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20. The sign of Addition is an erect cross; thus A+B implies the sum of A and B, and is called A plus B.
21. Subtraction is denoted by a single line; as A-B, which is read A minus B; A—B represents their difference, or the part of A remaining, when a part equal to B has been taken away from it.
In like manner, A-B+C, or A+C-B, signifies that A and C are to be added together, and that B is to be subtracted from their sum. 22. Multiplication is expressed by an oblique cross, by a point, or by simple apposition: thus, AXB, A. B, or AB, signifies that the quantity denoted by A is to be multiplied by the quantity denoted by B. The expression AB should not 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 multiplication of numbers cannot be expressed by simple apposition.
23. When any quantities are enclosed in a parenthesis, or have a line drawn over them, they are considered as one quantity with respect to other symbols: thus, the expression AX(B+C-D), or A×B+C—D, represents the product of A. by the quantity B+C-D. In like manner, (A+B)×(A—B+C), indicates the product of A+B by the quantity A-B+C.
24. The Co-efficient of a quantity is the number prefixed to it: thus, 2AB signifies that the line AB is to be taken 2 times; AB signifies the half of the line AB.
25. Division,' or the ratio of one quantity to another, is usually denoted by placing one of the two quantities over the other, in the form of a fraction:
thus, signifies the ratio or quotient arising from the division of the quantity A by B. In fact, this is division indicated.
26. The Square, Cube, &c. of a quantity, are expressed by placing a small figure at the right hand of the quantity: thus, the square of the line AB is denoted by AB2, the cube of the line AB is designated by AB3; and so on.
27. The Roots of quantities are expressed by means of the radical sign √, with the proper index annexed; thus, the square root of 5 is indicated √5; √(AB) means the square root of the product of A and B, or the mean proportional between them. The roots of quantities are sometimes expressed by means of fractional indices: thus, the cube root of AXBXC may be expressed by A×B×C, or or (AXBXC), and
28. Numbers in a parenthesis, such as (15. 1.), refers back to the number of the proposition and the Book in which it has been announced or demonstrated. The expression (15. 1.) denotes the fifteenth proposition, first book, and so on. In like manner, (3. Ax.) designates the third axiom; (2. Post.) the second postulate; (Def. 3.) the third definition, and so on.
29. The word, therefore, or hence, frequently occurs. Το express either of these words, the sign.. is generally used.
30. If the quotients of two pairs of numbers, or quantities, are equal, the C
quantities are said to be proportional: thus, if; then, A is to B as C to D. And the abbreviations of the proportion is, A : B :: C: D; it is sometimes written A: B⇒C: D.
1. "A POINT is that which has position, but not magnitude*." (See Notes.)
2. A line is length without breadth.
"COROLLARY. The extremities of a line are points; and the intersections "of one line with another are also points."
3. "If two lines are such that they cannot coincide in any two points, with"out coinciding altogether, each of them is called a straight line." "COR. Hence two straight lines cannot inclose a space. Neither can two "straight lines have a common segment; that is, they cannot coincide "in part, without coinciding altogether."
4. A superficies is that which has only length and breadth.
'COR. The extremities of a superficies are lines; and the intersections of one superficies with another are also lines."
5. A plane superficies is that in which any two points being taken, the straight line between them lies wholly in that superficies.
6. A plane rectilineal angle is the inclination of two straight lines to one another, which meet together, but are not in the same straight line.
N. B. 'When several angles are at one point B, any one of them is ex'pressed by three letters, of which the letter that is at the vertex of the angle, that is, at the point in which the straight lines that contain the angle 'meet one another, is put between the other two letters, and one of these 'two is somewhere upon one of those straight lines, and the other upon the 'other line: Thus the angle which is contained by the straight lines, AB, 'CB, is named the angle ABC, or CBA; that which is contained by AB,
*The definitions marked with inverted commas are different from those of Euclid.
'BD, is named the angle ABD, or DBA; and that which is contained by 'BD, CB, is called the angle DBC, or CBD; but, if there be only one an'gle at a point, it may be expressed by a letter placed at that point; as the 'angle at E.'
7. When a straight line standing on another straight line makes the adjacent angles equal to one another, each of the angles is called a right angle; and the straight line which stands on the other, is called a perpendicular to it.
8. An obtuse angle is that which is greater than a right angle.
9. An acute angle is that which is less than a right angle.
10. A figure is that which is enclosed by one or more boundaries.-The word area denotes the quantity of space contained in a figure, without any reference to the nature of the line or lines which bound it.
11. A circle is a plane figure contained by one line, which is called the circumference, and is such that all straight lines drawn from a certain point within the figure to the circumference, are equal to one another.
12. And this point is called the centre of the circle.
13. A diameter of a circle is a straight line drawn through the centre, and terminated both ways by the circumference.
14. A semicircle is the figure contained by a diameter and the part of the circumference cut off by the diameter.
15. Rectilineal figures are those which are contained by straight lines.
16. Trilateral figures, or triangles, by three straight lines.
17. Quadrilateral, by four straight lines.
18. Multilateral figures, or polygons, by more than four straight lines.
19. Of three sided figures, an equilateral triangle is that which has three equal sides.
20. An isosceles triangle is that which has only two sides equal.
21. A scalene triangle is that which has three unequal sides.
24. An acute angled triangle is that which has three acute angles.
25. Of four sided figures, a square is that which has all its sides equal and all its angles right angles.
26. An oblong is that which has all its angles right angles, but has not all its sides equal.
27. A rhombus is that which has all its sides equal, but its angles are not right angles.
28. A rhomboid is that which has its opposite sides equal to one another, but all its sides are not equal, nor its angles right angles.
29. All other four sided figures besides these, are called trapeziums.
30. Parallel straight lines are such as are in the same plane, and which, being produced ever so far both ways, do not meet.
1. LET it be granted that a straight line may be drawn from any one point to any other point.
2. That a terminated straight line may be produced to any length in a straight line.
3. And that a circle may be described from any centre, at any distance from that centre.
1. THINGS which are equal to the same thing are equal to one another.
2. If equals be added to equals, the wholes are equal.
3. If equals be taken from equals, the remainders are equal.
4. If equals be added to unequals, the wholes are unequal.
5. If equals be taken from unequals, the remainders are unequal.
6. Things which are doubles of the same thing, are equal to one another.
7. Things which are halves of the same thing, are equal to one another. 8. Magnitudes which coincide with one another, that is, which exactly fill the same space, are equal to one another.
9. The whole is greater than its part.
10. All right angles are equal to one another.
11. "Two straight lines which intersect one another, cannot be both "rallel to the same straight line."