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triangle is included within another, the sum of all the three sides of "the included triangle is less than the sum of all the three sides of the other," and this is without doubt true, though I think by no means self-evident. It must be acknowledged also, that it is not exactly Euclid's proposition, which, however, it comprehends under it, and is the general theorem, of which the other is only a particular case. Therefore, though M. Clairaut may be blamed for maintaining that to be an Axiom which requires demonstration, yet he is not to be accused of mistaking a false proposition for a true one.

PROP. XXII.

Thomas Simpson in his Elements has objected to Euclid's demonstration of this proposition, because it contains no proof, that the two circles made use of in the construction of the Problem must cut one another; and Dr. Simson, on the other hand, always unwilling to acknowledge the smallest blemish in the works of Euclid, contends, that the demonstration is perfect. The truth, however, certainly is, that the demonstration admits of some improvement; for the limitation that is made in the enunciation of any Problem ought always to be shewn to be necessarily connected with the construction of it, and this is what Euclid has neglected to do in the present instance. The defect may easily be supplied, and Dr. Simson himself has done it in effect in his note on this proposition, though he denies it to be necessary.

Because that of the three straight lines DF, FG, GH, any two are greater than the third, by hypothesis, FD is less than FG and GH, that is, than FH, and therefore the circle described from the centre F, with the distance FD must meet the line FE between F and H; and, for

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the like reason, the circle described from the centre G at the distance GH, must meet DG between D and G, and therefore, the one of these circles cannot be wholly within the other. Neither can the one be wholly without the other, because DF and GH are greater than FG ; the two circles must therefore intersect one another.

PROP. XXVII. and XXVIII.

EUCLID has been guilty of a slight inaccuracy in the enunciations of these propositions, by omitting the condition, that the two straight lines on which the third line falls, making the alternate angles, &c. equal, must be in the same plane, without which they cannot be parallel, as is evident from the definition of parallel lines. The only editor, I believe, who has remarked this omission, is M. DE FOIX, Duc DE CANDALLE, in his translation of the Elements published in 1566. How it has escaped the notice of subsequent commentators is not easily explained, unless because they thought it of little importance to correct an error by which nobody was likely to be misled.

PROP. XXIX.

The subject of parallel lines is one of the most difficult in the Elements of Geometry. It has accordingly been treated of in a great variety of different ways, of which, perhaps, there is none that can be said to have given entire satisfaction. The difficulty consists in converting the 27th and 28th of Euclid, or in demonstrating, that parallel straight lines, or such as do not meet one another, when they meet a third line, make the alternate angles with it equal, or, which comes to the same, are equally inclined to it, and make the exterior angle equal to the interior and opposite. In order to demonstrate this proposition, Euclid assumed it as an Axiom, that "if a straight line meet two straight "lines, so as to make the interior angles on the same side of it less than "two right angles, these straight lines being continually produced, "will at length meet on the side on which the angles are that are less "than two right angles." This proposition, however, is not self-evident, and ought the less to be received without proof, that, as Proclus has observed, the converse of it is a proposition thai confessedly requires to be demonstrated. For the converse of it is, that two straight lines which meet one another make the interior angles, with any third line, less than two right angles; or, in other words, that the two interior angles of any triangle are less than two right angles, which is the 17th of the First Book of the Elements: and it should seem, that a proposition can never rightly be taken for an Axiom, of which the converse requires a demonstration.

The methods by which Geometers have attempted to remove this blemish from the Elements are of three kinds. 1. by a new definition of parallel lines. 2. by introducing a new Axiom concerning parallel lines, more obvious than Euclid's. 3. By reasoning merely from the definition of parallels, and the properties of lines already demonstrated, without the assumption of any new Axiom.

1. One of the definitions that has been substituted for Euclid's is, that straight lines are parallel, which preserve always the same distance from one another, by the word distance being understood, a per

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pendicular drawn to one of the lines from any point whatever in the other. If these perpendiculars be every where of the same length the straight lines are called parallel. This is the definition given by Wolfius, by Boscovich, and by Thomas Simpson, in the first edition of his Elements It is however a faulty definition, for it conceals an Axiom in it, and takes for granted a property of straight lines, that ought either to be laid down as self-evident, or demonstrated, if possible; as a Theorem. Thus, if from the three points A, B, and C of the straight line AC, perpendiculars AD, BE, CF be drawn all equal to one another, it is implied in the definition, that the points D, E and F are in the same straight line, which, though it be true, it was not the business of the definition to inform us of. Two perpendiculars, as AD and CF, are alone sufficient to determine the position of the straight line DF, and therefore the definition ought to be, "that two straight lines are parallel, when there are two "points in the one, from which the perpendiculars drawn to the "other are equal, and on the same side of it."

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This is the definition of parallels which M. D'Alembert seems to prefer to all others; but he acknowledges, and very justly, that it still remains a matter of difficulty to demonstrate, that all the perpendiculars drawn from the one of these lines to the other are equal. (Encyclopedie, Art. Parallele.)

