F PROPOSITION XLVII. on the side which is opposite to the right angle, is equal to the Let ABC be a triangle having the right angle BAC. The square described on the side BC which is opposite to the right angle, is equal to the sum of the squares H described on the sides BA, AC, between which is the right angle. >K "1. 42. On BC describe* the square BCED; and on BA and AC, the squares BAGF, ACKH. +1.23. From A drawt AI perpendicular to BC, D! [INTERC.12. and prolongł it to an unlimited length; and Cor. 6. join AD, AE, CF, BK. *1.17.Cor.l. Because ABC and ACB are* acute angles, AI will fallt between +1.23.Cor.3. the extremities of BC. And because AIB, IBD are right angles, 1L27.Cor.1. AL and BD aref parallel; and in the same manner AL and CE are parallel. Wherefore AL lies between BD and CE, and is parallel to them; and the quadrilateral figures BL, CL *1. 32 bis. have their opposite sides parallel, and are* parallelograms. BeNom. + Hyp: cause the angle BAC ist a right angle, and the angle BAG Constr. is$ also a right angle, AC and AG are* in the same straight line. *3. For the like reason, AH and AB are in the same straight line. † I. 11. And because the angle DBC ist equal to the angle FBA (each of them being a right angle), add to each the angle ABC; thereINTERC. 1. fore the whole angle DBA ist equal to the whole angle CBF; Cor. 4. and because the two sides AB, BD are equal to the two FB, BC respectively, and the angle DBA is equal to the angle CBF, the • 1. 4. side DA is* equal to the side CF, and the triangle DBA is equal + L 41. to the triangle CBF. Now the parallelogram BL ist double of the triangle DBA, because they are on the same base BD, and between the same parallels BD, AL; and the square BAGF is double of the triangle CBF, because they are on the same base FB, and between the same parallels FB, GC; but the doubles of Interc.). equal things are equal to one another; therefore the parallel. Cor. 10. ogram BL is equal to the square BAGF. In the same way may be shown, that the parallelogram CL is equal to the square ACKH. Therefore, by adding equals to equals, the sum of the •INTERC. 1. parallelogram BL and the parallelogram CL, is* equal to the sum Cor. 5. of the square BAGF and the square ACKH. But the sum of the parallelogram BL and the parallelogram CL, is the square +INTERC. 1. BCED; therefore the square BCED ist equal to the sum of the square BAGF and the square ACKH. And by parity of reasoning, the like may be proyed in every other instance. Wherefore, universally, in any right-angled triangle, &c. Which was to be demonstrated. A. PROPOSITION XLVIII. angle, be equal to the sum of the squares described on the other Let ABC be a triangle, which is such that the square described on one of its sides BC, is equal to the sum of the squares described on BA and AC. The angle BAC is a right angle. #1. 10. From the point. A draw. AD at right angles to B . 1.3. AC; and make* AD equal to AB; and join DC. Constr. Because AD ist equal to AB, the square on AD is equal to 11. 43. Cor. the square on AB. To each add the square on AC ; and the sum *Interc. 1. of the square on AD and the square on AC, is* equal to the sum Cor. 4. of the square on AB and the square on AC. But the sum of the +I. 47. square on AD and the square on AC, ist equal to the square on Constr. DC; because the angle DAC is a right angle. And the sum of * Hyp. the square on AB and the square on AC is* equal to the square +Interc. 1. on BC. Therefore the square on DC ist equal to the square on Cor. 3. BC. And because the square on DC is equal to the square on 11. 44. BC, DC is equal to BC. But because in the triangles BAC, spectively, and also the third side BC is equal to the third side "I. 8. DC; the angle BAC is* equal to the angle DAC. But the + Constr. angle DAC ist a right angle; therefore the angle BAC, which is Interc. 1. equal to it, ist also a right angle. Cor.1. And by parity of reasoning, the like may be proved in every other instance. Wherefore, universally, if the square described on one of the sides of a triangle, &c. Which was to be demonstrated. END OF THE FIRST BOOK. 121 NOTES, FAMILIAR AND GEOMETRICAL. NOMENCLATURE I.-BOOK I. • The Way to improve our Knowledge, is not, I am sure, blindly, and with an implicit Faith, to receive and swallow Principles; but is, I think, to get and fix in our minds clear, distinct, and compleat Ideas, as far as they are to be had, and annex to them proper and constant Names.'-Locke on the Human Understanding. iv. 12. 6. NOM. VI.-BOOK I. It may occur here, that for some purposes requiring extreme accuracy of division, as for instance on the nonius and scales of astronomical instruments, the divisions might be constituted by alternate layers of two metals of different colours, so as to leave a visible line absolutely without breadth. The same result might in a certain degree be obtained, by painting or staining the alternate intervals of different colours. It is not known whether this has ever been put in practice. NOM. VII-BOOK I. The descriptions of a point and a line, as given by the ancients, might be supposed intended to perplex beginners. The description of the first, in particular, appears to approach as nearly as possible to a description of nothing. The difficulty is referable to a love for enigmatical forms; and vanishes on explanation. NOM. VIII.