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As is evident from what has been said, the whole intention in preparing the work has been to furnish the student with that kind of geometric knowledge which may enable him to take up most successfully the modern works on Analytical Geometry.
N. F. D.
1°. A statement which explains the sense in which some word or phrase is employed is a definition.
A definition may select some one meaning out of several attached to a common word, or it may introduce some technical term to be used in a particular sense.
Some terms, such as space, straight, direction, etc., which express elementary ideas cannot be defined.
2°. Def.-A Theorem is the formal statement of some mathematical relation.
A theorem may be stated for the purpose of being subsequently proved, or it may be deduced from some previous course of reasoning.
In the former case it is called a Proposition, that is, something proposed, and consists of (12) the statement or enunciation of the theorem, and (6) the argument or proof. The purpose of the argument is to show that the truth of the theorem depends upon that of some preceding theorem whose truth has already been established or admitted. Ex.
The sum of two odd numbers is an even number” is a theorem.
3o. A theorem so elementary as to be generally accepted as true without any formal proof, is an axiom.
Mathematical axioms are general or particular, that is, they apply to the whole science of mathematics, or have special applications to some department. The principal general axioms are :-i. The whole is equal to the sum of all its parts, and
therefore greater than any one of its parts. ii. Things equal to the same thing are equal to one
another. iii. If equals be added to equals the sums are equal. iv. If equals be taken from equals the remainders are
equal. v. If equals be added to or taken from unequals the
results are unequal. vi. If unequals be taken from equals the remainders are
unequal. vii. Equal multiples of equals are equal ; so also equal
submultiples of equals are equal. The axioms which belong particularly to geometry will occur in the sequel.
4°. The statement of any theorem may be put into the hypothetical form, of which the type is-
If A is B then C is D. The first part“ if A is B” is called the hypothesis, and the
“then C is D” is the conclusion. Ex. The theorem “The product of two odd numbers is an odd number" can be arranged thus :-
Hyp. If two numbers are each an odd number.
5o. The statement “If A is B then C is D” may be immediately put into the form
If C is not D then A is not B, which is called the contrapositive of the former.
The truth of a theorem establishes the truth of its contra