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THE USE OF THE

LOGA

R I T

H M S

OGARITHMS are a Series of Numbers, invented by Lord Napier, a Scotch Nobleman, by which the Works of Multipli cation may be performed by Addition; and the Operation of Divifion may be done by Subtraction; for if the Logarithm of the Multiplicand be added to the Logarithm of the Multiplier, that Sum will be the Logarithm of the Product; and if from the Logarithm of the Dividend you fubtract the Logarithm of the Divifor, the Remainder will be the Logarithm of the Quotient. Again, if the Logarithm of any Number be divided by 2, the Quotient will be the Logarithm of the Square Root of that Number; or if the Logarithm of any Number be divided by 3, the Quotient will be the Logarithm of the Cube Root of that Number.

In the firft Page of the Tables you will find the Logarithm of the Numbers from 1 to 120, which are continued to 10,000. To find the Logarithm of any Number proceed thus: 1ft. If the Number be lefs than 100, fuppofe 45, look for 45 in the Column marked N°. opposite to which is 1.65321, the Logarithm required. Again, fuppofe the Logarithm 741 is wanted, look for 741 in the Column marked N°: oppofite to that is 2.86982, the Logarithm; but if the Number confifts of more than 3 Places, fuppofe 5738, find the first 3 Figures 573 in the Column marked N°. and right oppofite, under 8, Itands 2.75876; but as the Number is above 3 Places of Figures, the Figure 2 or Index must be called 3, and then it will stand thus, 3.75876, the Logarithm required. On the contrary, when a Logarithm is given to find the Number belonging to it, look among the Logarithms for the nearest to it, oppofite to which in the Column of No. will be found the Number required. Suppofe, for Example, the Logarithm 2.75662 was given, I find the nearest to it in the Table is 2.75664; oppofite to that is 571, the Number required. Again, fuppofe the Logarithm 3.74763 was given to find its Number, looking in the Table I find neareft thereto under 3, and oppofite to 559; now because the Index is 3, I call it 5593, which is the Number required; for when the Number is under 10 the Index is always o; from 10 to 100 the Index is I; from 100 to 1000 the Index is 2 i from 1000 to 10,000 the Index is 3; from 10,000 to 100,000 the Index is 4, &c.

To find the Logarithm of an abfolute Number and a Decimal.

If the Decimal confifts of one Place, and the whole Number be under 100, look for the Whole Number and Decimal as if it was an abfolute Number in the Column of Numbers, oppofite to that will be the Logarithm required, obferving to feparate the laft Figure in

the Number for a Decimal, and to call the Index 1 if above 10 and under 100; but if under 10 the Index 0. Required the Logarithm of 57,5 oppofite to 575 is 2,75967, calling the Index 1 it will be 1.75967, the Logarithm of 57,5.

Having the Logarithm, the Number is found by the Inverfe of the above, as for Example:

Required the Number answering to the Logarithm 2,98798? Looking in the Table I find that Logarithm under (7) and oppofite 972, therefore 972,7 is the Number fought.

When there are two Places of Decimals, find the Whole Number, and the first Place of the Decimal in the Column of N°. as before, and the laft Figure in the Column marked 1|2|3| &c. above, and at the Angle of Meeting will be the Logarithm required. Example, Required the Logarithm of 44,49? look for 444, and Right oppofite, under 9, is 2.64826, making the Index I at will be 1,64826, the Logarithm of 44,49: having the Logarithm, the Number is found as before.

Example, Required the Number anfwering to the Logarithm 0.75450 looking in the Table I find that Logarithm under 2, and oppofite 568 in the Column of Numbers; now as the Index is (o) the 5 only is a Whole Number, and 68, fet before the 2 above, will make the Decimal 682, confequently the Number is 5,682.

To find the Logarithm of the Sine, Tangent, or Secant, belonging to any Number of Degrees and Minutes required.

If the required Degrees be less than 45, feek the Degrees on the Top, and the Minutes in the Left Hand Column Marked M, against which in the Column figned at the Top with the propofed Name, ftands the Sine, Tangent, and Secant required; but when the Degrees given are more than 45, feek the Degrees at the Bottom, and the Minutes in the Right Hand Column marked M at the Bottom, and the propofed Name at the Bottom. Here it may be obferved, that the Degrees at the Top and Minutes at the Left Hand Column, added to the Degrees at the Bottom and Minutes in the Right Hand Column, always make 90°; hence, if a Sine be looked for, the Complement to it will be found in the adjoining Column; the same of Tangents and Secants.

Example ft. Required the Logarithm of the Sine 28° 37'?

Find 28° at the Top of the Page, and in the Left Hand Column marked M at the Top find 37, against which in the Column marked with the Word Sine, ftands 9.68029, the Logarithm of the Sine of 28° 37' required; the fame may be obferved of Tangents and Secants. Example 2d. Required the Logarithm of the Tangent of 67° 45'? Find 67° at the Bottom of the Page, and 45 in the Right Hand Column marked M at the Bottom, against this in the Column marked Tangent at the Bottom, ftands 10.38816, which is the Logarithm required.

Having the Sine, Tangent, or Secant, the Co-Sine, Co-Tangent, and Co-Secant are always found in the adjoining Columns.

