of this fact itself. I can recommend it as a very fruitful subject of contemplation, which has led people to the most important discoveries.

The number of points on a piece of line is singly infinite. You under

stand all this now, excepting the word singly. And that is what I am going to explain. Let us consider what is the number of points on a piece of surface. It is at least infinite, for if you draw any line on the surface, all the points on that line must be reckoned, and there is an infinite number of them. But it is more than that. For when you have traced out a line by the continuous motion of a point, you can trace out the surface by the continuous motion of that line; so that first you have an infinite number of points on the line, and then an infinite number of these infinites. Thus you see that the number of points on a piece of surface is twice as infinite as the number of points on a piece of line; or, as we are accustomed to say, the former is doubly infinite, and the latter singly infinite. Let us next consider what is the number of points in a piece of solid space. First you trace out a line by the continuous motion of a point; that gives you a singly infinite number of points. Then you trace out a surface by the continuous motion of that line. This gives you a singly infinite number of such lines, and a doubly infinite number of points. Lastly, you trace out the solid by the continuous motion of the surface. The number of surfaces is then singly infinite. Of lines, there is an infinite number of such infinities; that is, the number of lines is doubly infinite. Of points, there is an infinite number of double infinites; so that the number of points in a piece of solid space is three times as infinite as the number of points in a line. This number is called triply infinite.

In how many directions can I look without moving my head? If I put myself in front of a wall, every point on the surface of the wall is in a definite direction from my eye, and every

direction leads to a definite point on the wall. Thus there are just as many directions as there are points on that surface; that is to say, a doubly infinite number of directions.

How many pairs of points are there on a piece of line? Let the first point move along the line; it will have a singly infinite number of positions. Select one of these, and then let the second point move along the line. It will have an infinite number of positions for each position of the other; thus altogether there will be a doubly infinite number of pairs. In the same way you will find that there is a triply infinite number of sets of three points, or of triads of points, on a piece of line.

All these things can be said in another way. Suppose that all you knew about a point was that it was on a certain line. That would not enable you to identify the point; for you would not know which it was out of a singly infinite number. The point might vary among all the points on the line, and still fulfil the condition of being a point on the line. Still it could only vary in that one way. Such a point is said to have one variation. It is able to move about, but only on a fixed line. But to tell you that the point is on a certain surface would be to tell you less than this, for you would have a doubly infinite number of points to choose from. Suppose the surface traced out by the motion of a line; then the point might lie on any position of the line, and anywhere on the line. It could move along the line, and then the line might move along the surface. Such a point is said to have two variations. If now you are told merely that the point is in a certain region of solid space, you have a triply infinite number of points to choose from, and the point is said to have three variations. It may move along a line, then the line may move on a surface, and then the surface may move in space. Now the three kinds of room are distinguished by the number of dimensions that they

have. Solid-room has three dimensions, length, breadth, and thickness. Surface-room has length and breadth. but no thickness. Line-room has no breadth or thickness, but only length. So we may now say that a point in space of three dimensions (solid-room) has three variations; a point in space of two dimensions (surface-room) has two variations; and a point in space of one dimension (line-room) has one variation.

You must not suppose, however, that the idea of a number of variations is confined to single points. A pair of points on a line has two variations, for the two points may move independently. A direction in which you can look has two variations; for it may take up a doubly infinite number of positions. And by and by we shall be able to see that a space has four variations; three of position and one of size. In order to identify a thing, you must be told as many facts about it as it has variations. Thus a point on a line is identified if you know one fact about it, say the distance from one end of the line. But to identify a point on the earth's surface you must know two things; for instance, the latitude and the longitude. And to identify a point in space you must know three things;

the latitude, the longitude, and the height.

