The Construction of the Wonderful Canon of Logarithms/The Construction

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The Construction of the Wonderful Canon of Logarithms
by John Napier, translated by William Rae Macdonald

The Construction

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3701791The Construction of the Wonderful Canon of Logarithms — The ConstructionWilliam Rae MacdonaldJohn Napier

THE CONSTRUCTION OF
THE WONDERFUL CANON
OF LOGARITHMS; (HEREIN
CALLED BY THE AUTHOR
THE ARTIFICIAL TABLE)
and their relations to
their natural
numbers.

1.

Logarithmic Table is a small table by the use of which we can obtain a knowledge of all geometrical dimensions and motions in space by a very easy calculation.

IT is deservedly called very small, because it does not exceed in size a table of sines; very easy, because by it all multiplications, divisions, and the more difficult extractions of roots are avoided; for by only a very few most easy additions, subtractions, and divisions by two, it measures quite generally all figures and motions.

1.‍It is picked out from numbers progressing in continuous proportion.

2.Of continuous progressions, an arithmetical is one which proceeds by equal intervals; a geometrical, one which advances by unequal and proportionally increasing or decreasing intervals.

Arithmetical progressions: 1, 2, 3, 4, 5, 6 7; &c.; or 2, 4, 6, 8, 10, 12, 14, 16, &c. Geometrical progressions: 1, 2, 4, 8, 16, 32, 64, &c.; or 243, 81, 27, 9, 3, 1.

3.In these progressions we require accuracy and ease in working. Accuracy ts obtained by taking large numbers for a basis; but large numbers are most easily made from small by adding cyphers.

Thus instead of 100000, which the less experienced make the greatest sine, the more learned put 10000000, whereby the difference of all sines is better expressed. Wherefore also we use the same for radius and for the greatest of our geometrical proportionals.

4.In computing tables, these large numbers may again be made still larger by placing a period after the number and adding cyphers.

Thus in commencing to compute, instead of 10000000 we put 10000000.0000000, lest the most minute error should become very large by frequent multiplication.

5.In numbers distinguished thus by a period in their midst, whatever is written after the period is a fraction, the denominator of which ts unity with as many cyphers after it as there are figures after the period.

Thus 10000000.04 is the same as

\scriptstyle 10000{\frac {4}{100}}

also 25.803 is the same as

\scriptstyle 25{\frac {803}{1000}}

; also 9999998 0005021 is the same as

\scriptstyle 9999998{\frac {5021}{1000000}}

, and so of others.

6.When the tables are computed, the fractions following the period may then be rejected without any sensible error. For in our large numbers, an error which does not exceed unity is insensible and as if it were none.

Thus in the completed table, instead of 9987643.8213051, which is $\scriptstyle 9987643{\frac {8213051}{10000000}}$ , we may put 9987643 without sensible error.

7.Besides this, there ts another rule for accuracy; that is to say, when an unknown or incommensurable quantity is included between numerical limits not differing by many units.

Thus if the diameter of a circle contain 497 parts, since it is not possible to ascertain precisely of how many parts the circumference consists, the more experienced, in accordance with the views of Archimedes, have enclosed it within limits, namely 1562 and 1561. Again, if the side of a square contain 1000 parts, the diagonal will be the square root of the number 2000000. Since this is an incommensurable number, we seek for its limits by extraction of the square root, namely 1415 the greater limit and 1414 the less limit, or more accurately $\scriptstyle 1414{\frac {604}{2828}}$ the greater, and $\scriptstyle 1414{\frac {604}{2829}}$ the less; for as we reduce the difference of the limits we increase the accuracy.

7.‍In place of the unknown quantities themselves, their limits are to be added, subtracted, multiplied, or divided, according as there may be need.

8.The two limits of one quantity are added to the two limits of another, when the less of the one is added to the less of the other, and the greater of the one to the greater of the other.

Thus let the line a b c be divided into two parts, ab and bc. Let ab lie between the limits 123.5 the greater and 123.2 the less. Also let bc lie between the limits 43.2 the greater and 43.1 the less. Then the greater being added to the greater and the less to the less, the whole line ac will lie between the limits 166.7 and 166.3.

9.The two limits of one quantity are multiplied into the two limits of another, when the less of the one is multiplied into the less of the other, and the greater of the one into the greater of the other.

Thus let one of the quantities ab lie between the limits 10.502 the greater and 10.500 the less. And let the other ac lie between the limits 3.216 the greater and 3.215 theless, Then 10.502 being multiplied into 3.216 and 10.500 into 3.215, the limits will become 33.774432 and 33.757500, between which the area of a b c d will lie.

10.Subtraction of limits ts performed by taking the greater limit of the less quantity from the less of the greater, and the less limit of the less quantity from the greater of the greater.

Thus, in the first figure, if from the limits of ac, which are 166.7 and 166.3, you subtract the limits of bc, which are 43.2 and 43.1, the limits of ab become 123.6 and 123.1, and not 123.5 and 123.2, For although the addition of the latter to 43.2 and 43.1 produced 166.7 and 166.3 (as in 8), yet the converse does not follow; for there may be some quantity between 166.7 and 166.3 from which if you subtract some other which is between 43.2 and 43.1, the remainder may not lie between 123.5 and 123.2, but it is impossible for it not to lie between the limits 123.6 and 123.1.

11.Division of limits is performed by dividing the greater limit of the dividend by the less of the divisor, and the less of the dividend by the greater of the divisor.

Thus, in the preceding figure, the rectangle a b c d lying between the limits 33.774432 and 33.757500 may be divided by the limits of a c, which are 3.216 and 3.215, when there will come out $\scriptstyle 10.505{\frac {857}{3215}}$ and $\scriptstyle 10.496{\frac {2364}{3216}}$ for the limits of a b, and not 10.502 and 10.500, for the same reason that we stated in the case of subtraction.

12.The vulgar fractions of the limits may be removed by adding unity to the greater limit.

Thus, instead of the preceding limits of a b, namely, $\scriptstyle 10.505{\frac {857}{3215}}$ and $\scriptstyle 10.496{\frac {2364}{3216}}$ , we may put 10.506 and 10.496.

12.‍Thus far concerning accuracy; what follows concerns ease in working.

13.The construction of every arithmetical progression ts easy; not so, however, of every geometrical progression.

