FOOTNOTES:

"A quantity of lifeWhich bleeds away, even as a form of wax,Resolveth from his figure 'gainst the fire?"

And Hollinshed tells us that "it was alleged against Dame Eleanor Cobham and her confederates that they had devised an image of wax, representing the king, which, by their sorcerie, by little and little consumed, intending thereby, in conclusion, to waste and destroy the king's person."

In these cases, however, the operator always depended upon certain occult or demoniacal influences, or, in other words, upon the art of magic, and therefore examples of this kind do not come within the scope of the present volume. In the case of the Powder of Sympathy the results were supposed to be due entirely to natural causes.

FOOTNOTES:[3]Touching the Cure of Wounds by the Powder of Sympathy. With Instructions how to make the said Powder. Rendered faithfully out of French into English by R. White, Gent. London, 1658.[4]Canto III. Stanza 23.

[3]Touching the Cure of Wounds by the Powder of Sympathy. With Instructions how to make the said Powder. Rendered faithfully out of French into English by R. White, Gent. London, 1658.

[4]Canto III. Stanza 23.

hissubject has now found its way not only into semi-scientific works but into our general literature and magazines. Even our novel-writers have used suggestions from this hypothesis as part of the machinery of their plots so that it properly finds a place amongst the subjects discussed in this volume.

Various attempts have been made to explain what is meant by "the fourth dimension," but it would seem that thus far the explanations which have been offered are, to most minds, vague and incomprehensible, this latter condition arising from the fact that the ordinary mind is utterly unable to conceive of any such thing as a dimension which cannot be defined in terms of the three with which we are already familiar. And I confess at the start that I labor under the superlative difficulty of not being able to form any conception of a fourth dimension, and for this incapacity my only consolation is, that in this respect I am not alone. I have conversed upon the subject with many able mathematicians and physicists, and in every case I found that they were in the same predicament as myself, and where I have met men who professed to think it easy to form a conception of a fourth dimension, I have found their ideas, not only in regard to the new hypothesis, but to its correlationswith generally accepted physical facts, to be nebulous and inaccurate.

It does not follow, however, that because myself and some others cannot form such a clear conception of a fourth dimension as we can of the third, that, therefore, the theory is erroneous and the alleged conditions non-existent. Some minds of great power and acuteness have been incapable of mastering certain branches of science. Thus Diderot, who was associated with d'Alembert, the famous mathematician, in the production of "L'Encyclopedie," and who was not only a man of acknowledged ability, but who, at one time, taught mathematics and wrote upon several mathematical subjects, seems to have been unable to master the elements of algebra. The following anecdote regarding his deficiency in this respect is given by Thiébault and indorsed by Professor De Morgan: At the invitation of the Empress, Catherine II, Diderot paid a visit to the Russian court. He was a brilliant conversationalist and being quite free with his opinions, he gave the younger members of the court circle a good deal of lively atheism. The Empress herself was very much amused, but some of her councillors suggested that it might be desirable to check these expositions of strange doctrines. As Catherine did not like to put a direct muzzle on her guest's tongue, the following plot was contrived. Diderot was informed that a learned mathematician was in possession of an algebraical demonstration of the existence of God and would give it to him before all the court if he desired to hear it. Diderot gladly consented, and although the name of the mathematician is not given, it is well known to have been Euler. He advanced toward Diderot, and said in French, gravely, and in a tone of perfect conviction: "Monsieur,(a + bn) / n = x, therefore, God exists; reply!" Diderot, to whom algebra was Hebrew, was embarrassed and disconcerted, while peals of laughter rose on all sides. He asked permission to return to France at once, which was granted.

Even such a mind as that of Buckle, who was generally acknowledged to be a keen-sighted thinker, could not form any idea of a geometrical line—that is, of a line without breadth or thickness, a conception which has been grasped clearly and accurately by thousands of school-boys. He therefore asserts, positively, that there are no lines without breadth, and comes to the following extraordinary conclusions:

"Since, however, the breadth of the faintest line is so slight as to be incapable of measurement, except by an instrument under the microscope, it follows that the assumption that there can be lines without breadth is so nearly true that our senses, when unassisted by art, can not detect the error. Formerly, and until the invention of the micrometer, in the seventeenth century, it was impossible to detect it at all. Hence, the conclusions of the geometrician approximate so closely to truth that we are justified in accepting them as true. The flaw is too minute to be perceived. But that there is a flaw appears to me certain. It appears certain that, whenever something is kept back in the premises, something must be wanting in the conclusion. In all such cases, the field of inquiry has not been entirely covered; and part of the preliminary facts being suppressed, it must, I think, be admitted that complete truth be unattainable, and that no problem in geometry has been exhaustively solved."[5]

The fallacy which underlies Mr. Buckle's contention is thus clearly exposed by the author of "The Natural History of Hell."

