More Heretical Remarks

BY S. CHEW.

(FIFTH PAPER.)

The quadrature number of any wholly obtusely angular n-gon is either equal to, or it is greater than the tang ratio which belongs with the n-gon, for quadrature number in the sense that we use it, is an expression of geometric mean proportion. The square ngon has tang ratio = 16 and has quadrature number which is also 16, hence at this point quadrature number = tang which supplies to this n-gon inscribed diameter2 area. But when we reach beyond, to the geometer's circle, where we were learned to respect the idea that 12.5664 inches of circumfererence contains 12.5664 square inches area, it is there made to appear that quadrature number (12.5664) of this n-gon represents geometric mean proportion between the geometer's standpoint of reasoning and the tang ratio of the circle. Because 16 12 5664 :: 12.5661: 9.86965056, hence the fourth term of this proportion is an expression of his 2, or tang ratio? which belongs with a circle. Here it will be observed that quadrature number 12.5664 is far greater than 9.869650, hence it seems reasonable that n-gons above mentioned, should have quadrature numbers which are equal to, or they are greater than their respective tang ratio squares.

We here digress for a few remarks which seem called for. Regarding the circle of science as it is now perfected with the modernly extended π-value, it is proper to say, that we propose henceforth to view this monument of mathematical infidelity, from the standpoint of the arithmetic n gons. To show that this position is tenable let us agree that 12% = 201; that 201 × 201 =40401, and further agree that 40401 - 4050 93951. Then we can better agree respecting the matters which follow. Therefore, let an n-gon be made whose quadrature number equals 12.5625, and be made with values as follows: Let sine2 = 2831230, express measures of the angle 25250625, for

88416 and tang2

2569666)

of an n-gon.

Then with a common dividend

4050

values which appear in sine and tang2, it is easy to convey my idea that 25250625 ÷ 2569666923881

= sine ratio; that 25250625 2531250 938 tang ratio2, and that 25250625 +38416 =6571131 = number of sides2 of the n-gon we now are considering, hence the n-gon will have 2513 sides. Thus it seems that an arithmetic n-gon whose quadrature number is less than 12.5664 will have less that 26 equal sides. Which of the two will the geometers be best able to spare? Will it be their point of equal perimeter and area at 12.5664, or will it be the shape of their n-gon, for it now appears that an arithmetic n-gon whose perimeter = area at 12.5664 (quadrature number 12.5664) will have less than 25 equal sides, because 12,5664 represents geometric mean proportion between tang ratio? (9.981826+) of such n-gon and the true value of x for its posi tion in the family of n-gons.

Respecting the familiar value, 3.141592, we have this to say: Let sine2535, and tang2=155, express measures of 9937928, angle of an n gon. Then having 98252800 for a common dividend, we derive the following conclusions respecting sine ratio2, tang ratio2, and number of sides of the n-gon to which they belong. With fractions reduced to their lowest, it appears that 98252800 ÷ 9953280 984 sine ratio2, and 98252800

÷ 9937928

15352 6400

=

=

91

101431

=

1 2 2 2 2 2 1 tang ratio?, while 98252800 ÷ number of sides of the n gon.

Given two divisors of 6400 which supply quotients whose difference = 1, such divisors are the right ones to employ, if we wish to construct an 80 equal unit-sided n-gon. Hence, the sine ratio, for the 80 sides =√984 = 3.141878+. But persons who were purposely learned to shun, consideration of the dividing line between sine and tangent, will talk by the hour, of the consolation of faith in believing that the length of its circumscribed arc is just equal to 3.141592 times diameter. Arithmetic teaches that sine diam2 = 984 × 648 648-819 6400, and that tang diam2 = 911!1111 × 647,648, -6400 × = perimeter, which belongs with an 80 equal unit-sided n-gon. Hence the combination of arithmetic with n-gons shows that quadrature has not yet begun. It may be observed that the

124

=

80-sided described n-gon is no kindred of the perverted idea of inscribed and circumscribed n-gons, but on the contrary its sides represent 80 straightened arcs of a circumference whose home is between inscribed and circumscribed diameters of the n.gon.