I dare say, now, that you are rather indignant at being kept so long making perfectly obvious remarks that are true of everything, you may think it is beneath the dignity of human nature to spend all this time in contemplating the size and shape of a piece of wood. Very well; it is written in the fifteenth chapter of the Koran that when Adam was created all the angels were commanded to worship him. But that Eblis, the chief of them, refused, saying, “Far be it from me that am a pure spirit to worship a creature of clay." And that for this refusal he was shut out for ever from Paradise. Now the doom of Eblis awaits you if you fail to give due reverence to these little obvious every-day things; things that are true of every stone that lies on the pavement, of every drop of rain that falls from heaven, of every breath of air that fans you. Like him, you will find with astonishment that the creature of clay which you despise is the Lord of Nature and the Measure of all things, for in every speck of dust that falls lie hid the laws of the universe; and there is not an hour that passes in which you do not hold the Infinite in your hand.

II.

WE may say in general of the different studies which together make up education, that in England some of them are alive, others have only an imperfect vitality, and others have no vitality at all. As an obvious illustration of the difference I may refer to the classical and the modern languages. At our great schools all is zeal and emulation where Latin and Greek are in question, but the French lesson is languid; and while all the classical knowledge acquired there is carried away to the university to receive further augmentation, the little knowledge of French that has been picked up is dropped again almost immediately.

The

When we inquire whence arises this difference, we discover two causes which may give vitality to a study. first and most obvious is its intrinsic importance. And yet that this cause does not operate so powerfully as we might expect is evident from the example just given. The advantage of knowing French is evident to every one, but the use of knowing Latin, though conceivably it may be greater, is at any rate not so evident; yet the study of Latin flourishes, and that of French does not. More effect seems to be produced by a certain extrinsic importance which is given to some studies either by accidental No. 239.-VOL. XL.

circumstances or by deliberate design. What are called the "bread-studies" never quite lose their vitality; thus there will never be any difficulty in keeping alive some sort of study of law so long as a number of people get a livelihood from it. And classics, though not in the same strict sense a bread-study, have taken the lead of all studies among us mainly in consequence of the endowments which have been attached to them by those who, on various grounds, were convinced of their value.

When a study has through one of these causes, or more than one, acquired vitality, the teacher of it has an easy task before him. He no longer throws away his expositions upon empty benches, or upon unwilling hearers whose attention he can see to be comfortably absorbed in their novel. What is no less important, on such a subject text-books, manuals, and helps of all kinds are continually issuing from the press, whereas if the subject wants vitality it is to little purpose that the teacher here and there strikes out a flash of interest; the awakened mind goes to sleep again, the newborn ardour dies for want of nutriment.

Now, of the study of history we may say that it is slowly emerging out of a state in which, except at certain points, it was not kept alive in any of these ways. As to its intrinsic

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importance, this could only be political, and there were few politicians indeed who would have recognised the importance of any historian except Hansard. It had few prizes at the universities, and there were few means of making a livelihood by it. But at some points it was warmed into life by contact with other studies. Their provinces had enclaves within its boundary, so that we might occasionally see a public which did not in general study history, profoundly interested in some controversy which was really historical. Church controversies kept alive an interest in one set of historical questions, and our organised classical education diffused a considerable interest in another set. It used to be said, though I fear without much justice, that Niebuhr's speculations excited more interest here than in Germany; at any rate Mr. Grote could not complain of want of appreciation. Moreover, a great nation like this cannot but feel a good deal of interest, because it feels a great deal of pride, in its own history. Certainly English history has not been able to compete for a moment with Greek and Roman either in schools or universities. I do not think it can be said that the highly-educated Englishman, as a rule, includes among his acquirements an accurate or intelligent knowledge of English history. But even the moderately-educated Englishman reads with interest whatever appears on the subject, if it is not too long and has but a reasonable seasoning of "pictorial writing." And in this country, as in other parts of Europe, there has been of late much diligence in exploring the national archives, and out of the newly-acquired materials solid historical works in no small number have been built.

