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to by Euclid. But it is given as having been given by him; and as being moreover likely enough to be practically useful.
PROP. XLI.-BOOK I. COR. Inserted as being the foundation of the mensuration of the area of rectilinear triangles in general.
PROP. XLIII.-BOOK I. This and the three Propositions next following it, are in substance from Bonnycastle's Geometry.
PROP. XLVI.-BOOK I. Wanted in the Proposition III. 15. of Simson, which teaches that of straight lines in a circle, that which is nearer to the centre is greater than that which is more remote.
OBSERVATIONS ON PROPOSITIONS OMITTED FOR INSERTION
IN OTHER PARTS OF THE ELEMENTS,
The Proposition which in Simson's Euclid is the 43rd of the First Book, would appear to be more properly placed in the Second Book, in some part antecedent to the Proposition which is the 4th of Simson.
The Propositions which in Simson's Euclid are the 42nd, 44th, and 45th of the First Book, should be placed in the Sixth Book, in some part antecedent to the 25th of the Sixth Book of Simson; and they would appear to be most conveniently introduced, next after the Proposition which in Simson's Euclid is the 18th of the Sixth Book.
These Propositions in their former place have always been felt to be a great impediment to learners in their progress towards the demonstration of the 47th Proposition, which may be considered as the crowning Proposition of the First Book ; and as such they have usually been passed over by intelligent teachers.
END OF THE NOTES.
Containing Notices of Methods at different times proposed for
getting over the difficulty in the Twelfth Axiom of Euclid.
The uses of such a Collection, are to throw light on the particu, lars that inust not be left unguarded in any attempt at solution, and to prevent future explorers from consuming their time unnecessarily in exhausted tracks.
1. The objection to Euclid's Axiom (independently of the objections common to all Axioms), is that there is no more reason why it should be taken for granted without proof, than numerous other propositions which are the subjects of formal demonstration, and the taking any one of which for granted would equally lead to the establishment of the matter in dispute.
2. Ptolemy the astronoıner, who wrote a treatise on Parallel Lines of which extracts are preserved by Proclus, proposed to prove that if a straight line cuts two parallel straight lines the two interior angles on each side are together equal to two right angles, by saying that if the interior angles on the one side are greater than two right angles, then because the lines on one side are* no more parallel than those on the other, the two interior angles on the other side must likewise be together greater than two right angles, and the whole greater than four, which is impossible ; and in the same way if they were supposed less. In which the palpable weakness is, that there is no proof, evidence, or cause of probability assigned, why parallelism should be connected with the angles on one side being together equal to those on the other ;, the very question in debate being, whether they may not be a little more than two right angles on one side and a little less on the other, and still the straight lines not meet.
3. Clavius announces that a line every point in which is equally distant from a straight line in the same plane, is a straight line;' upon taking which for granted, he finds himself able to infer the properties of Parallel Lines. And he supports it on the ground that because the acknowledged straight line is one which lies evenly [ex æquo] between its extreme points, the other line must do the same, or it would be impossible that it should be everywhere equidistant from
ουδέν γαρ μάλλον αι αζ γη παράλληλοι και αι ζδ ηβ.-Procli Comment. in Primum Euclidis Librum. Lib. 4.
It is but right to notice, that Proclus calls this weeganogiouds and deičews &c évesce; and Barocius the Venetian translator in 1560, notes it in the margin as Flagitiosa Ptolemæi ratiocinatio. Professor Playfair says it is curious to observe in Proclus's account an argument founded on the principle known to the moderns by the name of the sufficient reason (El.of Geo. p.405). If the allusion is to this part, the sufficient reason of the moderns must be something very feeble.
the first*. Which is only settling one unknown by a reference to another unknown.
4 & 5. In a tract printed in 1604 by Dr. Thomas Oliver of Bury, entitled De rectarum linearum parallelismo et concursu doctrina Geometrica (Mus. Brit.), two demonstrations are proposed; both of them depending on taking for granted, that if a perpendicular of fixed length moves along a straight line, its extremity describes a straight line.
6. Wolfius, Boscovich, Thomas Simpson in the first edition of his ‘Elements,' and Bonnycastle, alter the definition of parallels, and substitute in substance, that straight lines are parallel which preserve always the same distance from one another;' by distance being understood the length of the perpendicular drawn from a point in one of the straight lines to the other. Attempts to get rid of a difficulty by throwing it into the definition, are always to be suspected of introducing a theorem in disguise; and in the present instance, it is only the introduction of the proposition of Clavius. No evidence is adduced that straight lines in any assignable position, will always preserve the same distance from one another ; or that if a perpendicular of fixed length travels along a straight line keeping always at right angles to it, what the mathematicians call the locus of the distant extremity is necessarily a straight line at all.
7. D'Alembert proposed to define parallels as being straight lines one of which has two points equally distant from the other;' but acknowledged the absence of proof, that any other points besides these two would be equidistant in consequence.
8. Thomas Simpson in the second edition of his Elements' proposed that the Axiom should be, that . 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 anothert.'
• Nam si omnia puncta lineæ AB, æqualiter distant à rectâ DC, ex æquo sua interjacebit puncta, hoc est, nullum in eâ punctum intermedium ab ex. tremis sursùm, aut deorsùm, vel buc, atque illuc deflectendo subsultabit, nihilque in eâ flexuosum reperietur, sed æquabiliter semper inter sua puncta extendetur, quemadmodum recta DC. Alioquin non omnia ejus puncta æqualem à rectâ DC, distantiam haberent, quod est contra hypothesin. Neque verò cogitatione apprehendi potest aliam lineam præter rectam, posse habere omnia sua puncta à rectâ lineâ, quæ in eodem cum illâ plano existat, æqualiter distantia!--Clavii Opera. In Euclid. Lib. I. p.50.
