266 gallons ; and then filing the vessel with water, draws 
off the same quantity of liquid as before, and so on four times, when 
8 1 gallons of pure win£ was left. How much wine did he draw each 
tim^? Ans. 64, 48, 36, and 21 gallons. 

Let a=256. x= the number of gallons of wine drawn the 
first time ; then a — a:=the wine left. 

It is obvious that the wine drawn out the second time will be 
found by the following proportion : 

a : a — x : : x : > ^^ = wine in 2d drawing. 

a 

Then, (a — x) — ^ ^ =^ ^ = wine left after the second 

a 
drawing. 



a 



Agam, a : ^^ L : : x : ^ — —^ — = the third 

a a* 

drawing. 

Whence, (^rf)'_(^=?I"*=(f=f)!= the wine left after 
a a^ . a^ 

the third drawing. Whence we conclude that ' ^^ ^ would be 
the wine left after n drawings. 

After four drawings, ^ L =81 by conditions. 



ALGEBRA. 83 

(a— fl;)*=81-256-a2 
Square root, (a— )2=9'16o=9- 16-266 

Square root again, a — x=3 • 4 • 1 6 
That is, 16-16— a;=12- 16 

Or, 16-16— 12- 16=4- 16=64=^. 

(c) A and B have two rectangular tracts of land, their tenths 
heing as 7. to 6, and the difference between the areas is 150 acres ; 
B's being the greater. Jifbw had A's been as broad as B's, it would 
have been 672 rods long ; bid had B's been as broad as A's, 'it would 
have been 900 rods long. How many acres ivere there in each ? 
Ans. A's 2100 acres, Ws 2250 acres. 
Let 7a:= the length of ^'s lot in rods, 
And ?/= the breadth of the same, 
Then, lxy=^ tlie square rods in u4's tract. 
Again, let Qx=^ the length of i>'s tract in rods, 
And, v= the breadth of the same. 

Then, 6ya^= the square rods in ^*s tract. 

By the given conditions, 

6?;a;— 7a:y=150-160 (1) 

Now had ^'s been v in breadth, it Avould have been 672 rods 
long, therefore 

Ql^v=lxy (2) 

Also, dOQy=Qvx (3) by the last 

given condition. 

By multiplying (2) and (3), omitting common factors, 
• 112-900=7^-2 

Or, 16-900=a;2 

Whence, a-=4-30=120 

Substituting 900y for 6y.c in (1) and 120 for x, we shall hav€ 
900y— 840y=150-160 
Or, 60y=150-160 

y=400 

Lastly, IlL2^:12?=2100 .I's acres. 

160 



84 ROBINSOIf^S SEQUEL. 

(d) A and B engaged to work for a certain number of days. At 
the end of the time, A, who had been absent 4 days, received $18.75, 
while B, who had been absent 7 days, received only $12. Now, had 
B been absent 4 and A 7 days, each would have been entitled to the 
sam£ sum. ^ 

How many days were they engaged, and at what rate ? 
Ans. They were engaged for 19 days, A at $1.25, B a^ $1 per day. 
Let ^= the time or number of days. 

a:= the daily compensation of A. 
y= the " ** B. 

Then by the given conditions 

(^— 4)a:=18a (1) 

(/— 7)y=12 (2) 

{t--l)x={t-4)y (3) 

From (3) ar=' ly. This value put in (1) gives 

{t:^y=\^ (4) 

t—1 ^ ^ 

Dividing (4) by (2) gives 

(^— 4)2^18|_ 75 ^25 
{t—lf !¥ 12^ 4^ 

Square root =_ or t=\9 

t—1 4 

The value of t put in (1) and (2) gives. .ar=1.25, y=I, 



SECTION II. 

QUADRATIC EQUATIONS. 

The following are but hints to the solutions of Equations in 
Robinson's Algebra, University Edition., commencing at example 
10, page 167. 

(10.) Put (ar— 4)2=y. 

Then, ?=1+1? 

y y^ 

y'— 8y-|-16=0 or. ..y— 4=0, 



ALGEBRA, 85 

(11.) Multiply by 16. Rule 2. 

Then, 64x^-^16x^-{-l=39- 16+1=625 

8^e-|-l=25 a;«=3 a:=729. 

(12.) Add 5 to each member. 

Then (x^—Zx-\-5)-{-(x''—^x-\-5)^==16 

By substitution, y2_|_gy^9_25 .y=2 or — 8. 

Hence x^ — 2x-\-5=4r x=i. 

(13.) By (Art. 99) we have . 

361 19 ' 

-^=_-^ t=—6 t^=3Q 

19 !9 

361 19 ' 

-^— ^=db2.,. ar=152 or 76. 

19 

(14) Observe that 81^^ and — s^e both squares, and if these 

x^ 

are taken for the first and last terms of a binomial square, the 
middle term must be 9a;_.2=18. 

X 

This indicates to add one to each member. Then extract the 
square root 9x-|--=±10. Hence, x=l or — 1 

X 

(15.) The first member of (15) is the same as (14.) 
Hence, add unity to each member and extract square root ; 

I 29 

we then have 9ar-4--=— +4 

X X 

9x^-^ix=2Q Put x=-. 

9 

«2_4^^28- 9=252 

«— 2= ±16 x=<i. 



86 ROBINSON'S SEQUEL. 

(Art. 105) 

(4.) Multiply every term by x, and . 

ar4_|.8a;3-|_i9;r2_i2a;=o. 
Operate for square root thus ; 






Divide by {x^ — ^x),. aud 

ar2_4^4-3=0 a;=l or 3. 

But the factor a;^ — 4ar gives a:=0 or 4. 

(5.) a;^— 10a:3+35a:^— 60a;+24=0 (a;^— S^r 



2ar2_5ir) — 10a:3-|-35a:2 
— 10a:3-f-35a;2 



10ar2_50a;+24 
(ar2_5x)2+10{a;2— 5ar)+24=a 
If we add unity to each member, we shall have complete 
squares. Extract the square root, and 

x^ — 5a*= — 4 or — 6. 
From these two equations we find a;=l, 2, 3, or 4. 

(6.) By mere inspection we perceive that this equation can 
take the form (f —xf —{x'' —x)^\?>1. 

y^—y=\o'2. y=12or— 11. 

x^—x=\^ or— 11. 
If we take — 11, the value of x will become imaginary. 12 
gives a:=4 or — 3. 

(7.) This equation may be put into this form : 
(if—cy^ )-^(^f —cy)=c'^ 
from which the reduction is easy. 



ALGEBRA. 07- 

(Art. 107.) 

(3.) Taken from the work we have 

{a-|-l )x^ — a'^x=a^ 
Or, (a-\-l)x^={x+l)aK 

Both members are of exactly the same form, and of course 
the equation could not be verified unless xz=:a. 

EXAMPLES. 

(1.) x^'+Ux^SO. Multiply by 4, &c. 
4a;2-(-^+l 12 =329+121=441 
2x+U= ±21 x=5 or —16. 

(2.) Drop 2a; from each member, and divide by 3 ; then 

x—l x—2 

X — = 

x—3 ^ 2 

Clearing of fractions and 

2x^ —Sx—2x-{-2=x^ —3x—2x-{-6 

ir2_3a:=4. Put 2a=3. 
Hence, (Art. 106) x=4 or —1. 

(3.) Multiply the equation by 6x ; then 

fi'y2 

J^^-L.Gx-\-6=13x 
x+\^ ^ 

x+1^ 

6x''+6x-{-6=7x''+7x 
Hence x^-\-x=e ' x=2 or —3. 

(4.) Clearing of fractions 

70a;— 21a;2+72a;=500— 150a; 

21a;2— 292a;=— 500. 

Or, 21a;2— 42a;=250a;— 600. or 21a;(a;— 2)=250(a;— 2). 

(6.) Put ('?+y)=a;. Then 
\y / 

a;2-|-a;=30. Or, a;=5 or — 6. 
Now, (5+y^=5or— 6 



fr. ROBINSON'S SEQUEL. 

y»— 6y=— 6, or y^-\-ey= — 6 

2y— 6= del y=3 or «. 

(6.) Put a:^=y ; Then y^-\-'7y=44 

4y3_|_^^49=226 

2y+7=±15 y-=4or-.ll. 
a;=(4)^ or (—11)* 
(7) a;2+a;=42. Hence a;= 6 or —7. 
That is y* +11=36 or 49 y=5 or ^38. 

(8.) 11— f+!^=^ 

^ ^ ic— 7 3 

33a;— 23 1— 3a:— 21 =a;*— 7a; 
a;2— 37a;=— 252 

4x'^A+3V =1369—1008=361 
2ar— 37= ±19 a?=28 or 9. 

(9.) 3a;2— 9a:=84 

12 



36a;2— .^+81 = 12-84+81 = 1089 
6a;— 9= ±33. 

(10.) Clearing of fractions we have 

2a:+27a;=16— <i: 

3a?+2Va;=16 
Multiply by 12, &c. 

6Va;+2=±14 .a;=4 or 7f 



(11.) ?(^=li)+4^=26 

?(?^1I)+2.= 13 
ar— 3 ^ 



6x— 33+2a;*— 6a;=13a;— 39 
2a:2— 13a:=— 6 

16a;2— ^+132=169— 48=121 
4a;— 13= ±11 



ALGEBRA. - 89 



(12.) Multiply by x^ and we have 



9 



1 1^2 



9 
lla;2— 54a?=— 63 

Put a;=— ; then w2__54^^_693 

«2_54^_|.272_36 

u — 27==b6 w=33 or 21. 

(13.) Clearing of fractions we have 

_a;2=27a:— 28 

x''+27x=2Q. Put2a=27. Jj^-. 

<c2-f-2aa;=2a+l ' " ™ 

a;+a=±(a4-l) «=1 or —28. 

(14.) Given mx^ — 2mxjn=nx^ — mn, to find x. 
By transposition, ma:^ — 2mx J n-\-mn=nx^ 
Square root, Jmx — Jmnz=.±:.Jnx 

By transposition, {Jm^Jn)x—Jmn 



The following are not in Robinson's Algebra. They are too 
severe for learners in general, and are, therefore, not proper in an 
.elementary work. 

We commence again with No. 1. 

(1.) Given a;^-|- =~2^-|-a;°, to find the values of x. 
We observe that the lowest root of x is the 6th ; therefore, 

i. JL 2 ^ 7 i 

put a;*=y; then x^=^y^. x=y^ . x^=^y^ . x^=7/^. x^=y^*. 



90. ROBINSON'S SEQUEL. 

Then y^^-X-^z=?Lu.ys 

Whence, yi»=66+y9. 

Therefore, y^— i=±V- Or, y^ =8 or —7. 

y3=2 or (—7)3. 

2 

ar=y^=4, or ( — 7)^. Ans. 

2 

(2.) Given * /_L-4-^/i=5z!^l> to find the values of x. 

^ X X X 

Ans. x= 1 or — V 
Multiply by x, and then we shall have 

'(•Vi)-H(■^/i)— •• 

Placing the value of x under the radical signs, then 

. ya;2+3/a;2=3— A 
That is a;^+a:^=3— ir^ 



2a;3+a;3=3. 
Whence, a;3=il or — f. 

Whence, a;3_ior— |. Or, ar=:l or— ¥• Ans. 

(4.) Given ^ar— 1)'+ A— Ay=^ to find a;. 

Put F=(x—-y and ^ = A— i) ' ; then 

P+Q=x (1) 

Multiply this last equation by (P — Q), then 

ButP2_g2— (^_1). therefore, a; (P— §)=(«— 1) 
Or, P-^=l-i (2) 

X 

Add ( 1 ) and (2), then 2P= (^— ^) +^ 

That is, 2P=P2+1 

Or, P2_2P+1=0, or P— 1=0 

Whence, a?— 1=1, or x=^{l±:j5). 





ALGEBRA. 


(5.) Given (x^- 


->)*+ (••-: 


values of x. 





91 



We observe that this equation is in the same form as the pre- 
ceding, and would be identical if we changed x^ to x, aMo 1. 
Therefore the value of ar^ in this equation will be of the same 
form as the value of x in the last example, except it will contain 
the factor a^, because the square root has been once extracted : 

That is x^=—{lzh^5), 



=-(^)' 



But this conclusion is too summary to satisfy the young algebraist ; 
therefore it is proper to take some of the intermediate steps. 

then the equation becomes 

P-\-Q=~^ (2) 

xMultiply (2) by \P — Q), then we have 

a 
But the value of {P^—Q^) drawn from (1), is (a:^— a^); 

therefore ^ (P— ^) = x^—a^ 

a 

or i>_§=a-?l (3) 

X'' 

By adding equations (2) and (3), we find 

2i>=«-^+?! . (4) ■ 

x^ a 

Multiply this equation by a, then 

2aP= a''——+x^ 
x^ 

that is, 2aP=a^+P^ 

or 0=a2— 2aP+P2. 



92 ROBINSON'S SEQUEL. 

Square root, 0= a — P, or P = a 

From the first of equations (1), we find 



a « 
.x^ — — 



,2 



From this equation, we find x=^-=^a(. ^ -\ . 

(6.) Given x^{\+l-y^{^x^-\-x)=10 , to find the val- 
ues of X. 

Observe that (^x^+x) = ^x^{\-\-—). Put (14-J_)=y ; 

^x 3a; 

then the given equation becomes x^y^ — 3a;2y=70. 

— 9 289 
Completing the square, x^y^ — 3a;^y-|--= 



x^y ^=db — 

"^ 2 2 



ic2y=10, or — 7 

That is a;2 4-^=10, or —7, 

^3 



Whence x=d, or — V» or 1(1=^^—251). 

/7^ Given 5^-^J x_ ^^{x-2jx) W-2,x+4. 

^ *^ ^+27^ 6+V^ (^+2V^) (6+V:r) 

to find the value* of x. 

Multiply by (6+ ^a:), then 

x-\-2jx ^ ^ ^^ x+SLjx 

Multiply by (x-{-2jx), and we shall have 

9(S6—x)=23(x^—4x)-\-7x^—3x+4 
Reducing, 15x^—4Sx=160. Whence a;= 5, or — ff. 



(8.) Given x^ -^^^4-15=^^^1^^, to find the values 
^ ^ 2^ 16 x^ 



of 



ALGEBRA. 93 

By transposition, a;^-|-15-f- — = + — ■ 

Add 1 to each member, (see Robinson's Algebra, Art. 99,) 

then .>+16+^=?^^-+^+l 

^ ^a;2 16 '2^ 

By evolution, x+- = ± (~+l\ 

X \ 4 / 

O n» 

Taking the plus sign, _■' = — 1-1 . ( 1 ) 

Taking the minus sign, - = — — — 1. (2) 

X 4 

From (1) a;*+4ar=32. Whence a;=4 or —8. 

From (2) 92;2-)-4ic = — 32, and x^^'Z^^y.IzI} 

y 

( 9 . ) Given {x^^sy — 4a;2 = 1 60, to find the values of x. 
Subtract 20 from both sides, then 

' (a;24-5)2_4(a;24-5)=140. 

Whence, x^ +5—2=: ±12. 

Therefore, x= ±3 or ±<y — 15. 

( 10. ) Given — + - =^—1= , to find the values 

^ ^ (a;2_4j2^(^2_4) 25^2 

of x, 

rf-2 

Multiply by a; 2, and put _— r-==y : then 

2 \ a 351 

Whence y+3= db V • y= f or —V • 

mt, X • ic^ 9 x"" 39 

That IS =-, or = — — 

a;2__4 5 a.2_4 5 

x=: ±3, or a;= ±VH- 

(11.) Given (ar— 2)2— e^a: (a;— 2) = 24— Har+lS^a:, to 
find the values of x. 

Expanding and reducing, gives 

a:2— 6a;^a;=20— 10ar+3V^. 



94 ROBINSON'S SEQUEL. 

Add 9x to both sides, and the first member will be a square ; 
that is, 

X 2 —6x Jx-\-9x = 20— <i:+3 Jx. 
Or, (S^a;— a;)2=20— a;+37a; 

' Now put 3jx — x^=y ; then 

y2=20+y. 
Whence, y^=5 or — 4. 

Then x—3^x^^, . or —5. 

Whence, :c=16, or 1, or ^.^tzllz:! . 

2 

(12.) Given (4x+iy-]-4x^(4x+l)=19\2-^(l0x-{-3x^)io 

find the values of x. 

Add 4x to both sides ; then 

(4x-\-iy-\-4x^(4x+l)+4x^l9n—6x—3x^ 
That is [ {4x+\)-{-2^x]-^-{-Gx+Sjx=1912 

Or, [ 2{2x+Jx)+l Y+3{2x+Jx) = 1912. 

Now put y=2x-\-Jx ; then 

(2y+l) = +3y=1912 
Or, 4y2+7y=:1911 

64^2+^-1-49=1911 • 16+49=30626 

8y+7=±175 * 

y=21, or — -V- 
That is, 2^+7^=21, or — V- 



From these last, we find ^-=9, or V, or ^^"^J^JJl 



3;r 



(13.) Given 8a;2^13= -+ »j6x^-\-52x^ , to find the values 

of X. 

Double the equation and remove the factor x^ from under the 
radical sign ; then 

16a:2— 26=3^+20:^6^^+62', 



That is, 16.c2 = (3.r+26) + 2;r72(3x+26). 

Now put 9/= J (3x-\-26) ; then the equation becomes 
iex^^ij^+(2j2)xy 



ALGEBRA. 95 

Add 2a;^ to both members, and 

1 8a;2 = 2/2 _|_ 2 ^2 • iry+2a;« • 
By evolution, Sxj2=y^j2 (a) 

Or. 2xj2=y=^3x+26 

By squaring, 8x^=3x-\-26. 

This equation gives a; = 2, or — y . 
By taking the minus sign to the second member of (a), 

we would find x-= ^ . 

64 

(14.) Given 4x''-\^21x+Sx^ J7x^—5x=207'^^f-, to find 

o 

the values of x 

4x^ 
Transposing — and removing the factor x from under the 

radical sign, will give 

I6x^ 

-y-+21a;+8^77;c— 5=207 

Subtract 16 from each member, then 
16x^ 



3 



■(21a:— 16)+8a; 77a;— 5=192 



That is l^+3(7a:—5)4-8a:77a:— 6=64-3 

o 

Put y=:J{lx—5). 

Then 15^+32/2 +8ary=64 • 3. 

* 3 

Clearing of fractions, and changing terms, 
16a;2+24a:y+92/2=64-9 
By evolution, 4a;+3y = 8 • 3 = ±24 
That is 4a:+37(7a— 5) = ±24 

Taking the plus sign and transposing 4ar, we have 
37(7a:— 6)=(6— a;)4 

By squaring 9(7a:— 5) = (36— 12.r+a;2)16 

Beduced, 16a:2_-265a:=— 621 . 



96 ROBINSON'S SEQUEL. 

This equation gives x=3, or — V/. By taking — 24, we 
obtain ^^ '^3±3V(-2 567) 
32 

(15.) Given a^b^z^—4{ahyx^^=(a—^yxi', to find the 
values of x. 

Put P^^ajr (1) 

And Q^=xi^ (2) 

mfn 

Then P^Q^=x^ (3) 

And P^=a;2-«» (4) 

Substitute these quantities in the given equation, and 

aH^F''—4{ab)^FQ={a--by Q^ (6) 
Kow let P=iQ 

Then a'^bU^ Q'''-^4{ab)^iQ^={ar-by Q' 
Dividing by Q'^ gives 

a^Pt^—4(ab)H=(a—by=a''—2ab+b^ 
Add (4ab) to the first member to complete the square, (see 
Art. 99, Robinson's Algebra.) 

Then a''b^t''—4(ab)H-\-4ab=a^+2ab+b'' 
By evolution, abt — 2^(a5)=a-|-5, or — a — b 

Whence abt=:(a+2jab+b)=z(Ja^Jby 



Or, f- U<^+J ^r or,n(y-^irN^ 

ab ' ab 



^ . , P xTi "-" 

But ''= — = = icamn 

Q -i- 

* ar-2 » 



Therefore, .•^■=(Vl±^^ or, =1>=V*): 
ab ab 



Whence, 



JM^|s„ j-ws^js 



ALGEBRA. 97 



SECTION III. 



QUADRATIC EQUATIONS CONTAINING MORE THAN ONE 
UNKNOWN QUANTITY. 

We commence by showing the outlines of the solution of the 
(3), (4), (6), (6), (7), and (8) equations in Robinson's Alge- 
bra, Art. (Ill), page 182. 

(3.) Futx^=F, and y^=Q. Then the equations become 
F+Q=Q (1) 

Square (1) and we have P^-\-2FQ-\-Q''=64^ (3) 

Subtract (3) from (2), and we have P^ Q''—2FQ=195, 
Hence, P$=15or— 13. 

Now we have P-\-Q=8, and PQ=15, whence P=5 or 3, 

and ^=3 or 5. That is, x^=5 or 3, &c. 

(4.) Puta;^=P, and y^ = Q; then the equations become 
P^ + Q''+P+Q=26, andP$=8 

2PQ =16 

(>+^)2+(P+$)=42. Hence, P+Q=6. 

(5.) Put — =:u: then u^-\-4:U= — w==_ or — — . 
The remaining operation is obvious. 

(6.) Given y^ — 8a;^y=64, and y — 2a; ^y^ =4, to find a: and y. 

To both members of the first equation add 16x, and to the 
second add x, to complete the squares ; then extract square root, 
and we have 

y— 4a;2=4(a?+4)^ and y^—x^= (a;+4)^ 
Four times the last equation subtracted from the preceding, 
gives y — 4y2=0. Or, y=16. 



•8 



ROBINSON'S SEQUEL. 



(7.) Multiply in the first equation as indicated, and subtract 
the second equation ; we then have 

«+y+2^'y'=25 or x^+y^=:6 
But from the second equation we have 

(a;2_(_y2)a;2y2=3o. Hence, x^y^ = 6 

3. 2^ X JL 9 

(8.) Divide the first equation by y^, and x^=2y^, or y^=^x'^ 
This put in the second equation gives 

X z f 

a;T_16a;3+64=64— 28=36. 



We continue this section by adding other and more severe 
equations, commencing with number one. 

(1.) Given -I oI'^'^^^^^Ta I \ to find the values of 
^ ^ ( 28 — y = x-\'^»Jx J 

X and y. 

By adding the two equations, omitting 16 on both sides, gives 

Squaring, 144 — Mjy-{-y—\Qx (1) 

Multiply the first equation by 16, and substitute the value of 
— 16a; from (1), then we shall have 

16y— 16^^=266— 144+247y—y 
Whence, 17y— 40^^=112 

>/y=f^±V(H|-^+lff) = n±H = 4or-^. 
Therefore, y = 1 6 or |f ^ . 

These values of y put in the first equation, give 
x=Jy=A, or my. 



(2.) Given 



y J(^x-\-y) 17 1 to find the 

(x+yy'^ y ^^^Jix+y) [values of 

x=y^^2 J X and y. 



ALGEBRA. 99 

Mnltiply the first equation by >/(^+y)» *^^^ 
y _|_^+y^^7 

^+y y 4 

Clearing of fractions, and 

Reducing, 4a;2 ^^^xy-^-^y^ 

Adding, to both sides, (Robinson's Algebra, Art. 99) and 

4 4 

By evolution, :?^=±(— +3^/) 

Whence a;=3y or — fy. 

These values of a; put in the second equation, readily give 

«=6, or 3, or 9T3V(-n9) 
32 

, y=8,orl,or-^-^V(-"9_) 
8 

(3.) Given j a:+4^^+4y=21+8Vy+4V(a:y) \ ^ ^^ 
and t Jx-^Jy=Q ) 

the values of x and y. 

From the first, x--'^J(xy)-\-^y—2\-\^^Jy—Ajx 

That is {^Jy—JxY =21+4(2^y— 7«) 

Let P=^Jy — ^a: ; then 

.^ P^— 4P=21 

^^^■L -P=2 ± 726=7 or —3. 

tR^ Zjy—Jx=l or —3. But Jx^Q—Jy, 

Therefore Sjy-^=7 or —3. 

7y= V or 1. 

y = -If ^ or 1. 



(4.) Given i Sx+ljxy^+9x^y==.(x^x)y 1 

and ( 6x+y : y : : x+5 : 3 J- to fand the 

values of x and y. 

From the first 9a;rf-2 Jny^-^9x^y=:3xy—y 



100 ROBINSON'S SEQUEL. 

That is (y^9x)+2jxy {7/+9x)^' =^3xy 
Add xy to both members and extract square root, then 
Ji/+9x+J^=2jxy (1) 

Whence y-\-9x=xy (2) 

From the second — 2y-j-18a;=ary (3) 

By subtraction, 3y — 9a; =0 

Or, y=3x 

This value of y put in (2) gives 12^=3a;^. 
Or, x=4. Whence y=12. 

By taking the minus sign to the second member of (1), other 
values of x and y can be found. 



(6.) Given \ x-^-y^-J "7"^ — i to find the values 

and Ix^+rJiT'-' ^~^) °f^aidy- 
The first cleared of fractions is 

x^ — y^ — Jx^ — y^=iQ 

Whence, - Jx^—y^=3, or —2 

a?=±5, or =1=371 



(6.) Given ( J^x+xYJ^y-+J{\-xY+y-=4 ) ^ 
and ( (4_;p2-)2^j8_4y3 f 

the values of x and y. 
From the first 



Squaring, 

\^2x-{-x^-\-y''=\Q—^J(\—xY-\^^+\—^-\-x^-\-y'' 

Reducing, a;= 4— 2 /(I— arj^+y^ 

Transposing 4 and squaring, gives 

a;2_8:y_|-lC=4( 1— 2a:+a;2 +y2 ) 

Reducing, 12=3a;2+4y2 (l) 

That is 4— a;2=^y!., U—x^Y=:]^ 

Comparing this last result with the second equation, we per- 
ceive that 



ALGEBRA. 101 

l^'.+4y'=16 (2) 

Add I to both members, (Art. 99, Algebra,) then 

9 ^ ^ ^4 4 

T, 1 .• 4y2 3 9 

By evolution, _^_-|-_==±- 

3*22 

Whence 4y2=9, or — 18 

y=dbf, or ±fV— 2 

The value of 4y^, that is 9, put in (1), gives x=\. 



.2 .,2 



(7.) Given {^+^'^I^=^l-t-r^- 

^ x—Jx^'—y^ 4 x^Jx^'—y^ > to find 
and ( ^2 ^xy=52—Jx^+xy-\-4 ) 

the values of x and y. 

Add 4 to both members of the last equation, and transpose 
the radical, then 

(x^-^xy+4)+(x^+xy+4)^=56 
This is a quadratic, and 



Jx^ +xy+4+ ^ = ±V^F = ±V 
Whence, Jx"-\-xy-\-4=7, or — 8 

a;2-|-ary=45, or 60. (1) 

Now take the first equation, and multiply numerator and 
denominator of each of the literal fractions by its numerator, 
then 






Expanding and uniting, and we have 

4 

16a;2=25y2 



4a; 

Ax — ±: 5y, or y= ± — 

5 



10« ROBINSON'S SEQUEL. 

This value of y put in (1), gives 

ar2-|-!^'_=45, or 60. (2) 

43*2 

Also, a;2— Zf„=45, or 60, (3) 

6 

From (2), 9a;2=9*5-5. Or, a;= ±5. 

Or, 9a-2=25-12=25-4-3 

3ar=5-273. Or, a:=dbl0^i 
From (3), a;2=9-6-5. Or, a;== dbl5 

Or, ir2=300. Or, x= ±10^3. 

Here we have 8 different values of x, each of which being 

4a; 
substituted in y=rfc — , will give 8 different values to y. 
5 



(8.) Given f h+V^ \ -^ -^^ /_4^ 1 to find 
and I V_^<5^=^=y+i [andy. 



Clearing the first of fractions, gives 

x-\-y^+^Jx=:^xy^ (1) 

In the second equation, multiply the numerator and denomina- 
tor of the fraction by the numerator ; then 

Multiply by (y+1), then extract square root, and we shall 

have 

Jx+Jx—y—\^y+\ (2) 

Or, Jx—y—\={y+\)-^Jx 

By squaring, x—y—\=y^-\-9.y-\'\ — '^Jx{y-\-\)-\'X 
B^duced, (i=^(y''+y)+^y-\'^—^Jx{y-\-\) 

Dividing by (y+1), 0=y+2— 2V^ (3) 

As we can divide by the binomial (y+1) without a remainder, 

it follows, by the theory of equations that (y+1) contains a root, 

that is y-|- 1=0. y= — 1. 

Corresponding with this value of y, equation (3) or (2) will 

give the value of :r. 2jx=l, ar = ^. 



ALGEBRA. 108 

To find other values, we must continue the solutions. 
Return to equation (1) and extract the square root of both 
members, and we shall have 

Jx-{-y= ztyjx (4) 

From (3), 2jx=y+2 (6) 

Double (4), and 2jx-\-2y= :±S.Jx{y) (6) 

That is, y+2+23/ =y2 +2y 

Or, y^ — y=2' Whence, y=2, or — 1. 

The value — 1 we found before ; which shows two roots equal 
to — 1. The other value 2, put in (6), gives a;=4. If we 
take the Djinus sign in (6), we shall have 

y+2+2y=— 2/2— 2y 
Or, 2/2+5y=— 2 

Whence, y= — f =h^ ^17 



(9.) Given 
and 



2a;2 X _1 I ues of a; and y. 

The first equation can be put in this form 

The solution of this quadratic gives 



or 



Whence, ^=16, or ??-. y=16a;. Jy=^Jx. 

X \Q 

Substituting the values of y and Jy, in the second equation, 

we find 

x__^ X J_l 

8 127^ 3 



iH ROBINSON'S SEQUEL. 

The double is ^-Jjx=- 

Add 3^ to both members to complete the square, (Art. 99, 
Algebra,) then 

By evolution, ^ Jx — ^=±f 

Whence, x=4, or V > but y=16a;=64, or ^S■. 
If in the, second equation we write ^j^x for the value of y, 
and V J^ for the value of Jy, we shall find 



•*' 64 > ^^ 144' 



values of x and y. 

Transposing 2a:^y^ in the the first equation, and we have 

By evolution, x^ — ^y^ = ±(l-|-^y) (1) 

The second equation can be put in this form, 

(23,='+l )(*+!) (2) 

Taking the plus sign in (1), we can put it into this form 

x^—xy+y^=2y^+l (3) 

By the help of (3) we perceive the equal factors in (2). Sup- 
press them, and (2) becomes 

x-^y=x-\-l. Or y=l. 
This value of y put in (1), gives a?=2, or — 1. 



(11.) Given 
and 



£J^-40y^=136-y»J^'-!^ I u> find 
^ "3' Itheval- 

yyy ' y^ y J andy. 

It is obvious that the first equation can be put into this form 

a;2y2_8()y 2 =272— 2^^0:^—272 
By transposition 



ALGEBRA. 



106 



(a;2y2__272)+2y7a:2y2_272=80y2 

By adding y^ to both members, and extracting square root we 
have 

(a;2y 2_272) 2+y= ±9y 
Whence, x^'i/'^—272=e4y\ or lOOy^ (i) 

Clearing the second of the given equations of fractions, and 
reducing, we have 

x^y'^—SBxy^Se (2) 

Put 2a=35, (see Art. 106, Robinson's Algebra.) 
Then x^y^ — 2cui;y=2a-\-l 

Adding a^, and taking square root, gives 

xy — a=ztz(a-\-l) 
Whence, xy=z(2a-\-l)=36, or —1 (3) 

These values of xy put in (1), give 

64y2 =36 -36—272, or 6V = _271 
64y2 = i024, or 8y=±32. y=4, or —-4. 
These values of y put in (3), give x=9, or — 9. 
Again, by observing (1), we perceive that we may put 
100y2 = i024, or 10y= ±32. 2/=3.2, or — 3.2. 



(12.) Given 
and 



2y^—^Jx 



+ 2jy-—\Jx^ 



^Jx 



I V^+V8(2/->/^)— 4=y+l 



to find 
the val- 
ues of« 
and y. 



Put 7^2. 
Then 



[^x=P, in the first equation. 

^^ I 2P — ^v^ 
Jx^ 2 



4F^-]-iJx'F=3x 
By adding x to both members to complete the square, we have 

4F^-\.4jxrF+x=4x 

2F+Jx=zt:2jx 
Or, 2F=^x, orSjx 

Restoring the value of P, we find 



1^ ROBINSON'S SEQUEL. 



Whence, Ay^ — 16jx=x, or 9a; (1) 

From the second of the given equations, we have 



>/8(y-V^)-4=(y-V^)+l 
Squaring, Q(y-.Jx)—4=(y—Jxy+2(y—Jx)+l 

Whence, (v^J^) '— 6(y— V^) = —5 
And y—Jx—3=±2 (2) 

Taking the plus sign y — 6= Jx (3) 

Taking the minus sign y — 1= Jx (4) 

Substituting the values of Jx and x taken from (3) in (1), 
we have 

4y2_-16y+80=y2_i0y+25; or, V— 90y+225 

Whence, y=^l2±l,EI^^ or y^^^l^J^ 

^/3 5 

Taking the values of the same from (4), and substituting, as 
before, we have 

4y2— •16y+16=y2__2y+l, and 9y^—lQy-\-9 

Whence, . y=3, or :?, and y=l±>/^ 
3 6 

Substituting the values of y in (3) and (4), we have the val- 
ues of X. 



(13.) Given 
and 



^x^—Uy—U x^ ^fA 

5y+ ^ y-^^ 



to find the 
»• values of x 

and y. 

Multiply the first equation by 3, transpose, &c., and we have 



x^ ,2x Ix^.x^ y 

lSyY~^3y~4~^ 



V^!^1^14=(^-^_15y-14)-94 



Put P=Jx^ — 16y— 14 ; then we shall have 
Whence, P= ^^g. db H = ^^ , or — 9^ 



'»4ji. 



ALGEBRA. 107 

That is, a;2—15y— 14=100, or V//. 

Or, x^ = 15y+U4; and x^ = \5y+^^U^. (1) 

The second equation may be written thus, 



■Mn)-^i.-*i 



8y ' \ 3 ' 2/ >* 3y 

Uniting the fractions, and 



x^,/4x+3y\_^ I4x+3y 
Sy"* \ 6 / ^l 12y 
Dividing every term by 2y, and we have 

x^ / 4x-\-3y \ _x / ^rg-f3y y 
T62^~^\ 12y / 2y\ 12y / 

For the sake of perspicuity, put P= i — X-Jl j , then 

16y2 2y ^ 

By evolution, -^— P=0 

4y 

Whence, ^L=P^=i^±?^ 

16y2 12y 

Clearing of fractions, Zx^ = \Qxy-\-\'2.y^ 

Whence, Qar^— 48a;y=36y2 

Add 64y2 to both members, to complete the squares, then 

9a;2__48a;y+64y2 = 100^2 

By evolution, 3a; — 8y= ±10y 

Whence, a:=6y, and a;= — fy (2) 

Substituting the first of these values of x in equation (1), we 

have 

36y2_i5y=ii4 

By adding || to both members, (Art. 99, Algebra,) we shall 
have 

36y2_-16y+?.| = 114+f| = Hi^ 
By evolution, Qy — f = =h V 

Whence, y=2, or — 1|. 



108 ROBINSON'S SEQUEL. 

These values put in the first of equations (2), give 

a:=12, or — y. 
By taking the second set of equations in (1) and (2), we shall 
find other values of x and y. 



(14.) Given j ^^y — \z=^x^y — \y^ / to find the values 

and (a;2_3=a:2y 2 (x^_y\>^ ) oi x and y. 

Ans. ar=l. y=4. 
Put a;^=P, and y^= Q, and we have 

P3_3 ^PQ(^p_^Q) (2) 

Now put P=tj Q, and equation ( 1 ) becomes 
(471^+1) §«—16w(2' = 16. 
Conceive w to be a known quantity, then the last equation is 
quadratic, and a solution gives 

^3_4(2/i^ +2^+l) _ 4^ ^__ 4 

But from (2), ^^= - = - 

Put the two values of Q^ equal, and put n^ — n-\-l=E, (3) 
Then 1_ =t-^ . Whence, 2w = ^^ZI? . (4) 

But from (3) resolved as a quadratic, 

2w=l=fc7(4i?— 3) (6) 

From (4) and (5), 2E ±2i2^(4J!?— 3)=6i2— 3 

Or, ±:2EJ (4E—3) = 4JS— 3 

Put J(4E—3)=S. . 

Then ' S^:^2BS = 0. Or, S(Szt2E) = 0. 

This last equation may be verified by taking either factor equal 
to zero ; and as the first factor only gives a rational quantity, we 
take that which gives i2=f. 

By retracing, we easily find x and y. 



ALGEBRA. 109 

We now add a few unwrought examples for the benefit of those 
who may wish to test their own unaided powers in these difficult 
operations. 

None of these that follow are as severe as many of the prece- 
ding. 



(15.) Given (. /J-^~"^-f ./-^^— =2 ) to find the values 

and ( a:2_i8=;c(4y— 9) ) o^ ^ and y. 

Ans. x=6, or 3. 
y==3, or f . 

(16.) Given (,+,)_^(,rz,T) ^- 

and ( (a;2-f-3/)2_|_(.^_y) = 2x(x^-^y)-{-50e 
to find the values of x and y. 

Atis. x=5, or — ^/. 
y=3, or ~|^. 

(17.) Given ^^+^^^^^24 1 _ ^^ . 

^ ' ! a; ^y x-\-y 6 I ^^ ^^"- ^"® values 

^ 4ar2 \ 

and V^— y+^=9^(^_^-) of ar and y. 

^W5. ic=3, or Y J or t\» or jf . 
y=2, or — ^P; or I, or — '^4^. 



(18.) Given W6V^+6Vy+W^=9-Wy I ^ fi„d the 
and ( a; — y=l2 J 

values of x and y. 

^Tis. a;=16, or sjyuLP. 



y=4, or ^eV-'*, 



(19.) Given (x+JSy^—n+^x=7+22j-y^] to find the 

, i 73^71:^X7-^+2^ y values of 

and j^ V "^^ ^+^""^1::^ j X and y. 



^?w. ir=4. y=2. 



110 ROBINSON'S SEQUEL. 

(20.) Given j a;^— 2/^=3 ) 

and ( {x^+y*y+x'y''{x''—^'')^+x^-^^=32Q \ 
to find the values of x and y. 

Ans. a;=s= ±2, or ±J(—1). 
y=r ±1, or ±2V(— 1). 

(21.) Given f ^+J^+J_ Jx-^^ ^Q9_ ^ ,^ g^^ the val^ 
and I 2 r~2_-.l^ [ ^^s of iT and y. 

^«*. ar=9, or J^f^, or ^-f », or 16. 

y=4, or — V> or — V» or i. 



SECTION IV. 

PROBLEMS PRODUCING QUADRATIC EQUATIONS CONTAINING 
MORE THAN ONE UNKNOWN QUANTITY. 

The following outlines of operations, refer to problems in Rob- 
inson's Algebra, Chapter III, page 183. We pass on to the sixth 
problem, page 186, and only include those which serve to illus- 
trate brevity and elegance in operation. 

The figures in parenthesis refer to the number of the problem 
in the book. 

(6.) Let t =s the time (hours) he traveled, and r= his rate 
per hour ; then r^=36 (1) 

But if r becomes (r+l), t must become (t — 3), and then 

{r+l){t-S)^36 (2) 

Or, r/_|_jf_3r— 3=36 

ri ac36 

^3(r+l) 
Hence, ?-^+rs=12, and r=3. 

(7.) Let x= the number of children, 

and y= the original share of each. 
Then a:y=4680O (1) 



ALGEBRA. Ill 

(x—2) (y+1950)=46800 (2) 

ajy+1950a;—2y— 2- 1950=46800 
1950(a;— 2) = 2y 

Or, 975(x — 2)z=xi/=i46809 

By division, x^ — 2a;=48 x=B» 

(8.) Let x= the number of pieces. 
Then = the cost of each piece. 

X 

48a:— 51^=675 

X 

4Qx^—675x=675 
16x^—225x=225. 



(9.) Let xz= the purchase money. 

Then 12^= the cost, and 390—^^^'^= his whole gain. 
100 100 ^ 

Then 12^ : 390-12^ : : 100 : ^ 
100 100 12 

Product of extremes and means, 

^^^=39000— 104a? 
300 

— =3000— 8a: 
300 



Put a = 300 and divide by 2 ; then 



— =i5a — 4a: 
a 

x^-\-4ax=5a^ 

a:2_|_4aa:+4a2=9a2 

a:-|-2a=3a a;=a=»300. 

(10.) Put a;+y= the greater part, 

and X — y= the less part. 
Then 2a:=60, a:=30, and ar^—y 2 = 704. 

(11.) Let a;= the cost ; then 89 — a:= the whole gain. 

X : 39— a; : : 100 : x. Ans. ar=10. 



112 ROBINSON'S SEQUEL. 

(12.) Let (x — 20) = the number of persons relieved by A. 

Then x-\-20 = the number of persons relieved by B. 

1200 , c 1200 
+5=. 



a;-}-20 X — 20 

Divide by 5, and put a=240 ; then 

' « fl: « 



a:+20 a;— 20 

aa;— 20a+a;2— 400=aa?4-20a 

a:2=40a-i-400=40(a+10)=40-260 

Or, a;2=400-25 a?=20-5=100. 

Hence 80 is ^'s number, and 120 A's. 

(13.) Let x= the price of a dozen sherry 
and y= the price of a dozen claret. 

7a:+12y=50 (1) 

— = the number of dozen of sherry for 10£. 

X 
n 

-= the number of dozen of claret for 6j£. 



(2) 



By substitution, _Z2^4-12y=50 

70y-|-36y2 -f72y = 1 50y+60 • 6 

36y2_8y=300 

92/2— 2y=75. Hence, y=3. 

(14.) Let 19a:= the whole journey. 
Then x= £'s days, also his rate per day. 
Or x^ = £'s distance. 
Also, 7a;-j|-32= A's distance. 

a:2_|_7a:_j.32=19a; 

x^—12x=—S2. 

Hence, a?=8 or 4. 

And 19ir=162 or 76. 





y 


Then 


12=3+? 
X y 


Or, 


^ 10 \Qy 
36 3y+6 

y 



ALGEBRA. 



113 



If we put X for the whole journey, we shall obtain the 13th 
equation, (Art. 104.) 

(16.) Leta?= the bushels of wheat, 

and ar-4-16= the bushels of barley. 
24__ 24, 1 
~x .r+16'^4 

24a;+16 • 24=24a:+^J±l?.^ 

a;2-[-l6a;=16- 96=16- 16-6 
Put 2o=16. Then 2a-2a-6=24a2 

ar+a= ±:ba .a;=4a=32. 



(16.) A put in 4 horses, and B put in x horses. 

18 
Then — = the rate per head. 

X 



Hence, 



4»18 

X 

4»20 
x+2' 
4-18 4-20 



|-18= the price of the pasture. 
j-20= the price of the pasture. 



X 

36^ 

X 



x-\-2 ' 

:J^+1. 

x+2^ 



x=6. 



;» 



(17.) Let 4x= the price per yard, 

and 9x= the number of yards. 
36;r2^324 



x=3. 



(18.) Let 10x-\-y= the number. 



Then 

And 

From (1), 
From (2), 

By division, 
8 



xy 
10a:+y4-27=10y+a; 
10a:=(2a; — \)y 
x+3=y 

i^=2^-l 
a:+3 



(1) 
(2) 



/ ^ 



114 ' ROBINSON'S SEQUEL. 

2a;3_-5a;=3 «=3- 

(19.) Let (a;— y), x, and (a;+y— 6), represent the numbers. 

Then 3a?— 6=33, or x=lS. 

(ic— y)2=a;3__2a?y+y2 

x^=x^ 
(x^y—6y== x^ +2xy+y^-'12x^l2y+ S6 
3a;2_|^22/2— 12^— 122/+36=441 
By subtracting the value of 3a;2— 12a;+36, we have 

2y2 — 122/=64. Hence, ir*^. 

(26.) Let a;+y= the greater, and x—y= the less. 

Then (a;2— y2)(2a:2+22/2)=1248 (1) 

Or, , a;4_3^4^624 

Also, 4xy=:20 (2) 

^ 6 . 626 

Whence, y=-. y*=-i- 

a:*-^-=624 
a;* 
a;8— 624a;4 =626. Put 2a=624. 
Then x'^—2ax''+a^=a^+2a-\-l 

x'—a= ±(a+l) 
Whence, a;4=2a+l=625. a;2=d=26. 

a;=6, or — 6. 

From (2), y=l- 

(27.) Let x= A's stock. a=1000. Then a-^= B's stock. 

Observe that 780= the whole gain. 

Then 9a;+(6a— 6a;)=6a+3a; : 9x : : 780 : 1140— a:. 

Or, 2a+x : Sx : : 780 : 1140— a:. 

This proportion will produce a laborious equation to work 
through. Therefore we will try 2x to represent ^'s stock ; then 
9-2a;=18ic. (a— 2a;)6=6a— 12a?. 

18a;+(6a-12ar)=6a+6a; : 18a? : : 780 : 1140— 2a?. 

Reducing, gives us 

a^x : 3a? : : 390 : 670— a?. 
670a— aa?+670a?—a?2 = l 170a:. 



ALGEBRA. 116 

Whence, a?2+1600a^= 570000. 

a;24-1600:c+(800)2 = 1210000. 
a:4-800=1100. 

a;=300. 2^=600, ^'s stock. 
When the Algebra was first published, the 6 months in the 
problem was printed 8 months, by mistake. How could we dis- 
cover that mistake ? 

We look at the answer and see that the numbers 600 and 400, 
make the stated sum 1000 ; therefore we will assume that these 
three numbers are correct. We will now take m to represent 9, 
and n to represent ^'s time. Then the preceding proportion 
becomes 

^mx—^nx-\-7ia : ^mx : : 780 : 1140—22:. 
Also, ^mx — 9,nx-\-na : na—^nx : : 780 : 640-(-2a; — a. 
Now give to X its value 300, and to a its value 1000, and these 
proportions will give m=9, and ?^=6. 

(28.) Let «2_. half the number in the first 

Then SLx^is=^ the number in the first. 

And 4a;-|-4= the number in the second/ 

3(2a;2-j-4a:-[-4)= the number in the third, 
3(.'c2-[-2a!-f 2)-|-10=s the number in the fourth. 

Sum, ll(*'2-|-2a'+2)+10=!121, the given sum. 
Whence, a; ^-j- 2^+2= 101. 

Or, a-'2+2.r-f-l = 100. 

By evolution, x-^\= ±10, or .r=9, for the minus sign will 
not apply. Then 22'2 = 162, the number in the first, 

(31.) Let a:= the greater of the two numbers, 

and y= the less. 
Then per conditions, xy-s^^x"^ — y^ (1) 

And a-'2-|-2/2_^3_^3 ^gj 

From ( 1 ), x^ — ^y=y^ • 

Conceive y a known quantity and complete the square thus ; 

4a;2_4y.a:_j-y2_5^2 

2.r— 2/= ztj5'7/ 



116 ROBINSON'S SEQUEL. 

Or, 22:=(lrhV5)y. Let (l±V5)=a. 

Then x=^ (3) 

2 . 

Let this value of x be substituted in (2), and we have 

4 ^^ 8 ^ 

Dividing by y* and clearing of fractions, and 
2a2+8=(a3— 8)y 

Whence, y= ^— 

Buta2=6±275. aS^^iedrS^S. Therefore, 

^ 8±8V5 2\1±V5/ ^ 

This last operation may not be obvious to some ; it will be seen 
by multiplying (1±V^)' ^7 V^' ^^^* ^^' ^^^^ numerator in paren- 
thesis is ^5 times the denominator. 

To find X we must simply multiply y by |a, see (3) ; that is, 
ar=Kl±V5)>/5=i(V5±5). 



The following are not in Robinson''s Algebra, but selected from 
every source, — mostly from Bland^s Problems. 

(1.) The swn of two nwmhers is 2, and the sum of their fifth 
powers is 32. What are the numbers.^ 
Let x= one number, and y= the other. 
Then x+z/=2 (1) 

And a:5+y'=32 (2) 

As the 5th power of 2. is 32, therefore 
(x^y)^=x^-\-y^ 
That is, x^-^-dx'y+lOx^^j^+lOx'^y^-^-Bxy^+y^^x^-^-y^ 
Or, 5x*y+10x^y''+10x''y^+5xy^=0 ' 

By division, x^ -\-2x^ y-\-2xy^ -]-y^ =0 
That is, x^-\-y^-{-2xy(x-\-y)=0 

Dividing by {x-\-y), and we have 

g.2_^_^y2_^2xy=0 



ALGEBRA, 117 

Or, x'+x7/+y^=0 

From (1), x^+^xi/+y^ =4 

By subtraction, a;y =4 (3) 

Multiplying (1) by «, gives x'^-\-xy=^^ 

That is, x^—2x=z—4 

Whence, x=l±^—3 



Then y=lqr^— 3 

Here we have obtained two expressions, the sum of whose 5th 
powers is 32, but not two numbers. 

We have not so clear an idea of (1=1=^ — ^)> ^^ ^^ ^^^^ ^^ ^ 
itself. If we compare equations (1) and (3), we shall perceive an 
impossibility/; for two numbers whose sum is only 2, can never 
make a product of 4. In the same manner when a sum is but 
2, the sum of the 5th powers of any two of its parts, can never 
make 32. 

To test our quantities, we will verify (2) with them. To save 
trouble, we will put a=jj — 3, then a^ = — 3, a'^=9. 

y5 = (l— a)5 = l— 5a4-10a^ — 10a ^ +5a^— gs 
aj5-|-y5 =2-f-20a2_j_l0a* =2— 60+90=32. 

(2.) The fore wheels of a carriage make 6 revolziiions mx>re than 
the hind wheels in going 120 yards ; but if the periphery of each 
wheel be increased by one yard, then the fore-wheels will make only 4 
revolutions more than the hind wheels, in running over the same dis- 
tance. Required the circumference of each wheel ? 

Ans. Fore wheels, 4, hind wheels 5 yards. 
Let ir= the yards in the circumference of the larger wheels, 
and y= the jrards in the circumference of the smaller. ' 

Put a=120. 

Then per question, -=- — 6. ' (1) 

X y 

And _JL.=_^_4 (2) 

^+1 y-f 1 ^ ^ 

Clearing of fractions, ay=iax — ^xy. (3) 

ay-\-a-=ax-\-a — ^xy — ^x — Ay — 4. (4) 



ria ROBINSON'S SEQUEL. 

Suppressing a in both members of (4), and then subtracting it 
from (3), we have 

0=— 2xy+4;r+4y+4. (6) 

From (3), :r= -^= J^^. = ^^ 
a—Qy 12U— 6y 20— y 

From (6), 0:=?^:? 

Therefore, Hd=J^ 

y— 2 20— y 
20y— y 2 _j_2o_y = 1 Oj/2 _20y 

Whence, ll2/2_39y=20. 

If we work out this quadratic, we shall find y = 4 ; but the 
operation would be a little troublesome, because the numbers are 
prime to each other. 

In cases like these, when a practical operator is only in pursuit 
of results, he looks at the absolute term, (in this example, 20), 
and observes its factors, 2, 10, 4, 5, and conceives y to represent 
one of them ; and if it verifies the equation, then y is really that 
factor. 

I will now conceive y to be 4, and divide the first member by 
y, the second by 4 ; then 

' lly— 39=5 

Or, lly=44, or 2/=.4. 

Therefore, as this supposition verifies the equation, the suppo-' 
sition itself is truth. 

Now let. us suppose y to be 6 ; then operate as before, and 
lly— 39=4 
ny=43 

Now as y does not come out equal to 5, the supposition was not 

true. 

■c 1, 2O2/ 20-4 - 

l*or X, we have x=. ^-= =5. 

20— y 16 

(3.) A and B engaged to reap afield for ^24 ; and as A could 
reap it alone in 9 days, they promised to complete it in 5 days. 
Finding, however, that they were unable to finish it, they called in 
to assist them the last two days, in consequence of which, B received 



^ 



ALGEBRA. 119 

$1 less than he otherwise would have done. In what time could B (yr 

C alone have reaped the field? 
Let x= the number of days in which B ffould reap the field, 
and y= the number of days in which C could reap it. 
As A could do it in 9 days, for one day's work he should have 

I of the money ; and as B could do it in x days, for one day's 

work he should have - of the money. A and B then working 

X 

together one day would do --[-- o^ the work, and in 5 days they 

y X 

would do s/'l+i^ : - : : 24 : ?1?1= the number of dollars B 
\9 x/ X rc+9 

would have received had (7 not been called in. 

But as B can reap the field in x days, for one day's work he 

24 6*24 
should have — dollars, and for five day's work, dollars, the 

X X 

sum he did receive. 

Therefore, — =:1 

2^+9 X 

Whence, x''—'^lx=:^\m<^. 

Here, as we are only in pursuit of results, we try Inspection. 
We perceive that 10 for the value of x would not be large enough, 
and 20, too large ; and as 1080 terminates in a cipher, we will 
try dividing by 1 5 ; then 

«— 87=— 72. Whence, x=\b. Am, 

Also, a;=72 ; but this will not apply to the problem. 

Again, as A could do the work in 9 days, for one day's work 

he should have — dollars, and for 5 day's work, — dollars. 

9 "^9 

5*24 2*24 

B should have dollars, and C, dollars, and the sum 

15 ' ' ^ 

to the three is 24 ; therefore, 

5-24 , 5-24 , 2-24 



=24 



9 • 15 ' ^ 
Or, |-j_^_|_?=i. Whence, y=18, Ans, 



^ 



120 ROBINSON'S SEQUEL. 

(4.) Bacchus caught Silenus asleep hy the side of a full cash, 
and seized the opportunity of drinking, which he continued, for two-, 
thirds of the time Silenus would have taken to empty the whole cask. 
After that, Silenus awoke and drank what Bacchus left. Had they 
both drank together, it would have been emptied two hours sooner, and 
Bacchus would have drank only half what he left Silenus. Required 
the time in which each would have emptied the cask sepa/rately. 

Ans. Bacchus in 6 hours, and Silenus in 3 hours. 

Let a= the volume of the cask. 

a;= the time Bacchus would require to drink it alone. 
y= the time Silenus would require to drink it alone. 

Then -= the volume Bacchus drank per hour. 
And _= the volume Silenus drank per hour. 

y 

- . -^= the volume Bacchus drank ; then 
X 3 

(a — -^jz= the quantity left to Silenus ; and this quantity divid- 
ed by the volume Silenus drank per hour, will give the hours he 
employed in drinking. 

That is (a—?^\l , or (y^?^\ = the time Silenus drank. 

Had they both drank together, (J^) would express the 

\x-\-y/ 



I 



time. 




Now by the given conditions. 




3^* 3x »+y 


(1) 


And (" ''VY-_''y 
\2 Sx/a «+y 


(2) 


Reducing (2). and l ^-/^ 




Or, 3x'—2y'=5xy 




9a»— l&ry=6y» . 





2 

Whence, x=2y. 
This vahie put in (1), gives 



ALGEBRA. Ill 



3~^ 3 3~ 



(6.) A Banker has two hinds of money ; it takes a pieces of the 
Jirst to make a crown, and b pieces of the second to make the same 
sum. Some one offers him a crown for c pieces : how many of each 
kind shall he take ? 

Ans. Of the first kind l ^J^ of the second, i 4-- 

(6— a) (a— 6) 

This problem is more of a puzzle than most others, yet it is a 
fair scientific question. 

Let a;= the number of pieces of the kind a, 
and y= the number of pieces of the kind b. 
Then x-\-y=c. (1) 

As a pieces are worth 1 crowiji, one piece is worth -, and x 

a 

pieces are worth — 
a 

By a parity of reasoning, y pieces of the second are worth - 
and the worth of both together is just 1 crown ; therefore, 

?+|=l (2) 

a 

Whence, hx-\-ay=^ah 

From (1), hx-\-by=bc 

By subtraction, (a — b)y=(a — c)b. y=S^^ZZL. 

a — 5 



In 'like manner we find x=A ^'. 



Itt ROBINSON'S SEQUEL. 

(6.) A and B traveled on the same road, arid at the same time, 
from Huntington to London. At the bOth mile stone from London, 
A overtook a drove of geese which were proceeding at the rate of 3 
miles in 2 hours; and iwo hours afterwards, met a stage wagon, 
which was moving at the rate of 9 miles in 4 hours. B overtook the 
same drove of geese at the Abth mile stone, and met the same stage 
wagon exactly forty minutes before he came to the ^\st mile stone. 
Where was B when A reached London ? 

Ans. 25 miles from London. 

Let x=. the rate which A and B traveled per hour. 
Then 50 — 9,x= the distance from London where A met the stage. 

m h m 

3 : 2 : : 5 : y = ^li® hours required for the geese to travel 
6 miles. 

Then when B was 45 miles from London, A must have been 

50 — ) miles from the same place, and the distance between 



3 / 



( 

the two travelers must have been i — 5 \ miles, and the val- 
ue of this expressson is the answer demanded. 

Now let t be the hours elapsed between the times that A and B 
met the stage. 

The motion per hour for the stage was f miles. 

B met the stage / 314-— ) miles from London ; but A met it 
before, nearer to London by — miles. That is, A met the stage 

(31 -4-— — . J miles from London. We have before determined 
^3 4/ 

that A met the stage (50 — 2x) miles from London; therefore, 

31+?^— -5^=50— 2a: (1) 

^3 4 ^ ' 

Now after A met the stage he traveled in one direction, and the 

stage in another for t hours, before the stage met B, Then their 

distance asunder must have been l-^-\-tx\. But the distance 



ALGEBRA. 123 

the two travelers are asunder, has been expressed by ( — 5 ) 

Therefore, ^+to=i— — 5 (2) 



From (1), /= 


.32.-228 j.^^^ ^^ ^_40.-60 
27 ' ' 27+ 12a; 


Therefore, 


3ar— 228_40a;— 60 
27 27+12a; 


Or, 


8a;— 57__10a;— 15 



• 9 9+4aj 

Clearing of fractions, 

32a;2_228if+72a;— 513=90a;— 135 
Or, 16a;2— 1232^=189 V 

Here the obvious whole number factors of 189 are 3, 9, and 
21 ; and as we are only in pursuit of results, we will try one 
or two of them. 21 we perceive at once is too large, therefore, 
try 9 ; then 

16a;— 123=21 

16a;=144, or a;=9, a true result. 

Now because a;=9, — f — 6=25, the answer to the question. 



SECTION v. 

PROBLEMS IN" PROPORTION", AND IN" ARITHMETICAL, 
GEOMETRICAL AND HARMONICAL PROGRESSION". 

The problems contained in Robinson's Algebra are not written 
out ; they are only referred to by article, and number of the prob- 
lem, and a mere outline of the solution indicated. 

We commence with Art. 117, example 3. 

(3.) Let X — 3y, x — yy a;+y, and x-\-%y represent the numbers ; 
then 2y=4. 



124 ROBINSON'S SEQUEL. 

The product of the 1st and 4th, is 

a;2_9y2 . of the 2d and 3d, is (x^—y*). 



.a;4_9^22^2 

jc4_ioa:2y2^9y* = 176985 

9y*= 144 

a;4_40a;3 =176841 

(4.) The same notation a^ in the last example. 
2;r=8. x=4. x^—y^z=15. 

(6.) Let n= the number of days. 
Then L=\+{n—\)\=n. 

S=^{\-\-n) ^n=i the whole distance. 
Also, (n — 6)15 = the whole distance. 

«2__29^^_I30 w=9or20. 

9—6=3. 20—6=14. 

(6.) The first day he must pay l+^; i representing the in- 
terest of one dollar for one day. 

First day, \-\- i. 

2d day, 1+ 2e. 

3d day, l^- 3i. 

Last day, 1+60J. 

(2-|-6U)30= the whole sum to be paid ; but as this sum is to 
be paid in 60 equal payments, each payment must be 

^^ 1^ ^-= Ans. 81 and | of a cent, nearly. 

(7.) Let X — 3y, x—y, x-\-y, and x-\-^y represent the numbers ; 
then 2;r2+18y2=50 

2a;2-|- 2 y^=34 

16y2 = i6 y=l. 



ALGEBRA. t26 

GEOMETRICAL PROGRESSION- AIND HARMONICAL 
PROPORTION. 

(Art. 124.) 

(1.) Let X represent the mean sought. / 

18 

(2.) Let x= the number sought. Then, by harmonical pro- 
portion 234 : X : :. 90 : 144— x 
90a;=234-144— 234x 
324a:=234- 144. Hence, ar=104. 

(3.) Let x= the number sought. 

Then 24 : a? : : 8 : 4—x 

Or, • , 3 : a; : : 1 : 4—x x=3. 

(4.) Let x= the second. 

Then 16 : 2 : : 16— a: : 1 a;=8, 

(6.) Let x= the first number, and y= the ratio. 
Then x+xy-\-xy^ =nO (1) 

xy^—xz=i90 (2) 

By subtraction,, 2a;4-^y=120, or .t= 

90 
From (2), we hare x= 

-— =— L , or 4y2— 3y=10 y=9L 

(5.) Let X, xy, xy^ , and xy^ represent the numbers. 
Then ^y3 __ y2 _4 

xy-^-xy"" 1+y 3 
From this equation we perceive at once that y=2 ; then 

a:+2a;+4a:-f 8ar = 1 5ar=30 ar!= 2. 



t28 KOBINSON^S SEQUEL. 

(6.) Let X, xy, xy^, and xy^ represent the numbers^ 

a?+a-2/2 = i48 (1) 

a'y+3ry3=888 (2) 

Or, <l+y2) = 4-37 (3) 

^y(l+y') = 4-222 (4) 

Divide (4) by (3), and, y=6. 

(7.) Let x, J(xy), and y represent the numbers ; then 
«^+>/{^)+y=-14 (1) 

^And a;2-}-a;2/+y2 ^34 (2) ' 

Put a;-|-3/=5, and J{xy)=^p; 

Then a;2-|^5^-|-y^="^'^' — i^^» and equations (1) and (2) 

become s-|-jys=i4 (3) 

s2_jo2^84 (4) 

Divide (4) by (3), and we have s^—psnQ (6) 

Add (3) to (5), and divide by 2, and 5fc=10. 

Hence, . . . ^ » » v . , . . j9s=4. 

(8.) Let X, xy, xy^ , and a-y^ represent the numbers ; then 
xy^-^xy=^9.A 
xy^+x : xy^-{-xy : : 7 : 3 
Or, y^-\-l : e/^+y : : 7 : 3 

Divide the first couplet by (y-^l), and we have 
y'—y+l : y : : 7 : 3 
3y2_32^_|_3_.7y^ ^31- 32/2— .10y=t— 3. 

From this equation we have y=3, the ratio. 

(9.) Let X, xy, xy^ , and xy^ represent the numbers \ 
Then ar(l+y+y2+y3)=:y-|-l 

And «:=rV- Put (y+l)=^. 

Then j\{A+Ay^)=A 

A-^-Ay^^lOA. Ay^==9A, or » . . . .y=3. 

Hence, xV> tV> <^<^- ^^^ ^^^^^ numbers. 



ALGEBRA. tf7 

(10.) Let X, — —y and y represent the numbers ; then 

^+^+y=26 (1) 

And a?y=72 

Put a:-j-y=* ; then equation ( 1 ) becomes 

144. ^ 

s+ilZ=26, or s2_26s=— 144 s=.tl8. 

s 

(11.) Let a?, a;y, and xy^ represent the numbers ; 

Then x^y^^2\Q (0 '^k 

a;2_f-a;2y4^328 (2) ^ 

From (1) <Py=6, or x^= — 

From (2) 



1+y^ 



36_ 328 ^^9^ 82 , 

9y 4___82y2 = —9. Hence, ys=3i 

(12.) Let Xt Jxy, and y represent the numbers : then 

^+>/^+y=i3^ (1) 

{x+y)Jxy=^Q_ (2) 

a;+y=13— 7a;y • (3) 

_ 30 _ 

13 — Jxy=—=r- Hence J xy =3. 
Jxy 

(13.) Let X, — ^, and y represent the numbers ; then 

^+y=i8 (1) 

?^!=.676 (2) 

18 ^ ' 

^1=M. xy^l^ (3) 

o 

From (1) and (3), we find x and y. 



128 ROBINSON'S SEQUEL. 

(14.) Let X, xy, and xy^ represent the numbers ; then 
(^xy^ — xy) (xy — x) are the first diflferences, and 

xy^ — 2xy-\-x= 6 

xy'-\- xy-\-x=42 
Difference, Sxy =36 xy=l2 

(15.) Let a;, ^ , and y represent the numbers. If y is 
x+y 

supposed greater than x, then ( y — ^^ j i ^^ — x j are the 

1st differences, and y — . — ^-Ua;=2, the 2d differences.- 
x+y 

ajy=72. Put {x+y)=s; 

Then . s— l_if=2 

s 

s2_25+l=289 
«— 1=17. s=X'\-y=lS. 

(17.) Let a;^, xy, and y^ represent the numbers ; then 
a;a_|_a;y+y2=:3i, and ic 2+2/2 =26. 

(18.) Let X, xy, xy^ , xy^, xy^^ and xy^^ represent the num- 
bers. Then, by the conditions, we have 

x-\-xy-\-xy^-\-xy^-\-xy'^-\-xy^ = lB9=a (1) 

And xy-\-xy'^=54=:b (2) 

But equation ( 1 ) may be put into this form 

( 1+y+y ^ )^+( 1 +y+y' >y^ =« 

Or, x+xy^= -^— 

Multiply this last equation by y, and its first member will be 
the same as the first member of equation (2). Therefore, 
^^ = 6 : a quadratic from which we obtain y, the ratio. 

(19.) Take the same notation as for (18) ; then we have 
(a:-fary)+(ir+a:2/)y^ = 189— 36=:153=a. (1) 



ALGEBRA. 
And (x+xy)y''=36=b. 

Divide (1) by (2), and we have 

1+2^^^153^51^ Hence. 



y' 



36 12 



(2) 



129 



r=2. 



CHAPTER III.— PROPORTION. 



(5.) Let X and y represent the numbers ; then 



x—y : x-{-y 

x-\-y : xy 

From the first, 2a; : 2y 

ISy . Il2^ 

7 * 7 

y : lly^ 



2 
18 
11 

18 

1 



9 

77 
7, or x=yy, 

77 

77. y=7. 



(6.) Let a: and y represent the numbers. 

a;+4 : y+4 : : 3 : 4 (1) 

ar_4 : y— 4 : : 1 : 4 (2) 

From (2) we have 4x — 16=y — 4, or y=4x — 12. This value 
of y put in (1), gives 

a;+4 : 4x—S : : 3 : 4 
ar-|-4 : x—2 : : 3 : 1 

a;_^4=3ar— 6 a;=6. 

(7.) Let X and y represent the numbers ; 

Then x-\-y=l6 

And xy : ar^+y^ : : 15 : 34 

Double the first and third terms, then add and subtract, ( The- 



orem 4), and 2a?y 


x^+y' 


\ : 30 


: 34 


x'+^y+y' : 


x^—^y+y^ 


: 64 


4 


^+y 


x—y 


: 8 


: 2 


16 


x—y 


: 4 


1 



Or. 



«— y=4. 



180 ROBINSON'S SEQUEL. 

(8.) Let x= the gallons of rum. 
And j/= the gallons of brandy, 

x—1/ : y : : 100 : X 
B X — y : X : : 4 : y 

Product, (x—yY \ xy \ \ 400 : xy 

Dividing the second and fourth by xy, and 

(x—yY : 1 : : 400 : 1 
x—y : 1 : : 20 : 1, or x—y^10. 



(9.) Let x-\-y=i the greaternumber, 



And 


X — y=. the less. 


Then 


a;2_y2=:=320. 




(ar+y ) 3 =;^3_|.3^2y^3^y2 _j.y 3 




(ar_y)3 _^3_3^2y_j_3^2/2--3/^ 



(1) 



6a;2y-|-2y3== diflf. of the cubes. 
2y= difference. Cube of (2y)=8y3 
6a;2y-|-2y3 : Sy^ : : 61 : 1 

Sar'-f-y^— 244y2. 3^.2^2432^2^ 

This value of x'^ put in equation (1), gives 

80y2=320, or y=2. 



We now give additional problems. 

(1.) The sum of four whole numbers in arithmetical progression 
is 20, and the sum of their reciprocals is |f-. What are the numbers? 

Let X — 3y, x — y, ar-f-y, and x-\-3y be the numbers ; 
Then 4a;=20, and x=5. 

Affam + -4- + = — 

^ x—Sy x—y ' x-\-y x-\-3y 24 

Uniting the 1st and 4th, and the 2d and 3d, we have 



ALGEBRA. 131 

2x , 2a: 25 



Dividing by x=5, and then clearing of fractions, reduces the 
equation to 2x''—'2y''+2x^—lQi/^z=^^(x^—9y'') {x^'—y''). 

96^2_43o^2^5^4_50a;2y2_j_45y4 

Putting the value of a;^ in the 2d member, we have 

96^-2_480^- == 1 25;i;2 — 1 230y2 ^46y4 
Whence, 0=292;-— 770y--|-45y^ 

Dividing by 5, 0=29a?— 154y2_j_9y4 
Or, 0=145— 154y2_j_9y4 

Here the sum of the coefficients is the same in both members ; 
therefore one of the values of the unknown quantity is 1 ; and as 
y=l answers the conditions of the problem, we are not required 
to find the other roots. 

Hence the numbers are 2, 4, 6, and 8. 

(2.) Tke sum of six numbers in arUhm^tical 2>rog7'ession is 33, 
und the sum of their squares is 199. What are tke numbers ? 

Ans. 3, 4, 6, 6, 7, and 8. 

Let [x — y) represent the third term, and {x-\-y) the fourth 
term ; then 2y will be the common difference, and 

{x—Sy), (^•— 3y), {x—y), {x-\-y), {x-\-2>y), and {x+oy) 
will represent the numbers. 

Then 6.r=33, or 2.i = ll. (1) 

And 6a:2-|-7Q?/2zz=199. (2) 

From (1), 4a.'2 = 121, or 12^-2c=363, 
Double (2), and write 363 for \9^x^ , then we have 
363+140^2=398 
140y2=:35; y=J-. 

(3.) Find four numbers in proportion such tk^U their sum shall 
ie 20, the sum of' their squares 1 30, and the sum of their cities 
^80. Ans. 6, 9, 2, and 3. 

Let w, X, y, and s represent the numbers 



132 ROBINSON^S SEQUEL. 

Then because they are in proportion, 

wz=^xy ( I ) 

By conditions, ?tf-|-a;-)-y+2;=20=a. (2) 

And w^+x^-^-y^'+z^^^lZO, (3) 

And ^3_(_^3^y3_|.23^98o. (4) 

From (2) we have w-^-z^a — {x-\-y) (5) 

w^-^^xoz+z^ =a^—2a{x-\-y)-{-x^-]-2xy-^y^ 
Suppressing 2wz in the first member, and its equal 2xy in the 
second member, and adding (x^-\-y^) to both members, we have 

w^^x^^y''-\-z''=a''—2a(x+y)-\-2x^-\-2y-' 
That is 130=400— 2a(a:+y)+2a:2+2j^2 

Whence, a{x+y) = l35-\-x^+y^ (6) 

By cubing (5), we have 

w^-\-32vz(w-^z)+z^=za^—3a^(x-\-y)-\'3a(x'\'yy 
—{x^-\-Sxy{x-\-y)-\-y^) 
By transposing, and observing that Swz equal 3xy, we have 
(w^-\-x^-\-y^-\-z^)-\~3xy{w-^x-\-y-\-z)=a^ — 3a^(a?+y) 
^ -^Sa(x-{-yy 
That is, 9Q0-{-Saxy=za^—3a''(x-\-y)-j-3a(x-\-yy 
Dividing by 20, or by a, which is the same thing, and we have 

49+3x7j=a'—Sa{x+7/)-\.3{x+yy 
Or, 3xy=35\—3a{x-^y)-{-3{x+yy 

Dividing by 3, and expanding the last term, gives 

xy=z 1 1 7 — a{x-\-7/) ^x^ -{-2xy-\-y^ 
Or, a{x+y) = \\7-^x--\-xy-{^/ (7) 

By comparing (6) and (7), we perceive that 

iry+1 17=135 
Or, xy=lS (8) 

By the aid of (8), (6) becomes 

«2+2a:y+y2-|_135=20(ar+y)+3d 
Or, (x-{-7jy—20(x+y) = —99' 

Whence, (ar+y)— 10=±1 

Therefore, x-\-y=\l, or 9 



ALGEBRA. 13»' 

From this last equation and equation (8), we find x=9, or 6, 
and y=2 or 3. 

(4.) The sum of Jive numbers in geometrical progression is 31, 
and the sum of their squares, 341. What are the numbers? 

Am. 1, 2, 4, 8, and 16. 
Let a:, xy, xy^, xy^^ xy* represent the numbers. ^ ; 

Put a=31 lla=341 

Then x-\-xy-\-xy^ -\-^y ^ -\-^y ^ =^ ( 1 ) 

And a;2-|-ar2y2_|.a2^4_|_^2y6_|_^2y8_iia ^2) 

By the formula for the sum of a geometrical series, we have 



-X 



y-1 



(3) 



And ^''^'°~— =11«. (4) 

Dividing (4) by the square of (3), gives 

/ ylO— 1 \ / y_l y_ll 

Factoring the first fraction, 

(y^+i)(y^-i)(y-i)(y-i) _n 
(y+i)(y-i)(y'-i)(y'— 1) « 

Suppressing common factors. 






Divide the first fraction, numerator and denominator, by (y-f-1) 
and the second fraction, numerator and denominator, by {y — 1), 
and we then have 

j^'+y^+y'+y+i 31 

Clearing of fractions and reducing, 

Here we observe the same coefficients whether we begin to the 
right or the left of the expression. In such cases, divide by half the 
power of the unknown quantity. In this example, divide by y^ , 



■«v.l««^*i.'.,.j»«» 



1^ ROBINSON'S SEQUEL. 

Then lOy^— 2I2/+IO— il+i^=0 (6) 

y y"" 

Now put 21y+-=P (6) 

y 

Then 2/+^=^ 

y 21 

Squaring, y2^2+l=^ 

10_ 10P» 
2^ 441 
Comparing (5), (6), ?tnd (7), we perceive that 

441 
10P2— 441P=4410 

F^—^-=b. By putting 6=441, 

Tlien P^-lp 4-il=1^4^= W±00>. 

10 ' 400 400 ' 400 



10y^+20+^=J^_ (7) 



By evolution, F — — =J 

'' 91) M 



441-84} . 21-29 



20 ^l 400 20 

p__21 -21+21 •29 __21 • 60_105 
20 20' ~Y' 

Or, P=II?L.^=_1^ 

20 5 

Now from (6), y4--=-> <^r — - 

w2__%=^__i. y^2, or 1 
2 . 2 

To find a?, we must return to equation (3), and in place of y, 

put in its value 2, and x=\. 

Hence, the numbers are 1, 2, 4, 8, and 16. 

(5.) There are six numbers in geometrical progression ; the sum 
of the extremes is 99, and the sum of the four means is 90. Whai 
are the numbers .^ Ans. 3, 6, 12, 24, 48, 96. 



Let X, xg, xy^ ^ &c. represent the numbers. 



ALGEBRA. 136 

Then xi/^+x=99 (1) 

And ary^ -^-xy^+xy^ -[-a:y=90 (2) 

Dividing (1) by (2), gives 

Dividing numerator and denominator by (y+l)» then 

'y^+y 10 

Clearing of fractions, 

10y4_iOy3_|_iOy2_iOy_(-io=lly3_|_iiy 

Or, 10y4— 21y3_|-.l03/2_2iy_|_io=0 

This is the same equation as (5), in the preceding example ; 
therefore, as in that example, y=2. 

Now from equation (1), we have 33a:=99, or a; =3, the first 
number. 

(6.) The number of deaths in a besieged garrison atnounted to 6 
daily; and allowing for this diminution, their stock of provisions 
was sufficient to last 8 days. But on the evening of the sixth day 
100 men were killed in a sally, and afterwards, the mortality increas- 
ed to 10 daily. Supposing their stock of provisions unconsumed at 
the end of the sixth day, to support 6 men for 61 days ; it is requir- 
ed to find haw long it would support the garrison, and the number of 
men alive when the provisions were exhausted. 

Ans. 6 days, and 26 men aUve when the provisions were 
exhausted. 

Let x^= the number of men at first, 

and ji?= the amount of provisions consumed by each per day. 

By the question, we have a decreasing arithmetical series, 
whose common difference is 6*, and number of terms 8. 

For the amount of provisions at first in store, we have px for 
the first term, and {px — 42ji9) for the last term. Then the sum 
of the terms must be {^px — 168p). 

Hence, 8pa; — 168p= the amount of provisions at first. 
^px — 90p=: the provisions consumed in 6 days. 



Whence, 



2pa; — 78p=6jo-61, the provisions left. 



136 ROBINSON'S SEQUEL. 

During the 6 days, 36 men died, and 100 men were killed ; 
therefore, at the end of the sixth day but 86 men were alive. 
Now the mortality increased to 10 daily, and at the end of n 
days their provisions were exhausted. 

Here we have another decreasing arithmetical series, the first 
term 86, the common difference 10, and the number of terms n. 
The last term is, therefore, 

86— 10(7J— 1) 

First term 86 

Sum of the terms (2-86— lOw+10)?? 

This number of men would require 

(Q6n — 5n^-]-5n)p amount of provisions. 

Therefore, (91nr-5n'')p=6p'61 

Whence, Sw^— 91w=— 366 

100»2—^+(91)2=(91)2— 366-20=8281— 7320=961 
By evolution, 10/1—91= ±31 

Whence, w=6, or 12^; but the last number cannot apply to 
the problem. 

(7.) Out of a vessel cmdaining 24 gallons of pure mne, a vintner 

drew off at three successive times a certain number of gallons, which 

farmed an increasing arithmetical progression, in which the diff'er- 

en,ce hetwem the squares of the extremes was equal ^o 16 times the 

•mean, and filled up the vessel with water after each draught, till he 

fmind what he last drew off, reduced to one-sixth of its original 

strength. Required the number of gallons of pure udne drawn off 

each time. 

Ans. 12, 8, and 3^. 

Let X — y= the number of gallons first drawn. 

X = the number at second drawing. 
and a;+y= the number at the third drawing. 
By the first given condition 

Axy==\Qx. y=4. 
(a? — y), or (x — 4)= the pure wine at the first drawing. 
He then filled the cask with water, making 24 gallons of liquid, 
which .contained (24— a;+4), or (28— a;) gallons of wine. 



ALGEBRA. 137 

Liquid. Wine. Liquid. 

^28— -a; W 
Now by proportion, 24 : 28 — x : : ar : ^^ 1—=. the 

wine taken out at the second drawing. 

Therefore, ("28 — x) — i — ZZz)—z= the wine left after the second 
^ ^ 24 

drawing, which is i Li — ZI_Z, by reduction. 

For the wine taken out at the third drawing, we have the fol- 
lowing proportion ; 

24 • (24-a;) (28-a;) . . ^ , 4 . (24-a:)(28--a^)(a:+4) 
24 ' 24-24 

The number of gallons of liquid at the third drawing was 
(ar-[-4j^ but only one-sixth of it was wine ; therefore, 
(24— a ;) (28— a;) (a;-f-4) _ (a;+4) 
24-24 6 

Whence, (jj^) (24-.+4)^^ 

4-24 
Put (24— a;)=P; then 

P(P-|-4)=96 

F+SL^ ±10 
P=8 or— 12. 
That is, 24 — x=^^, or x=\Q ; the minus sign is not appUcable. 
Hence the wine at the first drawing was x — 4, or 12 ; at the 2d. 
(28— a;)a; ^^ g ^^^^ ^^ ^^^ 2^ ^+4 ^^ ^^ ^^^ answer.. 
24 6 ' 

(8.) There are four numbers in geometrical progression such that 
the sum of the extremes is 56, and the sum of the means 24. What 
are the numbers r' Ans. 2, 6, 18, 64. 

Let X, xy, xy^ , and xy^ represent the numbers. 
Then by the conditions, 

a;y_|_a;2/2 _24=3a . (1) 

X J^xy^=5Q=:la {%) 

Dividing (1) by (2), and 

y(i+y)-3 
i+y3 7 



\ 



i 



13a ROBINSON'S SEQUEL. 

Dividing numerator and denominator by (1+y), and we have 

y _3 



1— y+y^ 7 

Or, 3y2_i0y-|-3=0 

If y is a whole number it is a factor of 3 ; that is 1 or 3. It is 
obviously not 1 . Try 3 ; then 

3y-- 10+1=0, ory=3. 
This value put in (1), gives \2x=24. x=2. 

Another Solution. 

Let X, and y be the means, and ~, and ?L the extremes. 

y X 

Then ^+?^=7a (1) 

y X 

And a:+y=3a (2) 

From (1) a;3-f.y3_7(^^y ^3^ 

(2) cubed, x^+y^-\-Sxy(x+y)=27a^ (4) 

That is, 7axy-\-9axy=27a^ 

16xy=:27a^ (6) 

Square of (2) gives x^ -\-2xy-\-y^ =: 9a^ 

27a2 
4xy= — -- 



By subtraction, x^—2xy+y''= rr_ 

4 
a;— y=-|a (6) 

From (2) and (6), we readily find x and y. 



SECTION VI. 

SOLUTIONS OF EQUATIONS OF THE HIGHER DEGREES. 

We shall take equations and solve them. The most difficult 
equations in the common popular books will be selected, begin- 
ning with 

newton's method of approximation. 

(1.) Given x^-\-2x^ — 23ar=70, to find one value of x. 



ALGEBRA. ia9 

By trial we find that one value of x is between 5 and 6, nearer 
6 than 6 ; therefore, let a=5 and y=: the remaining part of the 
root. Then a:=a-|-y. 

Expand, neglecting all the terms- containing the powers of y 
after the first, and we shall have 

a:3=: a3-|-3a2y-|-&c. 
2a;2=2a2-j-4ay -\-kQ. 
—23x =— 23a— 23y 
By addition, 

In this last equation we observe Ihat a has the same powers and 
coefficients as x, and the coefficients to y may be found by the 
following 

Rule. Multij)ly each coefficient of x by its exponent, diminish 
each exponent hy unity, and change x to a. 

Now y= — it — .^ ? ~~— . Givinof a its value, 5, we have 
^ 3a^^4a—23 ° 

y=|| = .l-[-. Now make a=5.1, and substitute again in the 

preceding formula, we have a new value of y. 

2*629 

Thus y= = .03. Now make a=5. 13, and substitute 

^ 75-43 

again, aad our new value of y will be .0045784-- Hence, a-j-y 

ora.=:5.134578+. 

(2.) Given a;4—3a;2-|-75ar= 10000, to find one value of x. 
By trial we find x must be near ten. Hence, put a=10 and 
x=a-^y. Then by the preceding rule 

Now make a=10 — .11=9.89. If we have the patience to 
substitute this value for a in the equation, we shall have a new 
value to y, true to 6 or 7 places of decimals, and of course a 
value to X to the same degree of exactness. ^ 

(3.) Given 3a;*— 35a;3— 1 la;^— 14a;+30=0, to find one value 
of X. 

By trial we find that x mu;st be near 12. Let a=12, and 
x:=a-\-y. Then by the rule * 

\ 



140 ROBINSON'S SEQUEL. 

^ 12a3— 106a2_22a— 14 5338 

Hence, a;=12— .00112=11.99888. 



(4.) Given 5x^—3x^—2x=\560, to find x. 
We find by trial that one value of x is more tlian 7. Put 
«r=a-|-y, and a=7. Then by the rule 

156Q+2a+3a--5a3 _ ^_ ^ ^^^ 

15a^—Sa—2 689 ' 

Hence, ar=7.00867+ 

We give Newton's method on account of its simplicity in prin- 
ciple; it is easily understood, and can long be retained ; but its 
numerical application is laborious and tedious. 

A more modern, delicate, scientific, and practical method is 
Homer's, of Bath, England, first given to the world in 1819. The 
principle is that of transforming one equation into another whose 
roots shall be less in value by a given quantity, and again trans- 
forming that equation into another whose roots maybe still less, &c. 
The theory is fully explained in Robinson's Algebra, last two 
chapters. 

We give the following examples, commencing with quadratics. 
Some of the equations here solved are in the author's class book, 
and are numbered as in that work, pages 313 — 333. 

(6.) Given a;2-|-7a:=l 194. to find the values of x by Homer's 
method. 

We must first find an approximate value of x by trial ; but the 
inexperienced might be at a loss how to make the trial. We 
suggest this method ; separate the first member into factors thus : 

Here two factors of 1194 diflFer by 7. If the factors were equal, 
each one would be the square root of 1194. 

Now one of the fac tors is a little less than the square root of 
1194, and the other a little greater ; but we want the less factor. 

In short, the square root of 1194 is a liUte below the arithmetical 
mean between x and (x-\-l). 



ALGEBRA. 



141 



This principle will do for a guide, when the coefficient of x is 
small in relation to the absolute term. The square root of 1194 
is 34*5 ; from this we will subtract the half of 7, 3.5, and the ap- 
proximate value of X will be 31 ; therefore, r=31. a=7. 

T si 



r+s 


38 
31.2 


1194(31.231 
1178 

1600 


a+2r+5 

s+t 


69.2 
23 


1384 
21600 


a+2r+2s+^ 


69.43 
31 
69461 


20829 
77100 


The operation may 
thus 


now be 
69461) 


carried on as in simple division ; 

77100(111 

69461 ,i^ 

76390 

69461 



69290 
and the figures thus obtained annexed to the portion of the root 
already obtained. Hence, a:=31. 231111. 

As the sum of the two roots is equal to — 7, the other root will 
be— 38.231111. 



(7.) Given ic^— -2 l;r=2 1459 1760730, to find theA^alues of x. 
Conceive — 21a; not to exist ; then the value of x will be the 
square root of the absolute term ; but this term has six periods 
of two figures each, and the superior period is 21, the greatest 
square in this is 16, root 4 ; hence, x must be over 400000 ; take 
r=400000. a=— 21. 

-^+r =399979 

r+5 460000 



— a-|-2r-f-5 

s-{-t 



214591760730(400000=r 
1599916 60000=5 



859979 
63000 



5460016 
5159874 



3000= t 
200=u 



14S 



ROBINSON'S SEQUEL. 



-^+2r+25+i: 


922979 
3200 


3001420 
2768937 


60±=t) 
l=w 




926179 
250 


2324837 
1852368 






926429 
51 


4724793 
4632145 


ar=46325L 



926480 926480 

926480 
A.S the algebraic sum of the two roots must make 21, the other 
i-oot must be —463230. 

We can find the negative root directly as well as indirectly, by 
taking r minus ; then s, t, u, v, &c., will be minus. The divisors 
and quotients both being minus, their products will be plus. 

The following example is in direct contrast to the preceding. 

{a) Given x^ — 32141a;=131, to find the values of x. 

Here the coefficient of the first power of the unknown 'quantity 
is large, and the absolute term comparatively very small. The 
factors X and {x — 32141) are so very unequal, that a resort to the 
square root of the absolute term for an approximate value of x, as 
in the preceding equation, would be useless. In this, and in simi- 
lar cases, we can obtain an approximate value, by conceiving the 
absolute term to diminish to zero. Then 

«2-^32141a;=0. 

This equation will be verified by putting a;=0, and ar=32141 ; 
and from this consideration we conclude that one value of x in 
our equation must be very small, and the other, over 30000. 
Hence, put riti30000, and the solution is as follows. 

131(30000=r 



-.^ 


—2141 


—64230000 




r-H 


32000 


+64230131 


2000=* 


— a+2/--[-5 


29859 


59718000 




.+/ 


2100 


4512131 


100=< 


— <i+2r+2s-l-if 


31959 


3195900 


40=?« 


t^^u 


140 


1316231 


1=V 



ALGEBHA. 




32099 


1316231 
1283960 


41 


32271 


32140 


32140 



u» 



131 
Here we obserye that the last divisor is numerically the same 

as the coefficient to x, and the last dividend is 131, the same as 

the absolute term. 

Now if we divide 131 by 32140, we shall obtain decimal places 

in the root, and the positive value of x will be 32141 s^jIj, very 

nearly. 

The first four or five decimal places will be exactly : then the 

figures will be too large, because the divisor accurately corrected, 

will increase a little at every step. From this example, we learn 

that when we have an equation in the form 

x^ — ax= zhb, 
and a numerically greater than b, the positive value of x will be 

( «rt- ), very nearly, the value being a very little in excess of the 

true value, and if - is a small fraction, this approximate value 

of X will be sufficiently near to call it the true value. 
When the equation is in the form 

x^ +««= dr&, 
and a greater than b, then the negative value of x will be express- 
ed by — ( adz- j, very nearly, and if b is much greater than d, 
we may say accurately, in a practical point of view. 

If we take the equation x^ — ax=^b, and take a;=a-|--, and at- 

a 

tempt to verify the equation with this value, we shall have 

a2+26+^— a^— 6=5, Or, —=0 

a^ a^ ^ 

The error then, is — ; and thus we perceive that if - is a small 



^44 ROBINSON'S SEQUEL. 

fraction, the approximate value of x is really found ; but if h 
is greater than a, it can hardly be called an approximation, 

EXAMPLES. 

{b) Given x^ — 3165a:=632, to find the approidmate values 
of X. Ans. x=3165^\%%, or — gVeV 

(c) Given x' — 2178:ir= — 69, to find the approximate values 
of X, Ans. a;=2178— 2^8^ or 2ff t- 

(d) Given a;2-j-3116a;=141, to find the approximate values 
of X, Ans. x= -(31 I6+3VV6 ), or 3VVe , 

(e) Given a;^-|-591a;= — 71, to find the approximate values 
of X, Am. x= — 591-|-5Vi, or — /Jy- 

The foregoing values are so near the true values, that they 
would be taken for true values, in any practical application. 

(/) Given x^-\-'^^x= — 4, to find the values of x. 

Ans. x= — y-(-^= — 8 nearly. 
— 8 is the^ exact negative value of x, and |- is the positive val- 
ue. We get y\ for the approximate positive value. 

(ff) Given x^ — ^x= — 1, to find the approximate values of ar. 
Am. x=^ — |=T^ > the true value is 2. 

(A) Given a;^ — %^^= — 1, to find the approximate values of x. 
Ans. a:=2j^ — 2^6=^ nearly ; 5 is the true value. 
The other value is ^. 

When a and b are equal, or nearly equal, in the equation 
x^dzax=±.b, 
I it is most difficult to find the value of r, or the approximate val- 
ue of x\ 

Having now sufficiently explained the means of finding r in the 
different cases, we resume the appUcation of Horner's method of 
operation. 

(8.) Given Ix^ — 3a;=376, to find one value of ar. 



ALGEBRA. 145 

Or x^—^x=^^. TxLtx==}y. (Art. 166.) 

Then ^—^=?IA. Or, y2—3y=376X 7=2625. 
49 49 7 

In this equation we perceive that y must be more than the 

square root of 2625, that is, more than 50. Hence, put r=50. 

rs t 



T+S 


47 
52 


2625(62.766+ 
235 




99 

2.7 


275 
198 




1017 


7700 




75 


7119 




10246 


68100 

51225 

6875 



Hence x= — '- Zt=7.+ 

7 ^ 



(10.) Given x^ — y3ya;=8, to find x true to seven places of 
decimals. 

Put a;=-^ ; then a;^=-^-^, and the given equation is trans- 
11 121 ^ ^ 

formed into ^ — _^=8. 
121 121 

Or, y2_3y_9g8. .,-.% 

It is obvious that y must be between 30 and 40 ; therefore, 
r=30. 



-«+r 


27 
32 


968 ( 30. 
810 


_-a+2r+s 

s+t 


b9 
2.6 


158(2.64883626 
118 




61.6 
64 


4000 
3696 




6224 
48 


30400 
24896 



10 



146 



ROBINSON'S SEQUEL. 



62288 


660400 


88 


498304 


62|2|9|6|8 


5209600 




4983744 




226856 




186890 




38966 




37376 




1590 




1245 



346 



After the 4th decimal, the operation was carried on by con- 
tracted division, giving ^=32.64883625. 

But x= ^r of y ; therefore, ic=2.96807602. 



Put ar=i- ; then «* =-t— — , and the equation becomes 
^R (36)2 ^ 



(11.) Given 4a;^-j-^ar=i, to find one value of x. 
This equation is the same as x^-\-:J^x=^-^. 

; thena;*=J''' 
36 

(36)2~(36)2 20 

Or, y^+7y=G4.8 

It is obvious that the value of y is between 6 and 6 ; therefore 
r=s6. a=7. 



«+r 


12 


64.8 ( 5.277812946 


r+s 


5.2 


60 


a+2r+s 


17.2 


480- 


8+t 


2.7 


544 




17.47 


13600 




77 


12229 




17647 


137100 




78 


122829 




176648 


1427100 
1404384 



ALGEBRA. 

227160 
175548 



147 



17|6|614 



51612 
35108 
16504 
15788 



816 

702 

114 
Whence, ^=6.277812*946, or —12.277812946, 
And a;=0. 146605915, or —0,341050359, 



(12.) Given iX^-]--5X=^j, to find one value of x. 

Put x=l 
5 

4y_28 



2 , 4a: 28 
' 5 33 



Then 



r_+ 

25 ' 25 33 
Or, 5^2_j_4y_7_oj)^2l,21212121 

r=:3. a=4. 



u+r 


7. 


21.21212121, 


&c, (3.021186235, 


r+s 


3 


21 




a+2r+8 


10.0 


2121 




.+t 


02 


2002 






1002 


11921 






21 


10041 






10041 


188021 






11 


100421 






100421 


8760021 






18 


8033824 






10042|2|8 


626197 
602536 

23661 
20084 





3577 



148 ROBINSON'S SEQUEL. 

y=3.021 186235. x =0.60 4237 257. 



(13.) Given 116 — 3x^ — 7.c=0, to find one value of x. 
^3 3 3 



Then. 3^+!3?=ll^ 

9 9 3 

y2_j_7y=:345. r=16. a=7. 

a+r 22 345(15.40158. 

r-{s 15.4 330 

a-{-2r-^-s 37 A 1500 

s+t 40 1496 

aJ^2r+2s-\-t 37|801 40000 

37801 

2199 
1890 

309 

302 

^=15.40158. ar=5.13386. 

We will now apply Horner's method to cubic and the higher 
equations. For the theory, we must go to the class books. 

CUBIC EQUATIONS. 

The first example here, is the third on pag,e 319 of the author's 
class book. Hence, 

(3.) Given x^^2x^ — 23a;=70, to find one value of x. 

By trial we find x must be a little over 5 ; therefore, 

r=5, A=2, JB=— 23, iV=70 

B —23 

r(r+A) 35^ r 8t 

1st Divisor 12 ^ 70 ( 6.134 

r» 26j 60 



ALGEBRA. "lit 

B' 72 10000 

5(5+3r+^) ATi\ 7371 

2d Divisor 7371 \ 2629000 

»2 1 J 2276697 

B" 7543 352303000 

*(^Q-\-t)t ...4599 305649104 

3d Divisor 758899 46653896 

e 9 

B'" 763507 

615 76 

4tli Divisor 76412276 

16^ 

Common division will give three or four more figures to perfect 
accuracy. 

(4.) Given x^ — 17a;^-|-42a;=185, to find one value of x. 

Here ^=—17, ^=42, iV^=185, and we find by trial that x 
must be between 15 and 16 ; therefore, r=15. 

B 42 

r{r+A) —30] r st 

1st Divisor ~12 ^ 185 ( 15.02 

H 225 J • 18Q 

B' 207 5000000 

*(s+37-+^) __0] 4 154008 

2d Divisor 207 \ 2Q11) "845992 (407 

«2 oj 8298 

J5" 207 16192 

t{^Q+t) 7004 1 4539 

3d Divisor 2077004 1653.0 

Hence, ir=15.02407+ 

(5.) Given x^-\'X^ =500, to find one value of x. 
Here A=\, B=0, r=7. 

*Q represents the root as far as previously determined 



tSO ROBINSON'S SEQUEL. 

B 

r(r~{-A) 56^ 

1st Divisior 56 [ 600 (7.61 

r« 49J ' 392 

^ 161 108 

(Sr-\-s-\'A)s 1356 104736 

2(1 Divisor 17466 3264 

8' 36 1887181 

18848 ) 1376819 
238 1 

Sd Divisor 1887181 Continue by common division. 

1 

1889563 

(6.) Given x^-\-10x^-{-5x=2600, to find one value of ar. 
Here ^=10, B=5, r=ll. 

B 5 2600 ( U . 006 

r(r+A) ^3J ^ 2596 

1st Divisor 236 \ 4 

r^ j 2^ J 3529188216 

B' 588 470811784 

(3E-\-u)u 198036 Continue by common 

4th Divisor .588198036 division. 

(6.) Find one value of a: from 5x^ — 6x^-{-3x= — 85. 

As the result is negative, we will change the second and every 
alternate sign of the equation, (Art. 178), and find a value of x 
from the equation 5x^ -\-6x^ -\-3x=85. 

Use the formula of (Art. 194). c=5, A=6, -5=3, and by 

trial we find r=2. 

r s 

B 3 85(2.1 

{cr+A)r..,. .32^ 70 

1st Divisor 35 f 16 

or^ 20 J 9.066 

"87 5.935 

{^+C8+A)s ,_ZSb Continuing this, we shall find 

2d Divisor 90.65 the value of x to be 2.1 6399+, 

^* 5 and its sign changed will be the 

94.35 value of x in the original equa. 



ALGEBRA. '161 

(7.) Find x from the equation \2x^-\-x^ — 6a;=330. 
Here c=12, ^=1, jB=— 6, r=3. 

B ....—6 

(cr+A)r .ml »• «^ 

1st Divisor 106 f 330(3.036 

cr^ 108 J 31£ 

5'. 326 ir , 

(3ci2+cO^ 11208 97836 24 \, 

3d Divisor 3261208 2216376 

c^ 108 Continue thus. 

3272624 
In the same manner perform 8 and 9. 

(Art. 196.) Page 323. 

(3.) Extract the cube root of l-352-606-460'694'688 
For the sake of brevity, take r=ll, in place of 1. 

1st Divisor 121 rst' 

B'=^r^ 363 1-352-606-460-694-688 ( 110692. 

{pB+t)t 16525 1331 

2d Divisor. . . .3646526 21 605 460 
25 1 8 232 625 

3663075 3 372 835 594 

{ZB-\-u)u,,: . 298431 3 299 453 379 
3d Divisor...: 366605931 73 382 215 688 
81 73 382 215 688 

366904443 

(3i2+2;> 663544 

36691107844 

(4.) By the table of cubes which run to 8000, we perceive 
at once that r in this example is 17. 

