Book I. to both the parallels That they are equal is evident, otherwife the lines would meet by the Axiom: That they are per pendicular to both is demonftrated thus: If AC and BD, which are perpendicular to AB, and equal to one another, be not also perpendi C cular to CD, from C let CE be D E B equal. But AC is equal to BD, (by hypothefis), therefore BE and BD are equal, which is impoffible; BD is therefore at right angles to CD. Hence the propofition, that "if a straight line fall on two "parallel lines, it makes the alternate angles equal," is eafily derived. Let FH and GE be perpendicular to CD, then they will be parallel to one another, A E B propofition. Wherefore in the triangles EFG, EFH, the fides HE and EF are equal to the fides GF and FE, each to each, and also the third fide HF to the third fide EG, therefore the angle HEF is equal to the angle EFG, and they are alternate angles. Q. E. D. This method of treating the doctrine of parallel lines is extremely plain and concife, and perhaps does not admit of any improvement, except that of putting the Axiom on which it is founded into a more fimple form: It might, for example, be expreffed thus: "If two ftraight lines interfect one ano"ther, and if from any two points of the one, perpendicu"lars be drawn to the other; the perpendicular that is near "er to the point of interfection is lefs than that which is more Book I. "remote." The demonftration founded on this Axiom, and conducted in the manner just explained, I am difpofed to think preferable to any that has yet been given. If I have not followed this method, it is because I wished to preferve the text of Euclid with the least alteration poffible. I therefore affumed as an Axiom, a propofition that is not perhaps fo obvious a property of ftraight lines as that which has just been stated, but which is certainly much more fo than Euclid's, and one which I know from experience that beginners find no difficulty in comprehending, or in admitting to be true. By means of it the 29th, and alfo what was formerly the 12th Axiom, are demonftrated without changing any thing in the series of Euclid's propofitions. From the detail that has just been given, it is evident, that to demonstrate the properties of parallel lines without having recourfe to fome Axiom, or which is the fame thing, without affuming fome property of straight lines, not contained in the definition of a ftraight line, is ftill a defideratum in elementary geometry. And, if we confider how much skill and ingenuity have in the course of many ages been applied to this investigation; and also reflect, that the thing fought for belongs to the very rudiments of the fcience, and therefore, if it exists at all, can be at no great diftance, we shall be inclined to confider the difcovery of it, as a problem in geometry never likely to be refolved. At the fame time,.. it appears extraordinary, that the definition of a straight line,. if it is complete, fhould not lead us to the knowledge of every property of fuch a line, without the affumption of any thing not contained in it. Why ought not the propofition, for instance, that has been juft ftated as an Axiom, that "if two straight lines "interfect one another, and if from any two points in the one "perpendiculars be drawn to the other, the perpendicular nearer to the point of interfection will be less than that "which is more remote," to be capable of demonstration, or of being deduced from the definition of a ftraight line? If there be nothing obfcure or imperfect in our notions of a ftraight line, of a perpendicular, or of the interfection of two ftraight lines, from whence can it poffibly arife that we are unable to demonftrate this propofition? It was no way wonderful, that, when Euclid gave bis vague and obfcure defi Book 1. nition of a ftraight line, he fhould not be able to demonftrate even the most fimple property of rectilineal figures, without the affumption of the Axiom, that two ftraight lines cannot inclose a space. But, when the defects of this definition feem to be wholly corrected, we might certainly expect to be able to derive all the properties of straight lines directly from that definition, and yet the fact is, that we are not at all affifted by it, in the cafe before us. I confefs this is not eafily accounted for; but there are two confiderations, which, though they may not contain the folution of the paradox, will perhaps ferve to render it lefs wonderful. The firft is, that the definition given of an angle is certainly imperfect. An angle is the inclination of two lines; now, the word inclination is not much better understood than the word angle, fo that we have here the very fame defect that there is in Euclid's definition of a ftraight line. It is, at the fame time, difficult to conceive any way in which this definition can be amended; and it is, perhaps, on account of its imperfection that we are obliged to affume fome property of the lines fubtending an angle, or of two lines making angles with a third line, as an Axiom; juft as the imperfect definition of a ftraight line must be affifted by the affumption, that two ftraight lines cannot inclose a space. It must farther be remarked, that whatever be the fource of this difficulty, it is not the only one of the kind that we meet with in the Elements of Geometry. A fecond instance occurs, where a certain relation between the lengths of ftraight, and curve lines is affumed as an Axiom, without being logically deduced from our ideas, either of straightnefs, or of curvature. This is the Axiom on which Archimedes, and all the geometers after him have founded the comparison of the lengths of curves with the lengths of straight lines, and is the fame which is placed here at the beginning of the 8th Book. It would be in vain, I believe, that one would feek to give a rigorous demonstration of that propofition, yet it is of a nature purely elementary, though more complex, without doubt, than that which we have been confidering. There is a third propofition of this fort, relative to surfaces, which is alfo laid down by Archimedes, for the foundation of the comparison of curve fuperficies with plane figures, and of which no demonstration is given. These are the only three properties properties of geometrical magnitudes in the whole fcience, Book I. that are taken for granted without being deduced from the definitions; that it is impoffible to demonftrate any of them, is what no one will take upon him to affirm; but the many and powerful efforts made for that purpose, which they have already withstood, ought to deter any one from throwing away much of his time in fearching after fuch demonstra tions. THE BOOK II. PROP. VII. HE demonftration of this propofition ufually occafions Book II. fome difficulty to beginners, and on that account another is added, which is fomewhat fhorter. PROP. A and B. Thefe Theorems are added on account of their great use in geometry, and their clofe connection with the other propofitions which are the subject of this Book. Prop. A is an extenfion of the 9th and 10th. BOOK III. Book III. Book V. BOOK III. DEFINITIONS. 'HE definition which Euclid makes the firft of this Book " of which the diameters are equal." This is rejected from among the definitions, as being a Theorem, the truth of which is proved by fuppofing the circles applied to one ano. ther, so that their centres may coincide, for the whole of the one must then coincide with the whole of the other. The converse, viz. That circles which are equal have equal diameters, is proved in the fame way. The definition of the angle of a fegment is alfo omitted, because it is not a rectilineal angle, but one understood to be contained between a ftraight line and a portion of the circumference of a circle. For the fame reason, no notice is taken in the 16th propofition of the angle comprehended between the femicircle and diameter, which is faid by Euclid to be greater than any acute rectilineal angle. The reafon for thefe omiffions has already been affigned in the notes on the fifth definition of the first Book. THE HE fubject of proportion has been treated fo differently by those who have written on elementary geometry, and the method which Euclid has followed has been fo often, and fo inconfiderately cenfured, that in thefe notes it will not perhaps be more neceffary to account for the changes that I have made, than for those that I have not made. The changes are but few, and relate to the language, not to the effence of the demonftrations; |