individual: this process is exemplified in the definitions prefixed to the Elements of Euclid.

23. It may be noticed, that in laying down a definition there is no necessity to have recourse to the highest genus, or even to remote species; the proximate superior species may in all cases be taken for the genus, and as that is always supposed to be known, we have only to add to its name that of the specific difference.

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24. Thus, in defining a right angled triangle, I describe it to be a triangle having a right angle: triangle is the species or kind to which the figure belongs, and its having a right angle is the circumstance by which it differs from every other species of triangle. I do not say, " a right angled triangle is a being," or "a figure," or "a plain figure," these species are too remote; but I call it a "triangle," which is the proximate species to right angled triangle: now the idea of triangle being previously known, that of a right angled triangle will likewise be known by specifying that it has a right angle.

25. The obvious use of definitions is to fix our ideas, so that whenever a definition is repeated, the precise idea intended by it, and no other, may immediately occur to the mind; and whenever an idea is present to the mind, its definition may as readily

occur.

26. Adequate and precise definitions may then be considered as the true foundation of every system of instruction; for when our ideas are fitly represented by words whose signification is fixed, there can be no danger of mistake either in communicating or receiving knowledge.

27. There are some ideas of which the mind perceives their agreement or disagreement immediately, without the necessity of argument or proof; this necessary determination of the mind is called a JUDGMENT, and the evidence or certainty with which it spontaneously acquiesces in this determination, is called INTUITION; also the irresistible force with which the mind is impelled to its determination, is called INTUITIVE EVIDENCE.

28. The faculty by which we perceive the validity of selfevident truth, is called COMMON SENSE, which signifies "that instinctive persuasion of truth which arises from intuitive evi

See An Essay on the nature and immutability of Truth, by James Beattie, LL. D. P. 1. c. 1,

dence:" it is antecedent to science, and although no part of it, yet" it is the foundation of all reasoning."

29. There are some ideas of which the mind cannot perceive the agreement or disagreement, without the intervention of others, which the logicians call middle terms; the proper choice and management of these are the chief business of science.

30. These middle terms serve as a chain to connect two remote ideas, that is, to connect the subject of our inquiry with some self-evident truth: thus, suppose A and D to be two ideas, of which the truth of A is self-evident, but that of D not so; and let it be admitted that A and D cannot be brought together, so as to afford the requisite means of comparison for determining their relation; in this case I must seek for some intermediate ideas, the first of which is connected with A, the last with D, and the successive intervening ones with each other: let these be B and C, now if it be intuitively certain, that B agrees with A, that C agrees with B, and that D agrees with C, it follows with no less certainty that D agrees with A: this latter certainty is however not intuitive, but of the kind which is called demonstrable", and the process by which the mind becomes conscious of this demonstrable certainty is called REASONING, OF DEMONSTRATION.

31. Every well ordered system of science will therefore consist of DEFINITIONS and PROPOSITIONS: definitions are used to explain distinctly the meaning of the terms employed, and to limit and fix our ideas respecting them with absolute precision. That which affirms or denies any thing, is called a proposition: I am; the sun shines; vice will inevitably be punished; two and three are five, &c. are propositions.

32. Propositions are either self-evident, or demonstrable; and since there can be no evidence superior to intuition, it follows that self-evident propositions not only require no proof, as some have said, they admit of none i.

h Every step of a demonstration must follow from truths previously known with intuitive certainty; but the conclusion or thing to be proved, depending on a connected series of intuitions, and no less certain than each of the preceding steps, is nevertheless not dignified with the name of intuition; it is obtained (as we have noticed above) by demonstration.

i For every proposition is proved by means of others which are more evident than itself, but nothing can be more evident than that which is self-evident; wherefore a self-evident proposition can admit of no proof.

33. Demonstrable propositions are such as do not admit of a determination by any single effort of the mind; to arrive at a consciousness of their truth, we are obliged frequently (as we have observed above) to have recourse to several intermediate steps, the first of which rests with intuitive certainty on some self-evident truth, the rest with the same intuitive certainty de pend on each other in succession, and the proposition, or truth to be proved, depends with like intuitive certainty on the last of these; so that the thing to be proved must evidently be true, since it depends on a self-evident truth, which dependance is constituted and shewn by a series of truths following or flowing from each other with intuitive certainty.

34. Propositions are likewise divided into practical and theo retical. A practical proposition is that which proposes some operation, or is immediately directed to, and terminates in practice; thus, to draw a straight line, to describe a circle, to construct a triangle, &c. are practical propositions.

35. A theoretical proposition is that in which some truth is proposed for consideration, and which terminates in theory: thus, the whole is greater than its part; contentment is better than riches; two sides of a triangle are together greater than the third, &c. are theoretical propositions.

