Here, each term, except the first and last, is both antecedent and consequent. When such a series consists of three terms, the second is the MEAN PROPORTIONAL of the other two.

16. Proposition.-The product of the extremes of any proportion is equal to the product of the means. For any proportion, as

abcd,

is the equation of two fractions, and may be written,

Multiplying these equals by the product of the denominators, we have (7)

axd=bxc,

or the product of the extremes equal to the product of the means.

17. Corollary. The square of a mean proportional is equal to the product of the extremes. A mean proportional of two quantities is the square root of their product.

18. Proposition. When the product of two quantities is equal to the product of two others, either two may be the extremes and the other two the means of a proportion. Let ad=bc represent the equal products.

If we divide by b and d, we have

and then divide by a and c, we have

bad: c.

(3d.)

By similar divisions, the student may produce five other arrangements of the same quantities in proportion.

19. Proposition. The order of the terms may be changed without destroying the proportion, so long as the extremes remain extremes, or both become means.

Let a: b::c: d represent the given proportion. Then (16), we have aXd=bXc. Therefore (18), a and d be taken as either the extremes or the means of a new proportion.

may

20. When we say the first term is to the third as the second is to the fourth, the proportion is taken by alternation, as in the second case, Article 18.

When we say the second term is to the first as the fourth is to the third, the proportion is taken inversely, as in the third case.

21. Proposition-Ratios which are equal to the same ratio are equal to each other.

This is a case of the first axiom (6).

22. Proposition.—If two quantities have the same multiplier, the multiples will have the same ratio as the given quantities.

Let a and b represent any two quantities, and m any multiplier. Then the identical equation,

mxaxb=mxbxa,

gives the proportion,

m×a:m×b:: a b (18).

23. Proposition.-In a series of equal ratios, the sum of the antecedents is to the sum of the consequents as any antecedent is to its consequent.

Let a: b ::c:d: e:f gh, etc., represent the

a+c+e+g:b+d+f+h :: a: b.

This is called proportion by COMPOSITION.

24. Proposition. The difference between the first and second terms of a proportion is to the second, as the difference between the third and fourth is to the fourth.

may be written,

a-bb:: c- -d: d.

This is called proportion by DIVISION.

25. Proposition.-If four quantities are in proportion, their same powers are in proportion, also their same roots. Thus, if we have abcd,

then,

also,

a2 b2c2: d2;

:

va: √b:: √c: vd.

These principles are corollaries of the second general axiom (7), since a proportion is an equation.

CHAPTER II.

THE SUBJECT STATED.

26. We know that every material object occupies a portion of space, and has extent and form.

For example, this book occupies a certain space; it has a definite extent, and an exact form. These properties may be considered separate, or abstract from all others. If the book be removed, the space which it had occupied remains, and has these properties, extent and form, and none other.

27. Such a limited portion of space is called a solid. Be careful to distinguish the geometrical solid, which is a portion of space, from the solid body which occupies space.

Solids may be of all the varieties of extent and form that are found in nature or art, or that can be imagined.

28. The limit or boundary which separates a solid from the surrounding space is a surface. A surface is like a solid in having only these two properties, extent and form; but a surface differs from a solid in having no thickness or depth, so that a solid has one kind of extent which a surface has not.

As solids and surfaces have an abstract existence, without material bodies, so two solids may occupy the same space, entirely or partially. For example, the position which has been occupied by a book, may be now occupied by a block of wood. The solids represented

Geom.-2

by the book and block may occupy at once, to some extent, the same space. Their surfaces may meet or cut each other.

29. The limits or boundaries of a surface are lines. The intersection of two surfaces, being the limit of the parts into which each divides the other, is a line.

A line has these two properties only, extent and form; but a surface has one kind of extent which a line has not: a line differs from a surface in the same way that a surface does from a solid. A line has neither thickness nor breadth.

30. The ends or limits of a line are points. The intersections of lines are also points. A point is unlike either lines, surfaces, or solids, in this, that it has neither extent nor form.

31. As one line may be met by any number of others, and a surface cut by any number of others; so a line may have any number of points, and a surface any number of lines and points. And a solid may have any number of intersecting surfaces, with their lines and points.

DEFINITIONS.

32. These considerations have led to the following definitions:

A POINT has only position, without extent.

A LINE has length, without breadth or thickness.
A SURFACE has length and breadth, without thick-

ness.

A SOLID has length, breadth, and thickness.

33. A line may be measured only in one way, or, it may be said a line has only one dimension. A surface has two, and a solid has three dimensions. We can not