that consist wholly of the carcases of mammoths and other giant vertebrates, piled upon one another from the sea bottom to the surface, and so well preserved that their flesh is eaten by Esquimaux and arctic carnivora. Really, our friends Sternberg, Henry, Fairfield, Osburn, and other paleontologists should be informed by wire of these inexhaustible deposits, so they will not waste any more time on the barren deserts of Africa and our own West. The change of climate in polar regions, from tropic heat to arctic cold, was so sudden that many of these huge beasts were "frozen while peacefully grazing on the previously unfrosted vegetation." One is reminded of the naive remark of a four-year-old boy, after gazing on some good specimens of taxidermal skill-"How did they get them dead standing?"

The laws of the conservation of matter and force are points for attack upon prevailing theories. Inventions have been multiplied to secure perpetual motion, or to do work without the expenditure of an equivalent amount of energy. For twenty odd years a man by the name of Keeley induced the public, including many capitalists, to believe he had discovered an occult force hitherto unknown, that was utilized in his laboratories to run a machine, called after him the Keeley motor. He was constantly on the point of perfecting it so it could be put to practical use. For a quarter of a century hard-headed business men of New York, after a look into his laboratory at the motor buzzing away without any apparent expenditure of energy, invested thousands of dollars to enable him to complete his invention. After his death, an investigation of the laboratory revealed that the occult force was plain. compressed air contained in tubes concealed beneath the floor, and Keeley slipped into his place among the fakirs.

Last year the papers of the state contained descriptions of an application of a windmill to secure locomotion. Its inventor, a Kansan, gravely claimed his machine would run faster against the wind than with it, because the windmill would turn faster in such case. Mr. Chas. Trippler, of New York, the first man to manufacture liquid air in large quantities in this country, in an article in the McClure, several years ago, claimed he could run his machine with liquid air and at the same time produce more liquid air than he used for motive power. He has probably reread and meditated upon the law of conservation since that date.

Radio-activity is sometimes referred to as violating the law of conservation, because radium and other radio-active elements seem to give off energy and emanations without loss of matter or force. This is only an apparent contravention, however, because radium and all other radio-active substances do lose both matter and energy through their emanations, but so slowly as not to be easily detected. Radio-activity is now known to consist in a slow disintegration, a breaking up into less complex elemental substances of lower density.

There are other scientific pretensions which ought not to be euphemistically considered fallacies, because their promulgators are not self-deceived, except in so far as they think they can deceive all the scientific world all the time, for fraud, like murder, will out, sometime, somewhere. The less-pleasing term, fake or fraud, is more accurate and apt. The famous Cardiff giant is sufficiently distant, in scientific time at least, to excite only a reminiscent smile. It has had many successors, but none so successful. The widely-advertised Calaveras skull is not so ancient as not to cause a wry face and nausea even as "Poor Yorick's." That the cranium of a Digger Indian should have been accepted as that of Tertiary man, even by the elect, is not a pleasant thought. A most amusing instance of attempted fraud fell under my observation a few years ago. Stepping into a clothing establishment, my attention was called to a very fine display of sea life in a large wall case. There were seaweed, sponges, coral, flying-fish, etc.; and last, but not least by any means, a perfectly preserved specimen of a mermaid. There it was before my wondering eyes as plain as it ever manifested itself to the gaze of any mediæval seafaring man-half scaly fish, and half anthropoid ape. The mermaid thus was rescued from the castle of myths and handed over to the taxonomist for classification as best he may. The proprietor, in answer to my questions, very glibly and with seeming pride informed me those specimens came from near Los Angeles, and asserted that he had caught the mermaid himself.

