divisions are formed in a similar manner, by taking the logarithms of the intermediate numbers.

2. The line of sines is constructed by taking the arithmetical complements of the logarithmic sines from the same scale of equal parts from which the line of numbers was constructed, and setting them off from 90 backwards, or towards the left hand; thus the arithmetical complement of the logarithmic sines of 80°, 70°, 60°, 50°, 40°, 30°, 20°, 10°, are 7, 27, 116, 192, 301, 466, 760; and these taken from the same scale of equal parts from which the line of numbers was constructed, and set off from 90 towards the left hand, gives the sines of the above degrees respectively.

As the sine of 90° is equal to radius, and as the several arithmetical complements are what the sines of the arcs want of radius, the reason of setting off the numbers as above is obvious.

3. The line of Rhumbs is constructed in a similar manner to the line of sines, by taking the arithmetical complements of the logarithmic sines of the degrees and minutes which are contained in the several points and quarter-points of the compass, and setting them off from the right-hand end of the scale towards the left.

4. The line of tangents is constructed in the same way as the line of sines, by setting off the arithmetical complements of the tangents under 45° backwards, from 45° towards the left hand. For the tangent of 45° is equal to the radius, and the arithmetical complement of any logarithmic tangent under 45° is what the tangent wants of radius.

It has been before observed, that the division at 40 serves both for 40 and 50-that at 30, for 30 and 60, &c.; the reason of which cannot well be given here.

5. The line of tangent rhumbs is constructed as the line of tangents, by taking the arithmetical complements of the logarithmic tangents of the degrees and minutes contained in the first four points of the compass, and setting them off from the end of the line towards the left hand.

6. The line of versed sines is constructed by means of the logarithmic versed sines extending to 180°.

Take the logarithmic versed sines of the supplements of the arcs, and subtract the logarithm of 2 from them; the arithmetical complements of the remainders, taken from the scale of equal parts, and applied from the right hand towards the left, will give the divisions of the line of versed sines.

The reason of this construction, and the use of the line itself, may be best learned from the teacher. It is by far the most difficult line on the scale, and is very seldom understood.

The preceding lines are the principal ones on the plane scale; and to render them convenient for practice they are transferred from the figure in page 7, to another of convenient length and breadth. These lines are of use in Trigonometry, Navigation, Fortification, Dialing, and in almost all the practical branches of the mathematics.

On these lines all proportions are solved; and when four numbers are proportionals, the ratio subsisting between the two first terms is equal to that

between the two last: that is, the quotient of the first term by the second, is equal to the quotient of the third term by the fourth. Therefore, from the nature of logarithms, the difference between the first and second terms, on the scale, is equal to the difference between the third and fourth. And as four quantities are proportionals when taken alternately, the distance between the first and third is equal to the distance between the second and fourth. Hence the following General Rule :

The extent of the compasses from the first term to the second, will reach, in the same direction, from the third term to the fourth.

Or, the extent of the compasses from the first term to the third, will reach, in the same direction, from the second term to the fourth.

This rule is general, except the first two or the last two terms of the proportion are on the line of tangents, and neither of them under 45°. In this case the extent on the tangents is to be made in a contrary direction, as has been shewn in my Mensuration.

The reason is obvious, from the construction of the line of tangents. For, had the tangents above 45° been laid down in their proper direction, they would have extended beyond the length of the scale; they are therefore doubled backwards, as it were, upon the tangents below 45°, and consequently lie in a contrary direction to their natural order.

It is further to be observed, that if the two last terms of a proportion be on the line of tangents, and one of them greater and the other less than 45°, the extent from the first term to the second, will reach from the

third beyond the scale. Therefore, apply the extent between the two first terms from 45° backward upon the line of tangents, keeping the left-hand point of the compasses where it falls, and bringing the right-hand point from 45° to the third term of the proportion, this extent on the compasses, applied from 45° backward, will reach to the fourth term on the tangent required. For, had the line of tangents been continued forward beyond 45°, the divisions would have fallen above 45° forward, in the same manner as they fall under 45° backward.

To find the product of two numbers, extend the compasses from 1 to either of the numbers, and that extent will reach the same way from the other number to the product. Multiply 5 by 12; the extent from 1 to 5 will reach from 12 to 60.

To divide one number by another, extend the compasses from the divisor to 1, and that extent will reach the same way from the dividend to the quotient. Divide 60 by 5; extend the compasses from 5 to 1, and the same will reach from 60 to 12, the quotient.

To find a mean proportional between two given numbers, as suppose between 7 and 28: extend the compasses from 7 to 28, and bisect that extent, then its half will reach from 7 forward, or from 28 backwards, to 14, the mean proportional required.

therefore extend the compasses from 1 to 25, and half that extent will reach from 1 to 5, the root required.

To extract the cube root, or the 4th root, &c., extend the compasses from 1 to the given power, then take such part of it as is denoted by the index of the root, and that part will reach from 1 to the root sought.

THE SECTOR

Is another mathematical instrument of very great use in Geometry, Trigonometry, &c. This instrument possesses many advantages above the common scale, as it is adapted to all radii and all scales.

The construction of this instrument is founded on the 4th Proposition of the 6th Book of Euclid. It consists of two legs or rules, made of box or brass, representing the radii of the sector of a circle, moveable round a joint, the middle of which represents the centre, from whence several scales are drawn on the faces.

The scales usually set upon sectors, may be distinguished into single and double. The single scales are such as are set upon plain scales-the double scales are those which proceed from the centre; each of these being laid twice on the face of the instrument, viz. once on each leg. From these scales, dimensions or distances are to be taken, when the legs of the instrument are set in an angular position.

THE PROTRACTOR

Is another mathematical instrument, used in surveying, for laying down angles on paper.

The simplest protractor consists of a semicircular

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