TO shew the use of Logarithms in Trigonometry.
Let ABC be a plane triangle, right angled

at A, of which the hypothenuse BC is 368
feet, and the angle at B 42° 35', and let the
sides AC and AB be sought.

By the ad Cor. to 2d Prop. of Plane Trig. R, is to the fin. B, as BC to CA: confequent


A ly, from the nature of numbers, CA is found by multiplying BC by the fin. B, and dividing the product oy R; and this is done, by adding the logarithm of BC, and the artificial fine of B together, and subtracting the artificial radius from the sum, for the remainder will be the logarithm of CA. And in the same manner may BA be found.

To find CA.

To fin! BA. As the radius 10.0000000 As R.

10.0000000 To fin. B 42° 35' 9.8303717 To cos. B. 42° 35'

To cos. B. 42° 35' 9.8670512 So is BC 368 f. 2.5658478 So is CB. 368 2.5658478

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T. CA 249.01 f. 2.3962195 | To BA. 270.96

To BA. 270.96: 2.4328990

When the first term is not radius, instead of subtra&ting its logarithm, we often add what it wants of the radius to the other two, and take 10 from the index of the sum ; this want is easily got, by subtracting the right-hand figure of the logarithm from 10, and all the rest from 9; and it is called the Arithmetical Complement of the Logarithm.

And because the square of a number is obtained by adding its logarithm to itself, or by doubling it; the square root of a number will be got by dividing its logarithm by 2.

To illustrare this, let the three sides of a plane triangle ABC be, AB 228, AC 136, and BC 318 feet; and let the angle at A be required.

By the 8th Prop. of Plane Trig. the rectangle contained by AB, AC is to the rectangle contained by the fum of the three fides and its excess above BC, as the square of R to the square of the cofine of į A. And the rectangles are got by adding their logarithms, and the squares by doubling them. Therefore, add the three fides together, and from the sum subtract BC. Then add into one sum the Arithmetical complements of AB Rs


LOGAR. and AC, and the logarithms of the sum and remainder, and W the sum of these four will be the cofine of the angle A.

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Remainder 23 Logarithm. 1.3617278

2)19.4030085 Half the angle A 59° 43' 22" cofine 9.7015042

The angle A

118 36 44



G E O M E T R Y.

N order to determine the magnitudes of any kind of quantities, Part I.

there is some quantity of the same kind assumed, as fufficiently known, and the others are said to be known, when their relation to it is known.

To avoid fractions, this assumed quantity is commonly among the least that are in common use. But after it is assumed, any multiple or part of it may be assumed for the same purpose. Quantities thus assumed, are called Measures of other Quantities.

As measures are of general utility, it is often necessary to determine their form, or other circumstances, by means of which their magnitude may be known.

This Treatise is divided into three parts. The first treats of Lines and Angles; the second of Superficies; and the third of Solids.

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The Icast measure of lines with us, is an inch, 12 of which make a foot; and 3 feet make a yard. But distances are often measured by a chain of 22 yards, of which 80 make a mile. If the length of a pendulum, vibrating seconds at London, be divided into 313 equal parts, eight of these parts will make an inch.


R 2


The Scots inch is a little longer than the English inch, for 18; Scots inches make 186 English inches. There are 37 Scots inches in an ell, and 24 ells in a Scots chain, 80 of which make a Scots mile ; so that ss Scots miles are equal to 62 English.

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Plate I.


Fig. 1.

Fig. 2.

To describe the construction and ufe of the inftru

ments used for taking angles.
The Quadrant is the fourth part of a circle divided into
ninety degrees, and, if the limb be large enough, each degree
subdivided into quarters or minutes. To the radius which
passes through the ninetieth degree, two fights are adapted ; and
a thread with a plummet is hung from the centre. The use of
it is to take angles in a vertical plane.

The Theodolite is a circle divided into 360 degrees, with one or more indices fixed on its centre; these indices are fitted with Nonuis's divisions, for finding the parts of a degree. The use of it is to take angles in an horizontal plane. It has a great deal of apparatus, in order to adjust it, such as three staffs, with joints and screws, and a level, for supporting it and placing it horizontally; it has also a telescope for observing objects; and besides these, it has a vertical arch for shewing the inclination of lines to the horizon, and a compass for shewing their inclination to the meridian.

PROB. II. Fig. 3. Pl. I. To describe the construction of the Geometrical Fig. 3. Square.

T Square

This is a square with a line and plummet hanging from one of its angles A, and each of the sides BE and ED is divided into 100 equal parts: and there are two fights C and F fixed on the fide AD.

There is also an index GH, with fights, which, when there is occasion, can be joined to the instrument, and made to move about the centre A.

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Pl. I.


To describe the construction and use of a line of chords, and of a line of equal parts.

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It is often neceffary to lay down a figure on paper, like to Part I. another figure, for which purpose, these lines are required, in order to make their angles equal, and their sides proportionals,

The line of chords is made thus : Let BAC be a quarter of a Fig. 4. circle, and join BC, and divide the arch BC into go degrees : Then from C transfer the distances of the divisions to the straight line CB, and mark them with the degrees in the corresponding arch. The chord of 60 degrees being the fide of an infcribed hexagon, is equal to the radius of the circle, by cor. to the Igth of 4th B. of El.

If now the angle EDF is to be measured, take the chord of Fig. s.
60°, and from the centre D describe with that distance the
arch FG. Then, if the distance FG applied on the line of
chords, from C towards B, gives 25', this ihail be the measure
of the angle proposed.

When an obtuse angle KDE is to be measured, produce KD
to F, and measure its supplement FDE, and then KDE will be
But if an angle of 50° is to be made at a given point M, in

Fig. 6.
the line KL, from the centre M, with the distance MN, equal to
the chord of 60°, describe the arch NR. Take soo from the
line of chords, and make NR equal to it, and join MR; and
it is plain, that NMR is an angle of 50°.

The line of equal parts is made by taking any convenient Fig. 7. distance for one of the parts, and laying it on the line, from one end to the other ; after which, one of the parts is usually subdivided into 10 equal parts. From such a scale, any number less than 100, may be taken, by calling each of the greater divisions 10, and each of the less divifions one.

If upon one fide of the line, thus divided, there be drawn ten others, at equal distances from one another, perpendiculars to the divided line, through the larger divisions, will divide them all; and if one of the spaces between the perpendiculars of the two outermoft lines, be each divided into ten equal parts, and from the end of one of them a line be drawn to the first division of the other, and the rest of the divisions be joined in their order, a diagonal scale will be made, from which any number less than a thousand may be taken.

These lines, with several others, as lines of fines, tangents, and secants, are usually marked on the scales used by artists. But they are most convenient for use, when placed on a sector, which is a jointed scale, like a carpenter's rule ; for by means of it, an arch may be made of a given number of degrees to any radius, and distances may be taken greater or less, without altering their ratios; this is done, by opening the sector, and extending the compasses from the number on one of the legs, to the same number on the other,


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