"If it be conceded that lines have breadth, then all we have to do is to assign some definite breadth to each line—say the one-thousandth of an inch—and allow for it. But the lines of the geometer have no breadth. All the micrometers of which Mr. Buckle speaks depend, either directly or indirectly, upon lines for their graduations, and the positions of these lines are indicated by rulings or scratches. Now, in even the finest of these rulings, as, for example, those of Nobert or Fasoldt, where the ruling or scratching, together with its accompanying space, amounts to no more than the one hundred and fifty thousandth part of an inch, the scratch has a perceptible breadth. But this broad scratch is not the line recognized by the microscopist, to say nothing of the geometer. The true line is a line which lies in the very center of this scratch and it is certain that this central line has absolutely no breadth at all."[6]

It must be very evident that if Mr. Buckle's contention that geometrical lines have breadth were true, then some of the fundamental axioms of geometry must be false. It could no longer hold true that "the whole is equal to all its parts taken together," for if we divide a square or a circle into two parts by means of a line which has breadth, the two parts cannot be equal to the whole as it formerly was. As a matter of fact, Mr. Buckle's lines are saw-cuts, not geometrical lines. Geometrical points, lines, and surfaces, have no material existence and can have none. An ideal conception and a material existence are two very different things.

A very interesting book[7]has been written on the movements and feelings of the inhabitants of a world of two dimensions. Nevertheless, if we know anything at all, we know that such a world could not have any actual existenceand when we attempt to form any mental conception of it and its inhabitants, we are compelled to adopt, to a certain extent, the idea of the third dimension.

But at the same time we must remember that since the ordinary mechanic and the school-boy who has studied geometry, find no difficulty in conceiving of points without magnitude, lines without breadth, and surfaces without thickness—conceptions which seem to have been impossible to Buckle, a man of acknowledged ability—it may be possible that minds constituted slightly differently from that of myself and some others, might, perhaps, be able to form a conception of a fourth dimension.

Leaving out of consideration the speculations of those who have woven this idea into romances and day-dreams we find that the hypothesis of a fourth dimension has been presented by two very different classes of thinkers, and the discussion has been carried on from two very different standpoints.

The first suggestion of this hypothesis seems to have come from Kant and Gauss and to have had a purely metaphysical origin, for, although attempts have been made to trace the idea back to the famous phantoms of Plato, it is evident that the ideas then advanced had nothing in common with the modern theory of the existence of a fourth dimension. The first hint seems to have been a purely mathematical one and did not attract any very general attention. It was, however, seized upon by a certain branch of the transcendentalists, closely allied to the spiritualists, and was exploited by them as a possible explanation of some curious and mysterious phenomena and feats exhibited by certain Indian and European devotees. This may have been done merely for the purpose of mystifying and confoundingtheir adversaries by bringing forward a striking illustration of Hamlet's famous dictum—

"There are more things in heaven and earth, Horatio,Than are dreamt of in your philosophy."

A very fair statement of this view is thus given by Edward Carpenter:[8]

"There is another idea which modern science has been familiarizing us with, and which is bringing us towards the same conception—that, namely, of the fourth dimension. The supposition that the actual world has four space-dimensions instead of three makes many things conceivable which otherwise would be incredible. It makes it conceivable that apparently separate objects, e. g., distinct people, are really physically united; that things apparently sundered by enormous distances of space are really quite together; that a person or other object might pass in and out of a closed room without disturbance of walls, doors or windows, etc., and if this fourth dimension were to become a factor of our consciousness it is obvious that we should have means of knowledge which, to the ordinary sense, would appear simply miraculous. There is much, apparently, to suggest that the consciousness attained to by the Indian gñanis in their degree, and by hypnotic subjects in theirs, is of this fourth dimensional order."As a solid is related to its own surface, so, it would appear, is the cosmic consciousness related to the ordinary consciousness. The phases of the personal consciousness are but different facets of the other consciousness; and experiences which seem remote from each other in the individual are perhaps all equally near in the universal. Space itself, as we know it, may be practically annihilated in the consciousness of a larger space, of which it is but the superficies; and a person living in London may not unlikely find that he has a back door opening quite simply and unceremoniously out in Bombay."

"As a solid is related to its own surface, so, it would appear, is the cosmic consciousness related to the ordinary consciousness. The phases of the personal consciousness are but different facets of the other consciousness; and experiences which seem remote from each other in the individual are perhaps all equally near in the universal. Space itself, as we know it, may be practically annihilated in the consciousness of a larger space, of which it is but the superficies; and a person living in London may not unlikely find that he has a back door opening quite simply and unceremoniously out in Bombay."