1st Divisor 289 r s t 

3r2 =B' 867 6382674 ( 175.2 i 

{^RJ^s)s 2576 4913 

2d Divisor 89276 469674 

26 446376 



IdS! 



ROBINSON'S SEQUEL. 



91876 

(SB+ty 10604 

3d Divisor 9198004 

4 



9208612 



23299000 

18396008 
4902992 
Complete another divisor, then 
continue as in simple division. 



(6.) Find x from the equation a;3 = 16926.972604. 
For the sake of brevity, let r represent the value of the two 
superior digits. That is, let r=26. 



Ist Divisor.... 626 

£'=3r^ 1876 

(3r+s> 761 

2d Divisor... 188261 

8^ 1 

B" 189003 

(SM+ty 462 16 

3d Divisor.... 18946616 

36 

18990768 
7648 

4th Divisor 1899084348 



16926.972604 ( 26.16002649 
16626 



301 972 
1 88 261 

113 721604 
113 673096 



48408 000 000 
Common division. 
189.91 ) 48408 ( 2649 



37982 

10426 

9496 



931 

769 
172 



It is not important to show a solution to the remaining exam- 
ples under this article. 

From Robinson's Algebra, page 324. 



(1.) Given 


x^+x^+x^ 


—X 


=600, to find 


one vah 


By trial, we 


find r=4. 








1 1 


1 




—1 = 


=600 ( 4. 


4 


20 




84 


332 


6 


21 




83 


168 


4 


se 




228 


. 





ALGEBRA. 


1 9 


67 311 


4 
13 

17 


62 
109 


A new transformed equation is 


5^+17* 


3^109s^+31l5=168. 



IBS 



'' 3 11 — ^ 

17. 109. 311. =168.(0.4 

.4 6.96 46.384 142.9536 

17.4 115.96 357.384 25.0464 

4 7.12 49.232 



17.8 123.08 406.616 
4 7.28 



18.2 130.36 

4 

18.6 
The next transformed equation is 

t* +18.6^3 _|_130.36^2 _|_406.616^ =25.0464(0.06 
.06 1.12 7.888 24.8602 

.1862 



18.66 
.06 


131.48 
1.12 


414.504 
7.956 


18.72 
.06 


132.60 
1.13 


422.460 


18.78 
.06 


133.73 





18.84 
Several other decimal places of the root may be found by the 
following division, — the powers of u above the first being consid- 
ered valueless, 

423. ) 0.1862 ( 0.00044019 
1692 



i 1700 



1692 



800 
Hence the root is 4.46044019. 



164 ROBINSON'S SEQUEL. 

(2.) Given x^—5x^-}-0x''-^9x=2.Q, to find one value of x, 
r=0.3, found by trial. 



—6. 

0.3 

—4.7 


+0. 

—1.41 

—1.41 


+9. 
—.423 

8.677 


r 
=2.8 ( 0.3 
2.6731 

.2269 


.3 


—1.32 


—.819 




—4.4 


—2.73 


7-768 




.3 


—1.23 






—4.1 
.3 


—3.96 







—3.8 
54_3.8s3_3.96s2-|-7.768«=0.2269. 

-3.8 —3.96 +7.758 =0.2269(0.02 

.02 —.0766 —0.081 0.16364 



—3.78 —4.0366 7.677 .07336 

.02 .076 .082 



—3.76 —4.11 7.696 

The remaining figures may be found by division, thus : 

7.6 ) .07336 ( 0.00978 
676 



686 
625 



610 
Hence the approximate value of x is 0.32978. 

(3.) Given x* — 9x^ — 1 la:*— 20a;= — 4, to find one value of x. 

r 



1 


—9. 
.1 


—11. 

—.89 


—20. 
—1.189 


=-4(.l 
—2.1189 




—8.9 
.1 

8.8 
.1 


—11.89 
—.88 

—12.77 
—.87 


—21.189 
— 1.277 


—1.8811 




—22.466 






—8.7 
.1 


—13.64 






1 


—8.6 


—13.64 


—22.466 


—1.8811 (.07 &c. 



ALGEBRA. 166 



-8.6 —13.64 —22.466 =—1.8811 ( .07 

.07 —.597 —.9966 —1.642382 



—8.63 —14.237 —23.4626 — .238718 
.07 —.592 —1.0380 



—8.46 —14.829 —24.6006 
.07 —.587 



-8.39 —16.416 
.07 



-8.32 



1 —8.32 —15.416 —24.6 =—0.238718(0.009 
.009 —.0748 —.139 — .2217647 



■8.311 —15.4908 —24.639 — .0169633 
9 —.0747 —.140 



—8.302 —15M55 —24.779 
9 —.0746 



—8.293 —16.6401 
9 



-8.284 



—24.78 )— .0169633 ( 0.900684 
14868 
20953 
19824 



11290 
9912 
13780 

Whence, a;=0. 179684, nearly. 

(4.) Given x^=z5000, to find one value of a?, or we may say 
find the fifth root of 5000. 

Here all the coefficients are zero, except the first, and r=5. 



1^ ROBINSON'S SEQUEL. 









r 


1 








=6000 ( 6 


6 


26 


125 


625 3126 


5 


26 


126 


625 1876 


5 


60 


376 


2600 


10 


76 


500 


3126 


6 


75 


750 




16 


150 


1250 




6 


100 






20 


250 






5 








26 








1 26. 


250. 


1250. 


3125. =1875. ( .4 


.4 


10.16 


104.06 


541.624 1466.6496 


26.4 


260.16 


1354.06 


3666.624 408.3504 


.4 


10.32 


108.19 


584.900 


25.8 


270.48 


1462.25 


4251.524 


.4 


10.48 


112.38 




26.2 


280.96 


1574.63 




.4 


10.64 






26.6 


291.60 






.4 








27.0 








1 27. 


291.6 


1674.6 


4251.52 =408.3509(.09 


.09 


22.38 


28.26 


144.26 395.6202 


27.09 


313.98 


1602.86 


4395.78 12.7307 


.09 


24.46 


30.46 


146.998 


27.18 


338.44 


1633.32 


4542.778 



Now by common division, 



4642.7|78 ) 12.7307 ( .0028, nearly. 
9.0866 



3.64i52 
Whence, a;= 5.4928, nearly. 



ALGEBRA. 157 

It is practically useless to solve such equations as the preced- 
ing, because solutions are so simple and direct by logarithms. 

(5.) Given x^ = (-^) ' , or a:^ = ^! , to find one 



value of X, 



+1/ x^-{-2x^ + l 

a;5-|-2a;3+a;=64. r=2 

2 1 =64 ( 2. 

2 4 12 24 50 

2 6 12 25 14 

2 _8 28 80 

4 14 40 105 

2 12 52 

6 26 92 

2 16 

8 42 

_2 

10 

10. 42. 92. 105. =14(0.1 

.1 1.01 4.3 9.63 11.463 



10.1 43.01 96.3 114.63 2.537 

.1 1.02 4.403 10.07 



10.2 
.1 

10.3 
.1 


44.03 
1.03 

45.06 
1.04 


10.4 
.1 


46.1 


10.5 




10.5 

.02 


46.1 
.21 


10.52 
.02 


46.31 
.21 



100.703 124.70 

4.506 



105.209 



105.21 124.7 =2.537(0.02 

.926 2.1227 2.536454 



106.136 126.8227 .000546 

.93 2.141 



10.54 46.52 107.066 128.9637 

128.96 ) .00054600 ( 0.000004+ 
51584 
Whence, a?=2. 120004, nearly. 
By an exact solution, the last figure would be 3> in place of 4. 



158 ROBINSON'S SEQUEL. 

Observation. When we observe that the sum of the coeffi- 
cients in any equation is zero, we may be sure that unity is one 
root of the equation. 

Then we can depress the equation one degree by division. For 
example, we are sure that the equation 

has a root =1, because 1 — 7-|-7 — 1=0 ; and 1 put for x will 
neither increase nor decrease any of the terms. 

The equation x^-^-'ix^ — Ix^ — 8a?-}-12=0, also has one root 
= 1, for the same reason. 

lii Bland's problems I find the following problem, (page 426). 
One root of the equation x'^ — hx^ — ^a;-|-6=0, is 5; determine all 
the roots. Here we perceive that another root is 1 ; therefore, 
the equation is divisible by (x — 5) (if — 1), or by x^ — 6ar-|-6 ; thus 

x^^QxJ^b ) a;4— 5a;'— ar+5 ( aj^+ar-f-l 
a:4_6a;3_|_5a;2 

x^ — ^x^-\-bx 



x^—Qx-\-b 
a;2_6a?-f5 • 

Whence a:2-|-a;+l=0, and x— i(±V^^— 1)- 

To solve some of the following problems, it may be necessary 
to see how the roots combine to form the coefl&cients. 

We shall consider all the roots as positive; and represent them 
by a, 5, c, c?, e, &c. 

Then an equation of the second degree will be represented by 

a;2__^^_^j_^0. (1) 

An equation of the third degree, by 



x^-^-a 
—5 



x^-\-ab 
4-ac 



x-^abc^O> (2) 



--cb 
An equation of the fourth degree, by 



x^ 



ALGEBRA. 

rc-|-a5cc?=0. 



169 



—a 


;r3- 


-aS 


x^ — abc 


—5 




-(m 


-^bd 


— c 


_ 


-ad 


— acd 


— rf 


- 


-cb 


-—cbd 


_ 


-cd 






H 


-bd 





(3) 



Now^ let A=a+b-\-^+d. £=:a(b+c+d)-[^(c+d)-\-cd. 

C=a(bc+bd-{-cd)-{-cbd. J)=abcd. 

Then the equation of the fourth degree will become 

x*—Ax^-\-Bx^^Cx-]-D=0. (4) 

This equation multipled by (x — U), gives the representative 
of an equation of the fifth degree, as follows : 

-£JD=0 (5) 



x^—A 
—E 



a;4+ B 
\-EA 



^3_ C\x^J^ J) 
—EB\ -\-ED 



In these equations we observe that the coefficient of the high- 
est power of a: is 1 ; and that the coefficient of the next inferior 
power is the sum of all the roots, with their signs changed ; — ^the 
absolute term is the product of all the roots. 



EXAMPLES. 

(1.) The roots of the equation 

6a:*— 43a;3-j-107a;2— 108a;+36=0, 

are of the form a,b,~, ~ , find them. 
a b 

Divide the equation by 6, so that it may compare with the 
fundamental equations (3) or (4), then 

Now if the roots are of the form a, 5, _, -, we may take these 

a b 

symbols to represent the roots. 
Then a 



^+^!+^=13, and a6=6. 
^ ab 6 



That is, 6(«+5)-l-a2+52 =43. 

By adding 2a5=:12 to the last equation, we have 



160 ROBINSON'S SEQUEL. 

Whence, a-\-b=5 ; but ab=6. a=2. b=3. 
And the roots are 2, 3, f, |. 

(2.) The roots of the equation 

a;4 _i0a;3 +352:2— 50a;+24=0, 
are of the form (a+1), (a—1), (6+1), (5—1), find them. 
Here 2a+26=10. (a^— 1) (6^— 1)=24. 
That is, a+b=5. a^S^—a^— 62+1=24. 
But 2a6+a2+62==25. 

By addition, a'b''-\-2ab-Jf-l=49. 

a6+l=7, or a6=6. 

Buta+6=5; hence, a=2, 6=3, and the roots are 1, 2, 3, 
and 4. 

The roots of the following equations are in arithmetical pro- 
gression ; find them. 

1." a;3_6a;2— 4ar+24=0. 

2. x^—9x^+^3x—ie=0: 

3. x^—'6x^+llx—6=0. 

4. a;4— 8a;3+14a;2+8a;— 15=0. 

5. a;^+a;3— lla:2+9ar+18=0. 

We work out the fifth and last example ; it being the only 
difficult one. 

Let (a— 36), (a — 6), (a+6), and (a+36), represent the roots ; 
then 4a=— 1, and (a''—b^)(a^—9b^)=lS. 

That is, a^— 10a2&2_|_96^ = 18. Buta2=-i.V, «* = 2-k- 

Therefore, -1 1^+96^ = 18. 

266 16 ' 

_i___1062+14464=288. 
Or, 1446t— 1062=288— yV- 

Add j''^^ to both members to complete the square. 

Then 1446^--1062+yV_=288yV4=^Hf^ 
By evolution, 1262— y5_=2_o_3^fi|&oi 

Whence, 62=1.449209 



ALGEBRA. 161 

And 5=1.20383. But a=— 0.25. 

Therefore, a— 35=— 3.86149, log. 0.586761. 
a— 5=— 1.45383, log. 0.162515. 
a+ 5= +0.95383, log. —1.979458. 
a+35= +3.36149, log. 0.526510. 
Log. 18, 1.255244, nearly. 

The sum of these numbers is — 1, as it ought to be, and the 
product of the roots is 18. 

Negative numbers have no logarithms, because there are in 
fact no such numbers. The product of several numbers is 
numerically the same, whether the numbers be positive or neg- 
ative ; therefore, we took the logarithm of each root as though 
it were positive. The product in every case will be positive or 
negative, according as the number of minus factors is even or 
odd. 

The roots of the following equations are in geometrical progres- 
sion : find them. 

1. a;3— 7:r2+14;r— 8=0. 

2. a;3— 13a;2_|_39_^_27^0. 

3. a;3— 14ic2+56a:— 64=0. 

4. a;3—26a:2+156a;— 216=0. 
Let a, ar, and ar^ represent the three roots. 
Then a-\-ar-\-ar^=2Qy and a^r^=:9,\Q. 

From these equations we find a=2 and r=3, and the roots 
sought are 2, 6, and 18. 

Problems like the preceding are impractical, because there is 
no natural method of finding the form of roots, a priori, and to 
give the form, is nearly equivalent to giving the roots themselves. 



RECURRING EQUATIONS. 



A recnrrrng equation is one in which there is a symmetry among 
the coefficients ; — the terms which are equally distant from the 
extremes, have the same numerical coefficient. For example, 
11 



162 ROBINSON'S SEQUEL. 

is a recurring equation, for the coefficients are 1, — 11, +17, 
which recur in the inverse order 17, — 11, 1. 

Here it is obvious that x= — 1 will satisfy the equation ; and 
if we change the second and every alternate sign, then x= 1 will- 
satisfy the equation ; that is, 1 is a root of the equation 

x^+nx'-\-17x^^l7x^—Ux—l=0. 

Here the sum of the coefficients is 0, and consequently x=1 
must satisfy the equation. 

Now we arrive, at this general truth : 

A recurring equation of an odd decree will have either — 1 or +1 
/or one of its roots. 

It will have — 1, if the corresponding coefficients have like signs — 
and -|-1, if they have unlike signs. 

Hence, every recurring equation of an odd degree will be di- 
visible by (x-{-\) or (x — 1), and can thus be reduced to an equa- 
tion of an even degree, and one degree lower than the original 
equation. 

Every binomial eqtcation is also a recurring equation. 

Every recurring equation of an even degree above the second, 
can be depressed one half by the following artifice : 

Take the equation, 

x^+5x^-\-2x^+5x+l=z0. 

Divide every term by the square root of the highest power, in 
this example by ic^ ; then 

Then • (.»+^)+5(.+l)+2=0. 

Put x+l=y; thenx^+24-—=y\ 

X x^ 

Whence, y^-^oyz=0, an equation of only half the degree 
of the given equation. 

This can be verified by y=0, or y= — 5.. 



ALGEBRA. }6S 

Therefore, ar-|-_=0, or x4-^ = — 5. 

X X 

Whence, a?=±V— 1, or x=\{—b^J1\) 

Find all the roots of the equation x'= — 1=0, 

One is obviously one root, therefore divide by x — 1=0. 

Then x'-^x^-\-x''-\-x^\=^. Divide by a:2. 

x x^ 

Put :c+l=y; then x^ +2+^^=1/^ 
X x'^ 

Whence, y^-\-y — 1=0. 

y=-i+iV5, or y=-i_i75. ^ 

Put 2a=_i-(-i^5; then — (l+2a)=— ^— i^5. 

Now x-\--,=2a, and x-{- = — (l+2a) 

X X 

From the first, x=a-\-jJa^ — 1, or x=a-^Ja^ — 1 

That is, x=\{J^—l+J^^^^l0^2j5) 

Or, ^=J(^5— 1— VZTc^-SVS) 

Taking the second. 

Or, ^•=— KV^+i+V—i^+VsJ 

Given (^'^+1) (.^'- + l) (^+l)=30.r^ to find the values' of x. 
Multiply as indicated, the product is 

a; B -(-a;5 +;c ^ +22- 3 -[-a;2 -[-.r-{- 1 =30^-3 
Dividing by x^, and 

X x^ x^ 
0. (.3+^.)+(.+^)V(.+l)=30 

Put (-+;)=y. Then (.3+^)+3(.+»)=,» 



164 ROBINSON'S SEQUEL. 

Whence, y3_3^_|_y2_|_y_3Q 

Or, y3_|_y2_2y^30. 

The first attempt at solving this by Horner's method shows us 
that r is 3 exactly ; that is, y=3. 

Then ar+i =3, and rr=i(3db75.) 

The other values of y are imaginary. 

Given {x-\-y) {xy-{-\)=^\^xy (1) 

and (a:2-j-y^)(a;2^2_(_ij_.2O0^2y2 ^g^ 

to find the values of x and y. 

Multiply as indicated, then 

x''y-\-xy^-^x^y=nxy (3) 

a:4?/2_j_^2^4_j_^2_|_^2^2082;2y» (4) 

Divide (3) by xy, and (4) by x^y^ , then 

^+2/+-+-= 18 (5) 

y X 
And a;2-|-y2+_L-|-Jl.=208 (6) 

Now put P=^a;+1Y and q=(vJ^\. 

Then P_[-§=18 (7) 

And P2_|_^2^212' (8) 

From (7), P2_|.2P^_j.^2^324 (9) 

2P^=112 (10) 

(10) from (8) gives 

P~Q=±zlO 
Whence, P=14 or 4, and ^=4 or 14. 

That is, a;4-l=14, or 4. y+-=4, or 14. 

X y 

Whence, x=(7dt4jS) or (2±V3). 7j=(7^z4j3) or (2^:173). 
We conclude this subject by the following equations : 
Given 6a;5+ll«*— 88a:3— 88a;2+lla:-|*6=0, to find n^ j^qq^^^ 
This equation necessarily has one root equal to — 1 ; therefore 

we divide by (x-\-\), and we find 



ALGEBRA. 165 

Now 5x^^6x—94+--\-^=0. 

X x^ 

Or. 6(.= +^)+6(.+l)=94. 

If we put x-\--=7/, then x^-{-—=y^^2. 
X x^ 

And 5(y2— 2)+6y=94. 

^^5 5 

2 1 6y 104 

y+i=±V- y=4, or— V 
a;+l=4. Whence x=2zizjS. 

X 

1 26 
Or, x-\--=: — , whence x= ^, or — 5 ; and the five roots 

X 5 

are —1,-5, i, (2-j-^3), and (2— ^S.) 

Given -S ('^6i"ll^"^Ql^''M^^' 1- to find at least one of 
the* values of x and y. 

By division, the equations may be reduced to 

^='+^=2/+- (1) 

And 2,3+J_9(,+i) (2) 

Now put a;+-=P, and y4-_=^. 
X y 

Then a;3+J_=P3_3p^ and 2/'+— =^='— 3$. 

Substituting these results, and (1) and (2) become 

P^—3F=Q (3) 

Q3_sQ=9P (4) 

Assume F=nQ ; then (3) and (4) become 



164 ROBINSON'S SEQUEL. 

And g3_3^^9^^ 

Dividing by Q^ 

And ^2_3^9„ , (g^ 

From (5), n^Q^=i^n+\ (7) 

From (6), w^ ^2^(3^+1)3^3 (8) 
Dividing (8) by (7), gives 3n3 = l. 

Whence, 9?^=3(9)3 ; and substituting this in (6), and we 

Lave ^2^3(9)3-1-3. Or, ^=\/3( 9)^+3. 

Having the value of n and Q, we have F=^nQ. The values 
of ^ and P will give us x and y. 

^7^5. a:=^(V3+3)^'+(V3— 1)^ 



SECTION VII. 
mDETERMIIf ATE ANALYSIS. 



Preliminary to this subject it is proper to call to mind a few 
facts in the theory of numbers ; for the Indeterminate and Pio- 
phantine analysis is but an application of that theory. 

(1.) The sum of any number of even numbers is even. 

(2.) The sum of any even number of odd numbers is even* 

(3.) The sum of an even and an odd number is odd. 

(4.) The product of any number of factors, one of which is 
even, will be an even number ; but the product of any odd num- 
ber is odd ; hence, 

(5.) Every power of an even number is even, and every 
power of an odd number is an odd number ; hence, 

(6.) The sum and difference of any power and its root is an 
even number. For the power and its root will be both even, or 
both odd, and the sum or difference of either, in either case, is 
an even number. 



ALGEBRA. 187 



PROPOSITIOJS-S. 

1 . If an odd number divide an even number, it will divide the 
half of it. 

Every number is either odd or even — even numbers are in the 
form 2w, and odd numbers are in the form 9.n'-\-\. Now by- 
hypothesis, let 

-=0 and q a whole number. 

Then 27z=^(2//+l) 

It is obvious that one factor in the second member is odd, 
therefore the other factor q, must be even, otherwise the product 
2w could not be even ; hence, q may be expressed by 2q\ and 

2n=2g'(2n4-l). Then —~ ==q\ and the odd number 

^ ^ ^ ^ 2n+l ^ 

(2n'-\-l) divides half the even number 2n, which was to be dem- 
onstrated. 

2. ff a numher p divide each of two numbers a and b, it mil 
divide their sum and difference. 

a b , 

_z=q, _=g . 

F P 

That is q and q\ the quotients, are whole numbers by hypoth- 
esis. Now by addition we have ^ ' =q-\'q', and by subtrac- 

P 
.. a — b , 

tion, — —z=q — q. 

P 
But the sum of two whole numbers is a whole number; there- 
fore, ( ^i- j is a whole number : and it is obvious that^ ) 
is also a whole number. Q. E. D. 

3. If two members are prime to each other, their sum is prime to 
each of them. 

Let a and b be the two numbers prime to each other, (a+^) 
their sum, is prime to each of them. 



168 KOBINSON'S SEQUEL. 

For by the last proposition if («+&) and a have a common 
divisor, their difference b must have the same divisor ; but b is 
not divisible by a by the supposition ; therefore, if two numbers, 
&c. Q. E. D. 

Corollary. In the same manner it may be demonstrated that 
if a and b be prime to each other, their difference (a — b) will be 
prime to each of them. , '-^^L 

4. If two numbers arej^rime to each other y their sum and difference 
will have the common measure 2, hut no other, or their sum and dif- 
ference mil be prime to each other. 

Let a and b be prime to each other, their sum is (a-|-6), and 
difference, (a — b) ; and if these numbers have a common divisor, 
their sum 2a and difference 25 will have the same ; but the only 
common divisor to 2a and 26 is 2. 

If one of these numbers, a or b, be even, and the other odd, 
then (a-j-5) and (a — b) are both odd and prime to each other. 
For if («+5) and (a — b) are not prime, let them have the com- 
mon measure n. Then by proposition 2, n will be the common 
measure of their sum and difference ; that is, the common meas- 
ure of 2a and 25 ; but the only common measure of these num- 
bers is 2 ; therefore, (a-|-5) and (a — b) are prime to each other, 
or have the common measure 2. Q. E. D. 

5. If two numbers a and b be prime to each other, b being 
the greater, then b may always be represented by the formula 
|)=;aq-|-r, in which r is less than &, and prime to it. 

The formula arises from the actual division of b by a, the re- 
sult is q, the integer quotient, and the remainder, r ; that is, 

a a 

Multiplying this equation by a, gives the formula ; r is neces- 
sarily less than a, if we suppose q to be the greatest quotient. 

We are i\ow to sbow that r and a are prime to each other. If 
they are not prime to each other, they have a common measure. 



ALGEBRA. 169 

Let us suppose a common measure and reduce the fraction - by 

a 

it, givmg - ; then the equation becomes 
a 

a a a' 

But by hypothesis the fraction - is irrec?wa6^^. Yet admittinor 

a ° 

a common measure to -, we have the reduced fraction ^~r^_ 



which is absurd ; therefore, a commom measure to - is inadmis- 

a 

sible, and r and a are prime to each other. 

6, If ayiy niimher he prime to each of two others, it will be prime 
to their product. 

Let c be prime to both a and b ; then we are to show that c is 
also prime to ab. * 

By the hypothesis _ is an irreducible fraction ; multiply this 
fraction by h, and we shall have — .m 

Now if this last fraction is reducible, some common measure 
must exist between b and c ; but by hypothesis there is none ; 
therefore, ab and c are prime to each other. Q. E. D. 

Corollary 1. If a^=b, then -1-= — ; and if c is prime to a, 

c c 

it is prime to a^ , a^ , and any power of a. 

Corollary 2. If c is prime to any number of factors as a, 
b, d, e, &c. it will be prime to their product. 

7. If two numbers, a and b, be prime to each other, then mb 
divided by a, and m'b divided by a, will have different remainders 
for all values of m less than a. 

Let us admit that the two operations in division will produce 



% 



170' ROBINSON'S SEQUEL. 



the same remainder, — then by performing the division we shall 
have 

mh . r 

— =9'+- 

* a a 

And — = q-\-- 

a a 



By subtraction, -( m — vi j=Q — g' 

•Or, J= g-g' 



am — m 

But by hypothesis, . is irreducible, at the same time (m — mf) 
a 

is less than a, which is absurd ; therefore, the two divisions can- 
not have the same remainder. 



8. If two numbers, a and b, be prime to each other, the equation 
ax — by= ±1, is always jwssible in integers. That is, positive in- 
tegral values of x and j mag be found which will satisfy it. 

By the preceding proposition — =q-\-- , and as r may be of 

a a 

any value less than a, according to the magnitude of m, r=a — 1 
is possible. Whence, 

nih=aq-\-a — 1 
Or, mb-\-\ = {q-^\)a 

Now let y=w, and a?=(5'-|-l) ; then - ^ 
by-\-\=ax 

Or, ax — hyz=\ 

But w is a whole number, therefore its equal y is a whole 
number, and ($'+1) is a whole number, and consequently its 
equal a? is a whole number ; that is, ax — by= 1 is possible, x and 
y being whole numbers. 



ALGEBRA. 171 

We now come more directly to the indeterminate analysis. 

For a perfect and definite solution of a problem, there must bo 
as many independent equations as unknown quantities to be de- 
termined ; and when this is not the case, the problem is said to 
be indeterminate. 

For instance, a;-f-y=20. Here x may have any value what- 
ever, and ^he equation will give the corresponding value toy, 
and the number of solutions may be infinite; but if we restrict 
the values of x and y to whole numbers, then only 19 different 
solutions can be found; for x may be equal to 1, or 2, or 3, &c., 
to 19, and y will equal the remaining part of 20. 

In some cases, the number of solutions is unlimited or infinite. 
ax — hy=c, represents a general equation of the kind, and a solu- 
tion gives x= ' "^ in which y may be any whole number what- 
a 

ever that will make (c-(-5y)=a, or any multiple of a ; but num- 
berless such values of y may be found, and consequently number- 
less values of x. 

N". B. Such equations are generally restricted to the least values 
of X and y. 

Equations in the form ax-\-by=c, may be very limited in the 
number of their solutions, — may have only one solution, or a 
solution may be impossible, when x and y are restricted to whole 
numbers. 

A solution gives x=-^ ^ . 
a 

Now if c is very large, and b and a small, y may take a great 

number of integral values, before the numerator becomes so small 

as not to be divisible by a. 

If we make y=l, and then find that i \ is a proper frac- 
tion, a solution is impossible in integers. 

The equation ax-\-hy=^c is always possible in integers, when e 
is greater than {ab — a — b), and a and b prime to each other. 

The equation ax-\-by=c, is possible sometimes, or rather with 
tome numbers, when c is less than (a5 — a — b). For example, 
7a:-{-13y=71, is impossible in integers, because (7-13 — 20) is 



172 ROBINSON'S SEQUEL. 

not greater than 71. But 7x-|-13y=27, is possible, giving rr=2, 
and y=l. ' That is, 7x-\-13i/ iput equal to any whole number 
greater than 71, will admit of a solution in integers; and put 
equal some numbers less than 71, will admit of a solution. 

In all these equations a and b are supposed to be prime to each 
other. If they have a common divisor, that same divisor must 
divide c, .or a solution is impossible in integer numbers. 

For if «ir4-Jy=-, it is obvious that ax is a whole number, also 
n 

by is a whole number, and the sum of two whole numbers can 
never be equal to an irreducible fraction, as the preceding in- 
dicates. 

For a particular example, 6a;-|-9y=32, is impossible in whole 
numbers. Dividing by 3, and 2.c-l~%= V- ^^ ^ ^^ ^^ ^^ ^ 
whole number, 2a; must be a whole number, and 3y must also be 
a whole number ; but it is impossible for any two whole numbers 
to be equal to \^ . 

In cases where solutions are possible, our rules of operation 
rest entirely on the following facts : 

1st. A whole number added to a whole numtber ^ the sum is a whole 
number, 

2d. A whole number taken from a whole number, the remainder is 
a whole nurriher. 

3d. Multiply a whole number by a whole number, and the product 
is a whole number. 

EXAMPLES. 

( 1 .) Given 3a:-j-5y=35, to find the values of x and y in whole 
numbers. 

35— 5y 

3 

Because x must be a whole number, the fractional form ^ 

3 

must be a whole number, or (11— y)+ — ~- must be a whole 
number. But (11 — y) is obviously a whole number ; therefore, 
the other part, ( -ZIJ^. ) must be a whole number also. 



ALGEBRA. 173 

Again, as y is a whole number, -Z is in fact a wliole number, 

o 

which added to { -Zl-^ ), and the sum is -^-, a whole number. 
V 3 / 3 

In this last expression the coefficient of y is reduced to 1 under- 
stood, and the operation had that end in view. 

Put this last expression equal n ; that is any wliole number, or 
rather some whole number. 

^+l=n, 
3 

Or, y=3^^— 2. 

In this last equation we can take n=^0, 1, 2, 3, <fec., as far as 
such substitutions will correspond to the values of x. 

If we take w=0, then y= — 2, an inadmissible result ; for we 
demanded positive values. Therefore, we take 7^=l, then y=l, 
and a;=10. If 7i=2, then y=4, and a?=3. If ?^^3, y=7, and 
a;=0. Hence the last is not a full solution, and the equation only 
admits of two solutions ; namely, .r=10 or 3, and y=l or 4. 

(2.) Given 2>5x — 24y=68, to find the least values of x and y 
in whole numbers. 

We require the least values, because an unlimited number of 
solutions may be found. 

From the equation, x=^^'Ay=.l+^l±l^ . 
33 • ' 35 

Hence, '" -= some whole number: but — •^= some whole 
35 35 

number ; therefore,, by taking the difference of these two whole 

numbers we have — ^ — = some whole number. 
35 

Three times a whole number is a whole number ; therefore, 

33y— 99 33y— 29 ^ , , , \ 

— = — — 2= some whole number.* 

35 35 

— t. =iwh. Also, — ^-=iwh. 

35 35 

*Subsequently we shall put wh to represent the phrase, some whole number. 



174 ROBINSON'S SEQUEL. 

Whence, ^^y^^lyi=??.^^y+^=wh. 

35 35 35 

\ 35 / ^ "^ 35 

Having thus deprived y of its numerical coefficient, we may 

put ^-"t^-?=w. Whence, y=35w— 32. 

Taking w=l. y=3, and x= — 3" =4, the least possible 

35 

values of x and y in integers, as was required. 

The next values are ic=28, and y=38. 

(3.) A man proposed to lay out $500 for cows and sheep ; the 
cows at the rate of '^ 17 per head, and the sheep at ^2. How iimny 
of each could he purchase ? 

Let a;= the number of cows, and y= the number of sheep. 

Then 17a:-|-2y=500, is the only equation that can be obtained, 
and X and y must be whole numbers by the nature of the prob- 
lem — he could not purchase a part of a cow, or a part of a sheep. 

To find the least number of cows, transpose 17a;. 

Then y=250— 8a;— ^. 

Now as X and y must be whole numbers, _ must be a whole 
^ 2 

number, and the least number corresponding to x must be 2 ; 

corresponding to this, y=233. 

Under all suppositions, the number of cows must be divisible 
by 2. 

Now if the object was to purchase as many cows and as few 
sheep as possible, we would transpose the other term thus : 

:.=522=:?^=:29+ZlI?^ 
17 ^ 17 

Whence, !=I?^=im,'A. Or, ^Zl^l^wh ; to this add 

17 17 

III, and ^^==3+^±^ 
17 17 ^ 17 



ALGEBRA. 176 

Therefore JA^=zn, or y=\ln — 6. 

, 17 

Put w=l, then y=12, the smallest number of sheep. The 
corresponding value of a:=28. 

The number of cows may be any one of the even numbers 
from 2 to 28. 

(4.) A man wished to spend 100 dollars in cows, sheep, and 
geese ; cows a^ 10 dollars a piece, sheep at 2 dollars, and geese at 
25 cents, and the aggregate number of animals to he 100. How many 
must he purchase of each ? 

Let x=. the number of cows, y the sheep, and z the geese. 

Then 10a;+2y+-=100. (1) 

And a:+y4-2=100. (2) 

Clear equation (1) of fractions, and 

40a;4-8y-(-s=400. 
x-\. ^+0=100. 

39a;+72/ =300. 

ir= £l=7-[- — ^ = a whole number. 

39 ' 39 

Or, 6( — IZIJl )= H — ^= a whole number; add — ^ and 

\ 39 / 39 39 

4H-135 ^^ 40y+1350 H:24^ ^^^^^^ ^^^^^^^ 

39 39 '^ ~ 39 

Therefore, y+24_^ ^^^ y_39p_24_i5, 

09 r 

This value of y, gives ar=5. Hence, 2=80. 

If we take ^=2, we shall have y=z5A\ then a: will come 
a minus quantity, an inadmissible circumstance in any problem 
like this. Therefore, 5 cows, 15 sheep, and 80 geese, is the only 
solution. 

(5.) A person spent 28 shillings in ducks and geese ; for the 
geese he paid 4s. Ad. a piece, and for the ducks, 25. Qd. a piece. 
What number had he of each ? 

Let a;= the number of geese, and y the number of ducks. 



% 



176 ROBINSON'S SEQUEL. 

28-12. 



Then 52ar+302/=28-12. Or, 26x-\-l52/=:168. 

, 3— lire 



15 



\ Whence, =wh. But =wh. 

16 15 

By addition, C-^+^)4=wk. l^Hd?=.+f+15 
^ \ 15 / 15 ^15 

_JL — z=n. x=15n — 12. a;=3, when w=l. 
15 

Then ?/=8— 2=6. When n=2, x=lQ, and y=—20. But 

this is inadmissible; therefore, x=3 and y=6, is the only possible 

solution. 

(6.) Divide the number 100 into two such parts that one of them 
may be divisible by 7, the other by 11, 

Let 7x= one part, and lly the other part. 
Then 7a:+lly=100. 

_100— 7a;__ 1— 7a; 

^ IT"" """Tr" 

11 ^11 11 U ' 11 

^+?=?i. x=lln—3. 
11 

Now put ?z=l, then a;=8 and y=4. 56 and 44 are, therefore, 
the required parts. 

(7.) Mnd the least number which being divided by 6 shall leave 
the remainder 3, and being divided by 13 shall leave the remainder 2. 

Let iV= the number sought, and x and y the quotients arising 

from the divisions. 

•^V— 3 . JV—2 

Then =x, and =y 

6 13 ^ 

Whence ^^=6x+3, and iVi= 13y+2. 

Consequently, 6z-\-l = lSy. 

6 6 



ALGEBRA. HT 

Then ^~-=n, or y=z6n-\-l 

For the least values of y we must take w=0, then y= 1 and 
x=2. ButiV=6;r+3=15. V^ 

We may determine iV" more directly without the aid of x and ft 
y, thus : 

. = some whole number ; also, =wh. 

6 13 

As iV does not contain a coefficient to be worked off, we may 
7\r 3 

put =^, and iV^=6p-["2 i^ which any integral value put for 

6 

J) will satisfy the first whole number, but the other must be sat- 

isfied also ; then put the value of N in ; that is, 

^ 13 

^^ I-= some whole number. 
13 

13 13 13 13 13 

Whence, p=1i3q-\-2. Assume 2'=0, then ^=2 and 
JV=6i?+3=15, as before. 

(8.) What number is that which being divided by 11, leaves a 
remainder of 3, divided by 1 9, leaves a remainder of 5, and divided 
by 29, shall leave a remainder o/" 10 ? 

Let ^y be the required number, and x, y, and z the several 
quotients, and of course they must be whole numbers. 

Then llar+3=iV; and 19y4-5=iV, and 292+10=iV. 

Hence, ir=??!±I, and x='^?li^. 19^=29^+5 

29^+5^ . lOH-5 

19 ~ 19 * 

200+10 , 0+10 

! — =zwh, or -J — =n. z=19n — 10. 

19 19 

Any integer written in place of n will give integer values to z 
and y, but x, or JLL must be a whole number also. 

Hence such a value of n must be found as will make 
29(19w— 10)+7 , 



m ROBINSON'S SEQUEL. 

Or, 561^290+7^ 651.^-283 ^^^^ n-8^^^ 
11 11 ^ 11 

^ o 

Whence, . =p. w= 11^+8. 

For the least value of n put^=0 ; then w=8. 
But 2=19n—10=19'8— 10=162— 10=142. 
And iV^=29- 142+10=4128, the number required. 

(9.) Required the least nuv}her that can he divided by each of the 
nine digits, without remainders. 
Let xz=i the number. 

Then -, -, -, -, -, -, -, -, must all be whole numbers. 
2 3 4 6 6 7 8 9 

Now if we make - a whole number, _ and _ , the double and 
8 4 2 

C|[uadruple, will be whole numbers of course. Also, if - is a whole 

number, — will he a whole number. 
3 

Therefore, we have only to find such a value of x as will make 

_, _, _, _, -whole numbers. -, may also be cast out, on 
9 8 7 6 5 6 -^ 

consideration that 6=2-3, and 2*3 are factors, one of 9, the 

other of 8, in the preceding expressions. 

Hence we have only to find such a value of x as will make each 

of the fractions -, _, -, and -whole numbers; and as these 
9 8 7 6 

denominators contain no common factors, their product is the 
least number that will answer the condition. 
Whence, ir=72 • 36=2520. 

(10.) A market woman has some eggs, which when counted out 
by twos, threes, fours, and fives, still left one ; hut when counted hy 
sevens, there was none left. Wliat was the least possible number of 
eggs she could have had ? Ans. 301. 

X——\ X 1 X 1 X 1 1 X 

This problem requires , , , , and _ to be 

whole numbers. 



ALGEBRA. 1^9 

j is a whole number, / j , its double, must be a 

whole number of course ; hence, we have only to make 

, .JUL- , , and -- whole numbers. 

3 4 5 7 

Put the first expression equal to the whole number n ; then 

a:=3w-[-l. This will satisfy the first expression ; but the others 

must be satisfied also; therefore, substitute (3^-|-l) for x in 

them. 

Then — , — , and — X-, must be whole numbers. 
4 5 7 

VS/i 7 6w . n ^ n ■, 
5 5 ^5 5 

That is, n=bp ; and substituting this value of n in the other 
two expressions, we have 

— ^ , and ^~^ , to be made whole numbers. 
4 7 

]^P+lz=2pJf-Pj^ =:wh. Whence, put ^±1=^. p=lq--\. 

15/? . 
Finally, this value of p put in — ^ , gives 

4 

im , which must be made a whole number 

4 

before we can be sure of a result which will satisfy all the con- 
ditions, 

l^^~^=(26^-3)+?=?=...A. 

Whence, 'Zl~-?=/. y=:4«:+3. 

For the least value of q, put /=0 ; then $-=3. 

Butjt?=7^— 1=20. w=5p. Then?*=100. .i-=3;i-f 1=301. 

(11.) Required the year of the Christian Era in which the solar 
cycle was, or will be 15, the lunar cycle 12, and the Roman Indiction 
12. 

N. B. The operator must of course know the exact import of 
these terms, and the facts in the case, before he can be required 
to solve the problem. 



180 ROBINSON'S SEQUEL. 

The solar cycle is a period of 28 years, at the expiration of 
which, the days of the week return to the same days of the 
month, (provided a common centurial year has not intervened.) 

The first year of the Christian Era was the tenth of this cycle ; 
therefore, we must add 9 to the year and divide the sum by 28, 
and the remainder will be the number of the cycle. 

The lunar cycle, or Golden number, as it is sometimes called, 
is a period of 19 years, after which the eclipses return in the same 
order as in the previous 19 years. The first year of the. Christ- 
ian Era was the second of this period ; therefore, we must add 1 
to the year and divide by 19, and the remainder is the year 
of the lunar cycle. 

The Roman Indiction is not astronomic. It is a period o€ 15 
years, the first of our Era being the 4th of the Indiction ; there- 
fore, add 3 to the year and divide by 15, and the remainder is 
the Indiction. 

The reader is now prepared to solve this or any other similar 
problem. 

Let X represent the required year ; then the problem requires 
that 

a;4-9— 15 a;-fl— 12 a:-f-3— 12 

28 ' 19 ' 15 

/^ g ^ ] \ /J 9 

should be whole numbers ; that is, , , , must be 

28 19 15 

whole numbers. 

The first expression will be satisfied by putting it equal to any 
whole number n\ then a;=28w-f-6. 

But the other two expressions must be satisfied also. 

Therefore ^^MltU and ^Mlt:^ 

19 15 

Or, '^''^n—b ^^^ 28?i— g ^^^^ ^^ ^j^^j^ numbers. 

19 15 

Or. ^^~^ and ^^^~^ must be whole numbers. 

19 15 

157^_13n-3_27^+3_^^,;^ 7(2n+3)_^^^ 

15 15 15 15 

15n_ 14n+2 1_^^,;^ n-21 

16 16 * 16 



ALGEBRA. 181 

Whence, n=15p-\-21. Every expression is now satisfied, 
I — 
19" 



except ^ ; to satisfy this, write in the value of n ; then 



9(15;.+21)-5 _,^ 
19 

19 ^^ ^ 19 

Whence, ^+^i=t.A., and ^5H^?2=^+6+^-H6 
19 19 ^ ^ 19 

19 ^ ^ ^ 
We canhot take q less than 1 . The least value of p will then 
be 3. But ?2=15p+21=66. 

a;=28?i-|-6=28-66+6 = 1854. 
If no interruption was to be made by the centurial years, 
the coincidence of these cycles would not occur again until the 
year 9834, which we find by making q=2. 



SECTION VIII. 

TO DETERMINE THE NUMBER 'OF SOLUTIONS AN EQUATION 
IN THE FORM AX-{-BY=C WILL ADMIT OF. 

An equation in the form ax — hy=c, will admit of an unlimited 
number of solutions, because x and y can increase together; but 
not so with equations in the form ax-[-by=c ; for in them an in- 
crease of X will cause a decrease of y, and an increase of y, a 
decrease of x; but neither x nor y are permitted to fall below 1. 
If c is very large in relation to a and b, the equation may have 
a great number of solutions, and we are now about to show a 
summary method of determining the solutions in any given case. 

The equation ax' — hy'=\ is always possible in wjiole num- 
bers, (Prop. 8, sec. vii;) therefore, c times the equation is also 
possible ; that is, 

acx' — cby'z=c, is possible. 

But ax-\-by=^c, is a general equation. 

Put these two values of c equal to each other, then 



182 ROBINSON'S SEQUEL. 

(xx-{-byz=zacz — cby' ( 1 ) 

From (1), x=cx—(^yjh\b. (2) 

For the sake of perspicuit}-, put ^ "1"^=^. 

a 

Then (2), becomes x^cx' — hn (3) 

Multiply (3) by a, and substitute the value of ax in (1), 
Then acx — abm-\-by=acx' — cby' 

Whence, yz=iam — cy' (4) 

From (3), we find 7^=^:-— _ (6) 

h b 

From (4), we find m=^]^+l (6) 

a a 

Equations (5) and (6) show that m must be greater than ^ 

a 

and at the same time less than — 

b 

Therefore, the limits to m are found. 

Now let us observe equations (3) and (4) ; rr must be a whole 
number, and as c, x, and b are whole numbers, m must be taken 
in whole numbers, and it may be any whole number between 

a b 

The number of solutions will, therefore, correspond with the 

difference between the integral parts of the fractions — and ^ 

h a 

except when ^ is a whole number, in that case x becomes b, and - 
/ b b 

must be considered a fraction, and rejected. If, however, we in- 
tend to include among^ the integral values, this precaution need 
not be observed. 

EXAMPLES. 

(1.) Required the number of integral solutions of the equation, 
7a;+9y=100. 
' First find the least value of x' and y' in the equation 
7a;'— 9/= 1. 



ALGEBRA. 18S 

The result will be x'—4. y'=3. 

Then -'=l?2:!=44l 'l='^^,=42l 

b 9 9 a 7 7 

Disregarding the fractions, the difference of the integral parts 
is 2, showing two integral solutions to the equation. 

If we had taken the difference between ^^^ and ^^, thus ; 
B|ofli__2 7.oo_i|_i^ ^e might have come to the conclusion that 
the equation would admit of only one integral solution. 

The integral difference in this case is not the difference of the 
integrals. 

When the fractional part of — is not less than the fractional 

b 

part of —, but equal or greater than it, we can find the number 
a 

of solutions by taking the difference, thus — — ^=^i^f yj. 

b a ab 

But ax' — by'=\ ; therefore, — — Jl.= — 

b a ab 

Whence we conclude that — will in this case show the num- 
ab 

ber of solutions. — In all cases it will be the number, or one less. 

(2.) What number of integral solutions will the equation 
9;5+13y=2000 admit of? Am. 17. 

9-13=117)2000(17. 

That is, we are sure of 17 different solutions, and there may 
be 18. 

The equation bx-\-9y^=40 admits of no solution in whole num- 
bers, c, 40, is not divisible by 5* 9=45, that is, no unit in the 
quotient. Yet the equation 

5a;-|-9y=:37, admits of one solution. 

The auxiliary equation 5x — 9y'=\, gives a;'=2, and y'=l. 



Therefore, ^^J^=^. ^=?Z=7i 



■2. 

b 9 ^ a 5 '* 



Here the difference of the integrals is 1, indicating one solu- 
tion. In fact x=2, y=S. 

How many solutions will the equation 2x-\-5g=40 admit of? 



184 ROBINSON'S SEQUEL. 

The auxiliary equation, 2a;' — 6/=l, gives x'=S, y'=l. 

—=24. ^=20. Or, 4 solutions. 
^ b a 

But observe that — in this case, is a complete integral, 24 ; 
b 

according to previous considerations, we must deduct one, and 

the number of solutions are 3, as follows : x=5. 10. 16. y=6. 

4. 2, and no other solution can be found. 

(3.) What number of solutions in whole numbers can be found 
for the equation 3x-\-5y-\-7z=100. 

As X and y each cannot be less than one, z cannot be greater 
than ^^^^^~^=13}. That is, z cannot be greater than 13, in 
whole numbers. Now suppose 0=1, and the equation becomes 
3a:-|-5y=93. 

The number of solutions for this equation, found as previously- 
directed, is 6. That is j x = 26. 21. 16. 11. 6. 1. 
( y=z 3. 6. 9. 12. 15. 18. 

Now X and y can make these six changes, and z be constantly 
equal to 1, and satisfy the primitive equation. 

We may observe here that x diminishes from one solution to 
another by the coefficient of y, and y increases by the coefficient 
of X, but this is not a general principle. 

Take 2=2, and the equation becomes 3a;-|-5y=86. 

This equation has also 6 solutions, z being through all the 
changes of x and y equal to 2. 

Now take s=3, then the original equation is 3x-\-5y=^lQ, This 
equation has five solutions. 

Now take 0=4, then 3a;-|-5y=72. This equation has four 
solutions. 

Take 0=5, then 3a;-[-5y=66. This equation has four solutions. 

Take 0=6, then 3ar-}-6y=58. This equation has four solutions. 

In this manner, by taking equal to all the integers up to 12 in 
succession, we find 41 solutions to the primitive equation. 

(4.) Required some of the integral solutions to the equation 
14a;+19y+2l0=252. 



ALGEBRA. 186 

Here we observe that 14 and 21 and 252, are all divisible by 7, 
therefore y must be 7, or one of its multiples ; suppose it 7, and 
divide the whole by 7. 

Then ' 2^+19-1-32=36. 

Or, 2^+30=17 

Whence, x may equal 1, and 2=5, or ^=7 and 2=1. Or, 
fl;=4 and 2=3. 

ix—1. 4. 1. 
> Hence, \ y=7. 7. 7. 
(2 = 1. 3. 5. 
As these examples are of little practical utility, we give no 
more of thep. 



SECTION IX. 



DIOPHAI^TINE ANALYSIS. 



Diophantus was a Greek mathematician, who flourished in the 
early days of science : and the analysis that bears his name, 
mostly refers to squares and cubes. 

The object of this analysis is to assign such values to the un- 
known quantities in any algebraic expression, as to render the 
whole a square or a cube, as may be required. 

The first principles of this branch of science are very simple, 
but in their application, they expand into the region of impossi- 
bility. 

To Euler and Lagrange, we are indebted for most that has ap- 
peared on this subject. 

Case 1st. The most simple expression to be made a square, 
is of this form : '- 

ax-\-h. 

All we have to do, is to put this expression equal to smne square, 
say n^ ; then 

a;= , and n, a, and &may be taken at pleasure. 

a 

The result will give a value of x which will render (aa;+5), a 

square as required. 



.*.. 



• * . 



^w 



186 ROBINSON'S SEQUEL. 

I EXAMPLES. 

(1.) Three times a certain number increased by 10, is a square. 
What is the number ? 

Here a=3, ^=10, and fl;= — III — 

o 

If we put w=l, then x^= — 3, and ax-\-b^= — 9-}- 10=1, a square. 

If we put w=2, then ic= — 2, and aar-(-5=4, a square; and by 

taking different values for n, we can find as many squares as we 

please. 

(2.) Find such values of x as will render the following expres- 
sions squares : 

(9a:+9), (7;r+2), (3;r— 5), (2^^— i). 
All these expressions correspond to the general expression, 
[ax-^b.) 

Case 2d. To advance another step, we require such values 
of X as will render any expression in the form 

{ax^-\-bx) a square. 

Because a: is a factor in every term of the power, we will make 
it a factor in the root : that is, put the root equal nx ; then 
ax^ -\-bx=^n^ x^ . 
ax-\-b=n'^x. 

x=-A- 
n^ — a 

^ * EXAMPLES. 

(3.) Six tim£s the square of a certain number, added to five times 
the same number, is a square. What is the number ? 

Ans. x= , that is, the number is expressed by 

with the liberty to give any value to n that we please. 

If we make n=l, then a;= — 1, and (ax^-{-bx) will become 
1, a square as required. 

If we make w=2, then a;=_-|= — f, and ax^ -\-bx=6.%^ — ^^=2by 
a square as required, and many other results may be obtained. 







ALGEBRA. 187 

(4.) Find what values of x will render the following expressions 
squares : 

(5a?2— 3a;), {lx^—\bx), {\2x^—yx), {yx'^—^x.) 

(5.) Find such a number that if its square he divided by 12, and 
one-third of the number be taken from the quotient, the remainder will 
he a square numher. 

Ans. 16 is one number, and there are many other numbers 
that will correspond to the conditions. 

The practical utility of this analysis may be exemplified in 
forming examples in quadratic equations. 

Thus ax^ — hx=zN, is a quadratic, and we would assign such 
values to N, as will make the values of x commensurable quan- 
tities. 

h b^ h^ 

Completing the square, gives x^ — -x-\- =iV'4- . 

a 4a ^ 4a^ 

To make the values of x -commensurable, it is necessary to 

/ 5^ \ 
make the expression t A^-|- ) a perfect square, and this we 

can do by putting it equal to some definite square, (by case 1st a 
and b being known quantities,) and deducting the values of N. 

Case 3d. Expressions in the form [x^d=.ax-^h), can be made 
perfect squares, by putting them equal to the square of [x — n.) 

This hypothesis assumes [x — n) to be the square root of 
{x^diziax-\-b), and as this expression may be any number between 
zero and infinity, we now enquire whether it be possible that 
[x — n) can always represent the root, whatever it may be. We 
reply, it can. 

If X is large and n small, [x — n) will be large. If x=in, then 
[x — n) will be zero. If n is numerically greater than x, then 
(x — n) will be negative ; but the square of a negative quantity is 
positive ; therefore, n can be so assumed that [x — nY can 
equal to any positive quantity whatever. - ::^ 

That is, x'^±:ax-{-b=x'^—2nx-\-n^. "* 

Or, x:= 

2w±a 

An expression in which n may be taken of any value whatever, 

and we shall have a corresponding value of x. 

_ ^ '*•* i^ 



% 

♦ 




^^^*v, > 






188 ROBINSON'S SEQUEL. 

Case 4th. An expression in the form (ax^ zhhx-\^c^ ) , can be 
made a complete square, by assuming its square root to be 

(c — 71X.) 

Because c^ is in the power, c must be in the root, and it is obvi- 
ous that X must be a factor in the other part of the root. 
^' Whence, ax^ztbx-\-c'' =c'^—2cnx-{-n^x'^. 

•■^^ axd3=—2cn-\-n^x. 

2cn±:b 



EXAMPLE. 

What value shall be given to x to make 8x^-[-17x-|-4 a perfect 
square ? 

Here a=8, 5=17, c=2. 

Whence, :.= ^^+^^. 

If 7^=l, then x——'2>, and '8;c2-|-17;c+4=25, a square. If 
n=3, then 2;=29, and the value of the expression is 7225, the 
square of 85. 

Case 5th. An expression in the form {^ax"^ -\-bx-\-c) , in which 
neither the first nor the last term is a square, neither branch of the 
root can be taken, and the expression cannot be made a square, 
unless we can separate it into two rational factors, or unless we 
can find some square to subtract from it, which will leave a re- 
mainder susceptible of being separated into two rational factors. 

By placing the expression [ax^ -\-bx-\-c) equal to zero, and 
solving the quadratic, we shall have two factors of the expression, 
but whether rational or commensurable factors or not, is the subject 
of enquiry. 

, To find the factors which make the product ax^ -\-bx-\-c, i^Mt 

this expression equal to 0, and work out the values of x thus, 

ax^-^-bx-^c^O. Or, ax^-\-bx^= — c. Complete the square, and 

4:a^x^-{-Aabx-\-b^=b'^—^ac. 

Or, ^ax-\-b=±J{b^—'Aac.) 

Or ^=±iV(*'— 4ac)— A. 

2a^^ ^ 2a 






/ 



ALGEBRA. '* 189 

We now perceive that the values of x must be rational, previ- 
ded J{b^ — 4ac) is a complete square. If it be so, let 

J_ /(P_4ac) ^=m, and —}_J{b^—4ac)—^=n. 

Then the two values of a; are x=m and x=n, and (x — m)(x — n), 
are the factors which will give the expression ax^-^x-^-c. 

EXAMPLES. 

(1.) Find such a value ofxas. mil make 6x^-|-13x-|-6 a square. 

Here a=6, 5=13, c=z6. h^ = l69, 4ac=144, P—4ac=25, 
and J(b^—4ac)=5. Now 12a:+13=db5. a:=— |. Or, x=—^. 
Or, 3;i:+2=0, and 2a;+3=0. 

That is, (Sx-\-2) (2a:-|-3), will produce the expression 
6a;2+13:r-f-6. 

Now to find such values of x, as will make the expression a 
square, put 

(3a:+2) (2x-{-3)=n^3x-\-2y . 

That is, take either factor of the expression for a factor in its 
square root. 

Then 2x-{'S:=n^3x-{-2.) 

2^2—3 

x= 

2—3n^ 

Take n=l, then x=l, and the expression becomes , % 

6-|- 134-6=25, a square. 

Take n=2, then x= — i, and the expression becomes 1, a 
square. 

Take n=S, then x= — f, and the expression becomes -^j, a 
square. 

(2.) Find such a value of x as shall render the expression 
(\Sx^-{-15x-{-7), a square. 

Here as neither the first nor the last term is square, nor (b^ — 4ac) 
SL square, we cannot find the required values of x, unless we can 
find a square, which subtracted from the expression, will leave a 
remainder divisible into rational factors. But in this case, 4ac is 



190 * . ROBINSON'S SEQUEL. 

greater than b^, we must therefore subtract such a square as to 
diminish a and c, and increase 6. 

To accomplish this object, we will subtract the square of (a? — 1), 
and not the square of (x-\-\.) 

That is, from 13x^+15x-{-7y 

Subtract a;^— 2x-{-\. 

Difference, 12x''-{-\7x-{-e. 

In this last expression, a=\2, 5=17, and c=^6. ^ 

Hence, b'^-^4ac=2S9—2QS=l, a square. 

We are now sure the difference is divisible into rational factors, 
and to obtain the factors, we put 

\2x^-\-\lx-{-Q=0. 

A solution of the quadratic, gives x= — f , or x= — f . 

Whence, (3i;-|-2)=0, and (4.2;-}-3)=0, and our original ex- 
pression becomes 

(^_l)3_^(3ar+2) (4a?+3.) 

It is obvious that [x — 1 ) must be in the root, and one of the 
factors may be in tlie other branch of the root ; that is, put 

(a;— 1)2_{-(3^+2) (4;r+3) = [ [x—\)-\-n{3x-\-2) y. 

By reduction, 4x-\-3=:2n{x—\)-\-n^ (3x^2,) 

Or. x=^"+3--^-. 

Take w=l, then x=3, and \3x^ -\-\ 5X'\-1 ^\Q^ , a square. 

(3.) Find such a value of x as will render 14x^-|-5x — 39, a 
square. 

After a few trials this expression is found to be the same as 
(2,r— l)2+(5.c— 8) (2x-\-b.) Assuming its root to be 2a:— 1+ 
p(5x — 8), then by squaring the root, making it equal to its power, 

and reducinor, we find x=-^-^^^~^ . 
5p^-\-ip—2 

Assuming p= 1 , x= y , and the expression equals 36, a square. 

Other values can be found, by assuming different values to jo. 

(4.) Find such a value of x as will make 2x2-f"2^^~l~28, <? 
square. 



ALGEBRA. a jgj 

After a little inspection, we find this expression equal to 
(a:-|-4)2-f-(a;+l) (x-\-l2.) Now if we make 

After reduction, we shall find x= — Jl_" ^ . 

J92— 2/>— 1 

Assume ^=4, then x=4, and the original expression is 144, 

a square. 

If (x-\-4y-{-(x+l) (x+n) = [ (x+4)—p(x-\-12)y, we shall 

find x=: ±- ±- If we take p=l, x=%. If we take »=| 

then a;=8, and we might find many other values of x that would 
answer the required condition. 

Case 6th. Expressions in the form a^x'^-\-hx^-\-cx^-\-dx-\'€y 
can be rendered square, provided we can extract three terms of 
their square roots. 

Assume such terms as the whole root, making its square equal 
to the given expression, and the resulting value of x will make 
the whole expression a square. 

EXAMPLE. 

Find such a value of x as will make 4x''-|-4x^-(-4x^-|"2^ — ^'^ 
square. 

We commence by extracting the square root as far as three 
terms, and find them to be (2a;^-|-ar-|-|.) 

Therefore, 4x^ ^4x'^ ■\-4x^ -\-'lLx—%^{'ix'' -^-x-^-lY . 

Expanding and reducing, 2a; — 6= |4:-[-t6- 
And ' ;r=13|. 

Essentially the same method must be performed in other ex- 
amples under this form. ^ 

Case 7th. Find such a value of x as will make ax^-\'C, a 
square. 

Expressions in this form, where /;=0, and where neither a nor 
c are squares, nor {h^ — 4ac) a square, present impossible cases : 
unless we can first find by inspection, some simple value of x 
that will answer the condition. 

m 



Idt « ROBIKSON'S SEQUEL. 

EXAMPLE. 

Find such a value of x as will make 2x^ -|-2, a square. 

It is obvious that if x=\, the expression is a square. Now, 
having found that 1 will make the expression a square, we can 
find other values as follows : 

Leta:=l-[-y; then x^ =^\-\-2y-\-y^ , 

And 2x''+2=4+4y-\-2yK 

Here the original expression is transformed into another expres- 
sion, having a square for its first term. 

Now we must find such a value of y as shall make 4-|-4y-|-2y^, 
a square. 

Assume 4-[-42/-|-22/"=(2 — my)^=4 — 4my-\^m^y^. Or, 

4-|-2y= — 4m4-m^y. Hence, y=-^ — —-, ^ may be any num- 

m^ — 2 

ber greater than one. Put m=2. Then y=6, and a;=l-f-y=7, 
and the original expression, 2a?2-|-2=98-|-2=:100, a square. 

N. B. It often occurs incidentally in the solution of problems, 
that we must make a square of two other squares. This can be 
done thus: Let it be required to make a:^-|-y^, a square. 

Assume x=p^ — q^, and y=2j)q. 

Then x''=2)^—2p^q^-\-q\ 

And y^= 4p^q^, 

Add, and x^ -\-y^ ^^p'^ -\'2p^ q"^ -\-q^ y which is evidently a 
square, whatever be the values of p and q. We can, therefore, 
assume j9 and q at pleasure, provided^ be greater than q. 



% SECTION X. 

DOUBLE AND TRIPLE EQUALITIES. 

We have thus far confined our attention to finding a value of x 
that would render a single expression a square. Now we propose 
finding a value of x that will render several expressions squares 
at the same time. 



ALGEBRA. 193 

Case 1st. As a general expression for double equality, let it 
be required to find such a value of x, that will make (ax-\-b) and 
(ca:-[-c?), squares at the same time. 

X:=: . 

a 

^2 ^ 




Whence, -^ =- , or d^ — cb=ap^ — ad. 

a c 

Transposing cb, and multiplying by c, gives 
c^t-= acp ^ -\-c 2 d — acd. 

As the first member of this equation is square for all values 
of c and t, it is only requisite to find such a value of p, as to 
make the second member a square ; which can be done by some 
of the artifices heretofore explained. 

To illustrate, we give the following definite problem : 

The double of a certain number increased by 4, makes a square ; 

and Jive times the same number increased by. 1, makes a square. 

What is that number? 
Let X be the number ; then 



2ar+4=^2 



Whience, \ ^ 



Then 5t''—^0=2p''—^. 

And 25/2 = 10^2+90. 

The first member of this equation is a square, whatever be the 
value of f ; and all the conditions will be satisfied, provided we can 
find such a value of p as to make the second member a square. 

This expression corresponds to case 7, and we cannot proceed, 
unless we find by trial, by intuition as it were, some simple value 
of ^ that will make 10p2_|_9Q^ a, square; and we do perceive 
that jt?=l will make the whole expression 100, a square. 

Now if jt>=l will give a definite and positive value to ar, which 
will answer the required conditions, the problem is solved. If 
not, we must find other values of p. 
13 



sions 



fH ROBINSON'S SEQUEL. 

Here x=^ /and i(p=l, x=0, and the expressions, 2a:-|-4 

6 

and 5a:-|-I, become 4 and 1. Squares, it is true, which answer 

the technicalities, but not the spirit of the problem. 

To find another value of jt?, put jO= !-[-?• Then 

1 0^2 _^90= 1 00+20jt>+2^^ 

To make this a square, assume 

\00-{-20q-{-2q'' = (l0— nqy = 100— ^Onq+n^g'*. 

By reduction, q= — l_ir_''. Now n must be so taken, that n' 
n^ — 10 

will be greater than 10; take w=5 and §'=8, p=Q, then x=l6, 

and the original expressions, 2^-j-4=36, a square, and 5x-\-l=z 

81, a square. 

Case 2d. A double equality in the forpi ox'^-\-bx=Ci* and 

cx^-\-dx=0, may be resolved by making a:=--, then the expres- 

y 

will become -^(a-j~*y) ^^^ — (^+^y)» which must be 

made squares. 

But if we multiply a square hy a square^ or divide a square hy 
a square, the product or quotient will he square. 

Now as each of the preceding expressions are to be squares, 

and as they obviously have a square factor — , it is only necessary 

y^ 

to make a-\-by, and c-\-dy, squares, as in the first case. 

We may also take another course and assume ox^-\-bx=p^x^ , 

which gives x= , whic^h value put in the other expression, 

p^ — a 

and we have c( ) -\-d( ) = D . 

\p^—a/ ' \p^—a/ 

Multiplying this by the square [p^ — aY , and the expression 
becomes ch^ — dhd-\-abp^= some square, from which the value 
of p can be found, and afterwards x. 

EXAMPLE. 

Find a numher whose square increased by the number itself, and 
*Thi8 symbol is read a square. 



ALGEBRA. 196 

whose square diminished by the number itself, the sum and difference 
shall be squares. 

Let ar= the number ; then by the conditions, 

a;2-|-a:=n, and x^ — a;= some other square. 



Assume x= - ; then — ( 1 -f-y ) = D . 



The first members of these equations are obviously squares, 
provided the factors (l-(-y) and (1 — y) are squares. 

To make these factors squares, put 

l-|-y=jt?2, and 1 — y=^q^ • 

Whence, y==p^ — l,andt/=l — g*. 

p^z=<2.—q^. 

All the conditions will be satisfied, when we discover what 
value must be given to q to make (2 — q^), a square ; and q=\ 
satisfies that condition. 

This value oi q makes j9=l, and y=0. 

But ic=:_=_= infinity. 

y 
If X is infinite, x^ can neither be increased or diminished by 
adding and subtractings;; ihereioxQ x^-\-x=i\2, vm^ x^ — ^•=n, 
because x^ is obviously a square. 

But practically, we say that this value of q will not answer the 
conditions ; therefore, we will find other values as follows : 
Put q=\-\-t. 
Then 2— y^ ^i__2t—i'' =^(\~ntY = \—ZrU+n'' f" . 

Or, /=?l!^i. Take n=2, 

n^+\ 
Then^=|. ^=1+1=1. y=i_||===_|4. 

1 25 

But x=-= — -- , the number souofht. 

y 24' 

Those who desire a positive number, can take n negative. 

Case 3d. To resolve a triple equality. 

Equations in the form cx-\-hy=:it'^ , ax-\-dy—u^, eX'\-fy='8^ , 
can be resolved thus : 



J96 BOBINSON'S SEQUEL. 



By eliminating x, we find y= 
By eliminating y, we find x 



au 



ad — be 

ad — be 

Substituting these values of a; and y in the equation &c-[^y=5^, 
we shall have 

(af-be)u-+ { de-ef )e_^, . 
ad — be 
Assume e«=r±:^2;; then u^=^t^z^, and put this value in the 
above, and divide by t^ , then 

{af—bc)z^J^[de—cf )_ s2 
ad— be '¥' 

As the second member of this equation is a perfect square, all 
the conditions will be satisfied when we find a value of z that 
will render the first member a square. This, when possible^ can 
be done by case 7, of section viii. 

After z is found, t can be assumed of any covenient value 
whatever. When u and i are known, x and y will be known. 

We are now through with theory, — not that we have presented 
the whole, there are some cases in practice that no general rules 
will meet, and the operator must depend on his own judgment 
and penetration. 

Much, very much, will depend on the skill and foresight displayed 
at the commencement of a problem, by assuming convenient ex- 
pressions to satisfy one or two conditions at once, and the remain- 
ing conditions can be satisfied by some one of the preceding rules. 

EXAMPLES. 

(1.) It is required to find three nwmbers in arithmetical progres- 
sion, such that the sum of every two of them may be a square. 

Let X, x-\-y, and ar-|-2y, represent the numbers. 
Then by the general formula, 

^x^y=t^, 2x-\-2y=:u^, 2x-\-Sy—s^, 

^ _ ■ /2 y 21 2 2,11 

By extermmating x, we have -= ^ . 



ALGEBRA. 197 

Continuing thus according to the general equations, we must 
go through a long and troublesome process, and in conclusion we 
shall find the numbers to be 482, 3362, and 6242. 

Another Sdviion. 

Observing the remark immediately preceding the example, we 

put — — y, — , and — -|-y to represent the numbers. 

Ai ^ At 

Then {x^ — y), (^^+y), and x^ must be the squares. But x^ 
is a square for all values of x ; therefore, we have only to make 
squares of (x^ — y) and {x'^-^-y-) 

Let y=2x-\-l ; then x^ -\-y=x'^ -\-2x-\-l , a square for all values 
of X. Hence, all we have to do is to find a value of x that will 
make a square of the expression x^ — y, or x^ — 2x — 1. Assume 
the square root to be (x — n) ; then 

x^ — 2x — l=x'^ — 2nx-\-n^ • 

x=.^L+l_ 
2(n—l) 

Take w=ll, then a;=:6.l, and -1^:2 = 18.605, y=13.2, and this 
numbers are 5.405, 18.605, and 31.805. Various other numbers 
may be found, by giving different values to n. 

(2.) Find two numbers such that if to each, as also to their sum^ 
a given square el^ be added, the three sums shall all be squares. 

Let x^ — a^, and y- — a^ represent the numbers ; then the first 
conditions are satisfied. 

It now remains to make x^-{-y^ — 2a^-\-a^ a square, or, x^-\' 
y 2 — Qj2 __ [-] ^ Assume y^ — a^ ==:2ax-\-a ^ . This assumption will 
make the expression a square, whatever be the values of either 
a; or a. But the assumed equation gives y^ =^2ax-\-2a^ , a.nd as 
y^ is a square, we must find such values of x and a, as shall make 
2aa:-|-2a2, a square. Put x=-na. Then 2wa2-|-2a^ = n, or, 
a^ (2w+2)= D . Hence it is sufficient that we put 2w-|-2= some 
square. Therefore, assume 2w-|-2=l6. Hence, w=7 and ir=7a. 
Now take a equal to any number whatever. If a=l, a:=7, y=4, 
and 48 and 15 are the numbers, add 1 to each, and we have 49 
and 16, squares ; sum, 63-|-l=64, a square. 



198 ROBINSON'S SEQUEL. 

(3.) Find three square nunilers whose sum shall he a square. 

Letir*+y2_f-22_Q^ Assume y'^='^z. Then xf -{-2xz-^z^ 
is a square. But 2a:s=D. Let x=uz, then ^uz^ = {j, or 2u= 
□ = 16, u=S, x=Qz, z—\, x—^, y=4. 

Therefore 64+16+1=81=92. 

(4.) Fhui three square numbers in arithmetical progression. 

Let x^ — y, x^, and x^-\-y represent the numbers. Assume 
a.2__y2_|_i^ then the first and last will be squares, and it only re- 
mains to make (y^+i), a square. 

Therefore, put 3/2+1 =(y—j9) 2. Whence, y=£- !. 

2p 
Take^=l, then 2/=|, and y2+|=f|=a;2. 

Consequently, -g^, ||, and ^\ are the numbers ; but we can 
multiply them all by the same square number, 64, without chang- 
ing their arithmetical relation^ and their products will still be 
squares, 1, 25, and 49. Multiplying these numbers by any square 
number, will give other numbers that will answer the condition. 

(5.) Find two whole numbers, such that the sum and difference of 
their squares when diminished by unity, shall be a square. 
Let a:+l=: one number, and y= the other. 
Then by the conditions, x'^-\-y^-\-^x=.{2 (1) 

And a:2— y2+2ar=n (2) 

Assume 2a:=a2, and 3/^=2cu;; then (1) and (2) become 
(a;2+2aa;+a2) and (a:^— 2aa;+a2), 

obvious squares whatever may be the values of x and a. 

But the equations 9,x=ia^ , and y^ =2ax, must be satisfied. 
Take a=4, then x=S, a;+l=9, and 9 and 8 are the numbers 
required. 

(6.) FtTid three whole numbers, such that if to the squares of each, 
the product of the other two be added, the three sums shall be square*. 
Let a?, xy, and xv be the numbers. 
Then by the conditions, x^ -\-x^vy= □ . 
x^y^-{-x'^v=0. 
And x^v^^x^yz=[2. 



ALGEBRA. 199 

Omitting the common square factor x^, it will be sufficient to 
make squares of the following expressions : 

\-\-vy=U. 

Assuming y=4v-\-4 will make the first and third expressions 
square. 

Substituting the value of y^ in the second expression, we shall 
have 16y^-f-33y-j-16, which must be made a square. 

Whence, \ev^+^3v-\-16={4—pvy. 

Reduced, gives v= — X-^. Take ^=5. 

p'^ — 16 

Then v=\^. Now take x=9, and 9, 73, 328, will be the re- 
quired numbers. 

(7.) Find two whole numbers whose sum shall be an integral cube, 
and the sum of their squares increased by thrice their sum, shall he an 
integral square. 

Let x-^y=^n^, that is, some cube. Then x^-^y^-\-Zn^=z\2' 
Put 2;ry=37i3, then x^ -\-9,xy-^y^ is a square, whatever may be the 
values of x and y. But x and y must conform to the equations 
x-\-y,=n^, and 2xy=Sn^. Work out the value of x from thesQ 
equations, on the supposition that n is known, and we shall find 
2x=n^-{-J{n^—6n^}. 

Now a; will be rational, provided we can find such a value of n 
as shall render n^ — 6n^ a square, but if we add 9 to this, we 
perceive it must be a square, and we have two squares, which 
difi"er by 9. Therefore one must be 16, the other 25, as these 
are the only two integral squares which differ by 9. Hence, 
;i6_6^3_|_9^25. Or, ^3—3=5. n^ = ^, n=2, and x=6, 
y=2. 

(8.) Find three numbers such that their sums, and also the sum 
of every two of them, may all he squares. 

Let x^ — Ax= the first, 4x= second, and 2:r-}-l= third. By 
this notation, all the conditions will be satisfied, except the sum of 
the last two. That is 62"-|-l must be a square, but to have three 
different whole numbers, no square will answer under 1 2 1 ,.the square 



200 ROBINSON'S SEQUEL. 

of 11. Hence, put 6a;-[-l = 121. Or, a;=20. And the numbers 
will be 320, 80, and 41. 

(9.) Find two numbers such thai their difference may he equal to 
the difference of their squares, and the sum of their squares shall bf 
a square number. 

Let X and y be the numbers. Then x — yz=x^ — y^ . Divide by 
X — y, and l=x-\-y. Hence a?=l — y, and x^-\-y^ = \ — 22/-(-2y^. 
Which last expression, 1 — 2y-|-2y^, must be made a square. For 

this purpose put 1—22/4-2^2 =(1 — nyY . Hence, y=5i^IIl i. 

n^ — 2 

Take n any value to render y less than one, in order to make 

X positive. Take w=3, then y=y, and a;=f , the answer. 

The following are not difficult, and we leave them as a pleas- 
ant exercise for learners. 

(10.) Find three numbers in geometrical progression, such that if 
the rman be added to each of the extremes, the sums in both cases shall 
be squares. Ans. 6, 20, and 80. 

(11.) Find three numbers, such that their product increased by 
unity shall be a square, also the product of any two increased by unity, 
shall be a square. Ans. 1, 3, and 8. 

Assume 1 for the first number, and x and y for the other two. 

(12.) Find two numbers, such that if the square of each be added 
to their product, the sums shall be both squares. Ans. 9 and 16. 

(13.) Find three integral square numbers in harmonical propor- 
tion. Ans. 25, 49, and 1235. 

(14.) Find two numbers in the proportion of S to 15, and such 
that the sum of their squares shall be a square number. 

Ans. 156 and 255. Bonnycastle's answer is 476 and 1080. 

(15.) Find two numbers such that if each of them be added to their 
product, the sums shall be both square. Ans. ^ and |. 

We have given as much on this topic as will be profitable, save 
the following remote and partial application. 



ALGEBRA. 201 

EXAMPLES. 

(1.) Given -j ^2_ — 7 [ ^^ ^^^ *^® values of x and y. 

x" ={l—y.) y2^(74-a;.) Here (7— y) and {1+x), must be 
squares. x=2, and y=3, will evidently answer the conditions ; 
and as these values will verify the given equations, the solution 
is accomplished. 

(2.) Given | l^^Zu'^y^'TLixy'^d \ ^^ ^°^ ^^^^^^ ^^ ^ ^^^ ^' 
As 4 and 9 are squares, the first members are square in fact, 
though not in form. But we can make the first members square 
in form, by assuming 

2.c2 — 2,xy—0, and Sy^— 2.ry=0. 
Then y^=4 and a;2=9, or y=2 and a;=3 ; values which ver- 
ify all the equations, 

(3.) Find such integral values of x, y, and z, as will verify the 

equations 

x^+y^+xy=^l. 

And a;2-j-s^+^2=49. 

If we add xy to the first equation, and xz to the second, the 
first members will be square ; and, of course, the second mem- 
bers will be square in fact, though not in form. 

We have then to make ^l-\-xy, and ^^-\-xz, squares. 

To accomplish this, put 37-|-a^=49, or xy=\^ (1) 

And 49+^2=64, or ii:s= 15 (2) 

From (1), 2:=-; from (2), a:=— . 
y z 

Hence, 122=15?/, or 2=—. 

Take y=4, then 2=5, and a;=3 ; values which will verify the 
given equations. 

(4.) Find such integral values of y and z that will verify the 
equation y'^ -\-z^ -\-yz=Q\ . 

Add yz to both members, then put Q\-\-ys=^n^ . 

Now if we assume w=8, yz=^2>. 

But yz=i'^ will give y-|-2=8, and these two equations will not 
give integral values to y and z. Therefore, take ?z=9, then 
w=^=81, y2=20, y-|-^=9. Hence, 2=4 or 5, and y=5 or 4. 



202 ROBINSON'S SEQUEL. 

(5.) Given ■! a. ^ajx^ZL r f *^ ^°^ ^^ values of x and y. 

Put xy=^py transpose, &c. 

Then 4a;2 = 12+2/>. Ay^ = \Q—Qp. 

Now if we find such a value of j9 as will make (IS-f-Sp) and 
(16 — Qp), squares at the same time, it is highly probable that such 
a value will verify the original equations. It is obvious that ^=2, 
will make the expressions squares ; then 4a:^ = 16, a;=2 and 2/=l, 
and these values will verify all the equations. 



,^\ n- \ 6a;24-2v^ =5^4-12 ) to fin( 

(6.) Given j 3^,:[:2^^_3^?1 3 ^ ^^^^^ 



d one value of x 



This problem is under (Art. 1 10, alg.) 

Add the equations together, and reduce, and we have 
9.r2=y2_|_3a:y+9. 

The first member of this equation is a square ; therefore the 
second member is a square, but to make it a square in form, as 
well as in fact, we perceive it is only necessary to make a;=2. 
Then dx^ =:y^ -^-Qy-^-^ , and 3.r=y-|-3 ; whence y=3, and these 
values verify the given equations. 

This method of operation must be used with great caution, and 
taken for just what it is worth. 

2 2 

(7.) Given ar-|-y=35, and x^ — y^^=5, to find the values of x 
and y. 

Put x^=F, and y^=Q. 

Then P^+Q^=35, and F^—Q^=5. 

Or, P^=35—Q\ and F^=5+QK 

The equations can all be verified, provided we find can such a val- 
ue of Q that will make (35 — Q^) a cube, and (5-\-Q^), a square. 

We will try the next less integral cube below 35. That is, we 
will assume 35—^3^27. Then Q=2, and (5-{-Q^)=9, a 

square. Then P=3, and x^==3, or a;=27, and y=8. 

This problem was given in the first editions of Robinson's 
Algebra, page 147, under the head of pure equations, but it was 
out of place and is now changed. 



PART THIRD 



SECTION I. 



OEOMrETRY. 




D right 



Thirty-one of the following problems will be found in Robinson's 
Geometry, commencing on page 100. 

(1.) From two given points, draw two equal straight lines which 
shall meet in the same point in a line given in position. 

Let A and B be the two given 
points, taken at pleasure, and MO 
the line given in position. 

Join AB and bisect it in J). 
Draw J)U perpendicular to AB, to 
meet the line IIO in U. Join AH 
and BJEJ, the lines required. Be- 
cause AJ)=^DB, and DE com- 
mon to the two A's ADE, BDE 
angles, therefore AE=^BE. Q. E, 

N. B. For simple and obvious demonstrations, we shall not go 
through the steps in full, but refer to Robinson's Geometry for 
the proposition that applies. 

(2.) From two given points on the same side of a line given in 
position, to draw two lines which shall meet in that line and make 
equal angles with it. 

Let A and B be the two given points, 
and HO the line given in position. 

From one of the given points as B, 
let fall the perpendicular B 0, to the 
given line, and produce it to D, making 
0D=B0. 

Then join AD : this line will neces- 
sarily cut the hne HO in some point E. 
Join EB, and AE and EB are the re- 
quired lines. jLBEO=I^DEO, (Book 1, Th. 13.) L.AEH= 
L,DEO, (Th.3, Bookl.) Whence, L.BE 0=1^ A EH. Q.E.D. 

203 




204 



ROBINSON'S SEQUEL. 



(3.) If from any jiolnt without a circle, two straight lines he dravm 
to the concave part of the circumference, making equal angles with the 
line joining the same point and the center ; the parts of these lines 
which are intercejjted loithin the circle, are equal. 

Let A be the point without 
the circle. Join A C and draw 
any other hne to cut the ciroie 
as AD ; then draw AB so that 
the angle CAB=^ CAD. Then 
we are to show that FB^ED. 

The two A's, ABC and 
ADC, having two sides AC, 
CB, of the one, equal to A C, 
CD, of the other, and their re- 
spective angles at A equal, the 
two A's are equal. That is, 
AB=AD. For the same rea- 
son the two A's ACF, ACE 
are equal, and AF=AE. 

Whence, An—AF=zAD~AE, or BF=DE. Q. E. D. 




(4.) If a circle be described on the radius of another circle, any 
straight line drawn from the point where they meet, to the outer cir- 
cumference, is bisected by the interior one. 

Let A C be the radius 
of one circle and the di- 
ameter of another, as 
represented in the figure. 
From the pointof contact 
A, of the two circles, 
draw any line, as All; 
this line is bisected in D. 
Join DC and ffB. Then 
ADChemg in a semicircle, is a right angle; also, AJIB is a 
right angle, for the same reason: therefore, DC and HB are 
parallel. Whence, 

AD : Aff : : AC : AB 




GEOMETRY. 



205 



But as AB is double of A C, therefore All is double of AD, 
or ^^is bisected in I). Q. E. D. 



(5.) JF'rom two given points on the sante side of a line given in 
position, to draw two straight lin£S which shall contain a given angle, 
and be terminated in that line. 

Let A and B be the 
two given points and j0"(9 
the line given in posi- 
tion. For the sake of 
perspicuity, we will re- 
quire two lines drawn 
from the two points, A 
and B, to meet in HO, 
and make an angle of 
50°. Subtract 50 from 
1 80, and divide the re- 
mainder by 2, this pro- 
duces 65°. 

At A make the angle 
BAC=Qb°, and at B 
make the angle ABC=^Qb° ; these two lines will meet in C, 
making an angle of 50°. About the A ABC describe a circle, 
cutting HO in //and 0. Join AH, BH AHB is equal ACB, 
(th. 9, b. iii, scholium,) the angle required. 

Lines drawn from A and B, to meet the line in 0, would also 
answer the conditions, 

N. B. When the given angle is not sufficiently small to cause 
the angle C to fall below the line HO, the problem is impossible. 

(6.) If from amj point without a circle, lines he drawn touching it, 
the angle contained hy the tangents is double of the angle contained by 
the line joining the points of contact, and the diameter drawn through 
one of them. 

This problem requires no figure. Imagine a point without a 
circle, a line drawn from that point to the center of the circle, 
and lines drawn to touch the circle on each side. Join the points 




206 ROBINSON'S SEQUEL. 

of contact and the center of the circle. Thus we have two equal 
right angled triangles, having the same hypotenuse, the line from 
the given point without the circle to the center of the circle. 
With the correct figure in the mind, the truth of the proposition 
is obvious. 




(7.) If from any two points in the circumference of a circle, there 
he drawn two straight lines to a point in a tangent to thai circle, they 
ivillmake the greatest angle when drawn to the point of contact. 

Let A and B be 
the two points in the 
circle, and CD a tan- 
gent line. The prop- 
osition requires us to 
demonstrate that the 
angle A GB is greater 
than the angle ADB. 
ACB=AOB, {th.9, 
b. iii, sch.) But A OB is greater than ADB, (th. 11, b. i, cor. 
1), therefore, ACB is also greater than ADB. Q. E. D 

(8.) Fro?n a given point within a given circle, to draw a straight 
line which shall make with the circumference an angle less than any 
angle made by any other line drawn from that point. 

Let P be the given 
point within the circle, 
and C the center. Join 
PC. Through P draw 
APB at right angles to 
PC. Also, through P 
draw any other line as 
P G ; then we are to show 
that PBt is less than 
PGH. 

From C let fall the 
perpendicular CD on the 
chord FG. PC is the 
hypotenuse of the right 




'4i 



GEOMETRY. 207 

angled triangle PDC \ therefore, PC is greater than CD, con- 
sequently the chord FO is greater than the chord AB, (th. 3, 
b. iii.) and the arc OAF is greater than the arc BOA. The 
angle POHis, measured by half the arc OAF, and PBt is meas- 
ured by half the arc BOA ; therefore, the angle POH is greater 
than the angle PBt, or PBt is less than POH. Q. E. D. 

N. B. The angle which any chord makes with the circumfer- 
ence, is the same as between the chord and tangent, — because the 
circumference and tangent unite as they meet the chord. 

(9.) If two circles cut each other, the grectiest line that can he 
draion through the point of intersection, is that which is parallel to the 
line joining their centers. 




Let A and B be the center of two circles which intersect in 0. 
Through draw mn inclined to AB, — then we are to prove that 
mn is less than it would be if it were parallel to AB. Draw AC 
and BI) perpendicular to mn, then CD^=\mn. Draw (7^ paral- 
lel to AB, then CH=AB ; and CZT being the hypotenuse of the 
right angled A CDH, GH, or its equal AB, is greater than CB. 
Now conceive mn to revolve on the center 0, until CD becomes 
parallel to AB ; CD will then become equal to AB. But mn 
will be all the while double of CD : therefore, mn will be the 
greatest when parallel to AB. Q. E. D. 

(10.) Iffrofrti. any point within an equilateral triangle, perpendic- 
ulars he drawn to the sides, they are together, equal to a perpendicular 
drawn from any of the angles to the opp)osite side. 



208 



ROBINSON'S SEQUEL. 




Let ABC be the equilateral A, 
CD a perpendicular from one of the 
angles on the oposite side ; then the 
area of the A is expressed by \AB 
X CD. Let P be any point within 
the triangle, and from it let drop the 
three perpendiculars FG, PH, P Oy 

The area of the triangle APB is 
expressed by ^ABy^PG. The area 
of the A CPB is expressed by 
\CBxPO\ and the area of the A 
CPA is expressed by ^CAy^PH. By adding these three expres- 
sions together, (observing that CB and CA are each equal to 
AB,) we have for the area of the whole A ACB, \AB{PG+ 
PH-\-PO.) 

Therefore, \ ABX CD==^AB{PG+Pir+P 0.) 

Dividing by i^^, gives CD=PG-\-PH+P 0. Q. E. D. 

(11.) If the points , bisecting the sides of any triangle he joiried, 
the triangle so formed, will be one-fourth of the given triangle. 

If the points of bisection be joined, the triangle so formed will 
be similar to the given A, (th. 19, b. ii.) 

Then, the area of the given A will be to the area of the A 
formed by joining the bisecting points, as the square of a line is to 
the square of its half ; that is, 2^ to 1, or as 4 to 1. Hence the 
A cut off is I of the given A- Q. E. D. 

{\9..) The difference of the angles at the base of any triangle, is 
double the angle contain£d by a line drawn from the vertex perpen- 
dirular to the base, and another bisecting the angle at the vertex. 

Let ^^C be a A. Draw A3f bi- 
secting the vertical angle, and draw 
AD perpendicular to the base. 

The theorem requires us to prove that 
the diferenne between the angles B and 
Cis double of the angle MAD. 

By hypothesis, the angle CAM= 
MAB. That is, CAM=MAD-\-DAB, (1) 




GEOMETRY. 



209 



( C+CAM+MAJ) 
By (th. 11, b. i, cor. 4.) -j j^j^j)^j^ 



:90°. I (2) 

:90°. f (3) 

Therefore, B+DA£=C-{-CAM-\-MAD. (4) 

Taking the value of CAM irom. (1), and substituting it in (4), 
gives B+DAB= C-\-3fAI)+DAB+MAD. 

Reducing, (B—C)=2MAD. Q. E. D. 




(13.) If from the three angles of a triangle, lines be drawn to the 
middle of the op-ponte sideSy these lines will intersect each other in the 
same point. 

Let ABC be a A, bisect 
BCmE, AC'mF. 

Join AU and BF, and 
through their point of inter- 
section 0, draw the line 
CD. JVow if ice prove AD 
=DB, the theorem is true. 

Triangles whose bases are 
in the same line, and vertex in the same point, are to one another 
as their bases ; and when the bases are equal, the triangles are 
equal. For this reason the A AFO=AFCO, and the A COF 
== A FOB. 

Put A AFO=a; then A FCO=a. Also, put A COF=b, 
as represented in the figure. 

Because CB is bisected in F, the A ACF is half of the whole 
A ABC. Because ^C is bisected in F, the A BFC is half the 
whole A ABC. 

That is, 2«+6==25+a. 

Whence, a=b, and the four triangles above the point 

are equal to each other. 

Let the area of the A AD be represented by x, and the area 
oiDOBhjy. 

Now taking COD as the base of the triangles, we have 
2a : X : : CO : OD 



Also, 



25= 
14 



2a 



CO 



OD 



210 



ROBINSON'S SEQUEL. 



Whence, 
Therefore, 



AD=DB, 



2a : y. Or, .c=y. 
Q. E. I). 




(14.) The three straigJd lines which bisect the three angles of a 
triangle, meet in the saine point. 

Let ABO be the A, 
bisect two of the angles 
A and C — the bisecting 
lines will meet in the 
same point 0. Join OB; 
we are required to demon- 
strate that OB bisects the 
angle B. 

From 0, let fall the perpendiculars on to the sides. The two 
right angled A's A OH and A 00, are equal in all respects, be- 
cause they have the same hypotenuse A 0, and equal angles by 
construction. In the same manner we jirove that the A CGO 
= £\00L Whence, 00= OL But (7(9= 0//; therefore, 
0H= 01. 

Now in the two right angled triangles OHB and OIB, we have 
0H= 01, and OB common, therefore, the triangles are equal, 
'dridIIBO=OBL Q. E. D. 

(15.) The two triangles formed by drawing straight lines from 
any point within a. parallelogram to the extremities of the opposite 
sides, are together half the parallelogram. 

Let ABD (7 be a parallelogram, E any 
point within. 

We are to show that the triangles AUB, 
CJED, are together half the parallelogram. 

Through the point £ draw a line par- 
allel to AB or CD, thus forming two parallelograms. 

The A AUB is half the lower para,llelogram, and the A CUD 
is half the upper parallelogram ; therefore, the sum of the two 
A's is half the whole parallelogram. Q. E. D. 

(16.) The figure formed by joining the points of bisection of the 
aides of any trapezium, is a parallelogram. 




• • ^ .Ar 



GEOMETRY. 



211 



Let AB CD ho a trapezium. 
Draw the diagonals ^ (7, JBD. 
Bisect the sides in a, b, c, and 
(/. Join abed. We are to prove 
that this figure is a parallelo- 



ABD is a A whose sides are 
bisected in a and b ; therefore, 
tlie A Aba is equiangular to the A ABD, (th. 19, b ii), and ab 
is parallel to BD, and by (th. 18, b. ii), ab=\BD. In the same 
manner we can prove that dc is parallel to BD and equal to half 
of it. Consequently ab and dc are parallel and equal. There- 
fore, by (th. 23, b. i), the figure abed is a parallelogram. Q. E.D. 







(17.) If squares be described on the three sides of a right angled 
triangle, and the extremities of the adjacent sides be joined, the triangles 
so formed are equal to the given triangle, and to each other. 

LetJJ5(7be 

the given right 
angled triangle 
and construct 
the figure as 
here represent- 
ed. It is ob- 
vious that the 
vertical right 
angled A ^l-^ff^ 
is equal to 
ABC. 

Draw AV 
perpendicular 
to BC, and 
call it X. We 
now pro})ose to 
show that HO 
=^x. BD is 
produced to G, the angles VBOojidi ABffare right angles, and 




Il2 ROBINSON'S SEQUEL. 

from these equals take away the common part ABG; thus 
showino- th^t ABV=BBG. 

o 

The two right angled triangles AB V, JIB G are equal, because 
they have equal angles, and the hypotenuse AB= the hypote- 
nuse JIB, because they are sides of the same square. Therefore, 
ffG=A V, and if one is in value x, the other has the same value. 

Now we designate any side of the square on BC by a, then 
twice the area of the A AB C is ax, and the double area of the 
triangle HBD is obviously ax. 

Therefore, HBD is equal in area to ABC. 

In the same manner we can prove that FCE:=ABC. Q.E.D. 

(18.) If squares he described on the hypotenuse and sides of a right 

anffled tnangle, and the extremities of the sides of the former, and the 

adjacent sides of the others he joined, the sum of the squares of Che lines 

joining them ivill he equal to five times the square of the hypotenuse. 

(See figure to the last Theorem.) 

In the right angled triangle HGD, we have 

x'-+{BG+ay={HDY (1) 

In the right angled triangle PFE, we have 

x-+{PC^y={FEY (2) 

Expanding (1) and (2), and observing that GB-=BV, PC= 
CV, we shall have 

x^-\-{BVy-\-2a(BV)-{-a^=(IIDy (3) 

And x''-{-(FCy+2a{FC)+a-=:(FFy (4) 

By adding (3) and (4), and observing that x- -\-(B V)" =b^ , 
andic2+(PC)2=c2, then 

(J2^^2 )^2a(i? r+ VC)-{-2a^=(IIFy+(FCy 
That is, a2_j_2rt(o)_j_2«2^ 

Or, 5a^ = (IIDy+(FCy 

Scholium, The sum of the squares of the sides of the last 
figure is 8a ^. 

(19.) The vei'ticol angle of an ohlique-angled triangle, inscribed in a 
circle, is greater or less than a right angle, hy the angle contained be- 
tween the base and the diameter draumfrom the extremity of tJie base. 



* 



GEOMETRY. 



213 




Let AE C be a A liav- 
ing the angle A CB grea- 
ter than a right angle, 
and describe a circle a- 
bout it. From one ex- 
tremity of the base as B 
draw the diameter BD. 

The angle DBC is a 
right angle, because it is 
in a semicircle. The ver- 
tical angle A CB is grea- 
ter than a right angle 
hj ACB; but ACD is 
equal ABD, because 
each is measured by half the arc AI). Therefore ACB is greater 
than a right angle by ABD. 

Next let A'CB be the A ; the angle A'CB is less than a righ^ 
angle by the angle DCA'=DBA\ hecause each is measured by 
half the arc DA'. Therefore, the vertical angle, (fee. 

(20.) If the base of any triangle he bisected by the diameter of its 
circumscribing circle, and from the extremity of that diameter, a per- 
pendicidar be let fall upon the longer side, it ivill divide that side into 
segmerds, one of which will be equal half the sum, and the other half the 
difference of the sides. 
Let .42? (7 be the A, 
bisect its base by the 
diameter of the circle 
drawn at right angles 



to AB. 

From the center 
let fall Om at right 
angles to A C, it will 
then bisect ylC From 
the extremity of the 
diameter B, draw Bfh 
perpendicular to ^1 C, 
and consequently par- 
allel to Om. Produce 




214 ROBINSON'S SEQUEL. 

Hh to M and join ML. Complete and letter the figure as 
represented. 

The two triangles Aah and Hha are equiangular. The angle 
a is common to them, and each has a right angle by construction, 
therefore the angle H=^ the angle A. But equal angles at the 
circumference of the same circle subtend equal chords, (th. 2, b. 
iii ;} therefore CB^=ML. The angle HML is a right angle, be- 
cause it is in a semicircle, therefore ML is parallel to AC, andJfZ 
is bisected in n. 

Now Am^\A C. nL^md^ \ML=\B C. 

Therefore by addition, Am-\-md=\(AC-\-CB,) 

Or, Ad=\(AC-^CB.) Q. E. D. 

Cor. If Ad is the half sum of the sides, dc or Ah must be the 
half difference ; for the half sum and half difference make the 
greater of any two quantities. 



(21.) A straight line dv&wn frmi the vertex of an equilateral trian- 
gle, inscribed in a circle, to any point in the opposite circumference, is 
equal to the two lines together, which are dratvn from the extremities 
of the base to the same point. 

Let ^i>6'be the e- 
quilateral A in a cir- 
cle. Take I> any point 
in the arc between i> 
and C, and join A.D, 
BD, andi)a 

Designate each side 
of the given triangle 
by a. 

Now ABDC is a 
(juadrilateral in a cir- 
cle, AD is one diago- 
nal and BC iho, otlier, 
and by (th. 21, b. iii) 
\ve have 

u(AD)=a{BD)+a{DC,) 

Diyiding by a, and AD=:BD-^DC. Q. E. D. 




GEOMETRY. 215 

(22.) The straight line bisecting any angle of a triangle inscribed 
in a given circle, cvis the circumference in a "point lohich is equidistant 
from the extremities of the sides opposite to the bisected angle, and 
from the center of a circle inscribed in the triangle. 

(See the figure to the last Theorem.) 

The angle BAD is measured by half the arc BD, (th. 8, b.iii) 
and the angle DA C is measured by half the arc D C ; therefore, 
if BAD=DAC, the arc BD must equal the arc DC. 

(23.) If from the cerder of a circle a line be drawn to any point in 
the chord of an arc, the square of that line, together with the rectangle 
contained by the segments of the chord, will be equal to the square 
described on the radius. 

(See the figure to the 21st Theorem.) 

From the center draw F to any point in A 0, and through 
the point Fdraw nm at right angles to OV, and join Om ; then 
Vm is a right angled triangle. Therefore, ( F)^-|-( Vmy = 
{Om)\ But (Vmy = (nV) (Vm)=(AV) (VC), (th. 17, b. 
iii.) Therefore, by substitution, 

{ OVy-\-{AV) (VC)=( Omy . Q. E. D. 

(24.) If two poiMs be taken in the diameter of a circle, equidistant 
frmn the center, the sum of the squares of the two lines drawn frmn 
these points to any point in the circumference will be always the same. 

Let C be the cen- 
ter of a circle, and A 
any point in the cir- 
cumference. CA= 
r, the radius. 

Put AD=y, DO 
=ar, and CB and CG 
each =a. Then BD 
=(x — a), and DG 
=(x+u). 

Now in the triangle 
ADB we have y^-{-(x^ay=(ABy. 

