were Spirit and Substance. And these Twain are One. 13 And the Fternal thought and the Thought became the Word and the Word became the Act. And lo, the Son and the Daughter of God, Twain in One. 14 And the Twain with the Unity are One, yet Three in Manifestation; of Whom Each in Manifestation is Twain for the creation of all things. 15 And the Alohim breathed, and by the will of the Eternal came forth the Universe: even from the Life and Substanse of God. 16 Yea, by number, by weight, and by measure hath the Alohim created all things, and all things are made double, the one answering unto the other. MARYLAND FOLK LORE. If your fence is not erected in the right time of the moon it will fall down. Things that grow above the ground, like peas, must be planted in the light of the moon. If the first butterfly you see in the year is white, then you'll have white head (meaning prosperity) that year; if the butterfly is brown, you will have brown head. Bees will not stand habitual profane swearing among members of a family. Bees must always be told if there is a death in the family. It brings very bad luck to count your lambs in the spring. If your horse is hurt by a nail, if you find the nail and keep it well greased, the horse will recover. Eggs laid on Good Friday never get stale; butter made on Good Friday has medicinal quality. When cows become restive and cannot be quieted it is a sign that bad luck will come to their master. If it thunders on All Fool's Day, good crops will be gathered that year. In certain sections of the country, Dr. Minor said that the signs of the zodiac count for a good deal during the planting season. Among the sayings in this connection are: Corn must be planted under the Twins. Melons under the Crab. Plant no seed under the Scorpion. Surgical operations on animals must be performed under the Archer.-Baltimore Sun. Remarks on Tabular Sines. er side, be made. order a sine ratio2 BY S. CHEW. (FIRST PAPER.) 4 X 6 + 1 = Let a five-sided n-gon, having four equal sides and one shortNext, let the four longer sides tangent the circle and be 6 inches, and the shorter side When so made, the square of the tabulated 72 = , and the square of the tabular tan3. From these values we deduce in regular 82, and a tangent ratio2 = 153 which latter values enable us to determine the circumscribed and inscribed circle diameters of the n-gon that we are considering. As the perimeter of this is made 25: √625 the employment of sine ratio2 (82) as a divisor of perimeter2 (625) supplies dividend 76.5 for the square of diameter of its circumscribed circle. And in a similar manner 625 158 (tang. 2) supplies 40.5 for the square of diameter of its inscribed circle. To agree with the equation a2+ b22, therefore, the two divisors of perimeter2 here employed seem to be right. Since 4 expresses the number of sides of this n-gon, therefore, 4 X 41 1718 the square of its number of sides. An n-gon whose perimeter is a geometric mean proportional between the square of its side and the square of its number of sides, and between sine 2 and circumscribed circle diameter2, and also between the tangent ratio2 and inscribed circle diameter2, as it is in this case, it is not apt to be wrong in other respects. Our present purpose is, therefore, to speak of this ngon in connection with the tabulated sine of its angle. Now it seems that the √ = .685992 = sine of its angle is printed in our tables at the angle 43° 18′ 48′′. Now if this angle 43° 18′ 48′′ is four times taken its sum will = 173° 15′ 12′′. Subtracting 173° 15′ 12′′ from 180°, it leaves 6° 44′ 48". The remaining short side of our n gon = = of its longer side, and 72 therefore the sine of angle for this short side= √ √ = .685992÷6.114332. Hence, it seems plain, if there is no error in printing .685992 for a measure of the angle 43° 18′ 48′′, then there could be no error in printing .114332 to express sine of angle of 6° 44′ 48′′. However, by looking, it will be found in the table of printed sines that .114362 would fall before arriving at 6° 34', thus making it appear that the arc of circle of this n-gon contained only 359° 38'+. This seems short for an arc which should cover 360°. = 5508 Any one can map the above described n-gon in a circle whose diameter 76.5, and since 76.5 X X 36 = 8588 1= short side of n-gon, we may prudently discredit the validity of our tabled sines, respecting its angles. 