From repeated deception the scientist is learning extreme caution concerning alleged discoveries and revolutionary theories. There are many mysteries which science has never explained, and may never solve, but thanks to the patient investigator and the keen philosopher, there are some things we do know, even if seen as "through a glass darkly." One

of these certainties is that if the established facts and principles of modern science are ever overthrown it will be by the trained scientist with microscope, telescope and spectroscope, not by the ignoramus with a divining-rod. Russia, the typical military power, was defeated, not by the undisciplined hordes of Asia, but by Japan, with all the enginery of militarism, together with trained and disciplined men behind the guns. In like manner, scientific tenets can be disproved only by the rigid laboratory methods of present-day science.

HARMONIC FORMS.

(SECOND PAPER.)

By BERNARD B. SMYTH, Topeka.

Read in abstract before the Academy, at Emporia, Kan., November 30, 1907. CHAPTER I.-PERFECT SQUARES.

THE

HE requirements of a magic square are that all the columns, horizontal lines and two main diagonals of the proposed square should add equally.

In a harmonic square not only do the rectilineal lines and diagonals add equally, but the sum of the vertices of all possible regular quadrilateral figures, as squares, rectangles, rhombs, and rhomboids, add equally. Harmonic squares are possible only when the root of the square, or number of cells on a side, is divisible by 4.

The term perfect square is applied to a square which adds equally not only in all the rectilineal lines but also in all the diagonal lines, thus making in straight lines a number of sums equal to four times the number of cells on one side of the given square. Thus a square of 4 should give 16 equal sums; a square of 5 should give 20 equal sums; a square of 6 should give 24 equal sums, etc. But the perfect squares here shown are of a superior character, and not only add equally in the many ways shown, but also add equally in all possible quadrilateral figures in any part of the square when as many cells are included in the quadrilateral as the number of cells on a side of the square.

Perfect squares are now constructed of any number of cells on a side above 3. A perfect square of 3 is impossible, for the reason that the number of sums necessary to entitle a square of 3 to be called "perfect" is twelve, while the greatest number of equal sums that can be obtained from any three numbers of a regular series of nine numbers is eight, as shown in part I of this paper, published in volume XIV of these Transactions (1894), pages 47, 48.

To form any sort of a magic square the given series must be divided into as many sets as there are cells on one side of the square. Whatever arrangement be given to the first set must be followed absolutely by each of the other sets The members of the first set need not be taken consecutively from

the series, but may be taken in any order whatsoever. An unused number of one set may be taken as a member of another set.

In odd squares all sets must be placed parallel to the first. In even squares compensatory arrangements must be observed so as to preserve a rhythmic or balanced effect. The initial members of the several sets in a perfect square need not be equidistant mathematically. It is only necessary in odd squares that the initials of the several sets bear the same relation to each other in position as the second does to the first, and in even squares be opposed, so as to balance.

SECTION 1.-PERFECT SQUARE OF FOUR.

Perfect squares of four cells on each side may be formed, as are harmonic squares, according to certain schemes which are here shown. They may be formed without the aid of a visible scheme; but human ability to see a mental picture of all numbers in position before writing any is not great and the visible scheme is a great help in that direction.

Before showing any of the schemes a few definitions would seem to be desirable.

DEFINITIONS.

An adjacent number, line or column is the one next to it in the same half square, as first and second are adjacent to each other; third and fourth are adjacent. Second and third, though contiguous, are not adjacent. An adjacent quarter is one on the same side, whether vertically or horizontally.

An alternate number, cell, line or column is the second removed, or with one intervening, as first and third are alternate; second and fourth are alternate.

An opposite line or column is the one in the opposite part of the square that would come against it if we fold the square along its middle line, as first and fourth are opposite; second and third, though contiguous, are opposite. An opposite cell or quarter is the one diagonally or diametrically opposite.

A couplet is two numbers in succession in a series, consisting, in a series whose first term is 1 and whose common difference is 1, of an odd and an even number; and in any other series consists of two numbers side by side when the entire series is arranged in pairs from the beginning. The first member of each couplet may be called the antecedent or leader and the second the consequent or follower.

A pair is two numbers standing side by side in the same