NOTES TO PART I.

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NOTE A. The acknowledged superiority of Geometry as a science has tempted its professors to claim for it a perfection which can belong to nothing human. Their endeavours to exalt it into a perfectly abstract science have led to many absurdities. Thus, for example, the first principle with which the Elements of Euclid set out is either unmeaning or untrue: "A point is that which has no parts." If by the word that is to be understood a magnitude,—then it will follow from the definition in Euclid, that a point is a magnitude which has no magnitude. If by the word that is only meant (ENS) a thing,-then it will follow that Spirit, Motion, Volition, in short, whatever has no parts, is "a point." From the same desire to give an air of abstractness to this science, originated the definition (as it is called) of a Right or Straight line; which is said to be "that which lies evenly between its extreme points." This is a mere verbal illusion, teaching us nothing whatsoever; for we have just as clear an idea of a straight line as of an even line, of straightness as of evenness, which are indeed words of nearly the same meaning in this definition. So that the definition comes to this, viz. "A straight line is that which lies straightly between its extreme points," '—an evident tautology. The definition of a Plane Surface is open to still greater objections. Euclid defines it to be," that which lies evenly between its extreme right lines;' where, together with the illusion of the word evenly, it may be objected that there are many plane surfaces which are not bounded by right lines, as a circle, &c. R. Simson gives another definition, viz. "A plane superficies is that in which any two points being taken, the straight line between them lies wholly in that superficies." To which it may be objected, that, in the first place, this is not a definition, but a theorem (if properly expressed): and in the second place, that in a plane surface like that in the annexed plate, CDEFG would not, by this definition, be a plane surface, because the right line AB does not lie wholly in it. The truth is, that the ideas of a Point, a Right Line, and a Plane Surface, are what logicians call Simple Ideas, which cannot be defined. (See Locke's Essay, Book III. Chap. IV.) By our sight and touch we get the ideas of straight lines and flat surfaces, nor is any definition wanting to explain them; and from these sensible ideas our mathematical ones are wholly derived. We conceive straight lines and flat surfaces as perfectly straight and perfectly flat; which, though they do not exist in nature, may exist in our minds,—and these are our mathematical ideas of the said magnitudes. They are in fact only our ideas of sensation modified and refined; so true it is that Geometry, the purest and most abstract of the sciences, is primitively derived from the commonest and humblest source of information-the

senses.

C

G

D

A

E

B

F

The word "even" is so very general and vague, a smooth globe being called even, that it was thought prudent to add the words "straight" and "flat" to the description of a right line and plane surface, as perhaps

more intelligible though less refined. It is also to be observed, that the explanations of a right line and plane surface given in this Treatise are applicable to infinite as well as finite right lines and plane surfaces,— which the "definitions" in the Elements are not. Of a Mathematical Solid there is no definition given or attempted, in Euclid: his definition of a Solid" that which has length, breadth, and thickness," being evidently applicable to any solid. The perfect smoothness of its surface, or surfaces, is that which distinguishes a mathematical from a physical or common solid. Another difference, not essential however, is; that a mathematical solid is considered as the space which a solid would occupy, rather than as a mass; but the mathematical results are not at all affected by the way in which we choose to consider it.

Plane Geometry is, properly speaking, that science which treats of geometrical quantities lying in the same plane (as we said in our first Edition) but it is better perhaps to define it in its most usual, than its most philological acceptation.

NOTE B. In the subsequent parts of this Treatise we shall for brevity omit the words "plane" and rectilineal;" it being understood that all the lines and figures spoken of are rectilineal and plane, except the circle, which is plane, but not rectilineal.

The Work is divided into Lessons, not that there are any logical grounds for the division, but in order that each may be considered as a separate Study to be mastered completely before proceeding any farther. Every advantage, however, is taken of a change in the subject, to begin a new lesson.

NOTE C. Articles 1, 2, and 3 form Proposition IV. in the Elements of Euclid; but besides that they are really distinct theorems, their being crowded into one enunciation confuses the mind and oppresses the memory of a reader.

NOTE D. Euclid's enunciation and proof of this theorem are as follows:

The angles at the base of an equal-sided triangle are equal to one another; and if the equal sides be produced, the angles upon the other side of the base shall be also equal.

Let ABC be a triangle, of which the side AB is equal to ac, and let the straight lines, AB, AC, be produced to D and E: the angle ABC shall be equal to the angle ACB, and the angle CBD to the angle вCE.

