PROB. 4. From a point A in a given right line BC, to draw another line perpendicular to it. Fig. 5.

Take AE equal to AB; and from the points B and E, with any radius greater than AB, make the section D; draw DA, which will be perpendicular to BC.

PROB. 5. To make a right angle; or, from the end A of a given right line AB, to raise a perpendicular. Fig. 6.

Take the point C nearer to A than to B; about the center C, with the radius CA, defcribe the circle EAD; through the points E and C draw the line &D; and join DA, which will be perpendicular to AB; and BAD is a right angle.

Otherwife, Fig. 7.

About the center A, with any radius AE, defcribe the arc Enm; fet the fame radius from E to n and from n to m; from the points n and m inake the section D; and join AD, which will be the perpendicular required to be drawn.

PROB. 6. From a given point A, to let fall a perpendicular upon a given infinite straight line BC. Fig.

8.

About the given point A defcribe a circle cutting BC in E and F; from the points E and F make the fection and draw the line AD from A towards n; and AD is the perpendicular required.

n;

DEFIN. 7. A plain triangle is a figure contained by three straight lines.

PROB.

PROB. 7. To make a triangle whofe fides fhall be equal to three given right lines AB, BC, and CA; any two of these being greater than the third. Fig. 9.

Draw the line AB of the given length; and from B, with a radius equal to BC, defcribe an arc; and from Ą, with a radius equal to AC, defcribe another arc cutting the former in C; join AC, CB; and ABC is the triangle required.

Note 1. If the given lines are all equal to one another, the triangle fhall be equilateral.

2. If two of the given lines be equal, the triangle fhall be ifofceles.

3. If the given lines are unequal, the triangle shall be fcalene. The fame operation ferves for making every

kind.

DEFIN. 8. A square is a quadrilateral figure whose fides are equal, and its angles all right ones.

9. A rectangle is a quadrilateral figure, whofe oppofite fides are equal, and its angles all right ones. It is faid to be contained under two lines, namely, the length and breadth.

PROB. 8. To make a fquare on any given right line AB. Fig. 10.

Make the right angle BAD; take AD equal to AB; and from the points D and B, with a radius equal to AB, make the fection C; join DC and BC, and ABCD is a fquare.

PROB. 9. To make a rectangle of two given right lines AB and BC. Fig. 11.

Make the right angle ABC; take AB and BC of the given lengths; from A, with a radius equal to BC, describe an arc; and from C, with a radius equal to AB, describe another arc cutting the former in D; join AD and DC; and ABCD is the rectangle requi

red.

IO.

PROB. 10. To divide a given right lined angle ABC into two equal parts, or to bifect it. Fig. 12.

About the angular point B, with any radius Bd, deferibe the arc de; and from the points d and e make the fection F; join BF, and the Angle ABC is bifected by the right line BF.

N. B. If the angles ABF and CBF are bifected, the given angle will be divided into four equal parts; and each of these being again bifected, it will be divided into eight equal parts; and thus the bifection may be carried as far as you please: But an arch or angle cannot be divided into three, five, or feven equal parts, by Euclid's propofitions. This is to be done by trials,

OF PROPORTION.

DEFIN. 10. When we compare any quantity with another of the fame kind, in order to estimate the magnitude of the one in respect of the other; the comparison is made, either by confidering what part or parts the one is of the other, or how oft the one contains the other.

11. The quantity compared is called the antecedent, and the quantity to which it is compared is called the confequent.

12. The ratio of any two quantities is the number expreffing what part or parts the antecedent is of the confequent, or how often the antecedent contains the confequent. Thus, if any quantity A is compared to another B, the ratio is 1. The ratio of 6 to 12 is, and of 4 to 8 is 4.

A

13. When the ratio between any two quantities is equal to the ratio between any other two, these quantities are proportional.

14. In three quantities A, B, C; if the ratio of A to B be equal to the ratio of B to C, these three quanti• ties are proportional; expreffed thus, As A is to B, fo is B to C, or A: BB: C. Here A and C are called the extreme terms, and B is a mean proportional between A and C. Alfo C is a third proportional to A and B.

15. In four quantities A, B, C, D; if the ratio of A to B be equal to the ratio of C to D, the four quan tities are proportional; or A: B::C: D.

Here A and D are the two extreme terms, and B and C are the means; also D is a fourth proportional to A, B, C.

Examples in Numbers.

1. In the numbers 4, 6, 9, the ratio of 4 to 6 is equal to the ratio of 6 to 9; that is, 4; therefore 4:6::6:9, and 6 is a mean proportional between 4 and 9; alfo 9 is a third proportional to 4 and 6.

16 12

2. In the numbers 16, 32, 12, 24, the ratio of 16 to 32 is equal to the ratio of 12 to 24; that is, 1=1; therefore 16:32::12:24. Here the firft and third terms, viz 16 and 12, are antecedents, and the second and fourth terms, 32 and 24, are confequents.

In four proportional numbers, the product of the two extremes is equal to the product of the two means; Eucl B. 6. prop. 16.; thus, if 16:32::12:24, then 16X24 32X12=384.

When four quantities are proportional, several variations may be made in their order and magnitude, which geometers call Alternation, Inverfion, Compofi= tion, Divifion, and Converfion.

16. Alternate proportion is the comparing of the ancedent with the antecedent, and the confequent with the confequent.

Exam. Suppofe four proportional quantities are represented by the proportional numbers 18: 6:21:7, these will be alternately proportional, viz. as 18:21::6:7.

17. Inverse proportion is when the confequent is made an antecedent, and the antecedent a confequent; that is, if 18621: 7, then, by inverfion, as 6: 18:7:21.

18. Compounded proportion is when the fum of the antecedent and confequent is compared to the confequent. If 18:6:21:7, then, by compofition, as 24:6:28:7.

19. Divided ratio is when the difference between the antecedent and confequent is compared to the confe quent. If 18:6:: 21: 7, then, by division, as 12:6 ::

14:7

Converse