This is evident, as an arithmetical progression is very easily formed by addition or subtraction; but a geometrical progression is continued by very difficult multiplications, divisions, or extractions of roots.

13.‍Those geometrical progressions alone are carried on easily which arise by subtraction of an easy part of the number from the whole number.

14.We call easy parts of a number, any parts the denominators of which ave made up of unity and a number of cyphers such parts being obtained by rejecting as many of the figures at the end of the principal number as there are cyphers in the denominator.

Thus the tenth, hundredth, thousandth, 10000^th, 100000^th 1000000^th, 10000000^th parts are easily obtained, because the tenth part of any number is got by deleting its last figure, the hundredth its last two, the thousandth its last three figures, and so with the others, by always deleting as many of the figures at the end as there are cyphers in the denominator of the part. Thus the tenth part of 99321 is 9932, its hundredth part is 993, its thousandth 99, &c.

15.The half, twentieth, two hundredth, and other parts denoted by the number two and cyphers, are also tolerably easily obtained; by rejecting as many of the figures at the end of the principal number as there are cyphers in the denominator, and dividing the remainder by two.

Thus the 2000^th part of the number 9973218045 is 4986609, the 20000^th part is 498660.

16.Hence it follows that if from radius with seven cyphers added you subtract its 100000000^th part, and from the number thence arising tts 10000000^th part, and so on, a hundred numbers may very easily be continued geometrically in the proportion subsisting between radius and the sine less than it by unity, namely between 10000000 and 9999999; and this series of proportionals we name the First table.

First table.
10000000.0000000
1.0000000
9999999.0000000
.9999999
9999998.0000001
.9999998
9999997.0000003
.9999997
9999996.0000006
to be continued up to
9999900.0004950

Thus from radius, with seven cyphers added for greater accuracy, namely, 10000000.0000000, subtract 1.0000000, you get 9999999.0000000; from this subtract .9999999 you get 9999998.0000001; and proceed in this way, as shown at the side until you create a hundred proportionals, the last of which, if you have computed rightly, will be 9999900.0004950.

17.The Second table proceeds from radius with six cyphers added, through fifty other numbers decreasing proportionally in the proportion which zs easiest, and as near as possible to that subsisting between the first and last numbers of the First table.

Second table.
10000000.0000000
100.000000
9999900.0000000
99.999000
9999800.0010000
99.998000
9999700.0001000
99.997000
9999600.006000

Thus the first and last numbers of the First table are 10000000.00000000 and 9999900.0004950, in which proportion it is difficult to form fifty proportional numbers. A near and at the same time an easy proportion is 100000 to 99999, may be continued with sufficient exactness by adding six cyphers to radius and continually subtracting from each number its own 109900^th part in the manner shown at the side; and this table

&c up to

9995001.222927

contains, besides radius which is the first, fifty other proportional numbers, the last of which, if you have not erred, you will find to be 9995001.222927.

[This should be 9995001.224804—see note.]

18.The Third table consists of sixty-nine columns, and in each column are placed twenty-one numbers, proceeding in the proportion which ts easiest, and as near as possible to that subsisting between the first and last numbers of the Second table.

18.‍Whence its first column is very easily obtained from radius with five cyphers added, by subtracting tts 2000^th part, and so from the other numbers as they arise.

First column of Third table.
10000000.00000
5000.00000
9995000.00000
4995.00125
9985007.49875
4992.50374
9980014.99501
&c up to
9900473.57808

In forming this progression, as the proportion between 10000000.000000, the first of the Second table, and 9995001.222927, the last of the same, is troublesome; therefore compute the twenty-one numbers in the easy proportion of 10000 to 9995, which is sufficiently near to it; the last of these, if you have not erred, will be 9900473.57808.

From these numbers, when computed, the last figure of each may be rejected without sensible error, so that others may hereafter be more easily computed from them.

19.The first numbers of all the columns must proceed from radius with four cyphers added, in the proportion easiest and nearest to that subsisting between the first and the last numbers of the first column.

As the first and the last numbers of the first column are 10000000.0000 and 9900473.5780, the easiest proportion very near to this is 100 to 99. Accordingly sixty-eight numbers are to be continued from radius in the ratio of 100 to 99 by subtracting from each one of them its hundredth part.

20.In the same proportion a progression ts to be made from the second number of the first column through the second numbers in all the columns, and from the third through the third, and from the fourth through the fourth, and from the others respectively through the others.

Thus from any number in one column, by subtracting its hundredth part, the number of the same rank in the following column is made, and the numbers should be placed in order as follows:—

Proportionals of the Third Table.

First column.
10000000.0000
9995000.0000
9990002.5000
9985007.4987
9980014.9950
&c continuously to
9900473.57808

Second Column.
9900000.0000
9895050.0000
9890102.4750
9885157.4237
9880214.8451
&c continuously to
9801468.8423

Third Column. ‍
9801000.0000
9796099.5000
9791201.4503
9786305.8495
9781412.6967
&c descending to
9703454.1539

Thence 4th, 5th
&c., up to

&c., up to

finally to

69th Column. ‍
5048858.8900
5046334.4605
5043811.2932
5041289.3879
5038768.7435
&c descending to
4998609.4034

21.Thus, in the Third table, between radius and half radius, you have sixty-eight numbers interpolated, in the proportion of 100 to 99, and between each two of these you have one numbers interpolated in the proportion of 10000 to 9995; and again, in the Second table, between the be two of these, namely between 10000000 an 9995000, you have fifty numbers interpolated in the proportion of 100000 to 99999; and finally, in the First table, between the latter, you have a hundred numbers interpolated in the proportion of radius or 10000000 to 9999999; and since the difference of these is never more than unity, there is no need to divide it more minutely by interpolating means, whence these three tables, after they have been completed will suffice for computing a Logarithmic table.

21.‍Hitherto we have explained how we may most easily place in tables sines or natural numbers progressing im geometrical proportion.

22.It remains, in the Third table at least, to place beside the sines or natural numbers decreasing geometrically their logarithms or artificial numbers increasing arithmetically.

23.To increase arithmetiually is, in equal times, to be augmented by a quantity always the same.