+ It is impossible to avoid mentioning, as indicative of the spirit in which books on geometry have been written, that this author absolutely proposes that the Proposition which states the equality of triangles having two sides and the included angle equal respectively, should be introduced as an Axiom; but “if this Axiom should not appear sufficiently evident,” he directs one triangle to be applied to the other in the manner of Euclid (Elements of Geometry, by Thomas Simpson. First Edition, p. 8). Now what is this but saying, ' Persuade your scholar to believe without proof if you can ; but if you cannot, you may give him the proof.' The world may be challenged to show, in what way it is possible to arrive at any perception of the truth of the Proposition mentioned, except by directly or indirectly, formally or informally, going through something equivalent to Euclid's proof. On wbat conceivable principle, therefore, was the proof to be sunk, and the scholar invited to believe without? This is not teaching geometry, but teaching to do without geometry.
9. Robert Simson proposes that the Axiom should be, 'that a straight line cannot first come nearer to another straight line, and then go further from it, before it cuts it*.' By coming nearer or going from it, being understood the diminution or increase of the perpendicular from one to the other.
The objection to all these is, that no information has been given on the subject of the things termed straight lines, which points
* This and most of what has preceded, is in the Arabic. In a manuscript copy of Euclid in Arabic but in a Persian band, bought at Ahmedabad in 1817, the editor on the introduction of Euclid's Axiom breaks out as follows.
“I maintain that the last proposition is not among the universallyacknowledged truths, nor anything that is demonstrated in any other part of the science of geometry. The best way therefore would be that it should be put among the questions instead of the principles; and I shall demonstrate it in a suitable place. And I lay down for this purpose another proposition, which is, that straight lines in the same plane, if they are subject to an increase of distance on one side, will not be subject to a diminution of distance on that same side, and the contrary ; but will cut one another. And in the demonstration of this I shall employ another proposition, which Euclid has employed in the tenth book and elsewhere, which is, that of any two finite magnitudes of the same kind, the smallest by being doubled over and over will become greater than the greatest. And it will further require to be laid down, that one straight line cannot be in the same straight line with straight lines more than one that do not coincide with one another; and that the angle which is equal to a right angle, is a right angle.'
العلوم المتعارفة ولا مما اقول القضية الأخرة ليست من يتضح في غير علم الهندسة فاذن الأولي أن يترتب في المسايل دون المصادرات وانا ساوضعها في موضع يليق بها ووضعت بذلكك قضية اخري هي أن الخطوط المستقيمة الكاينة في سطح مستو أن كانت موضوعة علي التباعد في جهة فهي لا تكون موضوعة علي التقارب في تلك الجهة بعينها وبالعكس الا ان يتقاطعا واستعمل في بيانها قضية اخري قد استعملها اقليدس في المقالة العاشرة وغيرها وهي أن كل مقدارین معدودين من جنس واحد فان الأصغر منهما يصير بالتضعيف مرة بعد أخري أعظم من الأعظم ومما يجب ايضا ان يوضع ان الخط المستقيم الواحد لا يتصل على الاستقامة باكثر من
۔ بعضها ببعض وان الزاوية المساوية القايمة قایم
خط واحد مستقیم
to any reason why the perpendicular's growing smaller should be pecessarily followed by the meeting of the lines. It may be true; but the reason why, 'is not upon the record. If the answer is that straight line' means a certain figure familiar to the eye, viz. the figure taken by a plumb-line, and that it is accordant with experience that lines of this kind do not first approach and then not meet; -it would only be an extension of the same process to say it is accordant with experience that the rectangles under the segments in a circle are equal, for there is no doubt of everybody who has tried having found it true.
10. Varignon, Bezout, and others, propose to define parallels to be straight lines which are equally inclined to a third straight line, or in other words, make the exterior angle equal to the interior and opposite on the same side of the line. By which they either intend to take for granted the principal fact at issue, which is whether no straight lines but those that make such angles can fail to meet; or if their project is to admit none to be parallel lines of which it shall not be predicated that they make equal angles as above with some one straight line either expressed or understood, then they intend to take for granted that because they make equal angles with one straight line, they shall also do it with any other that shall in any way be drawn across them,-a thing utterly unestablished by any previous proof.
11. Professor Playfair proposes to employ as an Axiom, that 'two straight lines, which cut one another, cannot be both parallel to the same straight line ;' in which he had been preceded by Ludlam and others, and which he says is a proposition readily enough admitted as self-evident. The misfortune of which is, that instead of being self-evident, a man cannot see it if he tries. What he sees is, that he does not see it. He sees that a straight line's making certain angles with one of the parallels, causes it to meet the other; and he sees that by increasing the distance of the point of meeting, he can cause the angle with the first parallel to grow less and less. But if he feels a curiosity to know whether he can go on thus reducing the angle till he makes it less than any assignable magnitude, (or in other words whether there may not possibly be some angle so small as to fail of causing the straight line to reach the other parallel), he discovers that this is the very thing nature has denied to his sight; an odd thing certainly, to call self-evident.
12. The same objections appear to lie against Professor Leslie's proposed demonstration in p. 34 of his Rudiments of Plane Geometry.
13. Professor Playfair in the Notes to his · Elements of Geometry,' p. 409, has proposed another demonstration, founded on a remark
non causa pro causâ. It purports to collect the fact* that (on the sides being successively prolonged to the same hand) the exterior angles of a rectilinear triangle are together equal to four right angles, from the circumstance that a straight line carried round the perimeter of a triangle by being applied to all the sides in succession, is brought into its old situation again; the argument being, that because this line has made the sort of somerset it would do by being turned through four right angles about a fixed point, the exterior angles of the triangle have necessarily been equal to four right angles. The answer to which is, that there is no connexion between