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THE FIRST BOOK
THE ELEMENTS OF GEOMETRY.
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 mag
nitude. 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 excludes 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.
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 it 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,
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 its or 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 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 mctaphysical 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 to"tam toti in infinitum productam si bina puncta unius binis
alterius congruant, patet ex ipsa admodum clara rectitudinis 'idea quam habemus.' (Supplementum in lib. 3. $ 550.), Now that which Boscovich would consider as an inference from our idea of straightness seems itself to be the essence of that idea, and to afford the best criterion for judging whether any given line be straight or not. It may, however, be better to express the definition a little differently from definition III, as given above, thus : If there be two lines which cannot coincide in two points, without coinciding altogether, each of them is called a straight line.
This way of expressing the definition seems preferable to that in the text, as it has been objected, that in the form in which it stands there, it is a definition not of a straight line, but of straight lines. This objection has not much weight in it, yet it cannot be doubted that the definition is more simple in this latter form.
From this definition the axiom above-mentioned, viz. that two straight lines cannot inclose a space, follows as a necessary consequence. For, if two lines inclose a space, they must intersect each other in two points, and yet in the intermediate part must not coincide, and therefore by the definition they are not straight lines. It follows, in the same way, that two straight lines cannot have a common segment, or cannot coincide in part, without coinciding altogether.
The definition of a plane is given from Dr. Simson, Euclid's being liable to the same objections with his definition of a straight line, for he says, that “ a plane superficies is one “ which lies evenly between its extreme lines.” The defects of this definition are completely removed in that which Dr. Simson has given. Another definition, different from both, might have been adopted, viz. Those superficies are called plane, which are such that if three points of one coincide with three points of the other, the whole of one must coincide with the whole of the other. This definition, as it resembles that of a straight line already given, might, perhaps, have been introduced with some advantage ; but as the purposes of demonstration cannot be better answered than by that in the text, it has been thought best to make no farther alteration,
In Euclid's Elements the general definition of a plane angle is placed before that of a rectilineal angle, and is meant to comprehend those angles which are formed by the meeting of other lines than straight lines. A plane angle is said to be " the inclination of two lines to each other which meet to
gether, but are not in the same direction.” This definition is omitted here, because the angles formed by the meeting of curve lines, though they may become the subject of geometrical investigation, certainly do not belong to the Elements; for the angles that must first be considered are those made by the intersection of straight lines with one another. The angles formed by the contact or intersection of a straight line and a circle, or of two circles, or two curves of any kind, can produce nothing but perplexity to beginners, and cannot possibly be understood till the properties of rectilineal angles have been fully explained. On this ground, without contesting the arguments which Proclus uses in defence of this definition, I have omitted it. Whatever is not useful should, in explaining the elements of a science, be kept out of sight altogether; for, if it do not assist the progress of the understanding, it will certainly retard it.
AMONG the Axioms there have been made only two alterations. The tenth axiom in Euclid is, that two straight * lines cannot inclose a space;' which having become a corollary to our definition of a straight line, ceases of course to be ranked with self-evident propositions. It is therefore removed from among the axioms, and that which was before the eleventh is accounted the tenth.
The twelfth axiom of Euclid is, that If a straight line 'meet two straight lines so as to make the two interior an
gles on the same side of it taken together less than two right angles, these straight lines being continually produced will at length meet upon that side on which are the angles which are less than two right angles.' Instead of this proposition, which, though true, is by no means self-evident, another that appeared more obvious, and better entitled to be accounted an axiom, has been introduced, viz..that two straight lines which intersect each other cannot be both parallel to the same straight line.' On this subject, however, a fuller explanation is necessary, for which see the note on the 29th proposition.
PROP. IV. & VIII.
The fourth and eighth propositions of the first book are the foundation of all that follows with respect to the comparison of triangles. They are demonstrated by what is called the method of supraposition, that is, by laying one triangle upon the other, and proving that they must coincide. The demonstrations of the fourth and eighth propositions, as they are given by Euclid, are as certainly a process of pure reasoning, depending solely on the idea of equality, as established in the 8th axiom, as any thing in geometry. But if the removal of the triangle from its place be considered as creating a difficulty, and as inelegant, because it involves the idea of motion, which is not essential to geometry, this defect may be entirely