Problem 10, MMP

Struve:

… die Berechnung der Oberfläche einer Halbkugel …
O: die Oberfläche der Halbkugel.
d: den Durchmesser der “Mündung in Erhaltung”, d.h. der größten Kreises der Halbkugel.
O = [ (2d – 2/9 d) – 1/9 (2d – 2/9 d) ] * d
O = 1/2 d π d
x = π/4 = (8/9)², Annäherung
A: Halbkreisumfang.
A = (2d – 2/9 d) – 1/9 (2d – 2/9 d)
A = (2d – 2/9 d)(1 – 1/9)
A = 2d (1 – 1/9) (1 – 1/9)
A = 2d (8/9)²
A = 2d x

🇩🇪 Lise:

Information
- M = 4 1/2, Mündung, Diameter
- O = Oberfläche von Halbkugel (🇺🇲 surface area)
- O = 2 π r²
- r = Radius, M/2
- π/4 = (1 – 1/9)²
π = 4 (1 – 1/9)²
O = 2 π r²
O = 2 * 4 (1 – 1/9)² (M/2)²
O = 2 (1 – 1/9)² M²
O = 2M (1 – 1/9)² M
O = 2M (1 – 1/9) (1 – 1/9) M
O = (2M – 1/9 2M) (1 – 1/9) M
O = (2 * 4,5 – 1/9 * 2 * 4,5) (1 – 1/9) 4,5
O = (9 – 1/9 * 9) (1 – 1/9) 4,5
O = (9 – 1) (1 – 1/9) 4,5
O = 8 (1 – 1/9) 4,5
O = (8 – 8 * 1/9) 4,5
O = (8 – 8/9) 4,5
O = 7 1/9 * 4,5
O = 32

	🇬🇧 Peet	🇬🇧 Peet
1	Example of working out a semicircle.	Example of working out a semi-cylinder.
2	If they say to you, A semicircle <of 9> in diameter	If they say to you, A semicircle <of 9> in diameter
3	by 4 1/2 in height, pray	by 4 1/2 in height, pray
4	let me know its area. You are to	let me know its area. You are to
5	take a ninth of 9, since a semicircle	take a ninth of 9, since a semi-cylinder
6	is half a [circle], result 1.	is half a [cylinder], result 1.
7	Take the remainder, namely 8.	Take the remainder, namely 8.
8	You are to take a ninth of 8,	You are to take a ninth of 8,
9	result 2/3 + 1/6 + 1/18. You are to take	result 2/3 + 1/6 + 1/18. You are to take
10	the remainder of the 8 after (subtracting)	the remainder of the 8 after (subtraction of)
11	the 2/3 + 1/6 + 1/18, result 7 1/9.	the 2/3 + 1/6 + 1/18, result 7 1/9.
12	You are to take 7 1/9 4 1/2 times,	You are to take 7 1/9 4 1/2 times,
13	result 32. See, this is its area.	result 32. See, this is its area.
14	You will find it correct.	You will find it correct.

	🇩🇪 Struve	🇺🇲 Clagett
XVIII, 1	Form der Berechnung eines Korbes (nb.t),	Example of calculating a basket[assumed by Struve as hemispheric in shape]
2	wenn man dir nennt einen Korb mit einer Mündung (tp-rз)	If someone says to you: “A basket with a mouth opening
3	zu 4 1/2 in Erhaltung. O	of 4 1/2 (i.e., a diameter of this size) in good condition , oh
4	laß du mich wissen seine (Ober)fläche. Berech-	let me know its [surface] area .”
5	ne du 1/9 von 9, weil ja der Korb (nb.t)	[First] calculate 1/9 of 9, since the basket is
6	die Hälfte eines Eies ist. Es entsteht 1.	1/2 of an egg-shell (? inr?). The result is 1.
XIX, 1	Berechne du den Rest als 8.	Calculate the remainder as 8.
2	Berechne du 1/9 von 8.	Calculate 1/9 of 8.
3	Es entsteht 2/3 1/6 1/18. Berech-	The result is 2/3 1/6 1/18. Cal-
4	ne du den Rest von dieser 8 nach	culate the remainder from these 8 after
5	diesen 2/3 1/6 1/18. Es entsteh[t] 7 1/9.	taking away those 2/3 1/6 1/18. The result is 7 1/9.
XX, 1	Rechne du mit 7 1/9 4 1/2mal.	Reckon with 7 1/9 four and one-half times.
2	Es entsteht 32. Siehe: es ist seine (Ober)fläche.	The result is 32. Behold, this is its area.
3	Du hast richtig gefunden.	You will find that it is correct.