Returning to the consideration of quadrature numbers of n-gons whose homes are between a square and a circle, it is not a new claim to say of them, that they represent geometric mean proportion between respective tang ratio? and x. But the number value of x which is thrust upon the public, and in the name of education, is the number which is found at the point of maximum darkness, and it is the number 16. And it seems that our educators have taken the position that they will never change from the idea that √tang ratio X 16 = quadrature number of all regular n gons. Then gon now before us is regular to the extent that it is not defective in anything. Therefore, we apply their idea to this n-gon whose tang ratio? 1519 whose inscribed diam2 16, and whose perimeter = √1519 X 16, (if their idea possessed merit), then the area of this n-gon would be equal to its perimeter, because its perimeter (15,90990) represents a geometric mean proportion between its respective tang ratio and 16. Is the area (15,90990) thus assigned to this n-gon correct, or defective? If defective, what are our educators capable of doing that will correct it?

The idea of dissimilar commensurable n-gons has been so religiously excluded from their lives in the past, they would be unlikely to accept an idea which would show to themselves their appearance to others.

However, we who are the sufferers from false teaching, cannot afford to ignore the fact, that wholly obtuse angular n-gons have quadrature numbers which are equal to, or they are greater than their respective tangent ratio squares. But whether it is equal to, or it is greater, the quadrature number always represents a geometric mean when respective tang2 is made one term of the proportion. Therefore, the arithmetic ngon now before us (whose sine ratio? 816, whose tang ratio? number of sides? 2025, whose four longer side, each

122,

whose

=

=

==

√253.125

√24415,90990

and whose short side 15.62049 .28941+), has a quadrature number which is either 15.90990 perimeter, or it is equal to its tang ratio? =1519. However, it is proposed to establish this n-gon for a dividing line between their system of mixed mathematics for the colleges, and a system of plain arithmetic measures, or a practical geometry for the rest of the world.

Perimeter area = quadrature number under all circumstances, hence perimeter2 ÷ quadrature number = area that is contained under perimeter universally. Therefore, area or the the size of an n-gon is governed wholly by the quadrature number which belongs with the form that is being considered. But with respect to college sayings, which refer to the n-gon, we simply accept them at par.

The Devil and Evil Spirits,

ACCORDING TO THE ANCIENT HEBREW RABBIS.

By Sapere Aude, Fra. R. C.

Communicated by W. Wynn Westcott, W. B., D. P. A.

In a previous essay on "Angels," I have the general opinions of the Ancient Jews upon the good Spirits whom they affirmed to exist, and take a part in the affairs of men. As a warning to the curious, in this essay, there will be an attempt to elucidate the ideas respecting the beings who are of an evil nature, and have an evil and malicious effect on human life and progress. The Christian religion has inherited from the Jewish faith its belief in Satan as the Arch Devil of our environment, and it is also assumed that Satan works by means of his inferior malefic assistants, to lead man astray from the path of virtue. Our modern Christianity does not, however, specify names for any other individual devils, nor even for classes of them, as did the old Jewish Rabbis.

Hebrew Rabbinic theology teems with names of evil spirits, classing them, and narrating episodes in great variety of their malice and success in leading men into sin and danger. It also enters into minute details of the many forms of Hell, in which are their dwellings, and whence they energize to persecute wicked men who stray from the paths of duty and beneficence.

A general idea, however, existed that the just man who never trangressed the dictates of the Mosaic Law, had but little to fear from their attacks, for over such the angel guards prompted by Jehovah kept a strict guard. The oldest ceremonial books of the Old Testament do not seem to realize any personal great Enemy of Mankind in active opposition to God, but in the later volumes that idea takes form.

The captivity of the Jews seems to have tainted their faith, and the influence of the Persian dualism became then apparent. In the Book of Job there is a definite attribution to an Evil Spirit of a power to contest with God for influence over man. The notion is further extended in the prophecies of Zechariah. From that time Satan became not only an Adversary of Divine Power but was acknowledged as the "Tempter." Ahrimanes was the name of the Persians for the Great Evil Spirit, and he seems to have been a type of physical evil becoming moral