« ΠροηγούμενηΣυνέχεια »
or, as he calls them, Common Notions, applicable (with the exception of the eighth) to all kinds of magnitudes, and not necessarily restricted, as are the Postulates, to geometrical magnitudes.
AXIOMS. I. Things which are equal to the same thing are equal to one another.
II. If equals be added to equals, the wholes are equal.
III. If equals be taken from equals, the remainders are equal.
IV. If equals and unequals be added together, the wholes are unequal.
V. If equals be taken from unequals, or unequals from equals, the remainders are unequal.
VI. Things which are double of the same thing, or of equal things, are equal to one another.
VII. hings which are halves of the same thing, or of equal things, are equal to one another.
VIII. Magnitudes which coincide with one another are equal to one another.
IX. The whole is greater than its part.
With his Common Notions Euclid takes the ground of authority, saying in effect, "To my Postulates I request, to my Common Notions I claim, your assent.”
Euclid develops the science of Geometry in a series of Propositions, some of which are called Theorems and the rest Problems, though Euclid himself makes no such distinction.
By the name Theorem we understand a truth, capable of demonstration or proof by deduction from truths previously admitted or proved.
By the name Problem we understand a construction, capable of being effected by the employment of principles of construction previously admitted or proved.
A Corollary is a Theorem or Problem easily deduced from, or effected by means of, a Proposition to which it is attached.
We shall divide the First Book of the Elements into three sections. The reason for this division will appear in the course of the work.
It is well known that one of the chief difficulties with learners of Euclid is to distinguish between what is assumed, or given, and what has to be proved in some of the Propositions. To make the distinction clearer we shall put in italics the statements of what has to be done in a Problem, and what has to be proved in a Theorem. The last line in the proof of every Proposition states, that what had to be done or proved has been done or proved.
lett Q. E. F. at the end of a Problem stand for Quod erat faciendum.
The letters Q. E. D. at the end of a Theorem stand for Quod erat demonstrandum.
In the marginal references :
Book I. Proposition 1. Hyp. stands for Hypothesis, supposition, and refers to something granted, or assumed to be true.
On the Properties of Triangles.
PROPOSITION I. PROBLEM.
To describe an equilateral triangle on a given straight line.
Let AB be the given st. line.
::: A is the centre of o BCD,
Def. 13. Now :: AC, BC are each=AB, .. AC=BC.
Ax. 1. Thus AC, AB, BC are all equal, and an equilat. A ABC has been described on AB.
Q. E. F.
PROPOSITION II. PROBLEM. From a given point to draw a straight line equal to a given straight line.
Let A be the given pt., and BC the given st. line.
It is required to draw from A a st. line equal to BC.
From A to B draw the st. line AB.
Post. 1. On AB describe the equilat. A ABD.
I. 1. With centre B and distance BC describe O CGH. Post. 3.
Produce DB to meet the Oce CGH in G. With centre D and distance DG describe O GKL Post. 3.
Produce DA to meet the Oce GKL in L.
Then will AL=BC. For :: B is the centre of O CGH, .:. BC=BG.
Def. 13. And :: D is the centre of O GKL, .: DL=DG.
Def. 13. And parts of these, DA and DB, are equal. Def. 21.
.. remainder AL=remainder BG. Ax. 3. But BC=BG; .: AL=BC.
Thus from pt. A a st. line AL has been drawn=BC.
R. E, F.
PROPOSITION III. PROBLEM. From the greater of two given straight lines to cut off a part equal to the less.
Let AB be the greater of the two given st. lines AB, CD.
It is required to cut off from AB a part=CD.
I. 2. With centre A and distance A.E describe O EFH,
cutting AB in F. Then will AF=CD. For
:: A is the centre of O EFH,
.. AF=AE. But
AE=CD; .. AF=CD.
Ax. 1. Thus from AB a part AF has been cut off=CD.
Q. E, F. EXERCISES. 1. Shew that if straight lines be drawn from A and B in the diagram of Prop. I. to the other point in which the circles intersect, another equilateral triangle will be described on AB.
2. By a similar construction to that in Prop. 1. describe on a given straight line an isosceles triangle, whose equal sides shall be each equal to another given straight line.
3. Draw a figure for the case in Prop. II., in which the given point coincides with B.
4. By a construction similar to that in Prop. III. produce the less of two given straight lines that it may be equal to the greater.