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by ro, and the Quotient, added to the before-found Logarithm, will be the true Logarithm of the given Number, when the proper Index is prefixed to it. If it were required to find the Logarithm of 35274, look for the Logarithm of 3527, in the Manner before taught, and you will have .547406, the Logarithm of 3527, without the Index; but, for the Loga-" rithm of 35274, take the Difference in the laft Column marked D, where you will find 123, which is the Difference betwixt the Logarithm of 3527 and 3528; multiply this Difference by 4, divide the Product by 10, add the Quotient to the before-found Logarithm, and prefix the proper Index 4, you will have 4.547455, for the true Logarithm of 35274. Difference 123

Log. of 3527 is .547406
Part proportional add 49

•547455

4

4912

To find the Logarithm for 32716: The Logarithm of 3271, without the Index, is .514681, and the Difference in the laft Column under D, is 132, which multiplied by 6, and the Product divided by 10, the Quotient, added to the Loga. rithm (of 3271, gives the Logarithm of 32716, without the Index, as follows.:

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Therefore the proper Index 4, being prefixed, gives 4.514760, for the true Logarithm of 32716, the given Number.

In the fame Manner may a Logarithm be found for a Number of fix Places, as 149635, by firft finding the Logarithm of the firft four Figures 1496, and the Difference in the laft Column; this Difference multiplied by the two laft Figures 35, and the Product divided by 100, by cutting off the two laft Figures, the reft added to the Logarithm of 1496, and the proper Index 5 prefixed, you have the true Logarithm of the given Number.

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101185

To fnd the Number to a given Logarithm.

Look in the fecond Column of one of the left-hand Pages of the Table for the Logarithm, without the Index; if you find it there exactly, then the Number which ftands against it in the first Column, is the Number fought: If it is not to be found exactly in any of thefe Columns, look for the next lefs Logarithm, and caft your Eye ftraight along that Line to the other Columns; if it be found in any of thefe, the Figure at the Top of the Column in which it is found must be joined to the other three, and thefe four Figures are the Number fought, the Value of which will be determined by the Index.

Exam. I. To find the Number to the Logarithm 0.796574.

Exam. 2. To find the Number to the Logarithm 3.532409.

Answer 6.26.

Answer 3408.

Exam. 1. I find the given Logarithm in the fecond Column of one of the left-hand Pages, and against it the Number 626; but as the Index of the Logarithm is o, I thence know, that the Number, to which it belongs, confifts of but one Place of Integers, or is under 10; therefore I place a Dot between the first Figure 6, and the other two, marking them off for Decimals, fo that 6.26 is the Number fought,

Exam. 2. Not finding the Logarithm given in the fecond Column of any of the left-hand Pages, I obferve the next less is .531479; from which looking along the Line, I find my Logarithm exactly in the laft Column of Logarithms but one of the right-hand Page; and at the Top of that Column I fee 8, which must be annexed to the Figures in the first Column of the left-hand Page, against the Logarithm firft found, where we have 340; to which annexing 8, it makes 3408, for the Number to the Logarithm 3.532499 given.

Here it may be obferved, that had the Index been either more or less than 3,, the Number must have confifted of a greater or a lefs Number of integral Figures: Thus, if the Index had been 4, there must have been five integral Figures, by annexing a Cypher to thofe found; but had the Index been 2, the 8 must have been marked off by a Dot for a Decimal; and in the fame Manner the Numbers are found to the following Logarithms:

0.759668

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Where the Reader may obferve, that, when the Index is o, then the Number fought has only one Place of Integers; if the Index is 1, then the Number fought has two Places of Integers; if the Index is 2, the Number fought has three Places of Integers.

If the given Logarithm is neither to be found in the second Column of the left-hand Pages, nor in either of the other nine Columns, take the next lefs, and fubtract it from the given Logarithm; if the Difference be confiderable, annex 9 to it, and divide this Sum by the Difference in the laft Column under D, and the Quotient will be the fifth Figure, to be annexed to the four already found.

Exam. 1. To find the Number to the Logarithm 4.571090 The next lefs Log. (whofe Number is 3724)

Difference

To 81 annexing 9, it is 819.