Thus from the fixed point b let a line be produced indefinitely in the direction of d. Along this let the point a travel from b towards d, mov- ing according to this law, that in equal moments of time it is borne over the equal spaces b 1, 1 2, 2 3,3 4,4 5 c. Then we call this increase by b 1, b 2, b 3, b 4, b 5, &c, arithmetical. Again, let b 1 be represented in numbers by 10, b 2 by 20, b 3 by 30, b 4 by 40, b 5 by 50; then 10, 20, 30, 40, 50, &c., increase arithmetically, because we see they are always increased by an equal number in equal times.

24.To decrease geometrically ts this, that in equal times, First the whole quantity then each of its successive remainders ts diminished, always by a like proportional part.

Thus let the line T S be radius. Along this let the point G travel in the direction of S, so that in equal times it is borne from T to 1, which for example may be the tenth part of T S; and from 1 to 2, the tenth part of 1 S; and from 2 to 3, the tenth part of 2 S; and from 3 to 4, the tenth part of 3 S, and so on. Then the sines T S, 1 S, 2 S, 3 S, 4 S, &c., are said to decrease geometrically, because in equal times they are diminished by unequal spaces similarly proportioned. Let the sine T S be represented in numbers by 10000000, 1 S by 9000000, 2 S by 8100000, 3 S by 7290000, 4 S by 6561000; then these numbers are said to decrease geometrically, being diminished in equal times by a like proportion.

25.Whence a geometrically moving point approaching a fixed one has its velocities proportionate to its distances from the fixed one.

Thus, referring to the preceding figure, I say that when the geometrically moving point G is at T, its velocity is as the distance T S, and when G is at 1 its velocity is as 1 S, and when at 2 its velocity is as 2 S, and so of the others. Hence, whatever be the proportion of the distances T S, 1 S, 2 S, 3 S, 4 S, &c., to each other, that of the velocities of G at the points T, 1, 2, 3, 4, &c., to one another, will be the same.

For we observe that a moving point is declared more or less swift, according as it is seen to be borne over a greater or less space in equal times. Hence the ratio of the spaces traversed is necessarily the same as that of the velocities. But the ratio of the spaces traversed in equal times, T 1, T 2, T 3, T 4, T 5, &c., is that of the distances T S, 1 S, 2 S, 3 S, 4 S, &c^[1] Hence it follows that the ratio to one ano, of the distances of G from S, namely T S, 1 S, 2 S, 3 S, 4 S, &c., is the same as that of the velocities of G at the points T, 1, 2, 3, 4, &c., respectively.

26.The logarithm of a given sine ts that number which has increased avithmetically with the same velocity throughout as that with which radius began to decrease geometrically, and in the same time as radius has decreased to the given sine.

Let the line T S be radius, and d S a given sine in the same line; let g move geometrically from T to d in certain determinate moments of time. Again, let bi be another line, infinite towards i, along which, from b, let a move arithmetically with the same velocity as g had at first when at T; and from the fixed point b in the direction of i let a advance in just the same moments of time up to the point c. The number measuring the line b c is called the logarithm of the given sine d S.

27.Whence nothing is the logarithm of radius.

For, referring to the figure, when g is at T making its distance from S radius, the arithmetical point d beginning at b has never proceeded thence. Whence by the definition of distance nothing will be the logarithm of radius.

28.Whence also it follows that the logarithm of any given sine ts greater than the difference between radius and the given sine, and less than the difference between radius and the quantity which exceeds it in the ratio of radius to the given sine. And these differences are therefore called the limits of the logarithm.

Thus, the preceding figure being repeated, and S T being produced beyond T to 0, so that 0 S is to T S as T S to d S. I say that b c, the logarithm of the sine d S, is greater than T d and less than 0 T. For in the same time that g is borne from 0 to T, g is borne from T to d, because (by 24) 0 T is such a part of 0 S as T d is of T S, and in the same time (by the definition of a logarithm) is a borne from b to c; so that o T, T d, and b c are distances traversed in equal times. But since g when moving between T and 0 is swifter than at T, and between T and d slower, but at T is equally swift with a (by 26); it follows that 0 T the distance traversed by g moving swiftly is greater, and T d the distance traversed by g moving slowly is less, than b c the distance traversed by the point a with its medium motion, in just the same moments of time; the latter is, consequently, a certain mean between the two former.

Therefore 0 T is called the greater limit, and T d the less limit of the logarithm which b c represents.

29.Therefore to find the limits of the logarithm of a given sine.

By the preceding it is proved that the given sine being subtracted from radius the less limit remains, and that radius being multiplied into the less limit and the product divided by the given sine, the greater limit is produced, as in the following example.

30.Whence the first proportional of the First table, which is 9999999, has its logarithm between the limits 1.0000001 and 1.0000000.

For (by 29) subtract 9999999 from radius with cyphers added, there will remain unity with its own cyphers for the less limit; this unity with cyphers being multiplied into radius, divide by 9999999 and there will result 1.0000001 for the greater limit, or if you require greater accuracy 1.00000010000001.

31.The limits themselves differing insensibly, they or anything between them may be taken as the true logarithm.

Thus in the above example, the logarithm of the sine 9999999 was found to be either 1.0000000 or 1.00000010, or best of all 1.00000005. For since the limits themselves, 1.0000000 and 1.0000001, differ from each other by an insensible fraction like $\scriptstyle {\frac {1}{10000000}}$ therefore they and whatever is between them will differ still less from the true logarithm lying between these limits, and by a much more insensible error.

32.There being any number of sines decreasing from radius in geometrical proportion, of one of which the logarithm or its limits ts given, to find those of the others.

This necessarily follows from the definitions of arithmetical increase, of geometrical decrease, and of a logarithm. For by these definitions, as the sines decrease continually in geometrical proportion, so at the same time their logarithms increase by equal additions in continuous arithmetical progression. Wherefore to any sine in the decreasing geometrical progression there corresponds a logarithm in the increasing arithmetical progression, namely the first to the first, and the second to the second, and so on

So that, if the first logarithm corresponding to the first sine after radius be given, the second ae will be double of it, the third triple, and so of the others; until the logarithms of all sines be known, as the following example will show.

33.Hence the logarithms of all the proportional sines of the first table may be included between near limits, and consequently given with sufficient exactness.