A

In BD take any point F, and from AF cut off AG equal to AF; and join FC, GB. Because AF is equal to AG, and Ac to AB, the two sides FA, AC are equal to the two GA, AB respectively; and they contain the angle FAG common to the two triangles, AFC, AGB. Consequently [by ART. 1.] the base Fc is equal to the base GB; as also [by ART. 2.] the angle ACF to the angle ABG, and the angle AFC to the angle AGB. But because AF is equal to AG, and AB to Ac, the part BF is equal to the part CG; therefore, in the triangles, BFC, CGB, the sides BF, FC, are respectively equal to the sides CG, GB, and the angle BFC is equal to the angle CGB. Consequently [by ART. 2.] the angle FBC is equal to the angle GCB; but these are the angles below the base.

F

B

C

Again In the same triangles, BFC, CGB, the angles BCF, CBG are equal [by ART. 2]. Consequently, taking these equal angles respectively from the angles ACF, ABG, which were above shown to be equal, the remaining angles ACB, ABC are equal; and these are the angles at the base. This, &c.

The above demonstration has some advantages over that given in the text; but the number and confusion of the lines and angles necessarily employed in it, have long rendered it the most difficult proposition in all Euclid to a learner. The advantage of an easy solution was to be preferred before all other less essential ones.

NOTE E. In their proofs of indirect propositions, Geometers not only omit carrying on the hypothetic phraseology, but they universally pursue a method of demonstration which is erroneous and absurd. For, taking that as true which is not true, but only supposed so, they tell us to do that which cannot be done, instead of telling us only to suppose it done. Thus, in the present theorem, R. Simson says: "let AB be the greater, and from it cut off DB equal to ac,"-referring us at the same time to PROB. III. for the method of cutting off DB. Now AB is only supposed greater than ac, and DB is only supposed cut off equal to ac; but we cannot actually cut off DB equal to ac, inasmuch as the mere supposition shows that this cannot be done. It is therefore absurd to refer us to a problem for the method of doing that which we afterwards find could not have been done. To render all such indirect proofs valid, it is only necessary, that the contrary of whatever principle we wish to assume be supposed true, and that we argue legitimately from such a supposition. But we are never to speak of performing actual operations on the figure, as if our supposition were really true, for they cannot be performed.

NOTE F. By this proof we get rid of a clumsy proposition in the Elements which is indeed only used in order to prove this theorem, and is never afterwards employed.

The triangles ABC, DEF might be such, and applied in such a manner, that the line EB would fall outside them both, as in the annexed figure. But the demonstration given in the text would still hold good; for the angles DEF, DBF would then be the differences between DFB and FEB, DBE and FBE, and therefore equal, because the latter angles are respectively equal, by ART. 4. However, if we always conceive the triangles applied at their greatest sides, (or at any of their sides, if they are all equal), the line EB will always fall within the figure.

E

B

NOTE G. The definition of a Perpendicular given in Simson's Euclid begins, "When a straight line standing on another straight line makes the adjacent angles equal, &c.;" in which the bisection of an angle is plainly assumed, i. e. of an angle equal to two right angles.

It is of course perfectly optional to collect all the Definitions used in a work, and place them together at the beginning of it, or to introduce them according as they are wanted. But in Geometry especially there are three important advantages derivable from the latter method; so very important indeed, that it is surprising they have been overlooked by Editors in general. 1°. The memory is less burthened at the commencement

of the science; there will be less terror and disgust felt at undertaking it, when we have but a few preparatory definitions to remember, than when we have a great number to get by heart, many of which are not used for several pages. 2°. Definitions will be better understood when the learner has become gradually conversant with the ideas, phrases, and figures of Geometry. They will also be better fecollected on account of the associa tions connected with them as they stand in the body of a work. 3°. The main advantage is this: that by not introducing them till they are wanted, we are enabled, by the foregoing articles, to prove that our definitions are possible. Thus, in our definition of a perpendicular we can prove, that one right line standing on another may make the adjacent angles equal to each other, and therefore that we are not assuming what might be chimerical.

NOTE H. The construction given in our first Edition for this PROBLEM might have perplexed a beginner in some cases; we therefore substitute another.

NOTE I. We must join the points A and B, C and D, which lie towards the same hand; for if we joined the points A and D, C and B, the joining lines might not be equal, though ac and BD were parallel.