On the other hand, the mathematicians, looking at it as a purely speculative idea, have endeavored to arrive atdefinite conclusions in regard to what would be the condition of things if the universe really exists in a fourth, or even in some higher dimension. Professor W. W. R. Ball tells us that

"the conception of a world of more than three dimensions is facilitated by the fact that there is no difficulty in imagining a world confined to only two dimensions—which we may take for simplicity to be plane—though equally well it might be a spherical or other surface. We may picture the inhabitants of flatland as moving either on the surface of a plane or between two parallel and adjacent planes. They could move in any direction along the plane, but they could not move perpendicularly to it, and would have no consciousness that such a motion was possible. We may suppose them to have no thickness, in which case they would be mere geometrical abstractions; or we may think of them as having a small but uniform thickness, in which case they would be realities.""If an inhabitant of flatland was able to move in three dimensions, he would be credited with supernatural powers by those who were unable so to move; for he could appear or disappear at will; could (so far as they could tell) create matter or destroy it, and would be free from so many constraints to which the other inhabitants were subject that his actions would be inexplicable to them.""Our conscious life is in three dimensions, and naturally the idea occurs whether there may not be a fourth dimension. No inhabitant of flatland could realize what life in three dimensions would mean, though, if he evolved an analytical geometry applicable to the world in which he lived, he might be able to extend it so as to obtain results true of that world in three dimensions which would be to him unknown and inconceivable. Similarly we cannot realize what life in four dimensions is like, though we can use analytical geometry to obtain results true of that world, or even of worlds of higher dimensions. Moreover, the analogy of our position to the inhabitants of flatland enablesus to form some idea of how inhabitants of space of four dimensions would regard us.""If a finite solid was passed slowly through flatland, the inhabitants would be conscious only of that part of it which was in their plane. Thus they would see the shape of the object gradually change and ultimately vanish. In the same way, if a body of four dimensions was passed through our space, we should be conscious of it only as a solid body (namely, the section of the body by our space) whose form and appearance gradually changed and perhaps ultimately vanished. It has been suggested that the birth, growth, life, and death of animals, may be explained thus as the passage of finite four-dimensional bodies through our three-dimensional space."

"If an inhabitant of flatland was able to move in three dimensions, he would be credited with supernatural powers by those who were unable so to move; for he could appear or disappear at will; could (so far as they could tell) create matter or destroy it, and would be free from so many constraints to which the other inhabitants were subject that his actions would be inexplicable to them."

"Our conscious life is in three dimensions, and naturally the idea occurs whether there may not be a fourth dimension. No inhabitant of flatland could realize what life in three dimensions would mean, though, if he evolved an analytical geometry applicable to the world in which he lived, he might be able to extend it so as to obtain results true of that world in three dimensions which would be to him unknown and inconceivable. Similarly we cannot realize what life in four dimensions is like, though we can use analytical geometry to obtain results true of that world, or even of worlds of higher dimensions. Moreover, the analogy of our position to the inhabitants of flatland enablesus to form some idea of how inhabitants of space of four dimensions would regard us."

"If a finite solid was passed slowly through flatland, the inhabitants would be conscious only of that part of it which was in their plane. Thus they would see the shape of the object gradually change and ultimately vanish. In the same way, if a body of four dimensions was passed through our space, we should be conscious of it only as a solid body (namely, the section of the body by our space) whose form and appearance gradually changed and perhaps ultimately vanished. It has been suggested that the birth, growth, life, and death of animals, may be explained thus as the passage of finite four-dimensional bodies through our three-dimensional space."

Attempts have been made to construct drawings and models showing a four-dimensional body. The success of such attempts has not been very encouraging.

Investigators of this class look upon the actuality of a fourth dimension as an unsolved question, but they hold that, provided we could see our way clear to adopt it, it would open up wondrous possibilities in the way of explaining abstruse and hitherto inexplicable physical conditions and phenomena.

There is obviously no limit to such speculations, provided we assume the existence of such conditions as are needed for our purpose. Too often, however, those who indulge in such day-dreams begin by assuming the impossible, and end by imagining the absurd.

We have so little positive knowledge in regard to the ultimate constitution of matter and even in regard to the actual character of the objects around us, which are revealed to us through our senses, that the field in which our imagination may revel is boundless. Perhaps some day thehumanity of the present will merge itself into a new race, endowed with new senses, whose revelations are to us, for the present, at least, utterly inconceivable.

The possibility of such a development may be rendered more clear if we imagine the existence of a race devoid of the sense of hearing, and without the organs necessary to that sense. They certainly could form no idea of sound, far less could they enjoy music or oratory, such as afford us so much delight. And, if one or more of our race should visit these people, how very strange to them would appear those curious appendages, called ears, which project from the sides of our heads, and how inexplicable to them would be the movements and expressions of intelligence which we show when we talk or sing? It is certain that no development of the physical or mathematical sciences could give them any idea whatever of the sensations which sound, in its various modifications, imparts to us, and neither can any progress in that direction enable us to acquire any idea of the revelations which a new sense might open up to us. Nevertheless, it seems to me that the development of new senses and new sense organs is not only more likely to be possible, but that it is actually more probable, than any revelation in regard to a fourth dimension.

FOOTNOTES:[5]"History of Civilization in England." American edition, Vol. II, page 342.[6]"The Natural History of Hell," by John Phillipson, page 37.[7]"Flatland," by E. A. Abbott. London, 1884.[8]"From Adam's Peak to Elephanta—" page 160.

[5]"History of Civilization in England." American edition, Vol. II, page 342.

[6]"The Natural History of Hell," by John Phillipson, page 37.

[7]"Flatland," by E. A. Abbott. London, 1884.