Thus since (by 27) the logarithm of radius is 0, and (by 30) the logarithm of 9999999, the first sine after radius in the First table, lies between the limits 1.0000001 and 10000000; necessarily the logarithm of 9999998.0000001, the second sine after radius, will be contained between the double of these limits, namely between 2.0000002 and 2.0000000; and the logarithm of 9999997.0000003, the third will be between the triple of the same, namely between 3.0000003 and 3.0000000. And so with the others, always by equally increasing the limits by the limits of the first, until you have completed the limits of the logarithms of all the proportionals of the First table. You may in this way, if you please, continue the logarithms themselves in an exactly similar progression with little and insensible error; in which case the logarithm of radius will be 0, the logarithm of the first sine after radius (by 31) will be 1.00000005, of the second 2.00000010, of the third 3.00000015, and so of the rest.

34.The difference of the logarithms of radius and a given sine is the logarithm of the given sine itself.

This is evident, for (by 27) the logarithm of radius is nothing, and when nothing is subtracted from the logarithm of a given sine, the logarithm of the given sine necessarily remains entire.

35.The difference of the logarithms of two sines must be added to the logarithm of the greater that you may have the logarithm of the less, and subtracted from the logarithm of the less that you may have the logarithm of the greater.

Necessarily this is so, since the logarithms increase as the sines decrease, and the less logarithm is the logarithm of the greater sine, and the greater logarithm of the less sine. And therefore it is right to add the difference to the less logarithm, that you may have the greater logarithm though corresponding to the less sine, and on the other hand to subtract the difference from the greater logarithm that you may have the less logarithm though corresponding to the greater sine.

36.The logarithms of similarly proportioned sines are equi-different.

This necessarily follows from the definitions of a logarithm and of the two motions, For since by these definitions arithmetical increase always the same corresponds to geometrical decrease similarly proportioned, of necessity we conclude that equi-different logarithms and their limits correspond to similarly proportioned sines. As in the above example from the First table, since there is a like proportion between 9999999.0000000 the first proportional after radius, and 9999997.0000003 the third, to that which is between 9999996.0000006 the fourth and 9999994.0000015 the sixth; therefore 1.00000005 the logarithm of the first differs from 3.00000015 the logarithm of the third, by the same difference that 4.00000020 the logarithm of the fourth, differs from 6.00000030 the logarithm of the sixth proportional. Also there is the same ratio of equality between the differences of the respective limits of the logarithms, namely as the differences of the less among themselves, so also of the greater among themselves, of which logarithms the sines are similarly proportioned.

37.Of three sines continued in geometrical proportion, as the square of the mean equals the product of the extremes, so of their logarithms the double of the mean equals the sum of the extremes. Whence any two of these logarithms being given, the third becomes known.

Of the three sines, since the ratio between the first and the second is that between the second and the third, therefore (by 36), of their logarithms, the difference between the first and the second is that between the second and the third. For example, let the first logarithm be represented by the line b c, the second by the line b d, the third by the line b e, all placed in the one line b c d e, thru:—

and let the differences c d and d e be equal. Let b d, the mean of them, be doubled by producing the line from b beyond e to f, so that b f is double b d. Then b f is equal to both the lines b c of the first logarithm and b e of the third, for from the equals b d and d f take away the equals c d and d e, namely c d from b d and d e from d f, and there will remain b c and e f necessarily equal. Thus since the whole b f is equal to both b e and e f, therefore also it will be equal to both b e and b c, which was to be proved. Whence follows the rule, if of three logarithms you double the given mean, and from this subtract a given extreme, the remaining extreme sought for becomes known; and if you add the given extremes and divide the sum by two, the mean becomes known.

38.Of four geometrical proportionals, as the product of the means ts equal to the product of the extremes; so of their logarithms, the sum of the means ts equal to the sum of the extremes. Whence any three of these logarithms being given, the fourth becomes known.

Of the four proportionals, since the ratio between the first and second is that between the third and fourth; therefore of their logarithms (by 36), the difference between the first and second is that between the third and fourth. Hence let such quantities be taken in the line b f as that b a

may represent the first logarithm, b c the second, be the third, and b g the fourth, making the differences a c and e g equal, so that d placed in the middle of c e is of necessity also placed in the middle of a g. Then the sum of b c the second and b e the third is equal to the sum of b a the first and b g the fourth. For (by 37) the double of b d, which is b f, is equal to b c and b e together, because their differences from b d, namely c d and d e, are equal; for the same reason the same b f is also equal to b a and b g together, because their differences from b d, namely a d and d g, are also equal. Since, therefore, both the sum of b a and b g and the sum of b c and b e are equal to the double of b d, which is b f, therefore also they are equal to each other, which was to be proved. Whence follows the rule, of these four logarithms if you subtract a known mean from the sum of the known extremes, there is left the mean sought for; and if you subtract a known extreme from the sum of the known means, there is left the extreme sought for.

39.The difference of the logarithms of two sines lies between two limits; the greater limit being to radius as the difference of the sines to the less sine, and the less limit being to radius as the difference of the sines to the greater sine.

Let T S be radius, d S the greater of two given sines, and e S the less. Beyond S T let the distance T V be marked off by the point V, so that S T is to T V as e S, the less sine, is to d e, the difference of the sines, Again, on the other side of T, towards S, let the distance T c be marked off by the point c, so that T S is to T c as d S, the greater sine, is to d e, the difference of the sines, Then the difference of the logarithms of the sines d S and e S lies between the limits V T the greater and T c the less. For by hypothesis, e S is to d e as T S to T V, and d S is to d e as T S to T c; therefore, from the nature of proportionals, two conclusions follow:—

Firstly, that V S is to TS as T S to c S.

Secondly, that the ratio of T S to c S is the same as that of d S to e S. And therefore (by 36) the difference of the logarithms of the sines d S and e S is equal to the difference of the logarithms of the radius T S and the sine c S. But (by 34) this difference is the logarithm of the sine c S itself; and (by 28) this logarithm is included between the limits V T the greater and T c the less, because by the first conclusion above stated, V S greater than radius is to T S radius as T S is to c S. Whence, necessarily, the difference of the logarithms of the sines d S and e S lies between the limits V T the greater and T c the less, which was to be proved.

40.To find the limits of the difference of the logarithms of two given sines.

Since (by 39) the less sine is to the difference of the sines as radius to the greater limit of the difference of the logarithms; and the greater sine is to the difference of the sines as radius to the less limit of the difference of the logarithms; it follows, from the nature of proportionals, that radius being multiplied by the difference of the given sines and the product being divided by the less sine, the greater limit will be produced; and the product being divided by the greater sine, the less limit will be produced.