And in the triangle ADG, y^-^(x-^ay—(AGy 




216 



ROBINSON'S SEQUEL. 



By expanding and adding, we find 

The triangle -4i> (7 gives 2?/^ -\-2x^ —^r'^ ; therefore, 
2r^+2a^=(ABy-\-(AGy . 

Because the first member of this equation is the same for all 
values of x and y — that is, because it is invariable ; therefore the 
second member must also be invariable. Q. E. D. 



(25.) If on the diameter of a semicircle two equal circles be described^ 
and in ike space included by the three circumferences, a circle be in- 
scribed, its diameter will be two-thirds the diameter of either of the 
equal circles. 

It is sufficient to represent a portion of the figure. 

Let B be the cen- 
ter of the semicircle, 
and BA the diame- 
ter of one of the e- 
qual circles, and E 
the center of the cir- 
cle sought — BD be- 
ing at right angles to 
AB from the point B. 

Put CB=r, and 
DE=x. Then BD 
=2r, BE=:^2r-—x, 
and CE=r-\-x. 

Now in the right 
angled triangle BEC, 
we have 

That is. 

By expanding. 

Reducing, 

Whence, 




{CBY+{BEY=(CEY. 

r2J^[2r—xY={r+xy. 

7.2 +4r2 ^Arx+x"" =r^ J^^rx+x^ . 

x=§r. Q. E. D. 



(26.) If a perpendicular be drawn from the vertical angle of any 
triangle to the base, the difference of the squares of the sides is equal 
to the differeyice of the squares of the seginerUs of the base. 




GEOMETRY. 217 

Let ABC be any triangle. Let fall AD 
perpendicular to the base. Now the two 
right angled triangles give us 

(ADy-{-{BDy={ABy. 

And (ADy-\-(I)C)^={AC)K 

By subtraction, (BDy—{D Cy^jABy'—jA C) ^ . Q. E. D. 

By factoring, {BD-{-DC){BD—DC)={AB-\-AC){AB—AC.) 
By observing that (BD-\-J)C)=BC, and converting this equa- 
tion into a proportion, we have 

BC : (AB+AC) : : (AB—AC) : (BD—DQ.) 
(This is Prop. 6, Plane Trigonometry, page 149, Robinson's Geometry.) 

ScHo. This proportion is true whatever be the relation of AB 
to AC. It is true then when AB=AC. Making this supposition, 
then BD becomes equal to D C, and the proportion becomes 
BC : AB+AC : : : 0. 

Now (AB-^AC) being two sides of a triangle are greater than 
the third side BC ; therefore the last zero is greater than the first, an 
apparent absurdity. 

But this is no more than saying that zero divided by zero can 
have a positive quotient — for we can subtract zero from zero as 
many times as we please, and still have zero left. 

The proportion is obviously true, for times BC is equal to 
times {^AB-\-AC.) Indeed may be to 0, as a to any quantity 
Avhatever. 

( 27.) The square described on the side of an equilateral triangle is 
equal to three times the square of the radius of the circumscribing 
circle. 

Let ABC be the equilateral tri- 
angle. Let fall the perpendicular 
AE on the base ; it will bisect the 
base. Draw BD bisecting the an- 
gle at B. D will be the center of 
the circumscribing circle, and AD or 
BD will be the radius. 

We are to prove AD=BD^ and 
find the value of BD in terms of AB. 




218 KOBINSON'S SEQUEL. 

Each angle of an equilateral triangle is 60^, (^ of 180°.) 
If we bisect these, each division will be 30°. 
Hence BAD=^30°, and ABJ)=30° ; therefore, AD=BJ). 
Put AB=2a, then BE=a. Also put BD==x, then DE=lx* 
Now in the riifht an^^led trianojle BDE, we have 

Whence, ^a^^-dx"". But ^w" ={ABY . 

Therefore, {ABY^7>{BDY, Q. E. D. 

(28.) The sum of the sides of an isosceles triangle, is less than the 
sum of any other triangle on the same base, and between the same 
parallels. 

Let ABC be the isosceles tri- 
angle. AB=AC. Throuorh the 
point A draw GAH parallel to 
BC. 

Take G any other point on the 
line GH, and'draw i?6^and GO. 

We are to show that AB-\-AO 
are less than^C 6^-|- G C. Produce 
AB to D, making AJ)=AB, or 
AC. 

Then by reason of the parallels GH and B C, the angle BAH 
is equal to the angle ABO, and IIAC=ABC. 

Because AD=AC, the anole ADII= the anerle ACH. 

Whence the two triangles AD If and ACH, are equal in all 
respects, and GB is perpendicular to -DC; whence any point in 
the line GHis equally distant from the two points D and C. 

Now the straight hne BI)=BA-\-AC, and because I>G=GC, 
B G-\- GB= GB+ GC. But X> 6^+ GB, the two sides of a A are 
greater than the third side jDB ; therefore, GB-\-GC are greater 
than BD, that is, greater than B^A-fAC. Q. E. D. ' 

*This mjglit not be admitted, at tlie same time the reader would readily 
admit that BE was one-half AB. ABE is aright angled triangle, one angle 
being 30 deg. the side opposite that angle is half the hypotenuse, and this is 
a general truth. Now the angle DBE equals 30 deg., therefore DE is half 
BD. 




GEOMETRY. 



219 



GEOMETRICAL CONSTRUCTIONS. 



(29.) In any triangle, given one angle, a side adjacent to the given 
angle, and the difference of the other two sides, to construct the triangle. 

Let ^^ represent the giv- 
en side, and from one ex- 
tremity as Ay make the an- 
gle BAC= to the given 
angle, (prob. 5, b. iv.) 

Take AJ)= to the given 
difference of the sides, and 
join DB. From the point B make the angle DBO equal to the 
angle BDC, then CB=CD, smd AD is the given diflPerence of the 
sides, and ABC is the triangle required. 




(30.) In any triangle, given the base, the sum of the other two sides, 
and the angle opposite the base, to construct the triangle. 

Draw AC equal to 

the sum of the sides. 

From the point ^ as a 

center, with a radius e- 

qual to the given base 

AB, describe an arc as 

represented in the fig- 
ure. 

From the point C in 

the line A C, make the 

angle ACB equal to 

half the given angle. 
If the problem is 

possible, this line CB 

will cut the circular arc 
in two points, B and B'. From B and B' make the angles CBD 
and CB'D', each equal to the angle at C. Join AB, AB', and 
either A ABD or AB'D', fulfils the required conditions. 

For CD=DB, and CD'=B'D', (because they are sides of a 
A opposite equal angles,) therefore AD-{-DB=mA C ; also AD'-^- 




220 ROBINSON'S SEQUEL. 

D B'-^AC. The angle ADB is double the angle C, (th. 11, b. 
i,) therefore it is the angle required. 

' (31.) In any triangle, given the base, the angle opposite to the base, 
and the difference of the other two sides, to construct the triangle. 

Subtract the given angle from 180°, and divide the remainder 
by 2, designating the result by a. 

Draw an indefinite line as AC, (see figure to 29,) and take 
AD equal to the given difference of the sides. 

From the point D, make the angle CDB-=-a. 

From ji as a center, with a radius equal to the given base AB, 
strike an arc, cutting DB in B. 

At J5make the angle DBC=a; then DC==BC, and ABC 
will be the triangle required. 

(^^2.) In any triangle, given the base, the perpendicular, and the 
angle opposite to the base, to construct the triangle. 

Draw AB equal to the given 
base, and D C parallel to it at 
the given perpendicular dis- 
tance. 

On the other side of the base 
AB, make the angle BAG e- 
qual to j^art of the gi^i^en angle, 
and ABG equal to the reinain- 
ing part, thus forming the A 
AGB. About the A ABG, 
describe a circle cutting DCm 
the points D and C. Join A C, 
CB, and xiCB is the triangle required. 

The angle BCG=BAG, (th..9, b. iii, scho.), and the angle 
ACG=ABG. Therefore by addition, ACB=BAG+ABG; 
that is, ACB-= the given angle. 

The triangle ADB will also answer the conditions ; for ACB 
=ADB. 

(33.) In any triaiigle, given the base, the ratio of the two sides, 
a7id the line bisecting the vertical angle, to construct the triangle. 




GEOMETRY. 



221 




Draw the base EG, and bisect it in i). 
Draw DB at right angles to EG. 

Divide EO in the point /, so that ^/ shall 
be to IG in the ratio of EH to HG, 

Find IB of such a ralue that 

HI : GI : : EI : IB. 

The three first terms are given ; therefore 
the fourth is known. From / as a center, with the distance IB 
as radius, strike an arc, c^utting DB in B. Join BI and produce 
it to H, making ZST equal to the given distance. Join EH, HG^ 
and EHG is the A required. 

Because HIy^IB=EIy^IG, a circle which passes through the 
points E, B, and G, will also pass through the point H, and the 
angle EHI=IHG, and for that reason EH \ HG \ \ EI \ IG, 
as required. (See th. 25, b. ii.) 



(34.) To draw a straight line through any given j^oint within a 
triangle to meet the sides , or the sides produced, so that the given point 
shall bisect the line so drawn. 

Let ABO be the A, and 
/^ the given point within it. 

Through F it is required 
to dra2v the straight line gl, so 
thai gP shall be equal PL 

From P draw PH paral- 
lel to AB. Take gH=AH. 

Join gP and produce it to I, and gl is the line required. 
P^is parallel to Al, 

gH : HA : : gP : PL ^ 

But gH=HA ; therefore gP=Pl. 

ScHO. Had we taken Hg double of AH, then gP would have 
been double of PI, and we might have required gP to be any 
number of times PL 




Because 



(35.) Find the square roof of \3 or any other number, by a geO' 
metrical construction. 




222 ROBINSON'S SEQUEL. 

Divide the number into any two factors, (say 2 and 6},) add 
the factors to<rether, for the diameter of a circle. 

Take the half sum of the two factors for 
the radius of a circle, and describe the cir- 
cle as represented in the margin. 

Let AB be a diameter, and take AD for 
one factor, and BB for the other ; and 
through I), draw FJS at right angles to^^. 
JDU or DF represents the square root required. 

In the present example, if ^i>=2 and DB=Q^; then the 
length of DE applied to the same scale will show the square root 
of 13. Because ABxDB={DJSy. 

When the two factors are very nearly equal, D will be very 
near the center of the circle, and DF will be very nearly the ra- 
dius of the circle, — always a little less, unless the factors ar«^ 
absolutely equal ; in that case each one is a root. On this prin- 
ciple toe extracted square root in the first part of this volume. 

Observe the A F> CE. CE is the half sum of the two factors, 
and DC is their half difference. 

Also, DE is the sine of the arc AE^ and DC is the cosine of 
the same arc ; therefore, ive can if we desire it, bring in the aid of 
a. table of natural sines and cosities. 

But the tables of natural sines are adapted to radius unity ; lieie 
the radius is 4|, therefore to have corresponding values of CD 
and DEy we have this proportion, 

^ : \ : : ^ : .52941, 

The result of this proportion carried to the table of natural sine ; 
gives .848365 for the corresponding cosine, and this multiplied by 
4J, gives 3.605551 for the square root of 13. 

Another Construction. 

(36.) Let it be required to find the square root of 250, (or any 
other number,) by a geometrical construction. 



# 



GEOMETRY. 



223 




Divide the number into two factors. 
Let one factor be represented by AB, 
the other hy AC; BC being their 
diflference. On the difference as a 
diameter, describe a circle. 

From the extremity A, draw AD 
touching the circle. AD represents 
the square root required. 

By (th. 18, b. iii, scho. 1), 
ABXAC={AD)K 

Therefore AD is the square root 
of the product of the two factors 
AC 2iTidiAB. 

Reinark. When the two factors are nearly equal, the circle will 
be very small, and AD will be very nearly Ao. But xio is the 
half sum of the factors AB and A C, hence we know that the 
square root of the product of two factors is always a little less than 
their half sum, unless the factors are absolutely equal. 

In the proposed example, we divide 250 into the two factors, 
25 and 10 — their diflference is 15. Hence 7^ is the radius of the 
circle. Take 7|- from any scale of equal parts in the dividers, 
and describe a circle. 

Draw any diameter as B C, and produce it to A, making AB=^ 
10. From A draw^i) to touch the circle ; take that distance in 
the dividers and apply it to the scale, and the result will be the 
square root of 250. 

The practical difficulty in this construction is to decide exactly 
where the point D is, therefore the first method of construction 
is the best. 

Geometrical constructions are not to be relied upon for numer- 
ical accuracy, but they are invaluable to impress theory, and are 
sure guides to numerical operations. 

Scho. If it were required to make a square equal to a given 
rectangle, either of the two preceding constructions may be applied. 
Let ^C be one side of the rectangle, AB the other; then AD 
will be a side of the required square. 



224 ROBINSON'S SEQUEL. 

PROBLEMS. 

The following problems do not admit of geometrical construc- 
tions, in the sense of some of the preceding — they require alge- 
bra applied to geometry. 

We take the problems from Robinson's Geometry, pages 105 
to 109. For theory, the reader must look elsewhere. 

We omit the first two problems, and number them as they are 
numbered in the geometry. 

PKOBLEM 3. 

In a triangle J having given the sides about the vertical angle, and 
the line bisecting that angle and terminating in the base, to find the 
base. 

Let ABC he the A, and let a circle be 
circumscribed about it. Divide the arc AliJB 
into two equal parts at the point jE, and join 
£JC. This line bisects the vertical angle, 
(th. 9, b. iii, scho.) Join BU. 

Put AD=x,' JDB=g, AC=a, CB=b, 
CD=c, and DjE=w. The two A's, ADC 
and JiJB C, are equiangular ; from which we have, 

w-\-c : b : : a : c. Or, cw-^c^ =ab. (1) 

But as JtJC and AB are two chords that intersect each other in 
a circle, we have, cw=xy (th. 17, b. iii.) 

Therefore, xg-\-c'^=ab (2) 

But as CD bisects the vertical angle, we have, 

a : b : : X : y (th. 23, b. ii.) 

Or, x=^-l (3) 






Hence, -y' -\^^ zz=ab : or, y=J( 



b" ' ' " \j ^ 



And, x=?J6»-el* 

b^ a 

Now as X and y are determined, the base is determined. 
N. B. Observe that equation (2) is theorem 20, book Hi. 




GEOMETRY. 225 

PROBLEM 4. 

To determine a triangle, from the base, the line bisecting the ver- 
tical dngle, and the diameter of the circumscribing circle. 

Describe the circle on the given diameter 
AB, and divide it in two parts, in the point 
i>> so that ADxI>B shall be equal to the 
square of one-half the given base. 

Through D draw ED G at right angles to 
AB, and EG will be the given base of the A- 

Put AD=^n, DB:=m, AB=d, DG=b. 

Then, n-\-m=d, and nm=:b^ ; and these two equations will 
determine n and m ; and therefore, n and m we shall consider ijis 
known. 

Now suppose EHG to be the required A, and join TUB and 

HA. The two A's AHB, DBF, are equiani^ailar, and therefore, 

we have, 

AB : HB : : IB : DB. 

But BI is a given line, that we will represent by c ; and if we 
put IB=iv, we 8hall have IIB=c-\-w ; then the above proportion 
becomes, d : c-f-w : : w : m. 

Now w can be determined by a quadratic equation ; and there- 
fore IB is a known line. 

In the right angled A DBI, the hypotenuse IB, and the base 
DB, are known ; therefore, I>I is known, (th. 36, h. i); and if 
i)/is known. Eland IG are known. 

Lastly, let EH—x, IIG=y, and put EI=p, and IG=q. 

Then by theorem 20, book iii, pq-\-c^:=xy (1) 

But, X : y : : p : q (th. 25, b. ii.) ^ 

Or, x^^l (2) 

q 

And from equations (1) and (2) we can determine x and y, the 
sides of the A ; and thus the determination has been attained, 
carefully and easily, step by step. 

PROBLEM 5. 
Three equal circles touch each other externally, and thus inclose one 
acre of ground ; what is the diameter in rods of each of these circles ? 
15 



226 



ROBINSON'S SEQUEL. 




Draw tliree equal circles to touch each 
other externally, and join the three centers, 
thus forming a triangle. The lines joining 
the centers will pass through the points of 
contact, (th. 7, b. iii.) 

Let H represent the radius of these 
equal circles ; then it is obvious that each 
side of this A is equal to 2JR. The triangle is therefore equilat- 
eral, and it incloses the given area, and three equal sectors. 

As each sector is a third of two right angle*, the three sectors 
are, together, equal to a semicircle ; but the area of a semicircle, 

whose radius is M, is expressed by - -*^-- (th. 3, b, v, and th. 1, 

b. v); and the area of the whole triangle must be -f"^^^ * 

but the area of the A is also equal to Ji multiplied by the per- 
pendicular altitude, which is BJS. . 



Therefore, 
Or, 

Hence, 



i22j3=---+160. 

2 

723(2^3— rt)=320. 
320 



320 



2^3—3.1415926 0.3225 
i?= 31.48-1- rods for the result. 



:992.248. 



PROBLEM 6. 

In a right angled triangle, having given the hose and the sum of the 
perpendicular and hypotenuse, to find these two sides. 

Let ABC he the A. Put CB= 
h, AB+AC=a, AB=x ; then AC 
=a — X. 

By (th. 36, b. i), 

a'—b^ 



Whence, 



2a 




Now the numerical value of x being known, the triangle can 
be constructed geometrically. 



GEOMETRY. 



227 



PROBLEM 7. 

Given the base and altitude of a triangle^ to divide it into three 
equal parts, by lines parallel to the base. 

Let ^^ 6" represent the A. Conceive 
a perpendicular let drop from C to the 
base AB, and represent it by b. Put 
2a=AJB. Then ab= the area of the 
triangle. 

Let X be the distance from C to FD ; 
then by (th. 22, b. ii), we have, 

x^ : b^ : : ^ab : ab 
Whence, x ; b : i \ \ J^, 

If X represents the distance from C to GE, then 

x^ : b^ : : f«6 : ab. 
Or, X : b : : Ji t J3, ^=^ 

We perceive by this tliat the divisions of the perpendicular are 
independent of the base, and that we may divide the triangle into 
any required number of parts, m, n, p, <fec., equal or unequal. 




PROBLEM 8. 

In any equilateral triangle, given the length of the three perpendic- 
ulars drawn from any point within, ta the three sides, to determiiie 
the sides. 

Let ABC be the A- We have 
shown in this volume tliat CD= 
PG+PH-{-PO=a. 

Put AD or VB^x ; then BC^ 
2x. 

Then by th-e right angled A 
CDB, we have a~-|-.r2=4x'", or 

V3 




298 ROBINSON'S SEQUEL. 

PROBLEM 9. 

In a right angled triangle^ having given the base (3), and the dif- 
ference hettoeen the hypotenuse and perpendicular (1), to find these 
sides. 

(See figure to Problem 6.) 

Let (7j5=3. AO—AB^l. AB=x. Then ^(7=a:+l, and 
ara+9=aj2+2a'+L x=4. 

PROBLEM 10. 

In a right angled triangle, having given the hypotenuse (5), and 
thediference between the base and perpendicular (1), to determine both 
of these tivo sides. 

(See figure to Problem 6.) 

Let CB=x. AB=x-\-l. Then 

Or, 2x^+2x=24. Whence, x=3. AB=4. 

PROBLEM 11. 

Having given the area or measure of the space of a rectangle in- 
scribed in a given triangle, to determine the sides of the rectangle. 

When we say that a triangle is 
given, we mean that the base and 
perpendicular are given. 

Let ABC be the triangle, AB=h, 
CD=p, CI=x ; then ID=p—x. 

By proportional triangles we have 
GI : EF : : CD : AB 

That is, X : EF I : p : b. EF=—. 

P 

By the problem —(p—x\—a. The symbol a being the giv 
en area. 

Whence, x^—pxz=--^, x=^p:l ^\p^^?^. 




GEOMETRY. 229 



PROBLEM 12. 



In a triangle having given the ratio of the two sides, together with 
both the segments of the base, made by a perpendictdar from the vertical 
angle, to determine the sides of the triangle. 

Let ACB be the A, (see last figure.) AD=a, BDr=b, and 
CD=x. Then AC=Ja^-{-x^, and CB=Jb^+x^. 

The ratio oi AC to CB is given, and let that ratio be as 1 to 
r ; then 

Ja^+x^ : j¥+x^ : : 1 : r. 

Whence, «2_|_^2 . 52_j_^3 : : i : rK 

Or, b^+x^=a^r''+r''x'' 

Or, 



a^r^-^b^ 



But AC=:Ja^-\-x'^, and as x^ is now known, ^Cis known. 

PROBLEM 13. 
In any triangle having given the base, the sum of the other two sides 
and the length of a line drawn from the vertical angle to the middle of 
the base, to find the sides of the triangle. 

Let ADE be the A- Suppose C 
to be the middle of the base. 

Put AC=a, DO or CE=^b, AE 
=x, DA^AE=c ; then DA:=c—x. 

Now by (th. 39, b. i), we have 
(DAy+{AEY=^(ACy-\-2{DC)~ 

That is, c2— 2car+2a;2=2a2_|-262. 

Or, 4a;2—4ca:+c2=:4a2 +462—^2. 

Zx—c= ^4a2+4p— c2 . 

Whence x becomes known, and consequently the sides become 
known. 

PROBLEM 14. 
To determine a right angled triangle, having given the length of 
two lines dravm from the acute angles to the middle of the opposite 
sides. 




idO 



ROBINSON'S SEQUEL. 



Let ABC be the triangle. Letter 
it as represented — CE^=a, AI>==b, &c. 

Then \^^'+y'=^''\ 

By add. 5x^ +^'' =a^ ■\-b^ =din. 



3^ 



2 «2 



a* — m. 




x=Ja^ — m. y==.Jb^ — m. 
~T~ "~3~" 

PROBLEM 15. 

To determine a right angled triangle, having given the perimeter 
and the radius of its inscribed circle. 

Let ^^C be the 
A, OjE'the radius 
of the circle. 

It is obvious that 
AE=A£>, CF= 
CD. Put AE=x, 
CF^y, FB^T, 
^p=. the perime- 
ter. Then by the 
conditions, 
:t+y-fr=p (1) 

From the right 
angled A ABC, 
we have 

(x+rY+(y+TY=^(x+yy 
By reduction, 

rx-\-ry-\-r^ =xy 

That is, (x-^y-\'r)r^=^rp^=xy 

Equation (4) expresses the area of the triangle. 
From (1), x^'-^-^xy+y' 
From (4), Axy = 




(2) 
(4) 



'2 — =o2 — 2»r-4-r^ . 



Apr 
By subtraction, x^ —2xy-\-y^ =p^ — 6pr-|-r» . 



GEOMETRY. fftt 



Whence, x — y= ± Jp ^ — 6jt?r-[-r - . 

But x-{-y—p — r. 



Therefore, x=z^{p—r)±^Jp''—Qpr+r^. 

PROBLEM 16 

To determine a triangle , having given the base, the perpendicular, 
and the ratio of the two sides. 

Let ABO be the A- AB=b, 
CD=a, J)B=x. Then 



CB:=/a:^^+^. 
Let the given ratio of the sides be 
as fn to n ; then 




J(b — x)^-\-a^ : ^a^-\-x'^ : : m : n. 
This proportion will give the value of or, then AC and CB 
will be known. 

PROBLEM 17. 

To determine a, right angled triangle, having given the hypotenuse, 
and the side of the inscribed square. 

Let J[i) (7 be the A. (See last figure.) Put CI—x,IE=ia, 
A G=y, and A C=h. Then by proportional triangles, we have 
CI : lU : : JfJG : GA. 
That is, X : a : : a '. y. Whence, 2-5^=0^. 

In the right angled A's AGE, ECL we have 

AE^ Jf~+a^. EC= Jx^^a- . 
Observing that AE-\-EC=AC—b, and a~—xy, we perceive 
that 



Jx^-\-xy-\'Jy~-{-xy^h. 

Whence, »Jx+Jy=. A^^ . 

■_ 7 3 

By squaring, x-\-y-\-^Jxy—-,!—- . 

x+y ___ 

Put (a?+y)=-5, and observe that ^Jxy=^Za ; then 
52 -f. 2^5=^2 _ 



232 



ROBINSON'S SEQUEL. 



Whence, «+a= zt^a^-j-i^. 

Now having the value of (x-^-y), and (ary) the separate values 
of X and y can be determined, which is a solution of the problem. 

PROBLEM 18. 

To determine the radii of three equal circles, inscribed in a given, 
circle to touch each other, and also to touch the circumference of the 
given circle. 

Let AD 
B be the 
given cir- 
cle. Di- 
vide the 
circumfe- 
rence 360 
deg. into 
3 equal 
parts. BD 
is one of 
those parts 

120° ; then the arc ^i)=60°. A circle inscribed in the A 
COE, will be one of the equal circles required. 

Let A 0=a, AH=x, H being the center of the circle. From 
H, draw i/F" perpendicular to CO, then AH=:HV. 

Hence FIVz!=x, OH=a — i?-, and F= J- 0^, because the an- 
gle F^0=30°. (See prop. 1, plane trig., page 139.) 

Now by the right angled A VH, we have 
{OV)^-[-{VHY={OHy. 

That is, (^^i::fy+a:2=(a— ar)«. ^ 

Whence, a;={273— 3)a. 

PROBLEM 19. 

In a right angled triangle, hamng given the periimter, or sum of 
all the sides, and the perpendicular let fall from the right angle on the 
hypotenuse, to determine the triangle, that is, its sides. 




GEOMETRY. 



238 



Let ABC he the A, and represent its 
perimeter byjo. Put AI>=:a, AB:=x, 
A C=y. Then B C=zp—x—y, 

Because BA C is a right angle, 

x^-\-y^-=p^—2p{x-\-y)-\-x^-\.2xy+y 
And, a{p — x — y)=xy 

Reducing (1), ^p{^-\-y)='P^-\-^y 

Double (2), 'Hap — 2a{x-^^y) = '2.xy 

By subtraction, (2a-j-2p) (^+y) — ^op=p^ 




Whence, 



x-\-y 



_p^-\-2ap 



(1) 

(2) 
(3) 
(4) 
(6) 

(6) 



Because BC=p-x^, BC=p-t±^l= ^t.._ 

2«+%) 2rt+2/) 



OjB" 



From (2) we observe that xy 

^ ' 2a4-2p 

Equations (6) and (7), will readily give x and y. 



(7) 



PROBLEM 20. 

To detennine a right angled triangle, having given the hypotenuse 
and the difference of two lines, drawn from the two acute angles to 
the center of the inscribed circle. 

Let ABC be the 
A, the center of 
the inscribed circle; 
then A bisects the 
angle CAB, and 
CO bisects the an- 
gle C. 

The angle A OH, 
being the exterior 
angle of the trian- 
gle A C, it is e- 
qual to CAO-\- 
ACO,i\mi'is,AOH 
is equal to half the 
sum of the angles 
CAB, BCA, or to 45°. Produce CO to II; from A let fall All 




234 ROBINSON'S SEQUEL. 

perpendicular on CA. Now in the A A OH, because ^=90°, 
and ^0//=45°, OAII=45'', and consequently AJI= Off. 

Put AC=a, AO=x, OC=x+d, Off and Aff, each equal to 
y. Then Cff=x-\-i/+d. 

In the A AffO, we have 2y2_^2 ^jj 

In the A ^^C', we have 

(x+y+d)'-\.y^=.a^ (2) 

Expanding, 

x''-{-y^-{-d'+{2x-\-2d)?/-\-2dx+y'' =a^ (3) 

Substituting the value of y^ and y from (1), and 

2x^-{-(2x+2d)-^-+2dx=a''—d^. 

Or, 2x''+j2'x''-\-j2dx-{-2dv=a''^^\ 

Dividing by (2-[->/2), and we have 
2 I J «^ — <^^ 
^ 2+V2 

Whence, ar = — ^±^/m4-^^ 

PROBLEM 21 

To detei-mine a triangle, having given the base, the 2^^fp€ndicular, 
and the difference of the two sides. 

(See figure to Problem 19.) 
Let ABC be the A- Put BD=^x, DC^y, AC=z, AB= 
z+d, AD=a, BC^h. 

By the conditions, ar-|-y=6 (1) 

x^-\^^-=z'-\-2dz-{-d' (2) 

y'+a'=z^^ (3) 

By subtraction, x^—y^=2dz-\-d^ (4) 

Factoring, (^+y) (^ — y)=d(2z-\-^) 
That is, b(x—y)=d{2z+d) 

From this we have the proportion, 

b : (2z+d) : : d : (x—y) 

This proportion is the following rule given in trigonometry, viz : 

In any plane triangle, as the base is to the sum of the sides, so ii 
the difference of the sides to the difference of the segments of the base. 



GEOMETRY. t$6 

We return to the solution. From ( 1 ) we have 

rr=a — y, whence a;^ — y'^=:ar — 2ay. 
From (3), z=Jy^-\-a^. These values put in (4), give 
a^ — 2ay=2c?7y2+^+c?2 



Squaring, (^a^—d^Y—Aa{a^—d'' )2/+4a-r =^d''y''+^o''d^ 
Or, (a2— ^a^a_4^(„2_^2 jy_|_4(g2__^2 yf^^a^'d- 

4^ 2 ^2 

a^ — d^ — ^ay-\-^y^ ==- - 

a 2 — <;- 

a . , '. 2 ,o , 4«2(/2 5«2J2_^4 



ar^2y=±:dj^ 
^ a 






Whence, y=-=p-( I 

^ 2^2\ a^—d^ / 



PROBLEM 22. 

To determine a triangle, having given the base, the perpendicular, 
and the rectangle, or product of the two sides. 

(See figure to Problem 19.) 

Let ABGhQ the A. Put BD=x, DC^y, BC=b, AD=za, 
and the recangle, {AB) (AC)=c. 

Now in the right angled triangles, ADB, ADC, we have 





AB=Jx^--\-aK AC^Jy^+a\ 




Whence, 


Ux-+a-){Jy^^+a^^)==c 


(0 


And, 


x+y=l 


(2) 


From (1), 


x^y^+a^x'+y')+a'=c' 


(3) 


From (2), 


x^-\-y''=b''—tcy 


(4) 



This value substituted in (3), gives 

x^y^J^a^b^—'^a^xy-{-a^ =c* 
x^y^—^^xy+a^—c^—aH^ 



2A2 



xy-^a^ = ±:Jc^—aH 
xy=a''±:Jc^—a^b^ (6) 



236 



ROBINSON'S SEQUEL. 



From equations (2) and (5) the values of x and y can be de- 
termined. 



PEOBLEM 23. 

To determine a triangle, having given the length of the three lines 
dravmfrom the three angles to the middle of the opposite sides. 

Let ABC be the A. 
Bisect the sides AB in D, 
AC in F, CB'mE. 

Put AE=^a, BF=h, CD 
=e, AD=u, AF=x, BE 

Now by (th. 39, b. i), 
we have, 

x^+b^=4u^-\-4g^ 




(1) 
(2) 

(3) 



By addition, a^-\-b^-\-c^=7(x'--\-y'^+u'') 
Whence, ^a^^^-^-c'^^^ix^Jl^iy^+iu'- 
From (1), 2^2^c2 = 4a;2-f4y2 



(4) 



By subtraction, ^a^-{'b^-\-c^)—u^—c^=4u^ 
Or, 4a2_[-452_j_4(.2_7^^2_7c2^28w2 

4a3_J_462_3c2=:35«<2 



By inference 
And, 



t^-±J^' 


2+4*2- 


-3c2 


V 35 


^-±J4«- 


2+4c2- 


-362 


rt^ 


36 




V=r^J'^ 


2+462- 


-3a2 



36 



PROBLEM 24. 

In a triangle, having given the three sides, to find the radius of the 
inscribed circle. 



GEOMETRY. 



237 




Let ABC be the A. 
From the center of the cir- 
cle 0, let fall the perpen- 
diculars OG, OE, OD, 
on the sides. 

These perpendiculars are 
all eqiial, and each equal 
to the radius required. 

Let the side opposite to 
the angle A, be represent- 
ed by a, the side opposite B by h, and opposite C by c. Put OE, 
OD, (fee. equal to r. 

It is obvious that the double area of the A BOC is expressed 
by ar ; the double area of A OB by cr ; the double area of ^ C 
by hr; Therefore, the double area of ABC is (^a-{-h-\-c)r. 

From A let drop a perpendicular on BC, and call it x. 

Then cw= the double area of ABC. Consequently, 
{^a-\-b-\-c)r=iax ( 1 ) 

The perpendicular from A will divide the base BC into two 
segments, one of which is Jc^ — x'\ the other, Jb'^ — x^, and the 
sum of these is a ; therefore, 



(2) 



'^=a^—2aJb^—x^J^b^—x'- 
^aJb^—x^=a^-]-b^—c^ 



J^'' 



-X^=z 



2a 



Whence, 

Or, ar=^62 — m"^ 

This value of a; put in (1), gives 



Whence, 



__ajb''—m^ 
a-\-b-\-c 



the required result. 



PROBLEM 25. 
To determine a right angled triangle, having given the side of the 
inscribed square, and the radius of the inscribed circle. 




23^ ROBINSON'S SEQUEL. 

Lei O be the center of a cir- 
cle, OH or OL=ir, the given 
radius, BE or ED^=a, a side of 
the given square. 

BO'i^ the diagonal of r-, and 
BD is the diagonal of «-, and 
B OD is one continuous line. 

The point D of the given 
square may be in the circle, in 
that case the hypotenuse touch- 
es the circle and the square in 
the same point, and that point is the middle of the hypotenuse. 
If the point D is not on the circumference, it must be without, as 
liere represented. 

Draw Dt to touch the circle in t, and that line produced both 
ways will define the hypotenuse. 

^1(7 and ^C will meet if produced, and ABC \\\\\ be the tri- 
angle required. 

OB^J<Zr, BD=j2a, £>0=(a—r)j2, J)V=(a—r)j2-^r, 

Now as i) is a point without a circle, and Dt touching it, we 
have by (th. 18, b. iii), {Dty-==DVxDU; that is, 

(i)/)2 = [(a_r)72~r] [(a— >-)72-fr ]=2a=— 4ar-fr2. 



Whence, Dfz=jZa^ — 4ar-|-r2=:c. 

Because A is a point without a circle, and AH, At, lines drawn 
touching the circle, AH=At, (th. 18, b. iii, scho. 2.) 

Observe that KH=a — r=d. Put AH, At, each equal to x ; 
then in the A AXB we have AK=:x — d, AD==x-\-c, KD^c 

Whence, (A'-|-(-)^=a--j-(.tr — d)-. 

x^ +2CX+C'' =a2 J^x'^^Stdx-^d* 



26+2^ 
Now the value of x being known, AB is known, and all the 
sides of the A AKD. But the A AKD is proportional to the 
triangle ABC, and gives us 

AK : KD ', i AB : BC 



GEOMETRY. 



239 



The first three terms of this proportion being known, the last 
is known, and the triangle is fully determined. 

PROBLEM 26. 

To determine^ a triangle and the radius of the inscribed circle, hav- 
ing given the lengths of three lines dravm from the three angles to the 
center of that circle. 

Let ABC 
be the A, 
the center of 
the circle. 

Put^O= 
a, OB=ic, 

OC=:h. AO 

bisects the 
angle A. 

Produce 
AO to D, 
Then because 
the angle A is bisected, CD : DB : : AC : AB. 

Put AB=x, AC=^y, and let the ratio of AB to BD be n\ then 
nx=BD and ny=^ CD. 

Now as the angle C is bisected by (70, we have 
AC : CD : : AO : OB 

That is, y : ny : : a : OD 

Whence, OD—na. 

Because AD bisects the angle A, we have, (th. 20, b. iii), 




Also, 
And, 

From (1), 



xy=^a^ {\'\-nY -^-n'^xy 
nx^ =c^-\-na^ 
ny^=b^-\-na^ 



xy. 



a'^ilJ^ny _a^(\^n) 



\—n 



1— w2 
The product of (2) and (3), gives 

n^x^y'^—{c^-\-na^) {h^^rui^) 
Squaring (4), and multiplying the result by n'^, also gives 



(1) 

(2) 
(3) 

(4) 
(5) 



»««y^ 






(6) 



240 ROBINSON'S SEQUEL. 

Equating (5) and (6), gives 

This equation contains only one unknown quantity n, but it 
rises to the fourth power — hence this problem is not susceptible 
of a solution from this notation short of an equation of the fourth 
degree. 

In cases where a, b, and c are numerically given, the solution 
may be possible through an equation of the second or third degree. 

We perceive by the figure, that if b=:c, x must equal y. 

PROBLEM 27. 

To determine a right angled triangle^ having given the hypotenuse 
and the radius of the inscribed circle 

Let ABC be the 
A, £^0 the radius 
of the circle. AU 
=AI)= X, CD= 
CF=zy. Then AB 
=.r+r. BC=y-{-r. 

By the right an- 
gled triangle, 

={x+yy {\) 
x-\-y=a (2) 
Reducing (1), 
gives xy=^rx-\-ry 
+r-. 

That is, xy=ar-^r^ (3) 

From (2) and (3), x and y are easily found. 




In numerical problems, great advantage can be taken of mul- 
tiple numbers, the same as we have shown in common algebra. 
The following example will be sufficient. 

The sum of the two sides of a 2)lane triangle is 1 1 55, the perpen- 
dicular drawn from the angle included by these sides to the base, is 



GEOMETRY. 



241 




I I \ 

HUB 



300 ; the difference of the segments of the hose is 495. WIicU are the 
lengths of the three sides? Am. 945, 375, 780. 

Write the given numbers in order, thus, 300, 495, 1156. Di- 
vide them by 16, and their relation is 20, 33, 77. 

The two latter numbers have a common factor 1 1 , which call 
a. Put 5=20. 

Then the three given lines will be h, 3a, and 7a. 

Let CB=x,. then AC=7a — x. 
BD=y, then AD=y-\-3a. CD=.h. 
In the right angled A CDB, we 
have y2_|_j2^^2 (1) 

AD Q gives 

(y_[.3a)2+52==(7a— :r)2 (2) 
Expanding (1) and subtracting (2) from it, gives 
6ay+9a2 ==49^,2 — i4«a; 
3ay=20a2 — '^ax 
Divide by a and write h in the place of 20, then/ 

Sy=ab — Ix 
Squaring, 9y^=a''b''—14abx+49x^ 

From (1), 9y2=_962 + 9x^ 

By subtraction, 0=:(a^-\-9)b^—l4abx-{'40x' 
Divide by & (or 20), then 

0=(a^-\'9)b—14ax-\-2x^ 
4a;2— 28aa;=— 2a2 b—l 8b 

Add (49a2), 4x^-^2Qax+49a^=49a^—2an^l8b 

=9a2— .360=729 
By evolution, 2x — 7a = ±27 

2a:=77±27=50, or 104 
a:=25, or 54 -» 

Here 25 is the number that corresponds with the problem ; 
therefore, jB (7=25- 15=375. We multiply by 15, because we 
reduced the numbers in the first place by dividing by 15. 
16 



242 ROBINSON'S SEQUEL. 

SECTION II. 

TRIOONOJWETRir. 

We shall here attempt to show the most practical method of 
finding the circumference of a circle to radius unity ; and of 
finding the sines and cosines. 

The trigonometrical equations that we may call into immediate 
use, are the following : 

We number them as they are numbered in Robinson's Trigo- 
nometry. 

sin.(a-|-5)=sin.a cos.6-l-cos.a sm.h (7) 

sin.(a — J)=sin.acos.6 — cos.asin.6 (8) 

cos.(a-|-6)=cos.a cos.6 — sin.a sin.6 (9) 

cos.(a — J)=cos.a cos.5-|-sin.a sin.5 (10) 

sin.2a+cos.^a=l (1) 

sin.2a=2sin.a cos.a (30) 

Or, sin.a=2sin.^a cos.^a 

By problem 23, book iv. of Robinson's Geometry, we learn that 
if we divide the radius into extreme and mean ratio, and take the" 
greater segment, that segment will be the chord of 36°. 

Let 1 be the radius of a circle, and x the greater segment re- 
quired ; then 

1 : X : : X : 1 — x 

Whence, a:=—i-f ^^5=0.6180340, the chord of 36° in a 
circle whose radius is unity. 

We learn by theorem 6, book v, Robinson's Geometry, that 
when c represents any chord of a circle, and x a. chord of one- 
third of that arc, the following equation will exist : 

x^ — 3x= — c. 
Put c=0.6 18034000, and a solution of the equation gives the 
chord of 12°. Again, put c equal to the chord of 12°, and an- 
other application of the equation will give the chord of 4°, and 
thus by the successive application of this equation, we have found 
the following values : 



TRIGONOMETRY. 24» 

1. The chord of 36°=0.6 18034000. 

2. The chord of 12°=0.209056903. 

3. The chord of 4°=0.069798981. 

4. The chord of 80=0.023270628, 

By theorem 4, book v, we learn that if c represent the chord 
of any arc, the chord of hcdf that arc will be represented by 



Having the chord of 80' the preceding expression gives us the 
chord of 40', 20', and 10', as follows : 

Chord of 40'=0.01 16355131. 
Chord of 20'=0,0058 177679. 
Chord of 10'==0.0029088819. 

The chord of 10' so nearly coincides with the arc of 10', that 
for all practical purposes, we may consider the chord and arc 
the same ; then the semicircumference must be 1080 times 
0.0029088819, or 3.141592462. A more exact determination 
gives 3.141592653+ for the length of 180'', when the radius is 
unity. 

The chords of all arcs under 10' cun be found from that chord, 
directly/ hy division. 

As the sine of an arc is half the chord of double the arc, 
therefore, we can have the natural sine of 18° by dividing the 
chord of 36° by 2. 

Having the sine of any arc, we can find its cosine by the fol- 
lowing equation : 

cos. «s=^l — sin.^a 

When sin.^a is a very small fraction, as it is for all arcs under 
10', then ^1 — sin.^a is very nearly equal to (1 — ^sin*a). 

By the foregoing we find the following sims and cosines : 

sin. r=.000^.908881 cos. 1'=.9999999576 

sin. 2'== .00058 17762. cos. 2'=. 9999998802 

sin. 3'=.0008726643 cos. 3'=.9999996692 

sin. 4'=:.001 1635524 cos. 4'=,9999993231 



244 ROBINSON'S SEQUEL. 

sin. 5'=.001 4544405 cos. 5'=.9999989423 

sin. 6'=.0017453286 cos. 6'=.9999984769 

sin. 7'=.0020362167 cos. 7'=.9999979269 

sin. 8'=. 002327 1036 cos. 8 =.9999972926 

sin. 9=.0026179916 cos. 9'=.9999965731 

sin. 10'=.0029088789 cos. 10'=.9999957689 

sin. 20 =.0058177378 cos. 20'=. 9999830770 

sin, 30'=.0087265343 cos. 30'=.9999618877 

sin. 40'=.01 16352640 cos. 40'=.9999323090 

From the chords of 4°, 12°, and 36°, we readily find 

sin. 2°=. 0348995000 cos. 2°=.9993908139 

sin. 6°=. 1045284515 cos. 6°=. 994521 8389 

sin. 18°=.309017000O cos. 18°= .951 06466 19 

Having the foregoing sines and cosines, we can find the sines 
and cosines of certain other arcs as follows : 

Put 2a= to any arc whose sine is known, then we can obtain 
the sines and cosines of the half of 2a, or a, by the following 
general equations : 

cos.^a+sin.^a=l (1) 

2cos. a sin. a = sin. 2a ( 2) 

Now if we suppose 2a=18°, we have sin. 2a=.3090 170000 ; 
and by substituting this value of sin. 2a, and adding and subtract- 
ing the equations, we shall have 

cos. 2 a+2cos.a sin. a+sin.^ a== 1 .3090 1 70000 ( 3) 

and cos.^a— 2cos.asin.a+sin.2a=0.6909830000 (4) 
By extracting the square root, 

cos.a-f-sin.a=l. 1441228508 (5) 

cos.a— sin.a=0.8312532699 (6) 
By adding (6) and (6), and dividing by 2, we find 
cos.a=cos.9°=. 9876880603 



TRIGONOMElTlEtY: 246 

Subtracting (6) from (5), and dividing by 2, gives 
sin.a=sin.9°=. 1 564342904 

If we put 2a=6°, a like operation will give the cosine and sine 
of 3°. If we put 2a=2°, a like operation will give the cosine and 
sine of 1°, and so on. 

Again, we must not overlook the fact that the cosine of 2° is 
the same value as the sine of 88° ; therefore, if we put 2a=88°, 
an operation like the preceding will give us the cosine and sine of 
44°. Another operation will give us the cosine and sine of 22°, 
and still another of 11°, and so on. 

If we require the cosine and sine of any particular arc that we 
cannot arrive at, by any of these subdivisions, we may apply th§ 
following equations : 

sin. (a-(-6) =sin.a cos.6-j-cos.a sin.5 (7) 
sin. (a — &)=sin.acos.6 — cos.asin.6 (8) 

For example, if a=6° and 5=1°, and we have the cosine and 
sine of 6° and 1° ; then (7) will give us the sine of 7°, and equa- 
tion (8) will give us the sine of 5°. 

Whence by equations (1), (2), and (7), (8), the sine and co- 
sine of every degree of the quadrant can be obtained without the 
trouble of fractional parts of degrees. 

But there is a better method to continue the table after a begin- 
ning has been made, which we illustrate by the following example : 

Suppose we have the sine and cosine of 15° and 16°, and also 
the sine and cosine of each of the small arcs from zero to 5° or 
6°, and require the sine and cosine of 17° or of any other arc 
under 20°, we would operate as follows : 

Let the arc ^J9=15°, ^i>=2°; then 
^^=17°, ^6^=17°, i>6^=17°+16° 
=32°. 

Draw the chord BD. Now because an 
angle at the circumference is measured 
by half its subtended arc, therefore, 
the angle 7ii?i)=16°. The chord BD 
is double the sine of 1°; and it is obvi- 
ous that we have BD and all the angles of the small right an- 




246 ROBINSON'S SEQUEL. 

gled triangle nBD ; and if we compute Bn, and add it to DH, 
the sine of 15°, we shall have BJEy the sine of 17° ; and nD sub- 
tracted from CH\ will give the cosine of 17°. 
The computation is as follows : 

(We use the logarithmic sines and cosines, diminishing the indices by 10. 
to correspond with radius unity in the table of natural sines.) 

Log. sine 1° —2.241855 

Log. 2 , .301030 

Log. of ^i> 2.542885 —2.542885 

sine 16°... .—1.440338 cosine . . .—1.982842 
»i>.... 009621 —3.983223 %JS .033554— -2.525727 

, Nat. cos. 15° .... 965 93 Nat. sin. 15° .258820 

Nat. cos. 17°. . ..95631 Nat. sin. 17° .292374 

Thus we can go on and compute the sine and cosine of 19°. 

RemarJc. When the triangle nBD is taken sufficienly small, the 
chord BD is confounded with the arc, and the triangle is then 
called the differential triangle, and figures largely in the differen- 
tial calculus ; and by it we can readily compute the sine and cosine 
of (16° r), (15° 2'), &c., having the sine and cosine of the de- 
gree, whatever it may be. 

If we were making a table of sines and cosines for every min- 
ute of the quadrant, it would require too much labor to use the 
foregoing equations for every minute, we would use them for 
every degree, and then fill up the sines and cosines for the inter- 
mediate minutes by 

INTERPOLATION. 

In the appendix to Robinson's University Algebra, standard 
edition, is the following formula for inserting any intermediate 
term of a series : 

In this formula a is the first term of a series consisting of a, a^, 
^2, ttg, &c., terms, b is the first term of the first difference, c is 
the first term of the second difference, and so on. The interval 



TRIGONOMETRY. 



«47 



between two given numbers in the series is always to be taken as 
unity, therefore, % is a fractional part of that unit. 
The following example will clearly illustrate. 

1. Given the sines of 1°, 2°, 3°, 4°, 6°, and 6°, to find the sines 
of 1° 12', 2° 12', 3° 12', and 1° 24', 2° 24', or to find the sine of 
any arc between 1° and 3° by interpolation. 



(*«) 


Ist diff. 


Sddiflf. 


3d diflf. 


sin. 1°=.0174524035 


(-H) 


{-^) 


{-d) 


sin. 2°=.0348995000 


.0174470965 






sin. 3°=.O523359508 


.0174364508 


106457 




sin. 4°=.0697664685 


.0174205177 


159331 


52874 


fiin. 5°=.0871557450 


.0173992765 


212412 


53081 


sin. 6°=.1045284515 


.0173727165 


265600 


53188 



To interpolate 12', we must put n of the formula = ||. 
2 n — 1 4 n — 2 6 



Whence, 



And, 



10 

2 n — 1 
%= — n* : 

10 



10 



n'. 



1 n—\ 



10 
48 



2 100 2 3 1000 

The products will be positive or negative according to the rules 
of multiplication. For the sine of any arc between 1° and 2°, we 
take the first line of the column under a for the first term of the 
series, and the first line of the column under h for the first dif- 
ference, and so on. 

To find the sine of any arc between 2° and 3°, we must take 
the second line of the column under a for the first term of the 
series, and the second line of the column under h for the first 
difference, and so on. 

Whence, the following equations : 

sin. 1° 12'=.0174524034+T\(.0174470965)+^3.ir( 106457)— 
yf«~o( 52874) = .0209424308 

sin. 2° 12'=.0348995000+y2^(.0174364508)+yfxr( 159331)— 
tHttC 53081)=.0383878114 



248 ROBINSON'S SEQUEL. 

sin. 3° 12'=.0523359508+y\(.0174206177)+yf^(.212437)— 
i-UTr(-63213)=.0558214986 

If we put w=||=,V» we can find the sine of 1° 24', 2° 24', 
and 3° 24', in precisely the same manner. 

In short, if we put n equal any number of times gV» we can 
find the sine of the degree and that number of minutes, but it is 
best to be regular, and find the sines to 1° 12', 1° 24', 1° 36', 
and so on, and then interpolate again between the numbers thus 
found. 

Little attention has been paid to this subject of late, because the 
labor when once done, is done forever ; and it has all been done 
in the preceding age ; our object has been to present a systematic 
view of the whole matter, and show the student that the task of 
computing a trigonometrical table is not so great as is generally 



We have thus far computed natural sines and cosines, but we 
generally use logarithmic sines and cosines. 

To find the logarithmic sine, we simply take the logarithm of the 
natural sine from a table of the logarithms of numbers, increasing the 
index bg 10. 

After a few logarithmic sines have been found at equal inter- 
vals of arc, then the intermediate logarithms can be found directly 
by interpolation. 

To make a table of logarithmic sines true to six places of deci- 
mals, we must compute with at least eight decimal places ; and 
to make a table true to nine places of decimals, we must compute 
with twelve decimal places. 

To show the advantage of working on a large scale, we will 
require the log. sines of 1°, 2°, 3°, 4°, 5°, and 6°, true to nin£ 
places of decimals. 

The natural sines we already have, and the necessary tabl-es 
of logarithms are in the latter part of this volume, the same as are 
to be found in our Surveying and Navigation. 

Most operators would take out the logarithm of each natural sine 
separately, having no connection mth eojch other, but this would re- 
quire much unnecessary labor, and it is to explain the artifices, that 
we bring forward the example. 



TRIGONOMETRY. 249 

In the first place we will take the sine of 6°, that is, find the 
log. of the number .1045284515. 

Log. .104 —1.017033339299 

Factor, 1.005 Table A, .002166071750 

Log. prod., . 104520^7 01919941 1049 

1.00008. . . Table C, 34742166 

.104528 36160 W 019234153215 

o 

Dividing the given number by this number, we find another 
factor to be 1.0000008. The log. of this factor corresponds to 
8(c) in table C ; therefore, 

—1.019234153215 
347432 

Log. .1045284515= —1.019234500647, nearly. 

Add 10. 

Tabular log. sine 6°= 9.019234500647 

Having the logarithm of .1045284515, and requiring that of 
.0871557425, we first consider whether we cannot find some con- 
venient divisor to the first that will produce the second for a 
quotient, or produce a number very near the second. To find 
definitely what this divisor is, represent it by D ; then 

•^^^^^-=0.087155. Whence, i>=1.2, nearly. 

Log. .1045284515 —1.019234500647 

Divide by 1.2) log. 0.079181246048 

Gives .087107043 log. —2.940053254599 



Factors, 



1.0005 
1.00005 
1.000008 
1.0000008 



217099966 

Table C, 21714178 

3474352 

347435 



Prod, nearly = .0871657425 log. nearly=— 2.940295890530 

We found these factors by taking .0871557425 for a dividend, 
and .087107043 for a divisor; the quotient is 1.0005588, which 



260 ROBINSON'S SEQUEL. 

we directly separate into the single factors, 1 .0005, 1 .00006, <fec. 
(See Art. 14, Robinson's Surveying and Navigation.) 

We can easily find the logarithmic sine of 1°, because the 
number .0174524035 happens to be peculiarly favorable ; 174= 
87-2, and 174 multiplied by 1003, gives 174622. Whence, 
.87X.02X(1.003)=.0174622. 
.0174524036 



Again, 



'=1.00001166. 



Factors, 



.017422 

' Log. .02 
Log. .87 
Log. 1.003 
Log. 1.00001 
Log. 1.000001 
Locr, 1.0000006 



. . . . —2.301 029 995 644 
....—1.939 519 252 619 

001300 943 017 

Tabled 4 342 923(a) 

434 294(5) 

260 574 (6c) 

26 058(6rf) 



Log. 1.00000006 

Product nearly .0174524035 log. —2.241 855 255129 

Add 10. 
Tabular log. of 1°, therefore, is 



8.241 865 255129 

To find the logarithmic sine of 2°, we proceed thus : To the 
log. of the sine of 1°, add the log. of the number 2, then we shall 
have the logarithm of a number a little above the one required, 
which can be reduced by division. 

Log. .0174524035 —2.241855255129 

Add log. 2, .301029995644 

Log. .034904807 —2.642885250773 

Divide by 1.000152 Sub. log. 1.0001 43427277 

—2.642841822496 
• Alsosub.log. 1.00005 21714178 

—2.642820108318 

Also sub. log. 1.000002 .^ 868588 (2b) 

Log. .0348996= —2.542819239730 very 

Add 10. nearly. 

Tabular log. 3° (true to 10 places), = . . .8.5428192397 



TRIGONOMETRY. 261 

We found the divisor 1.000152 by the following equation. 
Calling D the divisor sought, then 

.034904807^ Qg^ggg^ Whence, i>=1.000152. 

By a little examination, we shall find that if we multiply the 
sine of 1° by 3, and divide the product by 1.000406, the quotient 
will be the sine of 3° very nearly. 

To the log. —2.241 856 255 129 

Add log. 3, 477 121 254 720 

—2.718 976 509 849 

Sub. log 1.0004 '.^ 173 690 053 

—2.718 802 819 796 
Also, sub. log. 1 .000006 2 605 764 (26) 

Log. sine of 3°= —2.718 800 214032 

AddJlO^ 

Tabular log. sine 3°= 8.718 800 214 032 

In a similar manner we can find the logarithmic sine of 4°. 

If it were our object to compute a table of logarithmic sines 
and cosines for every degree and minute of the quadrant, we 
would first compute each degree and half degree in natural num- 
bers — and take the logarithms of those numbers. Then we would 
interpolate for the intermediate logarithms. 



We now proceed to solve proUeTns m Trigonometry and Menswrc- 

tion. 

(Problems 1 and 2 are on page 167, Robinson's Geometry.) 

1. Given AB 428, the angle C 49° 16', and (AC+CB), 918, to 
find the other parts. 

Let ABC represent 
the A. Draw^I>=918. 
From D draw DB so 
thatthe angle ADB shall 
be half the angle ACB, 
that is, 24° 38'. From 
^ as a center with AB 
as a radius, strike an arc 




252 ROBINSON'S SEQUEL. 

cutting BDm B. From B make the angle I)BC=2^° 38': 

then A CB will be the A required. 
In the AADB we have AB : AD : : sin. D : sin. ABD, 
That is, 428 : 918 : : sin. 24° 38' : sin. ^^i>. 

Sin. 24° 38' 9.619938 

Log. 918 2.962843 

12.682781 

Log. 428 2.631444 

Sin. 63° 22' 48" or its supplement 116° 37' 12". .9.951337 

From this take DB C 24° 38' 

ABC= 9r°^59M2" 

Having now two angles of the A ABC, we have the third angle 
^=38° 44' 48", and with all the angles, and the side AB, we 
find A (7=564.49, and conseqently 5(7=354.51. 

(2.) Given a side and its opposite angle, and the difference of the 
other two sides, to construct the triangle and find the other parts. 

Let ABC be the A. ^(7=126, 5= 
29° 46', and AM, the diflference between 
AB and BC, =43. 

From 180° take 29° 46' and divide the 
remainder by 2. This gives the angle 
BMC or BCM. BMC taken from 180°, 
gives AMC. 

Now in the A AMC, we have the two sides AC, 126, AM, 

43, and the angle AMC, to find the angle A. The computation 

is as follows : 180°— 29° 46'=150° 14' ; half, =75° r=BMC. 

180°— 75° 7'= 104° 52,'= AMC. Now in the A AMC, we have 

AC : AM '. : sin. 104° 63' : &m. ACM 

Sin. 104° 63'=cos. 14° 63'. 

126 : 43 : : cos. 14° 53' : sm. ACM 

Cos. 14° 63'. 9.986180 

Log. 43 1.633468 

11.618648 

Log. 126 2.100371 

fiin. ^C7Jf=sin. 19° 22' 28" 9.518277 




i 



TRIGONOMETRY. 



253 



Whence, ^(7^=76° 7'+ 19° 22' 28"=94° 29' 28". Conse- 
quently ^=55° 51' 32". Now we have all the angles, and AC, 
of the A ABC. 

(3.) Two lines meet, making an angle of 50°. On one line are 
two objects, one 200, the other 500 yards from the angular point. 
Where abouts on the other line will these two objects appear under the 
greatest possible angle, and what unll that angle be? 

LetP^andPi) 
be the two lines, 
and A and B the 
two objects. 

Let i) be the re- 
quired point on the 
other line. Then 

pd=jTaxpb 




= J500X 200 = 
316.226+ yards; 
but this requires 
demonstration. 

If we make PD=JPAxPB, and then pass a circle through 
the points A, B, and D, PD will touch the circle in the point 
D. (Th. 18, b. 3, scho.) And because PD is a tangent, the 
angle ABB at the point of contact, is greater than any other an- 
gle AdB, on either side of D, (see th. 7, page 101 of this volume.) 
Or we may prove it here. AeB=ADB, (th. 9, b. iii, scho.); 
but AeB is greater than AdB, therefore, ADB is greater than 
AdB ; that is, greater than any angle drawn from any point be- 
tween P and D. The same demonstration will apply on the other 
side of D. 

The computation for the angle, is as follows : 

From D let drop the perpendicular DH, then in the A PDH, 
we have 

As radius, 10.000000 10.000000 ^ 

To PD. 2.600000 2.500000 * 



So is sine 50°, 
To DH, 242.24, 



9.884262 



cosme 



9.808067 
2.384252 PH, 203.26, 2.308067 

m 



254 



ROBINSON'S SEQUEL. 



From PH take PB, and we have JIB==3.26. From PA take 
PJI, and we have Aff=^296.S4. 
Now 



HB \ HD \ \ 


; R : 


: tang. ABD, 


AH '. HD '. \ 


: E ; 


: tang. BAD. 


12.384252. 




12.384252 


HB 0,513218 




AH 2.471800 



Tan. ABB 89° 14' 11.871034 tan. BAB 39° 16' 9.912462 

180°— (89° 14'+39° 16') =51° SO'=:ABB, the greatest angle 
required. 

At the point G on the line PG, the objects A and B would 
extend the greatest possible angle, and in that case also, PG= 
JPAxPB7~ But the angle A GB must be of such a value that 
AI)B+AGs=nO'' ; therefore, ^6^^=128° 30'. 

(The following are on page 174, Robinson's Geometry.) 
(3.) From an eminence q/* 268 feet in perpendicular height, ike 
angle of depression of the top of a steeple which stood on the same 
horizontal plane, was found to he 40° 3', and of the bottom 56° 1 8'. 
What was the height of the steeple? Ans. 1 1 7.8 feet. 

Let BC hQ the eminence 268 feet, 
and AD the steeple. Draw CE par- 
allel to the horizontal AB. Then 
JSCD=40'' 3', £CA=CAB=56° 18'. 
i)(7^=56° 18'— 40° 3'=16° 15'. 
2)^C=90°— 66° 18'=33° 42' 

In the A ABO, we have 
sin. Qe"" 18' : 268 : : sin. 90° : AC. 
^^_ 268X-g 
sin. 56° 18' 
In the A ADC, we have the supplement to the angle ADC 
equal to 16° 15' added to 33° 42', or 49° 57' ; therefore. 
As sin. ADC : AC : : sm. DCA : AD 

268Xi2 




That is, sin. 49° 67' : 



sin. 56° 18' 



sin. 16° 15' : ^i> 



TRIGONOMETRY. 256 

^7)^ 268'.a-sm. 16° 15^ _ 2.428135+1 0. +9.446893 __ 
sin. 49° 67' -sin. 66° 18' 9.883836+9.920099" ~" 

21.876028—19.803936=2.071093 
Log. ^i>=2.071093. Whence, ^2>=1 17.78 feet. 

(4.) From, the top of a mountain three miles in height, the visible 
horizon appeared depressed 2° 13' 27". Required the diameter of 
the earth, and the distance of the boundary of the visible horizon. 

Ans. Diameter of the earth 7968 miles, distance of the hori- 
zon 164.64 miles. 

Let AB represent the mountain, and 
AD the visible distance. AB produced 
will pass through the center of the earth 
at C. From D draw CD perpendicular 
to AD. Join BD. AD (7 is a right an- 
gled triangle. 

C^i>=90°— 2° 13' 27"=87° 46' 33". 
ACD=9P 13' 27". ADB=^ACD= 
r 6' 44". ^^i>=91° 6' 44". " 

No-jF in the A ABD, we have 

sin. 1° 6' 44" : 3 : : sin. 91° 6' 44" : AD. 

Sin. 91° 6' 44"=cos. 1° 6' 44" 9.999919 

Log 3 0.477121 

10.477030 

Sin. 1° 6' 44" 8.288029 

Log. 164.64 2.189001 

in the triangle AD C, we have 

sm.ACD : AD : : cos. A CD : CD 

Cos. -4Ci>=cos. 2° 13' 27" 9.999674 

AD ^ 2.18900 1 

12.188676 
Sin. ACD=sm. 2° 13' 27" 8.588932 

3.699743 

Double 0.301030 

Diameter log. 7958 miles, nearly 3.900773 




256 ROBINSON'S SEQUEL. 

(Several of tlie following problems in Mensuration are taken from the 
Surveying and Navigation, page 60.) 

(5.) Find the length of an arc of 30°, the radius being 9. 

When the radius is 1, an arc of 180°=3. 141592 ; therefore, 

3 141592 
an arc of 30° and radius 1 must be — , and this multiplied 

by 9 must be the required result. 

„ 3.141592-3 . „,c>«oo A 
Hence, =4.712388, Ans. 



(6.) Find the area of a circular sector whose arc is 18°, and ra- 
dius 1^. 

We must first find the length of the arc, as in the last problem, 
then multiply its half by the radius. 

•^ 141 fiQ9 

Whence, _LlZ:!^ixl8X ^=.3141592Xf=.235619=-» arc. 
180 

Therefore the area must be fX 0-2356 19=0.363427, Ans. 

The arc of 1° and radius unity is .0174533. 

Therefore, that of 9° is .0174533X9, and this multiplied by 

the square of the radius will give tlie true result. 

That is, .0174533X9X1=0.353403. 

(7.) Required the area of a sector whose radius is 26, and ar- 
147° 29'. 

.0 174533Xl47.4833x625 __g^^ 3^^^ 
2 

(8.) What is the length of a chord which cuts off one-third of the 
area from a circle whose diameter is 289 ? Ans. 278.6716. 

Like many problems in relation 
to the circle, this can be solved only by 
approximation . 

As circles are in all respects pro- 
portional to their radii, I will ope- 
rate on radius unity, and in conclu- 
sion, multiply by 2-|^, 

If the segment FFD=} of the 
whole circle, ABDE will equal | of 
the whole. Because ^ — |=i. 




TRIGONOMETRY. 267 

The space ABDE contains two equal sectors, DCB, ACE^ 
and the triande ^Ci). Put the arc BD=x, CB^\. Then 

o 

C6^=sin. X, OD=cos. x. 
The area of the two sectors together is x. 

The area of the triangle ECD is sin. x cos. x. 

Therefore, ir+sin. x cos. x=\7i. rt=3. 141 592653+ 

Double, 2a;-|-2sin. x cos. x=^7t. 

But sin. 2.2;=2sin. x cos. x. (See eq. (30), page 143, Geom.) 

Therefore, 2ar-[-sin. 2x^^7t (1) 

Here we have a correct and definite equation, but we cannot 

solve it, as it contains an arc and its sine, and they are not united. 

by any definite numerical law ; we must, therefore, resort to 

ajyproximation. 

We know that sin. 2x is not much less than 2x. 
Therefore, 4x=^7i is not far from the truth. 
Also, 2 sin. 2x=}7t is not far from the truth. — The one too 
small, the other too large. 

That is, x=z^^rt, approximately, and sin. 2a:=i7t, approximately. 

To find the arc BD approximately, we have this proportion : 

rt : j\7t : : 180° : Arc BD. Whence, ^i>= 15°. 

By the table of natural sines we find sin. 2ii; = sin. 31° 34' 
nearly. Or, ic=15° 47' nearly. 

}Ve nozv knoio that to make the area ABDE=}7t, the arc BD 
must be greater than 15° and less than 15° 47'. 

I will now suppose the arc BD=15° 20', and compute the area 
ABDE, corresponding to that supposition. 

For the numerical value of the arc 15° 20', we have the follow- 
nig proportion : 

180° : 15}j° : : 3.14159265 : Mq BD 

Or, 540 : 46 : : 3.14159265 : Arc^i)=0.2676175. 

The tables will give us (sin. 15° 20') (cos. J5° 20') thus : 
17 




258 ROBINSON'S SEQUEL. 

Sin. 15° 20' 9.422318 

Cos. 15° 20' 9.984259 

Sum less 20=log — 1.406577=0.2550200 nearly. 

jBi>=a;=0.2676175 nearly. 

Area ABDE= 0.5226375 nearly. 

But the required area of ABDE is \7t— 0.5235987 nearly. 

Hence, 15° 20' for ^i>, gives an area too small by 0.00096 12 

Now we wish to increase the area 

ABDE by the little narrow space 

EDdey and this is so narrow that 

Dd and Ee are in respect to practi- 
cal or numerical purposes, right lines, 

and EDcle is a trapezoid, and its par- 

alel sides may be taken as equal ; it 

is then practically a parallelogram 

whose area is given and its longer side 

equal to 2(cos. 15° 20'). 

Let y= the width of this parallelogram or trapezoid, (as we 

may call it either.) Then we shall have the following equation : 
2cos.(15° 20')y=0.0009612 
Or, cos.(15° 20')y=0.0004806 

That is, 0.9644^=0.0004806. Whence, y=0.000498 

That is, we must increase the natural sine of BJ) 15° 20', by 

0.000498. 

The natural sine of 15° 20' is 0.264434 

To which add 0.000498 

N. sin. of 15° 21' 47" cor. to sum 0.264932 

Thus we learn that the arc Bd corresponds to 15° 21' 47" as 

nearly as a table of natural sines computed to 6 decimal places 

will give it. 

Twice the cosine of 16° 21' 47", to a radius of ^(289) is the 

chord sought, which we compute as follows : 

Cos. 15° 21' 47" 9.984184 

Log. 289 2.460898 

Log. 278.67+ 2.445082 



TRIGONOMETRY. 



269 



(9.) WTiat is the radius of a circle whose center being taken in the 
circumference of another containing an acre, sliall ciU of half of its 
contents /* , 

This problem is the same for circles of every magnitude ; 
therefore, we will operate on a circle of radius unitt/. 

Let X represent the number 
of degrees in the arc AB, and 



180 



the length of each degree; 



TCX 




then "'*' represents the length 
180 ^ ^ 

of the arc AB. 

BF= sin. X. CF= cos. x. 
FA=\ — cQs.x. (ABy = 
(1 — cos.a;)2-|-sin.2^, or AB 
= jj2 — 2cos.iC, which equals 
the radius of the cutting circle. 

The area of the sector CBAD^ is measured by the arc 

AB'CA; that is, ---,' From this take the triangle CBD, or 
180 6 

sin. a: cos. a;, and the segment ABB will be left. That is, 

Segment ABFJ)=—— sin. x cos. x. (^=3.141592.) 

*Oiie reason for the appearance of this work is that it is required, because 
able mathematicians have written so obscurely. They seem to have written 
as I should, were I indifferent whether the reader, or rather the learner, un- 
derstood me or not. Do not the following extracted solutions justify this 
observation ? they are brief, to be sure, and no one sets a higher value on 
brevity than does the author of this work ; — but nothing is meritorious 
which is ^^anting in perspicuity. 

The following extracts are from the Mathematical Diary, published by 
James Ryan, 1825. 

Solution. — By Robert Adrain, LL. D. 

In a circle to radius unity, let 2« be the arc of which the chord is the re- 
quired radius, then ?r being the area of the given circle to radius unity, if we 
express analytically the area cut off by the radius sought and divide by 2, 
we obtain the transcendental equation 



(^^z\cos. 22+1 sin. 2^=1 



260 ROBINSON'S SEQUEL. 

Again, as x= tlie degrees in AB, (180 — x) = the degrees 
in BE. Because the angle BAH is at the circumference, it is 
measured by \ of (180— a;), or (90— iic^. 

Whence, the arc BJI=90 — ^ x, measured in degrees. 

For the length of the arc BH, we observe that 180° of the cir- 
cumference would be measured by 7t^2 — 2cos.a?. 

for 1 then, we have -^ _1_, this multiplied by the num- 
ber of degrees, (90-|)will produce (?.?/?^|^V90-?) 

for the linear measure of the arc BIT. 

This multiplied by the radius AB, or J2 — 2cos.ar, will give the 
area of the sector ABHD ; that is, 

Sector ABHD=^(^-i^^y\Uo-'^\ 

From this subtract the triangle ABD, which is measured by 
sin. a:(l — cos.a:), and we have the segment BIIDF. 
That is. 

Segment ^^i>^=<l:^e!:^(i??:^^^^sin.:r+sin..; cos.;r. 
90 \ 2 / ' 

But segment ABFD=i —- — sin. x cos. x. 
^ 180 

The sum of these two segments is the double circular space, 

ABHD required ; that is, 

Hence, «=35° 24', and therefore, if R=the radius of the given circle, the 
radius sought = 2r sin. (35°24')=1.158e. 

Solution. — By Dr. Henry J. Anderson. 
Let the radius of the given circle be represented by unity, and of the two 
portions of its circumference terminated at the intersections of the two circles 
let the greater be denoted by 2<}>. Then, by the rules of mensuration, it will 
be found that the two parts into which the given circle is divided are 
equal, each to 2<j) coe,^ i<p-^7r — <p — sin. <p. Putting this equal iv, the area 
of the semicircle, and transposing, we have 

sin. <t> — (2cos.^i4) — 1)=-, Or by trigo., sin.^ — ^cos.4)=-, 

whence, ^=109° 11' 17" and the required radius =1.15874. 

If the contents of the given circle be one acre, then the required radius will 
be 206.7336 links, or about 45.4814 yards. 



TRIGONOMETRY. 261 

The area ABHD=^—^m.x-\-—J(\—cos.x)(\^0.^-x)\ 
180 ' 180\^ ^^ ^/ 

This reduces to ( 1— cos. x-\- \ — sin. x. 

If we put this expression equal to the given quantity, _ we 

jit 

cannot resolve the equation, because it would contain the linear 
quantity x and the transcendental quantities sin. x and cos. x. 
Therefore if we solve the problem at all, we must do it indirectly, 
by approximation, as we are obliged to do with nearly all 
problems pertaining to the circle. 

This expression is a general one, and if we assume x any num- 
ber of degrees, we can readily obtain the corresponding value of 
the expression, and if any assumption corresponds to a given value, 
the problem is solved ; and if it nearly corresponds, we shall 
have nearly the radius required, which can be increased or de- 
creased, as we are about to explain. 

For the area ABHD to contain half of the circle, it is our 
judgment that the arc AB should contain about 76° ; therefore, 
we assume x=-lb^ and the expression becomes 

(^ _P , 10 COS.75\7t . ^e 
1 — COS.754- I — sin.75. 
^ 24 / 

By the table of natural sine^ we find 

(0.741 18+0.10787)7t— 0-96693 

The final result of this supposition is that the area ABHD=^ 
1.701370. But the half of the circle is 1.570796 ; therefore, we 
have taken x too great to obtain the area of half the circle. 

We will now take x=10°. 

Then A— cos.70+!^^^V— .sin.70° will be an expression 
for a less area than before. 

By log. log. 0.79099 —1.898175 

log. 3.1415926 0.497149 

2.48500 0.395324 

Nat. sin. 70^ 93969 

Area ABHD 1.54531 

Given result .1.57079 

Error too small 0.02548 



t62 ROBINSON'S SEQUEL. 

This error must be conceived to be a winding 'parallelogram, 
whose length is BHD. Dividing .02548 by BHBy will give the 
amount to be added to the radius AH. The radius AH or AB 
is 2sin.35°=l. 14716. 

The angle ^^li)=180— 70=1 10°. 

The Unear value of ^^i>=Qif!im^^i!il^??)li^. 

180 

Now the amount that the radius must be increased is expressed 

^^ (.02548)18 



(1.14716) (3.141592)11 

By logarithms. Log. .02548 —2.406199 

Log. 18 1.255273 

—1.661472 
Log. 1.14716 0.0596187 

Log. 3.141592 0.4971499 

Log. 11 1.0413927 

1.5981613 1.698161 



0.01 1 568 —2.06331 1 

Add... 1.14716 

1.158728= the required jradius AB, which will cut the 
circle into two equal parts. 

If the radius of the given circle is (a), in place of unity, then 
the radius of the cutting circle must be 
(1.15828)a 
To find the number «>f degrees and minutes m ABy divide 
1.15828 by 2, which gives .57914 for the sine of half AB, or 
35° 23' 30" or ^jB=70° 47'. 



The following theorems are extracted from pages 219 and 220 
of Robinson's Geometry, 

(1.) Show geometrically, that R(R-l-.cos. A) = 2 cos.' ^ A ; and 
that R(R— cos. A)=2 sin.'^ A. 




TRIGONOMETRY. S^5 

Let CB or CA represent the radius of 
a circle and call it E. Let the arc ^i>= 
Ay and draw the lines here represented. 

Then GD=^m.A, CG=cos.A, BG= 
E-]-Qos.A, OA—R — cos.^, AI=^m.\A, 
CI=co8.\A, jBi>=2cos.J^. 

From C draw CO perpendicular to 
JBD ; then JB 0= OD, and the two A's 
BOCy Bl) G, are equiangular ; therefore, 

BG : BD : : BO : BC. 

That is, jB-j-cos.^ : 2cos. ^-4 : : cos.^J^ : B. 
Whence, B(B+cos.A)=2cos.^ I A. Q. E. D. 

Again, by the similar A's AGDy ADB, we have 

AG : AD : : AD : AB, 

That is, i2— cos. A : 2sin. ^A : : 2sin.i^ : 2E. 
Whence, B(E—cos. A)=2sm.^A. Q, E. D. 

(2.) Show that R'sin. A=2sin.^ A cos.^ A. 
By similar triangles, we have 

AG I CI : : AD : DG. 

That is, E : cos. ^A : : 2sin. ^A : sin. A. 
Whence, E s'm. A=2sm.^Acos.^A. Q. E. D. 

(3.) Prove that tan. A-}-tan. B= ^ "^ — ^~t — I radius beinff unity. 

COS. A COS. B 

It is admitted that tan.^= — '- — , and tan. Bz 



By addition, tan. A-{-tsLn. B= 



COS.Jl COS. ^ 

sin A , sin.B 



COS. A cos.B 



sin, A cos. ^-f-cos. A sin. B sm.(A-\-B) n E D 

COS. -4 COS. ^ cos.-4cos. ^ 

(4.) Demonstrate geometrically y that R sec. 2 A = tan. A tan. 2 A 
+R2. 



264 



ROBINSON'S SEQUEL. 



Take CB radius, let the arc BD—9.A. 
Then ^^= tan. 2^, ^(7= sec. 2^. Draw 
CjE' bisecting the angle J^ OS, then BE= 
tan. -4. Also, i?-£'= tan. ^, because the 
two triangles CBE and CDS, are in all 
respects equal. 

Now by the similar A's ADE, ABO, 
we have this proportion, 

AD \ DE \ \ AB ', BO. 
That is, AD : tan. A : : tan. 2A : H 
Whence, AD • i2=tan. A • tan. 2 A 
By adding jK^ to both members and fac- 
toring, we have 

(AD-{-E)E= tan. A tan. 2^-|-i22 

But (^i>+i^)=^(7=sec. 2^; therefore, 
B sec. 2^=tan. A tan. 2A-\-R^ . 




Q. E. D. 



(5.) Show that in any plane triangle, the base is to the sum of the 
other two sides, as the sine of half the vertical angle is to the cosine 
of half the difference of the angles at the base. 

Let ABC be the A. Call AB the 
base, and produce A C the shorter side 
so that CD=: OB and OEz= OB. Then 
if be taken as the center of a circle 
and CB radius, that circle must pass 
through the points E, B, and D, and 
the angle EBD must, therefore, be a 
right angle. 

Because A CB is the exterior angle 
of the A ODB, and that A isosceles, 
the angle A CB must equal 2D, or the 
angle D is half the vertical angle. 

Because BA is the exterior angle of the A AEB, we have 
BAC=AEB+ABE (1) 

But AEB=CBE=CAB+ABE. This value of AEB, sub- 
stituted in ( 1 ), gives BA 0= CBA+ABE+ABE (2) 

Whence, BAC^CBA=^2ABE (3) 




TRIGONOMETRY. ; Ml 

This last equation shows us that ABE is half the difference of 
the angles at the base. 

Now in the A ABD, we have 

AB : AD : : sm.D : sin. ^^i9. 

But the sin. ABD=cos. ABE^ because the sum of these two 
angles make 90°. Hence the preceding proportion becomes 
AB : {AC-\-CB) : : ^m.\ACB : gob.\{BAC--CBA). Q.E.D. 

ScHO. 1 . The A AEB gives us this proportion, 
AB : AE : : sin. E : sin. ABE. 

Because the angles E and D together make 90°, sin.^=cos.i>. 

Hence, AB : AE : : cos. D : sin. ABE. 

That is, in relation to the triangle ABC, and generally, 

The base of any 'plane triangle, is to the difference of the other two 
sides, as the cosine of half the angle opposite to the base, is to the sine 
of half the difference of the other two angles.'^ 

ScHO. 2. Draw AH parallel to EB, and of course perpendic- 
ular to DB ; then we have 

DA '. AE \ '. JDH \ HB. 

If Affhe made radius, DR is tangent to the angle DAB, and 
BB is tangent to the angle BAB. 

Because ^^is parallel to EB, the angle BAB is equal to ABE; 
but ABE has been demonstrated to be equal to the half difference 
of the angles CAB, CBA ; therefore, DAB is the half sum of 
the same angles, for the half sum and half difference of any two 
quantities make the greater of the two. Therefore, the preceding 
proportion becomes the following theorem : 

As the sum of the sides is to their difference, so is the tangent of the 
half sum of the angles at the base, to the tangent of half their dif- 
ference. 

This theorem is demonstrated in some form in every treatise on 
plane trigonometry. It is the 7th prop., page 149, Robinson's 
Geometry 

(6.) The diference of two sides of a triangle, is to the difference of 
the segments of a third side, made by a perpendicular from the oppo- 
site angle, as the sine of half the vertical angle is to the cosine of half 
the difference of the angles at the base ; required the proof 

*This is theorem v, Robinson's Geometry, page 220. , 




«66 ROBINSON'S SEQUEL. 

Let ABC he the 
A. On the shor- 
ter side CB as ra- 
dius, describe a 
circle, cutting AB 
in F, AC in B, 
and produce AC 
to K Draw CD 
perpendicular to the 
base, then DB is one segment of the base, AD is the other, and 
AF is their difference. AJI is obviously the difference of the 
sides. 

Now in the A ABF, we have 

Aff : AF : : sin. AFff : sm. AHF (1) 

This proportion demonstrates the theorem, as will appear when 
we show the values of these angles. 

Because CHFB is a quadrilateral in a circle, the angles 
HFB+E^im". 

But BFB-\-AFir=lBO''. 

By subtraction, F—AFJI=0, or AFJI=F. 

In the same manner we prove that AIIF=ABF. 

Substituting these equals in proportion (1), it becomes 
AH : AF : : sin. ^^ : sin. ABF (2) 

The angle F is half the angle A CB, because A CB is at the 
center of the circle, and F at the circumference intercepting the 
same arc. 

Also, sin. ABF=cos. ABIT, because EBE is a right angle, and 
the sine of an arc over 90° is equal to the cosine of the excess over 90°. 

Again, BCF— the sum of the angles at the base. BHC, or 
its equal CBH, is half BCF\ therefore, CBH = the half sum 
of the angles at the base of the triangle A CB, and HBA is their 
half difference. Whence proportion (2) becomes 
Aff : AF : : 8m.^(ACB) : cos.:^( ABC—A). Q.E.D. 

ScHO. Because -4 is a point without a circle, &c. 

ABxAF=AFxAff. 
Whence, AB : AF : : Aff : AF, 



TRIGONOMETRY. «67 

The two A's ABE, AHF, have the common angle A, and the 
sides about the equal angle proportional, therefore, (th. 20, b. ii.) 
the two A's are similar, and AFH=E. AHF=ABE. 

(7.) Given the base, the difference of the other two sides, and the 
difference of the angles at the base, to construct the triangle. 
(See figure to Theorem 5.) 

Draw AB equal to the given base. From B on the opposite 
side of the base, make the angle ABE=z half the difference of 
the angles at the base. 

Take AE, the given difference of the sides, in the dividers ; 
put one foot on A, and strike an arc cutting BE in E. Join AE, 
and produce EA. 

Make the angle EBD=i90°. BD and EA produced will meet 
in D. Bisect ED in C, and join BG, and AOB will be the A 
required. 

N. B. This problem was Suggested bj the investigation of 
theorem v. 



(8.) Prove that smr\l ^ =tan.-\/-. 



Remark. The notation sin.~*w, signifies an arc of a circle whose 
radius is unity, and sine u, &c., &c. 

Hence the above proposition in plain English is this : 

The radius of a circle is unity, the sine of an arc in thai circle is 

^1 Prove that the tangent of the same arc must be - /- . 

^a-\-x ^a 

Let y= the cosine of the arc in question. 

Then ya+_^=l. Whence, y^^.JL-. 
a-\-x a-\'X 

But to every arc we have the following proportion : 
cos. : sin. : : 1 : tan. 



That is, J-^ : -/-^ : : 1 : tan. 
\a+x ^a+x 

Or, Ja : Jx : : I : tan.=^-. Q. E. D. 



268 ROBINSON'S SEQUEL. 

(9.) If tan.(a-j)^,_sin^c^ ^^ tan.a tan.4= tan.^c. 
tan. a sin.^a 

To perform the reduction, multiply by tan. a, and in the last 

term take its equal — '— ; then 
cos.a 

tan. (a — 5)= tan. a — _; '. . 

sin. a cos.a 

That is, ^1l^J?IL*_=tan.- ^'"'"^ 



l-|-tan.atan.5 sin.a cos.a 

tan.a-toii.J=tan.a+tan.=« tan.4-C!in:!£+E5:!i^iL?i^) 

\ sin.a cos.a / 

Whence, ( l+tau.=a) tan. j^sin.'c+sin.'otan^atoij 

sin.a cos.a 
Multiplying by cos.a, and observing that (l-|-tan.^a)=sec.^a, 
and cos.a sec. a=l ; then 

, , sin.^ c . sin.^ctan.atan.6 
sec. a tan.o= + ' 



sin.a sm.a 

Or, sec.a tan.6 sin.a=sin.^ c-|-sin.^ c tan.a tan.J. 

Take — '— for tan.a in the last term, then 
cos.a 

, , . . „ , sin. ^c sin.a tan. 5 

sec.atan.6 sm.a=sm.^c4- 

cos.a 

Or, (cos.a sec.a)tan.5 sin.a=sin.2c cos.a-j-sin.^c sin.a tan.6. 

Observing again that (cos.a sec.a) = l, we have 

tan.6 sin.a=sin.^c cos.a-j-sin.^c sin.a tan.6 

Divide each term by cos.a, and taking tan.a for ! — 1-, we find 

cos.a 

tan.a tan.J=sin.^c-{-sin.^ctan.a tan.^. 
(1 — sin.^c)tan.a tan.5=sin.^c. 
But (1 — sin.*c)=cos.2c, because sin.2c-|-cos.^c=l ; therefore, 
cos. ^ c tan.a tan.5=sin. ^ c 

tan.a tan.6= '- — ^=tan.^c. Q. E. D. 

cos.^c 




TRIGONOMETRY, v 269 

SECTION III. 

PROBLEMS IN SPHERICAL TRIGONOMETRY AND ASTRONOMY 

Let ABC he a right angled triangle, right 
angled at B. a the side opposite A, h the 
side opposite B, and c the side opposite G. 

Taking the complement of the oblique an- 
gles A and C, calling them A\ G\ and the 
complement of h calling it b\ 

Then Napier's Circular Parts give us the following equations. 
We retain the same numbers for the equations as in our Geometry, 
page 186. 

(11) i2 sin.c=tan.a tan. ^' (16) i? sin.^'=tan.2>' tan.c 

(12) i2sin.rt=tan.ctan.(7' (17) i2sin.^'=cos.a cos.C" 

(13) i2sin.a=cos.5'cos.^' (1^) i2sin.5'=cos.a cos.c 

(14) jRsimc=cos.5'cos.(7' (19) i2 sin. C"=tan.5' tan. a 

(15) i2sin.5'=tan.^'tan.(7' (20) J^ sin. (7'=cos.c cos.^' 

These equations are written in the present form to assist the 
memory, the second members being the products of two cosines 
or two tangents ; but in practice, we often modify an equation by 
taking sine for cosine, and cotangent for tangent, and the reverse. 

For instance, in equation (18), we invariably take cos.S for 
sin.i', it being the same, which saves the trouble of finding the com- 
plement to the hypotenuse. The same may be said of other com- 
plements. 

In all spherical triangles, right angled or oblique angled, the sine of 
the sides are to each other as the sines of the angles opposite to them. 

When two sides of a spherical triangle are given, there can be 
but one result, that is, there can be no ambiguity about the parts 
required ; but when only one side is given, and one of the ob- 
lique angles in a spherical triangle, the conditions correspond 
equally to two triangles, and the answer is said to be ambiguous. 
For a learner fully to comprehend this, it is necessary to learn to 
construct his triangles as follows : 

We shall illustrate by examples, beginning with the 10th ex- 




270 ROBINSON'S SEQUEL. 

ample, page 199, Robinson's Geometry, which will sufficiently 
illustrate several others. 

(1.) In the right angled triangle ABC, right angle at B, given 
AB 29° 12' 50" and the angle C 37° 26' 21", to find the other parts. 

To construct a spherical A, 
the operator should have a 
scale of chords and semitan- 
gents ; but he can do all with 
a ruler and dividers. 

Take 0(7 in the dividers 
equal to the chord of 60°, (or 
any distance if no scale is at 
hand), and from any point 
as a center describe the circle 
CHDh. Draw CD and Hh 
at right angles through the 
center. Each of the lines 
(7, ODy OH, as well as the curve HC, HD, &c. represent 90° 
on a sphere. OHC is a right angle, — that is, any line from to 
the circumference will make a right angle with the circumference- 

Now from C we propose to make the angle JICA=3'7° 26' 21". 
Divide the quadrant BJ) into degrees, beginning at II. Take 
IFF equal to 37° 26' 21" ; or if the scale is used, take the chord 
of 37° 26' 21" from the scale and apply it from II to F. Apply 
the ruler from C to P, and through the point n, where this line 
crosses ITO, describe the curve CnD. 

From If, set off IfQ=^29° 12' 50", apply the ruler between Q 
and C, and mark F where this line cuts HO. From the center 
0, with V radius, strike the arc VA. Lastly, through A and 
draw BAG through the center. The A ABC or its supple- 
ment DBA, is the one required. The side A C is measured by 
the arc, but neither A C nor the angle A can be measured instru- 
mentally. To measure sides, they must either be on the circum- 
ference or on the straight lines through the center. 

Remark. If the angle BAC had been given, we should call 
the triangle ADQ the supplemental triangle, for ^6^ is the sup- 



TRIGONOMETRY. 271 

plement to AB, AD to AC, and the angle ABO is supplemental 
to ADB or its equal BCA. 

When we have all the parts of the triangle ABC, we in effect 
have all the parts of the triangle DAB, also all parts of the A 
AD G and all parts of the triangle GA C. 

That is, when one spherical triangle is determined, we have 
three others, the whole four making up a hemisphere. 

For the numerical computation oi AC we take equation (14) 

modified thus : 

cir. jn c'.r. 7. -Ssin.c 19.688483 

sm. A C=sm. 6= — -, — — 

sm.(7 9.783843 

sin. 63° 24' 13" 9.904640 

To find BC or a, we take a modification of (18). 

E cos.5 19.775374 

cos.a= 

cos.c 9.940917 

cos. 46° 56' 2" 9.834457 

To find the angle A, we take a modification of equation (13). 

. . Bsin.a 19.863539 

sm.-4= — — — 

sm.b 9.904640 

sin. C6° 27' 60" • 9.958899 

Whence, 
^C=:53° 24' 13", BC=46° 55' 2", and the Z- A, 65° 27' 60". 
AD=126° 36' 47", ^2>=133° 4' 58", and theA BAD, 114° 32' 10" 

The same figure will sufficiently illustrate example 12, page 
199, Robinson's Geometry. 

(2.) In the right angled triangle ABC, given AB, 64° 21' 36", 
and the angle C, 61° 2' 16", to find the other parts. 

(14) sin.^C7=sin.6=^^^ l^M^nS 

^ ' sm. C 9.941976 

sin. 68° 16' 16" 9.967940 

Whence, -4(7 == 68° 15' 16", and ^i> = 111° 44' 44". The 

answers given in the book correspond to the triangle ADB, — • 

and those answers were given to exercise the judgment of the 

learner. 

The other parts are found as in the last example. 



272 ROBINSON'S SEQUEL. 

(3.) In the right amjled spherical triangle, given AB, 100® 10' 3" 
and the angle BCA, 90° 14' 20", to find the other parts. 

Because the sines, cosines, 
(kc, of the tables correspond 
to arcs under 90° ; therefore 
we will operate on the sup- 
plemental triangle, ADE. 
BC^DE. 180°— .^j5=79° 
49' 67"= JZ>=c. 

The angle ^7)^=90°— 
(14'20")=89°45'40". 

AC=h, and AB = h' in 
the equations. AB=^Cy AED 
=90°, ^i>^=:(7' = 89°46' 
40". 

To solve this A, we use equation (20). 

R cos. C 17.620026 




sin. A 



COS. c 9.246810 

sin. 1° 2ri2" 8.373216 



To compute AI>y we take equation (16) ; AB the supplement 
of AC=h. 

R COS. 5=cot. A cot. C. 

cot.^=l°21' 12" 11.626819 

cot.(7=89°46'40" 7.619860 

AB COS. 79° 60' 6" 9.246679 

AC 100° 9' 65" 

To find BE, or its equal BC, we take equation (13). 

i2sin.a= sin.6sin.-4. 

sin. 79° 60' 6" 9.993128 ' 

sin. 1° 21' 12" 8.373216 

BC, sin. 1° 19' 62" 8.366344 

These examples give a sufficient key to the solution of all other 
examples in right angled spherical trigonometry. * 



TRIGONOMETRY. 273 

We now turn to the application of spherical trigonometry — 
taking the examples from page 215, Geometry. 

MISCELLANEOUS ASTRONOMICAL PROBLEMS. 

(1.) In latitude 40° 48' north, the sun hore south 78° 16' west, at 
3h. 38m. P. M., apparent time. Required his altitude and declina- 
tion, making no allowance for refraction. 

Ans. The altitude, 36° 46', and declination, 15° 32' north. 

Let Hh be the horizon, 
Z the zenith of the observer, 
P the north pole, and PS 
a meridian through the sun. 

PZ is the co-latitude, 49° 
\9.\ and PS is the co-decli- 
nation or polar distance, one 
of the arcs sought. ZS is 
the co-altitude or ST is the 
altitude of the sun at the 
time of observation. 

The angle ZPS is found 
by reducing 3h. 38m. into 
degrees at the rate of 4m. to one degree ; hence, ZPS=54° 30' 

Because EZS=7Q° 16', PZS=101° 44'. From Z let fall the 
perpendicular Z^ on PS. Then in the right angled spherical 
A PZQ, equation (13) gives us* 

P sin. Z^=sin. PZsin. P. 

sin. PZ=sin. 49° 12' .9.879093 

sin. P =sin. 54° 30' 9.910686 

sin. Z^=sin. 38° 2' 33" 9.789779 

To obtain the angle PZQ, we apply equation (19), which gives 

P COS. PZQ=cot. PZ tan. ZQ, 
That is, i2cos. PZQ=t3,n. 40° 48' ta,n. 38° 2' 33". 

* To apply the equations ■witliout confusion, letter each right angled spher- 
ical triangle ABC, right angled at B, then A must be written in place of P ; 
nndwhen operating on ZSQ, write A in place of S, and C for the angle SZQ 
18 




io 



^4 . ROBINSON'S SEQUEL. 

9.936100 
9.893464 

PZQ= COS. 47° 30' 50' 9.829564 

PZS= 101° 44' 
SZQ= 54° 14' 10" 
To obtain ZS or its complement, we again apply (19) 

(19) E COS. SZQ=cot. ZStsLYL. ZQ. 
That is, i2cos. 54° 14' 10"=tan. STtan.38° 2' 33". 

i? COS. 54° 14' 10"= 19.766744 

tan. 38° 2' 33 " = 9. 893464 

tan. 36° 46', nearly, 9.873280 

To find PS, we take the following proportioif : 

sin. P : sin.Z^ : : sm. PZS : &m. PS 
That is, sin. 54° 30' : cos. 36° 46' : : sin. 101° 44' : sin. PS 

COS. 11° 44' 9.990829 

cos. 36° 46' .9.903676 

19.894505 

sin! 54° 30' 9.910686 

PS, 74° 28' sin 9.983819 

Whence, the sun's distance from the equator must have been 
15° 32' north. 

(2.) In north latitude, when the sun's declinatlo7i ?m5 14° 9.Q' north, 
his altitudes, at two different times on the same forenoon, were 43° 
7'-}-, (ind 67° 10'-(- ; and the change of his azimuth, in the inter' 
vol, 45° 2'. Required the latitude. Ans. 34° 20' north. 

Let PK be the earth's axis, ^^^^^^^ 
Qq the equator, and Bh the ho- ^KUBm^^ 

Also, let Z be the zenith of the ^■^•^^^ 
observer, Sm the first altitude, 
Tn the second, and the angle 
rZ>S=45° 2'. Our first opera- 
tion must be on the triangle ZTS. 
ZT=22° 50', Z^=46° 53', and 
we must find TS, and the Z_ TSZ- 




TRIGONOMETRY. 275 

From T, conceive TB let fall on ZS making two right angled 
A's ; and to avoid confusion in the figure, we will keep the arc 
TB in mind, anA not actually draw it. 

Then the A ZTB furnishes this proportion : 

R : sin. 22° 60' : : sin. 46° 2' : sin. TB=sm. 16° 66' 8" 

To find ZB we have the following proportion, (see p. 186 Geo.) 
R : COS. ZB : : cos. 16° 66' 8" : cos. 22° 60' 

Whence, we find Z5=16° 34' 20". Now in the right angled 
spherical A TBS, we have TB = 15° 56' 8", BS=46° 63'— 16° 
34' 20", or ^5=30° 18' 40" ; and TS is found from the following 
proportion : 

R : COS. 15° 56' 8" : : cos. 30° 18' 40" : cos. TS 

This gives TS=33° 53' 16". To find the angle TSZ, we have 
the proportion, sin. 33° 53' 1 6" : R : : s'm. TB 15° 66' 8" : sin. TSZ. 

Whence, the angle TSZ=29'' 30'. 

The next step is to operate on the isosceles spherical A PTS, 
We require the angle TSF. 

Conceive a meridian drawn bisecting the angle at P, it will 
also bisect the base TS, forming two equal right angled spherical 
triangles. 

Observe that P>S^=75° 40' and ^ TS=16° 56' 38". 

To find the angle TSF we apply equation (19), in which a= 
16° 56' 38", 5=75° 40', and the equation becomes 

R cos. TSP—coi. 76° 40' tan. 16° 56' 38" 

Whence, TSP=S5'' 31' 40", and PSZ=Q5° 31' 40"— 29° 30' 
=66° 1' 40". 

The third step is to operate on the A ZSP ; we now have its 
two sides ZS and SP, and the included angle. 

From Z conceive a perpendicular arc let fall on SP, calling it 
ZB ; then the right angled spherical triangle SZB, gives 

R : sin. ZS : : sin. Z SB : sin. ZB 
That is, R : sin. 46° 53' : : sin.66° 1'40" : sin.Z^=sin.37° 15'20" 
To find SB we have the following proportion, (see Geo. p. 185.) 

R : cos. SB : : cos. ZB : cos. ZS 
Thai is, R : cos. SB : : cos. 37° 15' 20" : cos. 46° 63' 



276 



ROBINSON'S SEQUEL. 



Whence SB=30'' 49' 40". Now from FS, 15" 40', take SB, 
30° 49' 40", and the diflference must be BP, 44° 50' 20". 

Lastly, to obtain PZ, and consequently Z Q the latitude, we have 
R : COS. ZB : cos. BP : cos. ZP==sin. ZQ 

That is, B : cos. 37° 16' 20" : : cos. 44° 50' 20" : sin. ZQ= 
sin. 34° 21' north. 

This computation differs one mile from the given answer, but 
any two operators will differ about this much, unless each observe 
the utmost nicety. 

This is a modification of latitude by double altitudes, but in 
real double altitudes the arc ^aS^ is measured from the elapsed 
time between the observations, and the angle TZS is not given. 



(3.) In latitude 16° 4' north, when the su7i's declination is 23° 2' 
north. Mequired the time in the afternoon, and the sun's altitude and 
hearing when his azimuth neither increases nor decreases. 

Ans. Time, 3h. 9m. 26s. P. M., altitude, 45° 1', and bearing 
north 73° 16' west. 

