nite Power which they dare not name, but which is nevertheless folding them round in its all-embracing purpose of wisdom and love, so that they live. and move and have their being in God, though they know it not! The glory of our life! The blood of God flows within our veins. We are of kin with God. We are heirs of God and joint heirs with Christ; endowed with his omnipotence; the divine Presence in us; the divine Life back of us; the universe streaming through us; God pulsating in our very breath. The aspiration of this thought of God! As Plotinus, one of the ancient opponents of the Church that stood between him and God, said, and so may we all say "I am striving to bring the God within me into harmony with the God without.' In this thought there is strength for all our needs-the resources of omnipotence! There is hope for all weary workers for humanity -the Lord God wills the perfection of the race! There is peace in our souls! Marcus Aurelius, with his half-open vision, could say, "All is well with me that is well with thee, O Universe!" But we take the more sacred term and say, "All is well with me that is well ith Thee, O God!" Our own sweet Southern poet, Sidney Lanier, once wrote: "I have a boy. Every day, when my work is done, I take him in my strong arms, and lift him up and pore into his face. The intense repose, penetrates somehow with a thrilling mystery of potential activity which dwells in his large open eyes, teaches me new things. I say to myself: 'Where are the strong arms in which I too might lay me and repose, and yet be full of the fire of life?' And always through the twilight came an swers from the other world: Master! Master! There is One in whose arms we rest- · God!'" 64 NOTES AND QUERIES AND HISTORIC MAGAZINE. A monthly journal of history, folk-lore, legends; science, art, literature; Masonry, mysticism, myths; metaphysics, psychics, theosophy; games, mathematics, education; occult, arcane, and recondite matters; scraps, odds, and ends gathered from Many a quaint and curious volume of forgotten lore." Many people know many things, no one everything." Commenced in 1882. Vol. XVIII for 1900. Vols. I-XVIII, 1882-1900 each fully indexed. Circulates in all parts of the world. $1.00 a year, in advance. Back volumes, and back numbers supplied. Sample number, five cents. Address S. C. & L. M. GOULD, Manchester, N. H. OBSERVATIONS Regarding The Logic of Prisms. BY S. CHEW. (FOURTH PAPER.) It seems that a 1 to 2 relation of sine ratio to tang ratio? as they apply to the n-gons may be likened to an eclipse of the moon. Their effect gradually moves on until it arrives at maximum, and then again as gradually moves off. Mathematically speaking however it also seems that the limits of contact and of departure of the ecilpsing force are of more importance than is the point of maximum darkness. In my third paper no reason was given why 8 and 16.2 were selected for the sine ratio? and the tang ratio of dissimilar n gons, aside from the fact that they were there employed to make n-gons for argument against an objectional equation. Hereafter we will use the same n-gons for a different purpose; use them to show the limits of contact and of departure. To locate these limits so they will not be disturbed, we here use some fiddle - string logic. It is well known that the law observed by the violin string is to vibrate inversely as its length. Some wise one of the past however discovered that the law of inverse number of vibrations as compared with its length, ceased at a point = 8 of the vibrating length of the string. Therefore until a musical string forgets its own law, the gauge points for the limits of diam? area ngons seem fixed. Starting at the maximum n-gon ( equilateral triangle) and moving on the line of extension of diameter of their inscribed circles, we meet with the first inscribed diam2 area ngon, formed by the action of tang ratio?2 16.2, and all n-gons having tang ratio squares not greater than 16.2 nor less than 16 belong to the class of inscribed diam2 areas. At the tang ratio2: 16, however we arrive at the point of maximum darkness, for the sine ratio which beongs with this n-gon = 8, and it seems evident that a darkness = which moved on by a ratio must again move off by its opposite ratio. Therefore all sine ratio squares not less than 8 nor greater than 8 also belong with the class of inscribed diam2 areas. Hence by respecting the law of geometry which governs the vibrating musical string we may determine the points of contact and of departure of effect produced by the 1 to 2 relation of sine ratio2 to tang ratio?. However we are not quite there yet, because the fundamental factor for the exact quadrature of the n-gons will always be the term which is now known by the name, tang ratio. To the n gon however whose sine' ratio=8 belongs the tang ratio2=1519. Therefore it seems that we may think as we will, but the fact will remain, that the quadrature numbers of all n-gons are equal to their respective tang ratio squares between (and including) the limits tang ratio2=15198 and tang ratio2 16.2, and that the areas of their respective n-gons are all of them equal to the square of diameter of their respective inscribed