Archimedes and His Mechanical Method

Transcription

1 chimedes and His Mechanical Method Histoical Contet: When: 87 -.C. Whee: Syacuse, Sicily (Geece) Who: chimedes Mathematics focus: Investigation of the use of mechanical pinciples to establish a fundamental mathematical elationship involving volumes. Suggested Readings: chimedes and his contibutions to mathematics and science: chimedes on mechanical and geometic methods: NCTM s Histoical Topics fo the Mathematics Classoom (969): Late Geek geomety (pp ) and chimedes and his anticipation of calculus (pp ) Key seach wods/phases: chimedes, Geek geomety, mechanical method, volume elationships, Eudous, Democitus, chimedes Code Poblem to Eploe: Investigate the special elationship involving volumes geneated by otations of geometical figues about an ais, namely V :V :V 3 :: ::3, whee V V 3 V Why This Poblem is Impotant: Establishes a fundamental mathematical elationship involving volumes. Povides insight into chimedes use of his mechanical method fo investigating and undestanding mathematical elationships.

2 Poblem Solving Epeiences: Fo 000 yeas, mathematicians and histoians wondeed how chimedes achieved his many esults; some even concluded that chimedes has delibeately hidden his pocedues. Then, in 906 in a emote monastey, J.L. Heibeg found a lost lette fom chimedes to Eatosthenes, in which he descibed his method. The lette was the eased pat of a palimpsest being used fo payes and to guide ituals. In this lette, chimedes descibed his method : Cetain theoems fist became clea to me by means of a mechanical method. Then, howeve, they had to be poved geometically since the method povided no eal poof. It is obviously easie to find a poof when we have aleady leaned something about the question by means of the method than it is to find one without such advance knowledge. To illustate his mechanical method, conside the poblem of elating the volume of a cone inscibed in a hemisphee, both shaing a common cicula base of adius : Though stange it may seem, chimedes solved this poblem via the Law of the Leve: leve is in equilibium iff the espective poducts of the two weights and thei distances (fulcum to suspicion point) ae equal, i.e. the moments a = b. a b chimedes agument was as follows: Constuct a cicle with cente, adius, then daw two pependicula diametes and CD Etend segment C to point E, the intesection with the pependicula line to segment at point. Repeat fo segment D to get intesection point F E C D F

3 . Eplain why E = F =. Complete ectangle EFGH, then daw line pependicula to at a andom point Z, intesecting the cicle at points R and S, segments E and F at T and U espectively, and segments HE and GF at points P and Q espectively. Label Z =, ZR = y. H P E R C y T Z U S D F G Q. Pove that + y = ()(). Hint: Daw segments R and R. Divide both sides of the equation by (), to get + y () =. π + πy (*) Multiply the left side of the equation by π /π to get = π () Etend to point W so that the length of segment W = (). H P E C R T y W Z U S D F G Q Visually otate the entie diagam 360 o aound ais W. 3. s a esult of the otation, what is epesented geometically by the otation of tiangle EF? The otation of the cicle with cente? The otation of ectangle EFGH? The otation of segment PQ? The otation of tiangle CD?

4 The otation of ac CD? nd in tems of these otations, what is the visual meaning of the epessions π, π y, and π () in the pevious equation? Recalling that point Z was a andom point on segment, the last equation Sumof aeas of cicles C and C ( o length Z) can be ewitten as =. ea of ciclec ( o length W ) This popotion suggests visually a leve W with fulcum at point. W Z 3 cicle C cicle C cicle C 3 nd, as abitay point Z moves fom to, we can e-think the popotion in tems of the cone, sphee, and cylinde being geneated by tiangle EF, the sphee + cone cicle, and ectangle EFGH espectively: =, whee length cylinde W is used because it is the distance to the cente of gavity of the geneated cylinde. u evised pictue is: W cone sphee cylinde 4. Show that (sphees) + (cones) = (cylinde). chimedes then used a elationship suggested by Democitus (430.C.), namely that fo a cone inscibed in a cylinde, both shaing a common cicula base of adius, the volume of the cone is one-thid the volume of the cylinde. 5. Use Democitus elationship to show that (sphees) = (cone). This equation elates the volume of the full sphee and the volume of the lage cone geneated by tiangle EF. Howeve, the desied equation needs to elate the volume of the hemisphee (geneated by ac CD) and the volume of the small cone inscibed in it (geneated by tiangle CD). 6. Show that (hemisphee) = (small cones), which is what chimedes was tying to pove. nd with Democitus elationship, chimedes established V :V :V 3 :: ::3; he late equested that this popotion and epesentative diagam be etched on his gavestone.

5 Such an ingenious use of the Law of the Leve! lso, you should see the seeds of calculus and using integation to find volumes of evolution. Etension and Reflection Questions: Etension : In the poof s step(*), epessions wee multiplied by π, something chimedes could not do as π was not yet a ecognized numbe and the fomula ea of a cicle = π was not known. Rathe, chimedes achieved the same esults using a Theoem poven by Eudous (370.C.): The aeas of two cicles ae in the same atio as the squaes of thei diametes, i.e. =. His poof uses a ( d) known fact : If simila polygons ae inscibed in cicles C and C, thei aeas ae in the same atio as the squaes of the cicles diametes (a vaiation of Euclid s Poposition 9, ook 6). s an ealy eample of poof by contadiction, Eudous poof begins: Suppose the statement is not tue, i.e. < o >. ( d ) ( d ) Without loss of geneality, assume <. ( d ) Constuct a egion R inteio to cicle C so that its aea S < and =. S ( d ) Cicle C Cicle C Region R ea ea S diamete d diamete d Finish this poof, by eaching a contadiction via the known fact. Etension : fte poving this volume elationship, chimedes continued: Fom this theoem, to the effect that a sphee is fou times as geat as the cone with a geat cicle of sphee as base and with height equal to the adius of the sphee, I conceived the notion that the suface of any sphee is fou times as geat as a geat cicle in it.

6 To undestand his easoning, fist conside a tiangle with base b and height h, whose aea = ½bh. Now, patition the base into a lage numbe of small sections of length b i such that b = b +b +.+b n. y connecting the opposite vete to the endpoints of each small section, n- tiangles ae fomed, all with height h. Then, we have = ½bh = ½( b +b +.+b n )h. h h b b b b 3 b 4 b 5 b 6 Use this idea to pove chimedes claim that the suface aea of any sphee is fou times as geat as the aea of a geat cicle in it. pen-ended Eploation: Use calculus and integation to pove these same elationships, namely V :V :V 3 :: ::3 fo volumes geneated by otations of these geometical figues about an ais: V V 3 V and the claim that the suface aea of any sphee is fou times as geat as the aea of a geat cicle in it.