Let Pp be the earth's axis, 
Hh the horizon, Qq the equator, 
QZ and Pp, each equal to 16° 
4' north, and Qd, qd, each e- 
qual to 23° 2' ; then the dotted 
curve dd represents the parallel 
of the sun's declination. 

Through Z and N an infinite 
number of vertical circles can 
be drawn, one of these will touch 
the curve dd ; let it l^e Z OK 

At the point where this circle touches the curve dd will be 
the position of the sun at the time required, and P OZ will be a 
right angled spherical A, right angled at 0. The problem re- 
quires the complement of ZO, and the time corresponding to the 
angle ZPO. 

In the spherical A P OZ, we have 

R : COS. PO : : cos. ZO : cos. PZ 

That is, R : sin. 23° 2' : : sin. altitude : sin. 16° 4' 




Whence, sin. alt. = 



TRIGONOMETRY, 
i? sin. 16° 4' 



277 



sin. 46° r nearly'. Ans. 



sin, 23° 2' 
To find the angle at P, we have the following proportion : 

COS. 16° 4' : R : : cos. 46° 1' : sin. P 

Whence, sin. P = sin. 47° 21' 30", and ZPO = 47° 21' 30", 
which being changed into time, at the rate of 16° to one hour, 
gives 3h. 9m. 26s. 

To find the angle PZ 0, we have this proportion : 
cos. 16° 4' : R '. : cos. 23° 2' : sin. PZO = sin. 73° 16' 

(4.) The sunset south-west ^ sovthy when his declination was 16° 
4' south. Required the latitude. Ans. 69° 1' north. 

Draw a circle as before. Let 
Hh be the horizon, Z the zenith, ^^^vn 
P the pole. The great circle 
PZH'i^ the meridian, and ZCN 
at right angles to it, and of 

coui'se east and west. Let BC ^^^S^^ff^BBi^^KKKUi 
be a portion of the equator, and ^^H^H|HBn^S^^HI 
B the arc of declination. The ^^|SHH^H|^H^SBI 
position on the horizon where ^^^|^8^^^H^|B^B^| 
the sun set is the arc 110=45° ^^^B^l^^^P^^Bs^^M 
—6° 37' 30"=39° 22' 30". 

Consequently, the arc 00=50° 37' 30". , 

In the right angled spherical triangle BOO, we have BC, BO 
given to find the angle BOO, which is the complement of the 
latitude, or the complemc'nt of the angle B CZ. 
To find the angle BOO, we apply equation (14). 

i? sin. ^ 0=sin. OCsin. J5(70 
That is, R sin. 16° 4'=sin. 50° 37' 30" sin. BOO 

Rsin. 16° 4' 19.442096 

sin. 50° 37' 30" 9.888184 

cos. 69° 1' nearly, 9.653912 

ScHO. The arc -6 (7 on the equator measures the angle -BP (7, 
corresponding to the time from 6 o'clock to sun rise or sun set. 




S78 ROBINSON'S SEQUEL. 

This arc is called the arc of ascensional difTerence in astronomy. 
The time of sun set is before six if the latitude is north and tlie 
declination south, as in this example, but after six, if the latitude 
and declination are both north or both south. 

To obtain this arc, the latitude and declination must be given ; 
that is, BO and the angle BCO, the complement of the latitude. 
Here we apply (12), that is, 

M sin. BO = tan. D tan. L 
an equation in which D represents the declination, and L the 
latitude. 

(5.) The altitude of the suUy when on the equator, was 14° 28'-]-, 
hearing east 22° 30' south. Required the latitude and time. 

Ans. Latitude 56° T, and time 7h. 46m. 12s. A. M. 

Let S be the position of the sun on the equator. (See the last 
figure.) Draw the arc ZS, and the right angled spherical A 
ZQS is the one we have to operate upon. 

Then ZS is the complement of the given altitude, and the an- 
gle QZS, is the complement of 22° 30'. The portion of the 
equator between Q and S, changed into time, will be the required 
time from noon, and the arc QZ will be the required latitude. 

First for the arc QS. 

R : sin. ZS : : sin. QZS : sin. QS 
That is, R : cos. 14° 28' : : cos. 22° 30' : sin. ^aS^ = 73° 27'38" 

But 73° 27' 38" at the rate of 4m. to one degree, corresponds 
to 4h. 13m. 48s. from noon, — and as the altitude was marked -|-, 
rising, it was before noon, or at 7h. 46m. 12s. in the morning. 

To find the arc QZ we have the following proportion : 

R : COS. 63° 27' 38" : : cos. ^Z : sin. 14° 28' 

Whence, cos. ^Z=cos. 66° 1' nearly, and 56° 1' is the latitude 
souffht. 

o 

(6.) The altitude oft/ie sun was 20° 41' at 2h. 20m. P. M. when 
his declination was 10° 28' south. Required his azimuth and the 
latitude. Ans. Azimuth south 37° 5' west, latitude 51° 58' north. 




TRIGONOMETRY. 279 

This problem furnishes the 

spherical A PZ 0, in which the ^H!|^^B9I 

side Z is the complement of ^B|fig8Q^&^^|B| 

20"" 41' or eO"" W, F0=90'' ^H^j^KKBrn 
-\-iO° 28' = 100° 28', and the ||yi^|^SIHBI 
angle ZFO is 2h. 20m., chang- 
ed into degrees at the rate of 
15° to one hour, or ZPO=35°. 
Now in the triangle ZFO, 
we have 

sin. ZO : sin. ZPO : : sin.PO : sin.PZO That is, 
cos. 20° 41' : sin. 35° : : cos. 10° 28' : sin. BZO = cos. 37° 5'. 
In the right angled spherical A B OZ, we apply equation (16). 
(16). R cos.37° 5'=tan. 20° 41' tan. BZ. 

i2cos. 37° 6' 19.901872 

tan. 20° 41' 9.576958 

tan. ^Z=tan. 64° 40' 40" 10.324914 

To find FB in the right angled A BFO, we apply the same 
equation, (16). R cos. 35°=tan. 10° 28' tan. FB. 

R cos. 35° 19.913365 

tan. 10° 28' 9.266555 

tan. 12° 42' 40" ..10.646810 

But FB is obviously greater than 90°, therefore the point B is 
12° 42' 40" below the equator, but from jB to Z is 64° 40' 40"; 
therefore from Z to the equator, or the latitude, is the difference 
between 64° 40' 40" and 12° 42' 40", or 51° 58' north. 

Ans. Lat. 51° 58' north. 

(7.) If in August 1840, Spica was observed to set 2h. 26m. 14s. 
hefore Arcturus, what was the latitude of the observer ? Taking no 
account of the height of the eye above the sea, nor of the effect of 
refraction. Ans. 36° 48' north. 

By a catalogue of the stars to be found in the author's Astron- 
omy, or in any copy of the English Nautical Almanac, we find the 
positions of these stars in 1 840 to have been as follows : ^ 

Spica, right ascension, 13h. 16m. 46s. Dec. 10° 19' 40" south. 

Arcturus, " *' 14h. 8m. 25s. Dec. 20° 1' 4" north. 



280 ROBINSON'S SEQUEL. 

Let L = the latitude sought. Put d=10° 19' 40", and D— 
20° r 4". 

The difference in right ascensions is 61m. 39s., and this would 
be about the time that Arcturus would set after Spica, provided 
the observer was near the equator or a little south of it ; but as 
the interval observed was 2h. 26m. 14s., the observer must have 
been a considerable distance in north latitude. In high southern 
latitudes Arcturus sets before Spica. 

When an observer is north of the equator, and the sun or star 
south of it, the sun or star will set within six hours after it comes 
to the meridian. 

When the observer and the object are both north of the equa- 
tor, the interval from the meridian to the horizon is greater than 
six hours. 

The difference between this interval and six hours, is called the 
ascensional difference, and it is measured in arc hj £0 in the 
figure to the 4th example. 

Now let X = the ascensional difference of Spica corresponding 
to the latitude £, and y = the ascensional difference correspond- 
ing to the same latitude ; then by the scholium to the 4th exam- 
ple, calhng radius unity, we shall have 

sin. a:=tan. L tan. d ( 1 ) 

sin. y^tan. L tan. J) (2) 

The star Spica came to the observer's meridian at a certain time 
that we may denote by M. 

Then Jlf-^/^e— — ) = the time Spica set. 

And M-{-51m. 39s.4-(6-|-^ )= the time Arcturus set. 

By subtracting the time Spica set from the time Arcturus set 
we shall obtain an expression equal to 2h. 26m. 14s. That is 

51m. 39s. +^+i^=2h. 26m. 14s. 

Ot, -^+I_=lh. 34m. 36s. (3) 

' 16^16 ^ ^ 

^ ar+y=16(lh. 34m. 36s.) (4) 

Equation (3) expresses time. Equation (4) expresses arc. 

When we divide arc by 16 we obtain time, one degree being 



TRIGONOMETRY. 281 

the unit for arc, and one hour the unit for time ; therefore, when 
we multiply time by 15 we obtain arc ; that is, Ih. multiplied by 
15 gives 15° ; hence (4) becomes 

a;4-y=23° 39'=a 
x—a—y (5) 

That is, the arc x is equal to the difference of the arcs a and y ; 
but to make use of these arcs and avail ourselves of equations 
(1) and (2), we must take the sines oi the arcs, (see equation 
(8), plane trigonometry) ; then (5) becomes 

sin. a:=sin. a cos. y — cos. a sin. y (6) 

Substituting the values of sin. x and sin. y from (1) and (2), 
(6) becomes 

tan. L tan. c?=sin.a cos. y — cos. a tan. L tan. D (7) 
Squaring (2), sin.2y=tan.^i/tan.2i). 
Subtracting each member from unity, and observing that (1 — 
sin.^y) equals cos.^y, then 

cos.^2/=l — tan. ^Z tan. 2 i). 
Or, cos. y= ^1— tan.^i/tan.^i). 

This value of cos.y put in (7), gives 
tan. Ztan. <^=sin. aj\ — tan.^Z tan.^i> — cos. «tan. Xtan. D (8) 
By transposition and division, 

/tan.rf+cos.atan^\ tan. i;=Vl-tan.=itan.=i) 
\ sin. a / 

Squaring, (^g^^lJ+g"^- "*'"'• -"Vtan.^X==l-tan.'X tan.'i) 
\ sin. a J 

Dividing by tan.^X and observing that _- = cot.^Z we 

lan. j-i 

have /tan^+cos^tan^\ =oot.=i-tan.^/) 

\ sin. a / 

Or, cot.'X=tan.'2)+('-ggl ^''"^- " ^.g^^-gV 

\ sin. a / 

=tan.^2)+C^+'^-V 
Vsin. a tan. a / 

We must now find the numerical value of the second member. 
Using logarithmic sines, cosines, tangents, (fee, we must diminish 
the indices by 10, because the equation refers to radius unity, 
log. tan. Z>.==— 1.561460. tan.-i>=— 1.122920=0.132712 num. 



1282 V ROBINSON'S SEQUEL. 

log. tan. d —1.260623 log. tan. i> —1.561460 

sin. a — 1.603305 tan. a — 1.641404 



0.45424 —1.657318 0.83188. . . . . . —1.920056 

0.45424-1-0.83188=1.28612 (1.28612)2 = 1.654105 
Whence, cot.2Z=0.132712-|^l. 654105= 1.786817 

Square root, ..cot. Z= 1.33672 

Taking the log. of this number, increasing its index by 10 will 
give the log. cot. in our tables. 

log. 1.33672=0.126076-f-10.=10.126076=cot. 36° 48' 

(8.) On the 14th of November, 1829, Merikar was observed to rise 
48m. 3s. before Aldebaran : what was the latitude of the observer ? 

Ans. 39° 34' north. 

The positions of these two stars in the heavens, Nov. 1829, 
were as follows : 

Menkar, right ascension, 2h. 53m. 21s. Dec. 3° 24' 52" north. 
Aldebaran, " 4h. 26m. 7s. Dec. 16° 19' 31" north. 

Aldebaran passes the meridian Ih. 32m. 46s. after Menkar. 
Now let M represent the time Menkar was on the meridian, then 
M-\-\\i. 32m. 46s. represents the time Aldebaran was on the 
meridian. Also, let x= the arc of ascensional diflference corres- 
ponding to the latitude and the star Menkar, and y that of the 
star Aldebaran. 

Then M—(q-\-—\ — the time Menkar 

And Jf-l-lh. 32m. 46s — ( 6-[- - ) = the time Aldebaran rose. 

Subtracting the upper from the lower, the difference must be 
«8m. 3s. ; that is, 

lh.-|-32m. 46s 'L-X-—=A^m, 3s. 

^ 16 ' 15 

Whence, -^— !_= —44m. 43s.= —0.74527. 

16 16 

That is, Ih. being the unit, 44m. 43s. = 0.74527 of an hour, 

and multiplying by 1 5, we shall have as many degrees of arc as 

we have units ; therefore, 

a;— y=— (0.74527)15=— 11° 10' 45"=— a. 

x=y—a. 

sin. a:=sin.^ cos.o — cos.y sin.a ( 1 ) 



rose. 



TRIGONOMETRY. 283 

Put c/=3° 24' 52", D=\6° 19' 31", and L= the required lat- 
itude. Then by scholium to the 4th example, 

sin. a;=tan.c? tan.Z. sin.y=tan.i> tan.Z. 
These values of sin.a: and sin.y, substituted in (1), give 
tsm.d tan.Z=cos.a tan.D tan.Z — cos.y sin.a ( 2) 
But sin.2y==tan.2i>tan.2i;, and 1— -sin.2y=l— tan.^i^tan.^Z. 
Or, cos.^y=l — ^tan.^*Z)tan.2j&. 

Or, cos.y=iJl — ^tan.^2>tan.^X. 

By substituting this value of cos.y in (2) and transposing, we 

find 

sin.a^l — tan.^i) tan.2j&=(cos.a tan.i) — tan.e?)tan.X 

Dividing by sin.a, and observing that -r-^= , we have 

"^ •' sm. a tan.a 

n — 1 — rm — rr /tan.i) tan.c?\ , r 

J\ — tan.^i) tan.2Z=l — ) tan.ii. 

\ tan.a sm.a/ 

Squaring and dividing by tan.^Z, and at the same time observ- 

insr that =cot.Z, and we shall have 

^ tan.Z 

cot.«Z-tan.«i>=f-^^-*_^V 
\ tan.a sin. a/ 

We will now find the numerical values of the known quantities. 

Log. tan.i). . .—1.466696 Log, tan.c?. . . — 2.776685 

Log. tan.a —1 .296 1 79 Log. sin.a 1 .2876 1 7 

Log. 1 .48089 . . . 0.170517 Log. 0.3076 . . . —1.488068 

tan.2i)=0.085778 1.48089—0.3076=1.17329 

Whence, cot.^ Z— 0.085778= (1.1 7329) 2. 

Or, cot.2ii= 1.462293. 

cot.Z= 1.20925. 
Log. cot.i;+10i=10.082785=cot. 39^ 34'. Ans, 

(9.) iw latitude 16° 40' north, when the sun's declination was 23° 
1 8' northy I observed him twice, in the same forenoon, bearing north 
68° 30' east. Required the times of observation, and his altitude at 
each time. 

Ans. Times 6h. 15m. 40s. A. M., and lOh. 32m. 48s. A. M., 
altitudes 9° 69' 36", and 68° 29' 42". 




2B4 ROBINSON'S SEQUEL. 

Let Z be the zenith, P the 
north pole, and the curve dd be 
the parallel of the sun's declina- 
tion along which it appears to 
revolve. Make the angle PTiS' 

equal to 68° 30' ; then the sun bii^^^^^^^^^^I 
was at S at the time of the first HV^^^BHHH^^^H 
observation, and at S' at the time IIS^^^^HH|B|HI 

the ^I^S^^BBB^BSfl 

In the spherical A PZS' there ^^^I^^H^bIB^^I 
is given PZ, PS' and the angle 
PZaS" ; also, in the A PZS' there is given PZ, PS, and the 
angle PZS. Observe that PSS' is an isosceles A- 

Describe the meridian PB bisectincr the anole S'PS, and then 
we have three right angled spherical triangles, BPS, BPS\ and 
BPZ ; taking the last, we have the following proportion : 

R : sin. PZ : : sin. PZB : sin. PB 
That is, P : cos. 16° 40' : : sin. 68° 30' : sin. P^=sin.63° 2' 30". 
To find ZB, we take the following proportion, (see page 185, 
observation 1, Robinson's Geometry) : 

P : coB.ZB : : cos. BP : cos. PZ 
That is, P : cos. Z^ : : cos. 63° 2' 30" : sin. 16° 40' 

P sin. 16° 40' 19.457584 

cos. 63° 2' 30" 9.656411 

cos. 50° 45' 48" 9.801173 

To find S'B, we have 

P : COS. S'B : : cos. 63° 2' 30" : sin. 23° 18' 

P sin. 23° 18' 19.597196 

COS. 63° 2' 30" 9.656411 

• COS. 29° 14' 38" 9.940786 

Observe that S'B=:BS ; therefore, Z;S'=50° 45' 48"+29° u' 
38"=80° 0' 26", and Z>S"=50° 45' 48"— 29° 14' 38"=21° 31' 10", 
the complement of the altitudes. Consequently the altitude at 
the first observation was 9° 59' 34", and at the second, 68°28'50". » 

* Our results differ a little from the given answer, owing, perhaps, to our 
not being minute in taking out the logarithms, or finding the nearest second 
corresponding to a given logarithm. — Experienced men on these matters do 
not pi'etend to work to seconds. 



TRIGONOMETRY. 



285 



To find the time from noon at the first observation, we have the 
following proportion : 

sin. PaS^ : sin-PZ^S^ : ": s'm. ZS : sin. ZFS. That is, 
cos.23° 18' : sin.68°30' : : sin.80° 0' 26" : sin.ZPAS^=sin.86°5'30" 

Had the angle been 90°, the time would have been just 6h. but 
the angle 3° 54' 30" less ; this corresponds to 15m. 38s. in time. 
Therefore, the time was 6h. 15m. 38s. For the time at the second 
observation, we have 
cos.23°18' : sin.68°30' : : sin,21°31'10" : sin.ZPAS"=sin.21°48'40" 

21° 48' 40"=lh. 31m. 14s. from noon, or lOh. 32m. 46s. ap- 
parent time in the morning. 

(10.) An observer in north latitude marked the time when the stars 
Megulus and Sjnca were eclipsed by a plumb line, that is, they were 
both in the same vertical plane passing through the zenith of the ob- 
server. One hour and ten minutes afterwards, Regulus was on the 
observer's meridian. What was the observer's latitude ? 

The positions of the stars in the heavens were 

Regulus, right ascension lOh. Om. 10s. Dec. 12° 43' north. 

Spica, *' " 13h. 17m. 2s. Dec. 10° 21' 20" south. 

Let R be the position of Regu- ' 
lus, S the position of Spica, P the 
pole, and Z the zenith. 

Then the side PaS'=100°21'20", 
PE=n° 17', and the angle BPS 
=3h. 16m. 52s., converted into de- 
grees ; that is, PPS=49'^ 13'. 

One hour and ten minutes re- 
duced to arc, give 17° 30'; but the 
stars revolve according to siderial, 
not solar time, and to reduce solar 
to siderial arc we must increase it by about its ^\-g th part ; this 
gives about 3' to add to 17° 30', making 17° 33' for the angle 
ZPR. Our ultimate object is to find PZ, the complement of 
the latitude. 

In the A PRS, we have the two sides PB, PS, and the in • 
eluded angle P, from which we must find PS and the angle SEP, 
and we can let a perpendicular fall from M on to the side PS and 




286 ROBINSON'S SEQUEL. 

solve it in the usual way ; but to show that a wide field is open 

for a bold operator ; we will put the unknown arc IiS=x, the 

side opposite Ii=r, and opposite S=Sy and apply one of the 

equations in formula (S), page 191, Robinson's Geometry. 

rpr - • D cos.a; — cos.r cos.s 

That IS, cos.i^= 

sm. r sin.5 

Whence, cos.P sin.r sin.s-j-cos.r cos.5=cos.a; 

We now apply this equation, recollecting that radius is unity, 

which will require us to diminish indices of the logarithms by 10. 

cos.P=cos. 49° 13' —1.815046 

sin.r=sin.lOO° 21' 20". . .—1.992068 —cos —1.254579* 

sin. s =sin.77° 17' —1^89214 cos —1.342679 

0.6268 —1.797123 .03956 . . .—2.597258 

cos.a;=0.6268— 0.03956=. 58724. 
Whence, by the table of natural cosines, we find a:=54°2'20". 
To find the angle SEP or ZEP, we have 
sin. 54° 2' 20" : sin. 49° 13' : : sin. 100° 21' 20" : sin. ZBP 
Whence, ZEP=66° 57' 30". 

Let fall the perpendicular PB on PZ produced, then the right 
angled spherical A PPP gives this proportion : 

P : sin. 77° 17' : : sin. 17° 33' : sm.PB=sm. 17° 6' 22" 
To find PP we have 

P : COS. PB : : cos. 17° 6' 22" : cos. 77° 17' 
Whence, P^=76° 41'. Now to find the angle BPP, we have 

sin. 77° 17' : P : : sin. 76° 41' : sin. ^i2P=sin. 86° 1' 
From PPB take PPZ, and ZPB will remain ; that is. 
From 86° 1' take 66° 57' 30", and ZPB=19° 3' 30". 
By the application of equation (12), we ^nd that 
P sin. 17° 6' 22"=tan. BZ cot. 19° 3' 30" 
Whence, ^Z=5° 48' And PZ=76° 41'— 5° 48'=70° 63'. 
The complement of 70° 53' is 19° 7', the latitude sought. 
By this example we perceive that by the means of a meridian 
line, a good watch, and a plumb line, any person having a knowl- 
edge of spherical trigonometry, and having a catalogue of the stars 
at hand, can determine his latitude by observation. 

♦Observe that r is greater than 90®, its cosine is therefore, negative in value, 
rendering the product cos. r cos. «, or .03956, negative. 



PART FOURTH. 

PHYSICAL. ASTRONOIWY. 

KEPLER'S LAWS. 

1 . The orbits of the planets are ellipses, of which the sun occupies 
one of the foci. 

2. The radius vector in each case describes areas about the focus 
which are proportional to the times. 

3. The squares of the times of revolution are to each other as the 
cuhes of the mean distances from the sun. 

The first of these is a mere fact drawn from observation. The 
second is also an observed fact — but susceptible of mathematical 
demonstration, under strict geometrical principles, and the law of 
inertia. The demonstration is to be found in Robinson's Astron- 
omy, and in various philosophical works. 

The third is also susceptible of demonstration by means of the 
calculus — and by simple geometrical proportion, if we suppose 
the orbits circular. 

We now propose to investigate and determine the relative times 
of revolutions of two bodies about the sun, on the supposition 
that they revolve in circles, (which is not far from the truth,) and 
are attracted towards the center inversely proportional to the 
squares of their distances. 

Let S be the center of the sun, AS 
the radius vector of one planet, and SV 
that of another. 

Let m be the mass of the sun, SA=^ri 

and S V=E. Then ~ is the force which 

is exerted on the planet at A, and -— is 

the force exerted on the other planet at V. 

If we take any small interval of time, 

say one minute, and let AI) represent the 

distance the first planet falls from the tan- 




288 ROBINSON'S SEQUEL. 

gent of its orbit in unity of time, and VH the distances the other 
falls in the same time, 

Then ^ ; ^ '. '. AD '. VH (1) 

That the planets may maintain themselves in their orbits, the 
first must run over the arc AB in the unit of time, and the sec- 
ond must run over the arc VF. But this interval or unit of time 
can be taken ever so short ; and when very short, as a minute or 
a second, AB and VF, may, yea must be considered straight lines, 
chords 'Comc'iding with the arc. 

But if we take any chord of an arc, as AB, and from one ex- 
tremity draw the diameter, and from the other let fall the perpen- 
dicular BD, we shall have 

AB : AB : : AB : 2r 

AB^ VF^ 
Whence, AI)= , and in like manner VIf= 

Substituting these equals in proportion (1 j, and dividing the 
first couplet by m, and multiplying the last couplet by 2, we have 

1 1 AB^ . VF^ 



W r R 



1 1 



Or, _ : _ : : AB^ : VF^ (2) 

r R 

Because the first planet is supposed to run along the arc AB, 

in one minute, the number of minutes it will require to make its 

revolution will be found by dividing the whole circumference by 

AB. The circumference is expressed by 2r7t, and put t to repre- 

sent the time of revolution ; then t = , or AB = - — In 

AB t 

the same manner if T represents the time of revolution of the 

272 Tt 
second planet, we must have VF=^——-. By squaring these ex- 
pressions and substituting the values of AB^ and VF^ in pro- 
portion (2), we have 



r 



R t^ T^ 



O 1 • JL • • l! • :?" 



ASTRONOMY. 

Multiply the first couplet by tR, then 
i2 : r : . l! : 



Or, 



R^ 



R^ 

1^2 



L.=.^, Whence, t^ : T" : 



289 



•3 : i23. 



This last proportion corresponds with Kepler's third law. 



The following propositions are to be found on page 146 of 
Robinson's Astronomy. The frequent requests we have received 
to demonstrate them, suggested the propriety of pubhshing the 
demonstrations in this connection. 

The propositions are as follows : 

(1.) If two comets m-ove in parabolic orbits, the areas described by 
them in the same time are proportional to the square roots of their 
perihelion distances. 

Conceive a comet to revolve in an 
ellipse, F' the position of the sun, and 
A'F' the perihelion distance. 

Let F'D=r, F'C=x, DC=y, and 
put t to represent the number of hours 
required by the comet to make a rev- 
olution. 

Now nry == the area of the ellipse. This area divided by t, 
will express the area described by the comet about the sun in one 
hour. Let that area be represented by a. 




Then 



nry _ 



a. Let R, x\ y\ T, and A, represent similar 
quantities pertaining to another orbit, and by parity of reasoning, 

T 

Whence, 

By squaring, 



ry^ 
t 
r^y^ 



T ' ' 

qi3 



By Kepler's 3d law, t^ : T^ 
19 



: R\ or t^ = 



(1) 



290 ROBINSON'S SEQUEL. 

The value of /' substituted in (1), and reduced, will give 

y- X y : : a^ : A^ (2) 

r xt 

By inspecting the right angled triangle F'CDy we readily per- 
ceive that y^=:r^ — x'=(r-\-x) (r — x). Similarly, y'^=(B-\-x') 

Now if we suppose the ellipse to be infinitely eccentric, (as we 
must when it becomes a parabola,) {r-\'x) = 2r nearly, and 
(r — x)^=A'F'z=p exactly, (calling p the perihelion distance of 
one comet, and P the perihelion distance of the other.) 

Similarly , {R+x')z=9,R, and (B—x')=F. 

Substituting these values in (2), we have 

r R 

Or, p '. P \ \ a^ \ A^ 

Or, Jp : JP : : a : A Q. E. D. 



, (2.) j^we sui^pose a planet moving in a circular orhit, whose radius 
is equal to the perihelion distance of a cornet moving in a parabola, 
the areas described by these two bodies, in the same time, will be to 
each other as 1 to the square root of 2. Thus are the motions of 
comets and planets cminected. 

Let S be the position of 
the sun, P the perihelion point 
of a comet revolving in an 
ellipse. 

Put SP=x, and let i=the 
time in which the planet 
would revolve in the circle, 
and T= the time required 
for the comet to revolve in 
the ellipse. 

By the first law of Kepler the same body describes equal areas 
in equal times ; therefore if we divide the area of the circle by 
the number of units in the time of revolution, we shall have the 
area described in one unit of time. 

The area of the circle is ^ar*, and this divided by <, gives 




ASTRONOMY. 291 

= the sector described by the planet in unity of time. Also, 

I 

= the sector described by the comet in the same time. 

Conceive these two sectors to commence on the line SP, then 

3 A Ti 

(sector in circle) : (sector in ellipse) : : — : -__ (1) 

f JL 

A and B are the semi-conjugate axes of the ellipse. 

By Kepler's third law, 

e : T^ : : x^ : A^ (2) 

Multiplying the last couplet of ( 1 ) by tT, gives 

(sector in circle) : (sector in ellipse) : : Tx^ : [tA)B 

By squaring, we have 
(sec.incircle)^ : (sec. in ellipse )2 :: {T''x^)x : (il^^^^^a ^3^ 

From (2) we find T^x^=.t^A^, and substituting the value of 
T^x^ in (3), we have 
(sec.incircle)2 : (sec.inellipse)2 :: t^'A^'x : (^M^)^^^ 

:\ Ax \ B^ (5) 

Observe the right angled triangle CSQ. SG=A, CS=A — x, 
OG=B. 

A''—(A—xy=B^ 
Or, 2JaJ— a;2=52 

Substituting this value of B^ in (5), and dividing the last 
couplet, gives 

(sector in circle)^ : (sector in ellipse) ^ : : A : 2A — x 

Dividing the last couplet by A, and extracting the square 
root, gives 

(sector in circle) : (sector in ellipse) : : 1 : ^2 — — (6) 

When the ellipse is very eccentric, A is very great in relation 

, X • 

to X, and the fraction, — is then very insignificant in value. As 

an ellipse becomes more and more eccentric, its curve approaches 
nearer and nearer to a. parabola, and when it becomes a parabola, 

A is infinite in respect to x, and the fraction — is then absolutely 

zero, and proportion (6) becomes 

(sector in circle) : (sector in parabola) : : I : J2 Q. E. D. 



292 ROBINSON'S SEQUEL. 

The following inquiry has frequently come to us. We now 
give it in the words of a correspondent. 

Mr. Robixsox : 

Dkar Sir. — On page 192, Art- 180 of your Astronomy, it is stated that 
because the mean radial force causes the moon to circulate at -I.- part 
greater distance from the earth than it otherwise would, its periodical revo- 
lution is increased by its 179th part. The question is, where does the fraction 
-K come from? 

RE PL y. 

The mean radial force acting in the direction of the radius 
vector does not prevent the moon from describing equal areas in 
equal times. Therefore the moon describes the same area with, 
as it would without this action ; but the radius is increased, and 
consequently the angular velocity diminished. 

We will now give the increase of radius, and require the cor- 
responding decrease of angular velocity, and we shall find the 
ratio of one will be double that of the other, on the condition that 
the increase or decrease of either, is small in relation to the whole- 

Let U be the angular 
point of two equal sectors, 
7' the radius of one, and A 
its arc. x its angle on the 
radius of unity. 

Let {r-\-h) be the radius 
of the other sector, A ^ its arc 
and y its angle. 

Then by reason of the two 
equal sectors, rA=(r-{-h)A^ (1) 

From one sector, \ : x : : r : A. Or, A=rx. 

From the other, 1 : y : : (r-\-h) : A^ Or, A ^=(r-\-h)y. 

Substituting the values of A and ^, in (1), we have 
r'x={r-\-kyj/ 

Or, X : y : : (r-]-hy : r^ 

Or, % \ y \ \ r-^-'lrh-^-h^ : r^ 

Because A is a very small fraction in relation to r, h^ can be 

omitted ; then 

X \ y \ \ r^-f-2rA : r^ 

Or, X \ y \ '. r -j-2 h r 




ASTRONOMY. 293 

This last proportion shows that if the radius r is increased by h, 
the angular velocity and consequently the periodic time must be di- 
minished by 2A. 

PROPOSITION . 

Given the position of the earth as seen from the sun, the position 
of any other planet as seen from the sun, to find the position of that 
planet as seen from the earth. 

The motion of the earth and planets being known, and the 

elements of their orbits, the astronomical tables give the position 

of the earth and any planet for any given instant of time.. The 

position of the planet from the earth must then be computed by 

plane trigonometry. But before we give a definite example, we 

adduce the following 

LEMMA. 

1 . In any plane triangle the greater of two sides is to the less, as 
radius to the tangerd of a certain angle. 

2. Radius is to the tangent of the difference between this angle and 
46°, as the tangent of half the sum of the angles at the base of the tri- 
angle is to the tangent of half their difference. 

To obtain that certain angle, we must place the two sides at 
right angles to each other. 

Let CA be the greater of two 
sides of a A, and CE a less 
side placed at right angles ; then 
CAE is the certain angle spoken 
of, less than 45°, and EAB is 
the difference between it and 
45°. 

From (7 as a center with the 
longer side as radius, describe the semicircle. Then DE = the 
sum of the sides, and EG their difference. Join DA, A G, and 
from E draw EB parallel to DA. DA G is & right angle because 
it is in a semicircle ; therefore, EB being parallel to DA; EBG 
is a right angle also. DA=zAG, and EBz=BG. 

Let a be the greater side of a triangle represented in magnitude 
but not in position, by CA, and c the shorter side, represented in 
magnitude by CE ; then it is obvious that 

a '. c '. '. R : tan. CAE 



■ 



294 



ROBINSON'S SEQUEL. 



This angle taken from the table and subtracted from 46° will 
give the angle EAB. By proportional triangles we have 



BE 
That is, a-f-c 
But. AB 

Whence, a-\-c 



EG 



EB 



AB 
AB 

B 

R 



BG^EB 
EB 

tan. EAB 
tan. EAB 



By proportion 7, page 149, Robinson's Geometry, we find that 
a-^c : a — c : : tan.^sum ang. atbase : tan. ^ their diflf. 

Therefore by comparison, 
R : tan.-£'^-5 :: tan. | sum ang. at base : tan.^ their difF. Q.E.D. 

The application of this proposition is very advantageous when 
the logarithms of the two sides of a triangle are given and not 
the sides themselves. It obviates the necessity of finding the 
numerical values of the sides. This proposition is almost solely 
used in Astronomy, and we give the following example as an 
illustration. 

In the Nautical Almanac for 1864, Ifirid that on the first day of 
April at noon, mean time at Greenwich, the sun's longitude is 11° 
26' 28", and the logarithm of the radius vector of the earth is 
0.0000224. At the same time the heliocerUric longitude of Jupiter 
is 283° 46' 7", soitth latitude 6' 41", and logarithm of its radius 
vector 0. 71 45152. Required the geoceyitric latitude and longitude 
of Jupiter, and the logarithm of its true distance from the earth. 

Let S be the sun, 'Y'=a= the line made by 
Aries and Libra in the plane of the ecliptic, 
/y^ <fcc., the direction of counting lon- 
gitude. 

Place the earth at E, so that the sun at 
S will appear to be in 11° 26' 28" of longi- 
tude. Then E, the earth, will appear from 
the sun to be in 191° 26' 28" of longitude. 
jS'jB' is a very little over a unit in distance, 
as we see by the log. 0.0000224. 

The longitude of Jupiter as seen from the 
sun is 283° 46' 7" ; hence, draw SI so that 
the angle JD=^/will be 103° 46' 7", and its distance from S to I 
a little over S times SE. The angle ESI will be 92° 19' 39", 
and log. of SI is given at 0.7145152. Our object is to find the 




ASTRONOMY. 9U 

position of the line BI, or Sh which is supposed parallel to HI, 
and the logarithm of the distance HI. 

Jupiter not being in the plane of the ecliptic, we must reduce it 
to that plane, by multiplying its distance by the cosine of its 
inclination. 

Thus, to the log 0.7145152 

Add cos. 6' 41" 9.9999998 

Log. of distance in the ecliptic, . . .^ 0.7145150 

Now by the first part of the Lemma, 

As 0.7145150 

To 0.0000224 

So is radius 1 0:0000000 

,To tan. 10° 55' 21" 9.2855074 

This arc from 45° gives 34° 4' 39". The angle U SI irom 180° 
gives 87° 40' 21" for the sum of the angles U and /. Their half 
sum is therefore, 43° 50' 10"5. Let their half difference be de- 
noted by X. Then by the last part of the Lemma, 

B : tan. 34° 4' 40" : : tan. 43° 50' 10"5 : tan. x 

tan. 34° 4' 39" 9.830254 

tan. 43° 50' 10"5 9.982352 

tan.a: 33° 0' 19" 9.812606 

The angle ^=43° 50' 10"+33° 0' 19"=76° 50' 29". 
The angle 7=43° 50' 10"— 33° 0' 19"=10° 49' 51". 
The angle ISk=\0° 49' 51"; therefore, the geocentric loftgi' 
tude of Jupiter is 283° 46' 7"+10° 49' 51" or 294° 35' 58". 
For the log. of JSI^re have the following proportion : 

sin./ : SB : : sin. ISU : UI 
Or, sin. 10° 49' 51" : SU : : sin. 92° 19' 39" : £1 

Log. sin. 92° 19' 39" 9.999641 

Log. SU 0.00002 24 

9.9996634 

Log. sin. 10° 49-51" 9.2739400 

Log. UI 0.7257234 

This result is the logarithm of the distance from U along in 
the plane of the ecliptic to the point where the perpendicular falls 
from Jupiter, — the hypotenusal or absolute distance is a little 



296 ROBINSON'S SEQUEL. 

greater, but it is hardly perceptible in this case, as Jupiter is so 
near the ecliptic. Indeed it would increase the last decimal fig- 
ure in the lo'garithm by 2, making it 6. 

Jupiter appears from the sun at this time to be 6' 41" south of 
the ecliptic, but from the earth, the angle between it and the 
ecliptic would not be so great, because EI is greater than SI. 
But to compute the geocentric latitude of Jupiter or any other 
planet exactly, we have the following principle : 

We refer in particular to this example, but the principle is 
general. 

Conceive the perpendicular distance of t\ie planet from the 
ecliptic to be represented by i>, and let this distance be made 
radius ; then IS will be the cotangent of the heliocentric latitude, 
and IE the cotangent of the geocentric latitude. 

Denote the geocentric latitude by x ; then 

D \ R \ \ IS \ cot. 6' 41" 
And D : E '. '. IE : cot. a; 

7-ET 

Whence, IS : cot 6' 41" : : IE : cot. ar=lr cot. 6' 41" 

That is, Erom the log. of the ijlaneCs distance from the earth, 
svhtract the log. of its distance from the sun, and to the difference add 
the log. cotangent of the heliocentric latitude, and the sum is the log. 
cot. of the planet's geocentric latitude. 

To apply this equation with accuracy, requires some little tact 
in using logarithms. Observe that cot. 6' 41" is the tan. of 6' 41", 
subtracted from 20.0000. To find the tan. of 6' 41" or 401", first 
find the tangent of 1", fhen add the log. of 401. 

tan. l'=60" *. . .6.463726 

sub. log.60 1.778151 

tan. 1" 4.685575 

log. 401 2.603144 

tan. 6' 41" 7.288719 

Log. -£^/— log. //S'=0.7257234— 0.7145150=0.0112084. 

The log. cot. must be increased by this quantity, therefore the 
log. tan. must be diminished by the same ; hence 

7.2887190 
0.0112084 



ASTRONOMY. ' 29^7 

Log. tan. of geocentric latitude is 7.2775106 

Subtract log. tan. of 1" 4.685575 

Log. of 390"8, or 6' 31" nearly, 2.5919356 

Thus we find the geocentric latitude of Jupiter to be 6' 31" 
south at this particular time. 

Having the planet's latitude and longitude, we can compute its 
corresponding right ascension and declination, and the following 
results will be obtained : 

Right ascension, 19h. 46m. 9s. South declination 21° 19' 41". 

SOLAR ECLIPSES. 

We will now show the computation to determine the times of 
beginning and end, and other circumstances attending a solar 
eclipse as seen from any assumed locality on the earth. No person 
can do this with any safety, depending on the rules of another, 
he must understand the nature and scope of the problem for him- 
self. It requires a general knowledge of astronomy and philoso- 
phy, and a familiar knowledge of both plane and spherical 
trigonometry. 

The mathematical philosophy of the subject is explained gen- 
erally on page 214 of Robinson's Geometry, and here we will 
illustrate it by an example. 

As near as we can determine by some rough projections, the 
eclipse of May 26, 1854, will* be nearly central and annular as 
seen from Burlington in Vermont. Curiosity has, therefore, led 
us to make minute calculations for that place. 

We take the elements from the English Nautical Almanac. — 
Let the reader observe that the elements here correspond to the 
mean time of conjunction in rif/ht ascension. The elements in 
Robinson's Astronomy correspond to conjunction in longitude, the 
difference is 8m. 37s. in time. 

1854, May, 26. 

Greenwich mean time (/ in R. A 8h. 55m. 43.8s. 

Sun and moon's R. A * 4h. 13m. 7.41s. 

Moon's Dechnation North, 21° 33' 3r'8 

* This was written in April, 1853, and therefore spoken of in the future 
tense. 



«98 



ROBINSON'S SEQUEL. 



Sun's Declination North, 21° 11' 16"8 

Moon's Horary motion in R. A 31' 18"9 

Sun's Horary motion in R. A 2' 31"8 

Moon's Horary motion in Declination N 8' 7"3 

Sun's Horary motion in Declination N 25"9 

Moon's Equatorial Horizontal parallax 54' 32"6 

Sun's ** ** ** 8"5 

Moon's semidiameter 14' 53"5. Sun's S. D. 15' 48"9. 
Lat. of Burlington 44° 28' N. West Long. 73° 14'=4h. 52m. 56s 

Greenwich mean time of q^ ^^' 55ra. 44s. 

Long, in time 4h. 52m. 66s. 



Mean time of (/ at Burlington . 
Equation of time, add 



4h. 2m. 48s. P.M. 
3m. 15s. 



Conjunction at B., apparent time. . . .4h. 6m. 3s.=61° 30' 45". 

As the earth is not a perfect sphere, (the equatorial diameter 
being the largest,) the equatorial horizontal parallax requires re- 
duction for other latitudes, and latitude itself requires a reduction 
at all points, except at the equator and the poles. 

The horizontal semidiameter of the moon requires .augmenta- 
tion, as the moon rises in altitude, for the nearer the moon is to 
the zenith of the observer, the nearer it is in absolute distance. 

The following tables correct the elements in these particulars* 

Reduction of the Parallax and also of the Latitude. 



Lat. 


Red. 
of par. 


Red. of 
Lat. 


Lat. 


Red. 
of par. 


Red. of 
Lat. 


Lat. 


Red. 
of par. 


Red. of 
Lat. 


o 


" 


/ // 


o 


" 1 


/ II 


o 


/' 


/ II 





0.0 


0.0 














3 


0.0 


1 11.8 


33 


3.3 


10 28.3 


63 


8.8 


9 18.3 


6 


0.1 


2 22.7 


36 


3.8 


10 64.3 


66 


9.2 


8 32.9 


9 


0.3 


3 32.1 


39 


4.4 


11 13.2 


69 


9.7 


7 42.0 


12 


0.5 


4 39.3, 


42 


4.9 


11 24.7 


72 


10.0 


6 45.9 


16 


0.7 


6 43.4 


46 


6.5 


1128.7 


75 


10.3 


6 46.4 


18 


1.0 


6 43.7 


48 


6.1 


1125.2 


78 


10.6 


4 41.0 


21 


1.4 


7 39.7 


51 


6.7 


11 14.1 


81 


10.8 


3 33.5 


24 


1.8 


8 30.7 


54 


7.2 


10 65.7 


84 


11.0 


2 23.7 


27 


2.3 


9 16.1 


67 


7.8 


10 30.0 


87 


11.1 


1 12.3 


30 


2.7 


9 55.4 


60 


8.3 


9 57.4 


90 


11.1 


0.0 



ASTRONOMY. 



299 





Atigmentaiion of the Mooti's Semi- diameter. 






Horizon. Semi-diameter. 


Alt. 


Horizon. Semi-diameter. 


Alt. 


f4'30" 
If 


16' 


16' 
It 


17' 


14'30" 


15' 


16' 

It 


\r 


— OT 


H 


o 


// 


// 


2 


0.6 


0.6 


0.7 


0.8 


42 


9.2 


9.8 


11.2 


12.6 


4 


1.0 


1.1 


1.3 


1.5 


45 


9.7 


10.4 


11.8 


13.3 


6 


1.6 


1.6 


1.9 


2.1 


48 


10.2 


10.9 


12.4 


14.0 


8 


2.0 


2.1 


2.4 


2.7 


51 


10.6 


11.4 


13.0 


14.7 


JO 


2.4 


2.6 


3.0 


3.4 


54 


11.1 


11.8 


13.5 


15.2 


12 


2.9 


3.1 


3.6 


4.0 


57 


11.5 


12.3 


14.0 


15.8 


14 


3.4 


3.6 


4.1 


4.7 


60 


11.8 


12.7 


14.4 


16.3 


16 


3.8 


4.1 


4.7 


5.3 


63 


12.2 


13.0 


14.9 


16.8 


18 


4.3 


4.6 


5.2 


5.9 


<o^ 


12.5 


13.4 


15.2 


17.2 


21 


4.9 


5.3 


6.0 


6.8 


69 


12.8 


13.7 


15.6 


17.6 


24 


SQ 


6.0 


6.8 


7.7 


72 


13.0 


13.9 


15.9 


17.9 


27 


6.2 


6.7 


7.6 


8.6 


75 


13.2 


14.1 


16.1 


18.2 


30 


6.9 


7.3 


8.4 


9.5 


78 


13.4 


14.3 


16.3 


18.4 


33 


7.5 


8.0 


9.1 


10.3 


81 


13.5 


14.4 


16.5 


18.6 


36 


8.1 


8.6 


9.5 


11.1 


84 


13.6 


14.5 


16.6 


18.7 


39 


8.6 


9.2 


10.5 


11.9 


90 


13.7 


14.6 


16.7 


18.8 



Equatorial horizontal parallax 54' 32"6 

Reduction for latitude 6"4 

Reduced h. p 64' 27"2 

Subtract sun's h. p 8"6 

Relative or effective h. p 64' 18"7=3268"7 

Lat. of Burlington 44° 28' 

Reduction 11' 27 ^ 

Reduced latitude 44° 16' 33" 

As the sun and moon are west of the meridian, therefore the 
moon is apparently thrown back by the effects of parallax, and 
consequently the beginning of the eclipse will not take place un- 
til about, or after, the time of conjunction. 

To decide this point, let m represent the place of the moon 
at 4h. 6m. 3s., and S the place of the sun at the same time. 
^m=22' 15", their difference in declination, mn the parallax in 
altitude, and n will be the apparent place of the moon. Our 



300 



ROBINSON'S SEQUEL. 




object is to find the dis- 
tance between S and n, 

and we accomplish it by 

the aid of the spherical 

triangle FZ?n*. We have 

PZ, Fm, and the angle 

ZPm. We must find Zm 

and the angle ZmP. 

We use the following 

equation taken from page 

209, Geometry, in which A 

represents the moon's alti- 
tude, L the latitude of the 

place, and D the moon's polar distance. 

p_sin. A — sin. Zcos. D 
cos. L sin. J) 
Whence, sin. ^=cos. Zm = cos. P cos. L sin. D-\-sin. L cos. i), 

cos. P=cos. 61° 30' 45" 9.678489 

cos. Z=cos. 44° 16' 33" 9.854910 sin 9.843917 

sin. i>=sin. 68° 26' 28". .. ..9.968853 cos 9.562944 

0.31787 —1.502252 .255192 —1.406861 

Whence, the natural sine of the moon's true altitude, or cos. 

Zm=0.31787+0.25519=.57306. 

* As the moon is at m, and the sun at <S'. on the same meridian PjS, which is 
61° 30' 45" west of Burlington, or 134° 44' 45" west of Greenwich; there- 
fore, this is the meridian on which the sun will be centrally eclipsed at appa- 
rent noon ; and the latitude will be such that mS must be the moon's parallax 
in altitude. To find that latitude is a very easy and interesting problem. 
Let the latitude be such that the apparent altitude of the moon shall be rep- 
resented by X. Then 3257" cos. ar=mS=1335". (Rad. 1.) 

1335\R 13.125481 

cos.j;=_ 

3258.7 3.513044 

COS. ar=sin. moon's app. zenith distance 24° 11* 5" sin 9.612437 

Moon's declination north 2 1° 33' 3 2" 

Lat. (apparent,) 45° 44'37" 

Reduction of Lat 11*29" 

True Lat 45° 33' north. 

Hence the sun will be centrally eclipsed at noon in longitude 134° 45' west 

and latitude 45° 33' north. 



ASTRONOMY. 301 

By taking out the corresponding arcs, we find 

Moon's true altitude=34° 58' 10" Zm=55° V 50". 
sin. Zm : sin. P : : sin. ZP : sin. Z»iP= Smn. 
Or, sin. 55° 1' 50" : sin. 61° 30' 45" :: cos.44° 16'33" : sin. /Smn 
==sin. 60° 10' 8". 

The next and most delicate step is to obtain mji, the parallax 
in altitude. 

If we use the true altitude of the moon for the apparent alti- 
tude, we can find the approximate value of mn as follows : (see 
page 201, Surveying and Navigation.) 

Horizontal parallax 3258"7 log 3.513044 

COS. 34° 58' 10" add .9.913526 

1st approx. value of 7nn 44' 30"=2670" 3.426570 

Moon's true alt 34° 58' 10" 

Sub. parallax in alt. ... 44' 30" 
Moon's appa. alt. nearly . .34° 13' 40" 

H. p. as before, , . . .3.513044 

COS. 34° 13' 40" , 9.917380 

2d approx. value of mn, 44' 54"=2694" 3.430424 

H. p. as before 3.513044 

Moon's appa. alt cos. 34° 1 3' 1 6" 9.91 7422 

True value of mn, 44'56"6=2694"6 .3.430466 

Now in the A Smn, (which we may conceive to be a plane 

triangle,)we have Sm=1335", mn=2694"G, and the angle Smn, 

50° 10' 8", to find Sn. 

If from n we conceive a perpendicular to be let fall on to the 

meridian Sm, and designate it by p, and the other side of the right 

angled triangle thus formed by q, then we shall have 

H : 2694.6 : : sin. 50° 10' 8" : ;> 
And i? : 2694.6 : : cos. 50° 10' 8" : q 

mn 3.430466 3.430466 

sin. 60° 10' 8" . 9.885322 cos 9.806537 

p 2069.2 3.315788 q 1726.8. . .3.237003~ 

Observe that p is the effect of parallax perpendicular to the 
lunar meridian at that time; and q is the parallax in declination. 



302 ROBINSON'S SEQUEL. 

Moon^s true declination north of the sun 1336" 

Moon's parallax in declination south 1725"8 

Moon's apparent dec. south of the sun 390^8 

The apparent distance between sun and moon is, therefore, 

V(2069.2)2+(390.8)2=2105"7 
Moon's S. D., 14' 53"5. Augmentation for altitude 8". Sun's 
S. D. 16'48"9. Sum = 30' 50"4=1860"4. But the distance 
(apparent) from center to center, we have just determined to be 
2106"7 ; therefore the distance from limb to limb must be 255"3, 
and the eclipse has not yet commenced, and cannot commence, 
until the moon gains 256" on the sun's motion, which will require 
more .than ten minutes of time. 

We now require the apparent 
distance between the centers of 
the sun and moon, ten or twelve 
minutes later, so as to get the ap- 
parent rate of approach. The rate ^^^ 
is continually changing, but du- ^Qf^ 
ring any short interval of ten or 
twelve minutes, it may be consid- 
ered uniform, without any sensible error. 

If we vary the time, the angle ZPS will vary 1° to 4 minutes, 
but in that variable time the moon will move from c to m, and 
the angle ZPm will vary, but not quite so much as ZPS. The 
question now is. If we make a small difference in the angle ZPm, 
what corresponding difference will it make in the arc Zm ; and this 
is a question in the differential calculus,'^ although we can work 
it out at large by spherical trigonometry. 

We will take the interval of 12m., then the angle ZPS will 
increase 3°. But in one hour the moon's motion in right ascen- 
sion exceeds, that of the sun 28' 47" ; this in 12m. will be 5' 45"4, 
therefore the angle ZPm varies in that interval of time, 2° 64' 
14"6 =2.90406, taking one degree as the unit. 

* The differential calculus is the science of minute variations, or of corres- 
ponding small differences — a science which owes its birth to the varying 
dfeements of astronomy. 




ASTRONOMY. 30S 

The equation as before is 

sin. ^=cos. Zw=cos. P cos. L sin. D-\-m!L, L cos. D 
But we have caused P to vary, while L and D remain constant. 
What variation will this give to the altitude A ? 
Taking the differential of the equation, we find 

* jM sin. P COS. L sin. B-dP 

cos. ^ 
But we have assumed c?P=2. 90405, while P, Z, B, and -4, in 
this equation, have the same values as before. That is, 

/>=61° 30' 45", i;=44° 16' 33", i>=68° 26' 28", and ^=34° 
58' 10". 

log. 2.90405 0.463000 

sin.P —1.943954 (radius unity.) 

COS. L — 1.854910 

sin. B —1.968853 

COS. complement A 0.086445 

rf^=2.0755 0.317162 

Thus we find that the moon changes its altitude at this time, 
in the interval of 12 minutes, .2° 4' 32", and because the second 
member of the last equation is minus, the altitude has diminished. 

Moon's altitude was 34° 58' 10" 

Variation 2° 4' 32" 

Moon's altitude at this time 32° 53' 38" 

The angle ZPm=61° 30' 45"+2° 54' 15"=64° 25'. 
cos. 32° 53' 38" : sin. 64° 25' : : cos. 44° 16' 33" : sin. ZmP=sin. 
60° 16' 30". 

To find mn, or the parallax in altitude. 

To log. of the horizontal parallax , 3.513044 

Add COS. 32° 53' 38" 9.924100 

Approximate value of mn 45' 36"=2736" 3.437144 

From the moon's true alt. 32° 53' 38" 

Subtract apparent paralla x 45' 36" 3.513044 

Moon's appa. alt. nearly 32° 8' 8" cos 9.927877 

True value of mn 2760" 3.440921 

As before, 

E : 2760 : : sin. 50° 16' 30" : p 
R : 2760 : : cos. 60° 16' 30" : q 



4 

?i04 ROBmSON'S SEQUEL. 

3.440921 3.440921 

sin. 50° 1 6' 30" ..9.885 996 cos 9.805420 

J^2123" 3.326917 q 1763"5.. . .3.246341 

During the 12 minutes the moon moves over the oblique stkuiU. 
arc C7n, (in relation to the sun, as conceived to be stationary,) 
which is 5' 45"4 in right ascension, or the difference between the 
two meridians PS and Pm on the equator, is 5' 45"4, or 345"4. 
The perpendicular distance at the point m is therefore found 
by multiplying 345"4 by the cosine of the moon's declination to 
radius unity. Therefore, 

Log. 345"4 2.538322 

Moon's dec. 21° 35' nearly, cos 9.968429 

Perpendicular dis. between PS and Pm 32r'2. ..2.506751 
During one hour the moon's relative motion in declinatian is 
7' 41"4. During 12 minutes it is therefore 92"2, which added to 
22' 15" or 1335" makes 1427"2 for the distance represented by 
ma. But q 1763"5 is the effect of parallax on the line or the ef- 
fect in declination, and it being greater than 1427"2, their differ- 
ence, 336"2 is the apparent distance in declination of n below S, 
or of the center of the moon below the center of the sun. 

Again, p is the parallax in right ascension, projecting the moon 
2123" west of its true place, while it is 321 "2 east of the sun ; 
therefore the apparent right ascension of the moon is 1801 "8 west 
of the sun. Consequently the apparent distance of the two 
centers is 

V(1801"8p+(336"2)2 = 1 833"2. 
But the semidiameter of the sun and the augmented semidiam- 
eter of the moon at this time amount to 1850"4, differing only 
17"2. The distance between the centers being less than the sum 
of the semidiameters, shows that the eclipse has already com- 
menced 

Twelve minutes before this time, the distance between the 

centers was ■* 2105"7 

Now it is 1833"2 

Moon's apparent motion in 12 minutes . . .'. 272"5 

or 22^7 in one minute. 

Then 22"7 : 17"2 : : 60s. : 46.4 seconds. 



ASTRONOMY. 306 

That is, the eclipse commences 11m. 14.6 sec. after the appa- 
rent time of conjunction at Burlington, or at 4h. 17m. 17.6 sec. 

If 6 seconds be taken from the sum of the semidiameters for 
irradiation and inflection, as most astronomers recommend, the 
eclipse will commence at 4h. 17m. 30s. 

THE POINT OF FIRST CONTACT. 

The point n, the apparent place of the center of the moon is 
nearly west of S and the angle ZmP=50°. Therefore the point 
of first contact, from the sun's vertex, must be (50"-}~^0°)» ^40° 
towards the right, but if viewed through an inverting telescope, 
the appearance will be directly opposite. 

GREATEST OBSCURATION. 

The time of greatest obscuration will take place not far from 
Ih. 20m. after conjunction at Burlington, or not far from 5h. 26m. 
3s. ; we will therefore compute the apparent distance between the 
two centers for this time. We could compute it by proportion, 
provided the apparent motion of th,e moon was uniform, and in a 
straight hne ; but that motion being neither uniform nor in a 
straight line, we are compelled to compute it by points to obtain 
any thing like accuracy. 

Using the last figure, the angle ZP^=5h. 26m. 3s. =81^ 30' 
45" ; but during Ih. 20m. the moon will gain 38' 22" in right as- 
cension ; therefore the angle ZPw=80° 62' 23". 

In Ih. 20m. the moon will increase her declination 10' 49", 
making it 21° 44' 21", or Pm=68° 15' 39", and am is now 32' 
30"= 1950". 
As before, 
sin. -4=cos. Z7»=cos. P cos. L sin. i)-|-sin. L cos. D. 

COS. P=cos. 80° 52' 23" 9.200404 

cos. Z=cos. 44° 16' 33" 9.854910 sin 9.843917 

sin. i>=sin. 68° 1 6' 39" .9.967959 cos 9.568656 

0.10555. —1.023273 .25856 —1.412571 

Nat. sin. ^=0. 10555+0.25856=.3641 1 . 

Whence, A, moon's true alt.=21° 21' 12". Zm=68° 38' 48". 
20 ' 



% 



306 ROBINSON'S SEQUEL. 

sin. 68° 38' 48" : sin. 80° 52' 23" ; : cos. 44° 16' 33" : sin. ZwP= 
sin. 49° 22' 45". 

To find mn. Moon's horizontal par. log 3.513044 

COS. 21° 21' 12" 9.969114 

Approximate value of mn 50' 35"=3035" 3.482168 

From moon's true alt. . .21° 21' 12" 

Take 50' 35" 3.513044 

Moon's appa. alt. nearly 20° 30' 37" cos 9.970630 

True value of mn 3046" 3.483674 

R : 3045" : : sin. 49° 22' 45" : p 
B : 3045" : : cos. 49° 22' 45" : q 

3.483674 3.483674 

sin. 49° 22' 45" 9.880265 cos 9.813620 

i?=2311"8 3.363939 ^=1982"6... .3.297294 

In the Ih. and 20m. which elapses after conjunction, the moon 
gains 38' 22" or 2302" in right ascension on the sun ; but this is 
arc on the equator, it is not perpendicular distance, the two me- 
ridians PS and Pm, drawn from m ; but that distance is required 
and it is found thus : 

Log. 2302 3.362105 

Add COS. of moon's declination 21° 44' 21" .9.9679 59 

wic 2138"5 3.330064 

p 231 1"8 

Moon apparently west . . 173"3 

Moon's declination north of sun am 1960" 

Moon's parallax in declination q 1982"6 

Moon apparently south of the sun 32"6 

Distance between centers= ^(iTS^p +(32"6) ^ = 1 76"3. 
We know by comparing this result with the last, that the gi-eatest 
obscuration or nearest approach of the centers, must take place 
about 7 minutes after this time. We will, therefore, differentiate 
for 10 minutes. 

In 10 minutes the sun's polar angle will increase from the 
meridian 2° 30' 

For the 3'^ motion in R. A. sub. 4' 47" 
The angle ZPm will increase 2° 26' 13 '=2°.4202. 



ASTRONOMY. 307 

As before, 

. _ sin. P COS. L sin. D ( g.4202) 
COS. -4 
An equation in which ^=21° 21' 12", P= 80° 62' 23", Z= 
44° 16' 33", and i>=68° 16' 39". 

sin. P —1.994465 (radius 1) 

COS. L —1.864910 

sin. i> —1.967959 

log. 2.4202 , 0.383861 

COS. complement A 0.030886 

(^-4=1.7062 0.232071 

The minus sign before the second member shows that this must 

be subtracted from A. A 21° 21' 12" 

1.7062= 1° 42' 22" 

Moon's true altitude at this time, 19° 38' 60" 

Log. Horizontal parallax 3.513044 

cos. 19° 38' 60" 9.973950 

51' 9" 3.486994 

Moon's app. alt. nearly 18° 47' 41" 

3.613044 

cos. 18° 47' 41" 9.976154 

True value of mn 3084"5 3.489198 

cos. 19° 38' 60" : sin. 83° 17' 36" : : cos. 44° 16' 33" : mi.ZmP 
=sin. 49° 1' 40". 

R : 3084"6 : : sin. 49° 1' 40" : ^ 

R : 3084"6 :: cos. 49° 1' 40" : g 

3.489198 3.489198 

8in.49° 1^ 40" 9.877978 cos . 9.816700 

p 2329". . . . 3.367176 g 2022"7 3.306898 

At the last point, am was 1950" which has increased 77" by the 
moon's motion ; therefore it is now 2027". - 

At the last point, Sa was 2302" of arc which has increased 
287", making 2589", which must be reduced to the arc of a great 
circle as before. 

f 



!► 

^ 



ROBINSON'S SEQUEL. 

Log. 2589 3.413132 

Moon's declination 21° 46' cos 9.967927 



Moon east of sun 2404"6 3.381069 

Parallax in R. A. p., .2329" 
Moon east of sun, apparently 76"6 

Moon north of sun 2027" 

Parallax in declination, q., ,, 2022"7 
Moon north of sun, apparently .... 4"3 




Distance between centers = ^(76"6)2-|- (4"3)=75''6, appa- 
rently. 

Now to find the nearest approach of the centers, the time of 
forming the ring, its continuance, &c., we have a very delicate 
and simple problem in plane geometry. 

Let S be the center of 
the sun. Take SV = 
173"3, F"u4=:32"6. Then 
^aS'=176"3. Also take 

then /Sf^=75"6, BD= 
173"3+75"5=248"8. i>^ =37" nearly ; then ^J5 the moon's 
apparent motion on the face of the sun during 10 minutes must be 
V(248"8)2-f(3rp =251 "5. 
Therefore the apparent motion per minute is 25"15. 
We must now find Sm, the distance between the two centers at 
the time of their nearest approach. In the triangle ABS, we 
have all the sides, therefore by (Prop. 6, page 149, Geom.), we 
have AB : AS-\-SB : : AS—SB : Am-^B 
That is, 26r'6 : 26r'9 : : 100.7 : 100.86 
^wi+7w^=251.6 
Am—mB=100M 

2mJ5= 150.64 m.B= 76.32. 
Whence, Am=n&'lB. In the right angled triangle BmS, we 
have 

Sm= J(75"6y—{75"32y =6"5 

Moon's semidiameter from giv^a elements 14' 63"6 

Augmentation for alt. 18° <"" (see table) 4^7 



ASTRONOMY. 309 

Augmented semidiameter 14' 58"2 

Sun's semidiameter from given elements 15' 48"9 

Diflference, 50"7 * 

It is obvious that the ring will form when the distance between 
the two centers comes within the difference of the semidiameters. 
Suppose it to form when the moon's center passes n ; then in the 
right angled triangle Smn, Sn=50"7. Sm—6"5. 
And mw=7(50"7)2— (6"6)2=50"28. 

An=Am — mw=126"9, and at the rate of 25"16 per minute, 
this will be passed over in 5m. 2.7 seconds, nm in 2 minutes very 
nearly, and an equal line on the other side of m in 2 minutes 
more. 
The appa. time the Q)'s center arrives at A is 5h. 26m. 3s. 

To which add 5m. 2.7s. 

Ring forms at 5h. 31m. 5.7s. 

Time of nearest approach 5h. 33m. 5.7s. 

Rupture of the ring 5h. 35m. 5.3s. 

At the time of nearest approach the breadth of the ring on the 
north limb of the sun will be ST'O, and on the south limb 18"8 ; 
but if the customary allowance be made for irradiation and inflec- 
tion, these quantities reduce to 31 "3 and 18"3, and the duration 
of the ring must be reduced from 3m. 59.6s. to 3m. 55.2s. 

THE END OF THE ECLIPSE. 

We know by the moon's apparent motion (25''15 per minute, 
which is continually increasing) that more than an hour will be 
required from the time of nearest approach, for the eclipse to pass 
ofif. We will therefore compute the apparent distance between 
the two centers, one hour and ten minutes after the moon passes A^ 
of the last figure. 

By referring back we shall find that the point A corresponds 
with 6h. 26m. 3s. or 81° 30' 45" for the angle ZP5. One hour 
and. ten minutes later will be 6h. 36m. 3s. and will correspond to 
99° 0' 45" for the angle ZPS. (See next figure.) 

* Astronomers recommend a diminution of 3" for the sun's semidiameter 
for irradiation, and a diminution of 2" of the moon's semidiameter for in/lio^ 
Hon, this would make 49"7 for their difference instead of 50"7. 



• ' 



310 ROBINSON'S SEQUEL. 

Butthe difference between the right ascensions of the sun and 
moon is now 1° 21' 66" ; therefore the angle ZFm=97° 38' 49". 
At 5h. 26m. 3s. the value of ma was 1950", in one hour and ten 
minutes it increased 540", it is now 2490". 

Sa in arc=l° 21' 66"= 49 16" which we reduce to distance. 

Log. 4916 3.691612 

Sun's dec. cos. 21° 12' 9.969667 

Sa 4583" 3.661179 

As before, 

sin. ^=cos. Zm=cos. P cos. L sin. D-j-sin. L cos. D 
cos. P=cos. 97° 38' 49". .—1.124088 
cos. L =cos. 44° 16' 33", .—1.854910 sin.. .—1.843917 
sin. D =sin. 68° 7' 13". .— 1.967531 cos. .—1.671327 

—0.088416* —2.946529 0.26015-1.415244 

sin. A or cos. Zm=0.26015 — 0.08842=. 17173 
Whence, ^=9° 53' 21". Zw=80° 6' 39". 
sin. 80° 6' 39" : sin. 97° 38' 49" : : cos. 44° 16' 33" : sin.Zwi'= 
sin. 46° 6'. 

Log. Horizontal parallax 3.613044 

cos. 9° 63' 21" 9.993500 

63' 30" 3.506544 



Moon's app. alt. 8° 69' 61" cos 9.994618 

3.613044 

mn 3.607662 

sin. 46° 5' : p 
cos. 46° 6' : g 

3.607662 

cos 9.841116 

p 2918.4 3.465192 q 2232.4 3.348778 

Sa 4583. 2490 

~1664.6 267.6 



H : 


3218 : 


B : 


3218 : 




3.507662 


'5'... 


..9.857530 



Distance between the centers = 7(1 664.6 )2-f-(267.6)» = 1674"5. 

* This number must be minus because cos. 97°^ is minus, the cosine of any 
arc over 90°, as far as 270°, is minus. 




m 



ASTROHOMY. Sll 

The sum of the semi- 
diameters is now 1 843" ; 
therefore the sun is still 
eclipsed, and will be un- 
til the apparent motion of 

the moon passes over ^V^^^^^^^^HB^ 
169", which will require 
a little over 6 minutes of 

time, we will therefore compute the apparent distance of the cen- 
ters for 8 minutes later. In 8m. the angle ZPS will increase 
2°, and the angle ZFm will increase 2°— (3'+50") or 1^ 66' 10". 
At the last operation ZFm was 97° 38' 49", now it must be 99° 
34' 59", say 99° 35'. 

If we take D as constant in the last equation, we shall find 
that all our logarithms will be the same except cos. F, and all we 
have to do is to add to log. — 2.946529 the difference between 
COS. F in the last operation and the cosine of 99° 35', or the sine of 
9° 35' for the log. of the first number composing the natural sine 
of^. 

That is, to —2.946529 

Add .097279 ' 

Number, —0.110604 —1.043808* 

sin. u4=0.26015— 0.1 10604=. 149546. Whence, ^=8° 36'.t 
COS. 8° 36' : sin. 99° 35' : : cos. 44° 16' 33" : sin. ZwP=sin. 
45° 33' 50". 

Log. Horizontal parallax 3.513044 

COS. ^ 8° 36' 9.995089 

Approx, val. of mn 53' 42" 3.508133 

Q)'s appa. alt. nearly 7° 42' 18" cos 9.996064 

3.513044 

True value of rm 3229"5 3.509108 

JR : 39.2d"5 : : sin. 45° 33' 50" : p 
R : 3229"5 : : cos. 45° 33' 50" : q 

♦Artifice should be employed to take out the number corresponding to this 
logarithm, such as is taught in the author's Surveying and Navigation. 

t Here we have jumped from one result to another, and did not obtain 
the difference between one result and another, as we do by the differential 
method. 



312 ROBINSON'S SEQUEL. 

3.509108 3.509108 

sin. 46° 33' 50\ ... 9.853717 cos 9.845168 

^ 2902"6 3.462825 q 2260.6 3.364276 

Before Sa was in arc 4916", increase in 8m. 230"; therefore it 
is now 6146" which must be reduced as before. 

Log. 6146. 3.711470 

cos. 21° 12'. . . ..9.969667 
Sa in space, . . .4797"6. . .3.681037 

p.... .2902"6 ma before was 2490 

Paral. in R. A — 1896 Licrease in 8m 62 

2662 

q 2260.6 

Moon apparently north of sun 291 .6 



Distance between the centers=^(1895)2-|-(291.5)2= 1917"3. 
This being greater than 1843" shows that the eclipse has passed 

oior. 

The distance between the centers now is 191 7"3 

Eight minutes ago the distance was 167^45 

Apparent motion of the moon in 8 minutes 242"8 

Corresponding motion for 1 minute 30"35. 
From the sum of the semidiameters 1843", subtract 1674"5, and 
we obtain 168"5 for the moon to pass over before the end of the 
eclipse. This at the rate of 30"36 per minute requires 5m. 33s. 

Hence, to 6h. 36m. 3s. appa. time. 

Add 5m. 33s. 

Eclipse ends 6h. 41m. 36s. appa. time. 

But if we reduce the semidiameters for irradiation and inflection 
6", then we must diminish the time of ending 10 seconds. 

We may now observe that the moon's apparent motion across 
the sun was at the rate of 22"7 per minute at the beginning of 
the eclipse, 25"15 at the time of nearest approach, and 30"35 per 
minute at the end. This variability of the apparent motion is 
owing to the varying effects of the moon's parallax correspond- 
ing to the different altitudes, and this makes the problem tedious, 
and throws over it an air of complexity. 
Since six o'clock the rate of the moon's motion from the sun 



ASTRONOMY. ^» 

increased very much, and any one can see the rationale of this by 
inspecting the projection on page 293 of Robinson's Astronomy. 

Along the mid- day hours the sun and moon have an apparent 
motion together, but with diflPerent velocities. As the time from 
noon increases, the sun's motion along the ellipse is slower, and 
the moon appears to run over it faster and faster. After 6 the 
sun's apparent motion is no longer with the moon's, hence a rapid 
increase in the moon's apparent motion. 

We have made the problem much longer than we should have 
done, had we simply been in pursuit of results. Our object has 
been to explain and illustrate the problem to a learner, through 
each consecutive step, and we have found the following 

SUMMARY. 

Appa. time Burlington, Vt. Mean time. 

^ Beginning of the eclipse, 4h. 17m. 17.6s. 4h. 14m. 2.5b. 

Formation of the ring, 5h. 31m. 5.7s. 5h. 27m. 50.6s. 

Time of nearest appr. of cen. 5h. 33m. 5.7s. 5h. 29m. 50.6s. 

Rupture of the ring, 5h. 35m. 5.3s. 5h. 31m. 50.2s. 

End of the eclipse, 6h. 41m. 36 s. 6h. 38m. 21 s. 

Duration of the ring 4 minutes nearly ; duration of the eclipse 

2h. 24m. 19s. When the ring is most perfect, its breadth on the 

north limb will be 31", and on the south limb 18". 



Not long since the author received the following request : we 
extract from the letter. 

** One request more. In your Astronomy, page 191, near the 
bottom, you say, (speaking of the radial force,) 'and the diminu- 
tion in the one case is double the amount of increase in the other, and 
by the application of the differential calculus, we learn the mean 
result for the entire revolution, is a diminution whose analytical 

rS 
expression is ; an expression which holds a very prominent 

place in the lunar theory.* 

r Sf 

Now my enquiry is, how can we obtain the expression for 



314 ROBINSON'S SEQUEL. 

the mean result ? What operation in the calculus shall we go 
through ? 

Yours, &c., Wm. T " 

To this we returned the following reply : 

On page 193 you will find the following expression, 

4^ rS ( S COS.' X— I) 

for the radial force corresponding to any angle x from the 
syzigies. 

We already know the value of this force at the syzigies and 
quadratures, and at these points the result has the same general 
form ; therefore the result for the entire quadrant, that is, the 
mean result for the whole quadrant, will be found by taking the an- 
gle ar=45°, and as the mean result for each quadrant is the same, 
this will be the mean result for the entire revolution. 

Whence, a;=46°, sin. a;=cosic, and 2cos.^a;=l, or cos.^a:=|. 

Or, (Scos.^ar — 1)=^ ; whence the above general expression 

becomes ~-. 
2a3 

To this was returned the following observation : 

*'I understand your explanation, it is very simple ; but why 
did you not make this explanation in the book, — and more than all, 
why do you call it an application of the differential calculus ? /can 
see no calculus in it. 

Yours, &c., Wm. T." 

To this we rejoined as follows : 

If the operation I sent you is not calculation, I know not what 
it is — it may therefore be called calculus ; ahd if in any operation 
small quantities may be omitted on account of their insignificance 
in relation to larger quantities, the small difference so omitted 
constitutes the differential calculus, and to obtain that geneial 
expression, you will see, by looking on page 193 of the Astronomy, 
that the powers of r above the first were omitted. 



CALCULUS. 316 

THE €AjL€UI.irS. 

DIFFERENTIAL CALCULUS. 

The differential calculus is a branch of Analytical Geometry. 
It k a science for computing the ratio of small diflferences. 

For example, the side of a square is increased by a very small 
quantity, what will be the corresponding increase of the square 
itself ? 

The side of a cube is increased or diminished by a quantity 
very small in relation to the side itself, — ^how much will this in- 
crease or diminish the cube ? 

The arc of a circle is increased or diminished by a quantity very 
small in comparison with itself, what effect will this have on the 
sine and cosine of the arc ? 

The sun's longitude increases a certain distance in 10 minutes, 
what is its corresponding change in declination ? Or, find the 
law of these corresponding changes, or differences — called differ- 
etUicds, These questions explain in part the object of the calculus. 

The calls of astronomy gave birth to this science, as we have 
before remarked. 

For the development of this science, see the various works upon 
it. We confine ourselves in this book to a few difficult or curious 
operations. 

We presume the reader is acquainted with all the rules of 
operation. 

EXAMFLSS. 

(1.) Differentiate the expression jY—x^. Ans. 



Jl—x' 



Put u=Jl—x^. Square, u^=:l—x^ ; then 

xdx 



2udu= — 2xdXf or du^=-d* J\ — x^=- 



Jl—x' 

(2.) Find the differential of the equation 

X 

u= 



516 ROBINSON'S SEQUEL. 

By the rule for diflferentiating a fraction, we have 

x'dx 



dx(x-\- J 1 — x^) — xdx- 



{x+j\-x-y 



J\-X-+-^^:^-_ 

dx (^x+JU^"")^ 
By multiplying numerator and denominator of the second 



member by ^1 — x^ y then multiplying the equation by dx, we 
have 

J dx 

du= 



{x+J\-^x^Y J\—x' 



(3.) Q\yenu^=la-\-Jb — —\ to find the differential of u. 

Put y=J^^ — — and extract the 4th root of the original equa- 
tion ; then 

I tt*~ du=dy 

du:=^dy{a-\-yy (1) 

cdx 



But y^ = h — ~ Whence, yc?y= — _. dy 



2 *- x^ " ^Ah-^— 



Substituting the values of y and £?y in (1), we have 






v- 



CALCULUS. 317 



/I \ X \ J\ X 

C4.) Given u=~ ■ ^ to find the differential of u* 

^ ^ Jl-\-x—Jl—x 

Reduce the second member by multiplying numerator and de- 
nominator of the fraction by the numerator ; then 



X 

Apply the rule to differentiate a fraction, and to differentiate the 
numerator, simply make use of the first example. 

— x^dx 



du 



J^Z^-d<^+J^-^') 



X' 



Dividing both members by dx, and changing signs, and 

~dx Jl-x- '^ X- ^^^ 



X^J\—X^ X^'Jl- 



n-\-J\—x^\ 

Whence, du= — I )dx 

\ x^J\—<c^ / 



(l\ \ X \ J\ x\ 
^~=---'^^=_- ) to find the differential 
J\J^x—J\—x/ 

of u.\ 

The differential of a logarithm is the differential of the quan- 
tity divided by the quantity. But we have just obtained the 
differential of the quantity in the 4th example, therefore all we 
have to do is to divide that result by the quantity itself. 

/1_L /l_a:2v X dx^_ 

* These examples are common to all or nearly all the works on the calculus ; 
they are in the works of La Croix, from which they have been extracted into 
other works. 

+ This example is worked differently in Davied' Calculus, page 63 ; the 
work is extracted from La Croix. 



S18 ROBINSON'S SEQUEL. 



(6.) Given «= log. (ar+^l-f-a;') to find the diflferential of u. 
By the rule for differentiating a logarithm, we have 



du= Jl+x' 



x+Jl+x' 
To simplify the operation, divide both members by dx, then 

II X 

du H 
dx 



-=. JXJ^ 



x^J\-\-x- 
Multiply numerator and denominator of the second member 

by V H-^ then 



Whence, c?«= 



dm ^i-far^+a; ^ 

dx 



(7.) Given «=_?_ log. {xJ^^\-\-J\^l^) to find the 

differential of u. 

To avoid the confusion of mind which naturally accompanies 
the imaginary symbol J — 1, I put a=^ — 1. Then, 



att=log. {aX'\-J\-\-x^) 

J xdx 
adx — 



aX'\-Jl — x'* 

dx . V- hlf-L (a»+Vl— «')n/1— «' 

ax-\'Jl — x^ 

Now divide the numerator in the second member by the first 
factor in the denominator, thus : 



Jl—^^'+ax ) aj\—x^—x ( « 
aJ~U^x^+a^x 



CALCULUS. 3je 

There is no remainder because a^a:= — Xy as will be obvious wben 
we consider a=J — 1 ; hence a^= — 1. 

Whence, <^^^^ — ^— . or du=: ^ 

dx J\—x^ Jl^x^ 

(8.) Given w=log. ( V^+^ H lfV to find the diflferential 

Reduce the fraction by multiplying numerator and denominator 
by the numerator, then 



«=log.(! 



That is, «=log. (x-\-Jl-\-x'), and this is the 6th example. 
Therefore, du= ^ . 

(9.) Differentiate the equation w=a;'"(log. a?)". 
Put log. x=z ; then u=x^z'^ . 
And du=mx'^^z''dx-\-nz'^'^x^dz. 

Because 2=log. x, dz= — Now by substitution, 

X 

du=mx"^^ (log. xydx-^n(\og.xy-^a^^dx 

Finally, ^=(mlog:a;+»>°*-i(log.a;)°->. 
dx 

(10.) mffereniiateu=^'^^"^^^Mf)+'^ 

As before, let 0=log.a; ; then the equation becomes 
_j:^z^ x*ZjX^ 

^ 4~ ~8"^32 
Then ^./ = ^^,,2^^xx'zdz_ x^zdx_ x^ dz.x^dx 

'2 2 8^8 

But dz= — , and z = log.a; ; substituting these values, the pre- 

X 

ceding equation becomes 

du=xH\09r ^y^^ji ^''(^^g'^)d x_ x^(\og.x)dx ___x^dx,x^dx 
V e- y -r 2 ^ 8^8 



Whence, du=x^{\og.xydx. 



320 ROBINSON'S SEQUEL. 

(11.) Differentiate u=\og.^x : that is to say, the logarithm of 
the logarithm of x. 

Put log. x=:z. Then w=log. z. 

And du=—. But dz=^. 

z as 

That is, du=—= -— 

zx a;(log.a:) 

(12.) Differentiate u=\og.^x. '^ 

By the aid of the previous example we learn that 

log.5a;=log.(log.''a;). 

Whence, du='iJ^?L^ (1) . 

log. 4:?; 

d ( log.*a;)=-i — 5.' — L, this substituted in (1) reduces it to 
log.^a; 

du= ^(l^g-^^) 
(log.'»a;)(log.3^) 

Another step gives du= _i— ^l_^i 

(I0g.^..)(l0g.=^2r)(l0g^a:) 

Using the result of the previous example, we finally have 

, dx 

du= 

(log.'*a:)(log.^a;)(log.^a;)(log.a;)a; 

(13.) Differentiate u=e (x — 1 ) . 

Here we must take the logarithm of each member, observing 
that the log. of e^ is simply x, because e is the base of the Na- 
perian system of logarithms, the system always used in such 
examples. 

log. w=a;-|-log.(a: — 1). 

du , , dx dx 

— =dx-\- = 

u X — 1 x — 1 



u 



_e\x-\). 



Whence, r:!!=_Jl_=lAlIZiZ=c\ Or, du^edx, 
dx X — 1 X — 1 

,(14.) Differentiate u=^e'{x'^—^x'^'\-^x—Q). 
log.w=^-|-log.(4;^ — 'ix^-\-^x — 6) 
du_. .Sx^dx — Gxdx-\-6dx 
w'"" x^-^x^'+ex—G 

du x^ 

udx^'x^-^Sx^-^-Qx^ 



CALCULUS. 321 

du x^u 



dx x^'—^x'^'+Qx—Q 
(15.) Differentiate u= 



Or, du^e^x^dx. 



\—x 

u(\ — x)^=e'x. log.w-|-log.(l — x)-=x-\-\q^.x. 

dxL dx 7 [ dx du 1 . x-\-\ \-\-x — x^ 

u 1 — X X udx 1 — X X (1 — x)x 

du_ {\^x—x ^) e^x _[\J^x —x'' )e^ ^^ Jw=(i+fr:^!if!^ 
dx {\—x)x \ — x ' {\—xY ' (1—^)^ 

(16.) Dffererdiate uz^e'log.x. 
Put y=log. X ; then u=e^y. 

\og.u=x-\-\og.y. -_=cZa;-f--- 
u y 

■r, . J dx T dy dx 
But dy= — , and ^ — 



X y X log. X 

^TTj, du y . dx du x\opi;.x-\-\ 

Whence, — =dx4- = — ^ L_ 

u x\og:.x udx a;locr.a; 



du (x]og.x-\-'i)u (x log. x-\-\ )f log. X 

dx X log. X X log. X 

(17.) Differentiate u=^ — - — 
Ande'+1 = $ [ (2) Q 

au^^JL-^^ (3) 

Differentiating (1) gives e^dx^^dP. (4) 

And (2) gives e'dx^=dQ (5) 

Whence. (e^-f- 1 )e'dx= QdP 

And, {e''-^\)e'dx=PdQ 

By subtraction, ^e^dx=QdP—PdQ 

Whence, du= ~-. 

21 



M ROBINSON'S SEQUEL. 



CIRCULAR FUNCTIONS. 

For the sake of reference we will here note down the differential 
expressions for trigonometrical lines. 

Let the radius of a circle be unity. Represent an arc by x, 
then its differential will be dx. 

d sm.x=cos.xdx (1) d cos.x= — sin.x dx (2) 

d ver. sin.a;=sin.a;G?a; (3) d sec.a;=-?^ — (4) 

cos.ar 

d i2ing.x'=—-— (5) c^ cot.=— __^_ (6) 
cos.^ar ^ ^ sin.2.r ^ ^ 

One great difficulty which troubles and perplexes the studeiit 
in the calculus, arises from the fact that only the abstract theory 
of the science has been hitherto brought to our notice, in our ele- 
mentary books. 

All can understand how these equations, (1), (2), (3), <fec., are 
obtained ; but what if we can ? says the inquiring student. 
What use are they ? What do we learn by them ? 

It is^ useless to answer these questions by words only, we must 
show the answer by the following 

EXAMPLES. 

Equation (1), for example, shows a general truth. It is true 
applied any where along the quadrant of a circle, x is any arc 
that we choose to assume, and dx must be an arc sufficiently 
small to be considered a straight line. 

If a;=20°, and dx=l', then the difference between the sine of 
20° and the sine of 20° 1', is d sin.a:=cos. 20°Xl'. 

COS. 20° 9.972986 

Log. sin. or arc of 1'= .6.463726 

Sum less 20= log. of 0.0002733 —4.436712 

To the natural sine of 20° 342020 

Add the differential .0002733 

Sum is the Nat. sine of 20° 1' 3422933 

Thus we might give examples without end. 



CALCULUS. 323 

Because the diflferential of a logarithm is the differential of the 
quantity divided by the quantity, therefore 

, . , d sin. X COS. x, . , 

d W. sm. a;= - = ax=iC,oi.x ax. 

sin. X sin. a: 

This result corresponds to the modulus of unity ; for the modu- 
lus of our common system we must multiply by 0.43429448=wz. 

For example, if we assume ar=25°, and also assume dx-=\', the 
differential, or the difference between the log. sine of 25° and the 
log. sine of 25° V is expressed by m cot. 25° XI'- 

Log.m —1.637784 

cot. 25° 0.331327 

Log. sine V, less 10 .—4.463726 

.0002709 —4.432837 

To the log. sine of 25° 9.625948 

Add the differential .000271 

Log. sine of 25° 1'= 9.626219 

We might assume dx=^' as well as V, without error as far as 
six places of decimals ; but it would not do to assume dx= any 
large number of minutes ; hence the differential calculus must 
be applied with judgment.* 

To show another example of the utility of the calculus, we will 
let E represent the obliquity of the ecliptic, L the longitude of 
the sun at any time whatever, and D its corresponding declina- 
tion, the radius of the sphere being unity. 

Then the following equation is general : 

sin. jD=sin. -£'sin. X (1) 

This equation represents the sine of the sun's declination at any 
point whatever, along the ecliptic. Because sin. Z is 1 at the 
points where Z=90° or 270° ; therefore at these points sin. i>= 
sin. E, or J)=iE, as it should be. 

Now suppose that the sun changes its longitude 10', which we 
may call tlie {dL), or the differential of L, what will be the cor- 

* Those who are naturally more nice than wise, are commonly prejudiced 
against this science, and such frequently say it is no science at all ; how- 
ever, their objections are of no consequence. 



324 ROBINSON'S SEQUEL. 

responding change in its declination, or wliat will be the value of 
(dD)'! 

To answer this question we must take the differential of each 
member of the general equation, then we shall have 

COS. DdD=fim. U COS. LdL (2) 

^ Now whatever values we may assign to L and dL, equations 
(1) and (2) will always give D and dD at any point. 
For a definite example we give the following : 
What will be ihe differential in declination corresponding to the 
differential of 10' in longitude at 35° of longitude ? 

In other words, what will be the change in the sun's declina- 
tion'while it passes from longitude 35° to 35° 10' ? 
From ( 1 ) we' find D thus : 

"^ sin. E 9.599970 

sin. Z 35° 9.758591 

sin. B 13° 12' 5" 9.358561 

From (2), ^^^ sin. ^c o.^Zr/Z 

cos. 1) 

sin. E... 9.599970 

COS. Z 35° 9.913365 

c?Z= 10 log 1. 

COS. D, complement 0.011629 

Sum (less 20) 3.343 0.524964 

This is 3' 20"6 nearly, and if the sun's declination is 13° 12' 
5", when its longitude is 35°, the declination must be 13° 15' 
25"6 at the longitude 35° 10'. 

This is nc^ strictbj true, because the ratio of motion in declina- 
tion changes in a very slight degree between 35° and 35° 10'. 
But the ratio between the motion in longitude and the motion in 
declination, is strictly as 1 to .3343, at the beginning of the arc 
between 35° and 35° 10'. 

This ratio, is constantly changing, but still equation (2) always 
represents it. 

We now require the ratio of motion in declination when the 
sun's longitude is 90°, that is, Z=90° ; then cos". Z=0 ; and sub- 
stituting this value in (2) will cause the second member to dis- 
appear ; and 

cos. D dD=0 



CALCULUS. 



326 



Now one or the other of these factors must be zero ; but cos. D 
is evidently not zero ; therefore {dD) must equal 0, showing that 
there is no motion in declination exactly at that point. 

On the contrary, we may demand the sun's longitude when the 
motion in declination becomes zero. 

In other words, what will equation (2) show when (dD)=0 ? 

Then, "sin. ^cos. L dL=0. 

One of these three factors must be zero; but (sin. JE) cannot be 
zero, for it is a known constant quantity ; (dL) cannot equal zero in 
case the least possible time elapses, for the sun's apparent motion 
never ceases ; then (cos. L) must be zero, and Z=90°, or 270°. 

To show another example of the utility of the calculus, we 
present the problem that appears in another shape, on page 229, 
of our Surveying and Navigation, namely : 

Under ivhat circumstances will an error in the altitude of the sun, 
produce the least possible error in the time deduced therefrom ; the 
declination and latitude being constant quantities. 

Let P be the polar point, 
Hh the horizon, S the posi- 
tion of the sun, and Z the 
zenith of the observer. 

Let A = the altitude of 
the sun, D its decHnation, 
and X the latitude of the 
observer. 

Then by spherical trig- 
onometry (see page 214, 
Surveying and Navigation) 
we have 




COS. P= 



sin. A — sin. L cos. D 



COS. L sin. D 

Tht altitude A varies, and P the polar angle, or time from 
apparent noon, must vary in consequence of the variations of A ; 
and^if A is not accurately taken, P will not be accurate. 

In short, a differential to A, will produce a differential in P. 



326 ROBINSON'S SEQUEL. 

Therefore we must diflferentiate the equation, taking A and P as 
variables, and L and D constants. 

Whence, —sin. P dP=.^^^-—- ( 1 ) 

cos.L sin.i> 

But in the triangle PZS, we have 

cos. A ^ : sin. P : : cos. D : sin. SZP, or sin. Z 

f. . sin. P COS. D ,a\ 

Or, cos. A=^ (2) 

sin.Z ^ ^ 

Substituting this value of cos. ^ in (1), and dividing by sine 

P, we obtain 

- dP= ^?.^;A^___x^^ 

cos. Z sin. D sin. Z 

Or, —dP=^- ^.^ , (3) 

cos. L tan. D sin. Z 

Now it is obvious that the numerical value of (dP), or error 
in time, will be least when the denominator of the fraction in the 
second member is greatest, and that will be greatest when sin. Z 
is greatest, which is the case whenever Z=90°, which is when 
the sun is east or west of the observer. 

The sign before {dP) being minus, shows that when the altitude 
A increases, the angle P decreases. 

If we make dA^=0, equation (3) Avill give dP=0, showing 
that if there be no error in A, there will be none in P. 

We give a few more examples in circular functions. 

(1.) Let Z be an arc or angle whose radius is wiity and cosine 
(mx) : we require the differential of the arc in terms ofmx. 

In other words, differentiate Z=cos.~*(77iar). 

The rules of operation are all comprised in the following 
equations : 

d sin.a:=cos. x dx ( 1 ) d cos.a:= — sin.a; dx (2) 

" <ftang.a;=-^ (3) d cot.x=—-^— (4) 
"" cos.^a; ^ ^ sm.^a; ^ ^ 

in which x represents the arc to radius unity. 

Because to all arcs sin.^-[-cos.^=l. If cos. =nix, the sine= 
Jl—m^x''. 

The differential of the first member of our equation is evi- 



CALCULUS. 327 

dently dZ ; that of the second member is <f(cos.-»m.r), which bv 
equation (2) gives dZ=z ~^ ^-_ 

(2.) Differentiate Z=sin.~^(ma;). 

If {mx) is the sine of an arc the cosine is ^i — m^x^ . 

Whence, dZ= -J^J^ 

(3.) Differentiate the equation 

Z=sin.-(_X^S 

If ( — ^- ) represents the sine of an arc, the cosine of the 

same arc must be H 2y^ 

We can perform this operation the most clearly, by observing 
the following proportion : 

The differential of any arc, is to the differential of its cosine (taken 
negatively) y as radius (or unity) is to the sine of the same arc. 

This proportion is drawn from the consideration that if x rep- 
resents an arc, its differential is dx and its cosine is (cos. a;) and 
the differential of cos.a: by (2) is — sin.a; dx, and taken negatively, 
it is sin. a: dx ; and obviously 

dx : sin.a; dx : : \ : sin.ar 

Applying this proportion to the example before us, we have 

dZ : ^dj'^l : : 1 : ^^ («) 

The difficulty, (all there is of difficulty), is in taking the dif- 
ferential of the second term. 



Put JLJ^=Q; then (dQ) will be the differential value 
of the second term in the proportion. 

Differentiating the first member as a fraction, we have 



328 ROBINSON'S SEQUEL. 

—Aydy(\—y^ )+^ydy{ i-2y^ ) _^ ^^^ 
{\—y^Y 

Reducing, =5(iz-ll)±(il=V)= ^ 

{^-y'Y ydy 

That is, ,_:=V=^ 
(1—2/2)2 ydy 



Whence, -c^(2=_J^_ =-_J^>/lzll_, 
By substituting this value, proportion (a), becomes 



rfZ 



(1— y')Vi— 2y' VI— y' 

Or, dZ : M^-y') : : 1 : 1 

Whence, dZ^ ^^ 



(\-.y^) J\—Oiy^ 



(4.) Differentiate the equation Z=sin.~^(2w^l — ^m^.) 
If 2w^l — w^ is the sine of an arc, the cosine of the same arc 
must be (1— 2«^2). 

By the proportion observed in the last example, 

dZ : ^d{\—2u^) : : 1 : ^ujv^^ 
dZ : ' ^udu 

Whence dZ=. 



That is, dZ : ' ^udu : : 1 : ^ujX-^w" 

9,du 



(5.) Differentiate the equation w=cos. x sin. 2x. 
Regarding the second member as a product, and observing the 
dififerentials for sines and cosines, we have 

du= — sin. X sin. 2xdx-\- 2 cos.^2a; cos. xdx 

Whence, _if =(cos. 2ar cos. x — sin. 2a: sin. a;)-|-cos.ar cos.2;r. 

dx 

By observing equation (9), page 141, Robinson's Geometry, 

we perceive that the quantity in parenthesis is the same as cos. 

(te-\-x)y or COS. 3a; ; therefore, 

c?w=(cos. 3ar-}-cos 2a; COS. ar)c3?a;. 



CALCULUS. 389 

(6.) Differentiate the equation w=(taii. «)". 

Put tan. x==y ; then w=2/°, and duz=^ny^^dy (a) 

But v=tan.a;'; therefore c/y=(^ (tan. 0?)= , see (3). 

cos.^a; 

Whence (a) becomes du=. -^ '--l- 

cos.^a; 

Sin W/j* 
(7.) Differentiate the equation u=--—-^ — ---. 

Let sin. nx^=P, and sin. x^ Q. 

p 
Then the equation becomes w= — 

Whence, du^^^^J^H.^^^ 

Dividing numerator and denominator by Q'^S and we have 

Because P=sin. nx, dP=n cos. nx dx, and because 
^=sin. a; dQ=GOS.xdx. 
Now by substituting the values of P, Q, dP and dQ, (a) becomes 

7 (n sin. X cos. nx — n sin. 7ix cos. x) dx 

("sin. xy"*'' 
That is, by equation (8), page 141, Geometry^ 

, nsin.(nx — x)dx 

"^ (ibT^'^T 

(8.) Differentiate the equation «=log.(cos. x-\-J — 1 sin. x. 
The second member being a logarithm, its differential is the 
differential of the quantity divided by the quantity. That is, 

7 — sin. X dx-\-J — 1 cos. x dx. 

J — 1 sin.a:-|-cos.a; 



Or, 



du — sin. x-^-J — 1 COS. a; . — r- 

^^ J — 1 sin. ic-f-cos.ir 



Whence, du=J — \dx. 

(9.) Differentiate the equation z<=sin. 



Jl+: 



TT 



33G ROBINSON'S SEQUEL. 

X 

must be ; and we must have 



If the sine of an arc is the cosine of the same arc 



/ 1 



=) ,, 



X 



That is, du : ^dx^l+^x^ : : 1 : 



^x 

— - : : 1 : 1 (;w=- 

1+x^ l-\-x^ 



Orv ^.. : -J^- : : 1 : 1 du= ^"^ 




LUKAR OBSERVATIONS. 

The differential calculus may be used with great facility and 
success in clearing lunar distances from the effects of parallax and 
refraction. 

Let S'm' = the apparent central distance 
between the sun and moon, or the moon and 
a star, SS' is the refraction of the sun or star, 
and it is sufficiently small to be taken as the 
differential of the altitude. Also, m'm is the 
correction for the moon's apparent altitude, 
and we may call it the differential of the moon's 
altitude. 

The observed triangle is ZS'm'. Let aS^ represent the altitude 
of the sun or star, m the altitude of the moon, and x=S'm'y the 
observed distance. 

Now by the fundamental equation of spherical trigonometry, 

(see Geometry, page 191), we have 

^ COS. ir — sin. /S sin. m * ^^k 
COS. Z= -— (1) 

COS. O^COS. Wl 

We now take the differential of this equation, observing that Z 
is constant, and that x varies only on account of the variations 
of m and S, 

* Observing that the sine of an altitude is the same as tlie cosine of the 
corresponding zenith distance. 



CALCULUS. sfl 

First clear of fractions, tlien differentiate ; then we shall have 
— cos,Z(sin./S'cos.m dS-\-cos.Ssin.m dm)= — sin.a; dx — cos.S sin. 
m dS — cos.w sin.S dm. 

Changing all the signs, and substituting the value of cos. Z 
from (1), reduces to 
/cos.arsin. S — sin.^AS^sin.mX ^cr i /cos.arsin.m — sin.^msin.^S'^ 



',\y^_./cos,x sin.m — sin.^m sin.^S'X , 
/ \ • cos.m / 



\ cos. S 

=sin.a; t/ar-j-cos./S^sin.m dS-]-CQS.7/i sm.Sdm. 

Transposing and uniting the coefficients of dS and of dm, will 
give 

(cos.a; sin. S — sin. ^ S sin.m — cos. ^ S sm.m)dS 

Ccos.a: sin.m — sin. ^m sin. /S' — cos.^m sin./S^ )c?m . , 

^ 1 — = sm.a; ax 

COS. m 

Observing that sin. 2 /S'4-cos.^/S'=l, andsin.^m-[-cos.^m=l, and 
changing the order of the terms, we perceive that 

\ cos.m / \ COS. S / 

Here we should observe that (dm) is an elevation of the moon's 
apparent altitude, and (dS) is a depression of the sun or star's 
apparent altitude, therefore if we take (dm) positive, (dS) must be 
taken negative. Therefore, 

^^_/c_os^^^^i^Illn^y^_/^^ (2) 

\ sin. a; cos.m / \ sin. a; cos. aS / 

This is the final equation, (dx) representing the quantity be- 
tween the true and apparent distance. 

