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EUCLID's

ELEMENTS.

BOOK I.

"A

DEFINITIONS.

POINT is that which has no parts 2. 2. A line is length without breadth . 3. The extremes of a line are points. 4. A right line is that which lies evenly between its points .

5. A fuperficies is that which has only length and

breadth 4.

6. The bounds of a fuperficies are lines.

a It may perhaps be as well to fay, A point is that which is lefs than any affignable or even conceiveable magnitude.

may

b A line be conceived to be generated or produced by the motion of a point.

c. Some call a right line the.fhortcft of all lines that have the fame extreme points or bounds; others, that it is that line whose parts all tend the fame way, or lie in the fame direction, or of which no point in it is raised or depressed.

d Do not those describe a superficies beft, who say it is what is generated by the motion of a line, not moving endways, or in the direction of itself?

← This, as well as the third definition, of the extremes of a line, are not fo much definitions, as neceffary confequences and common notions, arifing from the definitions of a line and a fuperficies, there being no new geometrical term defined in either of them.

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24. Of three-fided figures, that is an equilateral triangle, which has three equal fides.

25. That an ifofceles triangle, which has but two equal fides.

26. That a scalene triangle, which has three unequal fides ".

27. Moreover, amongst three-fided figures, that is a right-angled triangle, which has a right angle.

28. That an obtufe-angled triangle, which has an obtuse angle.

29. And that an acute-angled triangle, which has three acute angles.

30. Amongft four-fided figures, that is a square, whofe fides are equal, and its angles right angles.

31. That an oblong, which is right-angled, but not equal-fided.

32. That a rhombus, which is equal-fided, but not rightangled.

33. That a rhomboides, whofe oppofite fides and angles are equal, but is neither equal-fided nor right-angled 9. 34. All four-fided figures befides thefe may be called trapeziums.

35. Parallels are right lines, which being in the fame plane, and produced infinitely either way, will not meet one another either way 1.

P This definition perhaps might as well have been omitted, for I fee no great ufe thereof.

9 This definition of a rhomboides, and that before it of a rhombus, are of no great confequence.

Some call parallels right lines thofe in one plane, neither inclining nor reclining, but having all the perpendiculars equal, that are drawn from the points of either of the lines to the other line.---Others call them equidifant right lines in one plane. ---Others fay, they are fuch that tend to a point infinitely diftant--Others, that as a perpendicular is the shorteft of all right lines drawn from a given point in a plane to a given right line in that plane, fo the longeft right line that can be drawn from that point to [or rather towards] that given right line, is parallel to it. But Euclid's definition of parallels is in general the beft; though in fome particular inftances, thefe others are not without their use. --Parallels must both lie in the fame plane, for otherwife, two right lines in different planes, the one in POSTU

POSTULATES or PETITIONS.

1. Grant that a right line may. point to another point.

be drawn from any one

2. That a finite right line may be continued directly forwards.

3. That a circle may be defcribed from any centre to any distance.

COMMON NOTIONS or AXIOMS.

1. Things equal to the fame thing, are equal to one another.

2. If equal things be added to equal things, the wholes are equal.

3. If equal things be taken away from equal things, the remainders are equal.

4. If equal things be added to unequal things, the wholes are unequal.

5. If equal things be taken away from unequal things, the remainders are unequal.

6. Things which are double to the fame thing, are equal to one another.

7. Things which are halves of the fame thing, are equal to one another.

8. Things which mutually agree with one another, are equal to one another s.

9. The whole is greater than a part of it.

IO. *All right angles are equal to one another.

II. If a right line falling upon two right lines does make the internal angles on the fame fide lefs than two right an

one plane above, and the other in a plane bencath, may be infinitely produced both ways, and never meet, and yet not be fuch as Euclid calls parallels.

--

This eighth axiom is univerfally convertible, although Proclus, Borelius, Taquet, &c. fay it is not. See Barrow's learned arguments upon this fubject in his 13th Mathematical Lecture.

* In fome books the 10th and 11th axioms are reckoned amongst the Poftulates,

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gles, thofe right lines, being infinitely produced, do meet on that fide where the angles are lefs than two right angles', 12. Two right lines do not comprehend a space.

Α

c

E
F

t All these axioms are fo evident, when the words by which they are declared are underfood, that no body can even deny their truth if he would: But I think the word equal fhould have been firft defined; which is, that thofe things are equal, that actually or potentially poffefs the fame fpace, or actually or potentially agree together; that is, actual or poffible congruency is equality. And notwithstanding the fault that is generally found with the eleventh axiom, viz. that it is rather a demonftrable propofition than an axiom; I was always, and ever fhall be, of opinion, that it is fo clear, when underfood, that no demonftration is required to make it more evident; and really think, those who have demonftrated it, have been only trifling; and instead of making it more evident, have more obfcured it.. -Since two right lines A B, CD in the fame plane, muft neceffarily meet or be parallel; and if they meet, it B muft either be to the right or left of the right line E F falling upon D them, what man of common fenfe would fay they would meet to the left? If both the angles E, F on the right were less than both the angles A E F, CFE on the left; and fince two different right lines cannot be drawn from the fame point out of a given right line parallel to that given right line; and fince it is demonftrated at prop. 28. lib. 1. If right lines which are parallel, that is, never meet, make the two inward angles on the fame fide of a right line, falling upon them, both together equal to two right angles, certainly and moft evidently, when two fuch angles are lefs than two right angles, the right lines which are croffed by the third right line (with which they make these angles) will not be parallel, and fo muft neceffarily meet. Wherefore those who will not have this to be an axiom, muft at least, I think, grant it to be a corollary or confequence arifing from prop. 28. lib. 1. requiring no other proof to make its evidence more clearly appear, than that propofition; and by means of fuch a corollary the 29th propofition may be demonftrated.

PROPO

PROPOSITION I.

PROBLEM.

To conftitute an equilateral triangle upon a given finite

L

A B.

right line.

ETA B be a given finite right line: it is required to conftitute an equilateral triangle upon the right line

C

From the centre A, with the distance A B, let the circle BCD be described [by poftul. 3.] and again from the centre B, with the distance B A, let the circle ACE be defcribed [by poftul. 3.] and from the point c, in which the circles cut one another, let the right lines CA, CB be drawn to the points A and B [by poftul. 1.].

D

B E

But

Therefore because the point A is the centre of the circle CD B, the right line AC [by def. 15.] will be equal to A B : Again, because the point B is the centre of the circle C A E, the right line B C will be equal to the right line B A. it has been proved, that the right line AC is equal to the right line A B; therefore each of the lines AC, CB, is equal to the right line A B. But things that are equal to one and the fame thing, are alfo equal to one another [axiom 1.]. Therefore the right line AC is equal to the right line C B; wherefore the three right lines AC, A B, BC are equal to one another.

And therefore the triangle ACB is equilateral [by def. 24.]. And also conftituted upon the given finite right line A B. Which was to be done.

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To put a right line at a given point equal to a given right line.

Let the given point be A, and the given right line be BC; it is required to put a right line at the point A equal to the given right line B C.

Draw [by poft. 1.] a right line AB from the point A to the point B; and upon the fame conftitute [by prop. 1.]

B 4

the

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