DISCOVERY OF THE TRUE PI-VALUE. The city was wrapped in quiet. Prude citizens slumbered in the embrace of night. The finger of a clock pointed to VI; and the wintery morn of the 8th of January, 1881, longed to be unfolded from its twilight shroud so that it could pose in modest robe of dawn. Awake, alone and in silence, a worker, absorbed in depths of thought, transfixedly gazed on a few figures which among others stood cut in bold relief, breathing, as it were, secrets of the mystic shrine. More and more these figurcs appeared alive and more and more forcibly were the numbers 4 and 5 impressed. Then recollections of 2, 8, and 9 swept through the agitated brain, and the fraction sought for more than two thousand years by many more thousand savants was found at last. Like a luminous star the discovery lit up the clouded record of mathematical research and spread joy in the mind of the man who first was permitted to break the seal and use the key which God alone had used before. But, "Can it be true?" Perhaps it is but a wild phantasy, born of a too zealous desire to succeed. A dizzy reel! Then a chilling tremor swept through the frame of the man and flushed the cheeks with a crimson blush, the blush of departing hope. A sickly smile of growing doubt cast shadows where just before the mien was heavenly lit.

How could one mortal hope to have found in labyrinthine maze the way to link knowledge divine to human understanding, while countless authorities, high and low, proclaimed such a find impossible. Yet, the humble worker's mind was stirred by God like faith, and boldly he strove to convince himself that he was but an instrument made fit, by accident, or by design of Jehovah's will to act as mediator between sophistry and science.

Thus the night passed in hope and fear, and the early dawn found the discoverer of perfect transported into dreams in which he saw the glory and felt the bliss of sublime victory.

CHARLES DEMEDICI, New York City.

"The Statistician and Economist for 1899-1900,'" pp. 497 to 508, contains a concise account of the discoveries, experiments, and propositions of Charles de Medici, accompanied with a full

page, full-length portrait of the man. The article is entitled "An Epitome of the New Geometry and the Commensurational Arithmetic." He has published treatises on these discoveries some of which we will mention, together with other works: Rational Mathematics. Geometry. Sections A, B, C, D, in four volumes. Published by A Lovell & Co., 3 East 14th St., New York. Price, 25, 60, and $1.25. Bonnd in boards.

Commensuration. Groundwork of Classification; a Panorama of Evolution; an Exposition of Darwinism and Theology Conciliated. Quarto, illustrated. New York, 1880.

A Trip Through the Universe. A Metaphysical Panosphy. Three books complete in one volume. Mind, Matter, Spirit. Relation of the World to the Universe. Boston, 1870.

The Pantagraph; a Manual for Tutors and Students who teach or study Logic, Ethics, and Metaphysics. Phila., 1874. The Normal System of Weights and Measures. One cube of pure water the basis. A Basic Classificotion. New York, 1880. Mathematical Commensuration; the Principles of its Theory and Practice.

The Solving Triangle and Protractor. An instrument which squares the circle, cubes the sphere, and rectifies the curve. Of use to the draughtsman, mechanic, architect, surveyor, navigator, astronomer, and metrologists. Price, 15 cents.

The Surd Law; its Principle and its Applaction. (In press). A Manual of Monologic; devised to simplify discussion, prevent wrangling, and regulate debate. (In press.)

He has published mady other monographs on mathematic, philosophic, and analytic subjects in books, in pamphlets, and charts, which he has liberally distributed to those in search for knowledge. He is the promulgator of the Montheon School, and its demonstrator, at 15 Astor Place, Room 45, New York. Address, 21 West 24d Street, New York City.

Squaring the CIRCLE. Proving the squaring of the circle requires: I. Geometric construction which produces a mean proportional square between the square on diameter and a square of equal perimeter to the circumference of the circle. 2d. A π-value or ratio between diameter and circumference of

circles which will arithmetically prove an equation between the three given squares obtained by construction. 3. Mechanical contrivances which show that two planes represented by plates of similar metal and exactly equal thickness, one in the form of the circle, the other in the form of the square, which are exactly balanced when placed in analytical scales. 4. It is required to be shown that a cylinder constructed on the circle's plane and a parallelog constructed on the plane of equal square to circle, both of the same altitude, balance each other in bulk and weight, and no matter what the altitude may be. 5. It must be shown by geometric construction that the side of the square of equal perimeter to the circle's circumference equals the subtending chord of two-seventh of circumference. It must also be shown that the side of the equal square to circle equals the subtending chord of six-seventeenth of circumference. -Charles DeMedici, in "Statistician and Economist," p. 499. 1899-1900.