Sometimes (dx) is positive and sometimes negative, according 
as the differential coefficient, or quantities in parenthesis are posi- 
tive or negative. When the differential coefficient of (dS) is 
negative, that term becomes positive, because (dS) is negative, 
and the product of two negatives is positive. 

When the altitude of the sun or star is greater than 60°, the 
corresponding refraction ((iS) will be a very small quantity, which 
can never be augmented by its coefficient ; therefore in that case 
the value of (dx) will be sufficiently represented by 

/c_os^^n.m-sin^5y^ (3) 

\ sin.a: cos.m / 



% 



332 ROBINSON'S SEQUEL. 

Equation (2) will solve any example that may be prepared. 
We will solve one or two of those found on page 227, of our 
Surveying and Navigation. 

For the first example there found, in which S=Q6° 3', m=39° 
18', x==46° 45', and the mpon's horizontal parallax 53' 51", ex- 
pression (3) will be sufficient. 

We must first find (dm), which is the parallax in altitude 
diminished by the refraction. 

53' 51 "=3231" log 3.509337 

cos. m 39° 18' 9.880651 

log. 2499" 3..397988 

39° 18' Refraction,... 69" 

2430"=(Zm 
For the coefficient, we operate thus : (radius unity.) 

sin. m.. — 1.801665 cos. 7W. .—1.888651 
cos. a;.. — 1.835807 sin. a:... — 1.862353 
log. 4340 ..—1.637472 —1.751004 

Nat. sin. aS' 99762 [sub. the upper.] 

-56362* log — -1 .750975 

^—1.999971 

c?m=2430 log -. 3.385606 

dx=40' 29"=.2429" 3.385577 

X orthe app. dis. 46° 45' 
True distance 46° 4' 31" 

The answer in the book is 46° 4' 25". Our omission of the 
second term in equation (2) might have produced an error of 4", 
not more, still making a difference of 2", but this is of no conse- 
quence in itself. Different operators may work the same exam- 
ple by the same or different methods, and they will rarely produce 
results within 5" of each other ; and as no observations can be 
relied on within that limit, such results in a practical point of 
view are said to agree. 

We now take the 6th example from page 227, Surveying and 
Navigation. 

* Because this quantity is minus, (dx) must be minus, and therefore we 
subtract it from the apparent distance to find the true distance. 



CALCULUS. 3to 

Given sun's app. alt. 8° 26', Q)'s app. alt. 19° 24', horizontal 
parallax 67' 14". Apparent central distance 120° 18' 46", to find 
the true distance. Ans. 120° 1' 46".. 

Here S=^° 26', m=19° 24', and ar=120° 18' 46". 

Horizontal parallax 57' 14"=3434" 3.535800 

cos.m 19° 24' 9.974 614 

Parallax in altitude 3239" 3.51041 4 

Refraction ^^ 161" 

dm=: . .T:T3078" dS=:6' 10"=370". 

(Because x is greater than 90° its cosine will be minus, which 
will render the differential coefficient of dm minus.) 

sin.m... — 1.521349 cos.m... — 1.974614 
COS. a;...— 1 .703045 sin. a: —1.936152 

—.16763 —1.224394 —1..9 10766 den. 

Bin.S .1466 6 

—.31429 log — 1.497340 num 

--1. 586574 

c?m=3078" log 3.488269 

First term of (dx) —1188" 3.074843 

sin. S. . .—1.166307 cos. S . . . — 1.995278 
cos. ar. . .—1.703045 sin. .r —1.936152 



—.07402 —2.869352 —1.931430 

sin.wi .33216 

—.40618 -1.608736 

—1.677306 

— c?iS^=370" 2.568202 

Second term of (dx) + 176" 2.245508 

—1188" 



dx= —1012"= 16' 52" 

Apparent central distance, 120° 18' 46" 

True distance, 1 20° 1' 54" 

The result differs 8" from the given answer as determined by 
other methods, which arises from taking 3078" as a differential ; 
it is a large arc, rather too large to be taken for a differential arc. 



334 ROBINSON'S SEQUEL. 

MAXIMA AND MINIMA. 

The differential of any quantity is a general expression for a 
small increase of the quantity ; but if the quantity is already a 
maximum, an increase is impossible, and the expression for an in- 
crease must be zero. 

A decreasing differential is a general expression for a small 
relative decrease of any quantity ; but when the expression is 
already a minimum, it can have no further decrease, and the 
expression for such decrease must therefore be zero. Hence, in 
cases of a maximum or minimum, we must put the differeniial 
of the quantity equal to zero, thus forming a new equation, which 
equation generally gives the results sought. 

For example, the following equation always unites the declina- 
nation of the sun with its longitude : 

sin. i)=sin. E sin. L ( 1 ) 

Here E is the obliquity of the ecliptic, and is a constant quan- 
tity. L is any longitude, and D the corresponding declination. 
Taking the differential of this equation, we find 

COS. D dl)=sin. E COS. LdL (2) 

If we now assume the condition that D is a maximum, it is the 
same as to assume that (dD)=0, which makes 
sin. E COS. L dL=0 

Here is the product of three factors, one of which must equal 
0. Sine E is of known value, not equal to zero ; therefore cos. 
LdL=0. If cos.Z=0, Z=90°. If dL=0, L=0. Substitu- 
ting these values of L in equation (1), we have sin. i>=sin. ^, 
or JJ=E, when D is a maximum, and sin. i>=0, or D=0, when 
i> is a minimum ; which are obvious results. 

Again, the general value of the differential of the sine of any 
arc whose length is x and radius unity, is cos. x dx. 

But if the sine is a maximum, it can have no differential, ex- 
cept in form. That is, cos. x=0, or dx=0 ; whence a:=90°, or 
x=0. Showing that when the arc is 90°, the sine is a maximum, 
"when 0, the sine is 0, a minimum. 

For another iWMsirsiiion, I propose to divide the number a into 
(wo suck parts that ike product of tke parts skall be the greatest pos- 
sible. • K 



CALCULUS. 335 

At first I will simply get an expression for any indefinite rect- 
angle. — That is, if x= one part, (a — x) will equal the other part, 
and (ax — x^) will be an expression for a rectangle which will be 
larger or smaller according lo the relative values of x and a. 

Taking the differential of the expression, we have adx — 2xdx. 

Now if I assume (cix — x"^) to be a maximum, it cannot in- 
crease, therefore its differential must be zero, or 

adx — 2xdx=^0. Or, a — 2a;=0, or, x=^a, 
which is the value of x when the product is a minimum, and 
may be verified by trial. 

EXAMPLES. 

(1.) Required the greatest rectangle that can he inscribed in the 
quadrant of a given circle. 

It is evident that one extremity of the diagonal of the rectangle 
must be at the center of the circle, and the other extremity at 
some point on the arc of the quadrant. 

Let a = the diagonal or radius of the circle, and x = the arc 
from one extremity of the quadrant to the point in which the rect- 
angle meets the curve. Then a sin .r= one side of the rectangle : 
acos. ar= the adjacent side, and the area of the rectangle is 
a^sin. iccos. x. 

The problem requires that this expression should be a maxi- 
mum, which is the same thing as requiring that its differential 
expression should be zero. 

Hence, a'^cos. x dx cos. x — a^sin. x dx sin. a;=0. 

Dividing by a^dx, and cos.^a; — sin.^;i:=0. 

Or, COS. a;=:sin. a;. 

But the arc which has its cosine equal to its sine is 45*^, which 
shows that the diagonal of the rectangle bisects the quadrant, 
and the rectangle is in fact a square. . 

We observe* that a^ disappears in the result, and this shows 
that the problem is independent of the radius, and equally ap- 
plies to all circles. 

In short, constant factors in a maximum may he cast out hy divi- 
sion hefore we take the diferential. ' 

(2.) Required the greatest possible rectangle that can he inscribed 
in a {fiven parabola. 



m 



336 ROBINSON'S SEQUEL. 

Put VD=a, VB=x, PB=.y. Then 
BD=a — X, 2t/(a — a:) = maximum, and y^ 
=: 2px, by the equation of the curve. 

Taking the differentials, we have 

dy{a^x)—ydx=0. Or, dy^^^ 

a — X 

ydy=pdx. Or, dy^=^^J^ 

y 

Whence, ^ =±. Or, y-=iap — px. 

a — X y 

That is, 2px^=ap — px. 9.x=a — x. x^^a. 

This result shows that from the vertex, J- of the given distance 
is the point through which to draw one side of the maximum 
rectangle. 

^ (3.) Problem 3 on page 253 of this work, is a beautiful exam- 
ple to show the power and utility of the calcidus. 

In fact it was the result of the calculus that pointed out the geo- 
metrical construction. 

Let AP=a, PB=^b, PD=x, and call the angle APD=P. 

Then I)JI=xsm.P, PB'=xcos.P, All^a—x cos. P, HB= 
X COS. P±ib, according as IT falls between A and B, or between 
B and P. Corresponding to our figure, HB=x cos.P — b. 

In the triangle AHD, we have 

1 : i2.ii.ADH 

1 4. AnzT « — ^ COS.P 
1 : isin. AI>Ii= 







DH 


: HA 


Or, 


X sin 


.P : a- 


—X cos.P 


Also, 


icsin 


.P : X cos.P— b 



1 : tsin. ffDB= 



X sin.P 
X cos.P — b 



X sin. P 

Adding these two tangents according to the mathematical law 

of the sum of tangents, expressed in equation (28), page 143 of 

Robinson's Geometry, will give 

, a — b 

X sin.P 

tan. ADB= 5-^ 5 — 7\ 

^ a — X cos.P /x cos.P — b \ 

X sin.P \ X sin.P ^ 
(a — b) X sin.P 



ar^sin.^P — (a — x cos.P) (;c cos.P — b 



CALCULUS. 337 



t^n.ADB=—^-^^J^^''J^^ 



x"- sin.2P+a6— (a4-6) x cos.P-f a;^ cos.^P 

(a — i)a;sin.P 

x^-\-ab — (a-f-^) a: cos.P 
By the requirement of the problem, the angle ADB must be a 
maximum ; but the angle will be a maximum when its tangent is 
a maximum. Hence we must put the differential of the ex- 



-^ 1 equal to zero. 



pression 

x'^-\-ab — (ci-\-b) X cos.P 

By omitting the constant factor (a — b) sin. P, and dividing nu- 
merator and denominator by x, we shall have only 

1 

r+— — (a+i) cos. P to differentiate. 

— dx-\- — dx 
^x^ 
Whence. -; ; " r-^=0. 



(x-{J'l-^{a-Yb) cos. pV 



Or, a;2 =«6. 



This equation directed us to make FD = J 500 -200 as was 
done on page 263. 

(4.) The difference of arc between the sun's right ascension and its 
longitude gives rise to one part of the equation of time. What is the 
sun's right ascension when this part of the equation is a maximum, and 
what is the maximum value ? 

Let Z= the sun's longitude, x=. the corresponding right ascen- 
sion, ^= the obliquity of the echptic ; then by spherical trigo- 
nometry we have 

I'cos. -£'=:cot. Z tan. a:= — 1- fl) 

tan.Z ^ ^ 

(See equation (16) Robinson's Geometry, page 186, also (5), page 139.) 

But, tan.(Z-^) = A^"^i^:i (2) 

^ ^ 1+tan.Ztan.a; ^ ^ 

(See page 143, equation (29), Geometry.) 

The problem requires that the differential of {L — x)^ or of 

tan. (Z — x) should be put equal to zero. 

22 



338 ROBINSON'S SEQUEL. 

Substituting the value of tan. L taken from (1) in equation (2), 

we have 

tan. « . 
— tan. a; 

cos.^ (1— cos.^)tan.a; 

tan. li/— a;)= -: — ^ =^^ 

j I tan.^a; cos. ^+tan.2a? 

COS. ^ 

1^, tan. X . 1 

Whence, =maximum, or 

COS. jE'-j-tan.^a; cos. -£'cot.a;-J-tan.a; 

= maximum. 

But this fraction is obviously a maximum when its denomii).ator 
is a minimum ; therefore, 

cos. ^ cot. a:-|-tan. a;=minimum. 

Taking the differential, we find 

— COS. JEJdx, dx ^ sin.^ar j-, 
-j- =0 — — = cos. -A 

sin.^a: cos.^a; cos.^a; 

tan. a:=^cos. ^ (3) 

» This value of tan. x put in ( 1 ) gives 

tan.Z = >/^^ (4) 

cos. ^ 

Equations (3) and (4) correspond to radius unity, but we can 
use the logarithmic table if we add 10 to the index of cos. ^ be- 
fore we take the root, thus : 

jE'=23° 27' 30" cos. +10 19. 962535 ( 2 

tan. re 43° 45' 50" 9.981267 

9.962535 

tan. Z 46° 14" 10" 10.018732 

Diff. of arc 2° 28' 20"=9m. 54.6s. 

(5.) What must he the inclination of the roof of a huUding 
that the water will run off in tlve least possible time ? Ans. 45°. 
Let a= the base of a roof, x= its perpendicular altitude : then 

Ja^+x^ will equal its length, and J^ will equal the time re- 
quired for water to fall through the height. 

But the time down an inclined plane is to the time through its 
perpendicular, as the length of the plane is to its height. 

Let <= the time down the plane ; then 



CALCULUS, 



339 



4 



Or, 



^=. 



J9^ 






But the problem requires that ^ should be a minimum ; its 
square must therefore be a minimum ; hence, 

— ! = a mmimum. 

X 

It will be observed that ^ is the force of gravity, and being a 
constant factor in the last expression, it was omitted. 
Taking the differential, we have 

'ix^ dx—dx{a^ J^x"" )_^ 

x^=:a^^ -or a;=a. 
The perpendicular being equal to the base, shovrs that the in- 
clination required must be 45°. 

(6.) Within a triangle is a given point P, the distance to the nearest 
mngle A is given, and the line AP divides the angle A into two an- 
gles m and n, of which m is greater than n. 

It is required to find the line EF drawn through the point P, so 
that the triangle AEF shall he the least p>ossiMe, 

Let AP==a, AF=x, AE=y. The 
angle PAF=^m, FAE=^n. 

The area of the A ABF=ax sin. m. 

The area of the A APF=ag sin. n. 

By the conditions, 

ax sin. m-|-ay sija. ?*= minimum* 
^ Also, by the conditions, 

arysin. (m-\-n)= minimum. 

From tlie first minimum, 

sin. mdx-^sm. ndg:=^0 

From the second, xdg-^-gdx^O 

From(l), 




(1) 

(2) 



From {2}, 



sin.m 
dx=:= — -dy 

y 



340 ROBINSON'S SEQUEL. 

Whence, y sin. n = x sin. m. 

Or, ay sin. n=.ax sin. m. 

This last equation shows that the line EPF mustjbe so drawn 
that the triangle APF will be equal to the triangle APE. 

We presume that the foregoing examples are sufficient to illus- 
trate the power and utility of the calculus in respect to maxima and 
minima. 



INTEGRAL CALCULUS. 

The integral calculus is the converse of the differential. In 
the differential cakulus we give the integral and require the dif- 
ferential. In the integral calculus we give the differential quan- 
tity, and require the integral from which the dijBferential was 
derived. Hence all our rules of operation msust have reference to 
the differential rules inversed. 

We cannot investigate the rules of operation in this work, but 
suppose the reader already acquainted with them. 

Many persons can operate ta some extent in the integral cal- 
culus without any distinct idea of what the integral calculus is, 
and this vagueness can never be fully driven away, except by 
close attention to the application and utility of the science. 

For example. — If we have the differential of a circular arc, by 
integration, we shall have the arc itself. 

If we have the differential o-f a circular segment, by integration 
we shall have the segment itself. 

If we have the differential quantity of a cone, by integrating 
that quantity we shall have the solidity of the cone itself. 

If we have the differential of a logarithm, by integration we 
shall have the logarithm itself. 

Thus we might go through the chapter. 

Because the integral is the opposite of the differential, the inex- 
perienced might conclude that one operation would be obvious 
from the other. 

In some instances the converse is obvious, but not generally so. 
We must not conclude that it is as easy to move in one direc- 
tion as in the opposite. — It is not as easy to ascend as to descend 
a plane — not so easy for a vessel to move against the stream as 



CALCULUS. 341 

with it. It is not so easy to find the cube root of a number as it 
is to cube the root when found. 

The sign for the differential is (d). The sign for integration 
is ( f). Hence, J'du=u. The two signs destroy one another, 
and give the quantity u. 

If we take the differential of a;* we shall have 4x^dx. There- 
fore if we must integrate 4x^dx, we must frame a rule of opera- 
tion that will give x^^y which is the following : 

Add unity to the index, divide hj the index so iTicreased, and take 
away dx. 

The differential of - is ^ conversely then the 

the integral of is . But to integrate this by 

the above rule appears at first sight impossible, nevertheless it 

can be accomplished by the following artifice : 

Put 1 — x^=y. Then xdx=^ — \dy. 

A J xdx dy 1 -1-7 

And, _=_-_JL=_iy Uy 

To the second member of this equation the rule will apply. 
Hence, J — ^ _=y~2__ — ______ . ^^2^^ 

The differential of xy is {xdy-\-ydx) ; therefore, 

J {xdy-\-ydx)=^xy . But the inquiry is, how we shall effect 
the integration under the rule. 

Here y is equal to, or greater, or less than x. Therefore we 
may assume that y=ax, a being a constant quantity. Whence, 
dy=:-adx, xdy=axdx. 

And, f (xdy-\-ydx)= C (axdx-\-axdx)=: f^axdx==^ax^. 

But ax^=ax'X, and because y=ax, ax^ =xy. Ans. 

Then we perceive that the actual integration was performed by 
the rule ; but the reader must not infer that the rule will apply to 
all cases — -far from it.* 

* The subject of integration requires the keenest algebraical talent, and fe"W 
persons are skilful algebraists in the highest sense of that term, who have 
aot been severely disciplined iu integration. 



» f 



342 ROBINSON'S SEQUEL. 

We give a few examples under this rule. 

(1.) Given the diferenilal {^x^ydy -\-^^^xdx), to find its 
i$Uegr(d. Ans. x^y^ . 

Here we may assume y=£ax ; then dyz^etdx, ydy=.a^xdx. 
Whence, J ( ^"^ ydy-\-^y^ xdx) =J' ( 2a-x^dx-{-2a''x^dx) = 

C Aa^x^dx. 
To the second member the rule applies ; that is, 

C Aa^x^dx=a^x'^=za^x^ ^x^z=iy^x^ . Ans. 

(2.) Inte^e^te (J!+y'i^+(^!+^--M 

Ans. JV+^ Ja^'^^. 
Assume j¥^+^=P, and Ja^+P==Q. 
Then ydy=FdF, and xdx=QdQ. 

Substituting these values, and the given expression becomes 
F ^QdQ - j-PQ^dP 

That is, PdQ-\-QdP. 

' But, J{PdQ-{-QdP)=PQ. 



That is, J^^+y^ X Ja'^+x'' . Ans. 

(3.) Inte^mte _— 3^y+3c/^ ^^^ (a— y+0)^ 

Put (a_^-|^s)*=zP ; then a— .y+2=/'^ 

And —dy+dz=4P^dP. 

Whence. __3^+3^ =3P^ JP. 

4(a-y+z)^ 
And, r_Z±.^i'±?^=P3=(«-y+2)'. ^«*. 

Observatton. The differential of - is VJ^^J . therefore. 



y y2 



. X 



the integral of this last expression is — But how shall we inte- 

y 

grate this, provided we did not know the integral ? 

The numerator would be the differential of the product xy, if 
the sign between the terms in the numerator were plus. 



CALCULUS. 843 

Let us put y^=ax. Then ydx=:^axdx, and xdi/=axdXf and the 

yclx — xdv , ax dx — ax dx j /. 

expression f-— ^ becomes or , and our ef- 

^ y^ a^x^ a^x^ 

fort fails. Now let us examine the cause of the failure. The 

product xy can represent any magnitude whatever, and if we 

put y=ax, then xy becomes ax^ ; and because x is variable, ax^ 

is still capable of representing any magnitude whatever. But iu 

the fractional expression -, if y=-ax, and a be regarded as con- 

y 

V X \ X 

stant, _ becomes — , or _, and in that case _ can only represent 
y ax a y 

* 1 X 

the ever constant fraction - ; but - must be capable of repre- 
a y 

senting any fraction whatever ; therefore we cannot put y = ax, 

unless we regard a as variable. 

Therefore to integrate the expression tL_^ ?L , put y = tx ; 

both t and x being variable. 

Then ydx=ixdx, dy:=tdx-\-xdt. 

xdy=txdx-\-x ^ dt. 
Whence, ydx — xdy=. — x^dt 

J ,, . r x^dt dt 

and the expression becomes — or— — 

t X z 

That is, jyl^HpL^J^t-^dt^t-^ hythe rule. 
Whence, the required integral is - ; but y ^=ix ; therefore, 

t y 

This branch of the subject may be treated as follows, provided 
the operator is cautious, and does not assume too much : 

When we differentiate a product as xy, we assume x as con- 
stant and y variable ; and then y constant and x variable, and 
thus we get two partial diflferentials. 

Now either one of these integrated on the supposition thai the 
letter which is affixed to (^d) is the variable one, and all others con- 



344 ROBINSON'S SEQUEL. 

stant, will give the true integral. Thus the diflferential of xy is 

xdy-\-ydx. 
Now integrate xdy on the supposition that x is constant and y 
variable, and we have xy for the integral. ^Iso, Cydx=xy. It 
would therefore appear that ^xy is the whole integral, provided 
we did not know, a priori, that xy is the integral. 

ffence, when we integrate two differential expressions, on the sup- 
position that the letter not affected with the differential sign (d), is 
constant, and find two equal integrals, we must take but one of them. 

The same principle holds good in relation to the three or more 
letters. The diflferential of xyz is 

xydz-\-xzdy-\-yzdx. 

Now if we integrate this expression on the supposition that xy 
is constant in the first term, xz constant in the second, and yz 
constant in the third, we shall have 

xyz-\-xyz-\-xyz. 
Here are three equal integrals, but we must take but one of these 
for the whole integral, because the differential was eflfected by 
three distinct suppositions. 

The diflferential of - is yJf^Z^l^J^—xy-^dy, 

y y"" y 

Integrating each of these expressions on the supposition that y 
is constant in the first, and x constant in the second, we have 

x.x 

y y 

but we must only take one of these for the integral, for the same 
reason as before. 

EXAMPLES. 

(1.) Integrate (Sxy^y'^ )dx-\-(3x^ —'2xy)dy 

Ans. Sx^y — y^x. 
We integrate the first part on the supposition that y is constant, 
and the second on the supposition that x is constant, and we obtain 

3x^ij—y^x-{-3x^y—y^x, 
and because we make two distinct suppositions, we divide by 2. 
Then test the result by taking the diflferential. 



CALCULUS. 346 

(2.) Integrate {'iy''x-\'^^)dx-{-{^x^y-\-^xy'^'\'^y^)dy. 
Integrating each term, we obtain 

y^x^-{-Sy^x-\-x^y^^^xy^-\-^y\ 
Here we find two terms equal to aj^y^, and two terms equal to 
^xy^f and one term Sy* ; hence I will take 

for the integral sought — which I find to be true by taking the 
differential. 

To integrate the varied expressions in the form ' 
x'^{a-\-bx'^ydx, 
we must resort to the established formulas, explained in elaborate 
works, which of course we cannot touch upon in a work like this. 

Because c? log. ar= Therefore, f =.\og.x-\-c {a) 

X ^ X 

** d sin. x=cos.xdx. " f cos. xdx=sm. x-\-c (b) 

** d cos. a;= — sin. xdx. " C — sin.icc?ar=cos.a;-j-c (c) 

'* .<^tan.x=_^-. » ** r_^_=tan.a;+c {d) 

cos.^a; *^ cos.^a; 

Each of these formula is a fundamental rule for integration. 
It is not necessary for us to explain the constard c. 

APPLICATION" OF THE INTEGRAL CALCULUS. 

We shall explain the application and utility of this science by 
examples. 

If x represents an arc of a circle whose radius is unity, and y 
the sine of the same arc ; then ^1 — y'^ will represent the cosine, 
and equation {b) above will become 

The integral of the first member will give the arc^ but it will 
be numerically indefinite, unless we can integrate the second 
member, and know the value of y corresponding to some definite 
value of X. 



346 ROBINSON'S SEQUEL. 

We cannot integrate the second member in finite terms, there- 
fore we must develop it in a series, and integrate term by term, 
and if the series is suflficiently converging, the value of x can be 
known to any required degree of approximation. 

' — L 

:=(1 — y^) ^dy. The binomial, expanded by the bi- 



/I- 

nomial theorem, produces 

Multiplying each term by dy, and integrating, we obtain 

, l-y3 , 1 3 ys , 1 3 5 y^ , 1 3 5 7 v% « , 
'2-3 245'2467'24689' ' 

This equation is true for all values of x. It is true then when 
a;=0 ; but if we make a:=0, y must equal at the same time. 
Therefore, if we make the supposition that x=0, the last equa- 
tion will become 0=0-|-c, or c=0. By some such special con- 
sideration, the value of c can be determined in almost every 
problem, although it is indeterminate in the abstract. 

Now the value of y is known to be \ when x, the arc, equals 
30°; therefore. 

The arc of 30==l+iA+llll+M-l-l+^Li:^lI^_ 
2 ' 3.2* ' 4.5.2« ' 4.6.7.28 ' 4.6.8.9.2»'> 

+ &C. 

Multiplying by 6 and taking ten terms of the series, we shall 
have the value of a semicircle to radius unity. That is, we shall 
have 

7t=3.1415926, 

which is true to the last figure. 

Thus we perceive that one operation in the integral calculus 
brings a result requiring many operations in common geometry. 

SURFACES AND SOLIDS. 

In general terms, aydx will represent the differential of any 
plane surface, and if so, Caydx-\-c, will represent any surface ; 



I 



CALCULUS. 347 

but we can find the integral only when we know some relation 
between x and y. 

Also, ay^dx will represent the differential of any solid ; there- 
fore Cay^dx-\-c will represent the solid itself ; but we can find 
this integral only when we have some relation between x and y. 

EXAMPLES, 

Suppose x to represent the perpendicular of any triangle, and 
y its base ; then if x increases downwards by dx, ydx will be the 
differential of the triangle, the angles remaining constant. 

Therefore, Cydx will be the area of the triangle itself. This 
integral will require no correction, for when x=^0, y=0. 

The area being a triangle, we have a relation between x and y, 
for no triangle can exist without this numerical relation. 

Suppose we measure one unit down the base, and through that 
point draw a line parallel to the base, and find the length of this 
to be a units. Then whatever be the magnitudes of x and y, this 
relation will be constant, and a will be greater or less according 
to the angles of the triangle. 

That is, X : y : : 1 : a. Or, y=ax. 

Consequently, Cydx^zfcixdx^ =— .ar=:-iL 

That is, the area of any triangle is half the product of its base and 
altitude. 

Let VCI be any area. VC= x, CI=y, 

CD = dx, then the space CDRI=i ydx, the 

differential of the area. 

If VCI represents a portion of a parabola, 
1. I 
then y^=z<ipx. Or, ?/=(2^)2a;2. 

Whence, Jydx=J{'^pYx^dx=%(9.pyx^. 

But Ty=(<ipyx^; therefore VCIi^^VOIg. 
If VCI is a portion of a circular area of which r is the radius, 
. Then (^r--^Y -\-y^ ^^r^ , and y^^^Sra;— a:^. 
Or. y=sj^rx — x^. 




348 ROBINSON'S SEQUEL. 



Hence, fydy=f J^rx — x^dx=. the area of this semi-segment. 

We cannot integrate this in finite terms, Ave can only approx- 
imate to the integral by expanding the binomial, multiplying each 
term by dx, and then integrating each term separately. 

After integration, if we take a;=2r, the result will be the area of 
a semicircle whose radius is r. 

SOLIDS. 

(1.) To find the solidity of a cone. 

If X represent the perpendicular altitude of an upright cone and 
y the radius of its base, then Tty^ will equal the area of the base, 
and if x be increased by dx, rty^dx will be the differential of the 
cone. Consequently, J'Tty^dx will be the solidity of the cone. 

As any cone whatever must have some constant ratio between 
its perpendicular and base ; therefore, 

x : y : : \ : a. Or, y=ax. 

Whence, CTty^dx^:: C7ia^x^dx=^^Tta^x^=:^x'7ta^x^=\x-7<.y^ . 

That is, the solidity of a cone is equal to the area of the base mul- 
tiplied by one-third of the altitude. 

N. B. The integral required no correction because x and y 
vanish together. 

(2.) To find the solidity of a paraboloid. 

Let VCI revolve on the axis VC (see last figure,) it will de- 
scribe a paraboloid of which jty^dx is the difi'erential. y^=^2px. 

Therefore, Cny^dx^ C^7tpxdx=pTCX^ . 

To form a correct idea of this solid, we must observe that jty^ 
z=.^piix. Consequently 2/)7ta; is the area of the circle described 
by the revolution of y, and therefore ^pnx' is the solidity of the 
cyhnder which would just circumscribe the paraboloid, and hence 
we perceive that the piraboloid is Just half of its circumscribing 
cylinder. 

(3.) To find the solidity of a sphere. ■ 

Let FC/ revolve on the axis VC as before; now on the suppo- 
sition that FC/is the arc of a circle whose radius is /*, 



CALCULUS. 34i> 

Then (r — x)'^-\-y'^=r^. y^z=2rx — ar^. 
Then J'Tti/^dx=ftJ^{2rx^x^)dx=:7i(rx^—^—\-\-c. 

This integral requires no correction, because when x=0, y=0 
and then the area equals 0, and c=0. 

This integral represents the true value of any segment corres- 
ponding to any assumed yalue of x between x=0 and x=2r. 

If x=2r the segment will comprise the whole sphere. 

Then n{ rx^ — — )==7t{ 4r^ — )= 

\ 3/ \ 3/3 

This corresponds to theorem 17, book vii. Geometry. 
(4.) The differential of is conversely. 

Iniegi'cUe . Ans. . 

Put n-{'l=m, then n=m — 1, and n — l=m — 2. 

With these substitutions the expression to be integrated is 

(m—\)x''-^dx ^^ (m—\) fx'^^dx(\-^)-'^. 
But (l+.)-"=l-..+..('!!±i).^-..(-t^) (^^).3+ 

Multiply the second member by x'^'^dx, then it becomes 

-f- (fee. ■ 

Now integrate each term separately, and the result will be 

^::ii^x-jL.^x -^-^f !!H:i^ V"''-^+ &c. <fec. 

m—l '2 2 \ 3 / ' 

Multiply each term by (m — 1), observing that m — 1 equals n, 

Thenx''^nx-+n(py-^^-n(^'^^(^^^ 

Replacing the value of m, the series becomes 



SfiO ROBINSON'S SEQUEL. 

.._„.... +«(:«±l)x- -„(«-ti) (!±?).-3+ &c. 

By factoring, a;"A— 7w;+«'!^«i— fi-^±i.!H::?a;3+ <fec. ) 



. Ans, 



(l+x)' 



We close this volume by giving the two following integrations, 
which troubled us very much some years ago. They are from 
Poisson's Mecanique ; the first on page 223, vol. 1 , the second is 
on page 406 of the same volume. 

Poisson gives the equation 

Then simply says, the integral complete is 

y=c &m.lxJ—-\-f V c and /being arbitrary constants. 

How did he obtain the integral? 

Divide both members of the equation by — &, and then multi- 
ply both by 2c?y. Then we shall have 

The first member is the differential of -r_, dx^ being constant. 

dx^ 

The second member is easily integrated. 

Then ^^L.c^—?-y\ 

dx^ h h^ 

We add ( — -c^ J for the arbitrary constant, for ( ._ cM niay 

represent any quantity as well as c alone, and we place it first, 
because the other term is minus. Taking the square root, we 
h«v«, 

t-rA'-'-y 



CALCULUS. 351 



2 



Integrating both sides, and 

The first member of this equation is an arc of a circle whose 

sine is ^- and radius unity. 
c 

Let AE be that arc ; then 

J)I!z=zl. DJS=&m, AH =sin. {x^^-{-f\ 
Hence, ?.=sin. ^a:^_-+/j 

Integrate dx — ^anu:dQ=~cos.QdQ. 

This corressponds to the general formula, 

di/-\-Pydx= Qdx, 
Assume y=zz.ef-^^^ (1) 

Then theory gives the following formula for the result : 

y^e-P'^ { Jef'^ . Qdx+c) (F) 

To apply this formula to our equation, we must make 

ic=y, P=— 2am, dx^dQ, ^=:?^cos. ^, Fdx = —^a7ndQ, 

a 

rFdx=—2amQ, 

Differentiating (1), substituting the result in the general f6r- 

mula and reducing, we find dz=e^ ^^""^—cos. QdQ (2) 

a 

To integrate this last equation, we must use the following 

formula : 

rudv=uv — fvdu (3) 

« It will be a good exercise for a learner to differentiate this equation twice, 
and see if it returns to the original. 



3^ ROBINSON'S SEQUEL. 

a a 

These values put in formula (3), give 

By applying the same formula, 

if we compare (2), (4), and (5), we shall perceive that the 
ftrst member of (4) is z, and the last term of (5) is also z. 
Therefore, (4) becomes 

e=e-2-^'<i^^sin.^— e2^™^'^4^mcos.^— 4a2m20 



e 
Or. 0= 



-SaJnQ 



/ _^cos. Q — ^gm cos. Q \ 



l+4a2m2 
This value of z put in the general formula (■^), gives 



Let us now reverse the operation and differentiate this equation. 

Thus, dx=d,{ce-^-^ )+_^^_^.^^?__c?^+.i^«^:^?_cf$ 

To differentiate the first term of the second member, we put 

«=re2amQ Then log. w=log. c+2a7W^log. e 
Observing that log.e=l, and differentiating this last equation 
we have —.^=^amdQ, ov du^=^amce'^*^dQ. 



If «=e-2»«»Q log.M=— 2amQ ^==— SamrfQ. 

u 



Whence, </«=— SamCe-zamQ^/Q, 



CALCULUS. 363 

Whence ^ =2a7n^^ -"'^4. g^cos.(g 4^m sin. ^ 



Dividing by 2am, 



._-^^2amQi 2^ sj n. ^ _ 4 ^m COS. ^ 
a(l-l-4a2m2) (l+4a2m2) 



By subtraction, 



dx _ 2^cos. ^ j^ 4ffmcos.Q 



-x= 



9,atndQ 2a2m(l+4a2m2) ' (l+4a2w2) 



That is __^_— a;= ^ff cos.Q , Ssfa'^m^ cos.Q 

' 2amdQ 2a^m(\^4a^m^y9.a''m(\-\-4a''m'') 

__ (l-(-4a^»yi2)2y cos.^_ 2^ cos.Q 

Therefore, c^ — 2amxdQz=.Rcos.QdQ, the original equation. 

a 



23 



LOGARITHMIC TABLES; 



ALSO A TABLE OF THB 



TRIGONOMETRICAL LINES; 

AND OTHER NECESSARY TABLES. 











' 






LOGARITHMS OF 


NUMBERS 








FROM 










1 TO 10000* 






N. 


Log. 


N. 


Log. 


N. 


Log. 


N. 


Log. 






1 


000000 


26 


1 414973 


51 


1 707570 


76 


1 880814 






2 


301030 


27 


1 431364 


52 


1 716003 


77 


1 886491 






3 


477121 


28 


1 447158 


53 


1 724276 


78 


1 892095 






4 


602060 


29 


1 432398 


64 


1 732394 


79 


1 897627 






5 


698970 


30 


1 477121 


65 


1 740363 


80 


1 903090 






6 


778151 


31 


1 491362 


56 


1 748188 


81 


1 908485 






7 


845098 


32 


1 505150 


57 


1 755875 


82 


1 913814 






8 


903090 


33 


1 518514 


58 


1 763428 


83 


1 919078 






9 


954243 


34 


1 531479 


69 


1 770852 


84 


1 924279 






10 


1 000000 


35 


1 544068 


60 


1 778151 


85 


1 929419 






11 


1 041393 


36 


1 556303 


61 


1 785330 


86 


1 934498 






12 


1 079181 


37 


1 568202 


62 


1 792392 


87 


1 939519 






13 


1 113943 


38 . 


1 579784 


63 


1 799341 


88 


1 944483 






14 


1 146128 


39 


1 591065 


64 


1 806180 


89 


1 949390 






15 


1 176091 


40 


1 602060 


65 


1 812913 


90 


1 954243 






16 


1 204120 


41 


1 612784 


66 


1 819544 


91 


1 959041 






17 


1 £30449 


42 


1 623249 


67 


1 826075 


92 


1 963788 






18 


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 322219 


46 


1 662768 


71 , 


1 851258 


96 


1 982271 






22 


1 342423 


47 • 


1 672098 


72 


1 857333 


97 


1 986772 






23 


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 875661 


100 


2 000000 






N 


. B. In the following table, in the last ni 


ne columns of each page, where 






thel 
intrc 
and 
thel 


irst or leading figures change from 9's 
>d«ced instead of the O's through the re 
to indicate that from thence the corre 
irst column stands in the next lower I 


to O's, points or dots are now 
.st of the line, to catch Ihe eye, 
spending natural numbers in 
ine, and its annexed first two 






figUJ 


'es «f the Logarithms in the second co 


.ttmn 







LOGARITHMS OF NUMBERS. 3 






N. 





I 


2 


3 


4 


5 


6 


7 


8 


9 






100 


000000 


0434 


0868 


1301 


1734 


2166 


2698 


3029 


3461 


3891 






101 


4321 


4750 


5181 


5609 


6038 


6466 


6894 


7321 


7748 


8174 






102 


8600 


9026 


9461 


9876 


.300 


.724 


1147 


1570 


1993 


2415 






103 


012837 


3259 


3680: 


4100 


4521 


4940 


5360 


5779 


6197 


6616 






104 


7033 


7461 


7868 


8284 


8700 


9116 


9632 


9947 


.361 


.775 






105 


021189 


1603 


2016 


3428 


2841 


3252 


3664 


4075 


4486 


4896 






108 


5306 


5715 


6125 


6633 


6942 


7350 


7767 


8164 


8671 


8978 






107 


9384 


9789 


.195 


.600 


1004 


1408 


1812 


2216 


2619 


3021 






108 


033424 


3826 


4227 


4628 


502» 


5430 


5830 


6230 


6629 


7028 






109 


7426 


7825 


8223 


8620 


9017 


9414 


9811 


.207 


.602 


.998 






110 


041393 


1787 


2182 


2576 


2969 


3362 


3765 


4148 


4540 


4932 






111 


5323 


5714 


6106 


6496 


6886 


7276 


7664 


8053 


8442 


8830 






112 


9218 


9606 


9993 


.380 


.766 


1163 


1638 


1924 


2309 


2694 






113 


053078 


3463 


3846 


4230 


4613 


4996 


5378 


5760 


6142 


6524 






114 


6905 


7286 
1075 


7666" 


8046 


8426 


8805 


9186 


9563 


9942 


.320 






115 


060698 


1452 


1829 


2206 


2582 


2958 


3333 


3709 


4083 






116 


4458 


4832 


6208 


5580 


6963 


6326 


6699 


7071 


7443 


7815 






117 


• sise 


8557 


8328 


92S8 


S6C8 


..38 


.407 


.776 


1146 


1614 






118 


071882 


2250 


2617 


2985 


3362 


3718 


4086 


4451 


4816 


6182 






119 


6647 


5912 


6276 


6640 


7004 


7368 


7731 


8094 


8457 


8819 






120 


9181 


9543 


9904 


.266 


.626 


.987 


1347 


1707 


2067 


2426 






121 


082V85 


3144 


3503 


3861 


4219 


4676 


4934 


5291 


5647 


6004 






122 


6360 


6716. 


7071 


7426 


7781 


8136 


8490 


8846 


9198 


9562 






123 


9905 


.268 


.611 


.963 


1315 


1667 


2018 


2370 


2721 


3071 






124 


093422 


3772 


4122 


4471 


4820 


5169 


5618 


5866 


6216 


6662 






125 


6910 


7257 


7604 


7951 


8298 


8644 


8990 


9335 


9681 


0026 






126 


100371 


071& 


1059 


1403 


1747 


2091 


2434 


2777 


3119 


3462 






127 


3804 


4146 


4487 


4828 


6169 


5510 


5851 


6191 


6631 


6871 






128 


7210 


7549 


7888 


8227 


8565 


8903 


9241 


9679 


9916 


.263 






129 


110590 


0926 


1263 


1599 


1934 


2270 


2606 


2940 


3275 


3609 






130 


3943 


4277 


4611 


4944 


5278 


5611 


5943 


6276 


6608 


6940 






131 


7271 


7603 


7934 


8265 


8596 


8926 


9256 


9586 


9915 


0246 






132 


120574 


0903 


1231 


1560 


1888 


2216 


2544 


2871 


3198 


3525 






133 


3852 


4178 


4604 


4830 


6166 


6481 


5806 


6131 


6466 


6781 






134 


7105 


7429 


7753 


8076 


8399 


8722 


9045 


9368 


9690 


..12 






135 


130334 


0655 


0977 


1298 


1619 


1939 


2260 


2580 


2900 


3219 






136 


3539 


3858 


4177 


4496 


4814 


5133 


5451 


5769 


6086 


6403 






137 


6721 


7037 


7354 


7671 


7987 


8303 


8618 


8934 


9249 


9564 






138 


9879 


.194 


.508 


.822 


1136 


1450 


1763 


2076 


2389 


2702 






139 


143015 


3327 


3639 


3951 


4263 


4574 


4886 


5196 


6607 


6818 






140 


6128 


6438 


6748 


7068 


7367 


7676 


7985 


.8294 


8603 


8911 






141 


9219 


9527 


9836 


.142 


.449 


,756 


1063 


1370 


1676 


1982 






142 


152288 


2594 


2900 


3206 


2610 


3815 


4120 


4424 


4728 


6032 






143 


5336 


5640 


5943 


6246 


6549 


6852 


7164 


7457 


7769 


8061 






144 


8362 


8664 


8965 


9266 


9567 


9868 


.168 


.469 


.769 


1068 






145 


161368 


1667 


1967 


2266 


2664 


2863 


3161 


3460 


3758 


4055 






146 


4353 


4650 


4947 


5244 


5641 


5838 


6134 


6430 


6726 


7022 






147 


7317 


7613 


7908 


8203 


8497 


8792 


9086 


9380 


9674 


9968 






148 


170262 


0565 


0848 


1141 


1434 


1726 


2019 


2311 


2603 


2895 






149 


3186 


3478 


3769 


4060 


4351 


4641 


4932 


5222 


6612 


6802 







4 


LOGARITHMS 








N. 





1 


2 


3 


4 


5 


6 


7 


8 


9 






150 


176091 


6381 


6670 


6959 


7248 


7536 


7825 


8113 


8401 


8689 






151 


8977 


9264 


9552 


9839 


.126 


.413 


.699 


.985 


1272 


1568 






152 


181844 


2129 


2415 


2700 


2985 


3270 


3565 


3839 


4123 


4407 






153 


4691 


4975 


5259 


5642 


5825 


6108 


6391 


6674 


6956 


7239 






164 


7621 


7803 


8084 


8366 


8647 


8928 


9209 


9490 


9771 


..51 






155 


190332 


0612 


0892 


1171 


1451 


1730 


2010 


2289 


2567 


2846 






156 


3125 


3403 


3681 


3959 


4237 


4614 


4792 


5069 


5346 


6623 






157 


5899 


6176 


6453 


6729 


7005 


7281 


7556 


7832 


8107 


8382 






158 


8657 


8932 


9206 


9481 


9765 


..29 


.303 


.677 


.860 


1124 






159 


201397 


1670 


1943 


2216 


2488 


2761 


3033 


3305 


3677 


3848 






160 


4120 


4391 


4663 


4934 


5204 


5476 


5746 


6016 


6286 


6566 






161 


6826 


7096 


7365 


7634 


7904 


8173 


8441 


8710 


8979 


9247 






162 


9515 


9783 


..51 


.319 


.586 


.853 


1121 


1388 


1654 


1921 






163 


212188 


2454 


2720 


2986 


3252 


3518 


3783 


4049 


4314 


4679 . 






164 


4844 


5109 


5373 


5638 


5902 


6166 


6430 


6694 


6957 


7221 






165 


7484 


7747 


8010 


8273 


8536 


8798 


9060 


9323 


9585 


9846 






166 


220108 


0370 


0631 


0892 


1163 


1414 


1676 


1936 


2196 


2456 






167 


2716 


29/6 


3236 


3496 


3765 


4016 


4274 


4633 


4792 


5051 






168 


5309 


5568 


5S26 


6084 


6342 


6600 


6858 


7115 


7372 


7630 






169 


7887 


8144 


8400 


8667 


8913 


9170 


9426 


9682 


9938 


.193 






170 


230449 


0704 


0960 


1215 


1470 


1724 


1979 


2234 


2488 


2742 






171 


2996 


3250 


8504 


3767 


4011 


4264 


4517 


4770 


6023 


5276 






172 


5528 


5781 


6033 


6286 


6537 


6789 


7041 


7292 


7544 


7795 






173 


8046 


8297 


8548 


8799 


9049 


9299 


9550 


9800 


..50 


.300 






174 


240549 


0799 


1048 


1297 


1546 


1795 


2044 


2293 


2541 


2790 






175 


3038 


3283 


3534 


3782 


4030 


4277 


4525 


4772 


5019 


5266 






176 


5513 


5759 


6006 


6262 


6499 


6746 


6991 


7237 


7482 


7728 






177 


7973 


8219 


8464 


8709 


8954 


9198 


9443 


9687 


9932 


.176 






178 


250420 


0664 


0908 


1151 


1396 


1638 


1881 


2125 


2368 


2610 






179 


2863 


3096 


3338 


3580 


3822 


4064 


4306 


4548 


4790 


5031 






180 


5273 


5514 


5755 


5996 


6237 


6477 


6718 


6958 


7198 


7439 






181 


7679 


7918 


8168 


8398 


8637 


8877 


9116 


9356 


9594 


9833 






182 


260071 


0310 


0548 


0787 


1026 


1263 


1601 


1739 


1976 


2214 






183 


2451 


2688 


2926 


3162 


3399 


3686 


3873 


4109 


4346 


4582 






184 


4818 


5054 


6290 


6525 


5761 


5996 


6232 


6467 


6702 


6937 






185 


7172 


7406 


7641 


7875 


8110 


8344 


8678 


8812 


9046 


9279 






186 


9513 


9746 


9980 


.213 


.446 


.679 


.912 


1144 


1377 


1609 






187 


271842 


2074 


2306 


2638 


2770 


3001 


3233 


3464 


3696 


3927 






188 


4158 


4389 


4620 


4850 


5081 


5311 


5542 


5772 


6002 


6232 






189 


6462 


6692 


6921 


7161 


7380 


7609 


7838 


8067 


8296 


8526 






190 


8754 


8982 


9211 


9439 


9667 


9895 


.123 


.351 


.578 


.806 






191 


281033 


1261 


1488 


1715 


1942 


2169 


2396 


2622 


2849 


3075 






192 


3301 


3527 


3768 


3979 


A205 


4431 


4656 


4882 


5107 


5332 






193 


5557 


5782 


6007 


6232 


6466 


6681 


6905 


7130 


7354 


7578 






194 


7802 


8026 


8249 


8473 


8696 


8920 


9143 


9366 


9589 


9812 






195 


290035 


0257 


0480 


0702 


0925 


1147 


1369 


1591 


1813 


2034 






196 


2258 


2478 


2699 


2920 


3141 


3363 


3684 


3804 


4026 


4246 






197 


4466 


4687 


4907 


6127 


5347 


5667 


6787 


6007 


6226 


6446 






198 


6665 


6884 


7104 


7323 


7542 


7761 


7979 


8198 


8416 


8636 






199 


8853 


9071 


9289 


9507 


9725 


9943 


.161 


.378 


.596 


.813 









OF NUMBERS 5 






N. 





1 


2 


3 


4 


g 


6 


7 


8 


9 






200 


301030 


1247 


1464 


1681 


1898 


2114 


2331 


2547 


2764 


2980 






201 


3196 


3412 


3628 


3844 


4059 


4275 


4491 


4706 


4921 


5136 






202 


5351 


5566 


5781 


5996 


6211 


6425 


6639 


6864 


7068 


7282 






203 


7496 


7710 


7924 


8137 


8361 


S564 


8778 


8991 


9204 


9417 






204 


9630 


9843 


..56 


.268 


.481 


.693 


.906 


1118 


1330 


1542 






205 


311754 


1966 


2177 


2389 


2600 


2812 


3023 


3234 


3445 


3656 






206 


3867 


4078 


4289 


4499 


4710 


4920 


5130- 


5340 


5661 


5760 






207 


5970 


6180 


6390 


6599 


6809 


7018 


7227 


7436 


7646 


7854 






208 


8063 


8272 


8481 


8689 


8898 


9106 


9314 


9522 


9730 


9938 






209 


320146 


0354 


0562 


0769 


0977 


1184 


1391 


1698 


1805 


2012 






210 


2219 


2426 


2633 


2839 


3046 


3252 


3458 


3665 


3871 


4077 






211 


4282 


4488 


4694 


4899 


6105 


5310 


5516 


5721 


5926 


6131 






212 


6336 


6541 


6745 


6950 


7155 


7359 


7663 


7767 


7972 


8176 






213 


8380 


8583 


8787 


8991 


9194 


9398 


9601 


9805 


...8 


.211 






214 


330414 


0617 


0819 


1022 


1225 


1427 


1630 


1832 


2034 


2236 






215 


2438 


2640 


2842 


3044 


3246 


3447 


3649 


3850 


4051 


4253 






216 


4454 


4655 


4866 


5057 


5257 


5458 


6668 


6859 


6059 


6260 






217 


6460 


6660 


6860 


7060 


7260 


7459 


7659 


7858 


8058 


8257 






218 


8456 


8656 


8865 


9054 


9253 


9451 


9660 


9849 


..47 


.246 






219 


340444 


0642 


0841 


1039 


1237 


1435 


1632 


1830 


2028 


2225 






220 


2423 


2620 


2817 


3014 


3212 


3409 


3606 


3802 


3999 


4196 






221 


4392 


4589 


4785 


4981 


5178 


5374 


5570 


6766 


5962 


6167 






222 


6353 


6549 


6744 


6939 


7135 


7330 


7625 


7720 


7915 


8110 






223 


8305 


8500 


8694 


8889 


9083 


9278 


9472 


9666 


9860 


..54 






224 


350248 


0442 


0636 


^829 


1«23 


1216 


1410 


1603 


1796 


1989 






225 


2183 


2375 


2568 


2761 


2954 


3147 


3339 


3532 


3724 


3916 






226 


4108 


4301 


4493 


4685 


4876 


5068 


5260 


5452 


5643 


5834 






227 


6026 


6217 


6408 


6599 


6790 


6981 


7172 


7363 


7654 


7744 






228 


7935 


8125 


8316 


8506 


8696 


8886 


9076 


9266 


9466 


9646 






229 


9835 


..25 


.215 


.404 


.593 


.783 


.972 


1161 


1350 


1539 






230 


361728 


1917 


2105 


2294 


2482 


2671 


2859 


3048 


3236 


3424 






231 


3612 


3800 


3988 


4176 


4363 


4561 


4739 


4926 


5113 


5301 






232 


5488 


5676 


5862 


6049 


6236 


6423 


6610 


6796 


6983 


7169 






233 


7356 


7542 


7729 


7915 


8101 


8287 


8473 


8659 


8845 


9030 






234 


9216 


9401 


9587 


9772 


9968 


.143 


.328 


.513 


.698 


.883 






235 


371068 


1253 


1437 


1622 


1806 


1991 


2175 


2360 


2544 


2728 






236 


2912 


3096 


3280 


3464 


3647 


3831 


4016 


4198 


4382 


4565 






237 


4748 


4932 


6115 


5298 


5481 


6664 


5846 


6029 


6212 


6394 






238 


6577 


6759 


6942 


7124 


7306 


7488 


7670 


7852 


8034 


8216 






239 


8398 


8580 


8761 


8943 


9124 


9306 


9487 


9668 


9849 


..30 






240 


380211 


0392 


0573 


0754 


0934 


1115 


1296 


1476 


1656 


1837 






241 


2017 


2197 


2377 


2657 


2737 


2917 


3097 


3277 


3456 


3636 






242 


3815 


3995 


4174 


4353 


4533 


4712 


4891 


5070 


6249 1 


5428 






243 


5606 


5785 


5964 


6142 


63^1 


6499 


6677 


6856 


7034 


7212 






244 


7390 


7568 


7746 


7923 


8101 


8279 


8466 


8634 


8811 


8989 






245 


9166 


9343 


9520 


9698 


9875 


..51 


.228 


.405 


.582 


.759 






246 


390935 


1112 


1288 


1464 


1641 


1817 


1993 


2169 


2345 


2521 






247 


2697 


2873 


3048 


3224 


3400 


3575 


3751 


3926 


4101 


4277 






248 


4452 


4627 


4802 


4977 


5152 


5826 


5601 


6676 


5850 ! 


6025 






249 


6199 


6374 


6548 


6722 


6896 


7071 


7245 


7419 


7592 j 


7766 







6 


LOGARITHMS 






N. 





1 


2 3 


4 


5 


6 


7 


8 


9 






250 


397940 


8114 


8287 


8461 


8634 8808 | 


8981 


9154 


9328 


9501 






261 


9674 


9847 


..20 


,192 


.366 


,538 


.711 


.883 


1056 


1228 






252 


401401 


1573 


1745 


1917 


2089 


2261 


2433 


2605 


2777 


2949 






253 


3121 


3292 


3464 


8636 


3807 


3978 


4149 


4320 


4492 


4663 






254 


4834 


5006 


5176 


6346 


5517 


5688 


5858 


6029 


6199 


6370 






255 


6540 


6710 


6881 


7061 


7221 


7391 


7561 


7731 


7901 


8070 






266 


8240 


8410 


8679 


8749 


8918 


9087 


9257 


9426 


9595 


9764 






267 


9933 


.102 


.271 


.440 


.609 


.777 


.946 


1114 


1283 


1451 






258 


411620 


1788 


1956 


2124 


2293 


2461 


2629 


2796 


2964 


3132 






250 


3300 


3467 


3635 


3803 


8970 


4137 


4305 


4472 


4639 


4806 






260 


4973 


5140 


5307 


5474 


5641 


6808 


5974 


6141 


6308 


6474 






261 


6641 


6807 


6973 


7189 


7308 


7472 


7638 


7804 


7970 


8135 






262 


8301 


H467 


8633 


8798 


8964 


9129 


9295 


9460 


9625 


9791 






263 


9956 


.121 


.286 


.451 


.616 


,781 


.946 


1110 


1275 


1439 






264 


421604 


1788 


1933 


2097 


2261 


2426 


2590 


2764 


2918 


3082 






265 


3246 


3410 


3574 


3737 


3901 


4065 


4228 


4392 


4555 


4718 






266 


4882 


5045 


5208 


f-371 


C634 


5697 


5860 


6023 


6186 


6349 






267 


6511 


6874 


G83o 


6999 


7161 


7324 


7486 


7648 


7811 


7973 






268 


8135 


8297 


8459 


8621 


8783 


8944 


9106 


9268 


9429 


9591 






269 


9762 


9914 


..75 


; .236 


.398 


.559 


.720 


.881 


1042 


1203 






270 


431364 


1525 


1685 


1846 


2007 


2167 


2328 


2488 


2649 


2809 






271 


2969 


3130 


3290 


3450 


3610 


3770 


3930 


4090 


4249 


4409 






272 


4569 


4729 


4888 


5048 


5207 


5367 


5526 


5685 


5844 


6004 






273 


6163 


6322 


6481 


6640 


6800 


6957 


7116 


7275 


7433 


7592 






274 


7751 


7909 


8067 


8226 


,8384 


8542 


8701 


8859 


9017 


9175 






5i75 


9333 


9491 


9648 


9805 


9964 


.122 


.279 


.437 


.594 


.752 






276 


440909 


1066 


1224 


1381 


1538 


1695 


1862 


2009 


2166 


2323 






277 


2480 


2637 


2793 


2950 


3106 


3263 


3419 


3676 


3732 


3889 






278 


4045 


4201 


4357 


4513 


4669 


4825 


4981 


5137 


5293 


5449 






279 


5604. 


6760 


5915 


6071 


6226 


6382 


6637 


6692 


6848 


7003 






280 


7158 


7313 


7468 


7623 


7778 


7933 


8088 


8242 


8897 


8552 






281 


8706 


8861 


9015 


9170 


9324 


9478 


9633 


9787 


9941 


..95 






282 


450249 


0403 


0557 


0711 


0865 


1018 


1172 


1326 


1479 


1633 






283 


1786 


1940 


2093 


2247 


2400 


2553 


2706 


2859 


3012 


3165 






284 


3318 


3471 


3624 


3777 


3930 


4082 


4286 


4387 


4540 


4693 






285 


4845 


4997 


5150 


5302 


5454 


5603 


6758 


5910 


6062 


6214 






286 


6366 


6618 


6670 


6821 


6973 


7125 


7276 


7428 


7679 


7731 






287 


7882 


8033 


8184 


8336 


8487 


8638 


8789 


8-940 


9091 


9242 






288 


9392 


9543 


9694 


9845 


9995 


.146 


.296 


.447 


.597 


.748 






289 


460898 


1048 


1198 


1348 


1499 


1649 


1799 


1948 


2098 


2248 






290 


2398 


2548 


2697 


2847 


2997 


3146 


3296 


3445 


3594 


3744 






291 


3893 


4042 


4191 


4340 


4490 


4639 


4788 


4936 


5085 


5234 






292 


5383 


5532 


5680 


6829 


5977 


6126 


6274 


6423 


6671 


6719 






293 


6868 


7016 


7164 


7312 


7460 


7608 


7756 


7904 


8052 


8200 






294 


8347 


8495 


8643 


8790 


8938 


9085 


9283 


9380 


9627 


9676 






295 


9822 


9969 


.116 


.263 


.410 


.567 


.704 


.851 


.998 


1145 






296 


471292 


1438 


1585 


1732 


1878 


2025 


2171 


2318 


£464 


2610 






297 


2756 


2903 


3049 


3195 


3341 


3487 


3633 


8779 


8925 


4071 






298 


4216 


4362 


4508 


4(>53 


4799 


4944 


5090 


5235 


5381 


6526 






299 


5G71 


6816 


6962 


6107 


6252 


6397 


6542 


6687 


6832 


6976 







OF NUMBERS. 7 






N. 





I 


2 


3 


4 


6 


6 


7 


8 


9 






300 


477121 


7266 


7411 


7555 


7700 


7844 


7989 


8133 


8278 


8422 






301 


8566 


8711 


8855 


8999 


9143 


9287 


9481 


9575 


9719 


9863 






302 


480i)07 


0151 


0294 


0438 


0582 


0725 


0869 


1012 


1156 


1299 






303 


1443 


1686 


1729 


1872 


2016 


2159 


2302 


2445 


2588 


2731 






304 


2874 


3016 


3159 


3302 


3445 


3587 


8730 


3872 


4015 


4167 






305 


4300 


4442 


4585 


4727 


4869 


5011 


5153 


6295 


5437 


5579 






306 


6721 


5863 


6005 


6147 


6289 


6430 


6572 


6714 


6856 


6997 






307 


7138 


7280 


7421 


7563 


7701 


7845 


7986 


8127 


8269 


8410 






308 


8551 


8692 


8833 


8974 


9114 


9255 


9396 


9537 


9667 


9818 






309 


9959 


..99 


.239 


.380 


.520 


.661 


.801 


.941 


1081 


1222 






310 


491362 


1502 


1642 


1782 


1922 


2062 


2201 


2341 


2481 


2621 






311 


2760 


2900 


3040 


3179 


3319 


3458 


3697 


3737 


3876 


4015 






312 


4163 


4294 


4433 


4572 


4711 


4850 


4989 


5128 


5267 


6406 






313 


6544 


5683 


5822 


5960 


6099 


6238 


6376 


6515 


6653 


6791 






314 


6930 


7068 


7206 


7344 


7483 


7621 


7759 


7897 


8035 


8173 






315 


8311 


8448 


8586 


8724 


8862 


8999 


9137 


C275 


9412 


8560 






316 


9687 


9824 


9902 


..99 


.236 


.374 


.511 


.648 


.785 


.922 






317 


601059 


1196 


1333 


1470 


1607 


1744 


1880 


5017 


2154 


2291 






318 


2427 


2564 


2700 


2837 


2973 


3109 


324^i 


3382 


3518 


3655 






319 


3791 


3927 


4063 


4199 


4335 


4471 


4607 


4743 


1878 


6014 






320 


5150 


5283 


5421 


5557 


5093 


5828 


5964 


6093 


C234 


6370 






321 


6505 


6640 


6776 


6911 


7046 


7181 


7316 


7451 


7583 


7721 






322 


7856 


7991 


8126 


8260 


8395 


8530 


8664 


8799 


8934 


9008 






323 


9203 


9337 


9471 


9606 


9740 


9874 


...9 


.143 


.277 


.411 






324 


510546 


0679 


0813 


0947 


1081 


1215 


1349 


1482 


1616 


1750 






325 


1883 


2017 


2151 


2284 


2418 


2551 


2684 


2818 


2951 


3034 






326 


3218 


3351 


3484 


3617 


3750 


3883 


4016 


4149 


4282 


4414 






327 


4548 


4681 


4813 


4946 


6079 


5211 


5344 


5476 


5609 


5741 






328 


5874 


6006 


6139 


6271 


6403 


t635 


6668 


6800 


6932 


7064 






329 


7196 


7328 


7460 


7692 


7724 


'.855 


7987 


8119 


8251 


8382 






330 


8514 


8646 


8777 


8909 


9040 


9171 


9303 


9434 


9566 


9697 






331 


9828 


9959 


..90 


.221 


.353 


.484 


.615 


.745 


.876 


1007 






332 


521138 


1269 


1400 


1630 


1661 


1792 


1922 


2053 


2183 


2314 






333 


2444 


2575 


2705 


2835 


2966 


3096 


3226 


3356 


3486 


3616 






334 


3746 


3876 


4006 


4136 


4266 


4396 


4626 


4656 


4785 


4916 






335 


5045 


5174 


5304 


5434 


5563 


5693 


5822 


5951 


6081 


6210 






336 


6339 


6469 


6598 


6727 


6856 


6985 


7114 


7243 


7372 


7501 






337 


7630 


7759 


7888 


8016 


8146 


8274 


8402 


8531 


8660 


8788 






338 


8917 


9045 


9174 


9302 


9430 


9559 


9687 


9815 


9943 


..72 






339 


530200 


0328 


0466 


0584 


0712 


0840 


0968 


1096 


1223 


1L61 






340 


1479 


1607 


1734 


1862 


1960 


2117 


2245 


2372 


2500 


2f27 






341 


2754 


2882 


3009 


3136 


3264 


3391 


3518 


3645 


3772 


3899 






342 


4026 


4163 


4280 


4407 


4534 


4661 


4787 


4914 


6041 


5167 






343 


6294 


5421 


5547 


5674 


5800 


5927 


6053 


6180 


6306 


6432 






344 


6658 


6685 


6811 


6937 


7060 


7189 


7316 


7441 


7667 


7693 






345 


7819 


7945 


8071 


8197 


8322 


8448 


8574 


8699 


8825 


8951 






346 


9076 


9202 


9327 


9452 


9678 


9703 


9829 


9954 


..79 


.204 






347 


640329 


0465 


0580 


0705 


0830 


0956 


1080 


1205 


1330 


1454 






348 


1579 


1704 


1829 


1953 


2078 


2203 


2327 


2462 


2576 


2701 






349 


2825 


2960 


3074 


3199 


3323 


3447 


3571 


3696 


3820 


3944 







8 




LOGARITHMS 






N. 





1 


2 


3 


4 


5 


6 


7 


8 


9 






350 


544068 


4192 


4316 


4440 


4664 


4688 


4812 


4936 


5060 


5183 






351 


5307 


5431 


5555 


5678 


5805 


5925 


6049 


6172 


6296 


6419 






352 


6543 


6666 


6789 


6913 


7036 


7159 


7282 


7405 


7529 


7652 






363 


7775 


7898 


8021 


8144 


8267 


8389 


8612 


8636 


8758 


8881 






354 


9003 


9126 


9249 


9371 


9494 


9616 


9739 


9861 


9984 


.196 






355 


550228 


0351 


0473 


0595 


0717 


0840 


0962 


1084 


1206 


1328 






356 


1450 


1572 


1694 


1816 


1938 


2060 


2181 


2303 


2425 


2547 






357 


2668 


2790 


2911 


3033 


3155 


8276 


3393 


3519 


8640 


3762 






358 


3883 


4004 


4126 


4247 


4368 


4489 


4610 


4731 


4852 


4973 






359 


5094 


6216 


5346 


5467 


5578 


5699 


5820 


5940 


6061 


6182 






360 


6303 


6423 


6544 


6664 


6786 


6905 


7026 


7146 


7267 


7387 






361 


7507 


7G27 


7748 


7868 


7988 


8108 


8228 


8349 


8469 


8589 






362 


8709 


8829 


8948 


9068 


9188 


9308 


9428 


9548 


9667 


9787 






863 


9907 


. .26 


.146 


.266 


.886 


.504 


.624 


.743 


.863 


.982 






364 


561101 


1221 


1340 


1459 


1678 


1698 


1817 


1986 


2055 


2173 






365 


2293 


2412 


2531 


2650 


2769 


2887 


3006 


3125 


3244 


3362 






366 


8481 


3600 


3718 


3837 


3956 


4074 


4192 


4311 


4429 


4548 






367 


4666 


4784 


4903 


5021 


5139 


5267 


6376 


5494 


5612 


5780 






368 


5848 


5966 


6084 


6202 


6320 


6437 


6555 


6673 


6791 


6909 






369 


7026 


7144 


7262 


7879 


7497 


7614 


7732 


7849 


7967 


8084 






370 


8202 


8319 


8436 


8554 


8671 


8788 


8905 


9023 


9140 


9257 






371 


9374 


9491 


9608 


9725 


9882 


9959 


..76 


.198 


.309 


.426 






372 


570543 


0660 


0776 


0898 


1010 


1126 


1243 


1359 


1476 


1692 






373 


1709 


1825 


1942 


2058 


2174 


2291 


2407 


2523 


2639 


2755 






374 


2872 


2988 


3104 


3220 


3836 


8452 


3568 


3684 


8800 


3915 






375 


4031 


4147 


4263 


4379 


4494 


4610 


4726 


4841 


4957 


5072 






376 


5188 


5303 


6419 


5634 


5660 


5766 


5880 


5996 


6111 


6226 






377 


6341 


6457 


6572 


6687 


6802 


6917 


7032 


7147 


7262 


7377 






378 


7492 


7607 


7722 


7836 


7951 


8066 


8181 


8296 


8410 


8625 






379 


8639 


8764 


8868 


8983 


9097 


9212 


9826 


9441 


9555 


9669 






880 


9784 


9898 


..12 


.126 


.241 


.355 


.469 


.583 


.697 


.811 






381 


580926 


1039 


1163 


1267 


1381 


1496 


1608 


1722 


1836 


1950 






382 


2063 


2177 


2291 


2404 


2618 


2631 


2746 


2868 


2972 


8085 






383 


3199 


3312 


3426 


8639 


3652 


3766 


3879 


3992 


4105 


4218 






384 


4331 


4444 


4667 


4670 


4783 


4896 


5009 


5122 


5235 


5348 






385 


5461 


5574 


5686 


5799 


5912 


6024 


6137 


6250 


6862 


6475 






386 


6587 


6700 


6812 


6926 


7037 


7149 


7262 


7374 


7486 


7599 






887 


7711 


7823 


7935 


8047 


8160 


8272 


8384 


8496 


8608 


8720 






388 


8832 


8944 


9056 


9167 


9279 


9391 


9503 


9615 


9726 


9834 






389 


9950 


..61 


.178 


.284 


.396 


.507 


.619 


.730 


.842 


.953 






390 


591065 


1176 


1287 


1399 


1510 


1621 


1732 


1843 


1965 


2066 






391 


2177 


2288 


2399 


2510 


2621 


2732 


2843 


2964 


8064 


3175 






392 


3286 


3397 


8508 


3618 


3729 


3840 


3950 


4061 


4171 


4282 






393 


4393 


4603 


4614 


4724 


4834 


4945 


5055 


5165 


5276 


5386 






394 


5496 


5606 


5717 


5827 


6937 


6047 


6157 


6267 


6877 


6487 






395 


6507 


6707 


6817 


6927 


7037 


7146 


7256 


7866 


7476 


7586 






396 


7695 


7805 


7914 


8024 


8134 


8243 


8353 


8462 


8572 


8681 






397 


8791 


8900 


9009 


9119 


9228 


9337 


9446 


S556 


9666 


9774 






398 


9883 


9992 


.101 


.210 


.319 


.428 


.537 


.646 


.755 


.864 






399 


600973 


1082 


1191 


1299 


1408 


1617 


1626 


1734 


1843 


1951 







OF NUMBERS. 9 






N. 





1 


2 


3 


4 


5 


6 


7 


8 


9 






400 


602060 


2169 


2277 


2386 


2494 


2603 


2711 


2819 


2928 


3036 






401 


3144 


3253 


3361 


3469 


3573 


3686 


3794 


3902 


4010 


4118 






402 


4226 


4334 


4442 


4550 


4658 


4766 


4874 


4982 


5089 


5197 






403 


5305 


5413 


5521 


5628 


5736 


5844 


5951 


6059 


6166 


6274 






404 


6381 


6489 


6596 


6704 


6811 


6919 


7026 


7133 


7241 


7348 






405 


7455 


7562 


7669 


7777 


7884 


7991 


8098 


8205 


8312 


8419 






406 


8526 


8633 


8740 


8847 


8954 


9061 


9167 


9274 


9381 


9488 






407 


9594 


9701 


9808 


9914 


..21 


.128 


.234 


.341 


.447 


.554 






408 


610660 


0767 


0873 


0979 


1086 


1192 


1298 


1405 


1511 


1617 






409 


1723 


1829 


1936 


2042 


2148 


2254 


2360 


2466 


2572 


2678 






410 


2784 


2890 


2996 


3102 


3207 


3313 


3419 


3525 


3630 


3736 






411 


3842 


3947 


4053 


4159 


4264 


4370 


4475 


4581 


4686 


4792 






412 


4897 


5003 


5108 


5213 


5319 


5424 


5529 


5634 


5740 


5846 






413 


5950 


6055 


6160 


6265 


6370 


6476 


6581 


6686 


6790 


6895 






414 


7000 


7105 


7210 


7315 


7420 


7525 


7629 


7754 


7839 


7948 






415 


8048 


8153 


8257 


8362 


8466 


8571 


8676 


8780 


8884 


8989 






416 


9293 


9198 


9302 


9406 


9511 


9615 


9719 


9824 


9928 


..32 






417 


620136 


0140 


0344 


0448 


0552 


0656 


0760 


0864 


0068 


1072 






418 


1176 


1280 


1384 


1488 


1592 


1695 


1799 


1903 


2007 


2110 






419 


2214 


2318 


2421 


2525 


2628 


2732 


2835 


2939 


3042 


3146 






420 


3249 


3353 


3456 


3559 


3663 


3766 


3869 


3973 


4076 


4179 






421 


4282 


4385 


4488 


4591 


4695 


4798 


4901 


5004 


6107 


5210 






422 


5312 


5415 


5518 


5621 


6724 


5827 


5929 


6032 


6135 


6238 






423 


6340 


6443 


6546 


6648 


6751 


6853 


6956 


7058 


7161 


7263 






424 


7366 


7468 


7571 


7673 


7775 


7878 


7980 


8082 


8185 


8287 






425 


8389 


8491 


8593 


8695 


8797 


8900 


9002 


9104 


9206 


9308 






426 


9410 


9512 


9613 


9715 


9817 


9919 


..21 


.123 


.224 


.326 






427 


630428 


0530 


0631 


0733 


0835 


0936 


1038 


1139 


1241 


1342 






428 


1444 


1545 


1647 


1748 


1849 


1951 


2052 


2153 


2255 


2356 






429 


2457 


2559 


2660 


2761 


2862 


2963 


3064 


3165 


3266 


3367 






430 


3468 


3569 


3670 


3771 


3872 


3973 


4074 


4175 


4276 


4376 






431 


4477 


4578 


4679 


4779 


4880 


4981 


5081 


5182 


5283 


5383 






432 


5484 


5684 


6685 


5785 


5886 


5986 


6087 


6187 


6287 


6388 






433 


6488 


6588 


6688 


6789 


6889 


6989 


7089 


7189 


7290 


7390 






434 


7490 


7590 


7690 


7790 


7890 


7990 


8090 


8190 


8290 


8389 






435 


8489 


8589 


8689 


8789 


8888 


8988 


9088 


9188 


9287 


9387 






436 


9486 


9586 


9686 


9785 


9885 


9984 


..84 


.183 


.283 


.382 






437 


640481 


0581 


0680 


0779 


«879 


0978 


1077 


1177 


1276 


1375 






438 


1474 


1573 


1672 


1771 


1871 


1970 


2069 


2168 


2267 


2366 






439 


2465 


2563 


2662 


2761 


2860 


2959 


3058 


3156 


3255 


3354 






440 


3453 


3551 


3650 


3749 


3847 


3946 


4044 


4143 


4242 


4340 






441 


4439 


4537 


4636 


4734 


4832 


4931 


5029 


5127 


6226 


5324 






442 


5422 


5521 


5619 


5717 


5815 


5913 


6011 


6110 


6208 


6306 






443 


6404 


6502 


6600 


6698 


6796 


6894 


6992 


7089 


7187 


7286 






444 


7383 


7481 


7579 


7676 


7774 


7872 


7969 


8067 


8165 


8262 






445 


8360 


8458 


8555 


8653 


8750 


8848 


8945 


9043 


9140 


9237 






446 


9335 


9432 


9530 


9627 


9724 


9821 


9919 


..16 


.113 


.210 






447 


650308 


0405 


0502 


0599 


0696 


0793 


0890 


0987 


1084 


1181 






448 


1278 


1375 


1472 


1569 


1666 


1762 


1859 


1956 


2053 


2160 






449 


2246 


2343 


2440 


2530 


2633 


2730 


2826 


2923 


3019 


3116 
' 







10 


LOGARITHMS 






N. 





1 


2 


3 


4 


6 


6 


7 


8 


9 






450 


653213 


3309 


3405 


3502 


3598 


3695 


3791 


3888 


3984 


4080 






461 


4177 


4273 


4369 


4466 


4562 


4658 


4754 


4850 


4946 


6042 






453 


5138 


6235 


5331 


5427 


5626 


6619 


6715 


5810 


5906 


6002 






453 


6098 


6194 


6290 


6386 


6482 


6577 


6673 


6769 


6864 


6960 






454 


7056 


7162 


7247 


7343 


7438 


7634 


7629 


7725 


7820 


7916 






456 


8911 


8107 


8202 


8298 


8398 


&488 


8584 


8679 


8774 


8870 






456 


8966 


9060 


9156 


9250 


9346 


9441 


9536 


9631 


9726 


9821 






457 


9916 


. .11 


,106 


.201 


.296 


.391 


.486 


.581 


.676 


.771 






458 


660866 


0960 


1055 


1150 


1245 


1339 


1434 


1629 


1623 


1718 






469 


1813 


1907 


2002 


2096 


2191 


2286 


2380 


2475 


2569 


2663 






460 


2768 


2852 


2947 


3041 


3135 


3230 


3324 


3418 


3512 


3607 






461 


3701 


3796 


3889 


3983 


4078 


4172 


4266 


4360 


4454 


4648 






462 


4642 


4736 


48jO 


4924 


6018 


5112 


5206 


5299 


6393 


6487 






463 


6581 


6676 


67d9 


6862 


5956 


6050 


6143 


6237 


6331 


6424 






464 


6518 


6612 


6705 


6799 


6892 


6986 


7079 


7173 


7266 


7360 






465 


7453 


7546 


7640 


7733 


7826 


7920 


8013 


8106 


8199 


8293 






466 


8386 


8479 


8672 


8665 


8759 


8852 


8946 


9038 


9131 


9324 






467 


9317 


9410 


9503 


9596 


9689 


9782 


9876 


9967 


..60 


.153 






468 


670241 


0339 


0431 


0524 


0617 


0710 


0802 


0895 


0988 


1080 






469 


1173 


1265 


1358 


1451 


1643 


1636 


1728 


1821 


1913 


2005 






470. 


2098 


2190 


2283 


2375 


2467 


2560 


2652 


2744 


2836 


2929 






471 


3021 


3113 


3205 


3297 


3390 


3482 


3574 


3666 


3758 


3850 






472 


3942 


4034 


4126 


4218 


4310 


4402 


4494 


4686 


4677 


4769 






473 


4861 


4953 


5045 


6137 


5228 


5320 


5412 


5503 


5695 


5687 






474 


5778 


6870 


6962 


6053 


6146 


6236 


6328 


6419 


6611 


6602 






475 


6694 


6785 


6876 


6968 


7059 


7151 


7242 


7333 


7424 


7616 






476 


7607 


7698 


7789 


7881 


7972 


8063 


8154 


8245 


8336 


8427 






477 


8518 


8609 


8700 


8791 


8882 


8972 


9064 


9155 


9246 


9337 






478 


9428 


9519 


9610 


9700 


9791 


9882 


9973 


..63 


.164 


.245 






479 


680336 


0426 


0517 


0607 


0698 


0789 


0879 


0970 


1060 


1151 






48a 


1241 


1332 


1422 


1513 


1603 


1693 


1784 


1874 


1964 


2056 






481 


2145 


2235 


2326 


2416 


2506 


2696 


2686 


2777 


2867 


2957 






482 


3047 


3137 


3227 


3317 


3407 


3497 


3587 


3677 


3767 


3857 






483 


3947 


4037 


4127 


4217 


4307 


4396 


4486 


4576 


4666 


4766 






484 


4845 


4936 


5026 


5114 


5204 


5294 


6383 


5473 


6563 


6652 






485 


6742 


6831 


5921 


6010 


6100 


6189 


6279 


6368 


6458 


6647 






486 


6636 


6726 


6816 


6904 


6994 


7083 


7172 


7261 


7351 


7440 






487 


7629 


7618 


7707 


7796 


7886 


7976 


8064 


8153 


8242 


8331 






488 


8420 


8509 


8598 


8687 


8776 


8865 


8953 


9042 


9131 


9220 






489 


9309 


9398 


9486 


9576 


9664 


9753 


9841 


9930 


..19 


.107 






490 


690196 


0285 


0373 


0362 


0550 


0639 


0728 


0816 


0905 


0993 






491 


1081 


1170 


1258 


1347 


1435 


1524 


1612 


1700 


1789 


1877 






492 


1966 


2053 


2142 


2230 


2318 


2406 


2494 


2583 


2671 


2759 






493 


2847 


2935 


3023 


3111 


3199 


3287 


3375 


3463 


3551 


3639 






494 


3727 


3815 


3903 


3991 


4078 


4166 


4254 


4342 


4430 


4517 






495 


4605 


4693 


4781 


4868 


4956 


6044 


5131 


5210 


5307 


5394 






496 


5482 


6669 


6657 


6744 


5832 


6919 


6007 


6094 


6182 


6269 






497 


6356 


5444 


6531 


6618 


6706 


6793 


6880 


6968 


7056 


7142 






498 


7229 


7317 


7404 


7491 


7678 


7666 


7752 


7889 


7926 


8014 






499 


8101 


8188 


8275 


8362 


8449 


8535 


8622 


8709 


8796 


b«83 







OF NUMBERS. 11 






N. 





I 


2 


3 


4 


5 


6 


7 


8 


9 






500 


693970 


9057 


9144 


9231 


9317 


9404 


9491 


9578 


9664 


9751 






501 


9838 


9924 


..11 


..98 


.184 


.271 


.358 


.444 


.631 


.617 






502 


700704 


0790 


0877 


0963 


1050 


1136 


1222 


1309 


1396 


1482 






503 


1568 


1664 


1741 


1827 


1913 


1999 


2086 


2172 


2258 


2344 






504 


2431 


2617 


2603 


2689 


2775 


2861 


2947 


3033 


3119 


3206 






505 


3291 


3377 


3463 


3549 


8636 


3721 


3807 


3896 


3979 


4065 






508 


4151 


4236 


4322 


4408 


4494 


4579 


4666 


4751 


4837 


4922 






507 


5008 


5094 


5179 


5266 


6350 


6436 


6522 


6607 


5693 


5778 






508 


5864 


5949 


6035 


6120 


6206 


6291 


6376 


6462 


6647 


6632 






509 


6718 


6803 


6888 


6974 


7069 


7144 


7229 


7315 


7400 


7485 






610 


7570 


7655 


7740 


7826 


7910 


7996 


8081 


8166 


8261 


8336 






511 


8421 


8506 


8591 


8676 


8761 


8846 


8931 


9015 


9100 


9186 






512 


9270 


9356 


9440 


9524 


9609 


9694 


9779 


9863 


9948 


..33 






513 


710117 


0202 


6287 


0371 


0466 


0540 


0625 


0710 


0794 


0879 






514 


0963 


1048 


1132 


1217 


1301 


1386 


1470 


1554 


1639 


1723 






515' 


1807 


1892 


1976 


2660 


2144 


2229 


2313 


2397 


2481 


2566 






516 


2650 


2734 


2818 


2902 


2986 


3070 


3154 


3238 


3326 


3407 






517 


3491 


5576 


3659 


3742 


3826 


3910 


8994 


4078 


4162 


4246 






518 


4330 


4414 


4497 


4581 


4666 


4749 


4833 


4916 


5000 


5084 






519 


5167 


5251 


6336 


5418 


5602 


6586 


5669 


5763 


5836 


6920 






520 


6003 


6087 


6170 


6254 


6337 


6421 


6504 


6588 


6671 


6754 






521 


6838 


6921 


7004 


7688 


7171 


7254 


7338 


7421 


7504 


7587 






522 


7671 


7764 


7837 


7920 


8003 


8086 


8169 


8253 


8336 


8419 






523 


8502 


8585 


8668 


8761 


8834 


8917 


9000 


9083 


9165 


9248 






524 


9331 


9414 


9497 


9580 


9663 


9745 


9828 


9911 


9994 


..77 






525 


720159 


0242 


0326 


0407 


0490 


6573 


0655 


0738 


0821 


0903 






526 


0986 


1068 


1151 


1233 


1316 


1398 


1481 


1663 


1646 


1728 






527 


1811 


1893 


.975 


2058 


2140 


2222 


2305 


2387 


2469 


2552 






528 


2634 


3716 


2798 


2881 


2963 


3046 


3127 


3209 


8291 


3374 






529 


3456 


3538 


3620 


3702 


3784 


3866 


3948 


4030 


4112 


4194 






630 


4276 


4358 


4440 


4622 


4604 


4686 


4767 


4849 


4931 


5013 






531 


5096 


5176 


5268 


5340 


6422 


5503 


6585 


5667 


5748 


5830 






532 


5912 


6993 


6075 


6156 


6238 


6320 


6401 


6483 


6564 


6646 






533 


6727 


6809 


6890 


6972 


7653 


7134 


7216 


7297 


7379 


7460 






534 


7641 


7623 


7704 


7786 


7866 


7948 


8029 


8110 


8191 


8273 






535 


8354 


8435 


8516 


8597 


867S 


8759 


8841 


8922 


9003 


9084 






536 


9166 


9246 


9327 


9403 


9489 


9570 


9651 


9732 


9813 


9893 






537 


9974 


..65 


.136 


.217 


.298 


.378 


.469 


.440 


.621 


.702 






538 


730782 


0863 


0944 


1024 


1105 


1186 


1266 


1347 


1428 


1508 






539 


1689 


1669 


1750 


1830 


1911 


1991 


2072 


2152 


2233 


2313 






540 


2394 


2474 


2656 


2636 


2716 


2796 


2876 


2966 


3037 


3117 






541 


3197 


3278 


3368 


3438 


3518 


3598 


3679 


3769 


3839 


3919 






542 


3999 


4079 


4160 


4240 


4320 


4400 


4480 


4560 


4640 


4720 






543 


4800 


4880 


4960 


5040 


5120 


5200 


5279 


5359 


5439 


5619 






544 


6399 


6679 


5759 


5838 


5918 


5998 


6078 


6157 


6237 


6317 






545 


6397 


6476 


6656 


6636 


6715 


6796 


6874 


6954 


7034 


7113 






546 


7193 


7272 


7352 


7431 


7611 


7590 


7670 


7749 


7829 


7908 






547 


7987 


8067 


8146 


8226 


8305 


8384 


8463 


8543 


8622 


8701 






548 


8781 


8860 


8939 


9018 


9097 


9177 


9266 


9335 


9414 


•9493 






549 


9672 


9651 


9731 


9810 


9889 


9968 


..47 


.126 


.205 


.284 





12 


LOGARITHMS 




N. 





1 


2 


3 


4 


6 


6 


7 


8 


9 




650 


740363 


0442 


0521 


0560 


0678 


0757 


0836 


0915 


0994 


1073 




551 


1152 


1230 


1309 


1388 


1467 


1546 


1624 


1703 


1782 


1860 




552 


1939 


2018 


2096 


2175 


2254 


2332 


2411 


2489 


2568 


2646 




553 


2726 


28(M 


2882 


2961 


3039 


3118 


3196 


3276 


3353 


3431 




554 


3510 


3568 


3667 


3745 


3823 


3902 


8980 


4058 


4136 


4215 




555 


4293 


4371 


4449 


4528 


4606 


4684 


4762 


4840 


4919 


4997 




556 


5075 


5163 


5231 


5309 


5387 


6465 


5543 


6621 


6699 


6777 




657 


5855 


6933 


6011 


6089 


6167 


6245 


6323 


6401 


6479 


6656 




558 


6634 


6712 


6790 


6868 


6946 


7023 


7101 


7179 


7256 


7334 




559 


7412 


7489 


7567 


7646 


7722 


7800 


7878 


7956 


8033 


8110 




560 


8188 


8266 


8343 


8421 


8498 


8676 


8653 


8731 


8808 


8885 




561 


8963 


9040 


9118 


9196 


9272 


9360 


9427 


9504 


9582 


9659 




562 


9736 


9814 


9891 


9968 


..45 


.123 


.200 


.277 


.354 


.431 




663 


750508 


0586 


0663 


0740 


0817 


0894 


0971 


1048 


1126 


1202 




564 


1279 


1356 


1433 


1610 


1587 


1664 


1741 


1818 


1895 


1972 




565 


2048 


2125 


2202 


2279 


2356 


2433 


2609 


2586 


2663 


2740 




566 


2816 


2893 


2970 


3047 


3123 


3200 


3277 


3353 


3430 


3606 




567 


3582 


3660 


3736 


3813 


3889 


3966 


4042 


4119 


4196 


4272 




568 


4348 


4425 


4501 


4578 


4654 


4730 


4807 


4883 


4960 


5036 




569 


5112 


5189 


5266 


6341 


5417 


6494 


5670 


6646 


5722 


6799 




570 


5875 


6951 


6027 


6103 


6180 


6256 


6332 


6408 


6484 


6560 




571 


6636 


6712 


6788 


6864 


6940 


7016 


7092 


7168 


7244 


7320 




572 


7396 


7472 


7648 


7624 


7700 


7775 


7851 


7927 


8003 


8079 




573 


8155 


8230 


8306 


8382 


8458 


8533 


8609 


8685 


8761 


8836 




574 


8912 


8988 


9068 


9139 


9214 


9290 


9366 


9441 


9517 


9692 




575 


9668 


9743 


9819 


9894 


9970 


..45 


.121 


.196 


.272 


.347 




576 


760422 


0498 


0573 


0649 


0724 


0799 


0875 


0060 


1025 


1101 




&77 


1176 


1251 


1326 


1402 


1477 


1552 


1627 


1702 


1778 


1853 




578 


1938 


2003 


2078 


2163 


2228 


2303 


2378 


2453 


2529 


2604 




579 


2679 


2754 


2829 


2904 


2978 


3053 


3128 


2203 


3278 


3353 




580 


34S8 


3503 


3578 


3653 


3727 


3802 


3877 


3962 


4027 


4101 




581 


4176 


4261 


4326 


4400 


4476 


4650 


4624 


4699 


4774 


4848 




582 


4923 


4998 


5072 


5147 


5221 


5296 


6370 


5445 


6520 


6594 




583 


6669 


5743 


5818 


6892 


6956 


6041 


6115 


6190 


6264 


6338 




584 


6413 


6487 


6562 


6636 


6710 


6785 


6859 


6933 


7007 


7082 




585 


7156 


7230 


7304 


7379 


7453 


7527 


7601 


7675 


7749 


7823 




586 


T.89S 


7972 


80i6 


8120 


8194 


8268 


8342 


8416 


8490 


8564 




687 


8638 


8712 


8786 


8860 


8934 


9008 


9082 


9166 


9230 


9303 




588 


9377 


9461 


9525 


9699 


9673 


9746 


9820 


9894 


9968 


..42 




689 


770115 


0189 


0263 


0336 


0410 


0484 


0557 


0631 


0705 


0778 




590 


0852 


0926 


0999 


1073 


1146 


1220 


1293 


1367 


1440 


1514 




591 


1687 


1661 


1734 


1808 


1881 


1956 


2028 


2102 


2175 


2248 




592 


2322 


2395 


2468 


3542 


2615 


2688 


2762 


2836 


2908 


2981 




593 


3055 


3128 


3201 


3274 


3348 


3421 


3494 


3567 


3640 


3713 




594 


3786 


3860 


3933 


4006 


40?9 


4152 


4225 


4298 


4371 


4444 




595 


4617 


4590 


4663 


4736 


4809 


4882 


4966 


5028 


5100) 


6173 




596 


5246 


5319 


5392 


5465 


5538 


6610 


5683 


5756 


5829 


5902 




597 


5974 


6047 


6120 


6193 


6265 


6338 


6411 


6483 


6556 


6629 




698 


6701 


6774 


6846 


6919 


0992 


7064 


7137 


7209 


7282 


7354 




699 


7427 


7409 


7572 


7644 


7717 


7789 


7862 


7934 


8006 


8079 





^ 


OF NUMBERS. 13 




N. 





1 


2 


3 


4 


5 


6 


7 


8 


9 




600 


778151 


8224 


8296 


8368 


8441 


8513 


8585 


8658 


8730 


8802 




601 


8874 


8947 


9019 


9091 


9163 


9236 


9308 


9380 


9452 


9524 




602 


9596 


6669 


9741 


9813 


9885 


9967 


..29 


.101 


.173 


.245 




603 


780317 


0389 


0461 


0533 


0505 


0677 


0749 


0821 


0893 


0965 




604 


1037 


1109 


1181 


1253 


1324 


1396 


1468 


1540 


1612 


1684 




605 


1755 


1827 


1899 


1971 


2042 


2114 


2186 


2258 


2329 


2401 




606 


2473 


2644 


2616 


2688 


2759 


2831 


2902 


2974 


3046 


3117 




607 


3189 


3260 


3332 


3403 


8476 


3646 


3618 


8689 


3761 


3832 




608 


3904 


3975 


4046 


4118 


4189 


4261 


4332 


4403 


4475 


4546 




609 


4617 


4689 


4760 


4831 


4902 


4974 


5045 


5116 


5187 


5259 




610 


5330 


5401 


5472 


5543 


6615 


5686 


5757 


5828 


5899 


5970 




611 


6041 


6112 


6183 


6254 


6325 


6396 


6467 


6538 


6609 


6680 




612 


6761 


6822 


6893 


6964 


7035 


7106 


7177 


7248 


7319 


7390 




613 


7460 


7531 


7602 


7673 


7744 


7815 


7885 


7956 


8027 


8098 




614 


8168 


8239 


8310 


8381 


8451 


8522 


8593 


8663 


8734 


8804 




615 


8875 


8946 


9016 


9087 


9157 


9228 


9299 


9369 


9440 


9510 




616 


9581 


9651 


9722 


9792 


9863 


9933 


...4 


..74 


.144 


.216 




617 


790285 


0356 


0426 


0496 


0567 


0637 


0707 


0778 


0848 


0918 




618 


0988 


1059 


1129 


1199 


1269 


1340 


1410 


1480 


1550 


1620 




619 


1691 


1761 


1831 


1901 


1971 


2041 


2111 


2181 


2252 


2322 




620 


2392 


2462 


2532 


2602 


2672 


2742 


2812 


2882 


2952 


8022 




621 


8092 


3162 


3231 


3301 


8371 


3441 


3511 


3581 


8661 


3721 




622 


3790 


3860 


3930 


4000 


4070 


4139 


4209 


4279 


4849 


4418 




623 


4488 


4558 


4627 


4697 


4767 


4836 


4906 


4976 


5045 


5115 




624 


5186 


5254 


5324 


5393 


6463 


5532 


5602 


5672 


5741 


5811 




626 


5880 


5949 


6019 


6088 


6158 


6227 


6297 


6366 


6436 


6505 




626 


6574 


6644 


6713 


6782 


6852 


6921 


6990 


7060 


7129 


7198 




627 


7268 


7337 


7406 


7476 


7545 


7614 


7683 


7752 


7821 


7890 




628 


7960 


8029 


8098 


8167 


8236 


8305 


8874 


8443 


8513 


8582 




629 


8661 


8720 


8789 


8858 


8927 


8996 


9066 


6134 


9203 


9272 




630 


9341 


9409 


9478 


9547 


9610 


9685 


9754 


9823 


9892 


9961 




631 


800026 


0098 


0167 


0236 


0305 


0373 


0442 


0511 


0580 


0848 




632 


0717 


0786 


0854 


0923 


0992 


1061 


1129 


1198 


1266 


1335 




633 


1404 


1472 


1541 


1609 


1678 


1747 


1816 


1884 


1952 


2021 




634 


2089 


2158 


2226 


2295 


2363 


2432 


2500 


2668 


2637 


2705 




635 


2774 


2842 


2910 


2979 


3047 


3116 


3184 


8252 


3321 


3389 




636 


3457 


3526 


3594 


3662 


3730 


3798 


8867 


3935 


4003 


4071 




637 


4139 


4208 


4276 


4354 


4412 


4480 


4548 


4616 


4685 


4753 




638 


4821 


4889 


4957 


5025 


6093 


6161 


5229 


5297 


5365 


5433 1 




639 


5601 


5669 


5637 


6706 


6773 


6841 


5908 


5976 


6044 


6112 j 




640 


6180 


6248 


6316 


6384 


6451 


6519 


6587 


6655 


6723 


6790 ! 




641 


6858 


6926 


6994 


7061 


7129 


7157 


7264 


7332 


7400 


7467 ! 




642 


7635 


7603 


7670 


7738 


7808 


7873 


7941 


8008 


8076 


8143 1 




643 


8211 


8279 


8346 


8414 


8481 


8549 


8616 


8684 


8751 


8818 i 




644 


8886 


8963 


9021 


9088 


9166 


9223 


9290 


9358 


9426 


9492 




645 


9560 


9627 


9694 


9762 


9829 


9896 


9964 


..31 


..98 


.165 




646 


810238 


0300 


0367 


0434 


0501 


0596 


0636 


0703 


0770 


0837 




647 


0904 


0971 


1039 


1106 


1173 


1240 


1307 


1374 


1441 


1508 




648 


1676 


1642 


1709 


1776 


1843 


1910 


1977 


2044 


2111 


2178 




649 


2246 


2312 


2379 


2446 


2512 


2o79 


2648 


2713 


2780 


2847 



14 


LOGARITHMS 




N. 





I 


2 


3 


4 


5 


6 


7 


8 


9 




6B0 


812913 


2980 


3047 


3114 


3181 


3247 


3314 


3381 


3448 


3514 




651 


3581 


3648 


3714 


3781 


3848 


3914 


3981 


4048 


4114 


4181 




652 


4248 


4314 


4381 


4447 


4514 


4681 


4647 


4714 


4780 


4847 




663 


4913 


4980 


5046 


6113 


5179 


5246 


5312 


5378 


5445 


5511 




654 


5578 


5644 


5711 


5777 


5843 


5910 


5976 


6042 


6109 


6175 




665 


6241 


6308 


6374 


6440 


6506 


6573 


6639 


6705 


6771 


6838 




656 


6904 


6970 


7036 


7102 


7169 


7233 


7301 


7367 


7433 


7499 




657 


7565 


7631 


7698 


7764 


7830 


7896 


7962 


8028 


8094 


8160 




658 


8226 


8292 


8368 


8424 


8490 


8556 


8622 


8688 


8764 


8820 




659 


8885 


8951 


9017 


9083 


9149 


9215 


9281 


9346 


9412 


9478 




660 


9544 


9610 


9676 


9741 


9807 


9873 


9939 


...4 


..70 


.136 




661 


820201 


0267 


0333 


0399 


0464 


0530 


0695 


0661 


0727 


0792 




662 


0858 


0924 


0989 


1055 


1120 


1186 


1251 


1317 


1382 


1448 




663 


1514 


1579 


1646 


1710 


1775 


1841 


1906 


1972 


2037 


2103 




664 


2168 


2233 


2299 


2364 


2430 


2495 


2560 


2626 


2691 


2766 




666 


2822 


2887 


2962 


3018 


3083 


3148 


3213 


3279 


3344 


3409 




666 


3474 


3539 


3605 


3670 


3736 


3800 


3865 


3930 


3996 


4061 




667 


4126 


4191 


4266 


4321 


4386 


4451 


4516 


4681 


4646 


4711 




668 


4776 


4841 


4906 


4971 


6036 


5101 


6166 


5231 


5296 


5361 




669 


5426 


5491 


6656 


5621 


5686 


5751 


5815 


5880 


5945 


6010 




670 


6075 


6140 


6204 


6269 


6334 


6399 


6464 


6528 


6593 


6658 




671 


6723 


6787 


6862 


6917 


6981 


7046 


7111 


7175 


7240 


7305 




672 


7369 


7434 


7499 


7663 


7628 


7692 


7767 


7821 


7886 


7961 




673 


8015 


8080 


8144 


8209 


8273 


8338 


8402 


8467 


8531 


8595 




674 


8660 


8724 


8789 


8853 


8918 


8982 


9046 


9111 


9175 


9239 




675 


9304 


9368 


9432 


9497 


9661 


9625 


9690 


9764 


9818 


9882 




676 


9947 


..11 


..75 


.139 


.204 


.268 


.332 


.396 


.460 


.525 




677 


830589 


0653 


0717 


0781 


0845 


0909 


0973 


1037 


1102 


1166 




678 


1230 


1294 


1358 


1422 


1486 


1560 


1614 


1678 


1742 


1806 




679 


1870 


1934 


1998 


2062 


2126 


2189 


2253 


2317 


2381 


2445 




680 


2509 


2673 


2637 


2700 


2764 


2828 


2892 


2956 


3020 


3083 




681 


3147 


3211 


3276 


3338 


3402 


3466 


3530 


3593 


3657 


3721 




682 


3784 


3848 


3912 


3976 


4039 


4103 


4166 


4230 


4294 


4367 




683 


4421 


4484 


4548 


4611 


4675 


4739 


4802 


4866 


4929 


4993 




684 


5056 


5120 


5183 


6247 


5310 


5373 


5437 


5500 


6564 


5627 




685 


5691 


6754 


5817 


5881 


5944 


6007 


6071 


6134 


6197 


6261 




686 


6324 


6387 


6461 


6614 


6577 


6641 


6704 


6767 


6830 


6894 




687 


6957 


7020 


7083 


7146 


7210 


7273 


7336 


7399 


7462 


7626 




688 


7588 


7652 1 7715 


7778 


7841 


7904 


7967 


8030 


8093 


8166 




689 


8219 


8282 8345 


8408 


8471 


8534 


8597 


8660 


8723 


8786 




690 


8849 


8912 8975 


9038 


9109 


9164 


9227 


9289 


9352 


9416 




691 


9478 


9641 ! 9604 


9667 


9729 


9792 


9855 


9918 


9981 


..43 




692 


840106 


0169 j 0232 


0294 


0357 


0420 


0482 


0545 


0608 


0671 




693 


0733 


0796 0859 


0921 


0984 


1046 


1109 


1172 


1234 


1297 




694 


1359 


1422 1 1485 


1547 


1610 


1672 


1735 


1797 


1860 


1922 




695 


1985 


2047 2110 


2172 


2235 


2297 


2360 


2422 


2484 


2547 




696 


2609 


2672 2734 


2796 


2869 


2921 


2983 


3046 


3108 


3170 




697 


3233 


3295 3357 


3420 


3482 


3544 


3606 


3669 


3731 


3793 




698 


3855 


3918 3980 


4042 


4104 


4166 


4229 


4291 


4363 


4416 




699 


4477 


4539 4601 


4664 


4726 


4788 


4850 


4912 


4974 


6036 





OF NUMBERS. 15 




N. 