Example

THUS, let the greater of the given sines be 9999975.5000000, and the less 9999975.0000300, the difference of these .4999700 being multiplied into radius (cyphers to the eighth place after the point being first added to both for the purpose of demonstration, although otherwise seven are sufficient), if you divide the product by the greater sine, namely 9999975.5000000, there will come out for the less limit .49997122, with eight figures after che point; again, if you divide the product by the less sine, namely 9999975.0000300, there will come out for the greater limit .49997124; and, as already proved, the difference of the logarithms of the given sines lies between these. But since the extension of these fractions to the eighth figure beyond the point is greater accuracy than is required, especially as only seven figures are placed after the point in the sines; therefore, that eighth or last figure of both being deleted, then the two limits and also the difference itself of the logarithms will be denoted by the fraction .4999712 without even the smallest particle of sensible error.

41.To find the logarithms of sines or natural numbers not proportionals in the First table, but near or between them; or at least, to find limits to them separated by an insensible difference.

Write down the sine in the First table nearest to the given sine, whether less or greater. Seek out the limits of the table sine (by 33), and when found note them down. Then seek out the limits of the difference of the logarithms of the given sine and the table sine (by 40), either both limits or one or other of them, since they are almost equal, as is evident from the above example. Now these, or either of them, being found, add to them the limits above noted down, or else subtract (by 8, 10, and 35), according as the given sine is less or greater than the table sine. The numbers thence produced will be near limits between which is included the logarithm of the given sine.

Example.

Let the given sine be 9999975.5000000, to which the nearest sine in the table is 9999975. 0000300, less than the given sine. By 33 the limits of the logarithm of the latter are 25,0000025 and 25.0000000. Again (by 40), the difference of the logarithms of the given sine and the table sine is .4999712. By 35, subtract this from the above limits, which are the limits of the less sine, and there will come out 24.5000313 and 24.5000288, the required limits of the logarithm of the given sine 9999975.5000000. Accordingly the actual logarithm of the sine may be placed without sensible error in either of the limits, or best of all (by 31) in 24.5000300.

Another Example.

LET the given sine be 9999900.0000000, the table sine nearest it 9999900.0004950. By 33 the limits of the logarithm of the latter are .0000I100 and 100.0000000, Then (by 40) the difference of the logarithms of the sines will be .0004950. Add this (by 35) to the above limits and they become 100.0005050 for the greater limit, and 100.0004950 for the less limit, between which the required logarithm of the given sine is included.

42.Hence it follows that the logarithms of all the proportionals in the Second table may be found with sufficient exactness, or may be included between known limits differing by an insensible fraction.

Thus since the logarithm of the sine 9999900, the first proportional of the Second table, was shown in the precedes, example to lie between the limits 100.0005050 and 100.0004950; necessarily (by 32) the logarithm of the second proportional will lie between the limits 200.0010100 and 200,0009900; and the logarithm of the third proportional between the limits 300.0015150 and 300.0014850, &c. And finally, the logarithm of the last sine of the Second table, namely 9995001.222927, is included between the limits 5000.0252500 and 5000.0247500. Now, having all these limits, you will be able (by 31) to find the actual logarithms.

43.To find the logarithms of sines or natural numbers not proportionals in the Second table, but near or between them; or to include them between known limits differing by an insensible fraction.

Write down the sine in the Second table nearest the given sine, whether greater or less. By 42 find the limits of the logarithm of the table sine. Then by the rule of proportion seek for a fourth proportional, which shall be to radius as the less of the given and table sines is to the greater. This may be done in one way by multiplying the less sine into radius and dividing the product by the greater. Or, in an easier way, by multiplying the difference of the sines into radius, dividing this product by the greater sine, and sub- tracting the quotient from radius.

Now since (by 36) the logarithm of the fourth proportional differs from the logarithm of radius by as much as the logarithms of the given and table sines differ from each other; also, since (by 34) the former difference is the same as the loga- rithm of the fourth proportional itself; therefore (by 41) seek for the limits of the logarithm of the fourth proportional by aid of the First table; when found add them to the limits of the logarithm of the table sine, or else subtract them (by 8, 10, and 35), according as the table sine is greater or less than the given sine; and there will be brought out the limits of the logarithm of the given sine.

Example.

THUS, let the given sine be 9995000.000000, To this the nearest sine in the Second table is 9995001.222927, and (by 42) the limits of its logarithm are 5000.0252500 and 5000.0247500. Now seek for the fourth proportional by either of the methods above described; it will be 9999998. 7764614, and the limits of its logarithm found (by 41) from the First table will be 1.2235387 and 1.2235386. Add these limits to the former (by 8 and 35), and there will come out 5001.2487888 and 5001.2482886 as the limits of the logarithm of the given sine. Whence the number 5001.2485387, midway between them, is (by 31) taken most suitably, and with no sensible error, for the actual logarithm of the given sine 9995000.

44.Hence it follows that the logarithms of all the proportionals in the first column of the Third table may be found with sufficient exactness, or may be included between known limits differing by an insensible fraction.

For, since (by 43) the logarithm of 9995000, the first proportional after radius in the first column of the Third table, is 5001.2485387 with no sensible error; therefore (by 32) the logarithm of the second proportional, namely 9990002.5000, will be 10002.4970774; and so of the others, proceeding up to the last in the column, namely 9900473.57808 the logarithm of which, for a like reason, will be 100024.9707740, and its limits will be 100024.9657720 and 100024.9757760.

45.To find the logarithms of natural numbers or sines not proportionals in the first column of the Third table, but near or between them, or to include them between known limits differing by an insensible fraction.

Write down the sine in the first column of the Third table nearest the given sine, whether greater or less. By 44 seek for the limits of the logarithm of the table sine. Then, by one of the methods described in 43, seek for a fourth proportional, which shall be to radius as the less of the given and table sines is to the greater. Having found the fourth proportional, seek (by 43) for the limits of its logarithm from the Second table. When these are found, add them to the limits of the logarithm of the table sine found above, or else subtract them (by 8, 10, and 35), and the limits of the logarithm of the given sine will be brought out.

Example.