circles. And assuming the foregoing remarks are true, then, we already have a machine for squeezing the last drop of water from stock now held the world over in the equation in the pernicious equation. If we are able to find the quadrature numbers of a class (or series) of n gons by simply respecting the law which applies to the matters considered, perhaps we can go still further and find the quadrature numbers of other individual ngons which have tang ratio squares far removed from either limit described. But why is it considered usfsul employment of time if it is devoted to finding the quadrature numbers of the individual n-gons? Because it saves time in the future, for if we wish to place 99 square inches of area, and in the shape of a form numbered 18, we would simply write: 99 X 18 perimeter, and thus find the first important essential value to work with, and afterwards apply this perimeter to a right parallelo. gram proportioned as 1 to 2. But then, why should we say that a figure of geometry that is proportioned thus and so, is en dowed with a particular abstract number by means of which In the answer to this the thankfulness that = = its exact quradrature may be shown. question we, the plebians, should feel comes of independence, for here we are not even required to consult the oracles, because we are now nearing the border land of practical geometry. It seems that the quadrature numbers of n gons may be learned by studying the logic of their respective prisms. At present however this remark applies only to prisms whose side surfaces represent right angled parallelograms. Respecting this class of prisms it can be said, that if their solids = perimeter of their respective end planes, then the sum of their side surfaces is also perimeter2 solid of prism. Apparently these three terms of a prism are never equals unless area of their end planes = quadrature number. The solid of a prism however equals product of end plane area X length, hence the length of this class of prisms end plane area perimeter quadrature number for all forms having the same proportions and angularity as the end of the prism considered. To more clearly explain meaning we shall say: let the carpenter take a piece of timber that is nicely squared and planed to 3 inches by 6 inches. Next let him square an end nicely with its surfaces, and parallel with this end have him saw it off, so as to make us a prism 18 inches long. When the prism thus made is examined we can say that its cubic solid 324, and again that the square of its end plane perimeter = 324, and also say that the sum of its side surfaces=324. Therefore the quadrature number (18) of the form (or shape) of this prism applies not only to the surface of its end plane, but also to its solid, and side surface as well. And what is true of one prism, seems equally true of others of its class. The position taken in these papers is to be that each individual shape is numbered, by inexorable law. And that respect for the number upon which a forms rests, supplies a simple and exact means of squaring its similars. To again illustrate meaning by the formation of a prism, we shall say: let the carpenter make a second prism proportioned as follows: since it is to have five sides, let it have four contiguous sides of 31 inches = 2 inches, and thus form a prism each, and have its fifth side whose end plane perimeter = conditions of its sides are such ject to a circumscribed and an inscribed circle. conditions are such that an exact diameter for circles is required. To space 15 inches in a 360° arc is one thing, but to space 15 inches broken into thus and so parts in a 360° arc is quite different. = = = 8 13 each of these Therefore proper instructions regarding these circles, would assist the carpenter in his efforts to supply a workman like 4% sides prism. To give such instruction we shall say that perimeter28134 - circumscribed diam and that perimeter2÷ 143 inscribed diam2 of an n-gon of 4 sides. Hence 15 X 15= 22591784 (sine ratio2) = 26.3828125 = circumscribed diam circle of end plane of prism and again 15 X 15 = 225 143 (tang ratio2) 15.8203125 inscribed diam circle of end plane of prism required. And since 26.3828125 15.8203125 10.5625 = 31 × 31 = side2 of n gon, which is in harmony with aa × b2= c2, while the gauge point (1519) is also respected, we may combine the logic of the musical string with logic of prisms and say, that the quadrature number of the 413 sides n-gon 15, hence it is the same as the perimeter employed in illustration. And also say, that a prism having 4 sides with an end plane perimeter 15 inches and with length 15 inches will have a solid of 225 cubic inches and a sum of side surfaces = 225 square inches. Therefore it seems that entire surface of this class of prisms perimeter2+2 q. n. Equal perimeters under the two conditions of prism end planes here spoken of, would contain areas related inversely as 18 to 15, or related inversely as their respective quadrature numbers. 13 = = = = If one shall bear thee word that such a one hath spoken evil of thee, then do not defend thyself against his accusations but make answer, "He little knew my other vices. or he had not mentioned only these." Epictetus. |