[8]"From Adam's Peak to Elephanta—" page 160.

hefollowing is a curious illustration of the errors to which careless observers may be subject:

Draw a square, like Fig. 19, and divide the sides into 8 parts each. Join the points of division in opposite sides so as to divide the whole square into 64 small squares. Then draw the lines shown in black and cut up the drawing into four pieces. The lines indicating the cuts have been made quite heavy so as to show up clearly, but on the actual card they may be made quite light. Now, put the four pieces together, so as to form the rectangle shown in Fig. 20. Unless the scale, to which the drawing is made is quite large and the work very accurate, it will seem that the rectangle contains 5 squares one way and 13 the other which, when multiplied together, give 65 for the number of small squares, being an apparent gain of one square by the simple process of cutting.

Fig. 19.Fig. 20.

Fig. 19.

Fig. 20.

This paradox is very apt to puzzle those who are not familiar with accurate drawings. Of course, every person of common sense knows that the card or drawing is not made any larger by cutting it, but where does the 65th small square come from?

On careful examination it will be seen that the line AB, Fig. 20, is not quite straight and the three parts into which it is divided are thus enabled to gain enough to make one of the small squares. On a small scale this deviation from the straight line is not very obvious, but make a larger drawing, and make it carefully, and it will readily be seen how the trick is done.

thinkit was the elder Stephenson, the famous engineer, who told a man who claimed the honor of having invented a perpetual motion, that when he could lift himself over a fence by taking hold of his waist-band, he might hope to accomplish his object. And the query which serves as a title for this article has long been propounded as one of the physical impossibilities. And yet, perhaps, it might be possible to invent a waist-band or a boot-strap by which this apparently impossible feat might be accomplished!

Travelers in Mexico frequently bring home beans which jump about when laid on a table. They are well-known as "jumping beans" and have often been a puzzle to those who were not familiar with the facts in the case. Each bean contains the larva of a species of beetle and this affords a clue to the secret. But the question at once comes up: "How is the insect able to move, not only itself, but its house as well, without some purchase or direct contact with the table?"

The explanation is simple. The hollow bean is elastic and the insect has strength enough to bend it slightly; when the insect suddenly relaxes its effort and allows the bean to spring back to its former shape, the reaction on the table moves the bean. A man placed in a perfectly rigid box could never move himself by pressing on the sides, but if the box were elastic and could be bent by the strength of the man inside, it might be made to move.

A somewhat analogous result, but depending on different principles, is attained in certain curious boat races which are held at some English regattas and which is explained by Prof. W. W. Rouse Ball, in his "Mathematical Recreations and Problems." He says that it

"affords a somewhat curious illustration of the fact that commonly a boat is built so as to make the resistance to motion straight forward less than that to motion in the opposite direction."The only thing supplied to the crew is a coil of rope, and they have (without leaving the boat) to propel it from one point to another as rapidly as possible. The motion is given by tying one end of the rope to the afterthwart, and giving the other end a series of violent jerks in a direction parallel to the keel."The effect of each jerk is to compress the boat. Left to itself the boat tends to resume its original shape, but the resistance to the motion through the water of the stern is much greater than that of the bow, hence, on the whole, the motion is forwards. I am told that in still water a pace of two or three miles an hour can be thus attained."

"affords a somewhat curious illustration of the fact that commonly a boat is built so as to make the resistance to motion straight forward less than that to motion in the opposite direction.

"The only thing supplied to the crew is a coil of rope, and they have (without leaving the boat) to propel it from one point to another as rapidly as possible. The motion is given by tying one end of the rope to the afterthwart, and giving the other end a series of violent jerks in a direction parallel to the keel.

"The effect of each jerk is to compress the boat. Left to itself the boat tends to resume its original shape, but the resistance to the motion through the water of the stern is much greater than that of the bow, hence, on the whole, the motion is forwards. I am told that in still water a pace of two or three miles an hour can be thus attained."

neof the most interesting books in natural history is a work on "Insect Architecture," by Rennie. But if the architecture of insect homes is wonderful, the engineering displayed by these creatures is equally marvellous. Long before man had thought of the saw, the saw-fly had used the same tool, made after the same fashion, and used in the same way for the purpose of making slits in the branches of trees so that she might have a secure place in which to deposit her eggs. The carpenter bee, with only the tools which nature has given her, cuts a round hole, the full diameter of her body, through thick boards, and so makes a tunnel by which she can have a safe retreat, in which to rear her young. The tumble-bug, without derrick or machinery, rolls over large masses of dirt many times her own weight, and the sexton beetle will, in a few hours, bury beneath the ground the carcass of a comparatively large animal. All these feats require a degree of instinct which in a reasoning creature would be called engineering skill, but none of them are as wonderful as the feats performed by the spider. This extraordinary little animal has the faculty of propelling her threads directly against the wind, and by means of her slender cords she can haul up and suspend bodies which are many times her own weight.

Some years ago a paragraph went the rounds of the papers in which it was said that a spider had suspended an unfortunate mouse, raising it up from the ground, andleaving it to perish miserably between heaven and earth. Would-be philosophers made great fun of this statement, and ridiculed it unmercifully. I know not how true itwas, but I know that itmight have beentrue.