-BOOK I. It is not essential to a figure, that it should inclose a space. Parabolas, hyperbolas, and spirals, are figures; but they do not inclose a space. Figures of this last kind, may be advanced as examples of linear figures, as distinguished from superficial. NOM. XV AND XVI.-BOOK I. From Euclid's Book of Data, with slight alterations. NOM. XIX.-BOOK I. The term Corollary by no means implies the non-necessity of demonstration, as has been sometimes represented. On the contrary, there must be an assignable reason for Corollaries, as for every thing else. It is true the reason may sometimes be one that can hardly fail to present itself though not given; and sometimes Corollaries are little more than recapitulations, of truths which have been evolved in the course of demonstrating some ulterior truth. But with the express view of opposing the notion that Corollaries are of the nature of the imaginary things styled ‘selfevident truths, it has been determined to insert no Corollary without its being followed by a statement of the reason ; which is a demonstration in little. Those who will pursue this plan, will be surprised to find how often they have considerable difficulty in making out a clear statement of the reason of a Corollary, which at first sight seemed perfectly obvious and inevitable. The cause of which appears to be, that in all people the habit of judging by a sort of guess-work, useful enough for many of the purposes of common life but incompetent to the severe distinguishing of truth from falsehood, has greatly outrun the habit and power of rigid demonstration. The practical use of inserting subsidiary propositions under the form of Corollaries, is, first, to save room, and secondly, to render the connexion more visible between what may be called the leading Propositions. If every Corollary in the First Book of Euclid were presented as a distinct Proposition, it is easy to see how the connexion would be obscured. NOM. XX.-BOOK I. In an elementary science like geometry, it would appear to be unnecessary to call anything a Lemma. The term, however, is used by Euclid; and is frequently employed in the progress of the sciences. For example, in optics or astronomy, if the chain of argument is interrupted to introduce some preliminary proposition from geometry or algebra ; such interloping proposition is usefully distinguished as a 'Lemma.' NOM. XXVI.-BOOK I. As a help to clearness, the use of the word 'must' will be confined to the tracing of false or impossible consequences in the manner here described. PROPOSITION I.--INTERCALARY BOOK. Cor. 14 and 15. The substance of these two Corollaries is demonstrated by Euclid in Propositions I and V of the Fifth Book; but is previously assumed by him without proof, in the Proposition called by Simson the Twentieth of the Third Book, the object of which is to show that the angle at the centre of a circle is double of the angle at the circumference. COR. 16. If evidence is demanded of the existence of magnitudes which, being added together to any number however great, cannot surpass a given magnitude, the simplest instance that can be given is in the magnitudes in the series Ž + + $ + Io + &c. Suppose there is first given to a person the half of some particular thing; then the fourth (which is the half of what was left); then the eighth (which is the half of what was left, again); then the sixteenth ; and so on. It is plain that what this person has received, is increased every time. But it is also plain that it can never amount to the whole thing, or 1; for there will always be a piece left. Still less can it ever amount to more than one, as for instance to two. And if each successive magnitude, instead of being one half of the preceding, was in any other constantly diminished proportion, (as for instance 18% Othe), it would be equally true (though not so easily proved) that the sum would never surpass some given magnitude. Hence, to have shown that a magnitude receives perpetual additions, is never evidence that it will arrive at a certain specified amount. Both mathematicians and political economists have fallen into snares for want of attention to this. Cor. 18. Euclid's proof as given in his Tenth Book, and by Simson at the beginning of the Twelfth, establishes the fact on a magnitude vastly smaller than there is any necessity for, and thereby destroys the simplicity of analogy. For example, if C should require to be taken sixteen times in order to be greater than AB, Euclid instead of establishing that the sixteenth part of AB is less than €, establishes it of the 32768th part, being 2048 times less than necessary. |