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For, the first analogy gives

the second gives





AD: EH=BC: FG by multiplication,


Hence, Theor. I., we have

AE: BF:: CG: DH.

And the same reasoning would extend to any number of analogies.

Cor. 1. If the second analogy were the same as the first, we should have A2: B2:: C2: D2; hence the squares of proportional numbers are proportional. The same is evidently true of the cubes or any other powers.

Cor. 2. Suppose we have the continued proportion A: B:: B:C::C:D; then,

First. Having A:B::B: C,

and A: B:: A: B,

we shall have

A2: B2: BA: BC;

or (Cor. 2. Theor. 11.) A2: B2 :: A: C.

Hence in a continued proportion, the first is to the third as the square of the first is to the square of the second. The ratio which A bears to C is sometimes called the duplicate of that which it bears to B.

Secondly. Having A: B:: B: C,

A:B::C: D,

and A: B:: A: B,

we shall have A3: B3:: BCA: BCD;

or (Cor. 2. Theor. I.) A3: B3 :: A: D.

Hence in continued proportionals, the first is to the fourth, as the cube of the first is to the cube of the second. The ratio A3: B3, or A: D, is sometimes called the triplicate of A: B; A1: B the quadruplicate; and so on. The law which continued proportionals observe in regard to such ratios is now apparent.

By means of these Theorems, and their Corollaries, it is easy to demonstrate, or even to discover, all the most important facts connected with the doctrine of proportion. The facts given here will enable the student to go through these Elements, without any obstruction on that head.





I. GEOMETRY is the science which has for its object the measurement of space.

Space has three dimensions, length, breadth, and height.
II. A line is length without breadth.

The extremities of a line are called points: a point, therefore, occupies no space.

III. A straight line is the shortest distance from one point to another.

IV. Every line, which is not straight, or composed of straight lines, is a curve line.

Thus, AB is a straight line; ACDB is a broken line, or one composed of straight lines; and AEB is a curve line.

V. A surface is that which has length and breadth, without height or thickness.

VI. A plane is a surface, in which, if two points be assumed at will, and connected by a straight line, that line will lie wholly in the surface.

VII. Every surface, which is not plane, or composed of plane surfaces, is a curve surface.

VIII. A solid or body is that which combines all the three dimensions of space.

IX. When two straight lines, AB, AC, meet together, the quantity, greater or less, by which they are separated from each other in regard to their position, is called an angle; the point of intersection A is the vertex of the A angle; the lines AB, AC, are its sides.


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The angle is sometimes designated simply by the letter at the vertex A; sometimes by three letters BAC or CAB, the letter at the vertex being always placed in the middle.

Angles, like all other quantities, are susceptible of addition, subtraction, multiplication, and division. Thus the angle DCE (see Fig. to Prop. 4.) is the sum of the two angles, DCB, BCE; and the angle DCB is the difference of the two angles DCE, BCE.

X. When a straight line AB meets another straight line CD, so as to make the adjacent angles BAC, BAD equal to each other, each of those angles is called a right angle; and the line AB is said to be perpendicular to


XI. Every angle BAC, less than D right angle, is an acute angle; every angle DEF, greater than a right angle, is an obtuse angle.


XII. Two lines are said to be parallel, when, being situated in the same plane, they cannot meet, how far soever both of them be produced.

XIII. A plane figure is a plane terminated on all sides by lines.

If the lines are straight, the space they enclose is called a rectitioneal figure or polygon, and the lines themselves taken together form the contour or perimeter of the polygon.


XIV. The polygon of three sides, the simplest of all, is called a triangle; that of four sides, a quadilateral; that of five, a pentagon; that of six, a hexagon; and so on.


XV. An equilateral triangle is one which has its three sides equal; an isosceles triangle, one which has two of its sides equal; a scalene triangle, one which has its three sides unequal.

XVI. A right-angled triangle is one which has a right angle. The side opposite the right angle is called the hypotenuse. Thus, ABC is a triangle right-angled at A; the side BC is its hypotenuse.


XVII. Among quadrilaterals, we distinguish :

The square, which has its sides equal, and its angles right. (See Prop. 28. I.)

The rectangle, which has its angles right, without having its sides equal. (See the same Prop.)

The parallelogram, or rhomboid, which has its opposite sides parallel.

The lozenge, or rhombus, which has its sides equal, without having its angles right.

And, lastly, the trapezium, only two of whose sides are parallel.

XVIII. A diagonal is a line which joins the vertices of two angles not adjacent to each other. AC, in the diagram of Prop. 28, is a diagonal.

XIX. An equilateral polygon is one which has all its sides equal; an equiangular polygon, one which has all its angles equal.

XX. Two polygons are mutually equilateral, when they have their sides equal each to each, and placed in the same order; that is to say, when following their perimeters in the same direction, the first side of the one is equal to the first side of the other, the second of the one to the second of the other, the third to the third, and so on. The phrase, mutually equiangular, has a corresponding signification.

In both cases, the equal sides, or the equal angles, are named homologous sides or angles.

Note. In the first four books we shall treat exclusively of plane figures, that is, of figures traced on a plane surface.

Explanation of Terms and Signs.

An axiom is a self-evident proposition.

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

A problem is a question proposed, which requires a solution. A lemma is a subsidiary truth, employed for the demonstration of a theorem, or the solution of a problem.

The common name, proposition, is applied indifferently to theorems, problems, and lemmas.

A corollary is an obvious consequence deduced from one or several propositions.

A scholium is a remark on one or several preceding propositions, which tends to point out their connexion, their use, their restriction, or their extension.

An hypothesis is a supposition, made either in the enunciation of a proposition, or in the course of a demonstration.

The sign is the sign of equality; thus, the expression A=B, signifies that A is equal to B.

To signify that A is smaller than B, the expression A≤B is used.

To signify that A is greater than B, the expression AB is used.

The sign is pronounced plus: it indicates addition.

The sign is pronounced minus: it indicates subtraction. Thus, A+B represents the sum of the quantities A and B; A-B represents their difference, or what remains after B is taken away from A; and A-B+C, or A+ C-B, signifies, that A and C are to be added together, and that B is to be deducted from the whole.

The sign indicates multiplication; thus AxB represents the product of A and B. Instead of the sign x, 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 by 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 A × (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 enclosed within a parenthesis being considered as a single quantity.

A number placed before a line, or a quantity, serves as a multiplier to that line or quantity; thus, 3 AB signifies that the

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