Some years ago, in the village of Havana, in the State of New York, a spider entangled a milk-snake in her threads, and actually raised it some distance from the ground, and this, too, in spite of the struggles of the reptile, which was alive.

By what process of engineering did the comparatively small and feeble insect succeed in overcoming and lifting up by mechanical means, the mouse or the snake? The solution is easy enough if we only give the question a little thought.

The spider is furnished with one of the most efficient mechanical implements known to engineers, viz., a strong elastic thread. That the thread is strong is well known. Indeed, there are few substances that will support a greater strain than the silk of the silkworm, or the spider; careful experiment having shown that for equal sizes the strength of these fibers exceeds that of common iron. But notwithstanding its strength, the spider's thread alone would be useless as a mechanical power if it were not for its elasticity. The spider has no blocks or pulleys, and, therefore, it cannot cause the thread to divide up and run in different directions, but the elasticity of the thread more than makes up for this, and renders possible the lifting of an animal much heavier than a mouse or a snake. This may require a little explanation.

Let us suppose that a child can lift a six-pound weight one foot high and do this twenty times a minute. Furnish him with 350 rubber bands, each capable of pulling six pounds through one foot when stretched. Let these bandsbe attached to a wooden platform on which stand a pair of horses weighing 2,100 lbs., or rather more than a ton. If now the child will go to work and stretch these rubber bands, singly, hooking each one up, as it is stretched, in less than twenty minutes he will have raised the pair of horses one foot!

We thus see that the elasticity of the rubber bands enables the child to divide the weight of the horses into 350 pieces of six pounds each, and at the rate of a little less than one every three seconds, he lifts all these separate pieces one foot, so that the child easily lifts this enormous weight.

Each spider's thread acts like one of the elastic rubber bands. Let us suppose that the mouse or the snake weighed half an ounce and that each thread is capable of supporting a grain and a half. The spider would have to connect the mouse with the point from which it was to be suspended with 150 threads, and if the little quadruped was once swung off his feet, he would be powerless. By pulling successively on each thread and shortening it a little, the mouse or snake might be raised to any height within the capacity of the building or structure in which the work was done. So that to those who have ridiculed the story we may justly say: "There are more things in heaven and earth than are dreamed of inyourphilosophy."

What object the spider could have had in this work I am unable to see. It may have been a dread of the harm which the mouse or snake might work, or it may have been the hope that the decaying carcass would attract flies which would furnish food for the engineer. I can vouch for the truth of the snake story, however, and the object of this article is to explain and render credible a very extraordinary feat of insect engineering.

nthe twentieth chapter of II Kings, at the eleventh verse we read, that "Isaiah the prophet cried unto the Lord, and he brought the shadow ten degrees backward, by which it had gone down in the dial of Ahaz."

It is a curious fact, first pointed out by Nonez, the famous cosmographer and mathematician of the sixteenth century, but not generally known, that by tilting a sun-dial through the proper angle, the shadow at certain periods of the year can be made, for a short time, to move backwards on the dial. This was used by the French encyclopædists as a rationalistic explanation of the miracle which is related at the opening of this article.

The reader who is curious in such matters will find directions for constructing "a dial, for any latitude, on which the shadow shall retrograde or move backwards," in Ozanam's "Recreations in Science and Natural Philosophy," Riddle's edition, page 529. Professor Ball in his "Mathematical Recreations," page 214, gives a very clear explanation of the phenomenon. The subject is somewhat too technical for these pages.

everalyears ago a correspondent of "Truth" (London) gave the following simple directions for finding the points of the compass by means of the ordinary pocket watch: "Point the hour hand to the sun, and south is exactly half way between the hour hand and twelve on the watch, counting forward up to noon, but backward after the sun has passed the meridian."

Professor Ball, in his "Mathematical Recreations and Problems," gives more complete directions and explanations. He says:

"The position of the sun relative to the points of the compass determines the solar time. Conversely, if we take the time given by a watch as being the solar time (and it will differ from it only by a few minutes at the most), and we observe the position of the sun, we can find the points of the compass. To do this it is sufficient to point the hour-hand to the sun and then the direction which bisects the angle between the hour and the figure XII will point due south. For instance, if it is four o'clock in the afternoon, it is sufficient to point the hour-hand (which is then at the figure IIII) to the sun, and the figure II on the watch will indicate the direction of south. Again, if it is eight o'clock in the morning, we must point the hour-hand (which is then at the figure VIII) to the sun, and the figure X on the watch gives the south point of the compass."Between the hours of six in the morning and six in the evening the angle between the hour and XII, which must be bisected is less than 180 degrees, but at other times the angle to be bisected is greater than 180 degrees; or perhaps it is simpler to say that at other times the rule gives the north point and not the south point."The reason is as follows: At noon the sun is duesouth, and it makes one complete circuit round the points of the compass in 24 hours. The hour-hand of a watch also makes one complete circuit in 12 hours. Hence, if the watch is held with its face in the plane of the ecliptic, and the figure XII on the dial is pointed to the south, both the hour-hand and the sun will be in that direction at noon. Both move round in the same direction, but the angular velocity of the hour-hand is twice as great as that of the sun. Hence the rule. The greatest error due to the neglect of the equation of time is less than 2 degrees. Of course, in practice, most people would hold the face of the watch horizontal, and in our latitude (that of London) no serious error would thus be introduced."In the southern hemisphere, or in any tropical country where at noon the sun is due north, the rule will give the north point instead of the south."

