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To measure a given angle, ABC. fig. 5.
If the lines which include the angle, be not as, long as the chord, of 60 on your scale, produce them to that or a greater length, and between them so produced, with the chord of 60 from B, describe the arc ed; which distance e d, measured on the same line of chords, gives the quantity of the angle BAC 48 degrees, as required; this is plain from def. 19.
To make a triangle BCE equal to a given quadrilateral figure ABCD. fig. 6.
Draw the diagonal AC, and parallel to it (by prob. 8.) DE, meeting AB produced in E; then draw CE, and ECB will be the triangle required.
For the triangles ADC, AEC being upon the same base AC, and under the same parallel ED (by cor. to theo. 13.) will be equal, therefore if ABC be added to each, then ABCD=BEC.
To make a triangle DFH, equal to a given fivesided figure ABCDE. fig. 7.
Draw DA and DB, and also EH and CF, parallel to them by prob. 8.) meeting AB produced in
H and F; then draw DH, DF, and the triangle HDF is the one required.
For the triangle DEA-DHA, and DBCDFB (by cor. 10. theo. 13.) therefore by adding these equations, DEA + DBC=DHA + DFB if to each of these ADB be added; then DEA
+ ADB DBC = ABCDE =
ABCDE = (DHA + ABD + DFB) = DHF.
To project the line of chords, sines, tangents, and secants, to any radius. fig. 8.
On the line AB, let a semicircle ADB be described ; let CD be drawn perpendicular to the centre C, and the tangent BE perpendicular to the end of the diameter ; let the quadrants, AD, DB, be each divided into 9 equal parts, every one of which will be 10 degrees; if then from the centre C, lines be drawn through 10, 20, 30, 40, &c. the divisions of the quadrant BD, and continued to BF, we shall there have the tangents of 10, 20, 30, 40, &c. and the secants C 10, C20, C30, &c. are transferred to the line CD, produced by describing the arcs 10, 10 : 20, 20 : 30, 30, &c. If from 10, 20, 30, &c. the divisions of the quadrant BD, there be let fall perpendiculars, let these be transferred to the radius CD, and we shall have the sines of 10, 20, 30, &c. and if from A .we describe the arcs 10, 10 : 20, 20 : 30, 30, &c. from every division of the arc AD; we shall have a line of chords. The same way we may have the sine, tangent, &c. to every single degree on
the quadrant, by subdividing every of the 9 former divisions into 10 equal parts. By this method the sines, tangents, &c. may be drawn to any radius; and if, after they be transferred to lines on a rule, we shall have the scales of sines, tangents, &c. ready for use.
Concerning Scales of equal parts.
If an inch be divided into any assigned number of equal parts, and if these parts be continued on in a right line, and if the last of them be subdivided into 10 equal parts, and thence if the first divisions be numbered with 1, 2, 3, 4, &c. as far as the ruler upon which they are transferred will sadmit, the scale is completed.
These numbers, 1, 2, 3, 4, &c. usually stand for 10, 20, 30, 40, &c. and every one of the subdivisions is called 1: but if the numbers 1, 2, 3, 4, &c. be called 100, 200, 300, 400, &c. then every one of the subdivisions will be 10, and the units. must be guessed at.
On one side of most surveying scales, there are lines or scales, marked at the end with 50, 45, 40, 35, 30, 25, 20, 15, 10, and sometimes with other numbers; these are scales of so many parts to an inch (whether of feet, yards, perches, or miles) as the respective number at the end of each expresses; but in the surveying way, they are counted to be so many perches to an inch, and sometimes so many feet to an inch.
On the contrary side there are two scales, one of 10, and the other of 20; or one of 100, and the
other of 200; or one of 1000, and the other of 2000 parts to an inch, diagonally divided ; (a view of the scale will make all easy ;) the first of these surveyors call a scale of 10, and the other a scale of 20 perches to an inch ; and are thus counted : every large division is 10, every one of the subdivisions is i, and every one downwards is one tenth of a perch ; or sometimes thus, every large division is called 100, every subdivision 10, and every one downwards 1 : or again, frequently by navigators, every large division is called 1000, every subdivision 100, every one downwards 10, and the tenth part of the distance between the lines 1.
Hence it is easy to measure the length of any line, knowing the scale by which it was laid down; and on the contrary, to set off any given distance from any scale.
F to a series of numbers in geometrical progres
sion, whose common ratio is 10, and first term l; we annex another series of numbers in arithmetical progression, whose first term is 0, and common difference P: these latter numbers will be the logarithms of the former.
1 10 100 1000 10000
If several geometrical means be taken, and the like number of arithmetical ones, to the corresponding numbers, the latter will be the logarithms of the former.
The nature therefore of logarithms is such, that addition of them answers to the multiplication of their corresponding numbers; and subtraction to division; that is, when two numbers proposed are to be multiplied into each other, if we take the logarithms answering to those numbers, and add them together, the sum will be the logarithm answering to the number, which is the product of the two proposed numbers.