PROBLEM 3. To draw a perpendicular on a given line AB, at any given point C in that line.—11 E. 1, or 7 D. 5.

To do this mechanically, take a little square made in wood, or brafs, like a carC.32 penter's square, and place it on the line AB

at the point C, as represented by the shaded fquare, draw the perpendicular CD by the fide of it, and it is done; if you have not a fquare, the common little fcale, which is generally given in a pocket-cafe of inftruments, will answer the purpose, for small fchemes.

Corollary. To know if the square be true, turn it, as represented by the unshaded one C.31 FCD, and then, if both ways will form the fame right line, the fquare is true; if not, the angle formed between the two lines will be double the error.

C. 32

PROBLEM 4. To draw a perpendicular to a given line AB from any point D, without it. 12 E. 1, or 8 D. 1.

Slide the fquare along the line AB, till it touches the point D, then draw the line EC, and it will be the perpendicular required.

PROBLEM 5. To draw a line parallel to

another

another given line, at a given diftance.-13

D. 5·

For this purpose there is generally (but not always) in a pocket-cafe of inftruments a parallel ruler. Euclid's method of drawing parallel lines is true in theory but not in practice. The eafieft and beft methods, if you have not a parallel ruler, are those defcribed in this and the next Problem.

Let AB be the given line, HG the given C.33 distance; take the distance HG in your com- F. 1. paffes, and fetting one point in any convenient place C, with the other describe an arc ab; then removing the compasses, and placing one point at a convenient distance from C (the further the more accurate) viz. in D, with the fame extent describe the arc cd; then by the fide of a ruler draw the line EF to touch but not cut the arcs, and it is manifeftly the parallel line required.

PROBLEM 6. To draw a line through a given point C, parallel to a given line AB.31 E. 1, or 12 D. 5.

This is readily done by a parallel ruler, or without it, thus: With your compaffes take the nearest diftance between the point C. 33 C, and the line AB, which may be known F. 2.

C.34

by defcribing the arc ab; if it exactly touches the line AB, you have taken a right diftance; if not, you have taken too little or too much; when you have the true distance, placing one point of the compaffes on a convenient place E, in the line AB, defcribe the arc cd; then from C draw a line CD, to touch the arc cd; and it is evidently the line required.

PROBLEM 7. To make an angle equal to a given angle.-23 E. 1, or Problem 9 D. 5.

Let A be the given angle: Draw a line DE; about A and D, with any radius, defcribe the arcs ab, cd, then take with the compaffes the distance between the two points a, b, which fet off from d to c, on the arc de, and through the point c draw the line DF; then will the angle EDF be equal to the given angle BAC.

PROBLEM 8. To make an angle equal to any number of degrees.

For this purpose in the pocket-cafes is generally a femicircle in brass, named a protractor, divided into 180 degrees, and numbered forwards and backwards.

C.35 Figure 1 represents the manner of laying the protractor in the center at the point C,

and the diameter on the line AB; then with the point of a needle make a prick at the number of degrees to be set off; then through d draw the line CD, and the angle DCB is that required.

But if you have not a protractor, then from the line of chords on your scale (which you have in every case of instruments) take off 60 degrees in your compaffes, and setting one point in C, figure 2, defcribe the arc db, then take off from your scale the number of degrees you are to make your angle equal to, and putting one point of the compaffes in b, fet off on the arc bd that extent; draw the line CD, and the angle DCB will be that required.

N. B. Generally there is a needle for pricking off the degrees in the handle of your drawing pen. Surveyors have commonly a whole circle protractor, as being more convenient.

We have now given all the Problems of this nature, that are neceffary for our prefent purpofe; and therefore must refer those who are defirous of feeing a greater variety of useful Problems, with their demonftrations, to our Geometrician.

BOOK V, SECTION II.

We shall now proceed to give fome Problems of a different kind, defigned to shew a few instances of the ufes of Geometry in taking heights, and diftances, and other practical purposes.

ALTIMETRY,

Or taking Heights of Objects.

PROBLEM 9. To find the height of an object by the fhadow of a pole; made either by the fun or moon.

Take the board, and screw on the pillar, and put one end of the brass wire upright in the hole, to represent the pole; then, to make it more fimilar to real practice, place the board in the funfhine, or if by night, before a high candle, that the shadow both of the pole and pillar may be caft on the board, which represents the ground. Then measure the shadow of the pole, and the fhadow of the pillar, also the height of the pole, and Theorem 10 comes in to our aid; C. 36 for let BC in the scheme represent the height of the pole, AB the shadow of the pole, EF

the