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 B limb, commonly of brass, divided into 180°, subtended by a diameter, in the middle of which is a small hole, or notch, called the centre of the protractor. And for the convenience of reckoning both ways, the degrees are numbered from the left hand towards the right, and from the right towards the left. This instrument is rendered very convenient by transferring the divisions from the circumference to the edge of a ruler. To make any angle with the protractor, lay the diameter of the protractor along the given line, which is to form one side of the angle, and its centre at the angular point; then make a mark opposite the given degree of the angle found on the limb of the protractor, and removing the instrument, by a plain ruler laid over that point and the centre, draw a line, which will form the required angle. In the same way we can discover the number of degrees contained in any given angle. By means of this instrument, we can expeditiously draw a line perpendicular to another. The improved protractor, with a vernier, to shew minutes &c. will be described in the treatise on surveying. When accuracy is required, no instrumental calculation ought to be relied on, the most accurately constructed scales giving only approximate results. LOGARITHMS. Logarithms are a set of numbers, so contrived that the products in multiplication, and the quotients in division, are obtained by means of addition and subtraction only. Or, Logarithms are a series of numbers in arithmetical progression, corresponding to another series in geometrical progression, the arithmetical series being the indices or powers of a given quantity, as a base.* As 10 is the base of our present system of arithmetic, so is it also employed as the base of the logarithms generally used. On this scale all the common tables of logs. are constructed. If we assume a series of numbers in geometrical progression, proceeding from 1, the ratio being 10, and set over them a series of numbers in arithmetical progression, beginning with 0, the common difference being 1, the numbers in the arithmetical series will be the logs. of the corresponding numbers in the geometrical series. Thus 7, 8. 0, 1, 2, 3, 4, 5, 6, 1, 10, 100, 1000, 10000, 100000, 1000000, 10000000, 100000000. From this it appears that the numbers in the arith * The invention of Logarithms is due to Lord Napier, Baron of Merchiston, in Scotland. In 1614, the inventor published the first tables of these numbers. His Logarithms are of that form which we call Hyperbolic Logarithms. In this system 1 is the logarithm of 2.718282. We are indebted for the modern table to Brigg, in whose system the logarithm of 10 is 1. The modern tables most in use are those of Tayler's and Doctor Hutlon's. Various methods have been proposed to construct logarithms, but the most simple and expeditious is that which may be seen in Gregory's Philosophy of Arithmetic, which see. metical series, which are the logs. of the corresponding numbers beneath them, are the exponents of the different powers of 10. But the sum of two exponents is the exponent of the product arising from the multiplication of their corresponding natural numbers, (see Philosophy of Arithmetic.) Hence it appears, that if the logs. of any two numbers be added together, the sum will be the log. of the product of these numbers. In the foregoing series, if we add 2 and 4, the sum 6 will be the log. of 100 × 10000, (=1000000.) Now as the addition of any two logs. answers to the multiplication of their corresponding natural numbers, it is evident that the converse will hold true; that is, that the subtraction of any two logs. will answer to the division of their corresponding natural numbers. Resuming the last example, if we take 4 from 6, the remainder, 2, will be the log. of 1000000÷10000 (=100.) Again, as the sum of any two logs. is the log of the product of their corresponding natural numbers, if we suppose the two numbers equal, then double the log. of one of them must be the log. of their product, which is the second power of one of them; thus 1+1 (=2), is the log. of 10x 10 (=100); and for the same reason 1+1+1 (=3), is the log. of 10x 10x 10 (=1000.) Hence it is evident, that twice the log. of any number is the log. of the second power of that number; three times the log. of any number, will be the log. of the third power of that number, &c. Therefore the log. of any power of a number, is equal to as many times the log. of the number as is denoted by that power. Thus, the log. of 101 is 1× 4 (=4); the log. of 1002 is 2× 2(4), &c. Then, to find the second, third, or |