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certain specified properties, may be constructed by & combination of those elementary constructions to which we are limited by the postulates. In a theorem it is required to demonstrate by the direct or indirect application of those elementary truths which are laid down' in th axioms that figures having certain specified properties have also certain other properties. (Some remarks, in further explanation, will be most conveniently appended to the second and fifth propositions.)
Let it be observed, that the object of the problem is not to furnish us with practical rules for drawing figures on paper, &c., having the required properties (with such an approach to accuracy as the nature of the means employed would admit of), for, by the aid of compasses, and other mathematical instruments, this may be effected in a more convenient manner. Nor are they designed to help us in conceiving of the figures which it is their object to produce. For, if we wish to have a square, or an equilateral A, to reason about, there is nothing to hinder us from conceiving the figure at once. The object of the problems is purely thạt described in the preceding paragraph. Again, it is not the object of the theorems to convince us of the truth of what is proved in them. In a large number of cases, most people would never imagine it possible to doubt the truth of these propositions. The object is to show by what intermediate steps the truth to be proved must be connected with the primary axioms and definitions with which we start. Most beginners are surprised and tired by what appears to them the unnecessary tediousness of some of the demonstrations. This is only because they do not understand the real object of those demonstrations.
All right angles are equal. This
may be proved as follows: Let A B C and DEF be any two right Zs. Pro
duce the line C B to any point M, and the line FE
to any point H.
Suppose the figure MABC to be applied to the figure H D EF, so that the point B
may coincide with the point E, and the line M C lie along the line H F.
Let the figures lie on the same side of the common line :
Then the line B A must lie along the line ED; for if it do not, but lie in some other direction, such as E K, inasmuch as H EK and KEF would be the same as the <s MBA and A BC, we should have the <s H EK and KEF equal to one another (see the definition of right <s); and also, H ED and DEF equal to one another.
But as the 2 HEK is larger than the < HED, it must be larger than the < DEF, which is equal to HED.
But DEF is larger than KEF; much more, therefore, must HEK be larger than KEF, or HEK and KEF are unequal to each other. But this contradicts what was before known, that HEK and KEF, or (what are the same Zs) MBA and A B C are equal to each other.
Now a supposition, which of necessity leads to a contradiction of what is known to be true, must be a false supposition.
The supposition which led us to the above contradiction was, that the line B A did not coincide with the line E D when the figures were placed upon one another as described.
This supposition, then, is a false one. But, to show that it is false to suppose that BA and ED do not coincide, is equivalent to proving that it it true that
they do coincide, or, in other words, that the < ABC is equal to the ZDEF.
Problem, On a given finite right line to construct an equilateral triangle.*
The following postulates and axioms are made
use of in the construction and demonstration in
this proposition :1. Postulate I. A straight line may be drawn from
a given point to any other given point. 2. Postulate III. A circle may be described with a
given centre, and at any distance from that
centre. 3. Axiom I. Magnitudes which are equal to the
same, are equal to one another. 4. Axiom C. If part of one bounded figure (and
part only) be inside another bounded figure, the boundaries of those two figures must intersect
one another in two points at least. 5. Definition F. All radii of the same circle are
equal to one another. Let A B be the given finite right line.
With the centre A, and at the distance A B, describe the circle C BE. (Post. III.)
* By speaking of the line as “ given," it is meant that its position is given. By speaking of it as finite,” it is meant that its length is given. A is said to be constructed on a given finite right line when that line forms one of the sides of the A. Respecting the use of the marks A and Z, see the note on p. 9.
CAUTION.— Whenever any construction is required in the course of a proposition, the learner should never be allowed to draw the whole figure to begin with, but should be made to draw it step by step, as directed in the text. This is of vital importance.
With the centre B, and at the distance BA,
describe the circle A CF. (Post. III.)
Part of the circle
ECB is inside the A
F circle ACF, and part
outside, therefore these circles must intersectin two points. Call these points C and D.
Take either of these points (say C), and join A and O by the right line A C,* and C and B by the right line C B. (Post. I.)
We thus get a A ACB, and that this A is a A such as was required, may be proved as follows :
The lines A B and AC are radii of the same circle ECB, and therefore are equal to one another (Definition F); and the lines B C and B A are radii of the same circle A CF, and therefore are equal to one another.
But since the lines A C and B C are both equal to the line A B, they are equal to one another; or, in other words, ACB is an equilateral A, and is constructed on the line A B.
It would have done equally well if we had taken the point D, and joined D and A, and D and B. The demonstration would have been word for word the same as above, only substituting D for C.
PROPOSITION II. From a given point to draw a straight line equal to a given finite straight line.
* Two points are said to be joined by a right line when a right line is drawn from one to the other.
For the construction in this proposition we must
be able 1. To join two given points by a right line.
(Post. I.) 2. To produce a given right line to any length.
(Post. II.) 3. To describe a circle with a given centre and at
a given distance from that centre. (Post. III.) 4. On a given finite right line to construct an
equilateral A.* (Prop. I.) In proving that the construction furnishes us with
a line drawn as required, we must know1. That radii of the same circle are equal to one
another. (Def. XIV.) 2. That magnitudes which are equal to the same,
are equal to each other. (Axiom I.) 3. That if equals be taken from equals, the re
mainders are equal. (Axiom III.) Let A be the given point, and B C the given right line.
Join A and one end (say C) of the line BC, by the line CA. (Post. I.)
On C A construct an equilateral A ADC. (Prop. I.)
With C as centre, and at the distance CB, describe a circle. (Post. III.)
Produce the line D O till it cuts the circumference of this circle in the point E. (Post. II.)
* In the construction, in propositions, use is frequently made of complex constructions, the mode of performing which has been shown in some previous proposition. Strictly speaking, all the details of these constructions ought to be gone through for the case in hand; but, to save time and avoid confusion, it is usual simply to suppose that these details have been gone through, and that we have got the required result. In any given case, it would be possible to analyse the construction, however complex, into its ultimate elements, namely the postulates.