.571009

81

Tab. Diff. 117) 819 (7, which annexed to 3724, before

819

(found, gives 37247 Answer.

Exam. 2. To find the Number to the Log. 2.694210 The next lefs Log. (whofe Number is 4945)

694166

44

To 44 annexing 9,

it is 449.

Tab. Diff. 88 (449 (5, which joined to 4945, the Number 440 (before found, the Anf. is 49455.

To find a Number confifting of fix Places to a given Logarithm.

Rule. Firft find the next lefs Logarithm in the Table, as alfo the Difference in the laft Column under D; then take out the four Figures belonging to the Logarithm in the Table, and,

in order to find the two Figures to be annexed to these to complete the Number, fubtract the Logarithm found in the Table from the given Logarithmn; to this Difference annex 99, and divide it by the Difference found in the last Column under D, and the Quotient will be the Figures to be annexed to the four before found.

But it is here to be obferved, that in thefe Tables, the decimal Part of the Logarithms confifting of but fix Places, a Number of fix Places cannot be found fufficiently exact, but only in the former Part of the Table, before the Logarithm of 4000, because the Differences are afterwards too fmall to calculate Numbers to fix Places.

To find a Number to fix Places to the Logarithm 5.454306: The next lefs Logarithm is .454236, whofe Number is 2846, and the tabular Difference 153. The given Log. is .454306 The next lefs

Tabular Diff. 153 ( 7199 ( 47 612

•454235

The Difference is

71

1079

1071

The Quotient 47, being annexed to the four Figures found in the Table, gives 284647, for the true Number to the Logarithm 5.454306: The Index being 5, fhews the whole to be Integers.

To find a Number to fix Places to the Logarithm 3.241877: The next lefs Logarithm in the Table is .241795, whofe Number is 1745, and the Difference in the laft Column is 249. The given Log. is .241877 Tabular Diff. 249) 8299 (33 The next lefs

.241795

82

747

829

747

The Quotient 33, annexed to 1745, makes 174533, for the Number to the given Logarithm: But, as the Index is but 3, there can be no more than four Places of Integers; so that 1745.33 is the true Number to the given Logarithm.

The Defcription and Ufe of the Table of Sines and Tangents.

The Table of Sines and Tangents has the 90 Degrees of the Quadrant, difpofed in the following Manner: At the Top of the first two Pages is placed o Degree, at the Top of the

two

two next Degree, and fo proceeding on to 44 Degrees at the Top of the two laft Pages; and then beginning with 45 Degrees at the Bottom of the two laft Pages, and proceeding back again to 89 Degrees at the Bottom of the two first Pages.

The Minutes are contained in the firft and laft Column of each Page, marked with the Letter M, for Minutes: In the firft Column to the left Hand of each Page, the Minutes increase as they defcend; on the left-hand Page, from 0 to 30, and on the right-hand Page, from 30 to 60: But in the laft Column of each Page, the Minutes increase as they ascend; on the right-hand Page, from 0 to 30, and on the left-hand Page, from 30 to 60: Between thefe Columns, in each Page, are four other Columns, marked at the Top, Sine, C-fine, Tangent, Cotangent, and at the Bottom, Co-fine, Sine, Co-tangent, Tangenti The Degrees, Minutes, Seconds, &c. are ufually wrote in this Manner, 26°. 15'. 3c", which are to be read, 26 Degrees, 15 Minutes, 30 Seconds.

To find the Logarithmic Sine of 23°. 21.

Find the 23° on the Top of the Table, and in the Column to the left Hand, marked M, the Numbers beginning at o, and increafing as they defcend, thus, 0, 1, 2, 3, to 30, find the 21'; against which, in the Column over which is the Word Sine, you have, for the Sine of 23°. 21', the Logarithm 9.598075.

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To find the Sine of 24°. 41': Having found the 24° at the Top of the Table, as before, the Minutes, being above 30, must be fought in the first Column of the right-hand Page, marked M, where the Numbers, beginning at 30, increase as they defcend thus, 30, 31, 32, to 60; amongst which feek for 41, and right against it, in the Column marked Sine at the Top, you have the Sine of 24°. 41', 9.620763.

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But, when the Number of Degrees exceed 45, you must look for them at the Bottom of the Page, and for the Mi

I

nutes

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