36. Propositions, both practical and theoretical, are either self-evident or demonstrable.

37. A self-evident practical proposition is named by Euclid a POSTULATE, and a self-evident theoretical proposition, an AXIOM. 38. A demonstrable practical proposition is called a PROBLEM, and a demonstrable theoretical proposition, a THEOREM.

39. Hence, postulates and axioms being intuitive truths or maxims of common sense, admit of no demonstration; but problems and theorems not being self-evident, therefore require to be demonstrated.

40. Definitions, postulates, and axioms, are the sole principles on which demonstration is founded: this foundation, narrow and slight as it may seem, is continually extended and strengthened by the constant accession of new materials; for every truth, as soon as it is demonstrated, becomes a principle of equal force and validity with truths which are self-evident, and reasoning may be built on it with the same degree of certainty as on them: thus reasoning, by its progress, continually in

creases its basis, and the powers of the mind, ample as they are, must hence be inadequate to the use of all that vast accumulation and variety of means, provided for their employment.

41. When from the examination and comparison of two known truths a third follows as an evident consequence, the known truths are called PREMISES, the truth derived an INFERENCE, and the act of deriving it from the premises is called DRAWING, Or MAKING AN INFERENCE.

Thus, if two and two be equal to four, and three and one be equal to four, these being the premises, it follows as an inference that two and two, and three and one, are equal to the same (viz. to four): now, since things that are equal to the same are equal to one another, it follows as a further inference, that two and two are equal to three and one.

42. This example will furnish a general, although necessarily an imperfect, notion of Euclid's method of proving his propositions: his demonstrations are nothing more than a regular and well connected chain of successive intuitive inferences, the first of which is drawn from self-evident premises, and the last the thing which was proposed to be proved.

43. Hence, although demonstration is necessarily founded on self-evident truth, it is not at all necessary in every case that we should have recourse to first principles, for this would make demonstration a most unwieldy machine, requiring too much labour to be of extensive use: every inference fairly drawn from self-evident principles is of equal validity with intuitive truth, and may be employed for the same purposes; thus Euclid, in his demonstrations, makes use of the truths he has before demonstrated with a confidence as well founded as though they were self-evident, and merely refers you to the proposition where the truth in question is proved. This saves a great deal of trouble, for truths once established may with the strictest propriety be employed as principles for the proof and discovery of others.

44. It frequently happens in the course of a demonstration, that an inference presents itself, which is useful in other cases, although not immediately so with respect to the proposition under consideration; when such an inference is made, it is called a COROLLARY, and the act of making it DEDUCING A COROLLARY. 45. A LEMMA is a proposition not immediately connected with the subject in hand, but is assumed for the sake of shortening

the demonstration of one or more of the following propositions.

46. A SCHOLIUM is a note or observation, serving to confirm, explain, illustrate, or apply the subject to which it refers.

47. Euclid in bis Elements employs two methods for establishing the truth of what he intends to prove; namely, direct and indirect, both proceeding by a series of inferences in the manner explained above. Art. 41, 42.

48. A DIRECT DEMONSTRATION is that which proceeds from intuitive or demonstrated truths, by a chain of successive inferences, the last of which is the thing to be proved.

49. AN INDIRECT or APOLOGICAL DEMONSTRATION, or as it is frequently named, REDUCTIO AD ABSURDUM, consists in assuming as true a proposition which directly contradicts the one we mean to prove; and proceeding on this assumption by a train of reasoning in all respects like that employed in the direct method, we at length deduce an inference which contradicts some self-evident or demonstrated truth, and is therefore absurd and false; consequently the proposition assumed must be false, and therefore the proposition we intended to prove must by a necessary consequence be true, since two contrary propositions cannot be both true or both false at the same time *.

NOTES AND OBSERVATIONS ON SOME PARTS OF THE FIRST BOOK OF EUCLID'S ELEMENTS.

50. The first book of Euclid's Elements contains the principles of all the following books; it demonstrates some of the most general properties of straight lines, angles, triangles, parallel lines, parallelograms, and other rectilineal figures, and likewise the possibility and method of drawing those lines, angles, and figures. It begins with definitions, wherein the technical terms necessarily made use of in this book are explained, and our ideas

k Mathematical demonstrations "are nothing more than series of enthymemes; every thing is concluded by force of syllogism, only omitting the premises, which either occur of their own accord, or are recollected by means of quotations." This might easily be shewn by examples, but the necessary explanations, &c. would take up too much room. See on this subject The Elements of Logic, by William Duncan, Professor of Philosophy in the Marischal College of Aberdeen, 9th Ed. a book which ought to be recommended to the perusal of students in Geometry.