TREATISE OF THE SIBYLS. ing title of an old work, “A recorded so it will be in a have been previously given. A Treatise of the Sibils, so highly celebrated, as well us by the Ancient Heathens, as by the Holy Fathers of the Church; giving an Accompt of the Names, and Number of the Sibyls, of their Qualities,, the Form and Matter of their Verses; as also of their Books now Extant under their Names, and the Errours crept in Christian Religion, from the Impostures contained therein, Particularly, concerning the State of the Just, and Unjust after Death. Written originally by DAVID BLONDEL; Englished by J. D. London; printed by T. R. for the Authour, MDCLXI. Quarto; pp. 310; leather bound; chapters, LIII.

(Vol. VIII, p. 413.) The followTreatise of the Sibyls," should be place of reference with those that The book is before us :

THE SCIENTIFC SKELETON. By Samuel Blodget, author and publisher, Grafton, No. Dakota. Portrait of the author. Pp. 106. 16mo. This book is worth reading, and should be read by persons who think. The author says:

"There is life within life, there is circulation within circulation. Each blood globule has a distinct circulation within itself. They have their periods of revolving around their common center the same as the planets. The period of a globule is very close to uniform, the same as a planet. The same law gives to each its reepective period, the law of life. They both have eccentricities in their movements, but the vital principle knows what it is about."

The Pernicious Equation.

BY S. CHEW.

(THIRD PAPER.)

=

area of the regular

If it is limited,

We should not

The equation, half perimeter X radius n-gons, seems to contain both wheat and chaff. and confined to its proper place it is then true. utterly condemn an equation which rightly applies to the only n-gon whose quadrature is exactly known. And yet the probable reason why exact quadrature begins and ends with a soli. tary n-gon is, that mind has been cursed with that blight which reads half perimeter X radius = area of n-gon. In connection with the n-gon to which this equation belongs, there are other means of finding its area, and other matters of more importance than the equation. For instance: it seems to require a sine ratio2 = 8, and a tangent ratio2 16, (a 1 to 2 relation of these values), to enable us to form a figure whose area is equal to the square of diameter of its inscribed circle. Therefore, if a second and dissimilar n-gon has a sine ratio2 which equals half tangent ratio of a third dissimilar n-gon, then the area of these dissimilars is also equal to square of diameter of their respective inscribed circles. It seems we must pass from under (be freed from) the influence of this 1 to 2 relation of sine ratio to tangent ratio2 in order to escape from their effects. Thus it appears that a possibility is shown whereby we may construct a series of dissimilar n-gons, and class them diam2 n-gons, among which will be found one whose area = half perimeter X radius. We will preserve this one, for it is the wheat, while the same equation if applied to other forms seems to represent chaff. Mere opinion, however, proves nothing, so it is the writer's duty to show if there is a just foundation for his remarks. To do this we shall say that with

61

25

for its sine2, and The numbers em

ployed to express sine2 and tang2 rigidly fix three terms of the n-gon proposed, namely, its sine ratio?, its tang. ratio, and its number of sides2. The sine ratio2 (8) and its number of sides2 167 (2025) are the only values required for present discussion. For after the n-gon is made if the square of its of circumscribed diam3, it will also = 8 of it inWe next choose for its perimeter √262.44, 32.4 diam2 of its circumscribed

side scribed diam2. and say, 262.44816 circle. And again say, 262.44 × 122 ÷ 2025 = 15.8112 = side2 of an n-gon. And since 61 X 32.4 125 × 15.8112, no error appears respecting the sine of its angle. Then again diam2

chord2

=

=

=

inscribed diam2 of n-gon; therefore, 32.4 15.8112 16.5888 inscribed diam2 sought. While lastly, 61 × 16.5888 64 × 15.8112; hence, no error appears in connection with its tangent of angle. This n-gon when made has four equal surdroot sides whose extremities are connected under the arc by a short side which = 16.2 — 4 × √15.8112 = .29468014. From the foregoing description, that n-gon whose sine ratio2 = 8116 should be easily made. Here we leave this for the present and proceed to construct a second n-gon, still employing √262.44 = 16.2 for its perimeter, and will hereafter consider perimeter in terms of inches.

=

80

To make the new n-gon we say let its base line 4 inches, and its two slant sides 4.05 inches each, and its upper and parallel side 3.6 inches. This n-gon has four straight line sides, each of which perfectly tangents an inscribed circle. When the figure is thus made, survey it carefully and thereby observe that its inscribed diam2 16.2; and since perimeter2 ÷ diam2 = tang. ratio, it thus appears that its inscribed diam2 16.2. Here it would seem that

=

=

tang. ratio? perimeter

=

=

a third dissimilar n-gon is now described, whose tang ratio? twice sine ratio of the second dissimilar n-gon previously mentioned. Having described two dissimilar n-gons which stand upon opposite sides of the form which is numuered 16 and whose perimeter √262.44, there arises a question respecting how

=