With one foot on O extend the other to the 12th division on the scale you choose to adopt, and apply that distance from A as above directed, and it will give the length proposed.

2. To draw a line that shall represent 35 yards.

Let each primary division be considered as 10 yards, then will each subdivision represent 1 yard; apply the compasses from 3 backwards (to the left) to the 5th subdivision, and 35 subdivisions will be included between the points; apply this from the given point, and draw the line as before.

3. To draw a line equal to 263.

On the diagonal scale, let each primary division represent 100, then will each subdivision represent 10, and the distance which each diagonal slopes or the first parallel will be 1, on the second 2, on the third 3, and so on; therefore for 263 extend from the number 2 backwards to the sixth subdivision, on the third parallel, (viz. the 4th line downwards) and it will be the distance required.

257. To measure any straight line *.

RULE. Extend the compasses from one extremity of the given line to the other, and apply this distance to any convenient scale of equal parts, the number of parts intercepted between the points, will be the length required.

Note. If the sides of a rectilineal figure are to be measured, the same scale must be used for them all; and one scale must be used for each of two or more lines, when their relative length is required to be ascertained".

258. To bisect a given straight line AB.

By the word measure is meant the relative measure of a line, that is, the Jength of that line compared with the length of another line, both being measured from the same scale; if we call the subdivisions of the scale feet or yards, the line will represent a line of as many feet or yards as it contains such subdivisions; to find the absolute measure of a line in yards or feet, we must evidently apply a scale of actual yards or feet to it.

• Any scale of equal parts may be employed for this purpose, but it will be proper to choose one that will bring the proposed figure within the limits you intend it to occupy; every part (viz. every line) of the figure must be mea sured by one scale, and not one line of the figure by one scale, and another Jine by another.

259. From a given point D, in a given straight line AB, to erect

Lay the centre of the protractor on D, and let the 90 on its circumference exactly coincide with the given line; draw the line FD along the radius, and it will be the perpendicular required.

259.B. From a given point F, to let fall a perpendicular to a given straight line AB. See the preceding figure.

RULE I. In AB take any point E, join FE, and bisect it in C, (Art. 258.)

II. From C as a centre with the distance CF or CE, describe

If the points AC and BC be joined, this rule may be proved by Euclid 8.1.

"The proof of this rule depends on Euclid 31. 3. Of the various methods fer erecting a perpendicular, given by writers on Practical Geometry, this is the most simple and easy.

the circle EDF; join FD, and it will be the perpendicular required 3.

260. Through a given point E to draw a straight line parallel to a given straight line AB.

RULE I. Take any point Fin AB, and from E and F as centres, with the distance EF, describe the arcs EG, FH.

II. Take the distance EG in the compasses, and apply it from F to H on the arc FH.

с

E

HD

A

G

III. Through E and H draw the straight line CD, and it will be parallel to AB as was required ".

By the Parallel Ruler.

Lay the ruler so, that the edge of one of its parallels may exactly coincide with the line AB. Holding it steady in that position, move the other parallel up or down until it cut the point E, through which draw a line CED, and it will be parallel to AB.

If E be too near, or too distant for the extent of the ruler, first draw a line parallel to, and at any convenient distance from AB, to which draw a parallel through E as before, and it will be parallel to AB.

261. At a given point A, in a given straight line AB, to make an angle BAC, which shall measure any given number of degrees. RULE I. Extend the compasses from the beginning of the scale of chords (mark

ed C,) to the 60th degree, and from the given point 4, with this distance, describe an arc cutting AB (produced if necessary) in E.

F

B

E

II. Extend the compasses from the beginning of the scale of

This depends on Euclid 31.8.

y Since the arcs EG, HF are equal, the angles EFG, FEH at the centres are equal, (Euclid 27.3.) and therefore AB is parallel to CD, Euclid 27.1.

chords, to the number denoting the measure of the proposed angle, and from E as a centre, with this distance, cut the above arc in the point F.

III. Through F draw the straight line AB, and the angle BAC will be the angle required".

EXAMPLES-1. Let the angle proposed measure 30 degrees.

