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DEFINITION 1. B. I.
HAT is meant by a point, a line, and a surface,
may easily be understood, by considering the nature of a solid or body. The boundaries of any solid are not parts of it, and therefore have no thickness, their only dimensions are length and breadth, and therefore they are surfaces. The boundaries of a surface have only length, if they had breadth they must be parts, not boundaries; they are therefore lines. The boundaries of lines want even length, they have therefore no dimensions, and are points.
The learned Robert Simson gives a different definition of a plane surface. A plane superficies is that in which any two points being taken, the straight line between them lies wholly in that superficies : this indeed is a well known property of a plane; but as it can easily be deduced from Euclid's definition compared with that of a right line, I thought it better not to make any change. The young
Student should be aware that this definition, and also the fourth, are not definitions properly so called, for a right line and a plane surface are simple ideas, which only admit description. It might be objected indeed to this definition of a plane surface, that there are planes, a circle, an ellipse, &c. which are not terminated by right lines: but if it be once understood what a plane is which is terminated by right lines, there can be no difficulty in conceiving that any part of it is also a plane, though that part be bounded by a curve.
DEF. 8. B. I.
In the Greek copies this is the ninth definition, as it is preceded by a sort of general definition of an angle, the inclination of two lines to one another in a plane, which meet together, but are not in the same direction: This definition is omitted, as its only use appears to be in the doctrine of the angle of a semicircle and segment (Prop. 16. and 31. B. 3.) which ought entirely to be rejected, as producing nothing but paradoxes and disputes, wholly unworthy of geometers.
DEF. 9. B. I.
This definition, and the tenth, are not Euclid's, but are added, because the terms often occur, and require explanation.
DEF. 13. B. I.
The definition which followed this, namely, a term is the extremity of any thing, is omitted as useless; the definitions of a segment of a circle, and of an oblong, are also omitted, because the first is one of the definitions of B. 3. and the latter is one of those of B. 2. where it is called a rectangle. Also the definitions of a rhombus, which has its four sides equal, but not its angles, and of a rhomboid, which has its opposite sides equal, but not the adjacent, are omitted, as no 'mention is made of them in the Elements. Instead of them is substituted the definition of a parallelogram.
DEF. 17. B. I.
R. Simson has rightly omitted the words which are usually added to this definition, which also divides the circle into two equal parts, as not belonging to the definition, being only a corollary from it. It can be demonstrated by conceiving one of the parts into which the circle is divided to be applied to the other, for it is evident they must coincide on account of the equality of the radii. The same thing is easily deduced from Prop. 31. and 24. B. 3. from the first of which it follows that semicircles are similar segments, and from the other, that they are equal to one another.
It should be remarked that the radius with which the circle is described must be the interval between the centre and some other point, not between any two points different from the centre; as is evident from the constructions of Prop. 2. and 3. B. l. which would be unnecessary, if the description of a circle from any point with a radius equal to any given line were conceded.
AX. 11. B. I.
This axiom does not treat of angles adjacent to the sạme perpendicular, which are equal by def. 11, but of those which are formed by different perpendiculars with different right lines. By help of it we can demonstrate that two right lines have not a common Plate 7. segment. For if it be possible, let the lines BC and Fig. 3 BĎ have a common segment AB, and let BE be perpendicular to the line ABC, if it be also perpendicular to ABD, the angles CBE and DBE are equal (1), (1) A: 11. which is absurd; but if not, let BF be perpendicular to the line ABD, and the angles ABF and ABE'are equal (1), which is absurd.
R. Šimson demonstrates this in a corollary to prop. 11. B. 1, but his demonstration does not appear to me perfect, through B he draws BE perpendicular to AB and assumes that there can be but one perpendicular at that point; but this cannot be conceded, since to erect a perpendicular, the line AB must first be
produced, and if this can be done in different ways, there can be several perpendiculars at the point B, as is evident from the construction of prop. 11. B. 1. and therefore the whole demonstration fails.
AX. 12. B. I.
Although it be a generally received principle, that right lines which are inclined to each other must meet if produced, yet when there are lines which though constantly approaching to each other never meet, as a curve and its asymptote, this principle is too defective to be ranked among geometrical axioms. Many have endeavoured, but unsuccessfully, to demonstrate the doctrine of parallels without this axiom: each has assumed a principle not more evident than that which they rejected. For those who are dissatisfied with Euclid's method, a new demonstration of the doctrine is giv in N. Prop. 27. B. 1.
By some editors the last three axioms are placed among the postulates. Although it must be confessed that they differ considerably from the others, and depending as they do upon the definitions, can scarcely merit the name of common notions, yet following the authority of the best editors, I have placed them among the axioms; and if (as most geometers assert) postulates refer to problems, axioms to theorems, they are rightly placed there.
PROP. 1. B. I.
Euclid assumes in the construction of this problem, that the circles intersect, it is impossible but part of the circle BCD, whose centre A is in the periphery of the circle ACE, must be outside that circle, and part within it.
PROP. 2. B. I.
According to the different situations of the point A, this construction somewhat changes; if the given point be in the given line, it is unnecessary to draw the line upon which the equilateral triangle is to be described ; and if it be either extremity of the given line, the triangle also is unnecessary, and no part of the construction is required except the description of the circle ACF, any line drawn to the periphery of which is equal to the given line: the point may be also so given as to be the vertex of an equilateral triangle of which the given line is the base, in which case it is evident that the line drawn to the extremity of the given line solves the problem: also another line equal to the given one can be drawn at the opposite side, in which case the radius of the circle whose centre is in the vertex of the triangle, is only the produced part of the side: these different cases are designedly omitted by Euclid, who wrote for geometers, not mechanics, and was only anxious to make his constructions intelligible, wholly regardless of practice.
PROP. 3. B. I.
In this proposition, as in the preceding, the position of the point À changes the construction; if it be at the extremity of the given line, the description of the circle, whose radius is the given line, is sufficient; it is possible that the line drawn from the point A may coincide with AB, in which case the description of the circle is unnecessary.
PROP. 4. B. I.
Euclid has expressed this proposition in an absolute not an hypothetic form, he says the triangle ABC being applied to EFD, &c. But as learners are apt to understand by these words some mechanical application, I thought it better to use the hypothetic form.
PROP. 8. B. I.
In this proposition the equality only of the angles at the vertices B and F is demonstrated, for from it by prop. 4. can be deduced the equality of the other parts; it seemed useful however, to annex the scholium, that the same might also be shewn by this proposition.