Repository: UIHistories Project: Mathematical Models Construction and Use of Mathematical Models (Fehr & Hildebrandt) [PAGE 15]

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7* T r i g o n o m o t r y: To show the v a r i a t i o n in the t r l g o m o m o trie f u n c t i o n s in all four ouadr a n t s , the f o l l o w i n g m o d e l is m o s t convincing* On a h o a r d 30 n x 3 0 !T, m a r k the center and r>asto a 1© i n c h r a d i u s , c o m p l e t e 3 6 0 ° pr otractor*:-. L e t 4 m e a s u r i n g tapes 3 0 " l o n g , m a r k e d f r o m - 1 5 to + 1 5 b e p a s t e d p a r a l l e l to the 0 ° ~ 1 8 0 ° and 9 0 ° - 2 7 0 ° axes and tangont to the circle.- A t t a c h to • the center a rofcatablo m e t a l str ip m a r k e d f r o m 0 to 20 i n c h e s * On it,, at the 1 0 i n c h d i v i s i o n , a t a c h a f r e e l y h a n g i n g m e a s u r i n g t stick m a r & o d f r o m 0 to 1 0 inches • As the a r m r o t a t e s the v a r i a tion in all the f u n c t i o n s can be oasily f o l l o w e d * . (See Plato II,. no, 5) R e c r e a t i o n and H i s t o r y 1 . -Numerology: G e m a t r i a or a s s i g n i n g n u m e r i c a l v a l u e w o r d s is one of the m o s t a n c i e n t p r a c t i c e s * A simple b o a r d i l l u s t r a t e this Is m a d e f r o m at l e a s t 5 c o n c e n t r i c circles. D i v i d e e a c h c i r c l e Into two s e c t o r s , e a c h s l i g h t l y less t h a n a s e m i - c i r c l e , b y two I n t e r s e c t « Ing d i a m e t e r s . D i v i d e e a c h of these sectors into 26 c o r r e s p o n d i n g equal parts.- In the one sector w r i t e the a l p h a b e t , and in the other in r e v e r s e order the n u m e r a l s 1 to 2 6 . Place a d o u b l e s e c tor s h i e l d , s l o t t e d , b e t w e e n the sectors c o n t a i n ing the n u m e r a l s and l e t t e r s . A r r a n g i n g the c i r c l e s so that the l e t t e r s in the one slot spell any 5 (or l o s s ) l e t t e r w o r d , the n u m e r i c a l v a l u e s appear on the o p p o s i t e slot of the shield.. VIII.

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2* Magic Square Board: Divide a board Into 10 by 10 equal squares xvith a h o o k on each s q u a r e * Cut n u m b e r s f r o m 1 to 1 0 0 and p l ^ c o on separate cards w h o s e size is " the same as that of ' the square on the b o a r d . P e r f o r a t e to fit h o o k and to cover the square. T h e n a m a g i c square of any order 3 to 10 can be r e p r e sented. (See P l a t e L V , n o . 6 ) 3. Trisecting the Angle: There arc many mechanical dev i c e s for s o l v i n g or a p p r o x i m a t i n g the three famous p r o b l e m s of antiquity. A simple m o d e l for t r i s e c t i n g an angle Is to take four s t i c k s , 8 inches in l e n g t h , slotted t h r o u g h at the m i d d l e l o n g V ' W l s o , -and f a s t e n all four at one end to one b o l t * Take 4 sticks of 7 i n c h l e n g t h . B o l t two of t h e m t o gether at the end and f a s t e n b o l t t h r o u g h slot of sec cond arm.(b)., D o l i k e w i s e w i t h the otherjtwo and tho third a r m # ( c ) . C o n n e c t tho outer l o o s e ends succ ;St# slvely b y six equal sticks about 4 or 5 i n c h e s in length. Ifjtho center (0) is p l a c e d at the v o r t e x o f any a n g l e , and the outer slots along the s i d e s , " d r a w ing a p e n c i l p o i n t a l o n g the two inner slots t r i sects the a n g l e .

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