()
1998 10 5 5 http://math.ntnu.edu.tw/horng

n=1 (2) n=k

(John (Leonhand Euler1707-1783) (Blaise Pascal
1649 (Sa (Arithmetica Infinitorum, 1655)
x=0 x=x2y x= 0
2 2 2
2 2 2
+ + + + + + + +
= = + +

Wallis1616-1703) 1623-1662)
vilian)
x0, x0 2
HPM
= = + + +
= = + + + +
2 2 2 2
n n n n n n
+ + + + = +

39 3 0 0, 1, 8, 27, 64,…


= = + + =
= = + + + =
+ + + = = = +
+ + + =
= = + + + + =
= = + + + =
= = + + + + =
1 4 ……
64 0
1 1
k k k k
k k k kn
+ + + + =
+ + + + +
n
n
k
HPM

n>2 n n nx y z+ = n=4
1738 1747 1 4a b4+
1
abab
4a b+ 2 2a p q= −
p 2q
2 2b p= q 2 2p m n= + 2q mn= m n
2m x= 2n y= 2 2 4 4p m n x y= + = + 4 4a b+ 4 4x y+ 4a b4+
4 4x y+



Pascal’s triangle

(A treatise on the Arithmetical Triangle) 1654
19

12
1 G 1
3 A 1
6 P 1
7 V 1 7 28 84
8 1 8 36

3
EC E C 2 3
E C EH CRµ
4
1 26
1 2
2
Dμ D B 1 3B θ 2 2 θμ 3 1 Hδ
E C 2 3 D B 1 3 DB B (13) 3 DBE
E B 4 3
HPM
B θ 2 2 Bθ B (22) 2 BθC
C B 4 2 B E 4 3 C E 3 2 7


(generalizable example)(generic examples) 8


1870

18 19 20


1889
(Arithmetics Principia)
∈⊂
N (number)1 (unity)a+1 (the successor of a)
(axiom) 1. 1∈N 2. a∈N a=a 3. a, b∈N a=b b=a 4. a, b, c∈N a=b, b=c a=c 5. a=b, b∈N a∈N 6. a∈N a+1∈N 7. a, b∈N a=b a+1=b+1 8. a∈N a+1≠ 1 9. k class 1∈k x∈N, x∈k x+1∈k N k
5 (1, 8, 6, 7, 9)
HPM
1. 1 2. 1 3. a 4. a b a, b 5. S 1 x∈S x+1∈S S
5 (the axiom of mathematical induction)
(Richard Dedekind, 1831-1916) 1888
Was sind und was sollen die Zahlen?(What are the (natural) numbers and what do they mean?)
sstransform ( )sφ R
( )Nφ N 1
Nproper partSproper partS
NK 1 KKKN KN 11
N ′

5


1. The sum of two biquadratic numbers such as 4a b4+ cannot be a square number
unless one of the two biquadratic numbers vanishes. 2. …no matter how large the numbers a and b, then I can progressively find smaller
numbers a and b and at the end can reach the smallest integral numbers.
3. 1: ( 1) : (n n k kC C k n k+ )= + −
4. Although this proposition has an infinite number of cases I shall give for it a very short demonstration by supposing two lemmas
5. evident in itself 6. [I]f this proposition is true in an arbitrary base, it will necessarily be true in the
next base. 7. The proof can be given in the same way in all the other case
HPM
8. (generic examples)HPM
9. Katz A History of Mathematics pp. 664-p.665 Calinger Classics of Mathematics p. 634
(2003)--HPM 2000 Kline, M. 1983—
Calinger, R. ed. (1995), Classics of Mathematics, New Jersey: Prentice-Hall, Inc. Edwards, A. W. F. (1987), Pascal’s Arithmetical triangle, London: Charles Griffin & Company
Limited. Boyer, C.(1991), A History of Mathematics, Canada: John Wiley & Sons, Inc. Katz, V. J. (1993), A History of Mathematics, New York: HarperCollins College Publishers.

HPM

1

-Al-Karaj Abu Bekr ibn Muhammad ibn al-Husayn Al-Karaji, 953~
1029 - 953
Al-Fakhri

n 10
n=10 n=1
2
2 310
2 3 3(1 2 ... 10) 1 2 3 ... 10+ + + = + + + + 2(1 2 ... 10) (1 2 ... 9)+ + + = + + + +
1 2 ... 10AB AD= = + + + ' ' 10DDBB = = ' ' 'BCDD C B
= 2 2 2 29 1010(1 2 ... 9) 10 2 10 10 9 10 10 10 2
3 C'
3 3 3 3
+ + +
= + + + + +
= + + + + + + =
= + + + +
--
al-Bahir fi'l-jabr The brilliant in algebra
19 --
234
n = kk
n = k n =k+1
n -
n=4 --

4 4 4 3 3 2( ) 4 4 6a b a b ab a b a b+ = + + + + 2
3 2b
2 )
c=a+b 4c cc= 3 3 3 3 2( ) 3 3c a b a b ab a= + = + + +
4 3 3 3 2
3 3 2 2
4 3 3 4 2 2 3 3 2 2
4 4 3 3 2 2
( ) ( )( ) ( )( 3 3 ( ) ( ) ( )3 ( )3
3 3 3 3 4 4 6
a b a b a b a b a b a b ab a b a a b b a b ab a b a b
+ = + + = + + + +
= + + + + + + +
= + + + + + + +
= + + + +
( )r s t rt st+ = +
n=5 al-Samawale
-
-