"Between the hours of six in the morning and six in the evening the angle between the hour and XII, which must be bisected is less than 180 degrees, but at other times the angle to be bisected is greater than 180 degrees; or perhaps it is simpler to say that at other times the rule gives the north point and not the south point.

"The reason is as follows: At noon the sun is duesouth, and it makes one complete circuit round the points of the compass in 24 hours. The hour-hand of a watch also makes one complete circuit in 12 hours. Hence, if the watch is held with its face in the plane of the ecliptic, and the figure XII on the dial is pointed to the south, both the hour-hand and the sun will be in that direction at noon. Both move round in the same direction, but the angular velocity of the hour-hand is twice as great as that of the sun. Hence the rule. The greatest error due to the neglect of the equation of time is less than 2 degrees. Of course, in practice, most people would hold the face of the watch horizontal, and in our latitude (that of London) no serious error would thus be introduced.

"In the southern hemisphere, or in any tropical country where at noon the sun is due north, the rule will give the north point instead of the south."

inuteworks of art have always excited the curiosity and commanded the admiration of the average man. Consequently Cicero thought it worth while to record that the entire Iliad of Homer had been written upon parchment in characters so fine that the copy could be enclosed in a nutshell. This has always been regarded as a marvelous feat.

There is in the French Cabinet of Medals a seal, said to have belonged to Michael Angelo, the fabrication of which must date from a very remote epoch, and upon which fifteen figures have been engraved in a circular space of fourteen millimeters (.55 inch) in diameter. These figures cannot be distinguished by the naked eye.

The Ten Commandments have been engraved in characters so fine that they could be stamped upon one side of a nickle five-cent piece, and on several occasions the Lord's Prayer has been engraved on one side of a gold dollar, the diameter of which is six-tenths of an inch. I have also seen it written with a pen within a circle which measured four-tenths of an inch in diameter.

In the Harleian manuscript, 530, there is an account of a "rare piece of work, brought to pass by Peter Bales, an Englishman, and a clerk of the chancery." D'Israeli tells us that it was "The whole Bible in an English walnut, no bigger than a hen's egg. The nut holdeth the book: there are as many leaves in his little book as in the great Bible,and he hath written as much in one of his little leaves as a great leaf of the Bible."

By most people, such achievements are considered marvels of skill, and the newspaper accounts of them which are published always attract special attention. And it must be acknowledged that such work requires good eyes, steady nerves, and very delicate control of the muscles. But with ordinary writing materials there are certain mechanical limitations which must prevent even the most skilful from going very far in this direction. These limitations are imposed by the fiber or grain of the paper and the construction of the ordinary pen, neither of which can be carried beyond a certain very moderate degree of fineness. Of course, the paper that is chosen will be selected on account of its hard, even-grained surface, and the pen will be chosen on account of the quality of its material and its shape, and the point is always carefully dressed on a whetstone so as to have both halves of the nib equal in strength and length, and the ends smooth and delicate. When due preparation has been made, and when the eyes and nerves of the writer are in good condition, the smallness of the distinctly readable letters that may be produced is wonderful. And in this connection it is an interesting fact that in many mechanical operations, writing included, the hand is far more delicate than the eye. That which the unaided eye can see to write, the unaided eye can see to read, but the hand, without the assistance or guidance of the eye, can produce writing so minute that the best eyes cannot see to read it, and yet, when viewed under a microscope, it is found to compare favorably with the best writing of ordinary size. And those who are conversant with the more delicate operations of practical mechanics, know that this is no exceptionalcase. The only aid given by the eye in the case of such minute writing is the arrangement of the lines, otherwise the writing could be done as well with the eyes shut as open.

Since the mechanical limitations which we have noted prevent us from going very far with the instruments and materials mentioned, the next step is to adopt a finer surface and a sharper point. These conditions may be found in the fine glazed cards and the metal pencils or styles used by card writers. In these cards the surface is nearly homogeneous, that is to say, free from fibers, and the point of the metal pencil may be made as sharp as a needle, but to utilize these conditions to the fullest extent, it is necessary to aid the eye, and a magnifier is, therefore, brought into use. Under a powerful glass the hand may be so guided by the eye that the writing produced cannot be read by the unaided vision.

The specimens of fine writing thus far described have been produced directly by the hand under the guidance either of a magnifier or the simple sense of motion. Just how far it would be possible to go by these means has never been determined, so far as I know, but those who have examined the specimens of selected diatoms and insect scales in which objects that are utterly invisible to the naked eye are arranged with great accuracy so as to form the most beautiful figures, can readily believe that a combination of microscopical dexterity and skill in penmanship might easily go far beyond anything that has yet been accomplished in this direction, either in ancient or modern times.