THUS, let the given sine be 9900000. The proportional sine nearest it in the first column of the Third table is 9900473.57808. Of this (by 44) the limits of the logarithm are 100024.9657720 and 100024.9757760. Then the fourth proportional will be 9999521.6611850. Of this the limits the logarithm, deduced from the Second table (by 43), are 478.3502290 and 478.3502812. These limits (by 8 and 35) being added to the above limits of the logarithm of the table sine, there will come out the limits 100503. 3260572 and 100503.3160010, between which necessarily falls the logarithm sought for. Whence the number midway between them, which is 100503.3210291, may be put without sensible error for the true logarithm of the given sine 9900000.

46.Hence it follows that the logarithms of all the proportionals of the Third table may be given with sufficient exactness.

For, as (by 45) 100503.3210291 is the logarithm of the first sine in the second column, namely 9900000; and since the other first sines of the remaining columns progress in the same proportion, necessarily (by 32 and 36) the logarithms of these increase always by the same difference 100503.3210291, which is added to the logarithm last found, that the following may be made. Therefore, the first logarithms of all the columns being obtained in this way, and all the logarithms of the first column being obtained by 44, you may choose whether you prefer to build up, at one time, all the logarithms in the same column, by continuously adding 5001.2485387, the difference of the logarithms, to the last found logarithm in the column, that the next lower logarithm in the same column be made; or whether you prefer to compute, at one time, all the logarithms of the same rank, namely all the second logarithms in each of the columns, then all the third, then the fourth, and so the others, by continuously adding 100503.3210291 to the logarithm in one column, that the logarithm of the same rank in the next column be brought out. For by either method may be had the logarithms of all the proportionals in this table; the last of which is 6934250.8007528, corresponding to the sine 4998609.4034.

47.In the Third table, beside the natural numbers, are to be written their logarithms; so that the Third table, which after this we shall always call the Radical table, may be made complete and perfect.

This writing up of the table is to be done by arranging the columns in the number and order described (in 20 and 21), and by dividing each into two sections, the first of which should contain the geometrical proportionals we call sines and natural numbers, the second their logarithms progressing arithmetically by equal intervals.

The Radical Table.

First column.
Natural numbers	Logarithms.
10000000.0000	.0
9995000.0000	5001.2
9990002.5000	10002.5
9985007.4987	15003.7
9980014.9950	20005.0
&c., up to	up to
9900473.5780	100025.0

Second column.
Natural numbers	Logarithms.
9900000.0000	100503.3
9895050.0000	105504.6
9890102.4750	110505.8
9885157.4237	115507.1
9880214.8451	120508.3
up to	up to
9880214.8451	120508.3

and the others, up to	69th column
	Natural numbers	Logarithms.
	5048858.8900	6834225.8
	5046334.4605	6839227.1
	5043811.2932	6844228.3
	5041289.3879	6849229.6
	5038768.7435	6854230.8
	up to	up to
	4998609.4034	6934250.8

For shortness, however, two things should be borne in mind:—First, that in these logarithms it is enough to leave one figure after the point, the remaining six being now rejected, which, however, if you had neglected at the beginning, the error arising thence by frequent multiplications in the previous tables would have grown intolerable in the third. Secondly, If the second figure after the point exceed the number four, the first figure after the point, which alone is retained, is to be increased by unity: thus for 10002.48 it is more correct to put 10002.5 than 10002.4; and for 1000.35001 we more fitly put 1000.4 than 1000.3. Now, therefore, continue the Radical table in the manner which has been set forth.

48.The Radical table being now completed, we take the numbers wee the logarithmic table from it alone.

For as the first two tables were of service in the formation of the third, this third Radical table serves for the construction of the principal Logarithmic table, with great ease and no sensible error.

49.To find most easily the logarithms of sines greater than 9996700.

This is done simply by the subtraction of the given sine from radius. For (by 29) the logarithm of the sine 9996700 lies between the limits 3300 and 3301; and these limits, since they differ from each other by unity only, cannot differ from their true logarithm by any sensible error, that is to say, by an error greater than unity. Whence 3300, the less limit, which we obtain simply by subtraction, may be taken for the true logarithm. The method is necessarily the same for all sines greater than this.

50.To find the logarithms of all sines embraced within the limits of the Radical table.

Multiply the difference of the given sine and table sine nearest it by radius. Divide the product by the easiest divisor, which may be either the given sine or the table sine nearest it, or a sine between both, however placed. By 39 there will be produced either the greater or less limit of the difference of the logarithms, or else something intermediate, no one of which will differ by a sensible error from the true difference of the logarithms on account of the nearness of the numbers in the table. Wherefore (by 35), add the result, whatever it may be, to the logarithm of the table sine, if the given sine be less than the table sine; if not, subtract the result from the logarithm of the table sine, and there will be produced the required logarithm of the given sine.

Example.

THUS let the given sine be 7489557, of which the logarithm is required. The table sine nearest it is 7490786.6119. From this subtract the former with cyphers added thus, 7489557.0000, and there remains 1229.6119. This being multiplied by radius, divide by the easiest number, which may be either 7489557.0000 or 7490786.6119, or still better by something between them, such as 7490000, and by a most easy division there will be produced 1640.1. Since the given sine is less than the table sine, add this to the logarithm of the table sine, namely to 2889111.7, and there will result 2890751.8, which equals 2890751 $\scriptstyle {\tfrac {4}{5}}$ . But since the principal table admits neither fractions nor anything beyond the point, we put for it 2890752, which is the required logarithm.

Another Example.

LET the given sine be 7071068.0000. The table sine nearest it will be 7070084.4434. The difference of these is 983.5566. This being multiplied by radius, you most fitly divide the product by 7071000, which lies between the given and table sines, and there comes out 1390.9. Since the given sine exceeds the table sine, let this be subtracted from the logarithm of the table sine, namely from 3467125.4, which is given in the table, and there will remain 3465734.5. Wherefore 3465735 is assigned for the required logarithm of the given sine 7071068. Thus the liberty of choosing a divisor produces wonderful facility.

51.All sines in the proportion of two to one have 6931469.22 for the difference of their logarithms.

