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 ab signifies that a equals b.

2. To show that a is less than b, we write a<b; 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 ab 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 bought to be taken from the whole.

5. The sign indicates multiplication; thus a xb represents the product of a multiplied by b. 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 a × (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 + bxa — b c. is the same as above.

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.

ora

7. The square root is indicated as follows 2 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 3a or as.

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

the divisor, thus, when a is to be divided by b; and thus, when 5 is to be divided by 4.

9. Like quantities are such as b, 4b, 8b, and can be incorporated, because one b+ four times b + eight times b makes thirteen times b. Likewise 10 cx + 8 cx can be incorporated, which become 18 cx, and therefore are like quantities.

10. Unlike quantities are such as a+b+c, for a, b, c, would not make three times a, and therefore cannot be incorporated.

11. When we wish to show that 9 × 10 × 12, being numerical, are to be multiplied, we must either retain the x or dot.; but in order to represent a x cx d, we can write them thus a c d. When we write 10 c bx y, the number 10 is called the co-efficient of c b x y, or 10 c the co-efficient of b x y, &c.

12. Let it now be understood that a + b + c can be written in no other form, being unlike quantities; but a+a+a is 3a, which we read three times a. We must also understand that a + a a is equal to twice a minus one a. Similarly a × a xbx c is similar to a2bc, by which is meant a multiplied by a multiplied by b multiplied by c; likewise a 2a b2 indicates a2 X Na x b2. When we write (a + b)2 it indicates the square of a + b, as all within a parenthesis or beneath a vinculum is considered as a single quantity. One is to be considered the co-efficient of a when no figure is expressed, thus a means la, and when no sign is expressed plus is understood, thus a means + a, but when it should be minus it is always expressed. Again,

1

the expression a- 2 means a2.

means

1

a3, because ao always indicates 1, since la becomes 1.

3

2

ADDITION OF ALGEBRAIC QUANTITIES.

This rule means the incorporation of similar quantities having like signs, rather than addition. Ex. gr. + a +a+a— a mean that the quantities with like signs, +a+a+a, are to be added, which make 3a, and that the similar quantity a, with the unlike sign, is to be subtracted from their sum, prefixing to the remainder the sign of the greater quantity; therefore +a+ a + a a becomes + 2 a, or 2 a, as the sign + is understood when no other quantity stands before it.

RULE. Add all the co-efficients of similar quantities having like signs, and if there be unlike signs, add each and subtract the less sum from the greater, prefixing the sign of the latter before the common letter or letters.