EUCLID, BOOK XI.

DEF. 1. A straight line is perpendicular or at right angles to a plane, when it makes right angles with every straight line meeting it in that plane.

DEF. 2. A plane is parallel to another plane when they do not meet, though both are indefinitely pro

duced.

DEF. 3. A plane is parallel to a straight line when they do not meet, though both are indefinitely produced.

DEF. 4. A prism is a solid figure contained by two parallel planes, and by a number of other planes all parallel to one straight line, and cutting the first two planes so as to form polygons.

or

The first two planes are called the ends bases of the prism, and the straight line to which all the other planes are parallel is the length of the prism.

The following Lemmas will be taken taken for granted:

LEMMA 3. The arcs which subtend equal angles at the centers of two circles are as the radii of the circles.

Let the two circles be placed so that their centers coincide at C: and so that one of the lines CA containing the angle A Ca in one of the circles coincides with the corresponding line CB in the other circle. Then since the angles

B

A

at the center in the two circles are equal; the other lines containing the angles, namely Ca, Cb, will coincide. And it will be true that Aa Bb :: CA : СВ.

LEMMA 4. The area of a rectangle is equal to the product of the two sides.

If A, B be the two sides, the rectangle is = A × B. Сов. If B be the base and A the altitude of a triangle, the area of the triangle is = 4 × B.

==

LEMMA 5. If a prism be cut by planes perpendicular to its length at different points, the areas of the sections are all similar and equal.

LEMMA 6. The solid content of a prism is equal to the product of its length and of the area of a section perpendicular to the length.

If A be the area of the section and H the length, the solid content is = A × H. In this case, solid contents are measured by the number of times they contain a unit of solid content.

COR. In a uniform prism the weight is as the solid content; hence the weight of any portion of a uniform prism is proportional to its length.

LEMMA 7. If a prism be cut by two planes pass

ing through any point of its length, one of the planes being perpendicular to the length and the other oblique to it;

and if a line be drawn at the point, perpendicular to the oblique section and intercepted by a line perpendicular to the length; the oblique section is to the perpendicular section as the portion of the perpendicular line intercepted is to the portion of the length intercepted.

Let Ll, LM be the perpendicular and the oblique section of the prism, of which the length is QL; LK perpendicular to the section LM, and KH perpendicular to the length QL. Then area LM: area Ll :: KL : HL.

B

26

MECHANICS.

1.

BOOK I. STATICS.

DEFINITIONS AND FUNDAMENTAL NOTIONS.

MECHANICS is the science which treats of the laws of the motion and rest of bodies.

2. Any cause which moves or tends to move a body, or which changes or tends to change its motion, is called FORCE.

3. BODY OF MATTER is anything extended, and possessing the power of resisting the action of force.

A rigid body is one in which the force applied at one part of the body is transferred to another part, the relative positions of the parts of the body not being capable of any change.

4.

All bodies within our observation fall or tend to fall to the earth: and the force which they exert in consequence of this tendency, is called their WEIght.

wEIGHT.

5. Forces may produce either rest or motion in bodies. When forces produce rest, they balance each other; they are in equilibrium; they destroy each other's effects.

6. STATICS is the science which treats of the laws of forces in equilibrium.

7. Two directly opposite forces which balance each other are equal.

Forces are directly opposite when they act in the same straight line in opposite directions.

8.

Forces are capable of addition.

Thus, when

two men pull at a string in the same direction, their forces are added; and when two heavy bodies are put in the same vessel suspended by a string, their weights are added, and are supported by the string.

9. A force is twice as great as a given force, when it is the sum of two others, each equal to the given force; a force is three times as great, when it is the sum of three such forces; and so on.

10. Forces (in Statics) may be measured by the weights which they would support.

11.

The Quantities of Matter of bodies are measured by the proportion of their mechanical effect.

12. The quantities of matter of two bodies are as their weights at the same place.

13. The Density of a body is measured by the quantity of matter contained in a given space.

14. A LEVER is a rigid rod, moveable, in one plane, about a point, which is called the fulcrum or center of motion, by means of forces which tend to turn it round the fulcrum.

15. The portions of the rod between the fulcrum and the points where the forces are applied, are called the arms.

16. When the arms are two portions of the same straight line, the lever is called a straight lever; otherwise it is called a bent lever.

17. The lever is supposed to be without weight, unless the contrary be expressed.

B 2

AXIOMS.

1. In a system which is in equilibrium, there is at every point a reaction equal and opposite to the action.

2. If two equal forces act perpendicularly at the extremities of equal arms of a straight lever to turn it opposite ways, they will keep each other in equilibrium. If AC BC, and P and Q be two equal forces acting perpendicularly on CA and

=

CB at A and B, they will balance each other.

3. If forces keep each other in equilibrium, and if any force be added to one of them, it will preponderate.

4. If two equal weights balance each other upon a horizontal straight lever, the pressure upon the fulcrum is equal to the sum of the weights, whatever be the length of the lever.

If P, Q be two equal weights which balance each other upon the horizontal lever

AB, the pressure upon C is

P+Q.

5. If two equal weights be supported upon a

straight lever on two fulcrums

at equal distances from the

weights, the pressures upon the two fulcrums are together equal to the sum of the weights.

If P, Q be two equal weights which are supported upon the line AB on two fulcrums C, D, so that AC, BD are equal; the pressures upon C, D are together equal to the sum of the weights P+ Q.

6.

On the same suppositions, the pressures on the two fulcrums are equal.