For since the ratio of every sine to its half is the same as that of radius to 5000000, therefore (by 36) the difference of the logarithms of any sine and of its half is the same as the difference of the logarithms of radius and of its half 5000000. But (by 34) the difference of the logarithms of radius and of the sine 5000000 is the same as the logarithm itself of the sine 5000000, and this logarithm (by 50) will be 6931469.22. Therefore, also, 6931469.22 will be the difference of all logarithms whose sines are in the proportion of two to one. Consequently the double of it, namely 13862938.44, will be the difference of all logarithms whose sines are in the ratio of four to one; and the triple of it, namely 20794407.66, will be the difference of all logarithms whose sines are in the ratio of eight to one.

52.All sines in the proportion of ten to one have 23025842.34 for the difference of their logarithms.

For (by 50) the sine 8000000 will have. for its logarithm 2231434.68; and (by 51) the difference between the logarithms of the sine 8000000 and of its eighth part 1000000, will be 20794407.66; whence by addition will be produced 23025842.34 for the logarithm of the sine 1000000. And since radius is ten times this, all sines in the ratio of ten to one will have the same difference, 23025842.34, between their logarithms, for the reason and cause already stated (in 51) in reference to the proportion of two to one. And consequently the double of this logarithm, namely 46051684.68, will, as regards the difference of the logarithms, correspond

to the proportion of a hundred to one; and the triple of the same, namely 69077527.02, will be the difference of all logarithms whose sines are in the ratio of a thousand to one; and so of the ratio ten thousand to one, and of the others as below.

53.Whence all sines in a ratio Sea of the ratios two to one and ten to one, have the difference of their logarithms formed from the differences 6931469.22 and 23025842.34 in the way shown in the following

Short Table.

Given Proportions of Sines.		Corresponding Differences of Logarithm.

Two	to one.	6931469.22
Four	to one.”	13862938.44
Eight	to one.”	20794407.66
Ten	to one.”	23025842.34
20	to one.”	29957311.56
40	to one.”	36888780.78
80	to one.”	43820250.00
A hundred	to one.”	46051684.68
200	to one.”	52983153.90
400	to one.”	59914623.12
800	to one.”	66846092.34
A thousand	to one.”	69077527.02
2000	to one.”	76008996.24
4000	to one.”	82940465.46

Given Proportions of Sines.		Corresponding Differences of Logarithm.

8000	to one.	89871934.68
10000	to one.”	92103369.36
20000	to one.”	99034838.58
40000	to one.”	105966307.80
80000	to one.”	112897777.02
100000	to one.”	115129211.70
200000	to one.”	122060680.92
400000	to one.”	128992150.14
800000	to one.”	135923619.36
1000000	to one.”	138155054.04
2000000	to one.”	145086523.26
4000000	to one.”	152017992.48
8000000	to one.”	158949461.70
10000000	to one.”	161180896.38

54.To find the logarithms of all sines which are outside the limits of the Radical table.

This is easily done by multiplying the given sine by 2, 4, 8, 10, 20, 40, 80, 100, 200, or any other proportional number you please, contained in the short table, until you obtain a number within the limits of the Radical table. By 50 find the logarithm of this sine now contained in the table, and then add to it the logarithmic difference which the short table indicates as required by the preceding multiplication.

Example.

IT is required to find the logarithm of the sine 378064. Since this sine is outside the limits of the Racial table, let it be multiplied by some proportional number in the foregoing short table, as by 20, when it will become 7561280. As this now falls within the Radical table, seek for its logarithm (by 50) and you will obtain 2795444.9, to which add 29957311.56, the difference in the short table corresponding to the proportion of twenty to one, and you have 32752756.4. Wherefore 32752756 is the required logarithm of the given sine 378064.

55.As half radius is to the sine of half a given arc, so is the sine of the complement of the half arc to the sine of the whole arc.

Let a b be radius, and a b c its double, on which as diameter is described a semicircle. On this lay off the given arc a e, bisect it in d, and from e in the direction of c lay off e h, the complement of d e, half the given arc. Then h c is necessarily equal to e h, since the quadrant d e h must equal the remaining quadrant made up of the arcs a d and h e. Draw e i perpendicular to a ic, then e i is the sine of the arc a d e. Draw a e; its half, f e, is the sine of the arc d e, the half of the arc a d e. Draw e c; its half, e g, is the sine of the arc e h, and is therefore the sine of the complement of the are de. Finally, make a k half the radius a b. Then as a k is to e f, so is e g to e i. For the two triangles c e a and c i e are equi-angular, since i c e or a c e is common to both; and c i e and c e a are each a right angle, the former by hypothesis, the latter because it is in the circumference and occupies a semicircle. Hence a c, the hypotenuse of the triangle c e a, is to a e, its less side, as e c, the hypotenuse of the triangle c i e, is to e i its less side. And since a c, the whole, is to a e as e c, the whole, is to e i, it follows that a b, half of a c, is to a e as e g, half of e c,is to e i. And now, finally, since a b, the whole, is to a e, the whole, as e g is to e i, we necessarily conclude that a k, half of a b, is to f e, half of a e, as e g is to e i.

56.Double the logarithm of an arc of 45 degrees is the logarithm of half radius.

Referring to the preceding figure, let the case be such that a e and e c are equal. In that case i will fall on b, and e i will be radius; also e f and e g will be equal, each of them being the sine of 45 degrees. Now (by 55) the ratio a k, half radius, to e f, a sine of 45 degrees, is likewise the ratio of e g, also a sine of 45 degrees, to e i, now radius. Consequently (by 37) double the logarithm of the sine of 45 degrees is equal to the logarithms of the extremes, namely radius and its half. But the sum of the logarithms of both these is the logarithm of half radius only, because (by 27) the logarithm of radius is nothing. Necessarily, therefore, the double of the logarithm of an arc of 45 degrees is the logarithm of half radius.

57.The sum of the logarithms of half radius and any given arc is equal to the sum of the logarithms of half the arc and the complement. of the half arc. Whence the logarithm of the half arc may be found if the logarithms of the other three be given.

Since (by 55) half radius is to the sine of half the given arc as the sine of the complement of that half arc is to the sine of the whole arc, therefore (by 38) the sum of the logarithms of the two extremes, namely half radius and the whole arc, will be equal to the sum of the logarithms of the means, namely the half arc and the complement of the half arc. Whence, also (by 38), if you add the logarithm of half radius, found by 51 or 56, to the given logarithm of the whole arc, and subtract the given logarithm of the complement of the half arc, there will remain the required logarithm of the half arc.

Example.