1 


3 


3 


4 


5 


6 


7 


8 


9 




j 700 


845098 


5160 


6222 


5284 


5346 


5408 


6470 


6532 


5694 


6666 




701 


6718 


5780 


6842 


6904 


5966 


6028 


6090 


6151 


6213 


6'276 




702 


6337 


6399 


6461 


6623 


6585 


6646 


6708 


6770 


6832 


6894 




703 


6955 


7017 


7079 


7141 


7202 


7264 


7326 


7388 


7449 


7511 




704 


7673 


7634 


7676 


7768 


7819 


7831 


7943 


8004 


8066 


8128 




1 705 


8189 


8251 


8312 


8374 


8435 


8497 


8559 


8620 


8682 


8743 




1 706 


8805 


8866 


8928 


8989 


9051 


9112 


9174 


9235 


9297 


9358 




707 


9419 


9481 


9542 


9604 


9665 


9726 


9788 


9849 


9911 


9972 




703 


850033 


0095 


0156 


0217 


0279 


0340 


0401 


0462 


0524 


0585 




703 


0646 


0707 


0769 


0830 


0891 


0952 


1014 


1076 


1136 


1197 




710 


1258 


1320 


1381 


1442 


1503 


1564 


1625 


1686 


1747 


1809 




f 711 


1870 


1931 


1992 


2053 


2114 


2175 


2236 


2297 


2358 


2419 




712 


2480 


2541 


2602 


2668 


2734 


2786 


2846 


2907 


2968 


3029 




713 


3090 


3150 


3211 


3272 


3333 


3394 


3455 


3516 


3577 


3637 




714 


3698 


3759 


3820 


3881 


3941 


4002 


4063 


4124 


4185 


4245 




715 


4306 


4367 


4428 


4488 


4549 


4610 


4670 


4731 


4792 


4852 




716 


4913 


4974 


6034 


5095 


5156 


6216 


5277 


6337 


6898 


6469 




717 


5519 


5680 


6640 


6701 


5761 


5822 


63S2 


5943 


6003 


6064 




718 


6124 


6185 


6245 


6306 


6366 


6427 


6487 


6548 


6608 


6668 




719 


6729 


6789 


6860 


6910 


6970 


7031 


T091 


7152 


7212 


7272 




720 


7332 


7393 


7453 


7513 


7574 


7634 


7694 


7755 


7815 


7876 




721 


7936 


7995 


8p56 


8116 


8176 


8236 


8297 


8357 


8417 


8477 




722 


8537 


8597 


8657 


8718 


8778 


8838 


8898 


8958 


9018 


9078 




723 


9138 


9198 


9258 


9318 


9379 


9439 


9499 


95.39 


9619 


9679 




724 


9739 


9799 


9859 


9918 


9978 


..38 


..98 


.168 


.218 


.278 




725 


860338 


0398 


0458 


0518 


0578 


0637 


0697 


0757 


0817 


0877 




72(5 


0937 


0995 


1056 


1116 


1176 


1236 


1296 


1356 


1415 


1476 




727 


1634 


1594 


1654 


1714 


1773 


1833 


1893 


1952 


2012 


2072 




728 


2131 


2191 


2251 


2310 


2370 


2430 


2489 


2549 


2608 


2668 




729 


2728 


2787 


2847 


2906 


2966 


3026 


3085 


3144 


3204 


3263 




730 


3323 


3382 


3442 


3501 


3561 


3620 


3680 


3739 


3799 


3868 




731 


3917 


3977 


4036 


4096 


4155 


4214 


4274 


4333 


4392 


4452 




732 


4511 


4570 


4630 


4689 


4148 


4808 


4867 


4926 


4985 


6046 




733 


5104 


6163 


5222 


6282 


5341 


5400 


5469 


5519 


5578 


6687 




734 


5696 


5756 


6814 


6874 


5933 


5992 


6051 


6110 


6169 


6228 




735 


6287 


6346 


6405 


6465 


6624 


6583 


6642 


6701 


6760 


6819 




736 


6878 


6937 


6996 


7055 


7114 


7173 


7232 


7291 


7350 


7409 




737 


7467 


7526 


7586 


7644 


7703 


7762 


7821 


7880 


7939 


7998 




738 


8056 


8115 


8174 


8233 


8292 


8350 


8409 


8468 


8527 


858<> 




739 


8644 


8703 


8762 


8821 


8879 


8988 


8997 


9056 


9114 


9173 




740 


9232 


9290 


9349 


9408 


9466 


9525 


9684 


9642 


9701 


9760 




741 


9818 


9877 


9936 


9994 


..53 


.111 


.170 


.228 


.287 


.345 




742 


870404 


0462 


0521 


0679 


0638 


0696 


0765 


0813 


0872 


0930 




743 


0989 


1047 


1106 


1164 


1223 


1281 


1339 


1398 


1466 


1616 




744 


1673 


1631 


1690 


1748 


1803 


1865 


1928 


1981 


2040 


2098 




746 


2156 


2215 


2273 


2331 


2389 


2448 


2506 


2564 


2622 


2681 




746 


2739 


2797 


2865 


2913 


2972 


3030 


8088 


3146 


3204 


3262 




747 


3321 


3379 


8437 


8495 


3553 


3611 


3669 


3727 


3785 


3844 




748 


8902 


8960 


4018 


4076 


4134 


4192 


4260 


4308 


4360 


4424 




749 


4482 


4540 


4698 


4656 


4714 


4772 


4830 


4888 


4945 


5003 









24 



16 


LOGARITHMS 




N. 





1 


2 


3 4 


6 


6 


7 


8 


9 




750 


875061 


5119 


6177 


6235 


6293 


6351 


6409 


5466 


5524 


5582 




751 


6640 


5698 


6756 


5813 


5871 


6929 


5987 


6045 


6102 


6160 




762 


6218 


6276 


6333 


6391 


6449 


6507 


6564 


6622 


6680 


6737 




763 


6795 


6853 


6910 


6968 


7026 


7083 


7141 


7199 


7256 


7314 




764 


7371 


7429 


7487 


7644 


7602 


7659 


7717 


7774 


7832 


7889 




756 


7947 


8004 


8062 


8119 


8177 


8234 


8292 


8349 


8407 


8464 




766 


8522 


8579 


8637 


8694 


8752 


8809 


886(5 


8924 


8931 


9039 




757 


9096 


9153 


9211 


9268 


9325 


9383 


9440 


9497 


9555 


9612 




758 


9669 


9726 


9784 


9841 


9898 


9956 


..13 


..70 


.127 


.185 




769 


880242 


0299 


0356 


0413 


0471 


0528 


0580 


0642 


0699 


0756 




760 


0814 


0871 


0928 


0985 


1042 


1099 


1156 


1213 


1271 


1328 




761 


1385 


14^42 


1499 


1556 


1613 


1670 


1727 


1784 


1841 


1898 




762 


1955 


2012 


2069 


2126 


2183 


2240 


2297 


2354 


2411 


2468 




763 


2525 


2581 


2638 


2695 


2752 


2809 


2866 


2923 


2980 


3037 




764 


3093 


3150 


3207 


8264 


3321 


3377 


3434 


3491 


3548 


3606 




765 


3661 


3718 


3775 


3832 


3888 


3945 


4002 


4059 


4115 


4172 




766 


4229 


4285 


4342 


4399 


4455 


4512 


4569 


4625 


4682 


4739 




767 


4795 


4852 


4909 


4965 


5022 


5078 


6135 


5192 


5248 


6305 




768 


5361 


5418 


5474 


5531 


5587 


5644 


5700 


5757 


5813 


5870 




769 


6926 


5983 


6039 


6096 


6152 


6209 


6266 


6321 


6378 


6434 




770 


6491 


6547 


6604 


6660 


6716 


6773 


6829 


6886 


6942 


6998 




771 


7054 


7111 


7167 


7233 


7280 


7336 


7392 


7449 


7506 


7561 




772 


7617 


7674 


7730 


7786 


7842 


7898 


7955 


8011 


8067 


8123 




773 


8179 


8236 


8292 


8348 


8404 


8460 


8616 


8573 


8629 


8655 




774 


8741 


8797 


8863 


,8909 


8965 


9021 


9077 


9134 


9190 


9246 




776 


9302 


9358 


9414 


9470 


9526 


9582 


9638 


9694 


9760 


9806 




776 


9862 


9918 


0974 


..30 


..86 


.141 


.197 


.253 


.309 


.866 




777 


890421 


0477 


0533 


0589 


0646 


0700 


0756 


0812 


0868 


0924 




778 


0980 


1035 


1091 


1147 


1203 


1259 


1314 


1370 


1426 


1482 




779 


1537 


1593 


1649 


1705 


1760 


1816 


1872 


1928 


1983 


2039 




780 


2095 


2150 


2206 


2262 


2317 


2373 


2429 


2484 


2540 


2695 




781 


2651 


2707 


2762 


2818 


2873 


2929 


2985 


8040 


3096 


3151 




783 


3207 


3262 


3318 


3373 


3429 


3484 


3540 


3595 


3651 


3706 




783 


3762 


8817 


3873 


3928 


3984 


4039 


4094 


4150 


4205 


4261 




784 


4316 


4371 


4427 


4482 


4538 


4593 


4648 


4704 


4769 


4814 




785 


4870 


4925 


4980 


5036 


6091 


5146 


5201 


6257 


6312 


5367 




788 


6423 6478 


5533 


5588 


5644 


6699 


5754 


5809 


6864 


5920 




787 


5975 


6030 


6085 


6140 


6195 


6251 


6306 


6361 


6416 


6471 




788 


6526 


6581 


6636 


6692 


6747 


6802 


6857 


6912 


6967 


7022 




789 


7077 


7132 


7187 


7242 


7297 


7362 


7407 


7462 


7617 


7672 




790 


7627 


7683 


7737 


7792 


7847 


7902 


7957 


8012 


8067 


8122 




791 


8176 


8231 


8286 


8341 


8396 


8451 


8606 


8561 


8615 


8670 




792 


8725 


8780 


8835 


8890 


8944 


8999 


9054 


9109 


9164 


9218 




793 


9273 


9328 


9383 


9437 


9492 


9647 


9602 


9656 


9711 


9766 




794 


9821 


9875 


9930 


9985 


.,39 


..94 


.149 


.203 


.258 


.312 




795 


900367 


0422 


0476 


0531 


0586 


0640 


0695 


0749 


0804 


0859 




796 


0913 


0968 


1022 


1077 


1131 


1186 


1240 


1295 


1349 


1404 




797 


1458 


1513 


1567 


1622 


1676 


1736 


1786 


1840 


I8y4 


1948 




798 


2003 


2057 


2112 


2166 


2221 


2275 


2329 


2384 


2438 


2492 




799 


2547 


2601 


2655 


2710 


2764 


2818 


2873 


2927 


2981 


3036 









OF NUMBERS 


17 






N. 




903090 


1 


2 


3 


4 


6 


6 


7 


8 


9 






800 


3144 


3199 


3253 


3307 


3361 


3416 


3470 


3524 


3578 






801 


3633 


3687 


3741 


3795 


3849 


3904 


3958 


4012 


4066 


4120 






802 


4174 


4229 


4283 


4337 


4391 


4445 


4499 


4553 


4607 


4661 






803 


4716 


4770 


4824 


4878 


4932 


4986 


5040 


5094 


6148 


5202 






804 


5358 


5310 


6364 


5418 


5472 


5526 


5580 


6634 


5688 


5742 






805 


5796 


5860 


5904 


5958 


6012 


6066 


6119 


6173 


6227 


6281 






806 


6335 


6389 


6443 


6497 


6551 


6604 


6658 


6712 


6766 


6820 






807 


6874 


6927 


6981 


7035 


7089 


7143 


7196 


7250 


7304 


7358 






808 


7411 


7465 


7519 


7573 


7626 


7680 


7734 


7787 


7841 


7895 






809 


7949 


8002 


8056 


8110 


8163 


8217 


8270 


8324 


8378 


8431 






810 


8485 


8539 


8592 


8646 


8699 


8753 


8807 


8860 


8914 


8967 






811 


9021 


9074 


9128 


9181 


9235 


9289 


9342 


9396 


9449 


9503 






812 


9656 


9610 


9663 


9716 


9770 


9823 


9877 


9930 


9984 


..37 






813 


910091 


0144 


0197 


0251 


0304 


0358 


0411 


0464 


0518 


0671 






814 


0624 


0378 


0731 


0784 


0838 


0891 


0944 


0998 


1051 


1104 






815 


1158 


1211 


1264 


1317 


1371 


1424 


1477 


1530 


1584 


1637 






816 


1690 


1743 


1797 


1850 


1903 


1956 


2009 


2063 


2115 


2169 






817 


2222 


2275 


2323 


2381 


2435 


2488 


2541 


2594 


2645 


2700 






818 


2753 


2808 


2859 


2913 


2966 


3019 


3072 


3125 


3178 


3231 






819 


3284 


3337 


3390 


3443 


3496 


3549 


3602 


3655 


3708 


3761 






820 


3814 


3867 


3920 


3973 


4026 


4079 


4132 


4184 


4237 


4290 






821 


4343 


4398 


4449 


4502 


4555 


4608 


4660 


4713 


4766 


4819 






822 


4872 


4925 


4977 


5030 


5083 


5136 


5189 


5241 


5694 


5347 






823 


5400 


5453 


5505 


5658 


5611 


5664 


5716 


5769 


5822 


5876 






824 


5927 


5980 


6033 


6085 


6138 


6191 


6243 


6296 


6349 


6401 






825 


6454 


6507 


6559 


6612 


6664 


6717 


6770 


6822 


6875 


6927 






826 


6980 


7033 


7085 


7138 


7190 


7243 


7296 


7348 


7400 


7463 






827 


7506 


7568 


7611 


7663 


7716 


7768 


7820 


7873 


7925 


7978 






828 


8030 


8083 


8185 


8188 


8240 


8293 


8345 


8397 


8450 


8502 






829 


8655 


8607 


8859 


8712 


8764 


8816 


8869 


8921 


8973 


9026 






830 


9078 


9130 


9183 


9235 


9287 


9340 


9392 


9444 


9496 


9549 






831 


9601 


9663 


9706 


9768 


9810 


9862 


9914 


9967 


..19 


..71 






832 


920123 


0176 


0228 


0280 


0332 


0384 


0436 


0489 


0541 


0593 






833 


0845 


0697 


0749 


0801 


0853 


0908 


0958 


1010 


1062 


1114 






834 


1166 


1218 


1270 


1322 


1374 


1426 


1478 


1530 


1582 


1634 






835 


1686 


1738 


1790 


1842 


1894 


1946 


1998 


2050 


2102 


2164 






836 


2206 


2258 


2310 


2362 


2414 


2466 


2518 


2570 


2622 


2674 






837 


2725 


2777 


2829 


2881 


2933 


2985 


3037 


3089 


3140 


3192 






838 


3244 


3296 


3348 


3399 


3451 


3503 


3656 


3607 


3658 


3710 






889 


3762 


3814 


3865 


3917 


3969 


4021 


4072 


4124 


4147 


4228 






840 


4279 


4331 


4383 


4434 


4486 


4538 


4589 


4641 


4693 


4744 






841 


4796 


4848 


4899 


4951 


5003 


5054 


5108 


5157 


5209 


5261 






842 


5312 


5364 


5415 


5467 


5518 


5570 


5621 


5673 


5725 


5776 






843 


5828 


5874 


5931 


5982 


6034 


6085 


6137 


6188 


6240 


6291 






844 


6342 


6394 


6445 


6497 


6548 


6600 


6651 


6702 


6754 


6805 






845 


6857 


6908 


6959 


7011 


7062 


7114 


7165 


7216 


7268 


7319 






846 


7370 


7422 


7473 


7524 


7576 


7627 


7678 


7730 


7783 


7832 






847 


7883 


7935 


7986 


8037 


8088 


8140 


8191 


8242 


8293 


8345 






848 


8396 


8447 


8498 


8549 


8601 


8652 


8703 


8754 


8805 


8857 






849 


8908 


8959 


9010 


9081 


9112 


9163 


9216 


9266 


9317 


9368 
J 







18 


I^OGARITHMS 






N. 
850 




923419 


I 1 2 


3 


4 


5 


6 


7 


8 


9 


. 




9473 9521 


9572 


9623 


9674 


9725 


9776 


9827 


9879 






851 


9930 


9981 


..32 


..83 


.W4 


.185 


.236 


.287 


.338 


.389 






86fi 


98<>i40 


0491 


0542 


0592 


0643 


0694 


0746 


0796 


0847 


0898 






853 


• 0949 


1000 


1051 


1102 


1153 


1204 


1254 


1305 


1366 


1407 






864 


1458 


1509 


1660 


1610 


1661 


1712 


1763 


1814 


1865 


1915 






855 


1966 


2017 


2068 


2118 


2169 


2220 


2271 


2322 


2372 


2423 






856 


2474 


2624 2676 


2626 


2677 


2727 


2778 


2829 


2879 


2930 






857 


2981 


3061 


3082 


3133 


3183 


3234 


3285 


3335 


3386 


3437 






858 


•3487 


3538 


3589 


3639 


3690 


3740 


3791 


3841 


3892 


3943 






869 


3993 


4044 


4094 


4145 


4195 


4246 


4269 


4347 


4397 


4448 






880 


4498* 


4549 


4699 


4650 


4700 


4751 


4801 


4852 


4902 


4950 






861 


5003 


5054 


5104 


6154 


6205 


6265 


5306 


6856 


6406 


6457 






862 


6507 


5558 


6608 


6658 


6709 


5759 


6809 


5860 


6910 


5960 






863 


6011 


6061 


6111 


6162 


6212 


6262 


6313 


6363 


6413 


6463 






864 


6514 


6564 


6614 


6665 


6715 


6766 


6815 


6865 


6916 


6966 






866 


7016 


7066 


7117 


71-67 


7217 


7267 


7317 


7367 


7418 


74G8 






866 


7518 


7568 


76lg 


7668 


7718 


7769 


7819 


7869 


7919 


7969 






867 


8019 


8069 


8119 


8169 


8:^19 


8269 


8320 


8370 


8420 


8470 






868 


8620 


8570 


8620 


8670 


8720 


8770 


8820 


8870 


8919 


8970 






869 


9020 


9070 


9120 


9170 


9220 


9270 


9320 


9369 


9419 


9469 






870 


9519 


9569 


9616 


9669 


9719 


9769 


9819 


9869 


9918 


9968 






871 


940018 


0068 


0118 


0168 


0218 


0267 


0317 


0367 


0417 


0467 






872 


0516 


0566 i 0616 


0866 


0/16 


0765 


0815 


0865 


0916 


0964 






873 


1014 


1064 


1114 


1163 


1213 


1263 


1313 


1362 


1412 


1462 






874 


1511 


1561 


1611 


1660 


1710 


1760 


1809 


1859 


1909 


1968 






875 


2008 


2068 


2107 


2157 


2207 


2256 


2306 


,2365 


2405 


2465 






876 


2504 


2554 


2603 


2653 


2702 


2762 


2801 


2851 


2901 


2950 i 






877 


3000 


3049 


3099 


3148 


3198 


3247 


3297 


3346 


3396 


3445 1 






878 


3495 


3644 


3693 


3643 


3692 


3742 


3791 


3841 


3890 


3939 i 






879 


3989 


4038 


4088 


4137 


4186 


4236 


4286 


4336 


4384 


4433 i 

1 






880 


4483 


4532 


4581 


4631 


4680 


4729 


4779 


4828 


4877 


4927 






881 


4976 


6026 


5074 


5124 


5173 


6222 


5272 


6321 


5370 


5419 






882 


5469 


6518 


5667 


6616 


5666 


5715 


6764 


6813 


6862 


6912 






883 


5961 


6010 


6059 


6108 


6157 


6207 


6256 


6305 


6364 


6403 






884 


6452 


6501 


6551 


6600 


6649 


6698 


6747 


6796 


6845 


6894 






885 


6943 


6992 


7041 


7090 


7140 


7189 


7238 


7287 


7336 


7385 






886 


7434 


7488 


7532 


7681 


7630 


7679 


7728 


7777 


7826 


7876 






887 


7924 


7973 


8022 


8070 


8119 


8168 


8217 


8266 


8315 


8365 






888 


8413 


8462 


8511 


8560 


8609 


8657 


8706 


8765 


8804 


8863 






889 


8902 


8951 


8999 


9048 


9097 


9146 


9195 


9244 


9292 


9341 






899 


9390 


9439 


9488 


9636 


9586 


9634 


9683 


9731 


W80 


9829 






891 


9878 


9926 


9975 


..24 


..73 


.121 


.170 


.219 


.267 


.316 






892 


950865 


0414 


0462 


0611 


0560 


0608 


0667 


0706 


0754 


0803 






893 


0851 


0900 


0949 


0997 


1046 


1095 


1143 


1192 


1240 


1289 






894 


1338 


1386 


1436 


1483 


1632 


1580 


1629 


1677 


1726 


1776 






895 


1823 


1872 


1920 


1969 


2017 


2066 


2114 


2163 


2211 


2260 






896 


2308 


2366 


2405 


2463 


2502 


2550 


2699 


2647 


6696 


2744 






897 


2792 


2841 


2889 


2938 


2986 


3034 


3083 


3131 


3180 


3228 






898 


3276 


3325 


3373 


3421 


3470 


3518 


3566 


3616 


3663 


3711 i 






899 


3760 


3808 


8856 


3905 


3963 


. 4001 


4019 


40JS 


4146 


4194 









OF NUMBERS. ]9 




N. 





1 


3 


3 


4 


5 


6 


7 


8 


' 9 




900 


954243 


4291 


4339 


4387 


4435 


4484 


4532 


4580 


4628 


4677 




901 


4726 


4773 


4821 


4869 


4918 


4966 


6014 


5062 


MTO 


5158 




902 


6207 


5256 


5303 


5351 


6399 


5447 


5496 


5643 


5592 


5640 




903 


6688 


5736 


5784, 


5832 


5880 


5928 


6976 


6024 


6072 


6120 




904 


6168 


6216 


6265 


6313 


6361 


6409 


6457 


6565 


6553 


6601 




905 


6649 


6697 


6745 


6793 


6840 


6888 


6936 


6984 


7032 


7080 




, 908 


7128 


7176 


7224 


7272 


7320 


7368 


7416 


7464 


7512 


7559 




907 


7607 


7655 


7703 


7751 


7799 


7847 


7894 


7942 


7990 


8088 




908 


8086 


8134 


8181 


8229 


8277 


8326 


8373 


8421 


.8468 


8516 




909 


8564 


8612^ 


8659 


8707 


8755 


8803 


8860 


8898 


8946 


8994 




910 


9041 


9089 


9137 


9185 


9232 


9280 


9328 


9375 


9423 


9471 




911 


9518 


9566 


9814 


9661 


9709 


9757 


9804 


9852 


9900 


9947 . 


. 


912 


9995 


..42 


..90 


.138 


.185 


.233 


.280 


.328 


.376 


.423 




913 


960471 


0518 


0566 


0613 


0661 


0709 


0766 


0804 


0861 


0899 




914 


0946 


0994 


1041 


1089 


1136 


1184 


1231 • 


1279 


1326 


1374 




915 


1421 


1469 


1616 


1563 


1611 


1658 


1706 


1753 


1801 


1848 




916 


1895 


1943 


1990 


2038 


2085 


2132 


2180 


2227 


2275 


2322 




917 


2369 


2417 


2464 


2511 


2559 


2608 


2653 


2701 


2748 


2795 




918 


2848 


2890 


2937 


2985 


3032 


3079 


3126 


3174 


3221 


3268 




919 


3316 


3363 


3410 


3457 


3604 


3552 


3699 


3646 


3693 


3741 




: 920 


3788 


3835 


3882' 


3929 


3977 


4024 


4071 


4118 


4165 


4212 ■ 




921 


4260 


4307 


4354 


4401 


4448 


4495 


4642 


4590 


4637 


4684 




922 


4731 


4778 


4825 


4872 


4919 


4966 


5013 


5061 


5108 


5155 




923 


5202 


5249 


5296 


5343 


6390 


5437 


5484 


5531 


5578 


5625 




924 


5672 


5719 


5766 


6813 


5860 


5907 


5954 


6001 


6048 


6095 




925 


6142 


6189 


6236 


6283 


6329 


6376 


6423 


6470 


6617 


6564 




926 


6611 


6658 


6705 


6762 


6799 


6845 


6892 


6939 


6986 


7033 




927 


7080 


7127 


7173 


7220 


7267 


7314 


7361 


7408 


7454 


7501 




928 


7548 


7595 


7642 


7688 


7735 


7782 


7829 


7875 


7922 


7969 




929 


8016 


8632 


8109 


8156 


8263 


8249 


8296 


8343 


8390 


8436 




930 


8483 


8530 


8576 


8623 


8670 


8716 


8763 


8810 


8856 


8903 




931 


8950 


8996 


9043 


9090 


.9136 


9183 


9229 


9276 


9323 


9369 




932 


9416 


9463 


9509 


9556 


9602 


9649 


9695 


9742 


9789 


9835 




933 


9882 


9928 


997^ 


..21 


..68 


.114 


.161 


.207 


.254 


.300 




934 


970347 


0393 


0440 


0486 


6533 


0579 


6626 


0672 


0719 


0765 




935 


0812 


0858 


0904 


0951 


0997 


1044 


1090 


1137 


1183 


1229 




936 


1276 


1322 


1369 


1415 


1461 


1508 


1564 


1601 


1647 


1693 




937 


1740 


1786 


1832 


1879 


1925 


1971 


2018 


2064 


2110 


2157 




938 


2203 


2249 


229S 


2342 


2388 


2434 


2481 


2627 


2573 


2619 




939 


2666 


2712 


2758 


2804 


2851 


2897 


2943 


2989 


3035 


3082 




940 


3128 


3174 


3220 


3266 


3313 


3359 


3405 


3451 


3497 


8543 




941 


8590 


3636 


3682 


3728 


3774 


3820 


8866 


3913 


3959 


4005 




942 


4051 


4097 


4143 


4189 


4235 


4281 


4327 


4374 


4420 


4466 




943 


4512 


4558 


4604 


4650 


4696 


4742 


4788 


4834 


4880 


4926 




944 


4972 


5018 


5064 


5110 


5156 


5202 


5248 


5294 


5340 


5386 




945 


5432 


5478 


5524 


,5570 


5616 


5662 


5707 


5753 


5799 


5845 




946 


5891 


5937 


5983 


6029 


6075 


6121 


6167 


6212 


6258 


6304 




947 


6350 


6396 


6442 


6488 


6533 


6679 


6925 


6671 


6717 


6763 




948 


6803 


6854 


6900 


6946 


6992 


7037 


7083 


7129 


7175 


7220 




949 


7266 


7312 


7358 7403 


7449 


7496 


7541 


7586 


7632 


7678 


' 





2a 


1 
LOGARITHMS 






N. 





1 


S 


3 


4 


5 


6 


7 


8 


9 






950 


977724 


7769 


7815 


7861 


7906 


7952 


7998 


8043 


8089 


8135 






951 


8181 


8226 


8272 


8317 


8363 


8409 


8454 


8500 


854G 


8591 






96-2 


8637 


8683 


8728 


8774 


8819 


8866 


8911 


8956 


9002 


9047 






953 


9093 


9138 


9184 


9230 


9275 


9321 


9366 


9412 


9457 


9503 






964 


9548 


9594 


9639 


9685 


9730 


9776 


9821 


9867 


9912 


9968 






955 


980003 


0049 


0094 


0140 


0185 


0231 


0276 


0322 


0367 


0412 






956 


0458 


0503 


0549 


0594 


0640 


0686 


0730 


0776 


0821 


0867 






957 


0912 


0957 


1003 


1048 


1093 


1139 


1184 


1229 


1275 


1320 






958 


1366 


1411 


1456 


1501 


1547 


1592 


1637 


1683 


1728 


1773 






959 


1819 


1864 


1909 


1954 


2000 


2045 


2090 


2135 


2181 


2226 






9G0 


2271 


2316 


2362 


2407 


2462 


2497 


2543 


2588 


2633 


2678 






961 


2723 


2769 


2814 


2859 


2904 


2949 


2994 


3(H0 


3085 


3130 






962 


3176 


3220 


3265 


3310 


3356 


3401 


3446 


3491 


3536 


3581 






963 


3626 


3671 


3716 


3762 


3807 


3852 


3897 


3942 


3987 


4032 






964 


4077 


4122' 


4167 


4212 


4257 


4302 


4347 


4392 


4437 


4482 






965 


4527 


4572 


4617 


4662 


4707 


4752 


4797 


4842 


4887 


4932 I 






966 


4977 


5GG2 


5067 


6112 


6157 


5202 


5247 


6292 


6337 


5382 






967 


B426 


5471 


5516 


6561 


5606 


5651 


6699 


5741 


6786 


5830 






968 


5876 


5920 


6965 


6010 


6056 


6100 


6144 


6189 


6234 


6279 






969 


6324 


6369 


6413 


6458 


6503 


6548 


6593 


6637 


6682 


6727 






970 


6773 


6817 


6861 


6906 


6951 


6996 


7040 


7085 


7130 


7175 






971 


7219 


7264 


7300 


7353 


7398 


7443 


7488 


7532 


7577 


7622 






972 


7666 


7711 


7756 


7800 


7845 


7890 


7934 


7979 


8024 


8068 






973 


8U3 


8157 


820Q 


8247 


8291 


8336 


8381 


8425 


8470 


8514 






974 


8559 


8604 


8648 


8693 


8737 


8782 


8826 


8871 


8916 


8960 






975 


90a5 


9049 


9093 


9138 


9183 


9227 


9272 


9316 


9361 


9405 






&76 


9450 


9494 


9539 


9583 


9628 


9672 


9717 


9761 


9806 


9850 






977 


9895 


9939 


9983 


..28 


..72 


.117 


.161 


.206 


.250 


.294 1 






978 


990339 


0383 


0428 


0472 


0516 


0561 


0605 


0650 


0694 


0738 1 






979 


0783 


0827 


0871 


0916 


0960 


1004 


1049 


1093 


1137 


1182 I 






980 


1226 


1270 


1315 


1359 


1403 


1448 


1492 


1536 


1580 


1625 j 






981 


1669 


1713 


1758 


1802 


1846 


1890 


1935 


1979 


2023 


3067 1 






982 


2111 


2156 


2200 


2244 


2288 


2333 


2377 


2421 


2465 


2509 1 






983 


2554 


2698 


2642 


: 2686 


2730 


2774 


2819 


2863 


2907 


2951 i 






984 


2995 


3039 


3083 


3127 


3172 


3216 


3260 


3304 


3348 


3392 1 

j 






985 


3436 


3480 


3524 


3568 


3613 


3657 


3701 


3745 


3789 


3833 j 






986 


3877 


3921 


3965 


4009 


4053 


4097 


4141 


4185 


4229 


4273 i 






987 


4317 


4361 


4405 


4449 


4493 


4537 


4581 


4625 


4669 


4713 






988 


4757 


4801 


4845 


4886 


4933 


,4977 


5021 


6065 


5108 


5152 






989 


6196 


6240 


6284 


5328 


5372 


;6416 


5460 


6504 


6547 


6691 






990 


6635 


6679 


5723 


5767 


6811 


5854 


6898 


5942 


6986 


6030 






991 


6074 


6117 


6161 


6205 


6249 


;6293 


6337 


6380 


6424 


6468 






992 


6612 


6655 


6599 


6643 


6687 


6731 


6774 


6818 


6862 


6906 






993 


6949 


6993 


7037 


7080 


7124 


:7168 


7212 


7255 


7299 


7343 






994 


7386 


7430 


7474 


7517 


7661 


7605 


7648 


7692 


7736 


7779 1 

1 






995 


7823 


7867 


7910 


7954 


7998 


8041 


8085 


8129 


8172 


8216 






996 


8259 


8303 


8347 


8390 


8434 


!8477 


8521 


8564 


8608 


8652 






997 


8695 


8739 


8792 


8826 


8869 


8913 


8956 


9000 


9043 


9087 






998 


9131 


9174 


9218 


9261 


9305 


9348 


9392 


9435 


9479 


9622 






999 


9665 


9609 


9662 


9696 


9739 


9,783 


9826 


9870 


9913 

i 


9957 





♦if 





TABLE II. Log. Sines 


Hud Tangents, (0°) Natural Sines. *1 






T" 


Sine. 


D.IO" 


(Jos'iiie. 


in(F 


T«ng. ■ 


D.IO" 


Coiang. 
Infinite. 


N.8ine. 


N. COS. 











0.000000 




10.000000 




0.000000 




00000 


lOOOOO 


60 






1 


6.463726 




000000 




6.463726 




13.536274 


00029 


100000 


59 






2 


764756 




000000 




764756 




235244 


00058 


lOOOOU 


58 






3 


940847 




000000 




940847 




059153 


00037 


100000 


57 






4 


7.085786 




000000 




7.065786 




12.934214 


00116 


100000 


56 






5 


162696 




000000 




162696 




837804 


00145 


100000 


56 






6 


241877 




9.999999 




241878 




758122 


00175 


100000 


54 






7 


308824 




999999 




308825 




691175 


00204 


100000 


63 






8 


366816 




999999 




366817 




633183 


00233 


I 00000 


62 






9 


417968 




999999 




417970 




582030 


00262 


100000 


51 






10 


463725 




999998 




463727 




536273 


00291 


100000 


50 






11 


7.505118 




9.999998 




7.505120 




12.494880 


00320 


99999 


49 






12 


54290S 




999997 




542909 




457091 


00349 


99999 


48 






13 


677668 




999997 




577672 




422328 


00378 


99999 


47 






14 


609853 




999996 




609867 




390143 


00407 


99999 


46 






15 


639816 




999996 




639820 




360180 


00436 


99999 


45 






16 


667845 




999996 




667849 




332151 


00465 


99999 


44 






17 


694173 




999995 




694179 




305821 


00496 


99999 


43 






18 


718997 




999994 




719003 




280997 


00524 


99999 


42 






19 


742477 




999993 




742484 




257516 


00553 


99998 


41 






20 


764754 




999993 




764761 




236239 


00582 


99998 


40 






21 


7.785943" 




9.999992 




7.785951 




12.214049 


00611 


99998 


39 






22 


806146 




999991 




806155 




193845 


00640 


99998 


38 






23 


825451 




999990 




825460 




174540 


00669 


99998 


37 






24 


843934 




999989 




843944 




156056 


00698 


99998 


36 






25 


861663 




999988 




861674 




138326 


00727 


99997 


35 






26 


878695 




999988 




878708 




121292 


00756 


99997 


34 






27 


895085 




999987 




896099 




104901 


00785 


99997 


33 






28 


910879 




999986 




910894 




039106 


00814 


99997 


32 






29 


926119 




999985 




926134 




073866 


00844 


99996 


31 






30 


940842 




999983 




940858 




059142 


00873 


99996 


30 






31 


7.955082 




9.999982 




7.956100 


2298 
2227 
2161 
2098 
2039 
1983 
1930 
1880 
1833 
1787 
1744 
1703 
1664 
1627 
1591 
1557 
1524 
1493 
1463 
1434 
1406 
1379 
1353 
1328 
1304 
1281 
1259 
1238 
1217 


12.044900 


00902 


99996 


29 






32 


968870 


2298 


S99981 


0.2 


968889 


031111 


00931 


99996 


28 






33 


982233 


2227 


999980 


0.2 


982253 


017747 


00900 


99995 


27 






34 


995198 


2161 


999979 


0.2 


996219 


004781 


00989 


99995 


26 






35 


8.007787 


2098 


999977 


0-2 


8.007809 


11.992191 


01018 


99995 


25 






36 


020021 


2039 


999976 


0-2 
0-2 


020045 


979955 


01047 


99995 


24 






37 


031919 


1983 


999975 


031945 


968055 


01076 


99994 


23 






38 


043501 


1930 
1880 
1832 
1787 
1744 
1703 
1664 
1626 
1691 
1567 
1524 
1492 
1462 
1433 
1405 
1379 
1353 
1328 
1304 
1281 
1269 
1237 
1216 


999973 


0-2 
0-2 


043527 


956473 ' 


01105 


99994 


22 






39 


054781 


999972 


054809 


945191 1 
934194 1 


01134 


99994 


21 






40 


065776 


999971 


0*2 


065808 


01164 


99993 


20 






41 


8.076500 


9.999969 


0'2 
0-2 


8.076531 


11.923469 


01193 


99993 


19 






42 


086965 


999968 


086997 


913003 i 


01222 


99993 


18 






43 


097183 


999966 


0'2 


097217 


902783 1 


01251 


99992 


17 






44 


107167 


999964 


0'2 

o;3 


107202 


892797 1 


01280 


99992 


16 






45 


116926 


999963 


116963 


883037 


01309 


99991 


15 






46 


126471 


999961 


3 
0.3 
0.3 


126610 


873490 


01338 


99991 


14 






47 


136810 


999959 


135851 


864149 


01367 


1)9991 


13 






48 


144953 


999958 


144996 


855004 


01396 


99990 


12 






49 


153907 


999956 


0.3 
0.3 


163952 


846048 


01425 


99990 


11 






50 


162681 


999954 


162727 


837273 


01454 


99989 


10 






51 


8.171280 


9,999952 


0.3 
0.3 
0.3 
0.3 
0.3 
0.3 


8.171328 


11.828672 


01483 


99989 


9 






52 


179713 


999950 


179763 


820237 


01513 


99989 


8 






53 


187985 


999948 


188036 


811964 


01542 


99988 


7 






54 


196102 


999946 


196166 


803844 


01571 


99988 


6 






56 


204070 


999944 


204126 


795874' 


01600 


99987 


5 






56 


211895 


999942 


211953 


7&8047 


01629 


99987 


4 






57 


219581 


999940 


0.4 
0.4 


219641 


780359 j 


01658 


99986 


3 






58 


227134 


999938 


227195 


772805 ' 


01687 


9998() 


2 






59 


234557 


999936 


0.4 
0.4 


234621 


765379 


01716 


99985 


1 






60 


241855 


999934 


241921 


758079:, 01746 


99985 








Cosine. 




Sine. 




Coihmg:. 




Tnnar. ' N. cos. 


N. sine- 






89 Degrees. | 





22 



hog. Sines aud TangentB. (l") Natural Sines. TABLE H. 



D.IO' 



Cosine. 



D.IO' 



Tang. 



DIG" Coiang. 



N. sine. N. cos. 



10 

11 

12 

13 

14 

16 

16 

17 

18 

19 

20 

21 

22 

23 

24 

26 

26 

27 

28 

29 

30 

31 8 

32 

33 

34 

36 

36 

37 

38 

39 

40 

41 

42 

43 

44 

45 

46 

47 

48 

49 

60 

61 

62 

53 

54 

66 

56 

57 

68- 

69 

60 



.241856 jj g 

249033 „,, 

256094 I;'' 

26304-2 };°° 

2€9881 {^^ 

276(il4 {{r.'t 

283243 |iX« 

289773 V^l 

290207 \ali 

302546 \lf, 

308794 \Yl^ 

.314954 }^, ' 

321027 aoi 

327016 QOR 

332924 g^j 

338763 q-q 

344604 all 

350181 qo^ 

355783 Qoo 

361316 Qin 

366777 gg" 

.372171 S^o 

377499 S°? 

382762 ^^ 

387962 ^L 

393101 o2^ 

398179 o^ 

403199 007 

408161 ofo 

413068 or^ 

417919 oXX 

.422717 ?^J 

427462 7^2 

432156 7^f 

436800 7^R 

441394 7^0 

445941 III 

450440 740 

454893 7^ 

459301 ^tj 

463665 «on 

.467985 ':" 

472263 ;l-\f 

476498 gXX 

480693 ^^l 

484848 ^«fi 

488963 l^l 

493040 ^', 
497078! ^^5 
501080 i l^i 

506045 i °^^ 

.608974 I °5q 

612867 ! ^11 

516726 ^^^ 

620551 ; lii 

524343 2^^ 

528102 ^^^ 
531828 : fi,^ 

535623 ^}° 

539186 «i.^ 
542819 . ^"^ 



9.999934 
999932 
999929 
999927 
999925 
999922 
999920 
999918 
999916 
999913 
999910 

9.999907 
999905 
999902 



999897 
999894 
999891 
999888 



9.999879 
999876 
999873 
999870 
999867 
999864 
999861 
999858 
999854 
999851 

9.999848 
999844 
999841 
999838 
999834 
999831 
999827 
999823 
999820 
999816 

9.999812 
999809 
999805 
999801 
999797 
999793 
999790 
999786 
999782 
999778 

9.999774 
999769 
999765 
999761 
999757 
999753 
999748 
999744 
999740 
999735 



0.4 
0.4 
0.4 
0.4 
0.4 
0.4 
0.4 



0.5 
0.5 
0.6 
0.5 
0.6 
0.5 
0.5 
0.5 
0-6 
0-6 
0.6 
0.6 
0-6 
0.6 
0.6 
0.6 
0.6 
0.6 
0.6 
0-6 
0-6 
0.6 
0.6 
0-7 
0-7 
0.7 
0.7 
0.7 
0.7 



.241921 
249102 
266166 
263115 
269956 
276691 
283323 
289866 
296292 
302634 



.3160-46 
321122 
327114 
333026 



344610 
350289 
356895 
361430 



.372292 
377622 



8ine. 



388092 
393234 
398316 
403338 
408304 
413213 
418068 

.422869 
427618 
432315 
436962 
441560 
446110 
450613 
455070 
469481 
463849 

.468172 
472454 
476693 
480892 
485050 
489170 
493250 
497298 
501298 
505267 

.509200 
513098 
516961 
620790 
624586 
528349 
532080 
536779 
539447 
6 43084 

Cotarifr- 



1197 
1177 
1168 
1140 
1122 
1105 
1089 
1073 
1057 
1042 
1027 
1013 
999 
985 
972 
959 
946 
934 
922 
911 



879 
867 
857 
847 
837 
828 
818 
809 
800 
791 
783 
774 
766 
758 
750 
743 
736 
728 
720 
713 
707 
700 
693 
686 
680 
674 
668 
661 
655 
650 
644 
638 
633 
627 
622 
616 
611 
606 



11.758079 



743835 
736885 
730044 
723309 
716677 
710144 
703708 
697366 
691116 

11.684954 
678878 
672886 
666975 
661144 
665390 
649711 
644106 
638670 
633105 

11-627708 
622378 
617111 
611908 
606766 
601685 
596662 
591696 
686787 
581932 

11.577131 



567685 
563038 
568440 
553890 
549387 
544930 
540519 
636151 

11.531828 
527546 
523307 
519108 
614950 
610830 
506750 
502707 
498702 
494733 

11.490800 
486902 
483039 
479210 
476414 
471651 
467920 
464221 
460563 
456916 



01742 
01774 
01803 
01832 
01862 
01891 
01920 
01949 
01978 
02007 
02036 
02065 
02094 
02123 
02152 
02181 
02211 
02240 
02269 
02298 
02327 
02356 
02385 
02414 
02443 
02472 
02501 
02630 
02660 
02589 
02618 
02647 
02676 
02705 
02734 
02763 
02792 
02821 
02850 



99985 
99984 
99984 
99983 
99983 
99982 
99982 
99981 



99980 62 



99980 
99979 
99979 
99978 
99977 
99977 
99976 
99976 
99975 
99974 
99974 
99973 
99972 
99972 
99971 
99970 
99969 
99969 
99968 
99967 
99966 



99.9661 30 

99965 

99964 

99963 

99963 

99962 

99961 

99960 

99959 



02879 99959 



99958 
99957 
99956 
99955 
99954 
99953 
99952 
99952 
99951 
99950 
99949 
03228 99948 
03257 99947 
03286 99946 
03316 99945 
03346 99944 
03374 99943 
03403 99942 



02908 

02938 

02967 

02996 

03025 

03054 

03083 

03112 99952 

03141 

03170 

03199 



03432 
03461 
03490 



99941 
99940 
99939 



Tang. 



N. COS. N.8ine 



88 Degrees. 



TABLE II. Log. Sines and Tangents. (2°) Natural Sines. 



23 



8.542819 
54G4'22 
549995 
553539 
557054 
560540 
563999 
567431 
570836 
574214 
577666 

8.580892 
584193 
587469 
590721 
593948 
597152 
600332 
603489 
606623 
609734 

8.612823 
615891 
618937 
621962 
624965 
627948 
630911 
633854 
636776 
639680 

8.642563 
645428 
648274 
651102 
653911 
656702 
659475 
662230 
664968 
667689 

8.670393 
673080 
675751 
678405 
681043 
683665 
686272 



691438 
693998 
8.696543 
699073 
701589 
704090 
706577 
709049 
711507 
713952 
716383 
_718800 
Cosme. 



D. 10" 



600 
595 
691 
586 
581 
576 
572 
567 
563 
559 
554 
550 
546 
542 
538 
634 
530 
526 
522 
519 
515 
511 
508 
504 
501 
497 
494 
490 
487 
484 
481 
477 
474 
471 
468 
465 
462 
469 
456 
453 
451 
448 
445 
442 
440 
437 
434 
432 
429 
427 
424 
422 
419 
417 
414 
412 
410 
407 
405 
403 



Cosine. 

1.999735 
999731 
999726 
999722 
999717 
999713 
999708 
999704 



999694 



999685 
999680 
999675 
999670 
999665 
999660 
999655 
999550 
999645 
999640 
999635 
999629 
999324 
999619 
999614 
999608 
999603 
999597 
999592 
999586 
999581 
999575 
999570 
999564 
999558 
999553 
999547 
999541 
999535 
999529 
,999524 
999518 
999512 
999506 
999500 
999493 
999487 
999481 
999475 
999469 
,999463 
999456 
999450 
999443 
999437 
999431 
999424 
999418 
999411 
999404 



D. 10" 



Tang. 



0.7 
0.7 
0.7 
0-8 
0-8 
0-8 
0.8 
0.8 
0.8 
0.8 
0.8 
0.8 
0.8 
0.8 
0.8 
0.8 
0.8 
0.8 
0.8 
0.8 
0.9 
0.9 
0.9 
0.9 
0.9 
0.9 
0-9 
0.9 
0-9 
0.9 
0-9 
0.9 
0.9 
0.9 
0.9 
1.0 
1-0 
1.0 
1.0 
1-0 
1.0 
1.0 
1.0 
1-0 
1-0 
1.0 
1.0 
1.0 
1.0 
1.0 
1.0 



1.543084 
546691 
550268 
553817 
557335 
560828 
564291 
567727 
571137 
574520 
577877 

.581208 
584514 
587795 
591051 
594283 
597492 
600677 
603839 
606978 
610094 

.613189 
616262 
619313 
622343 
625352 
628340 
631308 
634256 
637184 
640093 

,642982 
645853 
648704 
651537 
664352 
657149 
659928 
662689 
665433 
668160 

.670870 
673563 
676239 
678900 
681544 
684172 
6-6784 
689381 
691963 
694529 

.697081 
699617 
702139 
704246 
707140 
709618 
702083 
714534 
716972 
719396 

Coians:. 



D. 10"i Coiang. |{N. sine. N. cos. 



602 
593 
591 
587 
582 
577 
573 
568 
564 
559 
555 
551 
547 
543 
539 
535 
631 
527 
523 
519 
516 
512 
508 
505 
501 
498 
495 
491 
488 
485 
482 
478 
475 
472 
469 
466 
463 
460 
457 
454 
453 
449 
446 
443 
442 
438 
485 
433 
430 
428 
425 
423 
420 
418 
415 
413 
411 
408 
406 
404 



11.456916 
453309 
449732 
446183 
442664 
439172 
435709 
432273 
428863 
425480 
422123 

11.418792 
415486 
412205 
408949 
405717 
402508 
399323 
396161 
393022 
389906 

11.386811 
383738 
380687 
377657 
374648 
371660 
368692 
365744 
362816 
359907 

11.357018 
354147 
351296 
348463 
345648 
342851 
340072 
337311 
334567 
331840 

11.329130 
326437 
323761 
321100 
318456 
315828 
313216 
310619 
308037 
305471 

11.302919 
300383 
297861 
296354 
292860 
290382 
287917 
285465 
283028 
280604 



03490 



03519 99938 



03548 



03577 99936 



03606 
1 03635 
1 03664 
■03693 
103723 
03752 
1 03781 
03810 
03839 
03868 
03897 
03926 
03955 
03984 
04013 
04042 
104071 
104100 
103129 
{04159 
04188 
04217 
04246 
04275 
04304 
04333 
04362 
04391 
04420 
04449 
04478 
04507 
04536 
04565 
04594 
04623 
04653 99892 
04682 99890 
0471199889 
04740 99888 
04769 99886 
04798 99885 



Tans 



04827 



04856 9988 



04885 
04914 
04943 
04972 
05001 
05030 
05059 
05088 
05117 
05146 
05176 
06205 
05234 



99939 



99937 



99935 
99934 
99933 
99932 
99931 
99930 
99929 
99927 
99926 
99926 
99924 
99923 
99922 
99921 
99919 
99918 
99917 
99916 
99915 
99913 
99912 
99911 
99910 
99909 
99907 
99906 
99906 
99904 
99902 
99901 
99900 



99897 
99896 
99894 



99883 



99881 
99879 
99878 
99876 
99876 
99873 
99872 
99870 
99869 
9y867 



99864 
99863 



N. COS. N.8ine 



87 Degrees. 



24 



iOg. Sines and Tangciiis. (3°; Natural Sines, TABLE II. 





2 

3 

4 

5 

6 

7 

8 

9 
10 
11 
•12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
28 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
60 
51 8 
52 
53 
54 
55 
56 
57 
58 
59 
60 



Cosine. 



1 

1 

1. 

1 

1 

1 

1.2 

1.2 

1.2 

1.2 

1.2 

1.2 

1.2 

1.2 

1.2 

1.2 

1.2 



Sine. |U. W Cosine. D71o 

.71880a 
721204 
723595 
725972 
728337 
730{>88 
733027 
735354 
737667 
739969 
742259 
.744536 
746802 
749055 
751297 
753528 
755747 
757955 
760151 
762337 
764511 
.766675 
768828 
770970 
773101 
776223 
777333 
779434 
781524 
783605 
785675 
.787736 
789787 
791828 
793859 
795881 
797894 
799897 
801892 
803876 
805852 
,807819 
809777 
811726 
813667 
815599 
817522 
819436 
821343 
823240 
825130 
,827011 
828884 
830749 
832607 
834456 
836297 
838130 
839956 
841774 
843586 



401 
398 
396 
394 
392 
390 
388 
386 
384 
382 
380 
378 
376 
374 
372 
370 
368 
366 
364 
362 
361 
359 
357 
355 
353 
352 
350 
348 
347 
345 
343 
342 
340 
339 
337 
336 
334 
332 
331 
329 



325 
323 
322 
320 
319 
318 
316 
316 
313 
312 
311 
309 
308 
307 
306 
304 
303 
302 



.999404 
999398 
999391 
999384 
999378 
999371 
999364 
999357 
999350 
999343 
999336 
.999329 
999322 
999315 
999308 
999301 
999294 
999286 
999279 
999272 
999265 
.999257 
999250 
999242 
999235 
999227 
999220 
999212 
999205 
999197 
999189 
.999181 
999174 
999166 
999158 
999150 
999142 
999134 
999126 
999118 
999110 
.999102 
999094 
999086 
999077 



999061 
999053 
999044 
999036 
999027 
.999019 
999010 
999002 
998993 
998984 
998976 
998967 



998950 
998941 



1.2 
1.3 
1.3 
1.3 
1.3 
1.3 
1.3 
1.3 
1.3 
1.3 
13 
1.3 
1.3 
1.3 
1.3 
1.3 
1.3 
1.3 
1.3 
1.3 



1.4 
1.4 
1.4 
1.4 
1.4 
1.4 
1.4 
1.4 
1.4 
1.4 
1.5 
1.5 
1.5 



lanj; 



.719396 
721806 
724204 
726588 
728959 
731317 
733663 
735996 
738317 
740326 
742922 
,745207 
747479 
749740 
751989 
754227 
766453 
758668 
760872 
763065 
765246 
767417 
769578 
771727 
773866 
775995 
778114 
780222 
782320 
784408 
786486 

8.788554 
790613 
79-2662 
794701 
793731 
798752 
80'J763 
802765 
804858 
80 )742 
808717 
810683 
81-2641 
814589 
816529 
818461 
820384 
822298 
824205 
826103 

8.827992 
829874 
831748 
833613 
835471 
837321 
839163 
840998 
842826 
844644 



Cotanp. 



l». !(/ 



402 
399 
397 
395 
393 
391 
389 
387 
385 
383 
381 
379 
377 
375 
373 
371 
3o9 
367 
365 
364 
362 
360 
358 
356 
355 
353 
351 
350 
348 
346 
345 
343 
341 
340 
338 
337 
335 
334 
332 
331 
329 
328 
326 
325 
323 
322 
320 
319 
318 
316 
316 
314 
312 
311 
310 
308 
307 
306 
304 
303 



Cotang. |(N. sine. 



11.2806041 
278194] 
276796 
273412 
271041 I 
268683 i 
266337 I 
264004 I 
261683 I 
259374; 1 05496 
257078 I 05524 



05234 
05263 
05292 
06321 
05350 
05379 
05408 
06437 
05466 



11.254793 
252521 
250260 
248011 
245773 
243547 
241332 
239128 
236935 
234754 

11.232583 
230422 
228273 
226134 
224005 
221886 
219778 
217680 
215592 
213514 

11.211446 
209387 
207338 
205299 
203269 
201248 
199237 
197235 
195242 
193258 

11.191283 
189317 
187359 
185411 
183471 
181639 
179616 
177702 
175796 
173897 

11.172008 
170126 
168252 
166387 
164529 
162679 
160837 
159002 
167175 
156366 



05553 

05582 

05611 

05640 

05669 

05698 

0572 

05766 

05785 

05814 

05844 

05873 

05902 

05931 

05960 



Tang. 



06018 
06047 
06076 
06105 
06134 
06163 
06192 
06221 
06250 
06279 
06308 
06337 
06366 
06395 
08424 
06463 
06482 
06511 
06540 
06569 
06598 
06627 
06656 
06685 
08714 
06743 
06773 
06802 
06831 
06860 
06889 
06918 
06947 
06976 
'iIn. cos 



N.cos. 



99863 

99861 

99860 

99858 

99857 

99855 

99854 

99852 

99851 

99849 

99847 

99846 

99844 

99842 

99841 

99839 

99838 

99836 

99834 

99833 

99831 

99829 

9982' 

99826 

99824 

99822 

99821 

99819 

99817 

99815 

99813 

99812 

99810 

99808 

99806 

99804 

99803 

99801 

99799 

99797 

99795 

99793 

99792 

99790 

99788 

99786 

99784 

99782 

99780 

99778 

99776 

99774 

99772 

99770 

99768 

99766 

99764 

99762 

99760 

99758 

99766 



.V.Bine 



86 Degrees. 



TABLE 11. 



.og. Sines and Tangents. (4°) Natural Sines. 



25 



D. W 

300 

299 
298 
297 
295 
294 
293 
292 
291 
2;)0 
288 
287 
286 
285 
284 
283 
282 
281 
279 
279 
277 
276 
275 
274 
273 
272 
271 
270 
269 
268 
267 
266 
265 
264 
263 
262 
261 
260 
259 
258 
257 
257 
256 
255 
254 
253 
252 
251 
250 
249 
249 
248 
247 
246 
245 
2'14 
243 
243 
242 
241 



Cosine. 

9.998941 
998932 
998923 
998914 
998905 
998896 
998887 
998878 
998869 
998860 
998851 

9.998841 
998832 
998823 
998813 
998804 
998795 
998785 
998776 
998766 
998757 

9.998747 
998738 
998728 
998718 
998708 



8.843586 
845387 
847183 
848971 
850751 
852525 
854291 
856049 
857801 
859546 
861283 

8.863014 
864738 
866455 
868165 
869868 
871565 
873255 
874938 
876615 
878285 

8.879949 
881607 
883258 
884S03 
886542 
888174 
889801 
891421 
893035 
894643 
.896246 
897842 
899432 
901017 
902596 
904169 
905736 
907297 
908853 
910404 

8.911949 
913488 
915022 
916550 
918073 
919591 
921103 
922610 
924112 
926609 

8.927100 
928587 
930068' 
931544 
933015 
934481 
935942 
937398 
938850 
940296 
Cosine. 



998689 
998679 
998669 
998659 
9.998649 
998639 
998629 
998619 
998609 



998589 
998578 
998568 
998558 
9.998648 
998537 
998627 
998616 
998506 
998495 
998485 
998474 
998464 
998453 
998442 
998431 
998421 
998410 
998399 



998377 
998366 
998355 
998344 



Sine. 



D. 10" 



Tang, 



D. 1 0" Cota ng. l|N. sine. N. cos 



884530 
886185 
887833 
889476 
891112 
892742 
894366 
895984 

.897596 
899203 
900803 
902398 
903987 
906570 
907147 
908719 
910285 
911846 

.913401 
914951 
916495 
918034 
919668 
921096 
922619 
924136 
925649 
927156 

.928658 
930155 
931647 
933134 
934616 
936093 
937565 



940494 
941952 
Cotang. 



302 
801 
299 
298 
297 
293 
295 
293 
292 
291 
290 
289 
288 
287 
285 
284 
283 
282 
281 
280 
279 
278 
277 
276 
275 
274 
278 
272 
271 
270 
269 
268 
267 
266 
265 
264 
263 
262 
261 
260 
259 
258 
267 
256 
256 
255 
264 
263 
252 
251 
250 
249 
249 
248 
247 
246 
245 
244 
244 
243 



11 



11.166366 
153646 
151740 
149943 
148164 
146372 
144697 
142829 
141068 
139314 
137667 
, 135827 
134094 
132368 
130649 
128936 
127230 
125531 
123838 
122151 
120471 
118798 
117131 
116470 
113815 
112167 
110524 
108888 
107258 
106634 
10-1016 
102404 
100797 
099197 
097602 
096013 
094430 
092863 
091281 
089715 
088154 

11.086599 
085049 
083606 
081966 
080432 
078904 
077381 
075864 
074361 
072844 

11.071342 



06976 
07005 
07034 
07063 
07092 
07121 
07150 
07179 
07208 
07237 
07266 
07295 
07324 
07363 
07382 
07411 
07440 
07469 
07498 
07527 
07556 
07586 



99756 
99754 
99752 
99750 
99748 
99746 
99744 
99742 
99740 
99738 
99736 
99734 
99731 
99729 
99727 
99725 
99723 
99721 
99719 
99716 
99714 
99712 



07614 99710 



1 07643 
07672 
07701 
07730 
07759 
07788 
07817 
07846 
07875 
07904 
07933 
07962 
07991 
08020 
08049 
08078 
08107 
08136 
08165 



08223 
08252 
08281 
08310 



108368 
08397 
08426 
08455 
1 08484 
108513 
1 08542 
i 08671 
108600 



068863 

066866 

065384 

063907 

06243511 08629 |9y62 

060968 

059606 

068048 



Tang. 



08658 
08687 



08716 99619 



N. co,«. -N.eine 



99708 
99705 
99703 
99701 
99699 
99696 
99694 
99692 
99689 
99687 
99685 
99683 
99680 
99678 
99676 
99673 
d9671 
99668 
99666 



08194 99664 



99661 
99659 
99657 
99654 
99652 
99649 
99647 
99644 
99642 
99639 
99637 
99635 
99632 
99630 



99625 
99622 



85 Degrees. 



26 



hog. Sines and Tangents. (5°) Natural Sines. TABLE II. 





1 

2 

3 

4 

6 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 

21 

22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44 

45 

46 

47 

48 

49 

56 

5] 

62 

63 

54 

65 

56 

57 

58 

69 

60 



Sine. 

8.940296 
941738 
943174 
944608 
946034 
947456 
948874 
950287 
95169S 
953100 
954499 
955894 
957284 
958670 
960052 
901429 
962801 
984170 
965534 
966893 
968249 

8.969600 
970947 
972289 
973628 
974962 
976293 
977619 
978941 
980259 
981573 

8.982883 
984189 
985491 
986789 



D. 10" 



989374 
990660 
991943 
993222 
994497 

B. 995768 
997036 
998299 
999560 

9.000816 
002069 
003318 
004563 
005805 
007044 

9.008278 
009510 
010737 
011962 
013182 
014400 
015613 
016824 
018031 
019235 
Cosine. 



240 

239 

239 

238 

237 

236 

235 

235 

234 

233 

232 

232 

231 

230 

229 

229 

228 

227 

227 

226 

225 

224 

224 

223 

222 

222 

221 

220 

220 

219 

218 

218 

217 

216 

216 

215 

214 

214 

213 

212 

212 

211 

211 

210 

209 

209 

208 

208 

207 

206 

206 

203 

205 

204 

203 

203 

202 

202 

201 

201 



Cosine. 



(9.998344 
998333 
998322 
998311 
998300 
998289 
998277 
998266 
998255 
998243 
998232 
9.998220 
998209 
998197 
998186 
998174 
998163 
998151 
998139 
998128 
998116 
9.998104 
998092 
998080 
998068 
998056 
998044 
998032 
998020 
998008 
997996 
9.997984 
997972 
997959 
997947 
997935 
997922 
997910 
997897 
997885 
997872 
,997860 
997847 
997835 
997822 
997809 
997797 
997784 
997771 
997758 
997745 
997732 
997719 
997706 
997693 
997680 
997667 
997654 
997641 
997628 
997614 



D. 10"j Tang. 



1.9 

1.9 

1.9 

1.9 

1.9 

1.9 

1.9 

1.9 

1.9 

1.9 

1.9 

1.9 

1.9 

1.9 

1.9 

1.9 

1.9 

1.9 

2.0 

2.0 

2,0 

2.0 

2.0 

2.0 

2.0 

2.0 

2.0 

2.0 

2.0 

2.0 

2.0 

2.0 

2.0 

2.0 

2.0 

2,1 

2.1 

2,1 

2.1 

2,1 

2.1 

2.1 

2.1 

2.1 

2.1 

2.1 

2.1 

2.1 

2.1 

2,1 

2.1 

2.1 

2.1 

2.1 

2.2 

2.2 

2.2 

2.2 

2.2 

2.2 



Sine. 



941952 
943404 
944852 
946295 
947734 
949168 
950597 
952021 
953441 
954856 
956267 
8.957674 
959075 
960473 
961866 
963255 
964639 
906019 
967394 
968766 
970133 
.971496 
972855 
974209 
975560 
976906 
978248 
979586, 
980921 
982251 
983577 
8.984899 
986217 
987532 
988842 
990149 
991451 
992750 
994045 
995337 
996624 
.997908 
999188 
,000465 
001738 
003007 
004272 
005534 
006792 
008047 
009298 
010546 
011790 
013031 
014268 
015502 
016732 
017959 
019183 
020403 
021620 



D. 10"! Cotang. 



Co tang. 



242 

241 

240 

240 

239 

238 

237 

237 

236 

235 

234 

234 

233 

232 

231 

231 

230 

229 

229 

228 

227 

226 

226 

225 

224 

224 

223 

222 

222 

221 

220 

220 

219 

218 

218 

217 

210 

216 

215 

215 

214 

213 

213 

212 

211 

211 

210 

210 

209 

208 

208 

207 

207 

206 

206 

205 

204 

204 

203 

203 



11.058048 
056596 
055148 
053705 i 
052266 I 
050832 I 
049403 i 
047979 
046559 I 
0451441 
043733 I 
11.042326! 
040925 i 
039527 
038134 
036745 
035861 
033981 
032606 
031234 
029867 
11.028504 
027145 
025791 
024440 
023094 
021752 
020414 
019079 
017749 
016423 
11.015101 
013783 
012468 
011158 
009851 
008549 
007250 
005955 
004663 
003376 
11.002092 
000812 
10.999535 
998262 I 
996993 I 
996728 I 
994466 I 
993208 I 



N. sine 



08716 
08745 
08774 
08803 
08831 
08860 



08918 
08947 
08976 
09005 
09034 
09063 
09092 
09121 
09150 
09179 
09208 
09237 
09266 
09295 
09324 
09353 



99619 60 



99617 
99614 
99612 
99609 
99607 
99604 
99602 
99599 
99596 
99594 
99591 
99588 
99586 
99583 
99580 
99578 
99575 
99572 
99570 
99567 
99564 
99562 



09382|99559 
0941199556 
09440J99553 
09469199551 
09498199548 
09527199545 
09556199542 
09585i99540 
99537 
99534 
99531 
99528 
99526 
99523 
99520 
99517 



09614 
09642 
09671 
09700 
09729 
06758 
09787 
09816 

09845|99514 

09874199511 

09903199508 

09932 j9950i 

09961 J99503 

09990199500 

10019J9949 

10048J99494 

10077199491 

10106199488 

991953 1110135 99485 

990702 1 110164199482 

10.989454 



988210 

686969 

986732 

984498 

983268 

983041 

980817 

979597 j 

978380 I 



i 10192 99479 
11022199470 
110250,99473 
110279,99470 
10308 99467 



Tang. 



10337 
10366 
10395 
10424 
10463 



99464 
99461 
99458 
99455 
99452 
N. COS. Njsine. 



59 
58 
57 
56 
55 
54 
53 
52 
51 
50 
49 
48 
47 
46 
45 
44 
43 
42 
41 
40 
39 
38 
37 
^6 
26 
34 
33 
32 
31 
30 
29 
28 
27 
26 
25 
24 
23 
22 
21 
20 
19 
18 
17 
16 
15 
14 
13 
12 
11 
10 
9 
8 
7 
6 
6 
4 



8\ Degrees. 



TABLE IT. 



Log. Sines and Tangents. (6"') Natural Sines. 



27 



60 



Sine. 

9.019235 
020436 
021632 
022825 
024016 
025203 
026386 
027567 
028744 
029918 
031089 

9.032257 
033421 
034582 
035741 
036896 
038048 
039197 
040342 
041485 
042625 
.043762 
044895 
046026 
047154 
048279 
049400 
050619 
051635 
052749 
053859 

9.054966 
056071 
057172 
058271 
059367 
060460 
061561 
062639 
063724 
064806 

9.066886 
066962 
068036 
069107 
070176 
071242 
072306 
073366 
074424 
075480 
076633 
077583 
078631 
079670 
080719 
081759 
082797 
083832 
084864 
086894 



I>. 10"[ Cosine. ID. 10' 



Cosine. 



200 
199 
199 
198 
198 
197 
197 
196 
196 
195 
196 
194 
194 
193 
192 
192 
191 
191 
19a 
190 
189 
189 
180 
188 
187 
187 
186 
186 
185 
185 
184 
184 
184 
183 
183 
182 
182 
181 
J81 
180 
180 
179 
179 
179 
178 
178 
177 
177 
176 
176 
175 
176 
176 
174 
174 
173 
173 
172 
172 
172 



.997614 
997601 
997588 
997574 
997661 
997647 
997534 
997520 
997507 
997493 
997480 

.997466 
997452 
997439 
997425 
997411 
997397 
997388 
997369 
997355 
997341 

.997327 
997313 
997299 
997285 
997271 
997267 
997242 
997228 
997214 
997199 

.997185 
997170 
997156 
997141 
997127 
997112 
997098 
997083 
997068 
997053 

.997039 
997024 
997009 
996994 
996979 
996964 
996949 
996934 
996919 
996904 

.996889 
996874 
996858 
996843 
996828 
996812 
996797 
996782 
996766 
996761 



Sine 



I 2.2 
I 2.2 
2.2 
2.2 
2.2 
I 2.2 
2.3 
I 2.3 
! 2.3 
2-3 
2-3 
2.3 
2-3 
2.3 
2.3 
2.3 
2-3 
2-3 
2-3 
2-3 
2.3 
2.4 
2-4 
2.4 
2.4 
2-4 
2.4 
2.4 
2.4 
2-4 



2 

2 

2 

2 

2 

2 

2 

2 

2.5 

2.5 

25 

25 

2-5 

2.6 

2.6 

2-5 

2-6 

2.5 

2.6 

2.6 

2.5 

2.6 

2.6 

2.6 

2.5 

2.5 

2.6 

2 6 

2.6 

2.6 



Tang. ,J>. W' 



9.021620 
022834 
024044 
025251 
026455 
027655 
028852 
030046 
031237 
032425 
033609 

9,034791 
035969 
037144 
038316 
039485 
040651 
041813 
042973 
044130 
045284 

9.046434 
047682 
04872/ 
049869 
051008 
052144 
053277 
054407 
056635 
056659 

9.057781 
058900 
060016 
061130 
062240 
063348 
064453 
066656 
066655 
067762 

9.068846 
069038 
071027 
072113 
073197 
074278 
075356 
076432 
077505 
078576 

9.079644 
080710 
081773 
082833 
083891 
084947 
086000 
087050 



202 
202 
201 
201 
200 
199 
199 
198 
198 
197 
197 
196 
196 
195 
195 
194 
194 
193 
193 
192 
192 
191 
191 
190 
190 
189 
189 
188 
188 
187 
187 
186 
186 
185 
186 
185 
184 
184 
183 
183 
182 
182 
181 
181 
181 
180 
180 
179 
179 
178 
178 
178 
177 
177 
176 
176 
175 
175 
176 
174 



99354 
99361 
99347 
99344 
99341 
99337 
99334 
99331 
99327 
99324 
99320 
99317 
99314 
99310 
99307 
99303 
99300 
99297 
99293 
99290 
99286 
99283 
99279 
99276 
99272 
99269 
992G5 
99262 
99258 
y9256 
I Tang. Il N. cos. N.sine. 



0453 
0482 
0511 
0540 
0569 
0597 
0626 
0655 
0684 
0713 
0742 
0771 
0800 
0829 
0858 
0887 
0916 
0945 
0973 
1002 
1031 
1060 
1089 
1118 
1147 
1176 
1205 
1234 
1263 
1291 



99452 
99449 
99446 
99443 
99440 
99437 
99434 
99431 
99428 
99424 
99421 
99418 
99415 
99412 
99409 
99406 
99402 
99399 
99396 
99393 
99390 



Cotang. t N. sine. N. ccs. 

10.978380! 

977166 I 

975956 I 

974749 I 

973645 I 

972345 j 

971148 i 

969954 

968763 I 

967676 i 

966391 I 
10.965209 

964031 

962856 

961684 

960516 

959349 

958187 

957027 

955870 

954716 
10.963566 

952418 

951273 

950131 

948992 

947856 

946723 

945693 

944465 

943341 
10.942219 

941100 

939984 

938870 

937760 

936652 

935647 

934444 

933345 I 

932248 
10.9311541 

930062 

928973 

927887 

926803 

926722 

924644 

923668 

922496 

921424 
10.920356 

91929a 

»18227 

917167 

916109 

915053 

914000 

912950 

911902 jl 

910856 



99383 
99380 
99377 
99374 
99370 
99367 
99364 
99360 
1320'99357 



1349 
1378 
1407 
1436 
1465 
1494 
1523 
1552 
1580 
1609 
1638 
1667 
1696 
1726 
1754 
1783 
1812 
1840 
1869 
1898 
1927 
1956 
1985 
2014 
2043 
2071 
2100 
2129 
2168 
2187 



28 



Log. Sines and Tangents. (7°) Natural Sines. TABLE II. 




1 
2 
3 
4 
5 
6 
7 
8 
9 
10 
11 
12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
60 
61 
62 
53 
54 
55 
56 
57 
58 
59 
60 



D. Ml Cosine 



9.085894 
086922 
087947 
088970 
089990 
091008 
092024 
093037 
094047 
09505G 
096062 
9.097065 
098036 
099065 
100052 
101056 
102048 
103037 
104025 
105010 
105992 
9.106973 
107951 
108927 
109901 
110873 
111842 
112809 
113774 
114737 
115698 
.116656 
117613 
118567 
119519 
120469 
121417 
122362 
123306 
124248 
125187 
9.126126 
127060 
127993 
128925 
129864 
130781 
131706 
132630 
133551 
134470 
9.135387 
136303 
137216 
138128 
139037 
139944 
140860 
141754 
142655 
143565 



Cosine. 



171 

171 

170 

170 

170 

169 

1&9 

168 

168 

168 

167 

167 

166 

166 

166 

165 

165 

164 

164 

164 

163 

163 

163 

162 

162 

162 

161 

161 

160 

160 

160 

159 

159 

159 

158 

158 

158 

157 

157 

167 

166 

156 

156 

155 

156 

154 

154 

154 

153 

163 

163 

152 

152 

152 

152 

151 

151 

151 

150 



996751 
996735 
996720 
996704 
996688 
996673 
996657 
996641 
996625 
996610 
996594 
9.996578 
9965G2 
996546 
996530 
996514 
996498 
996482 
996465 
996449 
996433 
9.996417 
996400 
996384 
996368 
996361 
996335 
996318 
996302 
996286 
996269 
9.996252 
996236 
996219 
996202 
996186 
996168 
996151 
996134 
996117 
996100 
.996083 
996066 
996049 
996032 
996015 
995998 
996980 
995963 
995946 
995928 
9.995911 
995894 
995876 
995869 
996841 
995823 
996806 
995788 
995771 
996753 



2.6 
2.6 
2.6 
2.6 
2.6 



6 

6 

6 

6 

6 

7 

7 

7 

2.7 

2.7 

2.7 

2.7 

2.7 

2.7 

2.7 

2.7 

2.7 

2.7 

2.7 

2.7 

2.7 

2.7 

2.8 

2.8 

2.8 



Sine. 



2.8 
2.8 
2.8 
2.8 
2.8 
2.8 
2.8 
2.8 
2.8 
2.8 
2.9 
2.9 
2.9 
2.9 
2.9 
2.9 
2.9 
2.9 
2.9 
2.9 
2.9 
2.9 
2.9 
2.9 
2.9 
2.9 
2.9 
2.9 
2.9 



Taiig: 



mrw 



9.089144 
090187 
091228 
092266 
093302 
094336 
095367 
096395 
097422 
098446 
099468 
9.100487 
101504 
102519 
103532 
104542 
106550 
106556 
107559 
108560 
109559 
9.110556 
111551 
112643 
113533 
114521 
115507 
116491 
117472 
118462 
119429 
9.120404 
121377 
122348 
123317 
124284 
126249 
126211 
127172 
128130 
129087 
1.130041 
130994 
131944 
132893 
133839 
134784 
136726 
136667 
137605 
138642 
.139476 
140409 
141340 
142269 
143196 
144121 
145044 
146966 
146885 
147803 
Cotang. 



174 

173 

173 

173 

172 

172 

171 

171 

171 

170 

170 

169 

169 

169 

168 

168 

168 

167 

167 

166 

166 

166 

165 

165 

165 

164 

164 

164 

163 

163 

162 

162 

162 

161 

161 

161 

160 

160 

160 

159 

159 

159 

168 

158 

168 

167 

157 

157 

156 

155 

156 

156 

165 

156 

154 

154 

154 

153 

153 

163 



Ootang. I'N. sine. N. con. 



10.910856 
909813 
908772 
907734 ! 
906698 
905664 i 
9046331 
903605 j 
902578 j 
901554 I 
900532 I 

10.899513 i 
898496 ! 
897481 I 
896468 
895458 I 
8944501 
893444 j 
892441 



12187 
12216 
12245 
12274 
12302 
12331 
12360 
12389 
12418 
12447 
12476 
12504 



99256 
99251 
99248 
99244 
99240 
99237 
99233 
99230 
99226 
99222 
99219 
99215 



12591 
12620 



12678 



12706 99189 



8914401 1 12735 
890441 1 112764 
10.889444 !j 12793 99178 
88844911 12822 99175 
887457 I i 12851 
886467 
885479 

884493 II 12937199160 
883509 ij 12966 



12633 99211 
12562 99208 



99204 
99200 



12649 99197 



99193 



99186 
99182 



99171 



12880 99167 
12905 99163 



882528 
881548 
880571 
10.879596 
878623 
877652 
876683 



99156 
! 12995 99152 



13024 
13053 
13081 



99148 
99144 
99141 



13110 99137 
13139 99133 



13168 99129 
875716 j 13197 99125 
874751 '113226 99122 
873789 i! 13264)99118 
872828 1 1 13283199114 
871870 :ll3G12 99110 
870913 |! 13341 99106 
10.86995911 13370 99102 
99098 
99094 
867107 I i 13456199091 
866161 1113485 



869.006 jilSG 99 

868056 113427 



865216! 
864274 
8633c!3 



13514 



13543 99079 
13572 99075 



862396 
861458 
10.860524 
859591 
868()60 
857731 
856804 
855879 
864956 

864034 j 13860 
8531 15 ''.13889 
862197 ii 13917 



13773 
13802 



Tang. 



?9087 
99083 



13600 99071 
13629199067 
13658 99063 
13687 99059 
1371C 99055 
13744,99051 



99047 
)9043 



13831 99039 
99035 
39031 
M9027 



N. cos. N.eine. 



82 Degrees. 



Log. Sines and Tanj^ents. (8°) Natural Sinca. 



29 



Bino. 



9.143555 
144453 
145349 
146243 
147136 
148026 
148915 
149802 
150686 
151569 
152451 

9.153330 
154208 
155083 
155957 
156830 
157700 
158569 
159435 
160301 
161164 

9.162025 
162885 
163743 
164600 
165454 
166307 
167159 
168008 
168856 
169702 

9.170547 
171389 
172230 
173070 
173908 
174744 
175578 
176411 
177242 
178072 

9.178900 
179726 
180551 
181374 
182196 
183016 
183834 
184651 
185466 
186280 

9.187092 
187903 
188712 
189519 
190325 
191130 
191933 
192734 
193534 
194332 
I Cosine. 



D. 10' 



150 

149 

149 

149 

148 

148 

148 

147 

147 

147 

147 

146 

146 

146 

145 

145 

145 

144 

144 

144 

144 

143 

143 

143 

142 

142 

142 

142 

141 

141 

141 

140 

140 

140 

140 

139 

139 

139 

139 

138 

138 

138 

137 

137 

137 

137 

136 

136 

136 

136 

135 

135 

135 

135 

134 

134 

134 

134 

133 

133 



Cosine. 



9.996753 
995735 
995717 
995699 
995681 
995664 
995646 
995628 
996610 
995591 
995573 
9.995555 
995537 
995519 
995501 
995482 
995464 
995446 
995427 
995409 
995390 
9.995372 
995353 
995334 
995316 
996297 
995278 
995260 
995241 
996222 
995203 
995184 
995165 
995146 
995127 
995108 
995089 
995070 
995051 
995032 
995013 
9.994993 
994974 
994955 
994935 
994916 
994896 
994877 
994867 
994838 
994818 
.994798 
994779 
994759 
994739 
994719 
994700 
994680 
994660 
994640 
994620 
Sine. 



D. 10" 



3.0 
3.0 
3.0 
3.0 
3.0 
3.0 
3.0 
3.0 
3.0 
3.0 
3.0 
3.0 
3.0 
3.0 
3.1 
3.1 
3.1 
3.1 
3.1 
3.1 
3.1 
3.1 
3.1 
3.1 
3.1 
3.1 
3.1 
3.1 
3.2 
3.2 
3.2 
3.2 
3.2 
8.2 
3.2 
3.2 
3.2 
3.2 
3.2 
3.2 
3.2 
3.2 
3,2 
3.2 
3.2 
3.3 
3.3 
3.3 
3.3 
3.3 



3.3 
3.3 
3.3 
3.3 
3.3 
3.3 
3.3 
3.3 
3.3 
3.3 



Tang. 



9.147803 

148718 
149632 
150644 
151464 
152363 
153269 
154174 
155077 
155978 
156877 
9.167775 
158671 
169666 
160467 
161347 
162236 
163123 
164008 
164892 
165774 
9.166654 
167532 
168409 
169284 
170157 
171029 
171899 
172767 
173634 
174499 

. 175362 
176224 
177084 
177942 
178799 
179655 
18060S 
181360 
182211 
183069 

. 183907 
184752 
185597 
186439 
187280 
188120 
188968 
189794 
190629 
191462 

. 192294 
193124 
193953 
194780 
196606 
196430 
197253 
198074 
198894 
199713 



D. 10' Cotang. |N. sine. N. cos, 



153 

152 
152 
152 
151 
151 
151 
160 
160 
150 
150 
149 
149 
149 
148 
148 
148 
148 
147 
147 
147 
146 
146 
146 
145 
145 
146 
145 
144 
144 
144 
144 
143 
143 
143 
142 
142 
142 
142 
141 
141 
141 
141 
140 
140 
140 
140 
139 
139 
139 
139 
138 
138 
138 
138 
137 
137 
137 
137 
136 



Co tang. 



10.852197 
851282 
850368 
849456 
848546 
847637 
846731 
845826 
844923 
844022 
843123 

10.842225 
841329 
840435 
839543 
838653 
837764 
836877 
835992 
836108 
834226 



13917 

13946 

13975 

14004 

14033 

14061 

14090 

14119 

14148 

14177 

14205 

14234 

14263 

14292 

14320 

14349 

14378 

1440 

14436 

14464 

14493 



10.8333461 114522 



832468 
831691 
830716 
829843 
828971 
828101 
827233 
826366 
825501 

10.824638 
823776 
822916 
822058 
821201 
820345 
819492 
818640 
817789 
816941 

10.816093 
815248 
814403 
813561 
812720 
811880 
811042 
810206 
809371 i 
808538 

10.8077061 
806876 
806047 I 
806220 I 
804394 1 
803570 i 
802747 
801926 I 
8011061 
800287 



14551 
14580 
14608 
14637 
14666 
14695 
14723 
14752 
14781 
14810 
14838 
14867 
14896 
14925 
14954 
14982 
16011 
16040 
15069 
15097 
15126 
15155 
i 15184 
16212 
15241 
16270 
15299 
15327 
16356 
15385 



Tang. 



15442 
15471 
15600 
16629 
15557 
16586 
16615 
15643 



99027 
99023 
99019 
99015 
99011 
99006 
99002 



98994 
98990 



98982 
98978 
98973 
98969 
98965 
98961 
98957 
98953 



98944 
98940 



98931 
98927 



98919 
98914 
98910 
98906 
98902 
98897 
98893 



98876 
98871 
98867 
98863 
98858 
98854 
98849 
98845 
98841 



98827 
98823 
98818 
98814 
98809 



15414 98806 



98800 
98796 
98791 
98787 
98782 
98778 
98773 
98769 
N. cos. N.siDe 



81 Degrees. 



30 



Log. Sines aud Tangents. (9°) Natural Sines. 



TABLE n. 




1 

2 
3 
4 
5 
6 
7 
8 
9 
10 
11 
12 
13 
14 
15 
IP) 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
BO 
51 
52 
53 
54 
55 
56 
57 
58 
59 
60 



Sine. 



9.194332 
195129 
1959-25 
196719 
197511 
198302 
199091 
199879 
200666 
201451 
202234 

9.203017 
203797 
204577 
205354 
206131 
208906 
207679 
203452 
209222 
209992 

9.210760 
211626 
212291 
213055 
213818 
214679 
215338 
216097 
216854 
217609 

9.218363 
219116 
219868 
220618 
221367 
222116 
222861 
223606 
224349 
225092 

9.225833 
226573 
227311 
228048 
228784 
229518 
230252 
230984 
231714 
232444 

9.233172 
233899 
234625 
235349 
236073 
236795 
237515 
238235 
238953 
239670 



D. ny 



Ck>8ine. 



133 

133 

132 

182 

132 

132 

131 

131 

131 

131 

130 

130 

130 

130 

129 

129 

129 

129 

128 

128 

128 

128 

127 

127 

127 

127 

127 

126 

126 

126 

126 

126 

125 

125 

125 

125 

124 

124 

124 

124 

123 

123 

123 

123 

123 

122 

122 

122 

122 

122 

121 

121 

121 

121 

120 

120 

120 

120 

120 

119 



Cosine. 

9.994620 
994600 
994580 
994560 
994540 
994519 
994499 
994479 
994459 
994438 
994418 
9.994397 
994377 
994357 
994336 
994316 
994295 
994274 
994254 
994233 
994212 
9.994191 
994171 
994150 
994129 
994108 
994087 
9940(36 
994045 
994024 
994003 
9.993981 
993960 
993939 
993918 
993896 
993875 
993854 
993832 
993811 
993789 
.993768 
993746 
993725 
993703 
993681 
993660 
993638 
993616 
993594 
993572 
.993550 
994528 
993506 
993484 
993462 
993440 
993418 
993396 
993374 
993351 



D. It)' 



Sine. 



3.3 
3.3 
3.3 
3.4 
3.4 
3.4 
3.4 
3.4 
3.4 
3.4 
3,4 
3.4 
3.4 
3.4 
3.4 
3.4 
3.4 
3,5 
3.5 
3.5 
3.5 
3.5 
3.5 
3.5 
3.5 
3.5 
3.5 
3.6 
3.5 
3.5 
3.5 
3.5 
3.5 
3.5 
3.5 
3.6 
3.6 
3.6 
3.6 
3.6 
3.6 
3.6 
3.6 
3.6 
3.6 
3.6 
3.6 
3.6 
3.6 
3.7 
3.7 
3.7 
3.7 
3-7 
3.7 



Taai;. 

3.199713 

200529 
201345 
202169 
202971 
203782 
204592 
205400 
206207 
207013 
207817 

). 208619 
209420 
210220 
211018 
211815 
212611 
213405 
214198 
214989 
215780 

). 216568 
217356 
218142 
218926 
219710 
220492 
221272 
222052 
222830 
223606 

1,224382 
225156 
225929 
226700 
227471 
228239 
229007 
229773 
230589 
231302 

'.232066 
232826 
233586 
234345 
235103 
235859 
236614 
237368 
238120 
238872 

.239622 
240371 
241118 
241865 
242610 
243354 
2440{}7 
244839 
245579 
246319 

CoUmj;. 



136 
136 
136 
135 
136 
135 
136 
134 
134 
134 
184 
133 
133 
133 
133 
133 
132 
132 
132 
132 
131 
131 
131 
131 
130 
130 
130 
130 
130 
129 
129 
129 
129 
129 
128 
128 
128 
128 
127 
127 
127 
127 
127 
126 
126 
126 
126 
126 
126 
126 
125 
125 
126 
124 
124 
124 
124 
124 
123 
123 



Cotang. 



N. mne.lN. cos 



10.800287 
799471 
798666 
797841 
797029 
796218 
796408 
794600 
793793 
792987 
792183 

10.791381 
790580 
789780 i 
788982 
788185 
787389 
786596 
785802 
735011 
784220 

10.783432 
782644 
781858 
781074 
780290 
779608 
778728 
777948 
777170 
776394 

10.775618 
774844 
774071 
773300 
772529 
771761 
770993 
770227 1 1 
769461 ! 



16643 

15672 

15701 

15730 

16758 

1578 

15816 

16846 

15873 

16902 

16931 

16959 



98769 
98764 
98760 
98766 
98751 
98746 
98741 
98737 
98732 
98728 
98723 
98718 



15988 98714 



16017 
16046 
16074 
16103 
16132 
16160 
16189 
16218 
16246 
16275 
16304 
16333 
16361 
16390 
16419 
16447 
16476 
16605 
16533 
16562 
13591 
16620 
16648 
16677 
16706 



98709 
98704 
98700 
98695 
98690 
98686 
98681 
98676 
98671 
98667 
98662 
98667 
98652 
98648 
98643 
98638 
98633 
98629 
98624 
98619 
98614 
98609 
98604 
98600 
98595 



10.767935! 
767174! 
766414 I 
765655 
764897 I 
764141 j 
763386 1 
762632 
761880 I 
761128 

10.760378 
759629 
758882 
758136 
757390 
756646 
755903 
756161 
764421 
763681 



16734 98590 
16763 98585 
16792 9S580 
16820,98576 



16849 
16878 
16906 
16935 
16964 
16992 
17021 
17050 



198570 
98565 
98661 
98556 
98551 
98546 
98541 
98536 



17078198531 
17107 98526 



Tang. 



98621 
98616 
98611 
98506 
98501 
98496 
98491 
98486 
98481 
N. COS. N.sine, 



17136 
17164 
17193 
17222 
17250 
17279 
17308 
17336 
17365 



80 T).vaveK. 



TAFLE II. 



Log. Sinc« and Tar.gcnt.s. (10«-') Kjjturjil Sines. 



.31 



8 
9 
10 
11 
I'i 
13 
14 
15 
16 
17 
18 
19 
20 

22 
23 
24 
25 
2o 
2/ 

I SO 
j3i 
i 32 
j 33 
j34 
I 35 
I 3;» 
! 3/ 
I 38 
I 39 
I 40 
41 
I 42 

i "^^ 
4i 

4t) 
47 
48 
4y 
50 
51 
52 
53 
54 
55 
5b 
67 
58 
69 
60 



Sine. 

K 239670 
240386 
241101 
241814 
242526 
243237 

- 243947 
244655 
245363 
246069 
246775 

). 247478 
248181 
248883 
249583 
250282 
250980 
251677 
252373 
253067 
253761 

). 254463 
255144 
255834 
256523 
257211 
257898 
258683 
259268 
259951 
260633 

>. 26 1314 
261994 
262b73 
263351 
264027 
264703 
265377 
266051 
266723 
267395 

1.268065 
2t>8 /34 
269402 
2/0069 
270735 
271400 
272064 
272726 
273388 
274049 

1.274708 
276367 
276024 
276681 
2^77337 
277991 
278644 
279297 
279948 
280599 

C().«inf'. 



D. 10" Cosiu 



119 
119 
119 
119 
118 
118 
118 
118 
118 
117 
117 
117 
117 
117 
116 
116 
116 
116 
116 
116 
115 
115 
115 
115 
115 
114 
114 
114 
114 
114 
113 
113 
113 
113 
113 
113 
112 
112 
112 
112 
112 
111 
111 
1.11 
HI 
111 
111 
110 
110 
110 
110 
110 
110 
109 
109 
109 
109 
109 
109 
108 



1.993351 
993329 
993307 
993285 
993262 
993240 
993217 
993195 
993172 
993149 
993127 

•.993104 
993031 
993059 
993036 
993013 
992990 
992967 
992944 
992921 
992898 

.992875 
992852 
992829 
992806 
992783 
992759 
992736 
992713 
992690 
992666 

.992643 
992619 
992596 
992672 
992549 
992525 
992601 
992478 
992454 
992430 

.992406 
992382 
992359 
992335 
992311 
992287 
992263 
992239 
992214 
992190 

.992166 
992142 
992117 
992093 
992059 
992044 
992020 
991996 
991971 
991947 



D. 10" 

3.7 
3.7 
3.7 
3.7 
3.7 
3.7 
3.8 



8 

8 

8 

8 

8 

8 

8 

8 

8 

3.8 

3.8 

3.8 

3.8 

3.8 

3.8 

3.8 

3.9 

3.9 

3.9 

3.9 

3.9 

3.9 

3.9 

3.9 

3.9 

3.9 

3.9 

3.9 

3.9 

,9 

,9 

,0 

,0 

,0 

,0 



3 
3 

4 

4 

4 

4 

4.0 

4.0 

4.0 



4.0 
4.1 



Tanir. 

9.246319 
247057 
247794 
248530 
249264 
249998 
260730 
261461 
252191 
252920 
263648 

9.254374 
265100 
255824 
256547 
257269 
257990 
258710 
269429 
260146 
260863 

9.261578 
262292 
263005 
263717 
264428 
265138 
265847 
266555 
267261 
267967 

9.268671 
269375 
270077 
270779 
271479 
272178 
272876 
273573 
274269 
274964 

9.275668 
276361 
277043 
277734 
278424 
279113 
279801 
280488 
281174 
281858 

9.282642 
283225 
283907 
284588 
285268 
285947 
286<>24 
287301 
287977 
288662 



Co tang. 



D. 1(»"| CoUm/. |.N..«ine.|N. M*, 



123 
128 
123 
122 
122 
122 
122 
122 
121 
121 
121 
121 
121 
120 
120 
120 
120 
120 
120 
119 
119 
119 
119 
119 
118 
118 
118 
118 
118 
118 
117 
117 
117 
117 
117 
116 
116 
116 
116 
116 
116 
116 
116 
116 
116 
115 
115 
114 
114 
114 
114 
114 
114 
113 
113 
113 
113 
113 
113 
112 



il0. 753681 
752943 
752206 
751470 
750736 
750002 
749270 
748539 
747809 
747080 
74(i352 

10.745626 
744900 
744176 
743453 
742731 
742010 



1736598481 
17393 98476 
17422.98471 
17451 984{)6 
17479 98461 
17508 98455 
1753798450 
17565 98445 
17594 98440 
17623 98435 
17651 198430 
17680 98425 



741290 
740571 
739854 
739137 

10.738422 
737708, 
736995 i; 
736283 , 
735572 |! 
7348621; 
734153 ] 
733445 I 
73273911 
732033'! 

10.731329' 
730625 
729923 
729221 
728521 ii 
727822 
727124 
726427 
725731 
725036 jj 

10.7243421 
723649 I 
722957" 



17708 
17737 
17766 
17794 
17823 
17852 
17880 
17909 
17937 
17966 
17995 
18023 
18052 
18081 
18109 
18138 
18166 
18195 
18224 
18252 
18281 
18309 



98420 
98414 
98409 
98404 
98399 
9831*4 
98389 
98383 
98378 
98373 
98368 
9b 36 2 
98367 
98362 
98347 
98341 
98336 
98331 
98326 
98320 
98316 
98310 



18338 98304 
18367 198299 
18395198294 



18424 
18452 
18481 



98288 
96283 
98277 



18509 98272 
18538 98267 




79 Dojfreos. 



25 



:« 



Log. Sines and Tangents. (11°) Natural Sines. 



TABLE II. 



blue. 



9.280599 
281248 
281897 
282544 
283190 
283836 
284480 
285124 
285766 
286408 
287048 

9.287687 
288326 
288964 
289600 
290236 
290870 
291504 
292187 
292768 
293399 

9.294029 
294658 
296286 
295913 
296539 
297164 
297788 
298412 
299034 
299655 

9.800276 
300895 
301514 
302132 
302748 
303364 
303979 
304593 
305207 
305819 

9.306430 
307041 
307650 
308259 
308867 
309474 
310080 
310686 
311289 
311893 

(9.312495 
313097 
313698 
314297 
314897 
315495 
316092 
316689 
317284 
317879 
Cosine. 



D. 10' 



108 
103 
108 
108 
108 
107 
107 
107 
107 
107 
107 
106 
106 
106 
106 
106 
106 
105 
105 
105 
105 
105 
105 
104 
104 
104 
104 
104 
104 
104 
103 
103 
103 
103 
103 
103 
102 
102 
102 
102 
102 
102 
102 
101 
101 
101 
101 
101 
101 
100 
100 
100 
100 
100 
100 
100 
100 
99 
99 
99 



D. 10 " 



1.991947 
991922 
991897 
991873 
991848 
991823 
991799 
991774 
991749 
991724 
991699 

1.991674 
991649 
991624 
991599 
991574 
991549 
991524 
991498 
991473 
991448 

1.991422 
991397 
991372 
991346 
991821 
991295 
991270 
991244 
991218 
991193 

1.991167 
991141 
991115 
991090 
991064 
991038 
991012 
990988 
990960 
990934 

' 990908 
990882 
990855 
990829 
990803 
990777 
990750 
990724 
990697 
990671 

1.990644 
990618 
990591 
990565 
990638 
990611 
990485 
990458 
990431 
990404 
Sine. 



4.1 
4.1 
4.1 
4.1 
4.1 
4.1 
4.1 
4.2 
4.2 
4.2 
4.2 
4.2 
4.2 
4.2 
4.2 
4.2 
4.2 
4.2 
4.2 
4.2 
4.2 
4.2 
4.2 
4.3 
4.3 
4.3 
4.3 
4.3 
4.3 
4.3 



3 

3 

3 

3 

3 

3 

3 

3 

4.3 

4.3 

4.4 

4.4 

4.4 

4.4 

4.4 

4.4 

4.4 

4.4 

4.4 

4.4 

4.4 

4.4 

4.4 

4.4 

4.4 

4.4 

4.5 

4.5 

4.5 

4.5 



.288662 
289326 
289999 
290671 
291342 
292013 
292682 
293350 
294017 
294684 
295349 

.296013 
296677 
297339 
298001 
298662 
299322 
299980 
300638 
301295 
301951 

.302607 
303261 
303914 
304567 
305218 
305869 
308519 
307168 
307815 
308463 

.309109 
309754 
310398 
311042 
311685 
312327 
312967 
313608 
314247 
314885 
9.315523 
316159 
316795 
317430 
318064 
318697 
319329 
319961 
320592 
321222 
321851 
322479 
323106 
323733 
324368 
324983 
325607 
326231 
326853 
327475 



Co tan R. 
Degrt^s. 