Having described the arc EF with the radius 60, extend the compasses from the beginning of the scale to 30; lay off this extent from E, and draw a line through the point marked with the compasses, and the angle of 30° will be made.

2. At the point A in AB make an angle measuring 150 degrees".

Here the proposed angle being greater than 90, it will be convenient to take it at twice; lay off 80° first, on EF; then from F, lay off 70° more; draw a line through the extremity of the 70, and it will make with AB an angle of 150 degrees.

BY THE PROTRACTOR.

Lay the central point on A, and the fiducial edge of the radius along AB, so that they exactly coincide; then with the pointer, make a fine dot, opposite the proposed degree (reckoning from the line AB) on the circumference; through A and this dot, draw a straight line, and it will make with AB the angle required.

If the circumference of a circle be divided into 360 equal parts called degrees, one sixth part of the circumference will measure 60 degrees, and its chord will be equal to the radius of the circle (Euclid 15. 4.) ; wherefore, if the first 60 degrees on any scale of chords be taken in the compasses, and a circle be described with that distance as radius, the chords on the scale, will be the proper measure for the chord of every arc of that circumference, as well as for the circumference itself; and since the arc intercepted between the legs of the angle, (being described from the angular point as a centre,) is the measure of the angle it subtends, (Euclid 33.6. Art. 236.) the rule is manifest. By this problem an angle may be made, equal to any given angle.

• To measure, or lay down, an angle greater than 90°, the arc must be taken in the compasses at twice; thus for 100o, take 60° first, and then 40°; or 50° first, and then the remaining 50°, &c. For an arc of 170o take 90° and 80°, or 50o, 50o, and 70o, viz. at three times, &c. &c. If two straight lines cut one another within a circle, their angle of inclination is measured by half the sum of the intercepted arcs; but if they cut without the circle, their angle of inclination is measured by half the difference of the intercepted arcs. See the note on Art. 243,

EXAMPLES. Make at given points, in given straight lines, the following angles, viz. of 20°, 35°, 45°, 58°, 90°, 160°, and 1710.

262. To measure a given angle BAC. See the preceding figure.

RULE I. From the angular point B as a centre, with 60° from the scale of chords as a radius, describe the arc EF, cutting the legs of the given angle (produced if necessary) in E and F.

II. Extend the compasses from E to F, and apply the extent to the scale of chords, so that one point of the compasses be on the beginning of the scale; then the number to which the other point reaches will denote the measure of the given angle .

EXAMPLE. To measure the angle BAC.

Having with the radius 60o described the arc EF, extend the compasses from E to F; then if this extent reaches from the beginning of the scale to 35o, the angle BAC measures 35 degrees,

BY THE PROTRACTOR.

Lay the fiducial edge on AB, so that the central notch may

b The reason of this rule will be evident from the preceding note. An ingenious method of measuring angles, by means of an undivided semicircle, and a pair of compasses, without the assistance of any scale whatever, was proposed by M. De Lagni, in the memoirs of the French Academy of Sciences; some account of his method may be found in Dr. Hutton's Mathematical Dictionary, under the word Goniometry. Thomas Fantet De Lagni was born at Lyons in the 17th century, and died in 1734; he was successively professor royal of Hydrography at Rochford, sub-director of the General Bank at Paris, and associate geometrician and pensioner in the Ancient Academy. De Lagni excelled in Arithmetic, Algebra, and Geometry, sciences which are indebted to him for improvements; he invented a binary Arithmetic, requiring only two figures for all its operations; likewise some convenient approximating theorems for the solution of higher equations, particularly the irreducible case in cubics. He gave a general theorem for the tangents of multiple-arcs, and determined the ratio of the circumference of a circle to its diameter to 120 places, which is the nearest approximation for the purpose, that has been made. Our author was particularly fond of calculating, and it may be truly said of him, that "He felt the ruling passion strong in death;" for on his death bed, when he was apparently insensible, one of his friends asked him, What is the square of 12? to which he immediately replied, 144; we regret, that the last moments of this ingenious man, were not employed on subjects of infinitely greater importance.