Another definition that has been given of parallels is, that they are lines which make equal angles with a third line, toward the same parts, or such as make the exterior angle equal to the interior and opposite. Varignon, Bezout, and several other mathematicians, have adopted this definition, which, it must be acknowledged, is a perfectly good one, if

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it be understood by it, that the two lines called parallel, are such as make equal angles with a certain third line, but not with any line that falls upon them. It remains, therefore, to be demonstrated, That if AB and CD make equal angles with GH, they will do so also with any other line whatsoever. The definition, therefore, must be thus understood, That parallel lines are such as make equal angles with a

certain third line, or, more simply, lines which are perpendicular to a given line. It must then be proved, 1. That straight lines which are equally inclined to a certain line or perpendicular to a certain line, must be equally inclined to all the other lines that fall upon them; and also, 2. That two straight lines which do not meet when produced, must make equal angles with any third line that meets them.

The demonstration of the first of these propositions is not at all facilitated by the new definition, unless it be previously shewn, that all the angles of a triangle are equal to two right angles.

The second proposition would hardly be necessary if the new definition were employed; for when it is required to draw a line that shall not meet a given line, this is done by drawing a line that shall have the same inclination to a third line that the first, or given line has. It is known that lines so drawn cannot meet. It would no doubt be an advantage to have a definition that is not founded on a condition purely negative.

2. As to the Mathematicians who have rejected Euclid's Axiom, and introduced another in its place, it is not necessary that much should be said. Clavius is one of the first in this class; the Axiom he assumes is, "That a line of which the points are all equidistant from a cer"tain straight line in the same plane with it, is itself a straight line." This proposition he does not, however, assume altogether, as he gives a kind of metaphysical proof of it, by which he endeavours to connect it with Euclid's definition of a straight line, with which proof at the same time he seems not very well satisfied. His reasoning, after this proposition is granted (though it ought not to be granted as an Axiom), is logical and conclusive, but is prolix and operose, so as to leave a strong suspicion that the road pursued is by no means the shortest possible,

The method pursued by Simson, in his Notes on the First Book of Euclid, is not very different from that of Clavius. He assumes this Axiom, "That a straight line cannot first come nearer to another "straight line, and then go farther from it without meeting it." (Notes, &c. English Edition.) By coming nearer is understood, conformably to a previous definition, the diminution of the perpendiculars drawn from the one line to the other. This Axiom is more readily assented to than that of Clavius, from which, however, it is not very different; but it is not very happily expressed, as the idea not merely of motion, but of time, seems to be involved in the notion of first coming nearer, and then going farther off. Even if this inaccuracy is passed over, the reasoning of Simson, like that of Clavius, is prolix, and evidently a circuitous method of coming at the truth.

Thomas Simpson, in the second edition of his Elements, has presented this Axiom in a simpler form. "If two points in a straight line are "posited at unequal distances from another straight line in the same "plane, those two lines being indefinitely produced on the side of the least distance will meet one another."

By help of this Axiom it is easy to prove, that if two straight lines

AB, CD are parallel, the perpendiculars to the one, terminated by the other, are all equal, and are also perpendicular to both the parallels. That they are equal is evident, otherwise the lines would meet by the Axiom. That they are perpendicular to both, is demonstrated thus :

C

D

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If AC and BD, which are perpendicular to AB, and equal to one another, be not also perpendicular to CD, from C let CE be drawn at right angles to BD. Then, because AB and CE are both perpendicular to BD, they are parallel, and therefore the perpendiculars AC and BE are equal. But AC is equal to BD, (by hypothesis,) therefore BE and BD are equal, which is impossible; BD is therefore at right angles to CD.

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B

Hence the proposition, that “if a straight line fall on two parallel lines, it makes the alternate angles equal," is easily derived. Let

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FH and GE be perpendicular to CD, then they will be parallel to one another, and also at right angles to AB, and therefore FG and HE are equal to one another, by the last proposition. Wherefore in the triangles EFG, EFH, the sides HE and EF are equal to the sides GF and FE, each to each, and also the third side HF to the third side 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 concise, and is perhaps as good as any that can be followed, when a new Axiom is assumed. In the text above, I have, however, followed a different method, employing as an Axiom, "That two straight lines, which cut one another, cannot be both parallel to the same straight line.” This Axiom has been assumed by others, particularly by Ludlam, in his very useful little tract, entitled Rudiments of Mathematics.

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It is a proposition readily enough admitted as self-evident, and leads to the demonstration of Euclid's 29th Proposition, even with more brevity than Simson's.

3. All the methods above enumerated leave the mind somewhat dissatisfied, as we naturally expect to discover the properties of parallel lines, as we do those of other geometric quantities, by comparing the definition of those lines, with the properties of straight lines already

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