112 
112 
112 
112 
112 
111 
111 
111 
111 
111 
111 
111 
110 
110 
110 
110 
110 
110 
109 
109 
109 
109 
109 
109 
109 
108 
108 
108 
108 
108 
108 
107 
107 
107 
107 
107 
107 
107 
106 
106 
106 
106 
106 
106 
lOJ 
105 
105 
105 
105 
105 
106 
105 
104 
104 
104 
104 
104 
104 
104 
104 



ix)t.ang. (N. sine. N. cos 



10.711348 
710674 
710001 
709329 
708658 
707987 
707318 
706650 
705983 
705316 
704651 

10.703987 
703323 
702661 
701999 
701338 
700678 
700020 
699362 
698705 
698049 

10-697393 
696739 
696086 
696433 
694782 
694131 
693481 
692832 
692185 
691537 

10-690891 
690246 
689802 
688958 
688315 
687673 
687033 



19081 
19109 
19138 
19167 
19195 
19224 
19252 
19281 
19309 
19338 
19366 
19396 
19423 
19452 
19481 
19509 
19638 
19566 
19595 
19623 
19662 
19680 
19709 
19737 
19766 
197y4 
19828 
19861 
19880 
19908 
19937 
19966 
19994 
20022 
20061 
20079 
20108 
20136 
6863921120166 






20193 
20222 
20260 
20279 
20307 
120336 
20364 
20393 
20421 
20460 



685753 
6861161 

10-684477! 
683841 
683205 ' 
682570 
681936 
681303 
680671 i 

680^)39 i 

679408 'j 20478 
678778 i 1 20507 

10.678149^120535 
6776211 20563 
676894 I ! 20692 
676267 i 1 20(i20 
675642 1 1 20649 
675017 j 20677 
674393!! 20706 
673769 1 1 20734 
673147 j 20763 
6725261120791 



Tang. 



98163 
98157 
98152 
98146 
98140 
98135 
98129 
98124 
98118 
98112 
98107 
98101 
98096 
98090 
98084 
98079 
)73 
98067 
98061 
98056 
98060 
98044 
98039 
98033 
98027 
98021 
98016 
98010 
98004 



979981 31 



97992 
97987 
97981 
97975 
97969 



97963 1 25 



97958 

97952 

97946 

97940 

97934 

97928 

97922 

97916117 

97910 10 



7905 
97899 
97893 
97887 
97881 
97876 
97869 
97863 
97867 
97851 
97846 
97839 
97833 
97827 
97821 
97815 



N. COS. IV.><in«'. 



TABLE II. 



Log. Bines and Tangents. (12°) Natural Sines. 



33 




N.sine.iN. cos, 



2079197815 
2082097809 
20848 97803 
20877 97797 
20905 97791 
20933 97784 
20962 97778 
2099097772 
21019 97766 
21047 97760 
21076 97754 
2110497748 
21132 97742 
2116197735 
•21189 97729 
21218 97723 
21246 97717 

121275 97711 
21303|97705 
2133197698 
21360,97692 
21388 97686 
2141797680 
21445 97673 
2147497667 
21502 97661 
21530 97655 
21559 97648 
2168797642 
2161697636 
21644|97630 
2167297623 
2170197617 
21729 97611 
21758J97604 
21786:97598 
2181497692 
21843197686 
21871 97579 
21899 97573 
21928 97566 
21956 97560 
2198597653 
22013 97547 

I 22041 97541 



22070 
22098 
22126 



97534 

97528 
621 



22156^7515 
22183 97508 
2221297602 
|22240'97496 
122268197489 
22297197483 
22325 ©7476 
22363 97470 
2238297463 
2241097457 
22438 97450 
22467 97444 
22495 97437 



I N. cos.lN.8ine 



77 Degree*, 



34 



Log, Sines and Tangento. (13°) Natural Sines. 



TABLE n. 



S ine. 

9,352088 
352635 
353181 
353726 
354271 
354815 
355358 
355901 
356443 
356984 
357524 

9.368064 
358603 
359141 
359678 
360216 
3&0752 
361287 
361822 
362356 



D. 10' 



9.363422 
363964 
364485 
366016 
365546 
866075 
366604 
367131 
367659 
3681 85 

9.368711 
369236 
369761 
370286 
370808 
371330 
371852 
372373 
372894 
373414 

9.373933 
374462 
374970 
375487 
376003 
376519 
377035 
377649 
378063 
S78577 

9.379089 
379601 J 
380113 
380624 
381134 
381643 
382162 
382661 
383168 
383676 
Cosine. ' 



Cosine. 



9,988724 
9^695 
988666 
988636 
988607 
988678 
988548 
988619 
988489 
988460 
98iW30 

9.988401 
988371 
988342 
988312 
968282 
988252 
988223 
988193 
988163 
988133 

». 988103 
988073 
988043 
988013 
987983 
987953 
987922 
987892 
987862 
987832 

9.987801 
987771 
987740 
987710 
987679 
987649 
987618 
987588 
9&7667 
987526 

9.987496 
987466 
987434 
987408 
987372 
987341 
987310 
987279 
987248 
987217 

9.987186 
987156 
987124 
987092 
987061 
987030 
986998 
986967 
986936 



D. 10' 



4.9 
4.9 
4.9 
4.9 
4.9 
4.9 
4.9 
4.9 
4.9 
4.9 
4.9 
4.9 
4.9 
4.9 
6.0 
6.0 
6.0 
6.0 
6.0 
6.0 
6.0 
6.0 
5.0 
6.0 
5.0 
5.0 
6,0 
6.0 
5.0 
5.0 
5.1 
6.1 
6,1 
6.1 
6.1 
6.1 
6.1 
5.1 
5.1 



&.2 
6,2 
6.2 
6.2 
6.2 
6.2 
6.2 
5.2 
5.2 
6.2 
5.2 
6.2 
5.2 
5.2 
5.2 
6.2 



T ang. 

.363364 
363940 
364515 
365090 
365664 
366237 
366810 
367382 
367953 
368624 



9.369663 
370232 
370799 
371367 
371933 
372499 
378064 
373629 
374193 
374766 

9.375319 
376881 
376442 
377003 
377563 
378122 
378681 
379239 
379797 
380354 
380910 
381466 
382020 
382575 
383129 
383682 
384234 
384786 
385337 



9.386438 
386987 
387536 
388084 
388631 
389178 
389724 
390270 
390816 
391360 

1.391903 
392447 
392989 
393531 
394073 
394614 
396154 
396694 
396233 
396771 
Cotanp. 



D. 10' 



96.0 
95.9 
95.8 
95.7 
96,6 
95,4 
95,3 
95.2 
95.1 
95.0 
94.9 
94.8 
94.6 
94.6 
94.4 
94.3 
94.2 
94.1 
94.0 
93.9 
93.8 
93.7 
93.5 
93.4 
93.3 
93.2 
93,1 
93.0 
92.9 
92.8 
92.7 
92.6 
y2 6 
92,4 
92,3 
92,2 
93.1 
92.0 
91,9 
91,8 
91.7 
91.5 
91,4 
91.3 
91.2 
91.1 
91.0 
90.9 
90.8 
90.7 
90.6 
90.5 
90.4 
90.3 
90.2 
90.1 
90.0 
89.9 
89.8 
89.7 



. Cotang. I N. sine 



Tang. 



! 22863197851 
! 22892197345 
973:^8 
97331 
97325 
97318 
311 
97304 
97298 
97291 
97284 
23176 97278 
97271 
97264 
97257 
23288 97251 
23316 97244 



23797 



23769 97134 



97127 



23825 97120 
97118 
9710t> 
97100 
97093 



97072 
97066 
24079 97 06;S 
97051 
24136 97044 
24164 97037 
24192 97030 



N. COS. N sine. '• 



76 Degrees. 



36 



Log^ Sines and Tangents. (15°) Natural Sines. 




1 

2 
3 
4 
6 
6 
7 
8 
9 

la 
11 

12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
21 
28 
29 
30 
31 
32 
33 
34 
35 
36 
87 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
64 
55 
56 
51 
58 
59 
60 



Sine. 

J. 412996 
413467 
413938 
414408 
414878 
415347 
415815 
416283 
416751 
417217 
417684 

). 418150 
418615 
419079 
419544 
420007 
420470 
420933 
421395 
421857 
422318 

). 422778 
423238 
4236^7 
424166 
424616, 
425073 
425530 
425987 
426443 
426899 

). 427354 
427809 
428263 
428717 
429 L70 
429623 
430a76 
430527 
430978 
431429 

>. 431879 
432329 
432778, 
433226 
433675 
434122 
434569 
435016 
435462 
435908 

11.436353 
436798, 
437242 
437686 
438129 
438572 
439014 
439456 
4C9897 
440338 
CoiJine, 



D. 10" I Ijosinc. 

k. 984944 
984910 
984876 
984842 
984808 
984774 
984740 
984706 
984672 
984637 
984603 
.984569 
984535 
984500 
984466 
984432 
984397 
984363 
984328 
984294 
984259 
.984224 
984190 
984155 
984120 
984085 
984050 
984016 
983981 
983946 
983911 
.983875 
983840 
983805 
9837'^0 
983755 
983700 
983664 
983629 
983594 
983558 
.983523 
983487 
983452 
983416 
983381 
983345 
983309 
983273 
983238 
983202 
.983166 
983130 
983094 
983058 
983022 



78.5 
78.4 
78.3 
78.3 
78.2 
78.1 
78.0 
77.9 
77.8 
77.7 
77.6 
77.5 
77.4 
77.3 
77.3 
77.2 
77.1 
77.0 
76.9 
76.8 
76.7 
76.7 
76.6 
76.5 
76.4 
76.3 
76.2 
76.1: 
76.0 
76.0 
75.9 
75.8 
75.7 
75.0 
75.5 
75.4 
75.3 
75.2 
75.2 
75.1 
75.. 0, 
74.9 
74.9 
74.8 
74.7 
74 6 
74.5 
74.4 
74.4 
74.3 
74.2 
74.1 
74. ft 
74.0 
73.9 
73.8 
73.7 
73.6 
73.6 
73.51 



982950 
982914 
982878 
982842 



Sine. 



JX W\ Tang. 



5.7 
6.7 
5.7 
6.7 
5.7 
5.7 
6.7 
5.7 
5.7 
5.7 
5.7 
6.7 
6.7 
5.7 
6.7 
5.8 
6.8 
5. .8 
5.8 
5v8 
5.8 
5.8 
5.8 
5.8 
6.8 
5.8 
5.8 
5.8 
5.8 
5.8 
5.8 
5.8 
6.9 
5.9 
5.9 
5.9 
5.9 
5.9 
6.9 
5.9 
5.9 
6.9 
6.9 
6.9 
5.9 
5.9 
5.9 
6.9 
6.0 
6.0 
6.0 
6.0 
6.0 

6. a 

6.0 
6.0 
6.0 
6.0 
6.0 
6.0 



9.428062 
428557 
429062 
429561 
430070 
430573 
431075 
431577 
432079 
432580 
433080 

9.433580 
434080 
434579 
435078 
435576 
436073 
436670 
437067 
437563 
438059 

9.438554 
439048 
439543 
440036 
440529 
441022 
441614 
442006 
442497 
442988 

9.443479 
443968 
444458 
444947 
445.435 
44a923 
446411 
446898 
447384 
447870 
1.448356 
448841 
449326 
449810 
46Q294 
460777 
45L260 
451743 
452226 
452706 
.453187 
453668 
454148 
464628 
465107 
465586 
456064 
456542 
457019 
457496 
Cotanjj. 



a iiy; 

84.2 
84.1 
84.0 
83.9 
83.8 
83.8 

83.7 

83.6 
83.5 
83.4 
83.3 
83.2 
83.2 
83.1 
83.0 
82.9 
82.8 
82.8 
82.7 
82.6 
82.6 
82.4 
82.3 
82.3 
82.2 
82.1 
82.0 
81.9 
81.9 
81.8 
81.7 
81.6 
81.6 
81.5 
81.4 
81.3 
81.2 
81.2 
81.1 
81.0 
80.9 
SO. 9 
80.8 
80.7 
8ft. 6 
80.6 
80.5 
80.. 4 
80.3 
80.2 
80.2 
80. 1 
80.0 
79.9 

9.9 
79.8 
79.7 

9.6 
79.6 
79.5 



Lotuup;. N. sine. N. COS. 

96593 
96585 
96578 
96570 
96562 
96665 
96547 
96640 
96532 
96524 



10.571948! 25882 
571443 i 269 10 
570938 ': 2593 
670434' 2596 
6r>9930;i25994 
569427 1 126022 
668925! 126050 
5684231126079 
567921 126107 
667420 ,26135 



5669201126163 96517 



10.566420: '26191 
5669201:26219 
5654311126247 
564922 1126275 
5644241126303 
663927 1 126331 
5634301126359 
562933 I 26387 
562437 [26416 
561941 ! 26443 

10.561446!! 26471 
560952 1 1 26500 
560457 1 26528 



26556 
S6584 
26612 
26640 
26668 
26696 



559964 
559471 
658978 
558486 
657994 
557503 

557012 1 1 26724 
10-.666521H 26762 
556032 1126780 
5555421126808 
555053!! 26836 
654566 
554077 
553589 
553102 
552616 
652130- 
0.. 561644 



551159 



26864 
26892 
26920 
26948 
26976 
27004 
27032 
37060 



650674 127088 96261 



650190 127116 
549706 127144 
549223 127172 
5487401127200 



648267 ; 27228 96222 



547776 ij 27256 
547294 1 1 27284 
10.5468131; 27312 
546332 
545852 
545372 
644893 
544414 
643936 
543458 
542981 
542504 



Tang. 



96509 
96502 
96494 
96486 
96479 
96471 
96463 
96456 
96448 
96440 
96433 
96425 
96417 
96410 
96402 
96394 
96386 
96379 
96371 
96363 
96355 
96347 
96340 
96332 
96324 
96316 
96308 
96301 
96293 
96285 
96277 
96269 



96253 
96246 
96238 
96230 



96214 

96206 

961.98 

7340 96190 

27368 96182 

27396 96174 

27424 96166 

27452 96158 

27480 96150 

27508 96142 

27536 96134 



27564 



96126 



N. cos.lN.sine. 



74 Degrees. 



TABLE II. 



Log. Sinefl and Taflgcnts. (16°) Natural Sines. 



37 



Hine. 

).4t0338 
440778 
441218 
441658 
442096 
442636 
442973 
443410 
443847 
444284 
444720 

). 445165 
445590 
446026 
446469 
446893 
447326 
447759 
448191 
448623 
449054 

). 449486 
449915 
450346 
450776 
461204 
451632 
452060 
452488 
452916 
453342 

). 453768 
454194 
454619 
455044 
455469 
456893 
456316 
456739 
467162 
457584 

3 468006 
458427 
458848 
459268 
459688 
460108 
460527 
460946 
461364 
461782 

9.462199 
462616 
463032 
463448 
463864 
464279 
464694 
465108 
466522 
465935 



U. 10" Coiiiio. D. lu 



Cosine. 



73,4 
73.3 
73.2 
73-1 
73.1 
73.0 
72.9 
72.8 
72.7 
72.7 
72.6 
72.6 
72.4 
72.3 
72.3 
72.2 
72.1 
72.0 
72.0 
71.9 
71.8 
71.7 
71.6 
71.6 
71.6 
71.4 
71.3 
71.3 
71.2 
71.1 
71.0 
71.0 
70.9 
70.8 
70.7 
70.7 
70.6 
70.5 
70.4 
70.4 
70.3 
70.2 
70.1 
70.1 
70.0 
69.9 
69.8 
69.8 
69.7 
69.6 
69.5 
69.5 
69.4 
69.3 
69.3 
69.2 
69.1 
69.0 
69.0 
68.9 



9.982842 
982805 
982769 
982733 
982696 
982660 
982624 
982587 
982551 
982514 
982477 
982441 
982404 
982367 
982331 
982294 
982257 
982220 
982183 
982146 
982109 

9.982072 
982036 
981998 
981961 
981924 
981886 
981849 
981812 
981774 
981737 

9.981699 
981662 
981625 
981587 
981649 
981512 
981474 
981436 
981399 
981361 

9.981323 
981285 
981247 
981209 
981171 
981133 
981096 
981057 
981019 
980^)81 
,980942 
980904 
980866 
980827 
980789 
980750 
980712 
980S73 
980635 
980596 
Sino. 



6.0 
6.0 
6.1 
6.1 
6.1 
6.1 
6.1 
6.1 
6.1 
6.1 
6.1 



6.2 
6.2 
6.2 
6.2 
6.2 
6.2 
6.2 
6.2 
6.2 
6.2 
6.2 
6.2 
6.2 
6.3 



6.3 
6.3 
6.3 
6.3 
6.3 
6.3 
6.3 
6.3 
6.3 
6.3 
6.3 
6.3 
6.3 
6.4 
6.4 
6.4 
6.4 
6.4 
6.4 
6.4 
6.4 
6.4 
6.4 
6.4 
6.4 
6.4 
6.4 



Tanj;. 

9.457496 
467973 
468449 
458925 
459400 
469875 
460349 
460823 
461297 
461770 
462242 
462714 
463186 
463658 
464129 
464699 
466069 
466539 
466008 
466476 
466946 
467413 
467880 
468347 
468814 
469280 
469746 
470211 
470676 
471141 
471605 
472068 
472632 
472995 
473457 
473919 
474381 
474842 
475303 
476763 
476223 

9 476683 
477142 
477601 
478059 
478517 
478976 
479432 
479889 
480345 
480801 
481257 
481712 
482167 
482621 
483076 
483529 
483982 
484436 
484887 
485339 



1>. lo' Coian;:. ; N.ainc. N. coa 



79.4 
79.3 
79.3 
79 2 
79.1 
79.0 
79.0 
78.9 



10.542504! 27564 
5420271' 27692 
641651 i 27620 



Cotan<r. 



78 

78 

78 

78 

78 

78 

78 

78 

78.3 

78.2 

78.1 

78.0 

78.0 

77.9 

77.8 

77.8 

77.7 

77.6 

77.5 

77-5 

77.4 

77.3 

77.3 

77.2 

77.1 

77.1 

77.0 



76.9 
76.8 
76.7 
76.7 
76.6 
76.5 
76.5 
76.4 
76.3 
76.3 
76.2 
76.1 
76.1 
76.0 
76.9 
75.9 
76.8 



75.3 



541075 

540600 

540126 

539651 

539177 

538703 

538230 ! 

537758 
10.637286 

536814 

536342 

535871 

635401 

534931 

534461 

538992 

533524 

533055 
10.532587 

632120 

531663 

531186 

530720 

530254 

529789 

529324 

528859 

628395 
10.527932 

627468 

627005 

526643 ! 

526.081 I 

625619 1 

5261581 

524697 

524237 

523777 
10 523317 

522858 

522399 

521941 

521483 . 

5210251 128847 

520568 112887 6 

520111 j 28903 

5196651128931 

619199 jl 28959 
10.518743 1 28987 

518288! 29015 

517833 {129042 

517379 H2907O 

516925 I i 29098 

516471 1 129126 

516018 1 29154 

515665 '29182 

615113 |l2920iJ 

5146611129247 



27648 
27676 
27704 
27731 
27759 
27787 
27815 
27843 
27871 
27899 
27927 
27955 
27983 
28011 
28039 
28067 
28095 
28123 
28150 
28178 
28206 
28234 
28262 
28290 
28318 
28346 
28374 
28402 
28429 
28467 
28485 
28513 
28541 
28569 
28597 
28625 
28652 
28680 
28708 
28736 
28764 
128792 
128820 95 



Tanp. 



N. COP. N.8in 



96126 
96118 
96110 
96102 
96094 
96086 
96078 
96070 
.%062 
96064 
96046 
96037 
96029 
96021 
96013 
96005 
d5997 
95989 
96981 
95972 
96964 
95956 
95948 
95940 
96931 
95923 
95915 
95907 
95898 
95890 
95882 
95874 
95865 
95857 
95849 
96841 
95832 
95824 
95816 
S6807 
95799 
96791 
95782 
95774 
95766 
95757 
95749 
95740 
95732 
95724 
95716 
95707 
95698 
95690 
96681 
96673 
95664 
95656 
95647 
95639 

966;;o 



73 D<'a;rce8. 



38 



Log. Sines and Tangents. (17°) Natural Sines. TABLE II. 



! Sine. p. 10" 



7 
8 
9 
10 
11 
12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
28 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
88 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
62 
53 
64 
55 
56 
57 
68 
69 
60 



.465935 
466348 
466761 
467173 
467585 
467996 
468407 
468817 
469227 
469037 
470046 
9.470455 
470803 
471271 
471679 
472086 
472492 
472898 
473304 
473710 
474115 
9.474519 
474923 
475327 
475730 
476133 
476536 
476938 
477340 
477741 
478142 
■478542 
478942 
479342 
479741 
480140 
480539 
480937 
481334 
481731 
482128 
9.482525 
482921 
483316 
483712 
484107 
484501 
484895 
485289 
485682 
486075 
,486407 
486860 
487251 
487643 
488034 
488424 
488814 
489204 
489593 
489982 



68.8 

68.8 

08.7 

68.6 

68,5 

68.5 

68.4 

68.3 

68.3 

68.2 

68.1 

68.0 

68.0 

67.9 

67.8 

67.8 

67.7 

67.6 

67.6 

67.5 

67.4 

67.4 

67.3 

67.2 

67.2 

67.1 

67.0 

66.9 

66.9 

60.8 

66.7 

60.7 

66.6 

66.5 

66.5 

66.4 

66.3 

66.3 

66.2 

66.1 

66.1 

66.0 

65.9 

65,9 

65.8 

65.7 

65.7 

65,6 

65.5 

65.5 

65.4 

66.3 

65.3 

65.2 

65.1 

65.1 

66.0 

65.0 

64.9 

64.8 



Cosine. 



Cosine. 

9.980596 
980558 
980519 
960480 
980442 
980403 
980364 
980325 
980286 
980247 
980208 
.980109 
980130 
980091 
980052 
980012 
979973 
979934 
979895 
979855 
979816 
9.979776 
979737 
979697 
979658 
979618 
979579 
979539 
979499 
979459 
979420 
.979380 
979340 
979300 
979260 
979220 
979180 
979140 
979100 
979059 
979019 
9.978979 
978939 
978898 
978858 
978817 
978777 
978736 
978696 
978655 
978616 
,978574 
978533 
978493 
978452 
978411 
978370 
978329 
978288 
978247 
978206 



D. 10" 



Sine. 



6.4 

6.4 

6.5 

6.5 

6.6 

6.5 

6.5 

6.5 

6.5 

6.6 

6.5 

6.5 

6.6 

6.5 

6.6 

6.5 

6.5 

6.6 

6.6 

6.6 

6.6 

6.6 

6.6 

6.6 

6.6 

6.6 

6.6 

6.6 



6.6 
6.6 
6.6 
6.7 
6.7 
6.7 
6.7 
6.7 
6.7 
6.7 
6.7 
6.7 
6.7 
6.7 
6.7 
6.7 
6.7 
6.7 
6.8 
6.8 
6.8 
6.8 
6.8 
6.8 
6,8 
6.8 
6.8 
6.8 
6.8 
6.8 



9.485339 
485791 
480242 
486693 
487143 
487593 
488043 
488492 
488941 
489390 
489838 
3.490286 
490733 
491180 
491627 
492073 
492519 
492965 
493410 
493854 
494299 

9.494743 
495186 
495630 
496073 
496515 
496957 
497399 
497841 
468282 
498722 

9.499163 
499603 
600042 
500481 
500920 
601359 
601797 
502235 
502672 
603109 
503546 
503982 
504418 
504854 
605289 
505724 
606169 
606593 
507027 
507460 
507893 
508326 
608769 
509191 
609622 
510054 
510486 
510916 
611346 
611776 



D. 10" 



Co tang. 



75.3 

75.2 

76.1 

76.1 

76.0 

74.9 

74.9 

74.8 

74.7 

74.7 

74.6 

74.6 

74.5 

74.4 

74,4 

74.3 

74,3 

74.2 

74.1 

74.0 

74.0 

74.0 

73.9 

73.8 

73.7 

73.7 

73.6 

73,6 

73.5 

73.4 

73.4 

73.3 

73.3 

73.2 

73.1 

73.1 

73.0 

73.0 

72.9 

72.8 

72,8 

72,7 

72.7 

72.6 

72.5 

72.6 

72.4 

72.4 

72.3 

72.2 

72.2 

72,1 

72,1 

72.0 

71.9 

71.9 

71,8 

71.8 

71.7 

71.6 



Cotang. 



10.614661 
614209 
613768 
613307 
612867 
512407 
611957 
611508 
611059 
510610 
510162 

10.509714 
609267 
608820 
608373 
607927 
607481 
507035 
506590 
606146 
505701 

10.505257 
604814 
604370 
603927 
503486 
603043 
602601 
602159 
501718 
501278 

10,500837 
600397 
499958 
499519 
499080 
498641 
498203 
497765 
497328 
496891 

10.496454 
496018 
495582 
495146 
494711 
494276 
493841 
493407 
492973 
492540 

10.492107; 
491674 
491241 
490809 
490378 
489946 
489516 
489084 
488054 
488224 



N. sine.fN. cos, 



29237 
29265 
29293 
29321 
29348 
29376 
29404 
29432 
29460 
29487 



29543 
29571 
29599 
29626 
29654 



29682 95493 



29710 
29737 
29766 
29793 
29821 



96630 

95622 
95613 
95605 
95696 
95588 
95679 
95671 
95562 
95654 



29616 96645 



95536 
96528 
95619 
95511 
95502 



95485 
95476 
95467 
96459 
95450 



29849 95441 



29876 
29904 
29932 
29960 



29987 95398 
30015 95389 
30043 95380 



30071 
30098 
30126 
30154 
30182 
30209 
30237 
30265 
30292 
30320 
30348 



95372 
S5363 
95354 
95345 
95337 
95328 
96319 
95310 
95301 
95293 
95284 



30376 95275 



30403 
30431 
30459 
30486 
30514 
30542 
30570 
30697 
30025 
30G53 



30763 
30791 



Tang. 11 N. com 



95433 
95424 
95415 
96407 



95260 
95257 
95248 
95240 
95231 
95222 
95213 
95204 
96196 
95186 



30080 95177 
30708 95168 
30730 95159 



95150 
195142 



30819 95133 
30840 95124 
30874 95115 
3090^ 95100 



60 
59 
58 
57 
66 
65 

64 

53 

52 

51 

50 

49 

48 

47 

46 

46 

44 

43 

42 

41 

40 

39 

38 

37 

36 

35 

34 
3 I 

32 

31 

30 

29 

28 

27 

20 

25 

24 

23 

22 

21 

20 

19 

18 I 

17 

10 

15 

14 

13 

12 

u 

10 
9 
8 
7 

5 
4 
3 

1 




72 Degrees. 



TABLE II. 



Log. Since and Tangeut*. (18®) ^'atu^al Sines. 



39 



Sint'. 

9.489982 
490371 
490759 
491147 
491535 
491922 
492308 
492695 
493081 
493466 
493861 

9.494236 
494621 
495005 
495388 
495772 
496154 
496537 
496919 
497301 
497682 

9.498034 
498444 
498825 
499204 
499584 
499963 
500342 
50U721 
501099 
501476 

9.501864 
602231 
502607 
502984 
603360 
503736 
504110 
604485 
604860 
605234 

9.505608 
505981 
603354 
506727 
507099 
507471 
507843 
508214 
508585 
508956 

9.509326 
509696 
510066 
610434 
510803 
511172 
511540 
611907 
512276 
612 642 
Cosine. 



D. W 



64.8 
64.8 
64.7 
64.6 
64.6 
64.6 
64.4 
64.4 
64,3 
64.2 
64.2 
64.1 
64.1 
64.0 
63.9 
63.9 
63.8 
63.7 
63.7 
63.6 
63.6 
63.6 
63.4 
63.4 
63.3 
63.2 
63.2 
63.1 
63.1 
63.0 
62.9 
62.9 
62.8 
62.8 
62.7 
62.6 
62.6 
62.5 
62.6 
62.4 
62.3 
62.3 
62.2 
62.2 
62.1 
62.0 
62.0 
61.9 
61,9 
61.8 
61.8 
61.7 
61.6 
61.6 
61.6 
61.5 
61.4 
61.3 
61.3 
61.2 



Cosine. 

1.978206 
978165 
978124 
978083 
978042 
978001 
977959 
977918 
977877 
977835 
977794 

1.977752 
977711 
977669 
977628 
977588 
977544 
977603 
977461 
977419 
977377 

1.977335 
977293 
977251 
977209 
977167 
977125 
977083 
977041 
976999 
970967 
6914 
976872 
976830 
976787 
976745 
976702 
976660 
976617 
976574 
976532 

•976489 
976446 
976404 
976361 
976318 
976275 
976232 
976189 
976146 
976103 

.976060 
976017 
975974 
975930 
975887 
975844 
975800 
975757 
976714 
975670 



9.9' 



Sine. 



P. 10' 

6.8 
6.8 
6.8 
6.9 
6.9 
6.9 
6.9 
6.9 
6.9 
6.9 
6.9 
6.9 
6.9 
6.9 



7.0 
7.0 



7.1 
7.1 
7.1 
7.1 

7.2 
7.2 
7.2 
7.2 
7.2 
7.2 
7.2 
7.2 
7.2 
7.2 
7.2 
7.2 
7.2 



JTuuu:. 

9.511776 
612206 
512636 
613064 
513493 
513921 
614349 
514777 
515204 
515631 
516057 
,516484 
516910 
517336 
517761 
518185 
518610 
519034 
519458 
519882 
520305 

9.520728 
621161 
521673 
521993 
622417 
522838 
523259 
523680 
524100 
524520 
524939 
525359 
526778 
626197 
526615 
527033 
527451 
527868 
628285 
528702 
629119 
629535 
529950 
530366 
530781 
631196 
531611 
532025 
532439 
532863 

9.533266 
633679 
634092 
634604 
534916 
636328 
635739 
636150 
536561 
636972 
Cotang. 



D. 10" 



71 

71 

71 

71 

71 

71 

71,3 

71,2 

71,2 

71.1 

71.0 

71.0 

70.9 

70,9 

70.8 

70,8 

70.7 

70.6 

70.6 

70.5 

70.6 

70.4 

70.3 

70.3 

70.3 

70.2 

70.2 

70.1 

70.1 

70,0 

69,9 



69,8 
69.7 
69,7 
69,6 
69,6 
69,5 
69,6 
69,4 
69,3 
69.3 
69.3 
69.2 
69,1 
69.1 
69 
69.0 
68.9 
68.9 
68 8 
68.8 



Cotang. i N. sine 



10.488224 
487794 
487365 



486507 
486079 
486651 
485223 
484796 
484369 
483943 

10.483516 
483090 
482665 ! 
482239 
481815 i 
481390 I 
480966 { 
480542 
480118 
479696 ' 

10.479272 
478849 
478427 
478005 
477583 
477162 
476741 
476320 
475900 
475480 

10.475061 
474641 
474222 
473803 
473385 
472967 
472549 
472132 
471715 
471298 

10.470881 
470465 
470050 
469634 ! 
469219 I 
468804 I 
468389 I 
467975 } 
467561 
467147 I 

10.466734 
466321 
465908 
465496 
465084 
464672 
464261 
463850 I 
463439 
463028 
"Twig. 



130902 
130929 
130957 
1 30985 
31012 
31040 
1 31068 
131095 
131123 
31161 
131178 
131206 
131233 
131261 
31289 
J31316 
31344 
31372 
31399 
31427 
31454 
31482 
31510 
31537 
31565 
31593 
31620 
31648]! 
31675 
31703 
31730 



N. COS 



95106 
95097 
96088 
96079 
96070 
95061 
95052 
95043 
96033 
95024 
95016 
95006 
94997 
94988 
94979 
94970 
94961 
94952 
94943 
94933 
94924 
94915 
94906 
94897 
94888 
94878 
94869 
94860 
94851 
94842 
94832 
3175894823 
31786 94814 
31813 948U5 I 27 



60 
59 

58 
57 
56 
55 
64 
63 
62 
51 
60 
49 
48 
47 
46 
45 
44 
43 
42 
41 
40 
39 
38 
37 
36 
36 
34 
33 
32 
31 
30 
29 



94571 
94661 
94552 
N. COS. N.6ine, 



71 Degrees. 



49 



Log. Sines and Tangents. (13°) Natural Sines. 



TABLE II. 





1 

2 
S 

4 
6 
6 

7 
8 
9 
10 
11 
12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
36 
36 
37 
88 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
57 
58 
59 
60 



3.512642 
61300D 
613375 
613741 
514107 
514472 
514837 
515202 
515566 
515930 
516294 

}. 516657 
517020 
517382 
517745 
618107 
618468 
618829 
619190 
619551 
519911 

). 520271 
520631 
520990 
621349 
521707 
622066 
622424 
522781 
523138 
523495 

). 523852 
524208 
624564 
524920 
626276 
525 'j30 
525984 
520339 
52e693 
527046 

I. 627400 
527753 
628106 
528458 
528810 
529161 
629513 
629864 
530215 
630565 

•.530916 
531265 
531614 
531963 
632312 
532661 
533009 
533357 
533704 
534062 

Cosine. 



61.2 
61.1 
61.1 
61.0 
60.9 
60.9 
60.8 
60.8 
60.7 
60.7 
60.6 
60.5 
60.5 
60.4 
60.4 
60.3 
60.3 
00.2 
00.1 
60.1 
60.0 
60.0 
59.9 
59.9 
59.8 



59 

59 

59 

69 

59 

59 

59 

69 

59 

59.3 

59.2 

59.1 

59.1 

59.0 

59.0 

58.9 

58.9 

68.8 

58.8 

58.7 

58 7 

58.6 

58.6 

58.5 

68 

58 

58 

58 

58 

58 

58 

58.1 

68.0 

58.0 

57.9 



OOlSiUO. 

.975670 
9756-27 
975583 
975539 
975496 
975452 
975408 
975365 
975321 
976277 
975233 
.975189 
975145 
975101 
976057 
975013 
974969 
974925 
974880 
974836 
974792 
.974748 
974703 
974659 
974614 
974570 
974525 
974481 
974436 
974391 
974347 
.974302 
974257 
974212 
974167 
974122 
974077 
974032 
973987 
973942 
973897 
.973852 
973807 
973761 
973716 
973671 
973625 
973580 
973536 
973489 
973444 
.973398 
973362 
973307 
973201 
973215 
973169 
973124 
973078 
973032 
972986 
"Shie^ 



7.3 
7.3 
7.3 
7.3 
7.3 
7.3 
7.3 
7.3 



7.3 
7.3 
7.3 
7.3 
7.4 
7.4 
7.4 
7.4 
7.4 
7.4 
7.4 
7.4 
7.4 
7.4 
7.4 
7.4 
7.4 
7.4 
7.6 
7.5 
7.6 
7.6 
7.5 
7.5 
7.6 
7.6 
7.5 
7.5 
7.5 
7.5 
7.6 
7.5 
7.6 
7.6 
7.6 
7.6 
7.6 
7.6 



7.7 



9.536972 
637382 
537792 
538202 
638611 
539020 
539429 
539837 
540245 
540653 
541061 

9.541468 
541875 
542281 
542688 
543094 
643499 
643J>05 
644310 
644715 
645119 

9.545524 
545928 
546331 
646735 
647138 
547540 
547943 
548345 
648747 
649149 

9.540650 
549951 
550352 
560752 
551162 
661552 
551962 
552351 j 
562750 
553149 

9.553548 
553946 
554344 
554741 
656139 
665536 
555933 
556329 
556725 
557121 

1.557517 
667913 
558308 
658702 
659097 
659491 
659885 
660279 
560673 
561066 

"CotangT 



68.4 

68.3 

68.8 

68.2 

68.2 

68.1 

68.1 

68.0 

68.0 

67.9 

67.9 

67.8 

67.8 

67.7 

67.7 

67.6 

67.6 

67.5 

67.5 

67.4 

67.4 

67.3 

67.3 

67.2 

67,2 

67.1 

67.1 

67.0 

67.0 

66. 

66. 

66. 

66.8 

66.7 

66.7 

66.6 

66.6 

66.6 

66.5 

66.5 

66.4 

66.4 

66.3 

66.3 

66.2 

66.2 

66.1 

66.1 

66.0 

66.0 

65.9 

66.9 

66.9 

65.8 

65.8 

65.7 

65.7 

66.6 

65.6 

65.6 



Cotan);. 



ijiN'. Binc.|;>i. COS. I 



32557 
32584 
32612 



10.463028 

462618 j 

462208 j 

4617981 1 32639 

461389! 32667 

460980' 32694 

460571 I 32722 

460163 I 

459765 i 

459347 I 

458939 i 
10.4585321 

458125 j 

467719 I 

457312 I 

456900 I 

450501 ! 

456095 ! 

455690 1 

455285 

454881 I 
10.454476! 

464072 j 

453669 I 

453265 ! 

452862 I 

452460 ! 

452057 

451655 

451253 

450861 
LO. 450450 

460049 ! 

449648 

449248 

448848 

448448 

448048 



32749 
32777 
32804 
32832 
32859 
32887 
32914 
32942 
32969 
32997 
33024 
33051 
33079 
33106 
33134 
.33161 
33189 
33216 
33244 
33271 
33298 
33326 
33353 
33381 
33408 
33436 
33463 
33490 
33518 
33645 
33573 



447649 1 1 33600 
447250 1 1 33627 
446851 1 1 33665 
33682 



10.446452 
446054 
445656 
445259 
444861 
444464 



83710 
33737 
33764 
33792 
33819 



444067 1133846 
443671 I [33874 
4432761133901 

33929 

3395b 

33983 

34011 

34038 

34065 

34093 

84120 

3414 

4393271134175 
438934 1134202 
Tanjr. !lN. cos. N.sine, 




94552 

94542 

94533 

94523 

94514 

94504 

94495 

94485 

94476 

94466 

94457 

94447 

94438 

94428 

94418 

94409 

94399 

94390 

94380 

94370 

94361 

94351 

94342 

94332 

94322 

94313 

94503 

94293 

94284 

94274 

94264 

94254 

94245 

94235 

94225 

94215 

94206 

94196 

94186 

94176 

94167 

94167 

94147 

94137 

94127 

94118 

94108 

94098 

94088 

94078 

94068 

94058 

94049 

94039 

94029 

94019 

94009 

93999 

93989 

93979 

93969 



70 Degre'js. 



TABLE II. 



Log. i;iire« and TanReiitn. (20<=>) >aluraJ Siuoh. 



41 



Sine. p. W 



534052 
534399 
534745 
535092 
535438 
535783 
536129 
55G474 
536818 
537163 
537507 
537851 
538194 
538538 
538880 
539223 
539565 
539907 
540249 
540590 
540931 
541272 
541618 
541953 
542293 
542632 
542971 
543310 
543649 
543987 
544325 
9,544663 
545000 
545338 
545674 
546011 
546347 
546683 
547019 
547354 
547689 

1.548024 
548359 
548693 
549027 
549360 
649693 
550026 
550359 
550692 
551024 

1.551356 
551687 
552018 
552349 
552680 
653010 
553341 
553670 
554000 
554329 
Cosine. 



57.8 
57.7 
67.7 
67-7 
67.6 
57.6 
57.5 
57.4 
67.4 
57.3 
57.3 
57.2 
57.2 
67.1 
57.1 
67.0 
57.0 
56.9 
56.9 
66.8 
56.8 
56.7 
66.7 
56.6 
56.6 
56.5 
56.6 
58.4 
66.4 
56.3 
56.3 
56.2 
66.2 
66.1 
56.1 
66.0 
56.0 
55.9 
55.9 
56.8 
55.8 
55.7 
65.7 
65.6 
65.6 
55.6 
65.5 
55.4 
55.4 
55.3 
55.3 
56.2 
55.2 
55.2 
55.1 
55.1 
55.0 
55.0 
54.9 
54.9 



Oosiui*. 

). 972986 
972940 
972894 
972848 
972802 
972755 
97270:1 
972663 
972617 
972570 
972524 

). 972478 
972431 
972385 
972338 
972291 
972245 
972198 
972161 
972105 
972058 

>. 972011 
971964 
971917 
971870 
971823 
971776 
971729 
971682 
971635 
971688 

(.971540 
971493 
971446 
971398 
971351 
971303 
971256 
971208 
971161 
971113 

>. 971 066 
971018 
970970 
970922 
970874 
970827 
970779 
970731 
970683 
970635 

1.970586 
970538 
970490 
970442 
970394 
970345 
970297 
970249 
970200 
970162 



D. 10' 



Sine. 



7.7 
7.7 
7.7 
7.7 
7.7 
7.7 
7.7 
7.7 
7.7 
7.7 
7.7 
7.7 
7.8 
7.8 
7.8 
7.8 
7.8 
7.8 
7.8 
7.8 
7.8 
7.8 
7.8 
7.8 
7.8 
7.8 
7.8 
7.9 
7.9 
7.9 
7.9 
7.9 
7.9 
7.9 
7,9 
7,9 
7.9 
7.9 
7.9 
7.9 
7.9 
8.0 
8.0 
8.0 
8.0 
8.0 
8.0 
8.0 
8,0 
8.0 
8.0 
8.0 
8,0 
8,0 
8,0 
8.0 
8,1 
8.1 
8.1 



JTunj^ 

9.561066 
661459 
561851 
662244 
662636 
563028 
563419 
663811 
664202 
564592 
564983 

9,565373 
565763 
566163 
666542 
566932 
667320 
567709 
668098 
568486 
568873 

9.569261 
569648 
670035 
570422 
570809 
571196 
571581 
571967 
672362 
572738 

9.573123 
573507 
573892 
574276 
574660 
576044 
675427 
676810 
576193 
576676 

9.576968 
577341 
677723 
678104 
578486 
578867 
579248 
679629 
580009 
580389 

9.580769 
681149 
581528 
581907 
582286 
582665 
583043 
583422 
583800 
584177 
Cotanp. 



D. 10 



65.5 
65.4 
65.4 
65 3 
66.3 
65.8 
65.2 
65.2 
65.1 
65.1 
65.0 
65.0 
64.9 
64,9 
64.9 
64.8 
64.8 
64.7 
64.7 
64.6 
64.6 
64.6 
64,5 
64,5 
64,4 
64,4 
64.3 
64.3 
64,2 
64.2 
64.2 
64.1 
64.1 
64.0 
64.0 
63,9 
68.9 
63,9 
63.8 
63.8 
63.7 
63.7 
63,6 
63.6 
63.6 
63,5 
63,6 
63,4 
63.4 
63,4 
63,3 
63,3 
63.2 
63,2 
63,2 
63,1 
63.1 
63.0 
63,0 
62.9 



t'otang. >. Kine. N, cos. 



10.438934 
438541 
438149 
437756 
437364 
436972 
436581 
436189 
436798 
435408 
435017 

10.434627 
434237 
433847 
433458 
433068 
432680 
482291 
431902 
431514 
431127 

1 a. 430739 
430362 
429965 
429678 
429191 
428806 
428419 
428033 
427648 
427262 

10.426877 
426493 
426108 
425724 
425340 
424956 
424573 
424190 
423807 
423424 

10.423041 
422659 
422277 
421896 
421614 
421183 
420762 
420371 
419991 
419611 

10.419231 
418851 
418472 
418093 
417714 
417336 
416957 
416578 
416200 
415823 



Tanp;. 



, 3420-: 
1 34229 
i 34257 
134284 
134311 
34339 
34366 
34393 
34421 
34448 
34475 
34503 
84530 
34657 
34584 
34612 
84639 
34666 
34694 
34721 
84748 
84776 
34803 
34830 
84857 
84884 
34912 
349G9 
34966 
34993 
35021 
35048 
85076 
35102 
85130 
8515 
35184 
35211 
35289 
35266 
35293 
35320 
3584 
35375 
35402 
35429 
35456 
35484 
35511 
35538 
35565 
3559i; 
35619 
3564'i 
85674 
35701 
3572b 
35755 
3578i. 
85810 
35837 



93077 
93667 
93657 
93647 
93637 
93626 
93616 
93606 
93596 
93585 
i)3576 
93565 
93555 
93644 
93534 
93524 
93514 
93503 
93493 
93483 
98472 
J3462 
93452 
93441 
93431 
93420 
98410 
93400 
93f,fc9 
93379 
98368 
93368 
i N. COP. N.sine. 



60 
I 59 
68 
57 
56 
55 
54 
63 
L2 
ol 
50 
49 
48 
47 
46 
45 
44 
43 
42 
41 
40 
39 
38 
37 
86 
85 
84 
33 
32 
31 
30 
29 
28 
27 
26 
25 
24 
23 
22 
21 
:.0 
19 
18 
17 
16 
15 
14 
13 
12 
11 
10 
9 
8 
7 
6 
5 
4 
3 
2 
1 




09 Degrcps. 



42 



Log. Sines and Tangents. (21") Natural Sines. 



TABLE IL 



Sine. 



9.554329 
554658 
554987 
555315 
555(J43 
555971 
556299 
556626 
556953 
557280 
557()06 

9.557932 
558258 
558583 
558909 
559234 
55y558 
559883 
560207 
560531 
560855 

9.561178 
561501 
561824- 
662146 
562468 
562790 
563112 
563433 
563755 
664075 

9.564396 
564716 
665036 
565356 
565676 
565995 
566314 
666632 
566951 
567269 

9.567587 
567904 
568222 
568539 
568856 
569172 
569488 
569804 
670120 
570435 

9.570751 
571066 
671380 
571695 
572009 
572323 
672636 
672950 
673263 
673575 
Cosine. 



D. 10"| Cosine. 



64.8 
64.8 
64.7 
54.7 
54.6 
64.6 
54.5 
54.5 
54.4 
54.4 
54.3 
54.3 
54.3 
54.2 
54.2 
54.1 
54,1 
54.0 
54.0 
53.9 
53.9 
53.8 
53.8 
63.7 
53.7 
53.6 



53 

53 

53 

53 

53 

63 

53 

53 

63 

53 

53.1 

53.1 

53,1 

63.0 

53.0 

52,9 

52.9 

52.8 

52.8 

62.8 

52.7 

52.7 

52.6 

52.6 

52.5 

52.6 

52.4 

62.4 

52.3 

52.3 

52.3 

52.2 

52.2 

52.1 



9.970152 
970103 
970055 
970006 
969957 
969909 
969860 
9698 U 
969762 
969714 
969665 

9.969616 
969567 
969518 
969469 
969420 
969370 
969321 
969272 
969223 
969173 

(9.969124 
969075 
969025 
968976 
968926 
968877 
968827 
968777 
968728 
968678 
968628 
968578 
968528 
968479 
968429 
968379 
968329 
968278 
968228 
968178 
968128 
968078 
968027 
967977 
967927 
967876 
957826 
967775 
967725 
967674 

9.967624 
967573 
967622 
967471 
967421 
967370 
967319 
967268 
967217 
967166 



D. 1U"| Tang. 



8 

8 

8.2 

8.2 

8,2 

8.2 

8.2 

8.2 

8.2 

8.2 



2 

2 
2 

2 
2 
2 
3 
8.3 
8.3 
8.3 
8.3 
8.3 
8.3 
8.3 
8.3 
8.3 
8.3 
8.3 
8.3 
8.3 
8.4 
8.4 
8.4 
8.4 
8.4 
8.4 
8.4 
8.4 
8.4 
8.4 
8.4 
8.4 
8.4 
8.4 
8.6 
8.6 
8.6 
8.6 
8.6 
8.6 
8.6 



9.584177 
584555 
684932 
585309 
686686 
686062 
686439 
586815 
687190 
587566 
587941 

9.588316 
688691 
5890d6 
689440 
689814 
590188 
590562 
690935 
691308 
591681 

9.692054 
59242*6 
592798 
593170 
593542 
593914 
594285 
594656 
595027 
595398 
595768 
596138 
696508 
596878 
597247 
697616 
597985 
698364 
598722 
699091 

9.599459 
599827 
600194 
600562 
600929 
601296 
601662 
602029 
602395 
602761 

1.603127 
603493 
603858 
604223 

. 604588 
604953 
605317 
605682 
60)046 
6 06410 

Cotang. 



D. 10"| Cotang. I N .sine.lN. cos. 



62.9 

62.9 

62.8 

62 

62.7 

62.7 

62 

62 

62 

62 

62 

62 

62 

62 

62 

62 

62.3 

62.2 

62.2 

62.2 

62.1 

62,1 

62.0 

62.0 

61.9 

61.9 

61.8 

61.8 

61.8 

61.7 

61.7 

61.7 

61.6 

61.6 

61.6 

61.6 

61.6 

61.5 

61.4 

61.4 

61.3 

61.3 

61.3 

61.2 

61.2 

61.1 

61.1 

61.1 

61,0 

61.0 

61.0 

60.9 

60.9 

60.9 

60,8 

60.8 

60.7 

60.7 

60.7 

60.6 



10.415823 
415445 
416068 
414691 
414314 
. 413938 
413561 
413185 
412810 
412434 
412059 

10.411684 
411309 
410934 
410560 
410186 
409812 
409438 
409065 
408692 
408319 

10.407946 
407574 
407202 
406829 
406458 
406086 
405715 
405344 
404973 
404602 

10.404232 
403862 
403492 
403122 
402753 
402384 
402015 
401646 
401278 
400909 

10.400541 
400173 
399806 
399438 
399071 
398704 
398338 
397971 
397605 
397239 '■' 

10.396873, 
396507 I 
396142 : 
395777 : 
395412 
395047 '■ 
394683 ; 
394318 : 
393954 
393590 ■ 



36837 93358 
36864193348 
35891 193337 



35918 
35945 
36973 
36000 
36027 
36054 
36081 
36108 
36135 
36162 



36190 93222 



36217 



93211 



i 36244 93201 



36271 



93190 



36298 93180 



36325 



93169 



36461 
36488 
36515 



36542 93084 
! 36569 93074 
ij 36596 93063 
1 136623 93052 



36731 

36758 
36785 



j 1 368 12 92978 
1 136839 92967 



! 136867 



36894 92946 



! 36921 



93327 
93316 
93306 
93295 
93286 
93274 
93264 
93253 
93243 
93232 



36352 93159 
36379^3148 
36406 93137 
36434 93127 



93116 
93106 
93095 I 



36650 93042 
36677193031 
36704 93020 



93010 
92999 
92988 



92966 



92935 



Tang. I N. i 



136948 92926 
! 36975 92913 
137002 92902 
137029 92892 
i 37056 92881 
' 37083 92870 
137110 92859 
: 37 137 92849 
37164 92838 
3719192827 
: 37218 92816 
37245 92805 
37272 92794 
i 37299 92784 
137326 92773 
37353 92762 
37380 92751 
137407 92740 
j 37434 92729 
13746192718 



Log. Sines and Tangents. (22'') Natural Sines. 



43 



Sine. 



9,573675 



574200 
674512 
574824 
576136 
575447 
575758 
576069 
576379 
576689 

►.576999 
677300 
577618 
577927 
578236 
578545 
678853 
579162 
579470 
679777 

1.580085 
580392 
580699 
581005 
681312 
681618 
681924 
682229 
582636 
582840 

.583145 
683449 
583754 
684058 
584361 
584665 
584968 
686272 
685574 
585877 

.586179 
686482 
686783 
587085 
587386 
687688 
687989 
588289 
588690 
688890 

i. 6891 90 
589489 
589789 
590088 
E90387 
590686 
590984 
691282 
591680 
591878 
Cosine. 



D7W> 



52.1 
52.0 
52.0 
51.9 
51.9 
61.9 
51.8 
51.8 
51.7 
51.7 
51.6 
51.6 
51.6 
61.6 
51.5 
61.4 
51.4 
51.3 
51.3 
51.3 
51.2 



51.2 
51.1 
61.1 
51.1 
51.0 
51.0 
60.9 
60.9 
50.9 
50.8 
50.8 
60.7 
60,7 
50.6 
50.6 
60.6 
60.6 
50.5 
50.4 
60.4 
60.3 
60.3 
60.3 
50.2 
60.2 
60.1 
50.1 
50.1 
60.0 
50.0 
49.9 
49.9 
49,9 
49.8 
49.8 
49.7 
49.7 
49.7 
49.6 



Cosine. 



.967166 
967115 
967064 
967013 
966961 
966910 
966859 
966808 
966756 
966705 
966653 
.966602 
966550 
966499 
966447 
966395 
966344 
966292 
966240 
966188 
966136 
.966085 
966033 
965981 
965928 
966876 
965824 
965772 
965720 
965668 
965815 
.965563 
966511 
965458 
966406 
965353 
965301 
965248 
965195 
965143 
965090 
965037 
964984 
964931 
964879 
964826 
964773 
964719 
964666 
964613 
964560 
.964607 
964464 
964400 
964347 
964294 
964240 
964187 
964133 
964080 
964026 



Sine. 



D. 10" Tan" 



8.5 
8,5 
8.5 

a,5 

8.5 
&.5 
8.6 
8.5 
8.6 
8.6 
8.6 
8.6 
8.6 
8.6 
8.6 
8.6 
8.6 
8.6 
8.6 
8.6 
8.6 
8.7 
8.7 
8.7 
8.7 
8.7 
8.7 
8.7 
8.7 
8.7 
8.7 
8.7 
8.7 
8.7 
8.7 
8.8 
8.8 
8.8 
8.8 
8.8 
8.8 
8.8 
8.8 
8.8 
8.8 
8.8 
8.8 
8.8 
8.9 
8.9 
8.9 
8.9 
8.9 
8.9 
8.9 
8.9 
8.9 
8.9 
8.9 
8.9 



). 606410 
606773 
607137 
607500 
607863 
608225 
608588 
608950 
609312 
609674 
610036 

). 610397 
610759 
611120 
611480 
611841 
612201 
612561 
612921 
613281 
613641 

>. 614000 
614369 
614718 
615077 
615435 
615793 
616161 
616509 
616867 
617224 

1.617582 
617939 
618295 
618652 
619008 
619364 
619721 
620076 
620432 
620787 

1.621142 
621497 
621852 
622207 
622561 
622915 
623269 
623623 
623976 
624330 

1.624683 
626036 
625388 
625741 
626093 
626445 
626797 
627149 
627601 
627852 

Co tang. 
67 Degrees. 



D. 10' 



60.6 
60.6 
60.5 
60.6 
60.4 
60.4 
60.4 
60.3 
60.3 
60.3 
60.2 
60.2 
60.2 
60.1 
60.1 
60.1 
60.0 
60.0 
60.0 
59.9 
59.9 
59.8 
59.8 
59.8 
59.7 
59.7 
69.7 
59.6 
69.6 
59.6 
59.5 
59.5 
69.5 
69.4 
59.4 
59.4 
59.3 
59.3 
59.3 
59.2 
59.2 
59.2 
59.1 
59.1 
59.0 
59.0 
59.0 
58.9 
68.9 
68.9 
58.8 
58.8 
58.8 
58.7 
58.7 
58.7 
58.6 
58.6 
58.6 
58.5 



Cotang. ; N . Hine.l N. 



10.393590 
393227 
392863 
392500 
392137 
391775 
391412 
391050 
390688 
390326 
389964 

10.389603 
389241 
388880 
388520 
388169 
387799 
387439 
387079 
386719 
386359 

10.386000 
385641 
385282 
384923 
384565 
384207 
383849 
383491 
383133 
382776 

10-382418 
382061 
381706 
381348 
380992 
380636 
380279 
379924 
379568 
379213 

10-378858 
378503 
378148 
377793 
377439 
371085 
376731 
376377 
376024 
375670 

10.375817 
374964 
374612 
374259 
373907 
373665 
373203 
372851 
372499 
372148 
f^iig." 



37461 
37488 
37515 
37542 
37569 
37695 
37622 
37649 
37676 
37703 
37730 
37757 
37784 



92718 
92707 
92697 
92686 
92675 
92664 
92653 
92642 
92631 
92620 
92609 
92598 
92587 



3781192676 



37838 
37865 
37892 
37919 
37946 
37973 
37999 
38026 
38053 
38080 
38107 
38134 
38161 
38188 



38241 
38268 
38295 
38822 
38349 
38376 
38403 
38430 



92565 
92554 
92543 
92532 
92521 
92510 
92499 
92488 
92477 
92466 
92456 j 36 



92444 
92432 
92421 



38215 92410 



92399 
92388 
92377 
92366 
92355 
92343 
92332 
92321 
38456 92310 
38483192299 
38510J92287 
38537 92276 
38564 92265 



38591 
38617 
38644 
38671 
38698 
38725 



92254 
92243 
92231 
92220 
92209 
92198 



38778 92175 
38805 92164 
38832 92152 
3885992141 
38886i92130 



38912 
38939 
38966 
38993 
39020 
39046 
39073 
N. CO?. 



02119 
92107 
92096 
92085 
92073 
92062 
[92050 
>^Bine. 



35 I 

34 

03 

32 

31 

30 

29 

28 

27 

26 

25 

24 

23 

22 

21 

20 

19 

18 

17 

16 

15 

14 

13 

12 

11 

10 

9 

8 i 

7 : 

6 

5 

4 , 

3 

2 

1 





44 



I>og. Sine« and Tangents. (23°) Natural Sines. 



TABLK II. 





1 
2 
3 
4 
5 
6 
7 
8 
9 
10 
11 
12 
13 
14 
15 
1() 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
I 31 
i 32 
33 
34 
85 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
57 
58 
59 
60 



S ine. 

9.591878 
592176 
592473 
592770 
693067 
593303 
593659 
593955 
594251 
594547 
594S42 

9.595137 
595432 
595727 
598021 
596315 
696609 
596903 
597196 
597490 
597783 

9.598075 
598368 
598660 
598952 
599244 
599536 
599827 
600118 
600409 
600700 

9.600990 
601280 
601570 
601860 
602150 
602439 
602728 
603017 
603395 
603594 

9.603882 
604170 
604457 
604745 
605032 
605319 
605606 
605892 
606179 
60S465 

9.606751 
607030 
607322 
607607 
607892 
608177 
608461 
608745 
609029 
609313 



D W- 



Cotinu 



49.6 
49.5 
49.6 
49.5 
49.4 
49.4 
49.3 
49.3 
49.8 
49.2 
49.2 
49.1 
49.1 
49.1 
49.0 
49.0 
48.9 
48.9 
48.9 
48.8 
48.8 
48.7 
48.7 
48.7 
48.6 
48.6 
48.6 
48.6 
48.6 
48.4 
48.4 
48.4 
48.3 
48.3 
48.2 
48.2 
48.2 
48.1 
48.1 
48.1 
48.0 
48.0 
47.9 
47.9 
47.9 
47 8 
47.8 
47.8 
47.7 
47.7 
47-6 
47.6 
47.6 
47-5 
47-6 
47.4 
47.4 
47.4 
47.3 
47.3 



Oosinc. 

1.964026 
963972 
963919 
963865 
963811 
963757 
963704 
963650 
963596 
963542 
963488 
.963434 
963379 
963325 
963271 
963217 
963163 
963108 
963054 
962999 
962945 
.962890 
962836 
962781 
962727 
962672 
962617 
962562 
962508 
962453 
962398 
.962343 
962288 
962233 
962178 
962123 
962037 
962012 
961957 
961902 
961846 
.961791 
961735 
961680 
961624 
961569 
961513 
961458 
961402 
961346 
961290 
.961235 
961179 
961123 
961037 
961011 
960955 
960899 
960843 
960786 
960730 
Sine. 



Ta ng. 

9.627852 
628203 
628564 
628906 
629255 
629606 
629956 
630306 
6306.56 
631005 
631355 

9.631704 
632053 
632401 
632750 
633098 
633447 
633795 
634143 
634490 
634838 
,635l8o 
635532 
635879 
636226 
636572 
636919 
637265 
637611 
637956 
638302 
638647 
638992 
639337 
639682 
640027 
640371 
640716 
641060 
641404 
641747 

9.642091 
642434 
642777 
643120 
643463 
643806 
644148 
644490 
644832 
645174 
.645516 
645857 
640199 
646540 
646881 
647222 
647662 
<347903 
648243 
648583 
Cotanii. 



^6 De^nvF. 



58.5 

58.5 

58.5 

58.4 

58.4 

68 

58 

58 

58 

58 

58 

58 

58.1 

68.1 

58.1 

58.0 

58.0 

58.0 

57.9 

57.9 

57.9 

57.8 

67.8 

67.8 

57.7 

67.7 

57.7 

57.7 

57.6 

57.6 

57.6 

57.5 

67.5 

57.5 

57.4 

67.4 

57.4 

67.3 

57.3 

67.3 

57.2 

57.2 

57.2 

57.2 

57.1 

57.1 

67.1 

57.0 

•57.0 

57.0 

66.9 

56.9 

66.9 

m.o 

56.8 
66.8 
56.8 
66.7 
56.7 
56.7 



Cotang. I N. sine. N. COS. 



10.372148 139078 92050 



371797 '139100 
3714461 [39127 
371095!' 39153 
370746 1139180 



i 39207 
1 39234 
! 39260 



370394 
370044 
369694 
369344 
368996 

368646 1 39341 

10.368296 '39367 

367947 i 39394 

367599! 139421 



92039 
92028 
92016 
92005 
91994 
91982 
91971 



39287 91959 

39314 91948 

91936 

91925 

91914 

91902 

367250 II 39448 91891 

366902 1 1 39474 '91879 



366553 r 39501 
366205 ii 39528 
366857 j 139555 
365610! 1 39581 



3651621139608 91822 



10.364815 '139636 
364468 1 139661 
364121 ! 
363774 ! 
363428 ! 
363081 I 
362735 i 
362389 ' 
362044 1 
361698 ! 



39741 



39795 



39848 
39875 



91868 
91856 
91845 
91833 



91810 

91799 

39688 91787 

39715 91775 



91764 



39768 91752 



91741 



39822 91729 



91718 
91706 



10.361353 !i 39902 91694 
361008 II 39928191683 
360G63 'I 39955 19 1671 
360318 l| 39982 91660 



91648 
91636 
[40062 91625 
,40088 91613 



359973! 1 40008 
359629 II 40035 
359284 
358940 
358596!|40115|91601 
358253 ij 40141 bloiiO 
1 0. 357909 I j 401 68 b 167 8 



357o66 ' 
867223 I 
356880 1 
356537 I 
366194! 14030) 



40195191566 
40221 191555 



40248 



191543 



40275 91531 
91519 



365862 1140328 9 16(j8 
36.5610 j 40355 91496 
355168 j 40381 



354826 
10.354484 
364143 
353801 
363460 
353119 
362778 
352438 
352097 
351757 
351417 



40408 
40434 



91484 
91472 
91461 



40461 91449 



40488 



40541 
40567 
40594 



40621 91378 



91366 
91355 
Tang, ji N. cop. N.siTic 



40647 
40674 



91437 



40514 91425 



91414 
91402 
91390 



60 

59 
58 
57 
66 
55 
54 
53 
52 
61 
50 
49 
48 
47 
46 
45 
44 
43 
42 
41 
40 
39 
38 
37 
36 
35 
34 
33 
32 
•■il 
fcO 
29 
•2b 
27 
26 
25 
'J 4 
23 
22 
21 
2U 
19 
18 
17 
16 
15 
14 
Ki 
12 

n 

10 
9 
8 
7 
6 
5 
4 

2 
1 




Log. Sines and Tangents. (2-40) Natural Sines. 



45 



Sine. 






9.609313 




609597 


2 


609880 


3 


610164 


4 


610447 


5 


610729 


6 


611012 


7 


611294 


8 


611576 


9 


611858 


10 


612140 


11 


9.612421 


12 


612702 


13 


612983 


14 


613264 


15 


613545 


16 


613825 


17 


614105 


18 


614385 


19 


614666 


20 


614944 


21 


9.615223 


22 


615502 


23 


615781 


24 


616060 


25 


616338 


26 


616616 


27 


616894 


28 


617172 


29 


617450 


30 


617727 


31 


9.618004 


32 


618281 


83 


618558 


34 


618834 


36 


619110 


36 


619386 


37 


619662 


38 


619938 


39 


620213 


40 


620488 


41 


9.620763 


42 


621038 


43 


621313 


44 


621587 


46 


621861 


46 


622135 


47 


622409 


48 


622682 


49 


622956 


50 


623229 


51 


9.623512 


52 


623774 


63 


624047 


54 


624319 


56 


624591 


56 


624863 


57 


626136 


68 


625406 


59 


625677 


60 


625948 



D. 10' 



47.3 
47.2 

47.2 
47.2 
47.1 
47.1 
47.0 
47.0 
47.0 
46.9 
46.9 
46.9 
46.8 
46.8 
46.7 
46.7 
46.7 
46.6 
46.6 
46.6 
46.6 
46.6 
46.5 
46.4 
46.4 
46.4 
46.3 
46.3 
46.2 
46.2 
46.2 
46.1 
46.1 
46.1 
46.0 
46.0 
46.0 
45.9 
45.9 
45.9 
45.8 
45.8 
45.7 
45.7 
46.7 
46.6 
46.6 
45.6 
46,5 
45.5 
46.6 
45.4 
45.4 
45.4 
45.3 
45.3 
46,3 
45.2 
46.2 
45.2 



Cosine. 

1.960730 
960674 
960618 
960561 
960505 
960448 
960392 
960336 
960279 
960222 
960165 

1.960109 
960052 
959996 
959938 
959882 
959825 
959768 
959711 
959664 
959596 

i. 959539 
969482 
959425 
959368 
959310 
959263 
969195 
969138 
959081 
969023 

.958966 



D. ic/ 



958850 
958792 
958734 
958577 
958619 
958661 
958603 
958445 
958387 
958329 
958271 
958213 
958154 
958096 
968038 
957979 
957921 
957863 
967804 
957746 
967687 
957628 
957570 
957511 
957462 
957393 
957335 
957276 



9.4 

9.4 

9.4 

9.4 

9.4 

4 

4 

4 

4 

4 

4 

6 

6 

5 

5 



9.6 
9.6 
9.5 
9.5 
9.5 
9.5 
9.5 
9.5 
9.6 
9.6 
9.6 
9.6 
9.6 
9.6 
9.6 
9.6 
9.6 
9.6 
9.6 
9.6 
9.6 
9.6 
9.6 
9.7 
9.7 
9.7 
9.7 
9.7 
9.7 
9.7 
9.7 
9.7 
9.7 
9.7 
9.7 
9.7 
9.8 
9.8 
9.8 
9.8 
9.8 
9.8 
9.8 
9.8 



Tanft. 



9.648583 
648923 
649263 
649602 
649942 
660281 
650620 
660959 
651297 
651636 
651974 

9.652312 
652650 
652988 
653326 
653663 
654000 
654337 
654174 
665011 
666348 
655684 
656020 
666356 
656692 
657028 
657364 
667699 
658034 
658369 
658704 

9.659039 
659373 
659708 
660042 
660376 
660710 
661043 
661377 
661710 
662043 

9.662376 
662709 
663042 
663375 
663707 
664039 
664371 
664703 
665035 
666366 

3,666697 
666029 
666360 
666691 
667021 
667352 
667682 
6()8013 
668343 
668672 



D. W 



66.6 
66.6 
66.6 
56 6 
66.5 
66.6 
59.6 
66.4 
56.4 
66.4 
66.3 
56.3 
56.3 
56.3 
56.2 
56.2 
66.2 
56.1 
56.1 
56.1 
66.1 
66.0 
56.0 
56.0 
56.9 
66.9 
55.9 
55.9 
65.8 
55.8 



Uotang. 



Cotang. 



65 

55 

65.6 

55.6 

55.6 

65.6 

55.5 

55 

55 

55 

56 

55 

55 

56 

56.3 

55.3 

55.2 

65.2 

55.2 

65.1 

55.1 

56.1 

56.1 

66.0 

55.0 

56.0 



10.351417 
351077 
350737 
350398 
350058 
349719 
349380 
349041 
348703 
348364 
348026 

10.347688 
347350 
347012 
346674 
346337 
346000 
345663 
345326 
344989 
344652 

10.344316 
343980 
343644 
343308 
342972 
342636 
342301 
341966 
341631 
341296 

10.340961 
340627 I 
340292 i 
339958 
339624 
339290 
338967 
338623 
338290 
337957 

10.337624 
337291 
336958 
336625 
336293 
336961 
336629 
335297 
334965 
334634 

10.334303 
333971 
333620 
333309 
332979 
332648 
332318 
331987 
331657 
331328 



N. sine. N. cos. 



40674 
40700 
40727 
40753 
40780 
40806. 
40833 
40860 
40886 
40913 



40966 
40992 
41019 
41045 
41072 
41098 



41161 
41178 
41204 
41231 
41257 



91365 
91343 
91331 
91319 
91307 
91295 
91283 
91272 
91260 
91248 
91236 
91224 
91212 
91200 
91188 
91176 
91164 



41125 91162 



91140 
91128 
91116 
91104 
91092 



41284 91080 
4131091068 
41337 "91056 



Tang. 



41363 
41390 
41416 
41443 
41469 
41496 



41522 90972 



41549 
41675 



90960 
90948 



41602 90936 
41628 90924 



41655 
41681 
41707 
41734 
41760 
41787 
41813 
41840 
41866 



41919 
41946 
41972 
41998 
42024 



91044 
91032 
91020 
91008 
90996 
90984 



90911 
90899 
90887 
90875 
90863 
90861 
90839 
90826 
90814 



41892 90802 



90790 
90778 
90766 
90753 
90741 



4205190729 
42077 90717 
4210490704 
42130190692 
42166J90680 
42183 90668 
42209 90656 
42235 90643 
42262 90631 
N. COS. N. sine. 



65 Degrees. 



46 



Log. Siucs and Tant^enu?. (ti6 ) N^axiiral Sines. TAULE II. 



Cotaug. I N .siuc. N 

10.331327! 
330998 
330Jei8 :' 
380339 
330009 
329(i80! 
329351;: 
32902311 
328694:; 

328037;; 
10.327709 I 

3273811 

327053;; 

32672611 

326398 ]> 

326071 ': 

325743 1 ; 

32541611 

325090! I 

3247631; 
10.324436;! 

3241 10 j I 

323784;; 

3234571; 

323131 i 

322806 ji 

322480 it 

32215411 

321829 li 

321504 
10.321179 

320854 

320529 

3202061! 

319880 i 

319556 I 

319232]! 

31890811 

318684!! 

31826011 
10.317937 i 

317613 

31729011 

316967;: 

316644/ 

316321 i! 

31599911 

3156761: 

3153541! 

31503211 
10.314710;; 

314388!; 

814066 i; 

313745!; 

313423;; 

3131021: 

312781 

31246a 

312139 

311818 



.Sina. 



D. 10" 



9.625948 

626219 

626490 

6267i)0 

627030 

627300 

627570 

627840 

628109 

628378 

628647 
9.628916 

629185 

629453 

629721 

629989 

630257 

630524 

630792 

631059 

631326 
9.631593 

631859 

632125 

632392 

632658 

632923 

633189 

633454 

6337191 1: 

633984 
9.634249 

634514 

634778 

635042 

635308 

635570 

635834 

636097 

636360 

636623 
9.636886 

637148 

637411 

637673 

637935 

638197 

638458 

638720 

638981 

639242 
9.639503 

639764 

640024 

640284 

640644 

640804 

641064 

641324 

641684 

641842 
Cosine. I 



45, 
45, 
45, 
45. 
45, 
45, 
44. 
44, 
44, 
44, 
44. 
44. 
44. 
44. 
44. 
44. 
44. 
44. 
44. 
44. 
44. 
44. 
44. 
44. 
44. 
4A. 
44. 
44. 
144. 



9.8 
9.8. 
9.& 
9,8 
9.8 
9.8 
9,9 
9.9 
9.9 
9.9 
9 9 



Cosine. |0. 10' 

9,957276 

9572 17 

967158 

957099 

957040 

956981 

956921 

956862 

956803 

956744 

956684 
9.956625 

956566 

9565 J6 

956447 

956387 

956327 

956268 

956208 

956148 

956089 
9.956029 

955969 

956909 

955849 

955789 

956729 

955669 

955609 

955548 

955488 
9.955428 

955368 

955807 

955247 

955186 

955126 

9550G6 

9550U5 

95494-4 

964883 
9.954823 

954762 

964701 

954640 

954579 

964518 

954457 

954396 

954335 

954274 
9.954213 

954152 

964090 

954029 

953968 

953906 

953845 

953783 

953722 

953660 

Sin-'. 



9 9 
9.9 
9 9 
9 9 
9.9 
9.9 
9.9 
10.0 
10.0 
10.0 
10.0 
10.0 
10.0 
10.0 
10.0 
10.0 
10.0 
10.0 
10.0 
10.0 
10.1 
10.1 
10.1 
10.1 
10.1 
10.1 
10.1 
10.1 
10.1 
10.1 
10.1 
10.1 
10.1 
10.1 
10.1 
10.2 
10.2 
10.2 
10.2 
10.2 
10.2 
10.2 
10.2 
10.2 
10.2 
10.2 
10.2 
10.2 
10.3 



Tan g. 

9.668673 
669002 
669332 
669661 
669991 
670320 
670649 
670977 
671306 
671634 
671963 

9.672291 
672619 
672947 
673274 
673602 
073929 
674257 
674584 
674910 
676237 

9.675664 
675890 
676216 
676543 
676859 
677194 
677520 
677846 
678171 
678496 

9.678821 
679146 
67947J 
„679795 
680120 
680444 
680768 
681092 
681416 
681740 

9.682063 
682387 
682710 
683083 
683356 
683679 
684001 
684324 
684646 
684968 

9.686290 
085612 
685934 
686256 
686577 
686898 
687219 
687640 
687861 
6 88182 
Co tan J. 
64 Dcgrwfl. 



D. 10" 



42262 
4228fc 
42816 
4-J841 
4236, 
4239-1 
424:.i0 
42446 
42478 
42499 
42525 
4255'..' 
42578 
42604 
42631 
42667 
42683 
42709 
4273() 
42762 
42788 
42815 
42841 
42867 
42894 
42920 
42946 
42972 
42999 
43025 
43051 
43077 
43104 
43130 
43166 
43182 
43209 
48235 
43261 
43287 
43313 
43340 
43366 
43392 
43418 
43445 
43471 
43497 
43528 
43549 
43575 
43602 
43G28 
43654 
43680 
43706 
43738 
43759 
43785 
43811 
43837 



Tane. 



90631 
^iHJlo 

J0594 

90669 
90557 
90545 
90532 
)0520 
90507 
90495 
90483 
90470 
90458 
90446 
90433 
90421 
90408 
90396 
90383 
90371 
90358 
90346 
90334 
90321 
9030y 
90296 
90284 
90271 
90259 
90246 
90233 
90221 
90208 
90196 
90188 
90171 
90158 
90146 
90188 
90120 
901 08 
90095 
90082 
90070 
90057 
90045 
90082 
90019 
90007 
89994 
89981 
89968 
89956 
89943 
89930 
89918 
89905 
89892 
89879 



Log. Sine* and Tangents. (26°) Natural Sines. 



47 



Sine. D. 10" Cosine. 



641842 
642101 
642360 
642618 
642877 
643135 
643393 
643660 
643908 
644166 
644423 
644680 
644936 
645193 
645450 
645706 
645962 
646218 
646474 
64G729 
646984 
647240 
647494 
647749 
648004 
648268 
648512 
648766 
649020 
649274 
649527 
649781 
650034 
650287 
650639 
650792 
651044 
651297 
651549 
651800 
652052 

9.652304 
652665 
652806 
653057 
653308 
653558 
653808 
654059 
654309 
654558 

9.654808 
665058 
655307 
655556 
655805 
656054 
656302 
656651 
656799 
657047 



43.1 
43.1 
43.1 
43.0 
43.0 
43.0 
43.0 
42.9 
42.9 
42.9 
42.8 
42.8 
42.8 
42.7 
42.7 
42.7 
42.6 
42.6 
42.6 
42.5 
42.5 
42.5 
42.4 
42.4 
42.4 
42.4 
4a. 3 
42.3 
42.3 
42.2 
42,2 
42.2 
42.2 
42.1 
42.1 
42.1 
42.0 
42.0 
42.0 
41.9 
41.9 
41.9 
41.8 
41.8 
41.8 
41.8 
41.7 
41.7 
41.7 
41.6 
41.6 
41.6 
41.6 
41.5 
41.5 
41.5 
41.4 
41.4 
41.4 
41.3 



Cosine, j 



26 



9.953660 
953599 
953537 
963475 
953413 
953352 
953290 
953228 
958166 
953104 
963042 

9.952980 
952918 
952855 
952793 
962731 
952669 
952606 
952544 
952481 
952419 

9.952356 
962294 
952231 
952168 
962106 
952043 
951980 
951917 
951854 
951791 

9.951728 
951665 
951602 
951539 
961476 
961412 
961349 
961286 
951222 
951169 
951096 
951032 
960968 
950905 
950841 
950778 
950714 
950650 
950586 
950522 
950458 
950394 
950330 
950366 
950202 
960188 
960074 
950010 
949945 
949881 
"sine. 



D. 10" Tang. 



10.3 
10.3 
10.3 
10.3 
10.3 
10.3 
10.3 



10 

10 

10 

10 

10 

10 

10 

10 

10 

10 

10 

10 

10 

10 

10 

10 

10 

10 

10 

10 

10 

10 

10.5 

10.6 

10.6 

10.5 

10.5 

10.6 

10.5 

10.5 

10.6 

10.6 

10.6 

10.6 

10.6 

10.6 

10.6 

10.6 

10.6 

10.6 

10.6 

10.6 

10.6 

10.7 

10.7 

10.7 

10.7 

10.7 

10.7 

10.7 

10.7 

10.7 

10.7 



688182 
688502 
688823 
689143 
689463 
689783 
690103 
690423 
690742 
691062 
691381 
691700 
692019 
692338 
692656 
692976 
693293 
693612 
693930 
694248 
694566 
694883 
695201 
695518 
695836 
696153 
696470 
696787 
697103 
697420 
697736 
698053 



698685 
699001 
699316 
699632 
699947 
700263 
700678 
700893 

.701208 
701623 
701837 
702162 
702466 
702780 
703096 
703409 
703723 
704036 

. 704350 
704663 
704977 
705290 
705603 
705916 
706228 
706541 
706854 
707166 



_10^ 



53.4 

53.4 

53.4 

53.3 

53.3 

53.3 

63 

63 

63 

63 

63 

53 

53 

53.1 

53.1 

53.1 

53.0 

53.0 

53.0 

63.0 

62.9 

52.9 

62.9 

52.9 

52.9 

52.8 

52.8 

52.8 

62.8 

52.7 

52.7 

52.7 

52.7 

52.6 

52.6 

62.(5 

52.6 

52.6 

52.6 

62.5 

62.6 

62.4 

62.4 

52.4 

52.4 

52.4 

52.3 

52.3 

52.3 

52.3 

52.2 

52.2 

52.2 

52.2 

52.2 

52.1 

52.1 

62.1 

52.1 

52.1 



Cclang 
Degrees. 



Cotong. j|N. sine 



10.311818 
311498 
311177 
310857 
310537 
310217 
309897 
309677 
309268 
308938 
308619 

10.308300 
307981 
307662 
307344 
307026 
306707 
306388 
306070 
305752 
305434 

10.305117 
304799 
?J 04482 
304164 
303847 
303580 
303213 
302897 
302680 
302264 

10-301947 
301631 
301315 
300999 
300684 
300368 
300058 
299737 
299422 
299107 

10-398792 
298477 
298163 
297848 
297684 
297220 
296906 
2S6691 
296277 
295964 

10.2^5660 
295337 
295028 
294710 
294397 
294084 
293772 
293469 
29ol46 
292884 



Tang 



43837 
43863 
43889 
43916 
43942 
43968 
43994 
44020 89790 



N^oos 

89879 
89867 
89854 
89841 
89828 
89816 
89803 



44046 
44072 
44098 
44124 
44151 
44177 
44208 
44229 
44265 
44281 
44307 
44333 
44359 
44885 
44411 
44437 
44464 
44490 
44516 
44642 
44668 
44694 
44620 
44646 
44672 
44698 
44724 
44750 
44776 
44802 
44828 
44854 
44880 
449U6 
44932 
44958 
44984 
45010 
45036 
145062 
45088 
46114 
45140 
45166 
45192 
45218 
45248 
45269 
45295 
45321 
45347 
45373 
45599 
N . «-os. N.sine. 



89777 
89764 
89762 
89739 
89726 
89718 
89700 
89687 
89674 
89662 
89649 
89636 
89623 
89610 
89597 
89534 
89571 
89558 
89545 
89532 
89519 
89506 
89493 
89480 
89467 
89454 
89441 
89428 
89415 
89402 
89889 
89376 
89363 
89860 
89337 
89324 
89811 
89298 
89285 
89272 
89259 
89246 
89232 
89219 
89206 
b9193 
89180 
89167 
89153 
89140 
89127 
89114 
89101 



48 



Log. Sines and Tangent*. (27^) Natural Sines. 



TABLE n. 



19.657047 
667295 
657642 
667790 
658037 
668284 
668531 
668778 
659025 
659271 
659517 

9.659763 
660009 
660256 
660601 
660746 
660991 
661236 
661481 
661726 
661970 

9.662214 
662469 
662703 
662946 
663190 
663433 
663677 
663920 
664163 
664406 

9.664648 
664891 
665133 
665375 
665617 
666859 
666100 
666342 
666583 
666824 

9.667065 
667305 
667546 
667786 
668027 
668267 
668506 
668746 
668986 
669225 
669464 
669703 
669942 
670181 
670419 
670668 
670896 
671134 
671372 
671609 



Cosine. 



D. 10 



41.3 
41.3 
41.2 
41.2 
41.2 
41.2 
41.1 
41.1 
41.1 
41.0 
41.0 
41.0 
40.9 
40.9 
40.9 
40.9 
40.8 
40.8 
40.8 
40.7 
40.7 
40.7 
40.7 
40.6 
40.6 
40.6 
40.5 
40.6 
40.5 
40.5 
40.4 
40.4 
40.4 
40.3 
40.3 
40.3 
40.2 
40.2 
40.2 
40.2 
40.1 
40.1 
40.1 
40.1 
40.0 
40 
40.0 
39.9 
39.9 
39.9 
39.9 
39.8 
39.8 
39.8 
39.7 
39.7 
39.7 
39.7 
39.6 
39.6 



.949881 
949816 
949762 
949688 
949623 
949558 
949494 
949429 
949364 
949300 
949235 

.949170 
949105 
949040 
948975 
948910 
948846 
948780 
948716 
948650 
948584 

.948619 
948454 
948388 
948323 
948267 
948192 
948126 
948060 
947995 
947929 

.947863 
947797 
947731 
947666 
947600 
947633 
947467 
947401 
947336 
947269 

.947203 
947136 
947070 
947004 
946937 
946871 
946804 
946738 
946671 
946604 

.946538 
946471 
946404 
946337 
946270 
946203 
946136 
946069 
946002 
945935 



Sine. 



). 707166 _» 

707478 IJ.' 

707790 IJ.' 

708102 It' 

708414 °f 

708726 °j 

709037 ?{• 

709349 °J 

709660 l\ 

709971 l\- 

710282 °J" 

>. 710593 l\- 

710904 l\' 

711216 °;- 

711525 °;- 
711836 l\- 
712146 °J- 
712456 l\- 
712766 l\- 
713076 ^}- 
713386 l\- 

>. 713696 l\- 
714006 l\- 
714314 l\- 
714624 l\- 
714933 °f- 
.715242 l\- 
715651 °J- 
716860 l\- 
716168 l\- 
716477 °} • 

1.716786 °)- 
717093 I j; • 
717401 1 1\ ■ 
717709 l\- 
718017 l\- 
718325 ^f- 
718633 l\- 
718940 l\- 
719248 ?;• 
719555 ^|- 

.719862 ^;- 
720169 °}- 
720476 °}- 
720783 l\- 
721089 ?}• 
721396 °\- 
721702 v.- 
722009 2|- 
722316 °}- 
722621 ?J- 

1.722927 °\- 

723232 °/;- 

723638 ;°"- 
723844^^. 

7241491^0- 
724454 I °" • 
724759 I °" • 
725066;°"- 
726369^2- 
726674 p"- 



Co tang. 
Degreea. 



Cotang. I N. sine. N. cos, 



10 



10 



10 



10 



.292834 
292622 
292210 
291898 
291686 
291274 
290963 
290651 
290340 
290029 
289718 
.289407 
289096 
288785 
288475 
288164 
287864 
287644 
287234 
286924 
286614 
.286304 
285996 
286686 
285376 
286067 
284758 
284449 
284140 
283832 
283523 
,283215 
282907 
282599 
282291 
281983 
281675 
281367 
281060 1 
280752 1 
280445 i 
280138 
279831 1 
279524 , 
279217 
278911 
278604 
278298 
277991 
277686 
277379 
277073 
276768 
276462 
276156 
275861 
275546 
276241 
274935 
274631 
274326 



Tiing. 



45399 
46426 
45451 
45477 
45503 
45529 
46664 
45680 
45606 
45632 
45668 
45684 
45710 
46736 
45762 
45787 
45813 
45839 
45865 
45891 
45917 
45942 
45968 
45994 
46020 
46046 
46072 
46097 
46123 
46149 
46175 
46201 
46226 
46262 
46278 
46304 
46330 
46356 
46381 
46407 
4<)433 
46458 
46484 
46510 
46536 
46561 
46587 
46613 
46639 
46664 
46690 
46716 
46742 
46767 
46793 
46819 
46844 
46870 
46896 
46921 
46947 



N. COF. .^.^iln 



89101 
89087 
89074 
89061 
89048 
89036 
89021 
89008 
88995 
88981 
88968 
88955 
88942 
88928 
88916 
88902 
88888 
88875 
88862 
88848 
88835 
88822 
88808 
88795 
88782 
88768 
88766 
88741 
88728 
88715 
88701 
88688 
88674 
88661 
88647 
88634 
88620 
88607 
88593 
88580 
88566 
88653 
88639 
88526 
88512 
88499 
88485 
88472 
88458 
88445 
88431 
88417 
88404 
88390 
88377 
88363 
88849 
88336 
88322 
88308 
88295 



TABLE II. 



Log. Sines and Tangents. (28°) Natural Smes. 



37 



60 



S ine. 

9.671609 
671847 
672034 
672321 
672558 
672795 
673032 
673268 
673505 
673741 
673977 

9.674213 
674448 
674684 
674919 
675155 
675390 
675624 
675859 
676094 
676328 

9.676562 
676796 
677030 
677264 
677498 
677731 
677964 
678197 
678430 
6786S3 

9.678895 
679128 
679360 
679592 
679824 
680056 
680288 
680519 
680750 
680982 

9.681213 
681443 
681674 
681905 
682135 
682365 
682595 
682825 
683055 
683284 
.683514 
683743 
683972 
684201 
684430 
684658 
684887 
685115 
685343 
685571 



D. 10" 



Cosino. 



Cosine. 



1.945935 
945868 
945800 
945733 
945666 
945598 
945531 
945464 
945396 
945328 
945261 
.945193 
945125 
945058 
944990 
944922 
944854 
944786 
944718 
944650 
944582 
.944514 
944446 
944377 
944309 
944241 
944172 
944104 
944036 
943967 
943899 
.943830 
943761 
943693 
943624 
943556 
943486 
943417 
943348 
943279 
943210 
.943141 
9-13072 
943003 
942934 
942864 
942795 
942726 
942656 
942587 
942517 
.942448 
942378 
942308 
942239 
942169 
942099 
942029 
941959 
941889 
941819 



D. 10' 



Sine. 



T ang. 

.725674 
725979 
726284 
726588 
726892 
727197 
727501 
727805 
728109 
728412 
728716 
.729020 
729323 
729626 
729929 
730233 
730535 
730838 
731141 
731444 
731746 
.732048 
732351 
782653 
732955 
733257 
733558 
733860 
734162 
734463 
734764 
.735066 
735367 
735668 
785969 
736269 
736570 
736871 
737171 
737471 
737771 
738071 
738371 
738671 
738971 
789271 
739570 
739870 
740169 
740468 
740767 
741066 
741365 
741664 
741962 
742261 
742559 
742858 
743156 
743454 
743752 



D, 10' 



Cotang. I N. sine.lN. cos 



Cotang. 



47690 
47716 
47741 



1 47767 87854 
47793 87840 
4781837826 



47844 
47869 



47920 
47946 
47971 
47997 



87896 
87882 
87868 



87812 
87798 



47895 87784 



87770 
87756 
87743 
87729 



48022(87715 
48048 87701 
i 48073 87687 
48099 87673 



48124 



87659 



4815087645 
48175 87631 



48201 



48303 
48328 
48354 



48430 
48456 
48481 



Tang. I N. coo. N.sine 



87617 



48226 87603 
48252|87589 
48277 87575 



87561 
87546 
87532 



48379 87518 
48405 87504 



87490 
87476 
87462 



37 



61 Degrees. 



50 



Log. Sines and Tangents. (29°) Natural Sines. 



TABLE n. 




1 

2 
3 
4 
6 
6 
7 
8 
9 
10 
11 
12 
13 
14 
16 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
i29 
30 
31 
32 
33 
34 
36 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
48 
47 
48 
49 
60 
51 
52 
63 
54 
56 
66 
57 
58 
69 
60 



D. W 



685571 
686799 
686027 
686254 
686482 
686709' 
686936 
687163 
687389 
687616 
687843 
9.688069 
688295 
688521 
688747 
688972 
689198 
689423 
689648 
689873 
690098 
9.690323 
690548 
690772 
690996 
691220 
691444 
691668 
691892 
692116 
692339 
9.692562 
692785 
693008 
693231. 
693463 
693676 
69389& 
694120 
694342 
694564 
9.694786 
695007 
696229 
695460 
696671 
695892 
696113 
696334 
696664 
696775 
9.6969y5 
697215 
697435 
697654 
697874 
698094 
698313 
698532 
698751 
698970 
Coaine. 



38.0 

37.9- 

37.9 

37.9 

37.9 

37.8 

37.8 

37.8 

37.8 

37.7 

37.7 

37.7 

37.7 

37.6 

37.6 

37.6 

37.6 

37.6 

37.5 

37,5 

37.5 

37.4 

37.4 

37.4 

37.4 

37.3 

37.3 

37,3 

37,3 

37.2 

37.2 

37.2 

37.1 

37.1 

37.1 

37.1 

37.0 

37,0 

37.0 

37.0 

36.9 

36.9 

36.9 

36,9 

36.8 

36.8 

36.8 

36.8 

36.7 

36.7 

36.7 

36.7 

36.6 

36.6 

36.6 

36.6 

36.6 

36.6 

36.6 

36,6 



Cosine. \D. IC/' 



9.941819 
941749 
941679 
941609 
941639 
941469 
941398 
941328 
941258 
941187 
941117 
941046 
940976 
940905 
940834 
940763 
940693 
940622 
940651 
940480 
940409 
9.940338 
940267 
940196 
940126 
940054 
939982 
939911 
939840* 
939768 
939697 
939625 
939654 
939482 
939410 
939339 
939267 
939195 
939123 
939052 
938980 
9.938908 
938836 
938763 
938691 
938619 
938647 
938476 
938402 
938330 
938268 
9G8185 
938113 
938040 
937967 
937895 
937822 
937749 
937676 
937604 
937531 



'11. 7 
11.7 
11.7 
11.7 
11.7 



Sine. 



11 

11 

11 

11 

11 

11 

11.8 

11.8 

11.8 

11.8 

11.8 

11.8 

11.8 

11.8 



11.8 
11.8 
11.3 
11.8 
11.9 
11.9 
11.9 
11.9 
11.9 
11.9 
11.9 
11.9 
11.9 
11.9 
11.9 
11.-9 
12.0 
12.0 
12.0 
12.0 
12.0- 
12.0 
12.0- 
12.0 
12.0 
12.0 
12.0 
12.0 
12.1 
12.1 
12,1 
12.1 
12.1 
12.1 
12.1 
12.1 
12.1 
12.1 
12.1 
12.1 



Tang. D. 10" 



9,743752 
744050 
744348 
744646 
744943 
746240 
745638 
745836' 
746132 
746429 
746726 

9.747023 
747319 
747616 
747913 
748209 
748505 
748801 
749097 
749393 
749689 

9.749985 
75(0281 
760676 
760872 
751167 
761462 
751767 
752062 
762347 
752642 

9.752937 
753231 
753626 
753820 
754116 
764409 
754703 
754997 
756291 
765685 

9,765878 
756172 
766465 
766759 
757062 
767346 
767638 
757931 
768224 
768617 

9', 768810 
769102 
759395 
759687 
759979 
760272 
760564 
760866 
761148 
7 61431 
Cotang. 



49.6 
49.6 
49.6 
49.6 
49.6 
49.5 
49.5 
49,5 
49,5 
49.5 
49.4 
49.4 
49.4 
49.4 
49^4 
49.3 
49.3 
49.3 
49,3 
49.3 
49.3 
4Q,2 
49.2 
49.2 
49.2 
49.2 
49.2 
49.1 
49.1 
49,1 
49,1 
49.1 
49.1 
49.0 
49.0 
49.0 
49.0 
49.0 
49.0 
48.9 
48,9 
48,9 
48.9 
48.9 
48.9 
48.8 
48 ..8 
48.8 
48.8 
48,8 
48.8 
48.7 
48.7 
48.7 
48,7 
48,7 
48.7 
48.6 
48.6 



Cotang. 



10.256248 
266950 
255652 
256356 
266067 
254760 
264462 
254166 
253868 
263571 
263274 

10.252977 
252681 
252384 
252087 
251791 
261496 
251199 
250903 
260607 
250311 

10.260015 
249719 
249424 
249128 
248833 
248638 
248243 
247948 
247653 
247358 

10.247063 
246769 
246474 
246180 
245886 
246691 
245297 
246003 



N,8ine, 



48481 
48606 
48532 
48667 
48583 
48608 
48634 
48659 
48684 



87462 
87448 
87434 
87420 
87406 
87391 
87377 
87363 
87349 



4871087335 



48736 
48761 
48786 
48811 
48837 
48862 
48888 
48913 



48964 
48989 
49014 



49066 
49090 
49116 
49141 
49166 
49192 
49217 
49242 
49268 
49293 
49318 
49344 
49369 



87321 
87306 
87292 
87278 
87264 
87260 
87235 
87221 
87207 
87193 
87178 
87164 



49040187150 



49419 
49446 



87136 
87121 
87107 
87093 
87079 
87064 
87050 
87036 
87021 
87007 
86993 
86978 
86964 
86949 
86935 
86921 



244709 4947086906 



244416 11 49495 
10.2441221! 49621 
243828 149646 
24363&i|49571 
243241 49596 



342948 
242655 
242362 
242069 
241776 
241483 

10.241190 
240898 
240606 
240313 
240021 
239728 
239436 
239144 
238852 
238661 

I Tang. 



1 49622 
49647 
49672 
49697 
49723 
49748 
49773 
49798 



86892 
86878 
86863 
86849 
86834 
86820 
86805 
86791 
86777 
86762 
86748 
86733 
86719 



4982486704 
49849j86690 
4987486675 



49899 
49924 



86661 
86646 



4995086632 



49976 
50000 



N. cofi. N.Hiiic 



86617 
86603 



60 
69 
58 
57 
66 
55 
54 
63 
52 
61 
60 
49 
48 
47 
46 
45 
44 
43 
42 
41 
40 
39 
38 
37 
36 
35 
34 
33 
32 
31 
30 
29 
28 
27 
26 
25 
24 
23 
22 
21 
20 
19 
18 
17 
16 
15 
14 
13 
12 
11 
10 
9 
8 
7 
6 
5 
4 
3 
2 
1 




60 Degreeg. 



TABLE n. 



Log. Sines and Taugeats. (30°) Natural Sines. 



51 



Sine. 



D. 10" 



9.698970 
699189 
699407 
699626 
699844 
700062 
700280 
700498 
700716 
700933 
701151 j 

9.701368 
701585 
701802 
702019 
702236 
702462 
702669 
702886 
703101 
703317 
703533 
703749 
703964 
704179 
704395 
T04610 
704825 
705040 
705254 
705469 

9.706683 
705898 
706112 
706326 
706539 
706753 
706967 
707180 
707393 
707606 
.707819 
708032 
708245 
708458 
70S670 
708882 
709094 
709308 
709518 
709730 
.709941 
710163 
710J64 
710576 
710786 
710967 
711208 
711419 
711629 
711839 
Cosine. 



36.4 
36.4 
36.4 
36.4 
36.3 
36.3 
36.3 
36.3 
36.3 
36.2 
36.2 
36.2 
36.2 
,1 



36 

36.1 

36.1 

36.1 

36.0 

36.0 

36.0 

36.0 

36.9 

35.9 

36.9 

36.9 

36.9 

36.8 

35.8 

36.8 

35.8 

35.7 

35.7 

35.7 

35.7 

36.6 

35.6 

35.6 

35.6 

35.5 

36.6 

35.6 

35.6 

35.4 

35.4 

35.4 

35.4 

35.3 

36.3 

35.3 

35.3 

36.3 

36.2 

35.2 

35.2 

36.2 

35.1 

35.1 

35.1 

35.1 

35.0 



Cosine. 



D. 10'' 



12 

12 

12 

12 

12 

12 

12 

12 

12 

12 

12 

12 

12.3 

12.3 

12.3 

12.3 

12.3 

12.3 

12,3 

12.3 

12.3 

12.3 

12.3 

12.3 

12.4 

12.4 

12.4 

12.4 

12.4 

12.4 

12.4 

12.4 

12.4 

12.4 

12.4 

12.4 

12.4 

12.5 

12.6 

12,6 

12,6 

12.5 

12.6 

12.5 



12 

12 

12 

12 

12 

12 

12 

12.6 

12.6 

12.6 

12.6 

12.6 

12.6 

12.6 

12.6 

12.6 



Tanp" 



.761439 
761731 
762023 
762314 
762G06 
762897 
763188 
763479 
763770 
764061 
764352 

.764643 
764933 
766224 
765614 
766806 
766095 
766385 
766676 
766965 
767265 

.767546 
767834 
768124 
768413 
768703 
768992 
769281 
769570 
769860 
770148 

.770437 
770726 
771015 
771303 
771692 
771880 
772168 
772457 
772745 
773033 

.773321 
773608 
773896 
774184 
774471 
774759 
775046 
775333 
775621 
776908 

.776195 
776482 
776769 
777065 
777342 
777628 
777915 
778201 
778487 
778774 



10' 



48.6 



48.6 



Cotang. 



