Line of Lines exemplified.
To divide a line of 5 inches into three equal parts.—Set the index to three on the scale of lines, then from any scale of equal parts take the extent of 5 between the longer legs, and the distance between the shorter will be one-third of that between the longer.
Line of Circles exemplified.
To inscribe an enneagon, or regular polygon of nine sides, in a circle.—Let the radius of a circle be 3 inches, it is required to inscribe therein an enneagon. Set the index to the ninth division, open the compasses and take the extent of the radius, 3 inches, between the longer legs ; then will the distance between the shorter be a side of the required polygon. And thus for any other number marked on these compasses.
Line of Planes exemplified.
To find the square root of a given number by the propor tional compasses ; say, that of four.—Shut the compasses and unscrew the nut, slide the centre along the groove till the index points to the number 4 upon the line of planes. Open the compasses, and from any scale of equal parts on the plane scale, or elsewhere, take 4 between the points of the longer legs; apply the points of the shorter legs upon the same scale, and the distance between them will be equal to the square root of the distance between the points of the longer legs, which in this case is 2, the square root of the given number 4.
Likewise, if the index be set to 9, the root of 9 will be 3.
To find a mean proportion between two given numbers.— Required the mean proportion between 2 and The way to find a mean proportion between two numbers is, to multi ply them together, and extract the square root of the product. Therefore open the compasses with the index set against 9, the product of the two given numbers, till the distance of the longer legs be equal to 9. taken from some settle of equal parts; then the distance between the points of the shorter lees will be equal to 3 of those parts, %% hich is the mean proportion to 2 and 41. Fur as : 3 : : 3 :41.
The use of the Line c?/ Solids exempli: To extract the cube root of a given number.—To find the cube root of S by the line or solids, shut the compasses, unscrew the nut. move the centre along the groove till the index points to 8 on the line of solids; ()pen the compasses. and take S between the long points from any scale of ;vial pacts; then the distance between the shorter legs will lie the cube root of the given number, which in this example is 2.
Again. iin by reversion, if the index is set to 2 on the line of lines, take by the shorter legs of the compass, then apply the longer legs to the scale, and it will Iii' found that they extend to 4. Then suppose the square of 3 to be wanted ;
fix the index to 3 ill the line of lines, take 3 by the shorter and by applying the longer points to the scale, it will f curd that. 9 !cuts ii ill be contained between thein, which is the square of 3.
The same may be also pointed out by the index itself, without referring to any scale. nut only with respect to the squares, but also of the cubes. Thus let it be required to find the square of 3 ; set the index to 3 on the line of' lines, and turn the other side of the compasses, where the index will be found, against 9. Again, if the index be set to 2 on the flue of lines, it will stand against 4 on the line of planes. and against S on the line of solids. This is the foundation of the construction of the line of planes and the line of solids : the line of lines being first constructed, the others will easily follow ; since the planes are only the squares, and the solids the cubes, of the distances or numbers on the line of lines. The proportional compasses might theref ire he made a very useful arithmetical instrument, provided each of the distances were !graduated, and the compasses sufficiently long to give the operation correctness.
The use of the protractor is to lay down an angle of any number of degrees. Thus, let it be required to lay down an angle of 25 degrees on the line A 13, Figure 16 : supposing the angular point to commence at A, lay the Pantry of the semicircle, which is shown by a short line, to the point A, and bring the diameter or edge upon the line; then mark the paper at 25 in the circumference, at c ; then a line drawn from A to c will firm an angle c A B, .)r 13 A c, degrees, generally malked thus 25°. Again, let it be required to make an angle with A n, Figure 17, equal to 90°, or to con stitute a right angle at the point B : bring the diameter or straight edge of the instrument upon the line A tt, and the centre upon the stint A ; then at the point 90, in the middle of the semi-eircumferonee, tn.ike a inark upon the paper at c, and join c B; then will A B c be the right angle required. Lastly, let it be required to find the quantity of any given angle : In the centre of the instrument upon the angular point• and the straight edge upon one of the lines, and the number against the other line will show the degrees contained in both.