Let this proposition be true in any one single instance; that is, n being some whole number, let 1, 3, 5 up to 2n+1 put together give (n+1). Then the next odd number being 2n+3, the sum of all the odd numbers up to 2n+3 will be (n+1)+2 n+3, or n'+4n+4, or (n+2). But n+2 is the half of the even number next following 2n+3: consequently, if the proposition be true of any one set of odd numbers, it is true of one more. But it is true of the first odd number 1, for this is the square of half the even number next following. Consequently, being true of 1, it is true of 1+3; being true of 1+3, it is true of 1+3+5; being true of 1+3+5, it is true of 1+3+5+7, and so on ad infinitum.

Example 2.-The formula x" - â', n being a whole number, is always algebraically divisible by x-a.

n-1

X"a"X"-an-1xan-x-a" =x(xn−1—an−1)+a1−1 (x−a) In this last expression the second term a (x-a) is obviously divisible by x-a: if then any one of the succession

But this is

x-a, x2-a3, x3—a3, x'—a1, &c. be divisible by x-a, so is the next. obviously true of the first, therefore it is true of the second; being true of the second, it is true of the third; and so on, ad infinitum.

There are cases in which the successive induction only brings any term within the general rule, when two, three, or more of the terms immediately preceding are brought within it. Thus

in the application of this method to the deduction of the well known consequence of

it can only be shown that any one case of this theorem is true, by showing that the preceding two cases are true; thus its truth, when n = 5 and n = 6, makes it necessarily follow when n = 7. In this case the two first instances must be established (when n=1 by hypothesis, and when n=2 by independent demonstration), which two establish the third, the second and third establish the fourth, and so on.

An instance of mathematical induction occurs in many equations of differences, in every recurring series, etc.

6. From Jevons' Elementary Lessons in Logic, pp. 208-28, ed. 1878.

To express the difference between knowledge derived deductively and that obtained inductively the Latin phrases à priori and d posteriori are often used. By A priori reasoning we mean argument based on truths previously known; A posteriori reasoning, on the contrary, proceeds to infer from the consequences of a general truth what that general truth is. Many philosophers consider that the mind is naturally in possession of certain

laws or truths which it must recognise in every act of thought; all such, if they exist, would be à priori truths. It cannot be doubted, for instance, that we must always recognise in thought the three Primary Laws of Thought considered in Lesson xiv. We have there an à priori knowledge that "matter cannot both have weight and be without weight, or that " every thing must be either self-luminous or not self-luminous. But there is no law of thought which can oblige us to think that matter has weight, and luminous ether has not weight; that Jupiter and Venus are not self-luminous, but that comets are to some extent self-luminous. These are

facts which are no doubt necessary consequences of the laws of nature and the general constitution of the world; but as we are not naturally acquainted with all the secrets of creation, we have to learn them by observation, or by the d posteriori method.

It is not however usual at the present time to restrict the name à priori to truths obtained altogether without recourse to observation. Knowledge may originally be of an à posteriori origin, and yet having been long in possession, and having acquired the greatest certainty, it may be the ground of deductions, and may then be said to give à priori knowledge. Thus it is now believed by all scientific men that force cannot be created or destroyed by any of the processes of nature. If this be true the force which disappears when a bullet strikes a target must be converted into something else, and on

à priori grounds we may assert that heat will be the result. It is true that we might easily learn the same truth à posteriori, by picking up portions of a bullet which has just struck a target and observing that they are warm. But there is a great advantage in à priori knowledge; we can often apply it in cases where experiment or observation would be difficult. If I lift a stone

and then drop it, the most delicate instruments could hardly show that the stone was heated by striking the earth; yet on à priori grounds I know that it must have been so, and can easily calculate the amount of heat produced. Similarly we know, without the trouble of observation, that the Falls of Niagara and all other waterfalls produce heat. This is fairly an instance of a priori knowledge because no one that I have heard of has tried the fact or proved it à posteriori; nevertheless the knowledge is originally founded on the experiments of Mr. Joule, who observed in certain well-chosen cases how much force is equivalent to a certain amount of heat. The reader, however, should take care not to confuse the meaning of à priori thus explained with that given to the words by the philosophers who hold the mind to be in the possession of knowledge independently of all obser

vation.

It is not difficult to see that the à priori method is equivalent to the synthetic method considered in intension, the à posteriori method of course being equivalent to the analytic method. But the same difference is really ex

pressed in the words deductive and inductive; and we shall frequently need to consider it in the following lessons.

PERFECT INDUCTION AND THE INDUCTIVE

SYLLOGISM.

WE have in previous lessons considered deductive reasoning, which consists in combining two or more general propositions synthetically, and thus arriving at a conclusion which is a proposition or truth of less generality than the premises, that is to say, it applies to fewer individual instances than the separate premises from which it was inferred. When I combine the general truth that" metals are good conductors of heat," with the truth that "aluminium is a metal," I am enabled by a syllogism in the mood Barbara to infer that "aluminium is a good conductor of heat. As this is a proposition concerning one metal only, it is evidently less general than the premise, which referred to all metals whatsoever. In induction, on the contrary, we proceed from less general, or even from individual facts, to more general propositions, truths, or, as we shall often call them, Laws of Nature. When it is known that Mercury moves in an elliptic orbit round the Sun, as also Venus, the Earth, Mars, Jupiter, &c., we are able to arrive at the simple and general truth that "all the planets move in elliptic orbits round the sun. This is an example of an inductive process of reasoning.

It is true that we may reason without ren