B

A

D

The points ▲ and care taken first in the line "not the greater," because if they were taken first in the greater line, it might not be possible to take two others" equally remote" in the lesser.

NOTE J. Geometers have long confessed and regretted the imperfection of their Science in the Doctrine of Parallels. This doctrine, at least as set forth in the Elements, has been justly denominated-" the disgrace of Euclid." It is the only vulnerable point in the Science; and by its fatal inaccuracy, that systematie chain of rigorous demonstration, which constitutes the beauty and excellence of Geometry, is unhappily broken. Many attempts have been made to remedy this imperfection, to supply this indispensable link; but they have all been unfortunate. R. Simson, the editor of the Euclid now in use, employs two definitions, an axiom, and five theorems, for this purpose, yet his failure is complete. That the Author of such an humble Treatise as the present should pretend to succeed, where so many and so great men have been unsuccessful, may reasonably be considered the very height of self-delusion and self-sufficiency: but he confines his pretensions to this, that his system of Parallels is simpler, shorter, and more strictly demonstrative, than that given in the common Euclid.

The objection to Euclid's system is this, viz.:-In order to demonstrate the properties of parallel right lines, he assumes as self-evident a principle which is not self-evident. This, in a Science which professes to ground itself wholly on self-evident principles, is the greatest imperfection next to a false assumption. The system brought forward in our Treatise, it is believed, assumes no principle but what is immediately self-evident to the most ordinary capacity, and is as plain to the understanding of every person as an axiom can possibly be.

It will easily be observed, by any one who compares them, how much more simply and directly the several results follow from the System now

brought forward, than from the one in common use. By the old method, also, Propositions xvI. and XVII. of Euclid [ARTICLES 36 and 38], which are properly but subordinate results of Prop. xXXII. [ARTICLES 35 and 37], were obliged to be introduced in order to prove Prop. XXXII.; a most unphilosophical process, inasmuch as the particular truths should always be deduced from the general one, and not vice versa. By our method, Prop. xXXII., which contains the most beautiful result and the most powerful theorem of all Geometry, might be introduced so early as the 15th Article, though we postpone it to the 35th, in order to a more perfect arrangement.

NOTE K. In this demonstration it is taken for granted, that as no other right line but CD can be parallel to EF, therefore CD is. This mode of proof common to Euclid, as in ART. 10 [Prop. xiv. Euc.] But it certainly is not a logical nor legitimate inference that CD is parallel to EF, only because no other line is; no more than it is a logical or legitimate inference in Art. 10, that BC is in a right line with BD, only because no other line is. But the validity of both these proofs (and of all others similar to them) depends on a self-evident truth in the mind of the reader, which, because it is so obvious, writers on Geometry do not think it necessary to mention. Thus, in ART. 10, it is self-evident, that there must be some line which, being added to BD, shall lie in the same right line with it; and as it is there proved that no other line except BC can be that line, we are warranted in concluding that BC must be it. In the present Article likewise, it is self-evident that there must be some line which, being drawn through the point B, shall lie parallel to EF; and as it is proved that no other line except CD can be this line, therefore CD must be it.

It may be asserted that this latter principle, taken as self-evident, is not so; and that consequently here our doctrine of parallels fails. Το this it may be answered, that we conceive the principle must be selfevident to any one who has a clear notion of right lines.

NOTE L. The element in our first Edition for which the present is substituted, will be found in ART. 128, demonstrated from such principles only as are already established.

NOTE M. AS BD is supposed not parallel to ac, if perpendiculars be dropped from в and D on AC, they would be unequal, by ART. 19. It will be most convenient therefore to draw BE from that vertex B, whose perpendicular distance from Ac is supposed the least; for in this case BE will meet the perpendicular from the other vertex, D, below that vertex, and therefore also one or both of the sides AD, CD.

NOTE N. The common definition of a square is" a four-sided figure, which has all its sides equal, and all its angles right angles." But this definition is in part superfluous; for if but one of the angles be a right angle, it will follow (as in the next ART.) that all the angles must be right angles. Therefore it sufficient to the definition of a square, that one of its angles be granted a right angle.

NOTE O. Demonstration of ART. 33.

By ART. preceding, AB is equal to AD, and also EF to EH.

Therefore,

as AD and EH are equal, AB and EF must be equal. Moreover, as the angles at A and E are right angles, by preceding ART., AB and EF are the altitudes of these parallelograms (DEF. XII).

is equal to EFGH.

This, &c.

Hence, by ART. 28, ABCD