The Logarithm of any Number of Degrees above 90° is found by fubtracting the given Degrees from 180°, and taking the Logarithin of the Remainder.

To find the Complement Arithmetic of any Logarithm given.

The Complement Arithmetic of a Logarithm is what it wants of 10.00000, or 20.00000, and is ufed to avoid Subtraction; for finding it this is the Rule; take the Refidue or Remainder of the first Figure to, and fo of the Reft until you come to the last Figure, of which take its Remainder under 10, and it is done.

Example ift. I would have the Complement Arithmetic of 9,62595?

For the First Figure 9 write o; for 6, 3; for 2, 7; for 5, 4; for q, o; and for the laft Figure 5 write 5; and fo you have 0.37405x for the Complement Arithmetic fought.

Example 2d. The Complement Arithmetic of 20.33133 ?

For o, (always rejecting the firft Figure, when there are two Figures in the Characteristic) write 9, and fo on as before directed, and then you will have 9.66867, which is the Complement Arithmetic of 20.33133.

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Comp. Arith. required 0.37405 | Comp. Arith. required 9.66867

The Logarithm for Degrees, Minutes, and Seconds is found by taking the Difference between the next greater and leffer Logarithm; and faying, as 60": is to that Difference :: fo is the Seconds given : to a fourth Number, which fourth Number being added to the Right Hand of the next lefs Logarithm gives the Logarithm required. By the Reverse of this, when the Logarithm is given, the Seconds may be found.

But if the given Seconds be,,, or nearly any other even Parts of a Minute, the like Parts may be taken of the Difference of the Logarithms, and added to the next lefs Logarithm, and that will be the Logarithm of the Degrees, Minutes, and Parts of a Minute required.

The fame may be observed of Natural Sines.

The

The following Things relating to Plane Trigonometry should be well understood.

Notwithstanding what has been faid in Geometry, it may not be

here improper to obferve,

A Plane Triangle is any three Corner'd Figure bounded by three Right Linés, and confifts of fix Parts, three Sides and three Angles.

The Angles of every Plane Triangle are measured by an Arch of a Circle, defcribed on their Angular Point with the Chord of 60 Degrees, and are faid

to be greater or lefs ac- D
cording to the Num-
ber of Degrees con-
tained between their
Legs; but the Sides or
Legs are always mea-
fured by a Scale of
equal Parts.

360th Part of the Cir

A Degree is the

cumference of any Cir

cle.

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All Circles, whether great or small, are divided into 360 equal Parts, called Degrees; and each Degree into 60 equal Parts, called Minutes; and each Minute into 60 equal Parts called Seconds, &c. A Semicircle is 180 Degrees, and a Quadrant contains 90 Degrees.

A Right Angle contains 90 Degrees; an Obtufe Angle more than 90; and an Acute Angle lefs than 90 Degrees.

The Three Angles of every Plane Triangle contain 180 Degrees.

In a Right Angled Triangle, the Right Angle contains 90 Degrees, and the two Acute Angles 90 Degrees between them; therefore, if one of the Acute Angles in a Right Angled Triangle be given, the other Angle is found by fubtracting the given Angle from 90 Degrees.

When one of the Angles in any Plane Triangle is given, the Sum of the other two Angles is found by fubtracting the given Angle from 180 Degrees; and if two Angles are given, the Third is found by fubtracting the Sum of the given Angles from 180 De

grees.

The Complement of an Angle is what it wants of go Degrees. The Supplement of an Angle is what it wants of 180 Degrees. The Chord of 60o, Sine of 90°, and Tangent of 45° are equal, and each of them is alfo equal to the Radius.

The Co-Sine, Co-Tangent, and Co-Secant of an Arch are the Right Sines, Tangents, and Secants of the Complement of that Arch.

Three Letters fignifying an Angle, the middle one fhews the Angular Point, the other two denote the Legs which include it.

Degrees are marked with a Cypher over them, and Minutes with a Dafh, alfo Sides or Angles given are marked with a Dash, but if required, they are marked with a Cypher.

The Logarithm of any Angle exceeding 90 Degrees is found by fubtracting the given Angle from 180 Degrees, and taking the Logarithm of the Remainder.

The Solution of the feveral Cafes in Plane Trigonometry depend on four Propofitions called AXIOMS, which should be got perfectly by Heart.

AXIOM I.

In any Right-Angled Plane Triangle, if the Hypothenufe, or longeft Side, be made the Radius of a Circle, the other two Sides, or Legs, will be the Sines of their oppofite Angles: But if either of the Legs, including the Right Angle, be made Radius, the other Leg becomes the Tangent of its oppofite Angle, and the Hypothenule the Secant of the fame Angle.*

For in the Triangle A B C, let A B be made the Radius of a Circle, and with one Foot of the Compaffes on A or B defcribe a Circle, it is plain that the Leg BC will be the Sine of the Angle A, and AC the Sine of the Angle B: But if A C be made Radius, BC becomes the Tangent of the Angle A, and B A the Secant of the fame Angle.

B

Again,

NOTE. As this Axiom fhews how to work any Proportion in Plane Traverfe, or Mercator's Sailing; therefore, the Terms of Plane Sailing may be applied to the Right-angled Triangle, and then proceed to Plane

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