Relay-Version: version B 2.10 5/3/83; site utzoo.UUCP Posting-Version: version B 2.10.1 6/24/83; site dciem.UUCP Path: utzoo!dciem!ntt From: ntt@dciem.UUCP (Mark Brader) Newsgroups: net.math,net.legal,net.misc Subject: State law defining Pi (part 3 of 3: annotated text of the bill) Message-ID: <816@dciem.UUCP> Date: Thu, 29-Mar-84 18:11:45 EST Article-I.D.: dciem.816 Posted: Thu Mar 29 18:11:45 1984 Date-Received: Thu, 29-Mar-84 22:23:02 EST References: <814@dciem.UUCP>, <815@dciem.UUCP> Organization: NTT Systems Inc., Toronto, Canada Lines: 131 See part 1 article for netnews References:. /* Following is the text of Indiana House Bill #246 of 1897, with my own anno- tations (in comment signs and exdented, like this text). In my annotations, A, r, d, c, and s are respectively the circle's area, radius, diameter, circumfer- ence, and side of the inscribed square. */ A bill for an act introducing a new mathematical truth and offered as a contribution to education to be used only by the State of Indiana free of cost by paying any royalties whatever on the same, provided it is ac- cepted and adopted by the official action of the leg- islature of 1897. /* You normally have to pay royalties on mathematical truths? The Pythagoras estate must be doing well by now... */ SECTION 1. Be it enacted by the General Assembly of the State of Indiana: It has been found that a circular area is to the square on a line equal to the quadrant of the cir- cumference, as the area of an equilateral rectangle is to the square on one side. /* The part after the last comma is a remarkable way of saying "as 1 is to 1". In other words, this says A = (c/4)^2, which is the same as A = (pi*r/2)^2 = (pi^2/4)*r^2 instead of the actual pi*r^2. */ The diameter employed as the linear unit according to the present rule in computing the circle's area is entirely wrong, as it represents the circle's area one and one-fifth times the area of a square whose perimeter is equal to the circumference of the circle. /* The formula A = pi*r^2 is interpreted as A = d*(c/4), which is correct. The author claims that the d factor should be c/4, so the area by the author's formula is to the area by the real formula as c/(4*d) = pi/4. Since he be- lieves pi = 3.2, this ratio is 3.2/4 = 4/5. Therefore the area by the author's rule is 1/5 smaller than the actual area. He apparently thinks this means the other area is 1/5 larger than his area, which of course would actually require the ratio to be 5/6. */ This is because one-fifth of the di- ameter fails to be represented four times in the circle's circumference. /* In other words, c = (1-1/5) * (4*d); consistent with pi = 3.2. */ For example: if we multiply the per- imeter of a square by one-fourth of any line one-fifth greater than one side, we can in like manner make the square's area to appear one fifth greater than the fact, as is done by taking the diameter for the linear unit instead of the quadrant of the circle's circumference. /* He says that if we consider the area of a square of side x to be (4*x)*(x/4) and we replace the second x by (1+1/5)*x, we get an area 1/5 too large, and this is analogous to using d in place of c/4 with the circle. */ SECTION 2. It is impossible to compute the area of a circle on the diameter as the linear unit without tresspassing upon the area outside the circle to the extent of in- cluding one-fifth more area than is contained within the circle's circumference, because the square on the diame- ter produces the side of a square which equals nine when the arc of ninety degrees equals eight. /* I can only assume that "nine" is a mistake for "ten". */ By taking the quadrant of the circle's circumference for the linear unit, we fulfill the requirements of both quadrature and rectification of the circle's circumference. /* Getting repetitive here... */ Furthermore, it has revealed the ra- tio of the chord and arc of ninety degrees, which is as seven to eight, and also the ratio of the diagonal and one side of a square which is as ten to seven, disclos- ing the fourth important fact, that the ratio of the di- ameter and circumference is as five-fourths to four; and because of these facts and the futher fact that the rule in present use fails to work both ways mathematically, it should be discarded as wholly wanting and misleading in its practical applications. /* The meat of the bill. He says that s/(c/4) = 7/8, and d/s = 10/7, there- fore d/c = (10/7)*(7/8)/4, which he reduces only as far as (5/4)/4. Of course this is 5/16, and gives pi = c/d = 16/5 = 3.2. It also implies that the square root of 2 is 10/7. */ SECTION 3. In further proof of the value of the author's pro- posed contribution to education, and offered as a gift to the State of Indiana, is the fact of his solutions of the trisection of the angle, duplication of the cube and quadrature of the circle having been already accepted as contributions to science by the American Mathematical Monthly, the leading exponent of mathematical thought in this country. /* Not bad, eh? Of course these are problems well known to have no solution within the usual constraints (compass and straightedge construction only), and the third one is essentially equivalent to the matter of the bill. I guess the A.M.M. had a policy of politely acknowledging crankish submissions, and the au- thor took that as acceptance. Ah well, I suppose that if this bill was enact- ed, then it would become true that the A.M.M. had accepted the solutions. */ And be it remembered that these not- ed problems had been long since given up by scientific bodies as unsolvable mysteries and above man's ability to comprehend. /* "Given up" is not the same as "proved insoluble"! Posted by Mark Brader; see part 2 article for source. Spelling is reproduced as in the source (I hope). */