HPM
x 2x 3x 4x 5x 6x 7x 8x 9x 10x 11x 12x
1 1 1 1 1 1 1 1 1 1 1 1 1 2 3 4 5 6 7 8 9 10 11 12 1 3 6 10 15 21 28 36 45 55 66 1 4 10 20 35 56 84 120 165 220 1 5 15 35 70 126 210 330 495
1 6 21 56 126 252 462 792
1 7 28 84 210 462 924
1 8 36 120 330 792
1 9 45 165 495
1 10 55 220



Bagnols-sur-Cèze
Jacob Staff 1321
Maasei HoshewThe Art of the Calculator
2
“rising step-by-step without end”
-nn-1 2 3 3(1 2 ... ) 1 2 ...n n3+ + + = + + + 2
31 2 ... ) 1 2 ...n n+ + + = + + + 2
2
3
42 1 1
(
2 3 3
2 3(1 2 ... ) (1 2 ... ( 1))n n n+ + + = + + + + − 41 2 3(1 2 ... ( 1)) ( 1) (1 2 ... ( 2)n n n+ + + − = − + + + + − …
21 1=

3
Francesco Maurolico1494~1575 1494 9 16
(Messina)


1575 Arithmeticorum
Libri Duo
11n 2n
n 11 2 3 ... ( 1) 2nt n= + + + + = +n n
n 1n nt t− = −
2 1 2n n nt t t n n−+ = − =




1. Rashed (1994) 62-63 2. Katz (1998) 304 3. 306
HPM
2003—HPM
Katz, Victor (1998), A History of Mathematics: An Introduction, New York: HarperCollins College Publishers
Rashed, Roshdi (1994), The Development of Arabic Mathematics: Between Arithmetic and Algebra. Dordrecht: Kluwer Academic Publishers.
http://www-history.mcs.st-and.ac.uk/history/Mathematicians/Al-Samawal.html http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Al-Karaji.html http://www.dm.unipi.it/pages/maurolic/edizioni/arithmet/ariduo/intro.htm http://math-doc.ujf-grenoble.fr/LiNuM/TM/Gallica/S083056.html http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Infinity.html

HPM

(Blaise Pascal1623~1662)
(1909 )Georges Vacca (Francesco Maurolico1494~1575)1Hans FreudenthalKokiti Hara 1953 1962
1970 Nahum L. Rabinovitch
(Levi ben Jerson1288~1344)
2
Victor J. KatzA History of Mathematics
-Judith Grabiner
Hussein Tahir 1995
(Pappus of Alexandria)(essential ideas)Jean Itard 1961 (Euclid)
-

(Collection)()

HΘ KΛ…… H Θ
AE AΓ H K AE AΓΘΛ AE
AΓ K …… AE AΓ
Pappus
HPM
H AΓ H Θ AΓ 2 Θ
K AΓ 3 K

nCn

Mathematicsn12
3 4
32736
2413 89
27 aba2b2a3b3……anbn……

n2 3 Q.E.D.

-

1858~1932)



(2005) (2005) (2005)

nk
nk1
HPM

Rashed
-
Rashed
1. Kline (1972)p. 272 2. Rashed (1994)pp. 62~63 3. Grabiner Grabiner (1999) 4. Heath “ad infinitum” Heath (1981)p. 376 Grabiner, Judith. (1999),
http://mathforum.org/epigone/historia_matematica/chysnermgou/Pine.GSO.3.96.99030511105 1.12553A-100000@calvin.pitzer.eduT
Heath, Thomas L. (1956), The Thirteen Books of Euclid’s Elements. New York: Dover Publications.
Heath, Thomas L. (1981), A History of Greek Mathematics. New York: Dover Publications. Katz, Victor J. (1993), A History of Mathematics. New York: HarperCollins College Publishers. Kline, Morris (1972), Mathematical Thought from Ancient to Modern Times. New York: Oxford
University Press. Rashed, Roshdi (1994), The Development of Arabic Mathematics: Between Arithmetic and
Algebra. Dordrecht: Kluwer Academic Publishers. Tahir, Hussein (1995), “Pappus and Mathematical Induction”, The Gazette of the Australian
Mathematical Society 22 (4): 166-167. Thomas, Ivor (1994), Greek Mathematics Works II: From Aristarchus to Pappus. London:
William Heinemann LTD. (2005),HPM 8(2/3) (2005),HPM 8(2/3) (2005),HPM 8(2/3) (2005),--
HPM 8(4) (2005).HPM 8(4)
1970
International Study Group on the Relations between the History and Pedagogy of Mathematics, HPM
HPM
HPM
HPM


HPM







HPM “”(, 2001) 6
68


, 2000
2 John Wallis(1616-1703)
3 ∑ =
n
5
(Peano axiom)
6
HPM
1, 3, 2, 6
4 6



HPM
HPM
HPM
29
8 HPM

HPM

HPM HPM

HPM
(2001)., (
NSC 89-2511-S-003-031-; NSC 89-3511-S-003-121) (2002).HPM 5(12)1-3 (2001).
239-253 (2003).HPM
6(2/3) (2000).
() (2004).
()
e-mail suhui_yu@yahoo.com.tw 2. 3.
e-mailsuhui_yu@yahoo.com.tw 4. http://math.ntnu.edu.tw/horng/letter/hpmletter.htm 5.
HPM ()