These theorems, on account of the facility with which Logarithms are applied to them, are the most convenient of any for resolving the two cases to which they refer. When A is a very obtuse-angle, the second theorem, which gives the value of the cosine of its half, is to be used; otherwise the first theorem, giving the value of the sine of its half its preferable. The same is to be observed with respect to the side c, the reason of which was explained, Plane Trig. Schol.

END OF SPHERICAL TRIGONOMETRY.

NOTES

ON THE

FIRST BOOK OF THE ELEMENTS.

DEFINITIONS.

I.

In the definitions a few changes have been made, of which it is necessary to give some account. One of these changes respects the first definition, that of a point, which Euclid has said to be, That which has no parts, or which has no magnitude. Now, it has been objected to this definition, that it contains only a negative, and that it is not convertible, as every good definition ought certainly to be. That it is not convertible is evident, for though every point is unextended, or without magnitude, yet every thing unextended or without magnitude, is not a point. To this it is impossible to reply, and therefore it becomes necessary to change the definition altogether, which is accordingly done here, a point being defined to be, that which has position but not magnitude. Here the affirmative part includes all that is essential to a point, and the negative part includes every thing that is not essential to it. I am indebted for this definition to a friend, by whose judicious and learned remarks I have often profited.

II.

After the second definition Euclid has introduced the following, "the "extremities of a line are points."

Now, this is certainly not a definition, but an inference from the definitions of a point and of a line. That which terminates a line can have no breadth, as the line in which it is has none; and it can have no length, as it would not then be a termination, but a part of that which is supposed to terminate. The termination of a line can therefore have no magnitude, and having necessarily position, it is a point. But as it is plain, that in all this we are drawing a consequence from two definitions already laid down, and not giving a new definition, I have taken the liberty of putting it down as a corollary to the second definition, and have added, that the intersections of one line with another are points, as this affords a good illustration of the nature of a point, and is an inference exactly of the same kind with the preceding. The same thing nearly has been done with the fourth definition, where that which Euclid gave as a separate definition is made a corollary to the

fourth, because it is in fact an inference deduced from comparing the definitions of a superficies and a line.

As it is impossible to explain the relation of a superficies, a line, and a point to one another, and to the solid in which they all originate, better than Dr. Simson has done, I shall here add, with very little change, the illustration given by that excellent Geometer.

"It is necessary to consider a solid, that is, a magnitude which has length, breadth, and thickness, in order to understand aright the definitions of a point, line and superficies; for these all arise from a solid, and exist in it; The boundary, or boundaries which contain a solid, are called superficies, or the boundary which is common to two solids which are contiguous, or which divides one solid into two contiguous parts, is called a superficies; Thus, if BCGF be one of the boundaries which contain the solid ABCDEFGH, or which is the common boundary of this solid, and the solid BKLCFNMG, and is therefore in the one as well as the other solid, it is called a superficies, and has no thickness; For if it have any, this thickness must either be a part of the thickness of the solid AG, or the solid BM, or a part of the thickness of each of them. It cannot be a part of the thickness of the solid BM; because, if this solid be removed from the solid AG, the superficies BCGF, the boundary of the solid AG, remains still the same as it was. Nor can it be a part of the thickness of the solid AG: because if this be removed from the solid BM, the superficies BCGF, the boundary of the solid BM, does nevertheless remain; therefore the superficies BCGF has no thickness, but only length and breadth.

H

G

M

"The boundary of a superficies is called a line; or a line is the common boundary of two superficies that are contiguous, or it is that which divides one superficies into two contiguous parts: Thus, if BC be one of the boundaries which contain the superficies ABCD, or which is the common boundary of this superficies, and of the superficies KBCL, which is contiguous to it, this boundary BC is called a line, and has no breadth; For, if it have any, this must be part either of the breadth of the superficies ABCD or of the superficies KBCL, or part of each of them. It is not part of the breadth of the superficies KBCL; for if this superficies be removed from the superficies ABCD, the line BC which is the boundary of the superficies ABCD remains the same as it Nor can the breadth that BC is supposed to have, be a part of the breadth of the superficies ABCD; because, if this be removed from the superficies KBCL, the line BC, which is the boundary of the superficies

was.

D

N

D

A

B

K

KBCL, does nevertheless remain: Therefore the line BC has no breadth. And because the line BC is in a superficies, and that a superficies has no thickness, as was shown; therefore a line has neither breadth nor thickness, but only length.

"The boundary of a line is called a point, or a point is a common boundary or extremity of two lines that are contiguous: Thus, if B be the ex

tremity of the line AB, or the common extremity of the two lines AB, KB, this extremity is called a point, and has no length: For if it have any, this

length must either be part of the

H

G

length of the line AB, or of the line

KB. It is not part of the length of KB; for if the line KB be removed from AB, the point B, which is the extremity of the line AB, remains the same as it was; Nor is it part of the length of the line AB; for if AB be removed from the line KB, the point B, which is the extremity of the line KB, does nevertheless remain : Therefore the point B has no length; And because a point is in a line, and

M

a line has neither breadth nor thickness, therefore a point has no length, breadth, nor thickness. And in this manner the definition of a point, line, and superficies are to be understood."

III.

Euclid has defined a straight line to be a line which (as we translate it) "lies evenly between its extreme points." This definition is obviously faulty, the word evenly standing as much in need of an explanation as the word straight, which it is intended to define. In the original, however, it must be confessed, that this inaccuracy is at least less striking than in our translation; for the word which we render evenly is 85108, equally, and is accordingly translated ex æquo, and equaliter by Commandine and Gregory. The definition, therefore, is, that a straight line is one which lies equally between its extreme points: and if by this we understand a line that lies between its extreme points so as to be related exactly alike to the space on the one side of it, and to the space on the other, we have a definition that is perhaps a little too metaphysical, but which certainly contains in it the essential character of a straight line. That Euclid took the definition in this sense, however, is not certain, because he has not attempted to deduce from it any property whatsoever of a straight line; and indeed, it should seem not easy to do so, without employing some reasonings of a more metaphysical kind than he has any where admitted into his Elements. To supply the defects of his definition, he has therefore introduced the Axiom, that two straight lines cannot inclose a space; on which Axiom it is, and not on his definition of a straight line, that his demonstrations are founded. As this manner of proceeding is certainly not so regular and scientific as that of laying down a definition, from which the properties of the thing defined may be logically deduced, I have substituted another definition of a straight line in the room of Euclid's. This definition of a straight line was suggested by a remark of Boscovich, who, in his Notes on the philosophical Poem of Professor Stay, says, "Rectam lineam rectæ congruere totam toti in infinitum productum si bina puncta unius binis al"terius congruant, patet ex ipsa admodum clara rectitudinis idea quam