But by means of a very simple mechanical arrangement, the motion of the hand in every direction may be accuratelyreduced or enlarged to almost any extent, and it thus becomes possible to form letters which are inconceivably small. The instrument by which this is accomplished is known as a pantagraph, and it has, within a few years, become quite popular as a means of reducing or enlarging pictures of various kinds, including crayon reproductions of photographs. Its construction and use are, therefore, very generally understood. It was by means of a very finely-made instrument embodying the principles of the pantagraph that the extraordinarily fine work which we are about to describe was accomplished.

It is obvious, however, that in order to produce very fine writing we must use a very fine pen or point and the finer the point the sooner does it wear out, so that in a very short time the lines which go to form the letters become thick and blurred and the work is rendered illegible. As a consequence of this, when the finest specimens of writing are required, it is necessary to abandon the use of ordinary points and surfaces and to resort to the use of the diamond for a pen, and glass for a surface upon which to write. One of the earliest attempts in this direction was that of M. Froment, of Paris, who engraved on glass, within a circle, the one-thirtieth of an inch in diameter, the Coat of Arms of England—lion, unicorn, and crown—with the following inscription, partly in Roman letters, partly in script: "Honi soit qui mal y pense, Her Most Gracious Majesty, Queen Victoria, and His Royal Highness, Prince Albert,Dieu et mon droit. Written on occasion of the Great Exhibition, by Froment, à Paris, 1851."

The late Dr. Barnard, President of Columbia College, had in his possession a copy of the device borne by the seal of Columbia College, New York, executed for him by M.Dumoulin-Froment, within a circle less than three one-hundredths of an inch in diameter, "in which are embraced four human figures and various other objects, together with inscriptions in Latin, Greek, and Hebrew, all clearly legible. In this device the rising sun is represented in the horizon, the diameter of the disk being about three one-thousandths of an inch. This disk has been cross-hatched by the draughtsman in the original design from which the copy was made; and the copy shows the marks of the cross-hatching with perfect distinctness. When this beautiful and delicate drawing is brought clearly out by a suitably adjusted illumination, the lines appear as if traced by a smooth point in a surface of opaque ice."

Lardner, in his book on the "Microscope," published in 1856, gives a wood cut which shows the first piece of engraving magnified 120 diameters, but he said that he was not at liberty to describe the method by which it was done. As happens in almost all such cases, however, the very secrecy with which the process was surrounded naturally stimulated others to rival or surpass it, and Mr. N. Peters, a London banker, turned his attention to the subject and soon invented a machine which produced results far exceeding anything that M. Froment had accomplished. On April 25, 1855, Mr. Farrants read before the Microscopical Society of London a full account of the Peters machine, with which the inventor had written the Lord's Prayer (in the ordinary writing character, without abbreviation or contraction of any kind), in a space not exceeding the one hundred and fifty-thousandth of a square inch. Seven years later, Mr. Farrants, as President of the Microscopical Society, described further improvements in the machine of Mr. Peters, and made the following statement: "TheLord's Prayer has been written and may be read in the one-three hundred and fifty-six thousandth of an English square inch. The measurements of one of these specimens was verified by Dr. Bowerbank, with a difference of not more than one five-millionth of an inch, and that difference, small as it is, arose from his not including the prolongation of the letterfin the sentence 'deliver us from evil'; so he made the area occupied by the writing less than that stated above."

Some idea of the minuteness of the characters in these specimens may be obtained from the statement that the whole Bible and Testament, in writing of the same size, might be placed twenty-two times on the surface of a square inch. The grounds for this startling assertion are as follows: "The Bible and Testament together, in the English language, are said to contain 3,566,480 letters. The number of letters in the Lord's Prayer, as written, ending in the sentence, 'deliver us from evil,' is 223, whence, as 3,566,480 divided by 223, is equal to 15,922, it appears that the Bible and Testament together contain the same number of letters as the Lord's Prayer written 16,000 times; if then the prayer were written in 1-16,000 of an inch, the Bible and Testament in writing of the same size would be contained by one square inch; but as 1-356,000th of an inch is one twenty-secondth part of 1-15,922 of an inch, it follows that the Bible and Testament, in writing of that size, would occupy less space than one twenty-secondth of a square inch."

It only now remains to be seen that, minute as are the letters written by this machine, they are characterized by a clearness and precision of form which proves that the moving parts of the machine, while possessing the utmostdelicacy of freedom, are absolutely destitute of shake, a union of requisites very difficult of fulfilment, but quite indispensable to the satisfactory performance of the apparatus.

I have no information in regard to the present whereabouts of any of the specimens turned out by Mr. Peters, and inquiry in London, among persons likely to know, has not supplied any information on the subject.

