SCHOL.-Let AZ be assumed to be parallel to CI, and thence that the two AB and BZ be in a direct line. This is manifest (by Prop. 14), for the angles CBA and CBZ, on each side of the right line CB, are right.

COR. 1.-Two sides of a right angled triangle being given, the remaining side can be found, for it is the square root of the sum or difference cf the squares of the given lines, according as the given sides be about the right angle, or not; if about it, take the sum, if not, take the difference.

COR. 2.-Given any number of squares, to find a square equal to all taken together.

Let the right lines A,

B and C be sides of the given squares, and constructa right angle FDE, and cut off right lines on the legs of it, equal to the given lines A and B, (by Prop. 3), and draw GH, and assume DI and

DK equal to GH, and to the given line C; the square of IK is equal to the squares of A, B and C: for it is equal to the squares of DI and DK (by Prop. 47), but the square of DI is equal to the squares of DG and DH (by Prop. 47), therefore the square of IK is equal to the squares of DK, DH and DG, or of C, B and A, which are equal to DK, DH and DG.

COR. 3. Given two unequal right lines AB and BC, to find a right line, whose square is equal to the excess of the square of the greater given line, above the square of the less.

Produce either of the given lines AB, that the produced BD may be equal to the other BC (by Prop. 3), from the centre B, with the greater given line as an interval, describe a semicircle AEF; through D draw DE perpendicular to AD, it will be the right line sought for. For draw BE. The square of BE, or of BA, it

being equal to BE, is equal to the squares of BD and DE (by Prop. 47); if therefore the square of BD be taken away from the square of BA, the remainder will be equal to the square of DE.

COR. 4.-If from any angle of a triangle ABC, a perpendicular be let fall on the opposite side, the difference of the squares of the sides AB and BC about that angle, will be equal to the difference of the squares of the segments, AD and DC, of the side on which the perpendicular falls.

square

For the of the side AB is equal to the squares of AD and DB (by Prop. 47), and the square of BC is equal to the squares of BD and DC (by Prop. 47), therefore the difference of the squares of AB and BC, is equal to the difference between the sum of the of AD and DB, and the sum of the squares of CD and DB (by Ax. 3); or taking away the common square of DB, is equal to the difference between the squares of AD and DC.

COR. 5.-The excesses

of the squares of each of the sides AB and BC, above the squares of each of the conterminous segments, AD and DC, of the A remaining side, are equal.

D

D A

squares

For it appears that the excess of each is equal to the square of the perpendicular (by Prop. 47).

PROPOSITION XLVIII. THEOREM.

If the square of one side (AC) of a triangle (ABC) be equal to the squares described upon the remaining sides (AB and BC), the angle (ABC) opposite to that side, will be a right angle. To AB one of the sides about the angle ABC, and at the extremity of it B, draw the perpendicular BD (by Schol. Prop. 11), and equal to the other side BC (by Prop. 3), and join AD. The square of AD is equal to the

squares of AB and BD (by Prop. 47), or to the squares of AB and BC, being equal to BD (by Constr.); but the squares of AB and BC are equal to the square of AC (by Hypoth.); therefore the square of AD is equal to the square of AC, and therefore the right lines themselves AD and AC are equal (by Cor. 2, Prop. 46); but DB and BC are also equal, and the side AB is common to both triangles, and therefore the angle ABC is equal to the angle ABD (by Prop. 8); but ABD is a right angle (by Constr.), therefore ABC is also a right angle.

QUESTIONS REFERRING TO THE FIRST BOOK.

What is a finite right line?

How do you bisect a given finite right line?
What are equal in an isosceles triangle?

If two sides of a triangle, and the angle contained by them, be equal to two sides and the contained angle of another, what will be also equal?

By what proposition can you divide an angle into 4 equal parts, 8, 16, &c.?

If one side of a triangle be produced, to what will the external angle be equal?

If you subtract the sum of two angles of a triangle from two right angles, to what will the remainder be equal?

ADDITIONAL DEFINITIONS,

OR EXPLANATION OF TERMS.

1. Geometry is a science which has for its object, the

measurement of extent.

2. Extent has three dimensions, length, breadth, and height.

3. The general name of proposition is indifferently ascribed to theorems, problems, and lemmas.

4. An axiom is a proposition, evident by itself.

5. A theorem is a truth which becomes evident by means of a reasoning called demonstration.

6. A problem is a question proposed, which requires a solution.

7. A lemma is a truth employed subsidiarily for the demonstration of a theorem, or the solution of a problem.

8. A corollary is a consequence deduced from one or more propositions.

9. A scholium is a remark on one or more preceding propositions, tending to make known their connection, utility, restriction, or extension.

10. Hypothesis is a supposition, made either in the enunciation, or in the course of a demonstration.

11. The enunciation of a proposition, is that part of it which gives a distinct notion of what we wish to signify or perform, ex. gr.-To cut a given right line into two equal parts, is the enunciation of the 10th of the 1st Book.

12. We say that lines are conterminous, when terminating in the same point; and homologous when having the same proportion as that of lines similarly situated.

DEFINITIONS OF ALGEBRA.

1. The mark = is the sign of equality; thus the expression a=b signifies that a equals b.

2. To show that a is less than b, we write ab; and to show that a is greater than b, we write a>b. 3. The positive sign + is called plus; it indicates addition.

4. The negative sign is called minus; it indicates subtraction: thus a + b represents the sum of the quantities a and b ; and a- b represents their difference or that which remains in taking b from a; in the same way a b+c, or a + c b, signifies that a and c ought to be joined together, and that 6 ought to be taken from the whole.

5. The sign x indicates multiplication; thus a x b represents the product of a multiplied by 6. We also indicate multiplication by a point, as a. b, or without a point, as a b, both of which indicate the same thing with a x b. The expression ax (b+c-d) represents the product of a by the quantity b + c-d. If it be necessary to multiply a + b by a b+c, we indicate the product thus (a+b) x (a − b + c); all × that which is contained within the parenthesis is considered as a single quantity. A vinculum is frequently used instead of the parenthesis, thus a + bx a +c. is the same as above.

b

6. The square of a is indicated by a2, being the same as a xa, or the second power of a; other powers are shown in the same manner, by placing a figure according to the power, as at a.

7. The square root is indicated as follows 2a ora or a, any one of which shows the square root of a; other roots are similarly represented, all the difference being that of the figure, which is called the Index; the cube or third root is thus indicated 3√ a or a3.

8. This mark signifies division, but the usual mode of expression in Algebra is by placing the dividend above