Cotang. j N. sine. N. cos 



10.238561 
238269 
237977 
237686 
237394 
237103 
236812 
236521 
236230 
235939 
235648 

10.235357 
235037 
234776 
234486 
234195 
233905 
233615 
233325 
233035 
232746 

10.232455 
232166 
231876 
231587 
231297 
231008 
230719 
230430 
230140 
229852 

10.229563 
229274 
228985 
228697 
228408 
228120 
227832 
227543 
227256 
22696.7 

10.226679 
226392 
226104 
225816 
226529 
225241 
224954 
224667 
224379 
224092 

10.223806 
223518 
223231 
222945 
222658 
222372 
222085 
221799 
221612 
221226 



150000 
50025 
60050 
50076 
50101 
60126 
60151 
50176 
60201 
50227 
50252 
50277 
50302 
50327 
50352 
50377 
50403 
1 50428 
'60453 
60478 
50603 
60528 



86588 
86573 
86559 
86544 
86630 
86616 
86601 
86486 
86471 
86457 
86442 
86427 
86413 
86398 
86384 



Tang. 



86354 
86340 
86325 
86310 
86295 



60653 86281 



60678 

50603 

50628 

50654 

50679 

50704 

60729 

50754 

60779 

60804 

60829 

.50864 

.5087: 

50904 

50929 

50954 

50979 

51004 

51029 

51054 

51079 

61104 

51129 

61154 

51179 

51204 

51229 

5126<i 

61279 

5J304 

51329 

51364 

51379 

51404 

51429 

51454 

61479 

61504 



86266 
86251 
86237 
86222 
86207 
86192 
86178 
86163 
86148 
86133 
86119 
86104 
986089 
86074 
86059 
86045 
86030 
86015 
86000 
85985 
85970 
85956 
85941 
86926 
85911 
85896 
85881 
85866 
85861 
85836 
85821 
85806 
85792 
85777 
86762 
86747 
85732 
85717 



N. COS. N.8ine 



59 Degrees. 



52 



Log. Sine8 and Taagenta. (31°) Xatural Sines. 



TABLE II. 



Sine. |D. 10" (Josine, 





1 

2 
3 
4 
5 
6 
7 
8 
9 

la 
11 

12 

13 

14 

15 

16 

17 

18 

19 

20 i 

21 

22 

23 

24 

25 

26 

27 

2S 

29 

30 i 

31 9 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 9 
42 
43 
44 
45 
46 
47 
48 
49 
50 
61 
62 
53 
54 
55 
56 
57 
68 
59 
60 



.711839 OK 

712050 ^? 

712260 r.~ 

712469 ^° 

712679 tri 

712889 ^] 

713098 ■^l 

713308 t>l 

713617 t: 

713726 ^l 

713936 tl 

.714144 tj 

714352 ll 

714561 tl 

714769 t; 

714978 ^2 

715186 '^2 

715394 ^^ 

7166Q2 ^2 

715809 "il 

716017 ^* 

.716224 fl 

716432 q^ 

716639 ^2 

716846 r: 

717053 ^^ 

717259 tl 

717466 ^t 

717673 ^;: 

717879 t: 
718086 ^^ 

.718291 "ij 
718497 ^J 
718703 ^^ 
718909 i '11 
719114 ^^ 
719320 7.1 
719525 ^;: 
719730 f: 
719936 ^^ 
720140 ^? 

.720346 ll 
720549 !^1' 
720754;^^' 
720958 I tl 
721162' 5; 
721366 1 tl 
721570 ^X 
721774 ■ it 
721978:^ 
722181'^^ 

. 722385 ^^t 
722588 ^^ 
722791!^^ 
722994^^ 
723197:^^ 
723400 „ 
723603 to 
723805^^ 
724007^^ 
724210! ^ 
Cosine. I 



9.933036 
932990 
932914 
932838 
932762 
932685 
932609 
932533 
932457 
932380 
932304 

9.932228 
932151 
932076 
931998 
931921 
931845 
931768 
931691 
931614 
931537 
931460 
931383 
931306 
931229 
931152 
931075 
930998 
930921 
930843 
930766 

9.930688 
930611 
930533 
930466 
930378 
930300 
930223 
930145 
930067 
929989 
929911 
929833 
929766 
929677 
929599 
929521 
929442 
929364 
929286 
929207 
929129 
929050 
928972 



928816 
928736 
928657 
928678 
928499 
928420 
Sine. 



D. 10" 



12.6 
12.7 

12.7 



12 

12 

12 

12 

12 

12 

12.7 

12.7 

12.7 

12.7 

12.8 

12.8 

12.8 

12.8 

12.8 

12.8 

12.8 

12.8 

12.8 

12.8 

12.8 

12.9 

12.9 

12.9 

12.9 

12.9 

12.9 

12.9 

12.9 

12.9 

12.9 

12.9 

12.9 

13.0 

13.0 

13.0 

13.0 

13.0 

13.0 

13.0 

13.0 

13.0 

13.0 

13.0 

13.0 

13.1 

13.1 

13.1 

13.1 

13.1 

13.1 

13.1 

13.1 

13.1 

13.1 

13.1 

13.1 



Tang. 



1.778774 
779060 
779346 
779632 
779918 
780203 
780489 
780775 
781060 
781346 
781631 

1.781916 
782201 
782486 
782771 
783053 
783341 
783626 
783910 
784195 
784479 

(.784764 
785048 
785332 
785616 
785900 
786184 
786468 
786752 
787036 
787319 

>. 787603 
787886 
788170- 
788453 
788736 
789019 
789302 
789685 
789868 
790151 

1.790433 
790716 
790999 
791281 
791663 
791846 
792128 
792410 
792692 
792974 

). 793256 
793638 
793819 
794101 
794383 
794664 
794946 
795227 
795508 
7 96789 
Cotang. 



D. ll>" Cotang. N.sine. N. cos 



47.7 

47.7 

47.6 

47.6 

47.6 

47.6 

47.6 

47.6 

47.6 

47.6 

47.6 

47.6 

47.6 

47.6 

47.6 

47.5 

47.5 

47.4 

47 

47 

47 

47 

47 

47 

47.3 

47.3 

47.3 

47.3 

47.3 

47.3 

47.2 

47.2 

47.2 

47.2 

47.2 

47.2 

47.2 

47.1 

47.1 

47.1 

47.1 

47.1 

47.1 

47.1 

47.1 

47.0 

47.0 

47.0 

47.0 

47.0 

47.0 

47.0 

46.9 

46.9 

46.9 

46.9 

46.9 

46.9 

46.9 

46.8 



10.221226 
220940 
220664 
220368 
220082 
219797 
2195U 
219226 
218940 
218654 
218369 

10.218084 
217799 
217514 
217229 
216944 
216669 
216374 
216090 
215805 
215521 

10.215236 
214952 
214668 
214384 
214100 
213816 
213532 
213248 
212964 
212681 

10.212397 
212114 
211830 
211647 
211264 
210981 
210698 
210416- 
210132 
209849 

10.209567 
209284 
209001 
208719 
208437 
208164 
207872 
207690 
207308 
207026 

10.206744 
206462 
206181 
205899 
205617 
205336 
205055 
204773 
204492 
204211 



51504 86717 
61529 86702 
61554 85687 
51579 85672 



61604 
51628 
61653 
51678 
! 61703 
161728 
1 61753 
'I 61778 
61803 
51828 
61852 
61877 
51902 
51927 
61952 



85667 
85642 
85627 
86612 
86697 
85682 
85567 
85551 
86536 
85521 
85606 
86491 
86476 
86461 
86446 



61977185431 

62002185416 

52026J85401 

52061 

62076 



86386 

85370 

52101^85365 



6Q126 
62161 
152175 
! 52200 
! 62226 
i 52260 
i 52276 



85340 j 36 
85325 I 34 
86310 33 



85294 
85279 
85264 
86249 
52299 85234 28 
62324J85218 27 
i 52349 85203 26 



i 52374 
62399 
; 62423 
152448 
i 52473 
I 62498 



85188 1 26 



86173 
86167 
85142 
85127 
86112 



I 52522 85096 

i 52547 '^ ■"' 
1 62572 



! 62597 
1 62621 
i 52646 
■62671 
! 52696 
i 62720 
i 62746 
I 52770 
i 52794 
162819 
62844 



52869 8488; 
52893 84866 
52918 84861 



85U81 
85066 
85U51 
86036 
85020 
85005 
84989 
84974 
84959 
84943 
849:^8 
84913 
84897 



! 62943 
j! 62967 
1162992 



Tang. 



84836 

84820 

84805 

N. cos.JN.sine. 



58 Degrees. 



TABLE n. 



Log. Sines and Tangents. (32°) Natural SincB. 



53 



Sine. 



9.724210 
724412 
724614 
724816 
725017 
725219 
725420 
725622 
725823 
726024 
726225 
726426 
726626 
726827 
727027 
727228 
727428 
727628 
727828 
728027 
728227 

9.728427 
728626 
728825 
729024 
729223 
729422 
729621 
729820 
730018 
730216 

9.730415 
730613 
730811 
731009 
731206 I 
731404 
731602 
731799 
731996 
732193 

9.732390 
732587 
732784 
732980 
733177 
733373 
733569 
733765 
733961 
734157 

9.734353 
734549 
734744 
734939 
735135 
735330 
735525 
735719 
735914 
736109 



D. 10" 



Ck)sine. 



Cosine. |D. ly^ 



9.928420 
928342 
928263 
928183 
928104 
928025 
927946 
927867 
927787 
927708 
927629 
927549 
927470 
927390 
927310 
927231 
927151 
927071 
926991 
92691 1 
926831. 

9.926751 
926671 
926591 
926511 
926431 
926351 
926270 
926190 
926110 
926029 
926949 
925868 
926788 
925707 
925626 
925545 
925465 
925384 
926303 
925222 
925141 
925060 
924979 
924897 
924816 
924735 
924654 
924572 
924491 
924409 

9.924328 
924246 
924164 
924083 
924001 
923919 
923837 
923755 
923673 
923591 



Sine. 



13 

13 

13 

13 

13 

13 

13 

13 

13 

13 

13 

13 

13 

13 

13 

13 

13 

13 

13 

13 

13 

13 

13.3 

13.3 

13 

13 

13 

13 

13 

13 

13 

13 

13 

13 

13 



13.4 



13 

13 

13 

13 

13 

13 

13 

13 

13 

13 

13 

13 

13.6 

13.6 

13.6 

13.6 

13.6 

13.6 

13.6 

13.6 

13.6 

13.6 

13.7 

13.7 



Tang. 



). 795789 
796070 
796351 
796632 
796913 
797194 
797475 
797765 
798036 
798316 
798596 

). 798877 
799157 
799437 
799717 
799997 
800277 
800557 
800836 
801116 
801396 

). 801676 
801956 
802234 
802613 
802792 
803072 
803361 
803630 
803908 
804187 

). 804466 
804745 
805023 
806302 
805680 
806859 
806137 
806415 
806693 
806971 

). 807249 
807527 
807805 
80S083 
808361 
808638 
808916 
809193 
809471 
809748 

). 810025 
810302 
810580 
810857 
811134 
811410 
811687 
811964 
812241 
812517 

Cotanc. 



D. 10" 



46.8 
46.8 
46.8 
46.8 
46.8 
46.8 
46.8 
46.8 
46.7 
46.7 
46.7 
46.7 
46.7 
46.7 
46.7 
46.6 
46.6 
46.6 
46.6 
46.6 
46.6 
46.6 
46.6 
46.5 
46.5 
46.5 
46.5 
46.5 
46.5 
46.5 
46.5 
46.4 
46.4 
46.4 
46.4 
46.4 
46.4 
46.4 
46.3 
46.3 
46.3 
46.3 
46.3 
46.3 
46.3 
46.3 
46.2 
46.2 
46.2 
46.2 
46.2 
46.2 
46.2 
46.2 
46.2 
46.1 
46.1 
46.1 
46.1 
46.1 



Cotang. I N. sine. N. cos 



10.204211 
203930 
203649 
203368 
203087 
202806 
202525 
202245 
201964 
201684 
201404 

10.201123 
200843 
200563 
200283 
200003 
199723 
199443 
199164 
198884 
198604 

10.198326 
198046 
197766 
197487 
197208 
196928 
196649 
196370 
196092 
195813 

10.195534 
195255 
194977 
194698 
194420 
194141 
193863 
193686 
193307 
193029 

10.192751 
1S2473 
192195 
191917 
191639 
191362 
191084 
190807 
190529 
190252 
189975 
189698 
189420 
189143 
188806 
188590 
188313 
188036 
187759 
187483 

"Tani?. 



52992 
53017 
53041 
53066 
53091 
53115 
53140 
153164 
53189 
53214 
53238 
53263 



53312 
53337 
53361 
63386 
53411 
53435 
1 53460 
1 53484 
63509 
1 63634 
1 53568 
i 63683 
1 53607 
53632 
53656 
53681 
53706 
63730 
63754 
53779 



10. 



84805 
84789 
84774 
84759 
84743 
84728 
84712 
84697 
84681 
84666 
84650 
84635 
84619 
84604 
84588 
84573 
84657 
84542 
84626 
84511 
84495 
84480 
84464 
84448 
84433 
84417 
84402 
84386 
84370 
84355 
84339 
84324 
84308 
53804 84292 
1 53828J84277 
i 63853184261 
1 53877[84245 
1 53902 84230 
! 63926 84214 
! 63951 84198 
I53975s84l82 
54000 84167 
I 64024|84151 
154049 84135 
1 54073,84120 
54097184104 
1 54122J84088 
i 54146184072 
1 54171 84057 
i 64195 84041 
1 54220,84025 
I 64244:84009 
154269183994 
54293 83978 
1 54317 83962 
54342,8S946 
! 54366 83930 
1 54G91 183915 
! 64415183899 
; 64440;83883 
15446483867 



60 
59 
58 
57 
66 
55 
54 
53 
62 
51 
50 
49 
48 
47 
46 
45 
44 
43 
42 
41 
40 
39 
38 
37 
36 
35 
34 
S3 
32 
31 
30 
29 
28 
27 
26 
26 
24 
23 
^2 
21 
20 
19 
18 
17 
16 
15 
14 
13 
12 
11 
10 
9 
8 
7 
6 
5 
4 
S 
2 
1 




N. COP. N.fiir.e. 



57 Degrees. 



54 



Log. Sines and Tangents. (33°) Natural Sines. TABLE IL 



J_ S ine. D. V^' Cosine. D. W Tang. D. 10" Cotang 



8 
9 
10 
11 
12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44 

45 

46 

47 

48 

49 

50 

51 

52 

53 

64 

55 

56 

57 

58 

59 

60 



9.736109 
736303 
736498 
736692 
736886 
737080 
737274 
737467 
737661 
737855 
738048 
9.738241 
738434 
738627 
738820 
739013 
739206 
739398 
739590 
739783 
739975 
740167 
740369 
740550 
740742 
740934 
741125 
741316 
741508 
741699 
741889 
9.742080 
742271 
742462 
742662 
742842 
743033 
743223 
743413 
743602 
743792 
19.743982 
744171 
744361 
744560 
744739 
744928 
745117 
745306 
745494 
745683 
9.745871 
746059 
746248 
746436 
746624 
746812 
746999 
747187 
747374 
747562 
Cosine. 



32.4 

32.4 

32.4 

32.3 

32.3 

32.3 

32.3 

32.3 

32.2 

32.2 

82.2 

32.2 

32.2 

32.1 

32.1 

32.1 

32.1 

32.1 

32.0 

32.0 

32.0 

32.0 

32.0 

31.9 

31.9 

31.9 

31.9 

31.9 

31.8 

31.8 

31.8 

31.8 

31 



31.7 
31.7 
31.7 
31.7 
31.7 
31.6 
31.6 
31.6 
31.6 
31.6 



31.5 
31.5 
31.5 
31.5 
31.5 
31.4 
31.4 
31.4 
31.4 
31.4 
31.3 
31.3 
31.3 
31.3 
31.3 
31.2 



31.2 



9.923591 
923509 
923427 
923345 
923263 
923181 
923098 
923016 
922933 
922851 
922768 
9.922686 
922603 
922520 
922438 
922355 
922272 
922189 
922106 
922023 
921940 
9.921857 
921774 
921691 
921607 
921624 
921441 
921357 
921274 
921190 
921107 
9.921023 
920939 
920856 
920772 
920688 
920604 
920520 
920436 
920362 
920268 
9.920184 
920099 
920015 
919931 
919846 
919762 
919677 
919593 
919508 
919424 
9.919339 
919254 
919169 
919086 
919000 
918915 
918830 
918745 
918659 
918574 



I Sine. 



13.7 

13.7 

13.7 

13.7 

13.7 

13.7 

13.7 

13.7 

13.7 

13.7 

13.8 

13.8 

13.8 

13.8 

13.8 

13.8 

13.8 

13.8 

13.8 

13.8 

13.8 

13.9 

13.9 

13.9 

13.9 

13.9 

13.9 

13.9 

13.9 

13.9 

13.9 

13.9 

14.0 

14.0 

14.0 

14.0 

14.0 

14.0 

14.0 

14.0 

14.0 

14.0 

14.0 

14.0 

14.1 

14.1 

14.1 

14.1 

14.1 

14.1 

14.1 

14.1 

14.1 

14.1 

14.1 

14.1 

14.2 

14.2 

14.2 

14.2 



9.812517 
812794 
813070 
813347 
813623 



814175 
814452 
814728 
815004 
815279 
9.816555 
815831 
816107 
816382 
816658 
816933 
817209 
817484 
817759 
818036 
.818310 
818685 
818860 
819135 
819410 
819684 
819959 
820234 
820608 
820783 
9.821067 
821332 
821606 
821880 
822154 
822429 
822703 
822977 
823250 
823524 
3.823798 
824072 
824345 
824619 
824893 
825166 
825439 
826713 
825986 
826259 
). 826632 
826805 
827078 
827351 
827624 
827897 
828170 
828442 
828715 
828987 
Cotang. 



46.1 

46.1 

46.1 

46.0 

46.0 

46.0 

46.0 

46 

46 

46 

46.0 

45.9 

45.9 

45.9 

45.9 

45.9 

45.9 

45.9 

45.9 

45.9 

45.8 

45.8 

45.8 

45.8 

45.8 

45.8 

45.8 

45.8 

45.8 

45.7 

45.7 

45.7 

45.7 

46.7 

45.7 

45.7 

45.7 

45.7 

45.6 

46.6 

45.6 

45.6 

46.6 

46.6 

45.6 

45.6 

45.6 

45.5 

45.5 

45.5 

45.5 

45.5 

45.5 

45.5 

45.5 

45.5 

45.4 

45.4 

45.4 

45.4 



10.187482 
187206 
186930 
186653 
186377 
186101 
185825 
185548 
185272 
184996 
184721 
10.184445 
184169 
183893 
183618 
183342 
183067 
182791 
182516 
182241 
181965 
10.181690 
181415 
181140 
180865 
180590 
180316 
180041 
179766 
179492 
179217 
10.178943 
178668 
178394 
178120 
177846 
177571 
177297 
177023 
176760 
176476 
10.176202 
175928 
176656 
175381 
175107 
174834 
174561 
174287 
174014 
173741 
10.173468 
173195 
172922 
172649 
172376 
172103 
171830 
171558 
171285 
171013^ 
Tai^. 



N. sine. N. cos. 



54464 
54488 
54513 
54537 
54661 



83867 
83851 
83835 
83819 
83804 



54586 83788 



54610 



83772 



54635 83756 
54659 83740 
54683 «3724 



54708 
64732 
54766 
54781 



54805 83645 
5482983629 
54854 83613 



54878 
54902 
54927 



83697 
83581 
83566 



54951 83549 



54975 
54999 
55024 



56097 
55121 
55145 
55169 
65194 
55218 
65242 



83708 
83692 
83676 
83660 



83533 
83517 
83601 



56048 83485 
55072 83469 



83463 
83437 
83421 
83406 
83389 
83373 
83356 



5626683340 



56291 
65315 
55339 



56412 
65436 
56460 
56484 
55509 
56533 
55657 
56581 



83324 



83292 



65363 83276 
83260 
83244 
83228 
83212 
83195 
83179 
83163 
83147 
83131 
56605 83116 
55630 ,'83098 
55654 83082 
56678|83066 
65702J83060 
55726 83034 
5675083017 



55871 
55895 
55919 



83001 
82S85 



55775 
66799 
55823 
55847 82953 



82936 
82920 
82904 



N. eos. N.sine, 



60 
59 
68 
67 
56 
65 
64 
53 
52 
61 
60 
49 
48 
47 
46 
45 
44 
43 
42 
41 
40 
39 
38 
37 
36 
35 
34 
33 
32 
31 
30 
29 
28 
27 
26 
25 
24 
23 
22 
21 
20 
19 
18 
17 
16 
16 
14 
13 
12 
11 
10 
9 
8 
7 
6 
5 
4 
3 
2 



56 Degrees. 



TABLE II. 



Log. Sines and Tangents. (34°) Natural Sinea- 



55 



I). 10" 




1 

2 
3 
4 
5 
6 
7 
8 
9 
10 
11 
12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
36 
36 
37 
38 
39 
40 
41 
42 
43 
44 
46 
46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
57 
58 
59 
60 



Sine. 

9.747562 
747749 
747936 
748123 
748310 
748497 
748683 
748870 
749056 
749243 
749426 

9.749615 
749801 
749987 
750172 
750358 
760543 
750729 
750914 
761099 
751284 

9.751469 
761654 
751839 
762023 
752208 
752392 
752576 
752760 
752944 
753128 

9.763312 
753495 
763679 
753862 
764046 
764229 
754412 
754595 I 
764778 '■ 
754960 I 

9.7661431 
766326 I 
755608 I 
765690 : 
755872 j 
756054 i 
756236 I 
756418 ! 
756600 i 
756782 ! 

9.756963! 
757144 I 
767326 
757507 I 
757688 I 
767869 : 
758050 : 
758230 i 
768411 ! 
_758591 ! 
Cosine. I 



31.2 
31.2 
31.2 
31.1 
31.1 
31.1 
31.1 
31.1 
31.0 
31.0 
31.0 
31.0 
31.0 
30.9 
30.9 
30.9 
30.9 
30.9 
30.8 
30.8 
30.8 
30.8 
30.8 
30.8 
30.7 
30.7 
30.7 
30.7 
30.7 
30.6 
30.6 
30.6 
30.6 
30.6 
30.5 
30.5 
30.6 
30.5 
30.6 
30.4 
30.4 



30 

30 

30 

30 

30 

30 

30.3 

30.3 

30.3 

30.2 

30.2 

30.2 

30.2 

30.2 

30-1 

30.1 

30.1 

30.1 

30.1 



Cosine. 

9.918674 
918489 
918404 
918318 
918233 
918147 
918082 
917976 
917891 
917805 
917719 

9.917634 
917648 
917462 
917376 
917290 
917204 
917118 
917032 
916946 
916859 
.916773 
916687 
916600 
916514 
916427 
916341 
916254 
916167 
916081 
915994 
.916907 
915820 
916733 
915646 
915559 
915472 
915385 
915297 
915210 
915123 
.915036 
914948 
914860 
914773 
914686 
914698 
914610 
914422 
914334 
914246 
9.914158 
91407.0 
913982 
913894 
913806 
913718 
913630 
913541 
913453 
913365 
Sine. 



D. 10" Tang. 



14.2 
14.2 
14.2 
14.2 
14.2 
14.2 
14.2 
14.3 
14.3 
14.3 
14.3 
14.3 
14.3 
14.3 
14.3 
14.3 



14 

14 

14 

14 

14 

14 

14 

14 

14 

14 

14.4 

14.4 

14 

14 

14 

14 

14 

14 

14 

14 

14 

14.5 

14.5 

14.5 

14.6 

14.6 

14.6 

14.6 

14.6 

14.6 

14.6 

14.6 

14.6 

14.6 

14.7 

14.7 

14. 

14. 

14. 

14. 

14. 

14. 

14. 



7 
7 
7 
7 
7 
7 
7 
14.7 



,828987 
829260 
829532 
829806 
830077 
830349 
830621 
830893 
831166 
831437 
831709 
831981 
832263 
832626 
832796 
833068 
833339 
833611 
833882 
834154 
834426 
834696 
834967 
836238 
835509 
835780 
836061 
836322 
836693 
836864 
837134 

9.837406 
837675 
837946 
838216 
838487 
838757 
839027 
839297 
839568 
839838 

9.840108 
840378 
840647 
840917 
841187 
841457 
841726 
841996 
842266 
842635 

9.842805 
843074 
843343 
843612 
843882 
844161 
844420 
844689 
844958 
846227 
Cotang. 



D. 10 



45.4 

45.4 

45.4 

45.4 

45.4 

45.3 

45.3 

46.3 

45.3 

45.3 

46.3 

45.3 

46.3 

45.3 

45.3 

46.2 

45.2 

45.2 

45.2 

45.2 

46.2 

45.2 

46.2 

46.2 

45.2 

45.1 

45.1 

46,] 

45.1 

45.1 

46.1 

46.1 

45.1 

45.1 

45.1 

46.0 

45.0 

45.0 

45.0 

45.0 

45.0 

46.0 

45.0 

45.0 

44.9 

44.9 

44.9 

44.9 

44.9 

44.9 

44.9 

44.9 

44.9 

44.9 

44.9 

44.8 

44.8 

44.8 

44.8 

44.8 



Cotang. IjN.sine 




55919 
56943 
55908 
56992 
56016 
56040 
56064 
56088 
56112 
56136 
66160 



10. 16-8019!! 56184 



N. COS.) 



167747 ! 

167475 i 

167204 

166932 

166661 

166389 

166118 

165846 

165576 

10.165304 
166033 
164762 
164491 
164220 
163949 
163678 
163407 
163136 
162866 

10.162595 
162325 
162054 
161784 
161513 
161243 
160973 



56208 
56232 
56256 
66280 
56305 
56329 
56353 
56377 
56401 
56425 
56449 
56473 
56497 
56621 
56545 
56569 
56593 
56617 
66641 
56665 
66689 
66713 
56736 
56760 
56784 
66808 



160703 !i568S2 
16043211 56856 
160162 1 156880 



10.159892 
169622 
159353 
159083 
158813 
168543 
158274 
168004 
157734 
157465 

10.157196 
166926 
166657 
156388 
156118 
155849 
155580 
155311 
155042 
154773 
Tang. 



56904 
56928 
56962 
56976 
57000 
67024 
57047 
67071 
57095 
57119 
57143 
67167 
67191 
57215 
57238 
57262 
67286 
67310 
57334 
57358 



82904 

82887 

82871 

82855 

82839 

82822 

82806 

82790 

82773 

82767 

82741 

82724 

82708 

82692 

82675 

82659 

82643 

82626 

82610 

82593 

82577 

82661 

82544 

82528 

82511 

82495 

82478 

82462 

82446 

82429 

82413 

82396 

82380 

82363 

82347 

82330 

82314 

82297 

82281 

82264 

82248 

82231 

82214 

82198 

82181 

82165 

82148 

82132 

82115 

82098 

82082 

82065 

82048 

82032 

82015 

81999 

81982 

81965 

81949 

81932 

81915 



N. COS. N.sine, 



56 



Log. Sines and Tangents. (35°) Natural Sines. 



TABLE II, 



9.758591 
758772 
758952 
759132 
759312 "V 
759492 i:{" • 
759672^"- 

759852 ;;;!• 
760031 :;^ • 

760211 .;,^ • 

760390 x;; • 

.760569 hfo 
760748 i^^ • 
760927 h;Q • 

761106 2^ • 
761285 ^;! • 
761464 ^g • 
7616421^^- 
761821 j^^ • 
761999 U;q' 
7621771^^* 

.762356 ^^• 
762534 f- 
762712 f- 
762889 ^^• 
763067 :;'• 
763245 ;^- 
763422 :;^- 
763600 Z,' 
763777 ^^• 
763954 ^^• 

.764131 f- 
764308 ^q 
764485 ^^• 
764662 ^^• 
764838 Z,' 
765016 '^• 
765191 f- 
765367 -^• 
765544 *^- 
^^^-20 f^- 

29. 
29. 
29. 
29. 
29. 
29. 
29. 
29. 
29. 
29. 
29. 
29. 
29 



765<isu 
.765896 
766072 
766247 
766423 
766598 
766774 
766949 
767124 
767300 
767476 
9.767649 
767824 
767999 
768173 
768348 29 
768522 TL 
768697 if' 
768871 Zx' 
769045 ti' 
769219 "'^ 



29, 



Oosilie. 

1.913365 
913276 
913187 
913099 
913010 
912922 
912833 
912744 
912655 
912566 
912477 

9.912388 
912299 
912210 
912121 
912031 
911942 
911853 
911763 
911674 
911584 
911495 
911405 
911315 
911226 
911136 
911046 
910956 
910866 
910776 
910d86 
910596 
910506 
910415 
910325 
910235 
910144 
910054 
909963 
909873 
909782 

9.909691 
909601 
909510 
909419 
909328 
909237 
909146 
909055 
908964 
908873 
908781 
908690 
9085y9 
908607 
908416 
908324 
908233 
908141 
908049 
907968 
"sine." 



4.8 



'i'au g. 

.845227 
845496 
845764 
846033 
846302 
846670 
846839 
847107 
847376 
847644 
847913 

1.848181 
848449 
848717 
848986 
849254 
849522 
849790 
850058 
850326 
850593 

►.850861 
851129 
851396 
851664 
851931 
852199 
852466 
852733 
853001 
853268 

». 853535 
853802 
854069 
854336 
864603 
854870 

■ 856137 
856404 
855671 
855938 

1.856204 
856471 
856737 
857004 
857270 
857637 
867803 
858069 
858336 
858602 

1.858868 
859134 
869400 
869666 
869932 
850198 
850464 
860730 
850995 
861261 

C!otang. 



D. 10" 

44.8 

44.8 

44.8 

44.8 

44.8 

44.7 

44.7 

44.7 

44.7 

44.7 

44.7 

44.7 

44.7 

44.7 

44.7 

44.7 

44.7 

44.6 

44.6 

44.6 

44.6 

44.6 

44.6 

44.6 

44.6 

44.6 

44.6 

44.6 

44.6 

44.5 

44.5 

44.5 

44.5 

44.5 

44.5 

44.6 

44.6 

44.5 

44.5 

44 

44 

44 

44 

44 

44 

44 

44.4 

44.4 

44.4 

44.4 

44.3 

44.3 

44.3 

44.3 

44.3 

44.3 

44.3 

44.3 

44.3 

44.3 



Cotang. I N. piue. N. coh 



10. 



10.164773 
154504 
164236 
153967 
153698 
153430 
153161 
152893 
152624 
152356 
15S;087 

10.151819 
151651 
161283 
151014 
150746 
150478 
150210 
149942 
149675 
149407 
149139 
148871 
148604 
148336 
14«069 
147801 
147534 
147267 
146999 
146732 

10.146465 
146198 
145931 
145664 
145397 
145130 
144863 
144696 
144329 
144062 

10.143796 
143529 
143263 
142996 
142730 
142463 
142197 
141931 
141664 
141398 

10-141132 
140866 
140600 
140334 
140068 
139802 
139536 
139270 
139006 
138739 



I 67358 
57381 
'167405 
1167429 
57453 
57477 
57601 
57624 
57648 
67572 
57596 
57619 
67643 



81915 
81899 
81882 
81865 
81848 
81832 
81815 
81798 
81782 
81765 
81748 
81731 
81714 



57667 81698 



57691 
57715 
57738 
57762 
57786 
57810 



81681 
81664 
81647 
81631 
81614 
81597 



67904 
67928 
67952 
67976 
57999 
58023 
68047 
58070 



68141 
68165 



58212 



58260 
58283 
58307 



58354 
58378 
68401 
68426 
58449 
58472 
58496 
58519 
68543 
58567 



67833 81580 
57867 81563 
57881 81546 



81530 
81513 
81496 
81479 
81462 
81445 
81428 
81412 



58094,81395 
68118 81378 



81361 
81344 



58189 81327 



81310 



58236 81293 



81276 
81259 
81242 



5833081225 



81208 
81191 
81174 
81167 
81140 
81123 
81106 
81089 
81072 
81055 



58614 
58637 
58661 
68684 
68708 
58731 



58779 



Tang. I N. cos. N.si 



5869081038 



81021 
81004 
80987 
80970 
80953 
80:^36 



58755 80919 



80902 



54 Degrees. 



TABLE II. Ix)g. Sines and Tangents. (36°) Natural Sines. 



57 



Sine. 

769219 
769393 
769566 
769740 
769913 
770087 
770260 
770433 
770608 
770779 
770952 
771125 
771298 
771470 
771643 
771816 
771987 
772159 
772331 
772503 
772675 
772847 
773018 
773190 
773361 
773633 
773704 
773875 
774046 
774217 
774388 
9.774558 
774729 
774899 
775070 
775240 
776410 
775580 
775750 
775920 
776090 
776259 
776429 
776598 
776768 
776937 
777106 
777^75 
777444 
777613 
777781 
777950 
778119 
778287 
778455 
778624 
778792 
778960 
779128 
779295 
779463 



Cosine. 



D. 10'' 



Co«ine. 

.907958 
907866 
907774 
907682 
907690 
907498 
907406 
907314 
907222 
907129 
907037 

1.906945 
906852 
906760 
906667 
906576 
906482 
906389 
906296 
906204 
906111 

.906018 
905925 
905832 
905739 
905645 
905552 
905459 
905366 
905272 
905179 
9.905085. 
904992 
904898 
904804 
904711 
904617 
904523 
904429 
904335 
904241 
904147 
904053 
903959 
903864 
903770 
903676 
903581 
903487 
903392 
903298 
903202 
903108 
903014 
902919 
902824 
a02729 
902634 
902539 
902444 
902349 



Sine. 



Tang. 

.861261 
8()1627 
861792 
862058 
862323 
862589 

■ 862864 
863119 
863385 
863650 
863915 

.864180 
864445 
864710 
864975 
866240 
865506 
865770 
866036 
866300 
866664 

.866829 
867094 
867368 
867623 
867887 
868162 
868416 
868680 
868945 
869209 

.869473 
869737 

,870001 
870265 
870629 
870793 
871067 
871321 
871585 
871849 

.872112 
872376 
872640 
872903 
873167 
&73430 
873694 
878967 
874220 
874484 

.874747 
876010 
87&273 
875536 
875800 
876063 
876326 
876589 
876861 
877114 



Cotang 



Cotang. I N. sine. N. cos 



58779 80902 
58802 80885 
58826 80867 
B8849 80860 



58873 
58896 
58920 
68943 
58967 



80833 
80816 
80799 
80782 
80765 



6899080748 



59412 
59436 
59459 
69482 
59606 
69529 
59552 
! 59576 



5931880507 
69342 80489 
6936580472 
5938980455 



80438 
80422 
80403 
80386 
80368 
80351 
80334 
80316 
80299 
80282 
80264 
80247 
80230 
80212 
80195 



69599 
1 59622 

59646 

5^669 

59693 

59716 

59739 

59763 80178 
J59786f80l60 
159809^ 

69832 
159856 
i 59879 
159902 



1257801159926 



80143 
80125 
80108 
80091 
80073 
80066 



126516!! 69949 80038 



10. 



Tang. 



'60065 
i 60089 
60112 
60136 
60158 
60182 



79961 
79934 
79916 
79899 
79881 
79864 
N.j<ine, 



53 Degrees. 



58 



Log. Sines and Tangents. (37°) Natural Sinea. 



TABLE n. 



Sine. 

9.779463 
779631 
779798 
779966 
780133 
780300 
780467 
780634 
780801 
780968 
781134 

9.781301 
781468 
781634 
781800 
781966 
782132 
782298 
782464 
782630 
782796 

9.782961 
783127 
783282 
783458 
783623 
783788 
783953 
784118 
784282 
784447 

9.784612 
784776 
784941 
785105 
785269 
785433 
785597 
785761 
785926 
786089 

9.786252 
786416 
786579 
786742 
786906 
787069 
787232 
787396 
787557 
787720 



D. 10" 



51 9.787883 



788046 
788208 
788370 
788532 
788694 
788856 
789018 
789180 
789342 



Cosine. 



Cosine. ID. 10"! Tang. 



.902349 
902253 
902158 
90-2063 
901967 
901872 
901776 
901681 
901585 
9014y0 
901394 
901298 
901202 
901106 
901010 
900914 
900818 
900722 
900626 
900529 
900433 

9.900337 
900242 
900144 
900047 
899951 
899854 
899757 
899660 
899504 
899467 

9,899370 
899273 
899176 
899078 
898981 
898884 
898787 
898689 
898592 
898494 

9.898397 
898299 
898202 
898104 
898006 
897908 
897810 
897712 
897614 
897516 
897418 
897320 
897222 
897123 
897025 
896926 
896828 
8%729 
896631 
896532 

1 Sine. 



9.877114 
877377 
877640 
877903 
878165 
878428 
878691 
878953 
879216 
879478 
879741 
880003 
880265 
880528 
880790 
881052 
881314 
881576 
881839 
882101 
882363 
882625 
882887 
883148 
883410 
883672 
883934 
884196 
884457 
884719 
884980 

9.885242 
885303 
885765 
886026 
886288 
886549 
886810 
887072 
887333 
887594 

). 887855 
888116 
888377 
888639 
888900 
889160 
889421 
889682 
889943 
890204 

1.890465 
890725 
890986 
891247 
891507 
891768 
892028 
892289 
892549 
8 92810 

Co tang. 



D. 10" 



43.8 

43.8 

43.8 

43*.8 

43.8 

43.8 

43.8 

43.7 

43.7 

43.7 

43 

43 

43 

43 

43 

43 

43 

43 

43 

43 

43 

43.6 

43.6 

43.6 

43.6 

43.6 

43.6 

43.6 

43.6 

43.6 

43.6 

43,6 

43.6 

43.6 

43.6 

43.6 

43.5 

43.6 

43.5 

43.5 

43.5 

43.5 

43.5 

43.6 

43.5 

43.5 

43.6 

43.5 

43.5 

43.5 

43.4 

43.4 

43,4 



43.4 



Co tang. 

10.122886 
122623 
122360 
122097 
121835 
121572 
121309 
121047 
120784 
120522 
120259 

10.119997 
119735 
119472 
119210 
118948 
118686 
118424 
118161 
117899 
117637 

10.117375 
117113 
116852 
116590 
1163281 
116066 I 
1158041 
115543 
115281 I 



116020 
10.114758 
114497 
114235 
113974 



N.sine. N. cos. 

60182 
60205 
60228 
60261 
60274 
60298 
60321 
60344 
60367 
60390 
60414 
60437 
60460 
60483 
G0506 
60529 
60553 
60576 
60599 
60622 
60645 
60668 
60691 
60714 
60738 
60761 
60784 
60807 
60830 
60853 



60876 
60899 
60922 
60945 
60968 



113712 ll 60991 
113451 61015 
113190 !; 61038 



79864 
79846 
79829 
79811 
79793 
79776 
79758 
79741 
79723 
79706 
79688 
79671 
79658 
79635 
79618 
79600 
79583 
79565 
79547 
79530 
79512 
79494 
79477 
79469 
79441 
79424 
79406 
79388 
79371 
79353 
79335' 
79318 
79300 
79282 
79264 
79247 
9229 
79211 



112928 i 61061 79193 



112667 1 61084 
112406 116110 



10.112145 i!61130 79140 



111884 



61153 



61176 
61199 
61 22-. 



111623 
111361 
111100 
110840 '161245 
110579; 161268 
110318 161291 
110057 161314 
109796 161337 
10.109533 161360 
109275:161383 
109014,161406 
108753 '61429 
108493 161451 
108232 161474 
107972 ;|61497 
107711 !|61520 
1074511 61543 
1071901! 61566 



Tang. 



II N. co«. N.sine 



79176 
79158 



9122 
79105 
79087 
79069 
79051 
79033 
?9016 
;8998 
78980 
78962 
78944 
78926 
78908 
78891 
78873 
78855 

8837 
78819 
78801 



52 Degrees. 



60 



Log. Sinos and Tangents. (39°) Natural Sines. 



TABLE n. 





1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

H 

12 

13 

14 

15 

16 

17 

18 

19 

20 

21 

22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 



9.798772 
799028 
799184 
799339 
799495 
799651 
799806 
799962 
800117 
800272 
800427 

9.800582 
800737 
800892 
801047 
801201 
801356 
801511 
801665 
801819 
801973 

J. 802128 



D. 10 



802282 
802436 
802689 
802743 
8028^7 
803050 
803204 
803367 
803611 
9.803664 
803817 
803970 
804123 
804276 
804428 
804581 
804734 
804886 
805039 
805191 
805343 
805495 
805647 
805799 
806951 
806103 
806254 
806406 
806557 
9.806709 
806860 
807011 
807163 
807314 
807466 
807615 
807766 
807917 
808067 
Cosine. 



26.0 

26.0 

26.0 

25.9 

25.9 

25.9 

26.9 

25.9 

25.9 

25.8 

25.8 

25.8 

25.8 

25.8 

25.8 

25.8 

25.7 

25.7 

25.7 

25.7 

25.7 

25.7 

25.6 

25.6 

25.6 

25.6 

25.6 

25.6 

25.6 

25.5 

25.5 

25.5 

25.5 

25.5 

25.5 

25.4 

25.4 

26.4 

25.4 

25.4 

25.4 



CxMiine. 



25 

25 

25 

25 

25 

25 

25 

25.3 

25.2 

25.2 

25.2 

25.2 

25.2 

25.2' 

25.2 

25.1 

25.1 

25.1 

25.1 



9.890503 
890400 
890298 
890195 
890093 
889990 
889888 
889785 
889682 
889579 
889477 
9.889374 
889271 
889168 
889064 
888961 
888858 
888755 
888651 
8385'18 
8SH444 
). 888341 
888237 
888134 
888030 
887926 . 
887822 
887718 
837614 
887510 
887406 
>. 887302 
887198 
887093 
886989 
88G885 
886780 
886676 
886571 
886466 
886362 
9.886257 
886152 
886047 
885942 
885837' 
885732 
885627 
885622 
885416 
885311 
9.885205 
885100 
884994 
884889 
884783 
884677 
884572 
884466 
884360 
884264 



D. 10" 



Sine. 



17.0 

17.1 

17.1 

17.1 

17.1 

17.1 

17.1 

17.1 

17.1 

17.1 

17.1 

17.2 

17.2 

17.2 

17.2 

17.2 

17.2 

17.2 

17.2 

17.2 

17.3 

17.3 

17.3 

17.3 

17.3 

17.3 

17.3 

17.3 

17.3 

17.3 

17.4 

17. .4 

17.4 

17.4 

17.4 

17.4 

17.4 

17.4 

17.4 

17.4 

17.5 

17.5 

17.6 

17.5 

17.6 

17.5 

17.6 

17.5 

17.6 

17.6 

17.6 

17.6 

17.6 

17.6 

17.6 

17.6 

17.6 

17.6 

17.6 

17.6 



Tang. 



9.908369 
90S628 
90S886 
909144 
909402 
903660 
909918 
910177 
910435 
910693 
910951 
9.911209 
911467 
911724 
911982 
912240 
912498 
912766 
913014 
913271 
913529 
9.913787 
914044 
914302 
914660 
914817 
916075 
915332 
916590 
916847 
916104 
9.916362 
916619 
916877 
917134 
917391 
917648 
917905 
918163 
918420 
918677 
9.918934 
919191 
919448 
919705 
919962 
920219 
920476 
920733 
920990 
921247 
). 921503 
921760 
922017 
922274 
922630 
922787 
923044 
923300 
923657 
923813 
Cotang. 



D. 10" 



43.0 

43.0 

43.0 

43.0 

43.0 

43.0 

43.0 

43.0 

43.0 

43.0 

43.0 

43.0 

43.0 

43.0 

43.0 

43.0 

43.0 

43.0 

42.9 

42.9 

42.9 

42.9 

42.9 

42.9 

42,9- 

42,9 

42.9 

42.9 

42,9 

42.9 

42.9 

42,9 

42.9 

42.9 

42.9 

42.9 

42.9 

42.9 

42.8 

42.8 

42.8 

42.8 

42.8 

42.8 

42.8 

42.8 

42.8 

42.8 

42.8 

42.8 

42.8 

42.8 

42.8 

42.8 

42.8 

42.8 

42.8 

42.8 

42.8 

42.7 



Cotaug. 



10.091631 
091372 
091114 
090856 
090598 
090340 
090082 

089823 

089505 

089307 

089049 
10.088791 

0S8533 

088276 

088018 

087760] 

087502 

087244 ! 

a8S986 

086729 

086471 1 
iO. 036213 

036956 

085698 

085440 

035183 

034925 

084668 

084410 

084163 

033896 
10.083638 

083381 

083123 

082866 

082609 

082352 

082096 

081837 

081680 

081323 
10-081066 
080809 
080552 
030295 
080038 
079781 
079524 
07926? 

079010 It 64033 
078753 j 1 64056 
10.0784971164078 
07824011 64100 
077983 |i6412G 
077726 

077470 !i 64167 
077213;; 64190 
0769561164212 
076700 1 j 64234 
07644311 64266 
0761871164279 



N. sine. N. cos. 

77716 

77696 

77678 

77660 

77641 

77623 

77605 

77586 

77668 

77550 

77531 

77513 

77494 

77476 

77458 

77439 

77421 

77402 

77384 

77366 

77347 

77329 

77310 

77292 

77273 

77255 

77236 

77218 

77199 

77181 

77162 

77144 

77125 

77107 

77088 

77070 

77051 

77033 

77014 

76996 

76977 

76959 



62932 

62955 

62977 

63000 

63022 

63045 

63068 

63090 

63113 

63135 

63158 

93180 

63203 

63225 

63248 

63271 

63293 

63316 

63338 

63361 

63383 

63406 

63428 

63451 

63473 

63496 

63518 

63540 

63663 

63585 

63608 

63630 

63653 

63675 

63698 

63720 

63742 

63765 

63787 

63810 

63832 

63854 

63877 76940 

63899 76921 



6392: 



60 

59 

58 

57 

66 

65 

64 

53 

52 

51 

50 

49 

48 

47 

46 

45 

44 

43 

42 

41 

40 

39 

38 

37 

36 

35 

34 

33 ! 

32 

31 

30 

29 

28 

27 

26 

25 

24 



76903 



63944 176884 

j 63966 76866 

1 6398:.' 76847 

164011,76828 

76810 

76791 

76772 

76754 

76736 

64145 76717 

76098 

6679 

76661 

76642 

76623 

76604 



Tang. U N. pou. N.Bine 



50 Degrees. 



TABLE n. 



Log. SineB and Tangents. (40*) Natural Sines. 



61 



dTio^ 



Sine. 



D. IC 



810017 
810167 
810316 
810465 
810614 
810763 
810912 
811061 

.811210 
811358 
811507 
811655 
811804 
811952 
812100 
812248 
812396 
812544 

.812692 
812840 
812988 
813135 
813283 
813430 
813578 
813725 
813872 
814019 

.814166 
814313 
814460 
814607 
814753 
814900 
815046 
815193 
815339 
815485 

1.816631 
815778 
815924 
816069 
816215 
816361 
816507 
816652 
816798 
816943 
Cosine. 



25.1 
25.1 
25.1 
25.0 
25.0 
25.0 
25.0 
25.0 
26.0 
24.9 
24.9 
24.9 
24.9 
24.9 
24.9 
24.8 
24.8 
24.8 
24.8 
24.8 
24.8 
24.8 
24.7 
24.7 
24.7 
24.7 
24.7 
24.7 
24.7 
24.6 
24.6 
24.6 
24.6 
24.6 
24.6 
24.6 
24.5 
24.6 
24.5 
24.5 
24.5 
24.5 
24.5 
24.4 
24.4 
24.4 



Coaiae. jD. 10" 



24 

24 

24 

24 

24 

24 

24,3 

24,3 

24.3 

24.3 

24.3 

24.2 

24.2 

24.2 



883723 
883617 
883510 
883404 
883297 
883191 

.883084 
882977 
882871 
882764 
882667 
882550 
882443 
882336 
882229 
882121 

.882014 
881907 
881799 
881692 
881584 
881477 
881369 
881261 
881153 
881046 

. 880938- 
880830 
880722 
880613 
880505 
880397 
880289 
880180 
880072 
879963 

.879855 
879746 
879637 
879529 
879420 
879311 
879202 
879093 
878984 
878875 

.878766 
878656 
878547 
878438 
878328 
878219 
878.09 
877999 
877890 
877780 
Sine. 



8.0 

8.0 

8.0 

8.0 

8.0 

8.0 

8.0 

8.0 

8. 

8. 

8. 

8. 

8. 

8. 

8. 

8. 

8. 

8. 

8.2 

8.2 

8.2 

8.2 

8.2 

8.2 

8.2 

8.2 

8.2 

8.3 

8.3 

8.3 

8.3 



Tang. 



1.923813 
924070 
924327 
924583 
924840 
925096 
925352 
925609 
926865 
926122 
926378 

I.926G34 
926890 
927147 
927403 
927669 
927915 
928171 
928427 
928683 
928940 

.929196 
929452 
929708 
929964 
930220 
930475 
930731 
930987 
931243 
931499 

'.931755 
932010 
932266 
932522 
932778 
933033 
933289 
933545 
933800 
934056 

.934311 
934567 
934823 
935078 
935333 
9365S0 
535844 
936100 
936356 
936610 

1.936866 
937121 
937376 
937632 
937887 
938142 
938398 
938653 
938908 
939163 

Cotang. 



Cotang. i N .«ine. N. cos. 



10.076187 
076930 
076673 
076417 
075160 
074904 
074648 
074391 
074135 
073878 
073622 

10.073366 
073110 
072863 
072597 
072341 
072085 
071829 
071573 
071317 
071060 

10.070804 
070548 
070292 
070036 
069780 
069526 
069269 
069013 
068767 
068601 

10.068245 
0679901 
067734 t 
067478 I 
067222 I 
066967 
066711 
066455 
066200 
065944 

10.065689 

065433 

065177 

064922 i 

064667 I 

064411 I 

064166 i 

063900 ! 

063646 i 

063390 I 

10.0631341 
062879 I 
062624 I 
062368 
062113 
061868 
061602 
061347 
061092 
060837 



64279 
64301 
64323 
64346 
64368 
64390 
64412. 
64436 
64457 
64479 
64501 
64624 
64646 
64668 
64690 
64612 
64636 
64657 
64679 
64701 
64723 
64746 
'64768 
64790 
64812 
64834 
64856 
64878 
64901 
64923 
64945 
64967 
64989 
65011 
65033 
65065 
65077 
65100 
65122 
65144 
65166 
65188 
65210 
65232 
65264 
66276 
66298 
65320 
65342 
65364 
65386 
65408 
65430 
65452 
66474 
65496 
65518 
65540 
65562 
66584 
66606 



Tang. 



76604 
76686 
76567 
76648 
76630 
76611 
76492 
76473 
76466 
76436 
76417 
76398 
6380 
76361 
76342 
76323 
76304 
76286 
76267 
76248 
76229 
76210 
76192 
76173 
76164 
76135 
76116 
76097 
76078 
76059 
76041 
76022 
76003 
76984 
75965 
75946 
76927 
76908 
76889 
76870 
76861 
75832 
75813 
75794 
76775 
75766 
75738 
76719 
75700 
756S0 
75661 
76642 
75623 
75604 
75585 
75666 
75547 
75528 
75609 
76490 
76471 



N. COS. N.sine. 



49 Degrees. 



Log. Sines and Tangents. (41°) Natural Sines. 



TABLE IL 



Sine. D. 10" C!osiue 



9.816948 
817088 
817233 
817379 
817524 
817668 
817813 
817958 
818103 
818247 
818392 

9.81S636 
818681 
818825 
818969 
819113 
819257 
819401 
819545 
819689 
819832 

9.819976 
820120 
820263 
820405 
820550 
820693 
820836 
820979 
821122 
821265 

9.821407 
821550 
821693 
821835 
821977 
822120 
822262 
822404 
822546 
822688 

9.822830 
822972 
823114 
823256 
823397 
823639 
823680 
823821 
823963 
824104 

9.824245 
824386 
824627 
82-1668 
82-1808 
824949 
826090 
825230 
825371 
825511 
Cosine. 



24.2 
24.2 
24.2 
24.2 
24.1 
24.1 
24.1 
24.1 
24.1 
24.1 
24.1 
24.0 
24.0 
24. 
24.0 
24.0 
24.0 
24.0 
23.9 
23.9 
23.9 
23.9 

23. y 

23.9 
23.9 
23.8 
23.8 
23.8 
23.8 
23.8 
23.8 



23 

23 

23 

23 

23 

23 

23 

23 

23 

23 

23 

23 

23.6 

23.6 

23.6 

23.6 

23.5 

23.6 

23.5 

23.6 

23.6 

23.6 

23.6 

23.4 

23.4 

23.4 

23.4 

23.4 

23.4 



.877780 
877670 
877560 
877450 
877340 
877230 
877120 
877010 
876899 
876789 
87667 
.876568 
876457 
876347 
876236 
876125 
876014 
876904 
875793 
876682 
876571 
,876469 
876348 
876237 
876126 
876014 
874903 
874791 
874680 
874568 
874456 
,874344 
874232 
874121 
874009 
873896 
873784 
873672 
873660 
873448 
873336 
873223 
873110 
872998 
872886 
872772 
872659 
872547 
872434 
872321 
872208 
872095 
871981 
871868 
871755 
871641 
871528 
871414 
871301 
871187 
871073 
"si]ie.~ 



9.87 



D. 10" 



18.4 



18 

18 

18 

18 

18 

18 

18 

18 

18 

18 

18 

18,6 

18.6 

18.6 

18.6 

18.6 

18.6 

18. & 

18.6 

18.6 

18.7 

18.7 

18.7 

18,7 

18.7 

18.7 

18.7 

18.7 

18.7 

18.7 

18.8 

18.8 

IS. 8 

18.8 

18.8 

18.8 

18.8 

18.8 

18.3 

18.9 

18.9 

18.9 

18.9 

18.9 

18.9 

18.9 

18.9 

18.9 



'I'ang. 

9.939163 
939418 
939673 
939928 
940183 
940438 
940694 
940949 
941204 
941458 
941714 

9.941968 
942223 
942478 
942733 
942988 
943243 
943498 
943752 
944007 
944262 

9.944517 
944771 
945026 
945281 
945535 
945790 
946045 
946299 
946554 
946808 

9.947053 
947318 
947572 
947826 
948081 
948336 
948590 
948844 
949099 
949353 

9.949607 
949862 
950116 
950370 
950625 
950879 
951133 
951388 
951642 
951896 

3.952150 
952405 
952659 
952913 
953167 
953421 
953675 
963929 
954183 
964437 
Co tang. 



D. 10" 



42.5 

42.6 

42.5 

42.6 

42.5 

42.6 

42 

42 

42 

42 

42 

42 

42 

42 

42.5 

42.6 

42.6 

42.6 

42.6 

42.6 

42.6 

42.6 

42.4 

42.4 

42.4 

42 

42 

42 

42 

42 

42 

42 

42.4 

42.4 

42.4 

42.4 

42.4 

42.4 

42.4 

42.4 

42.4 

42.4 

42 



Cotang. I iN. sine. 



42 

42 

42 

42 

42 

42.4 

42.4 

42.4 

42.4 

42.4 

42.4 

42.4 

42.3 

42.3 

42.3 

42.3 

42.3 



10.030837 i! 66606 
060582 j 66628 
060327 165650 
060072 I j 65672 
059817 !JG5694 
05956-2 i I 65716 
059306 jj 65738 
059051 65759 
058796 1 1 65781 
0585421165803 
0582S6!j65825 

10.0580321166847 
057777 1 1 65869 
067622! 1 65891 



057267 
057012 
056757 
056502 
056248 
055993 
055738 
10.055483 
055229 



66913 
65935 
65966 
65978 
66000 
66022 
66044 
66066 
66088 



054974 1 1 66109 
0547191166131 
054465 166153 



054210 
053965 
053701 
063446 
053192 
10.062937 
062682 
052428 



i 66176 
166197 
[66218 
166240 
1 66262 
66284 
1 66306 
166327 



052174 j 66349 
06191911 66371 
051664 1 1 66393 
0514101 66414 
051166 66436 
050901 66458 
050647 I 66480 

10 050393 66601 
050138 I j 66623 
049884 1166645 
049630 
049376 
049121 
048867 1 66632 
0486121 1 66653 
048358 166676 
048104 j 66697 

10.047850 1 66718 
047595 1 1 66740 
047341] 166762 
047087 '66783 



66666 
C6688 
66610 



046833 
046679 
046325 
046071 
045817 
045563 



66805 
66827 
66848 
66870 
66891 
66913 



75471 

76462 
?5433 
75414 
76396 
75375 
76366 
76337 
75318 
76299 
75280 
75261 
76241 
75222 
76203 
75184 
75166 
76146 
76126 
75107 
76088 
76069 
76050 
75030 
76011 
74992 
74973 
74963 
74934 
74915 
74896 
74876 
74857 
74838 
74818 
74799 
74780 
74760 
74741 
74722 
74703 
74683 
74663 
74644 
74625 
74606 
74586 
74667 
74548 
74622 
74509 
74489 
74470 
4461 
74431 
74412 
74392 
74373 
74353 
4334 
74314 



N. COS. N.sine 



48 Degrees. 



TABLE n. 



Log. Sines and Tangents. (42°)* Natural gines. 



63 



Sine. 



D. 10" Cosine. 



D. 10" 



Tang. 



D. 10' 



Cotang. ;N. Bine. IN. COB 



.825.^11 
826651 
825791 
825931 
826071 
826211 
826351 
826491 
826631 
826770 
826910 

.827049 
827189 
827328 
827467 
827606 
827745 
827884 
828023 
828162 
828301 

.828439 
828578 
828716 
828S55 
828993 
829131 
829269 
829407 
829545 
829683 

.829821 
829959 
830097 
830234 
830372 
830509 
830646 
830784 
830921 
831058 

.831195 
831332 
831469 
831606 
831742 
831879 ! 
832015 
832162 
882288 
832425 
832561 
832697 
832833 
832969 
833105 
833241 
833377 
833612 
833648 
833783 



23.4 
23.3 
23.3 
23.3 
23.3 
23.3 
33.3 
23.3 
23.3 
23.2 
23.2 
23.2 
23.2 
23.2 
23.2 
23.2 
23.2 
23.1 
23.1 
23.1 
23.1 
23.1 
23.1 
23.1 
23.0 
23.0 
23.0 
23.0 
23.0 
23.0 
23.0 
22.9 
22.9 
22.9 
22.9 
22.9 
22.9 
22.9 
22.9 
22.8 
22.8 
22.8 
22.8 
22.8 
22.8 
22.8 
22.8 
22.7 
23.7 
22.7 

:22.7 

122.7 

122.7 

1 22.7 

'22.6 

!22.6 

122.6 

122.6 

^22 

1 22.6 



Co.siiie. 



.871073 
870960 
870846 
870732 
870618 
87050-i 
870390 
870276 
870161 
870047 
869933 
.869818 
869704 
869589 
869474 
869360 
869246 
869130 
869015 
868900 
868785 
.868670 
868555 
868440 
868324 
868209 
868093 
867978 
867862 
867747 
867631 
.867515 
867399 
867283 
867167 
867051 
866935 
866819 
866703 
866586 
866470 
.866853 
866237 
866120 
866004 
865887 
865770 
865653 
865536 
865419 
865302 
1.865185 
865068 
864950 
864833 
864716 
864598 
864481 
864363 
864245 
864127 



19.0 
19.0 
19.0 
19.0 
19.0 
19.0 
19.0 
19.0 
19.0 
19.1 
19.1 
19.1 
19.1 
19.1 
19.1 
19.1 
19.1 
19.1 
19.2 
19.2 
19.2 
19.2 
19.2 
19.2 
19.2 
19.2 
19.2 
19.3 
19.3 
19.3 
19.3 
19.3 
19.3 
19.3 
19.3 
19.3 
19.4 
19.4 
19.4 
19.4 
19.4 
19.4 
19.1 
19.4 
19.5 
19.5 
19.6 
19.6 
19.5 
19.5 
19.5 
19.5 
19.5 
19.6 
19.6 
19.6 
19.6 
19.6 
19.6 
19.6 



.964437 
964691 
954945 
955200 
955454 
955707 
955961 
956215 
956469 
956723 
956977 

.957231 
957485 
957739 
957993 
958246 
958500 
958764 
959008 
959262 
959516 

.959769 
960023 
960277 
960531 
960784 
961038 
961291 
961545 
961799 
962052 

.962306 
962560 
962813 
963067 
963320 
963574 
963827 
964081 
964335 
964588 

.9t>4J542 
9(i5095 
965349 
965602 
965856 
966109 
966362 
966616 



967123 
.9S7376 
967629 
967883 
968136 
968389 
968643 
968896 
969149 
969403 
969656 



42.3 
42.3 
42.3 
42.3 
42.3 
42.3 
43.3 
42.3 
43.3 
42.3 
43.8 
43.3 
43.3 
43.3 
43.3 
43.3 
43.3 
43.3 
42.8 
42.3 
43.3 
43.3 
43.3 
43.3 
42.3 
42.3 
42.3 
42.3 
42.3 
42.3 
42.3 
42.3 
42.3 
42.3 
42.3 
42.3 



42 

42 

42 

42 

42 

42 

42 

42 

42 

43.2 

42.2 

42.2 

42.3 

42.2 

43.3 

42.2 

42.2 

42.3 

43.2 

43.3 

43.2 

42.2 

42.2 

42.2 



10.045563 
045309 
045065 
044800 
044546 
044293 
044039 
043785 
043531 
043277 
043023 

10.042769 
042615 
042261 
042007 
041764 
041500 
041246 
040992 
040738 
040484 

10.040231 
039977 
039723 
039469 
039216 
038962 
038709 
038466 
038201 
037948 

10.037694 
037440 
037187 
036933 
036680 
036426 
036173 
035919 
035665 
035412 

10.036168 
034905 
034651 
034398 
034145 
033891 
033638 
033384 
033131 
032877 

10.032624 
032371 
032117 
031864 
031611 
031357 
031104 
030851 
030597 
030344 



||66913|74314 
1 1 66935174295 
j! 66956174276 
''66978174266 
i! 66999 74237 



ii 67021 
167043 
!' 67064 
'i 67086 
ii 67107 
i! 67129 
;: 67151 
!| 67172 
i 1 67194 
67215 
1 1 67237 
i! 67258 



74217 
74198 
74178 
74169 
74139 
74120 
74100 
74080 
74061 
74041 
74022 
74002 



1 67280|73983 



Cotang. 



Tang. 



3963 
3944 
73924 
73904 
73885 
73865 
73846 
73826 
73806 
73787 
3767 
73747 
73728 
73708 
73688 
73669 
73649 
73629 
73610 
73590 
73570 
73551 
73531 
73611 
73491 
73472 
73452 
73432 
73413 
73393 
373 
73353 
73333 
73314 
73294 
73274 
73254 
73234 
73216 
73195 
73175 
73155 
73136 
N. CO?. N.Hin«, 



1167301 

I {67323 

167344 

i 1 67366 

167387 

167409 

67430 

! 67452 

67473 

i 167495 

i 1 67516 

167638 

1 167559 

i 1 67580 

67602 

ii 67623 

ij 67646 

i 167666 

67688 

67709 

67730 

67762 

67773 

67795 

67816 

I 67837 

! 67859 

167880 

167901 

I 67923 



67944 



1:67966 
1 1 67987 
1 1 68008 
i 68029 
ii 68051 
I 68072 
68093 
168115 
1 68136 
68157 
iG8179 
i 68200 



47 Degrees^ 



27 



64 



Log. Sines and Tangenta. (43°) Natural Sines. 



TABLE n. 



S ine. 
9.833783 



1 
2 
3 

4 

6 

6 

7 

6 

9 

10 

U 

12 

13 

14 

16 J 

16 

17 

18 

19 

20 

21 

22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44 

45 

46 

47 

48 

49 

50 

51 

52 

53 

54 

55 

56 

57 

58 

59 

60 



833919 
834054 
834189 
834325 
834460 
834595 
834730 
834865 
834999 
835134 

9,835269 
835403 
835538 
835672 
835807 
835941 
836075 
836209 
836343 
836477 

9,836611 
836745 
836878 
837012 
837146 
837279 
837412 
837646 
837679 
837812 

9,837945 
838078 
838211 
838344 
838477 
838610 
838742 
838875 
839007 
839140 

9,839272 
839404 
839536 
839668 
839800 
839932 
840064 
840196 
840328 
840459 

9,840591 
840722 
840854 
840986 
841116 
841247 
841378 
841509 
841640 
8 41771 
Cosine. 



D. ly^ l Cosine. i D. m Tang. D. 10" | Cotang 



22.6 
22.5 
22.5 
22.5 
22.5 
22.5 
22.5 
22.5 
22.5 
22.4 
22.4 
22.4 
22.4 
22.4 
22,4 
22.4 
22.4 
22.3 
22.3 
22.3 
22.3 
22.3 
22.3 
22.3 
22.2 
22.2 
22.2 
22.2 
22.2 
22.2 
22.2 
22.2 
22.1 
22,1 
22.1 
22.1 
22,1 
22,1 
22.1 
22.1 
22,0 
22.0 
22,0 
22.0 
22.0 
22,0 
22,0 
9 

21,9 
21.9 
21.9 
21.9 
21.9 
21.9 
21.9 
21.8 
21.8 
21.8 
21.8 
21.8 



1.864127 
864010 
863892 
863774 
863656 
863538 
863419 
863301 
863183 
863064 
862946 
.862827 
862709 
862590 
862471 
862353 
862234 
862115 
861996 
861877 
861758 
.861638 
861519 
861400 
861280 
861161 
861041 
860922 
860802 
860682 
860662 
.860442 
860322 
860202 
860082 
869962 
859842 
859721 
859601 
869480 
859360 
859239 
859119 
858998 
858877 
868756 
868636 
868514 
868393 
858272 
858151 
,858029 
857908 
857786 
857665 
857543 
857422 
857300 
857178 
857056 
856934 
Sine. 



19.6 

19.6 

19.7 

19.7 

19.7 

19,7 

19.7 

19.7 

19.7 

19.7 

19.8 

19.8 

19.8 

19.8 

19.8 

19.8 

19.8 

19.8 

19.8 

19.8 

19.9 

19.9 

19.9 

19.9 

19.9 

19.9 

19.9 

19.9 

19.9 

20.0 

20.0 

20.0 

20.0 

20.0 

20.0 

20.0 

20.0 

20.1 

20.1 

20,1 

20.1 

20,1 

20,1 

20.1 

20.1 

20,2 

20.2 

20.2 

20,2 

20,2 

20.2 

20.2 

20.2 

20,2 

20.3 

20.3 

20.3 

20.3 

20.3 

20.3 



970162 
970416 
970669 
970922 
971175 
971429 
971682 
971935 
972188 
9.972441 



9.969656 .„ o 
969909 ^Z-i 



972948 
973201 
973454 
973707 
973960 
974213 
974466 
974719 

9.974973 
975226 
976479 
976732 
975985 
976238 
976491 
976744 
976997 
977250 
.977503 
977766 
978009 
978262 
978615 
978768 
979021 
979274 
979527 
979780 

9.980033 
980286 
980538 
980791 
981044 
981297 
981560 
981803 
982056 
982309 

9.982562 
982814 
983067 
983320 
983573 
983826 
984079 
984331 
984584 
9 84837 
Cotang. 1 



42.2 

42.2 

42.2 

42.2 

42,2 

42,2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42,2 

42,2 

42.2 

42.2 

42.2 

42.2 

42.2 

42,2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42,2 

42,1 

42.1 

42.1 

42.1 

42.1 

42,1 

42,1 

42.1 

42,1 

42,1 

42,1 

42,1 

42,1 

42,1 

42.1 

42.1 



I- 

10.030344 
030091 
029838 
029584 
029331 
029078 
028825 
028571 
028318 
028066 
027812 

10.027559 
027306 
027062 
026799 
026546 
026293 
026040 
025787 
025534 
025281 

10.025027 
024774 
024521 
024268 
024015 
023762 
023609 
023266 
023003 
022750 

10.022497 
022244 
021991 
021738 
021485 
021232 
020979 
• 020726 
020473 
020220 

10.019967 
019714 
019462 
019209 
018956 
018703 
018450 
018197 
017944 
017691 I 

10.017438 
017186 
016933 
016680 
016427 
016174 
015921 
015669 
015416 
015163 
"fang. 



|N .sine. N. cos 



68200 
68221 
68242 
68264 
68285 
68306 
68327 
68349 
68370 
68391 
68412 
68434 
68455 
68476 
68497 
68618 
68639 
68561 
68582 
68603 
68624 
68645 
68666 



68709 
68730 
68751 
68772 
68793 
68814 
68836 
68857 
68878 
68899 
68920 
68941 
68962 



73135 
73116 
73096 
73076 
73056 
73036 
73016 
72996 
72976 
72957 
72937 
72917 
72897 
72877 
72857 
72837 
72817 
72797 
72777 
72757 
72737 
72717 
72697 
72677 
72657 
72637 
72617 
72697 
72677 
72557 
72537 
72517 
72497 
72477 
72457 
72437 
72417 



68983 72397 
6900472377 
6902672367 
69046 72337 



69067 
69088 
69109 
69130 
69151 
69172 



72317 
72297 
72277 
72257 
72236 
72216 



69193 72196 
169214172176 
69235 72156 
69256I721S6 
69277 i721 16 
69298172095 
69319 72075 
6934072055 
6936172035 
69382 72015 



69403 
69424 
69445 
69466 



71995 
71974 
71954 
71934 



N. cos.lN.sine. 



46 Degrees. 



TABLE IT. 



Log. Sines and Tangents. (44°) Natural Sines. 



65 



S ine. 

,841771 
841902 
842033 
842163 
842294 
842424 
842555 
842685 
842815 
842946 
843076 
.843206 
843336 
843466 
843595 
843725 
843855 
843984 
844114 
844243 
844372 
844502 
844631 
844760 
844889 
845018 
845147 
845276 
845405 
845533 
846662 
9.845790 
845919 
846047 
846175 
846304 
846432 
846560 
846688 
846816 
846944 

.847071 
847199 
847327 
847454 
847582 
847709 
847836 
847964 
848091 
848218 

.848346 
848472 
848599 
848726 
848852 
848979 
849106 
849232 



849486 



Cosine. 



D. 10" 



21.8 
21.8 
21.8 
21.7 
21,7 
21.7 
21.7 
21,7 
21.7 
21.7 
21.7 
21.6 
21,6 
21,6 
21.6 
21.6 
21.6 
21.6 
21.6 
21.5 
21.5 
21.5 
21.5 
21.5 
21.5 
21.5 
21.6 
21.4 
21.4 
21.4 
21.4 
21,4 
21.4 
21.4 
21.4 
21.4 
21.3 
21.3 
21.3 
21.3 
21.3 
21.3 
21.3 
21,3 
21.2 
21,2 
21.2 
21,2 
21,2 
21.2 
21.2 
21.2 
21.1 
21.1 
21.1 
21.1 
21.1 
21.1 
21.1 
21.1 



Cosine. 



9.866934 
866812 
866690 
856568 
856446 
866323 
866201 
866078 
855956 
866833 
855711 
855588 
855465 
855342 
865219 
855096 
854973 
854850 
854727 
854603 
854480 

9.854356 
854233 
854109 
853986 
853862 
863738 
853614 
853490 
853366 
853242 
853118 
852994 
852869 
862745 
862620 
852496 
852371 
852247 
852122 
861997 

9.851872 
851747 
851622 
851497 
851372 
851246 
851121 
850996 
850870 
860745 
850619 
850493 
850368 
850242 
850116 
849990 
849864 
849738 
849611 
849485 



Sine. 



D.10" 



20.3 
20.3 
20.4 
20.4 
20.4 
20.4 
20.4 
20.4 
20.4 
20.4 
20.6 
20.6 



20 

20 

20 

20 

20 

20 

20.6 

20.6 

20,6 

20.6 

20.6 

20,6 

20.6 

20.6 

20.6 

20.7 

20.7 

20,7 

20.7 

20.7 

20,7 

20.7 

20.7 

20.7 

20.8 

20,8 

20,8 

20.8 

20.8 

20.8 

20.8 

20.8 

20.9 

20.9 

20,9 

20.9 

20,9 

20,9 

20,9 

20.9 

21,0 

21.0 

21.0 

21.0 

21.0 

21.0 

21.0 

21.0 



Tang. 



984837 
985090 
985343 
985696 
985848 
986101 
980354 
986607 
986860 
987112 
987365 
987618 
987871 
988123 
988376 
988629 
988882 
989134 
989387 
989640 
989893 

9.990146 
990398 
990661 
990903 
991156 
991409 
991662 
991914 
992167 
992420 
992672 
992926 
993178 
993430 
993683 
993936 
994189 
994441 
994G94 
994947 

9.995199 
996452 
995705 
995967 
996210 
996463 
996716 
996968 
997221 
997473 
,997726 
997979 
998231 
998484 
998737 
998989 
999242 
999495 
999748 

10.0 00000 
Co tang. 



D. 10" 



Cotang. 



N. sine 



10.015163 
014910 
014667 
014404 
014162 
013899 
013646 
013393 
013140 
013888 
012636 

10.012382 
012129 
011877 
011624 
011371 
011118 
010866,1 
0106131 
010360,' 
010107 i 

10.009855 ii 
009602 
009349 
009097 
008844 
608691 
008338 
008086 
007833 
007580 

10-007328 
007075 
006822 
006570 
006317 
006064 
005811 i 
005559 
005306 
005063 

10.004801 
004648 
004296 
004043 
003790 
003537 
003285 
003032 
002779 i 
002527 ! 

10.002274 I 
002021 
001769 
001516 
001263 
001011 
000758 
000505 
000263 
000000 



69466 
69487 
69508 
69529 
69549 
69570 
69591 
69612 



69633 71772 
69654 71752 



Tang. 



69675 
69696 
69717 
69737 
69758 
69779 
69800 
69821 
69842 
698G2 
69883 
69904 
69925 
69946 
69966 
69987 
70008 
70029 
70049 
70070 
70091 
70112 
70132 
70153 
70174 
70196 
70215 
70236 
70257 
70277 
70298 
70319 
70339 
70360 
70381 
70401 
70422 
70443 
70463 
70484 
70505 
70525 
70546 
70567 
70587 
70608 
70628 
70649 
70670 
70691 



N. cos 



71934 
71914 
71894 
71873 
71853 
71833 
71813 
71792 



71732 
71711 
71691 
71671 
71650 
71630 
71610 
71590 
71569 
71549 
71529 
71508 
71488 
71468 
71447 
71427 
71407 
71386 
71366 
71345 
71325 
71305 
71284 
71264 
71243 
71223 
71203 
71182 
71162 
71141 
71121 
71100 
71080 
71059 
71039 
71019 
70998 
70978 
70957 
0937 
70916 
70896 
70876 
70856 
70834 
70813 
70793 
70772 
70752 
70731 



7071170711 



N. COS. N. mm- 



45 Degrees. 



66 






LOGARITHMS 
















TABLE HI 


. 












LOGARITHMS OF 


NUMBERS, 














FROM 1 TO 110, 










INCLUDING 


TWELVE DECIMAL PLACES 






N. 
I 


Log. 


N. 

36 


Log. 




0. 000 


000 


000 000 


1. 656 


302 


500 


767 




3 


0. 301 


029 


995 644 


37 


I. 568 


201 


724 


067 




3 


0. 477 


121 


254 720 


38 


1. 579 


783 


596 


617 




4 


0. 602 


059 


991 328 


39 


1. 591 


064 


607 


264 




6 


0. 698 


970 


004 336 


40 


1. 602 


059 


991 


328 




6 


0. 778 


151 


250 384 


41 


1. 612 


783 


846 


720 




7 


0. 845 


098 


040 014 


42 


I, 623 


249 


290 


398 




8 


0. 903 


089 


986 992 


43 


1. 633 


468 


465 


679 




9 


0. 954 


242 


509 440 


44 


1. 643 


452 


676 


486 




10 


1, 000 


000 


000 OOO 


45 


U 663 


212 


513 


775 




H 


1. 041 


392 


685 158 


46 


1. 662 


767 


831 


682 




12 


1. 079 


181 


246 048 


47 


1. 672 


097 


857 


936 




13 


1, 113 


943 


352 309 


48 


1. 681 


241 


237 


376 




14 


1. 146 


128 


035 678 


49 


1. 690 


196 


080 


028 




15 


1. 176 


091 


259 059 


60 


1. 698 


970 


004 


336 




16 


1. 204 


119 


982 656 


51 


1, 707 


570 


176 


098 




17 


1, 230 


448 


921 378 


52 


1. 716 


003 


243 


635 




18 


1. 255 


272 


505 103 


53 


1. 724 


275 


869 


601 




19 


1. 278 


753 


600 953 


54 


I. 732 


393 


769 


823 




20 


1, 301 


029 


995 664 


55 


1. 740 


362 


689 


494 




21 


1, 322 


219 


294 734 


56 


1 748 


188 


027 


006 




22 


1. 342 


422 


680 822 


67 


1. 756 


874 


855 


672 




23 


1. 361 


727 


836 076 


58 


1. 763 


427 


993 


663 




24 


1. 380 


211 


241 712 


69 


1. 770 


852 


Oil 


642 




25 


1. 397 


940 


008 672 


60 


1. 778 


161 


250 


384 




26 


1. 414 


973 


347 971 


61 


I. 785 


329 


835 


Oil 




27 


1. 431 


363 


764 159 


62 


1. 792 


391 


689 


492 




28 


1. 447 


158 


031 342 


63 


1.' 799 


340 


649 


464 




29 


1. 462 


397 


997 899 


64 


1, 806 


179 


973 


984 




30 


1. 477 


121 


254 720 


65 


1. 812 


913 


366 


643 




31 


1. 491 


361 


693 834 


66 


1. 819 


543 


935 


542 




32 


1. 505 


149 


978 320 


67 


1. 826 


074 


302 


701 




33 


1. 618 


513 


939 878 


68 


1. 832 


608 


912 


706 




34 


1. 531 


478 


917 042 


69 


1. 838 


849 


090 


737 




35 


1. 544 


068 


044 350 


70 


1. 845 


098 


040 


014 















OF NUMBERS. 








67 






■ N. 


Log. 


N. 


Log. 






71 




851 


258 


348 


719 


91 




959 


041 


392 


321 






72 




857 


332 


496 


431 


92 




968 


787 


827 


346 






73 




863 


322 


860 


120 


93 




968 


482 


948 


654 






74 




869 


231 


719 


731 


94 




973 


127 


853 


600 






75 




875 


om 


263 


392 


95 




977 


723 


605 


289 






76 




880 


813 


592 


281 


96 




982 


271 


233 


040 






77 




886 


490 


725 


172 


97 




986 


771 


734 


266 






78 




892 


094 


602 


690 


98 




991 


226 


075 


692 






79 




897 


627 


091 


290 


99 




995 


635 


194 


59S 






80 




903 


089 


986 


992 


100 


2. 


000 


000 


000 


000 






81 




908 


485 


018 


879 


101 


2, 


004 


321 


373 


783 






82 




9i3 


813 


852 


384 


102 


2. 


008 


600 


171 


762 






83 




9i9 


078 


092 


376 


103 


2. 


012 


837 


224 


705 






84 




924 


279 


286 


062 


104 


2. 


017 


033 


339 


299 






85 




929 


418 


925 


714 


105 


2. 


021 


189 


299 


070 






8G 




934 


498 


451 


244 


106 


2. 


025 


305 


865 


265 






87 




939 


519 


252 


619 


107 


2. 


029 


383 


777 


685 






88 




944 


482 


672 


150 


108 


2. 


033 


423 


755 


487 






89 




949 


390 


006 


645 


109 


•2. 


037 


426 


497 


941 






90 




954 


242 


509 


439 


110 


2. 


041 


392 


685 


158 






LO 


GARITHMS 


OF THE PRIME NUMBERS 














FROM IK 


) TO ] 


11^9. 














I 


NCLUDING 


TWELV] 


K DE 


CIMAL PLACES 








N. 






Log. 






N. 


Log. 






113 


2. 


053 


078 


443 


483 


197 


2. 


294 


466 


266 


162 






127 


2. 


103 


803 


720 


956 


199 


2. 


298 


853 


076 


410 






131 


2. 


117 


271 


295 


656 


211 


2. 


324 


282 


455 


298 






137 


2. 


136 


720 


567 


156 


223 


2. 


348 


304 


863 


222 






139 


2. 


143 


014 


8G0 


254 


227 


2. 


356 


025 


857 


189 






149 


3. 


173 


186 


268 


412 


229 


2. 


359 


835 


482 


343 






151 


2. 


178 


976 


947 


293 


233 


2. 


367 


355 


922 


471 






157 


2. 


195 


899 


653 


409 


239 


2. 


378 


397 


902 


352 






163 


2. 


212 


187 


604 


404 


241 


2. 


382 


017 


042 


576 






167 


2. 


222 


716 


471 


148 


251 


2. 


399 


673 


721 


509 






173 


2. 


238 


046 


103 


129 


257 


2. 


409 


933 


123 


332 






179 


2. 


252 


853 


030 


980 


263 


2. 


419 


955 


748 


490 






181 


2. 


257 


678 


574 


869 


269 


2. 


429 


752 


261 


993 






191 


2. 


281 


033 


367 


248 


271 


2. 


432 


969 


290 


877 






193 


2. 


285 


557 


309 


008 1 


277 


2. 


442 


479 


768 


999 





68 








LOGARITHMS 












N. 


Log. 


N. 


Log. 




281 


2. 


448 


706 


319 


906 


601 


2. 


778 


874 


471 


998 




283 


2. 


451 


786 


435 


523 


607 


2. 


783 


188 


691 


074 




293 


2. 


466 


867 


523 


562 


613 


2. 


787 


460 


556 


130 




307 


2. 


487 


138 


375 


477 


617 


2. 


790 


285 


164 


033 




311 


2. 


492 


760 


389 


026 


619 


2. 


791 


690 


648 


987 




313 


2. 


495 


544 


337 


650 


631 


2. 


800 


029 


359 


232 




317 


2. 


601 


069 


267 


324 


641 


2 


806 


868 


879 


634 




331 


2. 


519 


827 


993 


783 


643 


2! 


808 


210 


973 


921 




337 


2. 


627 


629 


883 


034 


647 


2. 


810 


904 


280 


666 




347 
349 


2. 


540 


329 


475 


079 


663 


2. 


814 


912 


981 


274 




2. 


642 


826 


426 


673 


659 


2. 


818 


885 


490 


409 




353 


2. 


647 


774 


138 


016 


661 


2. 


820 


201 


459 


485 




359 


2. 


655 


094 


447 


578 


673 


2- 


828 


015 


064 


225 




367 


2. 


664 


666 


064 


254 


677 


2. 


830 


588 


667 


946 




373 


2. 


571 


708 


831 


809 


683 


2. 


834 


420 


703 


630 




379 


2. 


678 


639 


209 


957 


691 


2. 


839 


477 


902 


551 




383 


2. 


583 


198 


773 


980 


701 


2. 


845 


718 


017 


237 




389 


2. 


589 


949 


601 


323 


709 


2. 


850 


646 


235 


112 




397 


2. 


598 


790 


506 


763 


719 


2. 


856 


728 


890 


383 




401 


2. 


603 


144 


372 


687 


727 


2- 


861 


634 


410 


855 




409 


2. 


611 


723 


296 


019 


733 


2. 


865 


103 


970 


639 




419 


2. 


623 


214 


m2 


971 


739 


2. 


868 


643 


643 


162 




421 


2. 


624 


282 


085 


835 


743 


9. 


870 


988 


813 


759 




431 


2. 


634 


477 


268 


999 


761 


2. 


876 


639 


937 


004 




433 


2^ 


636 


488 


016 


871 


767 


2. 


879 


095 


879 


497 




439 


2. 


642 


464 


520 


242 


761 


2. 


881 


384 


656 


769 




443 


2. 


646 


403 


726 


235 


769 


i'. 


885 


926 


339 


800 




449 


2. 


652 


246 


388 


777 


773 


2. 


888 


179 


493 


917 




467 


2. 


659. 


916 


200 


064 


787 


2. 


896 


974 


732 


358 




461 


2. 


663 


70& 


925 


389 


7a7 


2- 


901 


468 


321 


400 




■ 463 


2.. 


665 


580 


994 


012 


809 


2. 


9f77 


948 


459 


773 




467 


2. 


669 


317 


88S 


008 


; 811 


2. 


909 


OQO 


864 


210 




479 


2. 


680 


335 


513 


415 


821 


2I 


914 


343 


157 


120 




487 


2. 


687 


628 


961 


120 


823 


2. 


915 


39» 


835 


203 




491 


2. 


691 


081 


487 


026 


827 


2^ 


9^17 


505 


509 


487 




499 


2. 


698 


100 


545 


623 


829 


2. 


S.18. 


654 


530 


558 




503 


2. 


701 


567 


985 


083 


839* 


2. 


923 


761 


960 


830 


1 


609 


; 2.. 


-zoe 


717 


782 


345 


853 


2. 


980 


949 


CGI 


1G3 


1 


621 


2. 


716 


837 


623 


304 


857 


2. 


932 


980 


821 


917 




523 


2. 


718 


602 


688 


873 


859 


2. 


933 


903 


163 


838 




541 


2v 


733 


197 


26B 


134 


863 


2. 


936 


010 


794 


546 




547 


2. 


737 


987 


326 


358 


877 


2. 


942 


999 


593 


360 




557 


2. 


745 


855 


195 


li92 


881 


2. 


944 


975 


908 


412 




563 


2. 


750 


508 


395 


940 


: 883 


2. 


946 


960 


703 


512 




569 


2, 


755 


112 


178 


598 


887 


2. 


947 


923 


619 


839 




571 


2. 


756 


636 


108 


333 


907 


2. 


957 


607 


287 


059 




677 


2. 


761 


175 


813 


171 


911 


2. 


959 


518 


376 


972 




587 


2. 


768 


638 


004 


465 


919 


2. 


963 


315 


513 


6C0 




593 


2. 


773 


054 


693 


364 


929 


^. 


968 


015 


713 


997 




599 


2. 


777 


427 


303 


257 


937 


2. 


971 


739 


590 


780 


1 



OF NUMBERS. 



69 



941 


ii. 


947 


2. 


953 


2. 


967 


2. 


971 


2. 


977 


2. 


983 


2. 


991 


2. 


997 


2. 


1009 


3. 


1013 


3. 


1019 


3. 


1021 


3. 


1031 


3. 


1033 


3. 



Log. 



973 689 620 234 

976 349 979 055 

979 092 900 639 

985 426 474 084 

987 219 229 907 

989 894 559 717 

992 553 512 733 

996 073 604 003 

998 695 158 313 

003 891 170 203 

005 609 445 427 

008 174 244 007 

009 025 742 086 

013 258 660 430 

014 100 321 518 



N. 



Ix>g. 



1039 


3. 


016 


615 


647 


658 


1049 


3. 


020 


775 


488 


195 


1051 


3. 


021 


602 


716 


026 


1061 


3. 


025 


715 


383 


898 


1063 


3. 


026 


533 


264 


623 


1069 


3. 


028 


977 


705 


205 


1087 


3. 


036 


229 


513 


712 


1091 


3. 


037 


824 


749 


671 


1093 


3. 


038 


620 


157 


372 


1097 


3. 


040 


206 


627 


671 


1103 


3. 


042 


575 


612 


437 


1109 


3. 


044 


931 


546 


149 


1117 


3. 


048 


053 


173 


103 


1123 


3. 


050 


379 


756 


239 


1129 


3. 


052 


693 


942 


370 



It is not necessary to extend this table, as the loj^arithm of any 
one of the higher numbers can be readily computed by the fol- 
lowing formula, which may be found in any of the standard works 
on algebra, namely : 

Log. (2-|-i)=log. z-f 0.8685889638 I -i ) 

The result will be true to ten decimal places for all numbers 

over 1000, and true to twelve decimals for all numbers over 2000. 

The logarithms of composite numbers can be determined by 

the combination of logarithms already in the table, and the prime 

numbers from the formula. 

Thus, the number 3083 is a prime number, find its logarithm, 
true to ten places of decimals. 

We first find the logarithm of 3082. By factoring this num- 
ber, we find that it may be composed by the multiplication of 46 
into 67. 

Log. 46 1. 

Log. 67 1. 

Log. 3082 3. 

Log. 3083=3.4888321343 



662 757 8316 
826 074 3027 
488 832 1343 



Now 



0-8685889fi38 
61 6d 

We give a few additional prime numbers : 



1151 
1153 
1163 
1171 
1181 
1187 
1193 
1201 
1213 
1217 



1223 
1229 
1231 
1237 
1249 
1259 
1277 
1279 
1283 
1289 



1291 
1297 
1301 
1303 
1307 
1319 
1321 
1327 
1361 
1367 



1373 
1381 
1399 
1409 
1423 
1427 
1429 
1433 
1439 
1447 



1461 
1453 
1459 
1471 
1481 
1483 
1487 
1489 
1493 
1499 



1511 
1523 
1531 
1543 
1549 
1553 
1559 
1667 
1671 
1579 



70 


AUXILIARY LOGARITHMS. 




AUXILIARY LOGARITHMS*. 




N. 


Log. 


N. 


Log. 




1. 009 


0. 003 891 170 2031 




1. 0009 


0. 000 390 576 304] 






1. 008 


0. 003 461 627 188 




1. 0008 


0. 000 347 233 698 






1. 007 


0. 003 030 465 635 




1. 0007 


0. 000 303 836 798 






1. 006 


0. 002 597 985 739 




1. 0006 


0. 000 260 435 661 






1. 005 


0. 002 166 071 750 


[►A 


1. 0005 


0. 000 217 099 966 


Ib 




1. 004 


0. 001 733 722 804 




1. 0004 


0. 000 173 690 053 






1. 003 


0. 001 300 943 017 




1. 0003 


0. 000 130 268 803 






1. 002 


0. 000 867 721 529 




1. 0002 


0. 000 086 850 213 






1. 001 


0. 000 434 077 479 J 




1. 0001 


0. 000 043 427 277^ 


— 




N. 


Log. 






1. 00009 


0. 


000 039 084 741 1 








1. 00008 


0. 


000 034 742 166 








1. 00007 


0. 


000 030 399 546 








1 . 00006 


0. 


000 026 066 884 








1. 00005 


0. 


000 021 714 178 








1. 00004 


0. 


000 017 371 430 








1. 00003 


0. 


000 013 028 638 


C 






1. 00002 


0. 


000 008 685 802 






1. 00001 


0. 


000 004 342 923 (a) 








1. 000001 


0. 


000 000 434 294 (b) 








1. 0000001 


0. 


000 000 043 429 (c) 








1. 00000001 


0. 


000 000 004 343 (d) 








1. 000000001 


0. 


000 000 000 434 (e) 








1. 0000000001 


0. 


000 000 000 043 (f)J 






N» 


imber. 


Log. 




0. 43. 


12944819 


—1. 637 784 298 




This decimal number is the modulus of our system of logarithms. | 




Its loga 


rithm is very useful in correcting other logarithms, as may 




be seen 


in the Chapter on Logarithms. 





TAULfci V. TAULE VII. 


"! 


Dip of the Sea Horizon. Mean Refraction of Celestial Objects. 


'.^S 


b2 


\^'» 


g2 


Alt 


Rcfr. || Alt.|Uefr 


, Alt 


llffr.;| Alt. 


Refr. 


Alt. 


Refr 


»d5- 


o-o 


?.| 


§0 





/ rf\ 


f f 1 


"o 


/ II 


1 '^ 


/ 1 II 





—FT 


0^ 


5'S, 


55^ 


5- a 


c 


33 OlllO C 


5 16 


>( 


2 35 


32 C 


)1 30 


67 


24 


P^ 


H 


^2. 


P D* 


IC 


31 32 


IC 


6 10 


' 1( 


2 24 


40 1 29 


68 


23 




~t If 




„ 


2C 


29 60 


20 


5 05 


2( 


2 22 


33 C 


1 28 


69 


22 


1 


59 


38 


6 4 
6 18 
6 32 
6 45 

6 58 

7 10 
7 12 


3C 


28 23 


30 


5 00 


3( 


2 21 


2C 


1 26 


70 


21 


2 


I 24 
1 42 


41 

44 


4C 


27 OG 


40 


4 66 


4( 


2 29 


4C 


1 26 


71 


19 


3 


5C 


25 42 


60 


4 61 


5( 


2 28 


34 C 


1 24 


72 


18 


4 


1 58 

2 12 
2 25 


47 
50 
53 


1 C 


24 29 


11 


4 47 


21 ( 


2 27 


2C 


1 23 


73 


17 


6 


10 


23 20 


10 


4 43 


IC 


2 26 


4C 


1 22 


74 


16 


6 


2C 


22 15 


20 


4 39 


2C 


2 25 


35 C 


1 21 


76 


15 


7 


2 36 


66 


3C 


21 15 


30 


4 34 


3C 


2 24 


20 


1 20 


76 


14 


8 


2 47 


69 


7 24 


4C 


20 18 


40 


4 31 


4C 


2 23 


40 


1 19 


77 


13 


9 


2 57 


62 


' 745 


50 


19 25 


60 


4 27 


5C 


2 21 


36 


1 18 


78 


12 


10 


3 07 


66 


7 66 


2 


18 35 


12 


4 23 


2^ C 


2 20 


30 


1 17 


79 


11 


11 


3 16 


68 


8 07 


10 


17 48 


10 


4 20 


IC 


2 19 


37 


1 16 


80 


10 


12 


3 25 


71 


8 18 


20 


17 04 


20 


4 16 


20 


2 18 


30 


1 14 


81 


9 


13 


3 33 


74 


8 28 


30 


16 24 


30 


4 13 


30 


2 17 


38 


1 13 


82 


8 


14 


3 41 


77 


8 38 


40 


16 45 


40 


409 


40 


2 16 


30 


1 11 


83 


7 


15 


3 49 


80 


8 48 


50 


15 09 


60 


4 06 


50 


2 15 


39 


1 10 


34 


6 


16 


3 56 


83 


8 58 


3 


14 34 


13 


4 03 


23 


2 14 


30 


1 09 


85 


5 


17 


4 04 


86 


9 08 


10 


14 04 


10 


4 00 


10 


2 13 


400 


1 08 


86 


4 


18 


4 11 


89 


9 17 


20 


13 34 


20 


3 57 


20 


2 12 


30 


1 07 


87 


3 


19 


4 17 


92 


9 26 


30 


13 06 


30 


3 54 


30 


2 11 


41 


1 05 


88 


2. 


20 


4 24 


95 


9 36 


40 


12 40 


49 


3 61 


40 2 10 


30 


1 04 


89 


1 


21 


4 31 


98 


9 45 


60 


12 15 


60 


3 48 


50 2 09 


42 


1 03 


90 





22 


4 37 


JXl 


9 54 


4 


11 61 


14 


3 45 


24 02 08 


30 


1 02 






23 


4 43 


104 


10 02 


10 


11 29 


10 


3 43 


102 07 


43 


1 01 






24 


4 49 


107 


10 11 


20 


11 08 


20 


3 40 


202 06 


30 


1 00 






26 


4 55 


110 


10 19 


30 


10 48 


30 3 38 


30 


2 06 


44 


69 






26 


5 01 


III 


10 28 


40 


10 29 


40 


3 35 


40 


2 04 


80 


58 






27 


5 07 


116 


10 36 


50 


10 11 


60 


3 33 


60 


2 03 


45 


57 






28 
29 


5 13 
5 18' 


119 

122 


10 44 
10 62 


5 


9 64 


15 


3 30 


25 


2 02 


30 


66 






30 


5 241 


125 


11 00 


10 


9 38 


10 


3 28 


10 


2 01 


46 


55 






31 


5 29 1 


128 


11 08 


20 


9 23 


20 


3 26 


20 


2 00 


30 


54 






82 


5 34 


131 


11 16 


30 


9 08: 


30 


3 24 


30 


1 59 


47 


53 






33 


5 39 


134 


11 24 


40 


8 54 


40 


3 21 


40 


1 58 


30 


62 






34 


5 44 


137 


11 31 


50 


8 41 


50 


3 19 


50 


1 67 


48 


61 






35 


5 49I 


140 


11 RQ 


6 


8 28 


16 


3 17 


26 


1 66 


30 


60 








10 


8 16 
8 03 


10 
20 


3 16 
3 12 


10 
20 


1 65 
1 56 


49 
30 


49 
49 








20 






TABLE VI. 


30 


7 15 


30 


3 10 


30 


154 


60 48 






Dip of the Sea Horizon at 


40 


7 40! 


40 


l^ 


40 


1 63 


30 47 






different Distances from it. 


50 
7 


7 30: 
7 2O1 


60 
17 


3 06 
3 04 


50 
27 


1 62 
1 61 


51 
30 


46 
45 








10 


fT 1 li 


10 
20 
30 


3 03 
3 01 


15 
30 


1 50 
1 49 


52 
30 

53 


44 
) 44 






Dist. 


Hight of Eye in i't.l 


IV 1 xii 

20 T noJ 


in 

Miles. 


5 


10 


16 2 


0J25 


30 


30 
40 


6 63| 


2 69 


46 
28 


1 48 


43 








T 


~7~ 


~T ~ 


- — 


~ 


6 45| 


40 


2 67 


1 47 


30 


O 42 






i 


11 


22 


34 4 


5 '56 


68 


50 


6 37i 


60 


2 56 


15 


1 46 54 

1 45 56 


3 41 








6 




17 2 


2 '28 


34 


8 


6 29i 


18 


2 64l 


30 


} 40 






4 


8 


12 1 


5J19 


23 


10 


6 22| 


10 


2 52 


45 


1 44 56 


}38 






I 


4 


6 


9 li 


2!l5 


17 


20 


6 15 


20 


2 51i 


29 


1 42 57 


) 37 






U 


3 


5 


7 < 


) 12 


14 


30 


6 08 


30 


2 49 


20 


1 4ll58 


3 35 






H 


3 


4 


6 I 


I 9 


12 


40 


6 01 


40 


2 47 


40 


I 40|!59 


) 34 






2 


2 


3 


5 ( 


) 8 


10 


50 


5 56 


50' 


2 46 


30 


I 38|;60 


) 33 






2i 


2 


3 


6 ( 


5 7 


8 


9 


5 98 


19 0; 


2 44 


20 


I 37JI61 


) 32 






3 


2 


3 


4 I 


) 6 


7 


10 


5 42 


10 i 


243; 


40 


I 361^2 


) 80 






3i 


2 


3 


4 £ 


) 6 


6 


20 


6 46 


20^ 


2 41 


31 


I 35; 63 
I 33' 64 
I 32 165 0( 


) 29 






4 


2 


3 


4 ^ 


I 5 


6 


30 


6 41 


30 5 


J 40 


20 


)28 






5 


2 


3 


4 4 


I 5 


5 


40 


6 26 


40 S 


\ 38 


40 


) 26 






6 


2 


3 


4 4 


I 5 6| 


60 


6 20! 


60 2 37il32 0| 


I 3ll^6 0( 


) 25 




1 


1 



>w. 



1 



3f43 



37 



-.^ 






^'» ^^