16 Again, let a second n gon be made, having five equal sides made tangent to greatest inscribed circle, and one shorter side equal of the length of a full side. Let the perimeter of this n-gon 14 inches, so that 5 × 2+1 will express the length of each side in connection with its whole perimeter. When it is made, this 5 sided n-gon will have for the square of tabular sine of its angle, and tabular tangent of the same. From its sine ratio2 8317, and from its tang2 ratio 12. = 32 for square of tabusine232 113 we derive a we obtain tangent Having chosen 141 for our perimeter, we next 16 16 22.0703125 1 circumscribed say: 14 X 14 ÷ 8317 circle diameter of n gon, and 14 X 14, 12.5 15.8203125 inscribed circle diameter of n-gon; and as the difference between the two diameter squares here given 6.25 = square of its 2.5 side no error will be found in the n-gon respecting the equation a2+b2c2. Then, again, perimeter 14 is also a geometric mean between 6.25 and 31.640625, or between a square of its side and the square of its number of sides. Our next duty is to look at the sine of its angle and compare it with existing tables. The sine of its angle = √.532152 which is found in tables at the angle 32° 09′ 02′′. This angle 32° 09′ 02′′ taken five times would 160° 45′ 10′′ of angle, which being subtracted from 180° will leave 19° 15′ 50′′ of angle reserved for the short 1 side of the n-gon. The sine of this 19° 15′ 50′′ angle, however, seems to = √ of .532152 .332595 which is to be found in the books placed at the angle 19° 25′ 34′′. To think the sines .532152 and .332595 are true measures of the angles represented in our sine and cosine tables would require the circumscribed circle of a 5 sides n-gon to have 360° 19' of arc. This seems like to much arc; however, it is nearer 360 than the arc which appeared from the quoted angles of the 4 sides n-gon. Perhaps the writer is not as familiar with sines and cosines as to be able to use them in a scientific manner; but at the same time any one may draw a circle whose diameter = √22.0703125, and draw within five equal connecting chords which each square would equal 32 22.0703125 6.25, and still have arc enough left to connect their extremities with a 21% chord. What is there, however, to make this clear to the mind? It is the diameter of our circumscribed circle. Therefore, 45 X 2.5 ÷ 45√328 √22.0703125 = diameter of the circle that is called for. 113 = THE CURVE OF IMMORTALITY. A Mathematical Analogy to Death and the Resurrection. By a Septuagenarian. This is an appendix of 20 pages to "The Open Court" for May, 1900. The paper is a mathematical curiosity and one that will be read by a mathematician with delight. The author was for several years a University professor of astronomy, and afterwards editor of the scientific department of a Chicago daily for a third of a century. "It would be very easy," says the editor of the above monthly, " to construct or find an analogy in algebraical geometry for the Buddhist doctrine of the transmigration of the soul, or of the dogma of cycles of existence." "to The author has aimed at a mathematical treatment of the subject which would not be above the comprehension of an average graduate of a High Schcol. We cannot go into details of the mathematical reasoning of this paper, but will say that it harmonizes with the theosophic doctrine which declares that the vital principle, the essential Atman," which is held to be loaded with the "Kama," exists before the body is formed as well as persists after its dissolution. Starting with the stated conception that the state of the ether is essentially positive, and mult plying by the imaginay unit four times in succession, we shall reach the positive again, and may conceive the several results to correspond to conditions in the grand scheme of Nature, thus: corresponds to Ether correrponds to Inorganic matter. corresponds to the Unknown. (?) The last four expressions in the first column represent the "four roots of unity," being the analytical values of radius unity at angular distances of 90°, 180°, 270°, and 360° from the origin. A UNIVERSAL PERSONAL UNITY. The personal relation of Jesus to the human race. By George N Abbott. This is also another mathematical paper based on the " roots of unity" to demonstrate a theological theory premised in this proposition: Jesus' sonship to humanity is a normal and integral relation. 1. Every normal personality virtually contains the radical principle of every other such personality. 2. All the individual pure functions of the grand personal unit or idea are implicitly mutual functions of each other. 3. Every organically individualized rationality possesses as an essential constitutive element an incipient germ of every other such rationality whose existence is possible in the normal organic development of the rational. These are the formulæ and the reasoning from them results, so far as the analogy will apply, a suggestion of an infinite number of secondary persons essentially cognate with one of the primary. While divinity involves humanity in its highest potency, humanity involves, or implies, divinity in a minimum degree. The five fifth roots of unity are used to show the analogy and the imaginary character of the roots, excepting the first, is also significant, their real value lying not in themselves but in the parent unit. The link in the argument goes to show that there belongs in the normal constitution of every human being the Equivalent of a Ovum Divinum. (Compare Gen. iii, 15; xxii, 18; Gal. iii, 16.) |