But it may still be observed that the study of history, as such, is only beginning to show signs of vitality. A historical subject which is not classical or English or ecclesiastical hardly yet excites interest among us, from which it appears that we are in

terested in Greece or Rome or England or religion rather than in history. Hence it is that there is a great gap, not only in the historical knowledge of our educated class, but also in our historical literature. Modern continental history is very much neglected; no one thinks it necessary to pretend to any complete knowledge of that subject, and we have extremely few elaborate English books upon it. It seems to be supposed that no part of modern French or German history need be studied unless it is of the most thrilling interest. Books on the French Revolution and Frederick the Great have been well received, but they have been full of everything that is amazing and astonishing. I have been lately told by reviewers that it is doubtful whether the German War of Liberation is of sufficient interest to deserve careful study! Now we are not nearly so nice when the question is of one of those parts of history the study of which is really alive among us. We do not then think that the ordinary course of historical affairs is not worth attention, and that only what is exceptional and astonishing should be studied. In ancient history we follow with painful exactness the petty campaigns of the Peloponnesian War; we speculate with insatiable curiosity upon the original constitution of Rome. About the dullest periods in our own history what volumes we write and read! How eagerly we inquire who wrote Junius! But our curiosity is dormant where it has not been awakened by one of those accidental causes that I have mentioned. We write no elaborate histories of modern France or Germany or Russia, and we do not think that such histories ought to be written.

Here then I note one great deficiency; but there is another. I have spoken of English history as a subject which has some vitality. And yet it has not vitality in the same sense as Greek and Roman history. The interest in it does not with most people awaken till their education is over. schools it is almost as dead as the

In

modern languages, and so it has been till lately in the universities. Hence with most people the study of it is never more than an amusement of leisure, and accordingly it is pursued without rigorous method or purpose. The curious questions take precedence of the important ones; what is abstruse or technical is passed over lightly; and since amusement is the object, the selfdenial of sacrificing prejudices to better knowledge and of recognising unwelcome truths is little practised. And now the inquirer, being in this not too serious frame of mind, is exposed to a great temptation, which comes from party-spirit. He is, or fancies himself to be, a Whig or a Tory, a Conservative or a Liberal, and this fact has the greatest possible influence upon his studies.

Upon the direct effect of party organisation upon politics much has been written both favourably and unfavourably. After its first appearance, and through the reigns of William III. and Anne, it was supposed to be mischievous, but with the advent of the Hanover family it became supreme, and began to be pronounced beneficial.

After

being bitterly attacked by Bolingbroke, it was supposed to have been triumphantly vindicated by Burke, and since that vindication few doctrines have been more generally received among us than the indispensable use of party organisation, though occasionally a faint voice is heard suggesting that the system has its disadvantages, and may perhaps by this time have played its part. I have no intention of contributing here a single word to this controversy. But it is evident that the system must have an indirect as well as a direct tendency. The custom which enlists almost all intelligent Englishmen in every generation from youth to age in a political party, whether good or bad for its immediate purpose, must have further and very serious effects upon the national mind. If party-spirit make people one-sided, as is sometimes said, it must be a

rather serious matter to subject a whole nation deliberately and on principle to the influence of party-spirit. If the study of history be important, and that of the national history preeminently so, it is surely worth consideration whether our party-organisation is or is not unfavourable to the growth of a true and grand view of the national history. Not only in political action but in the study of English history we are all alike Big-endians and Little-endians!

and

The important point is not that we differ and form parties in politics-this would not be worth discussing because it is certainly unavoidable-but that we carry back our party differences into history. In practical politics we have a sensible rule not to disturb the settlement once fairly reached of a controverted question, Vestigia nulla retrorsum. If we could in some similar way limit our political controversies retrospectively, honestly differ about the questions of the day without allowing the dispute to spread back over all past history, no great harm would be done. The important point is that habit of generalising or idealising our partyquarrels which leads us to see them reflected in past history. It would not matter so much that we are all either Liberals or Conservatives, if we had not persuaded ourselves that this difference is but a transient phase of

an

eternal and necessary conflict between two different classes of men. But when we idealise our party-war and picture it as an Armageddon, or battle between the good and evil principles, between the children of light and the children of darkness, we are driven to assume that the Liberals and Conservatives of the present day answer to the Whigs and Tories of the Revolution, and these to the Cavaliers and Roundheads of the Civil War, and these again to the parties of Strafford and Eliot. We go further and assimilate religious parties to political. From the Reformation onwards we regard the Puritans