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29. An hypothesis is a supposition, made either in the enunciation of a proposition, or in the course of a demonstration.
30. Equality is denoted by the sign which is read, "is equal to," or "equals." Thus the expression A=B, signifies that A is equal to B.
31. To signify that A is less than B, the character < is used; thus A <B.
32. To signify that A is greater than B, the character > is used; thus A> B. In each case the quantity towards which the character opens,is greater than the other.
33. Addition is denoted by the sign +, which is called plus. Thus A+B, represents the sum of A and B, and is read, "A plus B."
34. Subtraction is denoted by the sign which is called minus. Thus A-B, represents their differences, or what remains after B is taken from A, and is read, “ A minus B."
35. Multiplication is denoted by the sign : thus, A×B represents the product of A and B. Instead of the sign ×, a point (.) is sometimes employed; thus, A.B is the same thing as A x B. The same product is also designated without any intermediate sign between the letters: as AB. But this expression cannot be employed, when there is any danger of confounding it with that of the line AB, the distance between the points A and B.
The expression Ax (B+C-D) represents the product of A by the quantity B+C-D. If A+B were to be multiplied by A- B+C, the product would be indicated thus, (A+B) × (A-B+C). Whatever is inclosed within a parenthesis is considered as a single quantity.
36. A number placed before a line, or a quantity, serves as a multiplier to that line or quantity, and is called a coefficient thus, 3 AB signifies that the line AB is taken three times; A signifies the half of the angle A.
37. The square of the line AB is designated by AB'; its cube by AB3. The small figure placed at the right hand is called the index or exponent. What is meant by the square and the cube of a line will be explained in its proper place. 38. The square root is denoted by the character √, which is called the radical sign. Thus √2 signifies the square-root of 2; and VAX B signifies the square-root of the product of A and B.
1. Quantities which are equal to the same quantity, are equal to each other.
2. If the same or equal quantities are added to equals, the sums will be equal.
3. If the same or equal quantities are subtracted from equals, the remainders will be equal.
4. If the same or equal quantities are added to unequals, the sums will be unequal.
5. If the same or equal quantities are subtracted from unequals, the remainders will be unequal.
6. If equal quantities are multiplied by the same or equal quantities, the products will be equal.
7. If equal quantities are divided by the same or equal quantities, the quotients will be equal.
8. If a quantity is both multiplied and divided by the same or an equal quantity, its value will not be altered.
9. The whole of a quantity is greater than a part.
10. The whole of a quantity is equal to the sum of all its parts. 11. All right-angles are equal to each other.
12. From one point to another only one straight line can be drawn.
13. Magnitudes, whether lines, surfaces, or solids, which coincide throughout their whole extent, i. e. which exactly fill the same space, are equal.
Every straight line CD, which meets another straight line AB, makes with it two adjacent angles ACD, BCD, the sum of which is equal to two right-angles.
At the point C erect CE perpendicular to AB. Then will the sum of the two angles ACD, DCB, be equal to the sum of the two angles ACE, ECB. (Ax. 13.) But ACE, ECB are two right-angles; (Def. 12;) hence the A sum of the two angles ACD, DCB, must be equal to two right-angles.
In the same manner it may be proved that every straight line which meets another, makes with it two adjacent angles, whose sum is equal to two right-angles. Hence, universally,
The sum of the two adjacent angles formed by the meeting of one straight line with another, is equal to two right-angles.
Cor. 1. If one of the angles ACD, BCD, is a rightangle, the other must be a right-angle also.
Cor. 2. If the line DE is perpendicular to AB, in like manner AB will be perpendicular to DE.
For, since DE is perpendicular to AB, the angle ACD must be equal to its adjacent one DCB, and both of them must be right-angles. (Def. 12.) But since ACD is a right-angle, its adja
cent one ACE must also be a right-angle; (Cor. 1;) hence the angle ACE=ACD; (Ax. 11;) consequently AB is perpendicular to DE.
PROPOSITION II. THEOREM.
Two straight lines, which have two points A and B common to both, coincide with each other throughout their whole extent, and form one and the same straight line.
First, since the points A and B are common to both lines, it is evident that the two lines must coincide entirely between A and B ; for otherwise there would be two straight lines between A and B, which is impossible. (Ax. 12.) Suppose, however, that on being produced, those lines begin to separate at C, the one becoming CD, the other CE. From the point C draw the line CF, making with CA the right-angle ACF. Now since ACD is a straight line, the angle FCD will be a right-angle; (Prop. 1. Cor. 1;) and since ACE is a straight line, the angle FCE will likewise be a rightangle; consequently, FCE=FCD. (Ax. 11.) But the part FCE cannot be equal to the whole FCD. (Ax. 9.) Hence the straight lines which have two points A and B common, cannot separate in any point of their production; therefore they form one and the same straight line. Hence, universally,
Any two straight lines which have two points common, must coincide throughout, and form one and the same straight line.
PROPOSITION III. THEOREM.
If the two adjacent angles, ACD, DCB, are together equal to two right-angles, the two exterior sides, AC, CB, will lie in the same straight line.
For, if CB is not the extension of AC, let CE be that extension: then the line ACE being straight, the sum of the angles ACD, DCE, will be
equal to two right-angles. (Prop. 1.) But, by hypothesis, the sum of the angles ACD, DCB, is also equal to two rightangles: therefore, ACD+DCE=ACD+DCB; (Ax. 1;) and taking away the angle ACD from each, there remains DCB=DCE, i. e. a part equal to the whole, which is impossible; (Ax. 9;) therefore, CB is the extension of AC. In like manner it may be proved universally,
If any two straight lines meet another straight line at the same point and make the sum of the two adjacent angles equal to two right-angles, these two straight lines will form one continued straight line.
The two straight lines AB, DE, which intersect each other at C, make the opposite or vertical angles, ACD, BCE equal.
For, since DE and AC are A
straight lines, the sum of the angles ACD, ACE, is equal
to two right-angles; (Prop. D
1;) and since AB and CE
are straight lines, the sum of the angles ACE, BCE, is also equal to two right-angles; hence the sum ACD+ACE=