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) If equals are added to unequals the sums are unequal, the greater sum being that which is obtained from the greater magnitude.
(8) If equals are taken from unequals the remainders are unequal, the greater remainder being that which is obtained from the greater magnitude.
(1) The doubles and halves of equals are equal.
A Theorem is the formal statement of a proposition that may be demonstrated from known propositions. These known propositions may themselves be Theorems or Axioms.
The two next pages, within brackets, may be omitted the first time of reading the subject.
[A Theorem consists of two parts, the hypothesis, or that which is assumed, and the conclusion, or that which is asserted to follow therefrom. Thus in the typical Theorem
If A is B, then C is D, (i) the hypothesis is that A is B, and the conclusion, that C is D.
From the truth conveyed in this Theorem it necessarily follows:
If C is not D, then A is not B, (ii). Two such Theorems as (i) and (ii) are said to be contrapositive, each of the other.
For example, if it were universally true that, If a man is a Spaniard, his hair is black; then it would follow that if his hair is not black, the man is not a Spaniard. Each of these statements is the contrapositive of the other.
Two Theorems are said to be converse, each of the other, when the hypothesis of each is the conclusion of the other.
If C is D, then A is B, (iii) is the converse of the typical Theorem (i). The contrapositive of the last Theorem, viz. :
If A is not B, then C is not D, (iv) is termed the obverse of the typical Theorem (i).
Sometimes the hypothesis of a Theorem is complex, i.e. consists of several distinct hypotheses; in this case every Theorem formed by interchanging the conclusion and one of the hypotheses is a converse of the original Theorem.
The truth of a converse is not a logical consequence of the truth of the original Theorem, but requires independent investigation.
Thus, supposing it were true that if a man is a Spaniard his hair is black; it does not follow that if a man's hair is black he is therefore a Spaniard : for he might be a Turk or of any other nation.
Hence the four associated Theorems (i) (ii) (iii) (iv) resolve themselves into two Theorems that are independent of one another, and two others that are always and necessarily true if the former are true; consequently it will never be necessary to demonstrate geometrically more than two of the four Theorems, care being taken that the two selected are not contrapositive each of the other.
Rule of Conversion. If of the hypotheses of a group of demonstrated Theorems it can be said that one must be true, and of the conclusions that no two can be true at the same time, then the converse of every Theorem of the group will necessarily be true.
Obs. The simplest example of such a group is presented when a Theorem and its obverse have been demonstrated, and the validity of the rule in this instance is obvious from the circumstance that the converse of each of two such Theorems is the contrapositive of the other. Another example, of frequent occurrence in the elements of Geometry, is of the following type:
If A is greater than B, C is greater than D.
If A is equal to B, C is equal to D.
If A is less than B, C is less than D. Three such Theorems having been demonstrated geometrically, the converse of each is always and necessarily true.
Rule of Identity. If there is but one A, and but one B; then from the fact that A is B it necessarily follows that B is A.
This is an important axiom in geometrical reasoning. De Morgan used to illustrate it by the following example:
Suppose that in a town there were only one post-office and only one grocer's: and that it was known that the post-office was the grocer's; then it would follow that the grocer's was the post-office.
This is called the axiom of the unique solution, or the rule of identity]
EXPLANATION OF TERMS AND SIGNS.
A Problem is a geometrical construction to be effected by the aid of certain instruments.
It has been universally agreed by Geometers to use only the ruler, i. e. a straight edge, not divided, and a pair of compasses, in the solution of Problems.
A Corollary is a geometrical truth easily deducible from a theorem.
Q. E. D. stands for quod erat demonstrandum, and is usu ally written at the end of a theorem to mark that the truth of the theorem has been proved.
The parts of a Theorem are the general enunciation of the hypothesis and the fact to be proved, or statement in general language; the particular enunciation, or statement of the hypothesis and the fact to be proved in the particular case examined; and the proof.
In the proof it is frequently necessary to draw certain lines, or to conceive them as drawn. This is called the construction.
A beginner often asks “What is the use of Geometry ?'
The following remarks may perhaps help to shew him part at least of the use of it.
What is Geometry? What is the object of the science ?
It is not measurement, because that may be done directly. If I want to find the height of a tower, I may go to the top, and let a string down to the bottom, and then measure the string ; but this is not geometry, though it is measurement Geometry is the science of indirect measurement; in which, for example, by measuring one line we learn the length of another. If I measure the length of the shadow of the tower, and also the length of a vertical stick and its shadow, and have proved by geometrical reasoning, that as the length of shadow of the stick is to the length of shadow of the tower, so is the height of the stick to the height of the tower, that is, measure the height of the tower indirectly, this is a geometrical operation.
Now it is plain that many measurements must be effected indirectly. How for example is the height of a mountain ascertained? Or how is the distance of the moon from the earth found out to be very nearly 238000 miles? How do we know approximately the size of the sun, or the weight of some of the stars, or the velocity of light? It is plain that these results must be obtained by indirect measurement; and some of
them are obtained by measurement extremely indirect and circuitous, and consisting of a very great number of successive steps of reasoning ; each result, as soon as it is obtained, serving as the starting-point from which fresh results are attainable.
Now Elementary Geometry gives the beginning of all such chains of reasoning. The theorems are results which follow from the axioms, and, in their turn, will serve as the foundations for fresh theorems arranged in a long chain until questions such as those above mentioned can be solved.
Every theorem therefore may be shewn to be a means of indirectly measuring some magnitude. In theorem 4, for example, it is proved that AOD=COB; that is, if AOD is accessible, and is measured (by an instrument suitable for measuring angles), then it is not necessary to measure COB, for you have proved that it will be the same as AOD.
Again in Theorem 7, let A be a post on one bank of a river, B, C two posts on the opposite bank; it is required to find the distance across the river from B to A.
Measure BC, (which you can do, as they are both on the same bank,) and put up two posts E, F in a field at the same distance apart that B is from C: measure the angle at B, that is how much, when standing at B, you must turn a line pointing at C till it points at A; and copy this angle at E: similarly measure the angle at C, and copy it at F. Then this theorem has proved that AB=DE; that is if you measure in the field ED, you will indirectly have measured AB.
Theorem 5 is of very great importance, and is a good illustration of indirect measurement. Suppose B and C are two points with an obstacle between them, a house or a hill for example; how is the distance from B to C to be measured ? This theorem tells you; you may think it out for yourself.
So with this clue to the practical application of the theorems it will be well to go through all of them ; finding out in each case what the magnitude is which is indirectly measured, or the result indirectly obtained ; and inventing practical questions to which each theorem could be applied.