LET there be given the logarithm of half radius (by 51) 6931469; also the arc 69 degrees 20 minutes, and its logarithm 665143. The half are is 34 degrees 40 minutes, whose logarithm is required. The complement of the half arc is 55 degrees 20 minutes, and its logarithm 1954370 is given, Wherefore add 6931469 to 665143, making 7596612, subtract 1954370, and there remains 5642242, the required logarithm of an arc of 34 degrees 40 minutes.

58.When the logarithms of all arcs not less than 45 degrees are given, the logarithms of all less arcs are very easily obtained.

From the logarithms of all arcs not less than 45 degrees, given by hypothesis, you can obtain (by 57) the logarithms of all the remaining arcs decreasing down to 22 degrees 30 minutes. From these, again, may be had in like manner the logarithms of arcs down to 11 degrees 15 minutes. And from these the logarithms of arcs down to 5 degrees 38 minutes. And so on, successively, down to 1 minute.

59.To form a logarithmic table.

Prepare forty-five pages, somewhat long in shape, so that besides margins at the top and bottom, they may hold sixty lines of figures. Divide each page into twenty equal spaces by horizontal lines, so that each space may hold three lines of figures. Then divide each page into seven columns by vertical lines, double lines being ruled between the second and third columns and between the fifth and sixth, but a single line only between the others.

Next write on the first page, at the top to the left, over the first three columns, “0 degrees”; and at the bottom to the right, under the last three columns, “89 degrees”. On the second page, above, to the left, “1 degree”; and below, to the right, “88 degrees”. On the third page; above, “2 degrees”; and below, “87 degrees” Proceed thus with the other pages, so that the number written above, added to that written below, may always make up a quadrant, less 1 degree or 89 degrees.

Then, on each page write, at the head of the first column, “Minutes of the degree written above”; at the head of the second column, “Sines of the arcs to the left”; at the head of the third column, “Logarithms of the arcs to the left”; at both the head and the foot of the third column, “Difference between the logarithms of the complementary arcs”; at the foot of the fifth column, “Logarithms of the arcs to the right”; at the foot of the sixth column, “Sines of the arcs to the right”; and at the foot of the seventh column, “Minutes of the degree written beneath”.

Then enter in the first column the numbers of minutes in ascending order from 0 to 60, and in the seventh column the number of minutes in descending order from 60 to 0; so that any pair of minutes placed opposite, in the first and seventh columns in the same line, may make up a whole degree or 60 minutes; for example, enter 0 opposite to 60, 1 to 59, 2 to 58, and 3 to 57, placing three numbers in each of the twenty intervals between the horizontal lines. In the second column enter the values of the sines corresponding to the degree at the top and the minutes in the same line to the left; also in the sixth column enter the values of the sines corresponding to the degree at the bottom and the minutes in the same line to the right. Reinhold’s common table of sines, or any other more exact, will supply you with these values.

Having done this, compute, by 49 and 50, the logarithms of all sines between radius and its half, and by 54, the logarithms of the other sines; however, you may, with both greater accuracy and facility, compute, by the same 49 and 50, the logarithms of all sines between radius and the sine of 45 degrees, and from these, by 58, you very readily obtain the logarithms of all remaining arcs less than 45 degrees, Having computed these by either method, enter in the third column the logarithms corresponding to the degree at the top and the minutes to the left, and to their sines in the same line at left side; similarly enter in the fifth column the logarithm corresponding to the degree at the bottom and the minutes to the peo and to their sines in the same line at right side.

Finally, to form the middle column, subtract each logarithm on the right from the logarithm on the left in the same line, and enter the difference in the same line, between both, until the whole is completed.

We have computed this Table to each minute of the quadrant, and we leave the more exact elaboration of it, as well as the emendation of the table of sines, to the learned to whom more leisure may be given.

Outline of the Construction, in another
form, of a Logarithmic Table.

60.SINCE the logarithms found by 54 sometimes differ from those found by 58 (for example, the logarithm of the sine 378064 ts 32752756 by the former, while by the latter it 1s 32752741), it would seem that the table of sines is in some places faulty. Wherefore I advise the learned, who perchance may have plenty of pupils and computors, to publish a table of sines more reliable and with larger numbers, tn which radius is made 100000000, that is with eight cyphers after the unit instead of seven only. Then, let the First table, like ours, contain a hundred numbers progressing in the proportion of the new radius to the a less than it by unity, namely of 100000000 to 99999999.

Let The Second table also contain a hundred numbers in the proportion of this new radius to the number less than it by a hundred, namely of 100000000 to 99999900.

Let the Third table, also called the Radical table, contain thirty-five columns with a hundred numbers in each column, and let the hundred numbers in each column progress in the proportion of ten thousand to the number less than it by unity, namely of 100000000 to 99990000.

Let the thirty-five proportamals standing first in all the columns, or occupying the second, third, or other rank, progress among themselves in the proportion of 100 to 99, or of the new radius 100000000 to 99000000.

In continuing these proportionals and finding their logarithms, let the other rules we have laid down be observed.

From the Radical table completed in this way, you will
find with great exactness (by 49 and 50) the logarithms of
all sines between radius and the sine of 45 degrees; from
the arc of 45 degrees doubled, you will find (by 56) the
logarithm of half radius; having obtained all these, you
will find the other logarithms by 58. Arrange
all these results as described in 59, and you
will produce a Table, certainly the
most excellent of all Mathe-
matical tables, and pre-
pared for the most
important
uses.

End of the Construction of the Logarithmic Table.

↑ It is evident that the ratio of the spaces traversed T 1, 1 2, 2 3, 3 4, 4 5, &c,, is that of the distances T S, 1 S, 2 S, 3 S, 4 S, &c, for when quantities are continued proportionally, their differences are also continued in the same proportion. Now the distances are by hypothesis a proportionally, and the spaces traversed are their differences, wherefore it is proved that the spaces traversed are continued in the same ratio as the distances.

[p42-1] It is evident that the ratio of the spaces traversed T 1, 1 2, 2 3, 3 4, 4 5, &c,, is that of the distances T S, 1 S, 2 S, 3 S, 4 S, &c, for when quantities are continued proportionally, their differences are also continued in the same proportion. Now the distances are by hypothesis a proportionally, and the spaces traversed are their differences, wherefore it is proved that the spaces traversed are continued in the same ratio as the distances.

[1]