There was, however, another micrographer, Mr. William Webb, of London, who succeeded in producing some marvellous results. Epigrams and also the Lord's Prayer written in the one-thousandth part of a square inch have been freely distributed. Mr. Webb also produced a few copies of the second chapter of the Gospel, according to St. John, written on the scale of the whole Bible, to a little more than three-quarters of a square inch, and of the Lord's Prayer written on the scale of the whole Bible eight times on a square inch. Mr. Webb died about fifteen years ago, and I believe he has had no successor in the art. Specimens of his work are quite scarce, most of them having found their way into the cabinets of public Museums and Societies, who are unwilling to part with them. The late Dr. Woodward, Director of the Army Medical Museum, Washington, D.C., procured two of them on special order for the Museum. Mr. Webb had brought out these fine writings as tests for certain qualities of the microscope, and it was to "serve as tests for high-power objectives" that Dr. Woodward procured the specimens now in the microscopical department of the Museum. I am so fortunate as to have in my possession two specimen's of Mr. Webb's work. One is an ordinary microscopical glass slide, three inches by one, and in the center is a square speck whichmeasures 1-45th of an inch on the side. Upon this square is written the whole of the second chapter of the Gospel according to St. John—the chapter which contains the account of the marriage in Cana of Galilee.

In order to estimate the space which the whole Bible would occupy if written on the same scale as this chapter, I have made the following calculation which, I think, will be more easily followed and checked by my readers, than that of Mr. Farrants.

The text of the old version of the Bible, as published in minion by the American Bible Society, contains 1272 pages, exclusive of title pages and blanks. Each page contains two columns of 58 lines each, making 116 lines to the page. This includes the headings of the chapters and the synopses of their contents, which are, therefore, thrown in to make good measure. We have, therefore, 1272 pages of 116 lines each, making a total of 147,552 lines.

The second chapter of St. John has 25 verses containing 95 lines, and is written on the 1-2025th of an inch, or, in other words, it would go 2025 times on a square inch. A square inch would, therefore, contain 95 × 2025 or 192,375 lines. This number (192,375), divided by the number of lines in the Bible (147,552), gives 1.307, which is the number of times the Bible might be written on a square inch in letters of the same size. In other words, the whole Bible might be written on .77 inch, or very little more than three-quarters of a square inch.

Perhaps the following gives a more impressive illustration: The United States silver quarter of a dollar is .95 inch in diameter, so that the surface of each side is .707 of a square inch. The whole Bible would, therefore, very nearly go onone side of a quarter of a dollar. If the blank spaces at the heads of the chapters and the synopses of contents were left out, it would easily go on one side.

The second specimen, which I have of Mr. Webb's writing, is a copy of the Lord's Prayer written on a scale of eight Bibles to the square inch. According to a statement kindly sent me by the superintendent of the United States Mint at Philadelphia, the diameter of the last issued gold dollar, and also of the silver half-dime, is six-tenths of an inch. This gives .2827+ of a square inch as the area of the surface of one side, and, therefore, the whole Bible might be written more than two and a quarter times on one side of either the gold dollar or the silver half dime.

Such numerical and space relations are far beyond the power of any ordinary mind to grasp. With the aid of a microscope we can see the object and compare with other magnifications the rate at which it is enlarged, and a person of even the most ordinary education can follow the calculation and understand why the statements are true, but the final result, like the duration of eternity or the immensity of space, conveys no definite idea to our minds.

But at the same time we must carefully distinguish between our want of power to grasp these ideas and our inability to form a conception of some inconceivable subject, such as a fourth dimension or the mode of action of a new sense.

Wonderful as these achievements are, there is another branch of the microscopic art which, from the practical applications that have been made of it, is even more interesting. This is the art of microphotography.

About the middle of the last century Mr. J. B. Dancer, of Manchester, England, produced certain minute photographsof well-known pictures and statues which commanded the universal attention of the microscopists of that day, and for a time formed the center of attraction at all microscopical exhibitions. They have now, however, become so common that they receive no special notice. Mr. Dancer and other artists also produced copies of the Lord's Prayer, the Creed, the Declaration of Independence, etc., on such a scale that the Lord's Prayer might be covered with the head of a common pin, and yet, when viewed under a very moderate magnifying power, every letter was clear and distinct. I have now before me a slip of glass, three inches long and one inch wide, in the center of which is an oval photograph which occupies less than the 1-200th of a square inch. This photograph contains the Declaration of Independence with the signatures of all the signers, surrounded by portraits of the Presidents and the seals of the original thirteen States. Under a moderate power every line is clear and distinct. In the same way copies of such famous pictures as Landseer's "Stag at Bay," although almost invisible to the naked eye, come out beautifully clear and distinct under the microscope, so that it has been suggested that one might have an extensive picture gallery in a small box, or pack away copies of all the books in the Congressional Library in a small hand-bag. With such means at our command, it would be a simple matter to condense a bulky dispatch into a few little films, which might be carried in a quill or concealed in ways which would have been impossible with the original. If Major André had been able to avail himself of this mode of reducing the bulk of the original papers, he might have carried, without danger of discovery, those reports which caused his capture and led to his death. Andhereafter the ordinary methods of searching suspected spies will have to be exchanged for one that is more efficient.

The most interesting application of microphotography, of which we have any record, occurred during the Franco-Prussian war in 1870-71.

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