Topics in the History ofMathematics

Workbook

Paul Yiu

Department of Mathematics

Florida Atlantic University

Last Update: July 27, 2005.

Student:

Summer 2005

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Problem A1.(a) To trisect a right angle; i.e., to divide it into three equal parts.(b) To make an angle equal to half a right angle.(c) Trisect two right angles.(d) Divide a right angle into four equal parts.(e) Divide a right angle into six equal parts.

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Problem A2.(a) To trisect a given straight line segment.(b) To trisect a triangle by lines drawn from the vertex to the base.

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Problem A3.Through a given point to draw a straight line which shall make equal angles withtwo straight lines given in position.

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Problem A4.From two given points on the same side of a line given in position, to draw twolines which shall meet in that line, and make equal angles with it.

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Problem A5.On a given line to describe an isosceles triangle, whose perpendicular height isequal to the base.

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Problem A6.Given the diagonal of a square to construct it.

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Problem A7.Given the base, the perpendicular height, and one of the angles at the base of atriangle, to construct it.

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Problem A8.Given the hypotenuse of a right-angled triangle, and the difference of the two acuteangles, to construct the triangle.

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Problem A9.To construct a right-angled triangle, having given the hypotenuse, and one of theacute angles equal to one third a right angle. Show that the side opposite to thegiven angle is equal to one half the hypotenuse.

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Problem A10.Given the two sides, and an angle opposite to one of the given sides of a triangle, toconstruct it. Show that, in general, there are two triangles answering the conditionsof the problem.

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Problem A11.Given the base, one of the other sides, and the perpendicular on the base of atriangle, to construct it.

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Problem A12.On a given base, to construct an isosceles triangle having the angle at the vertexequal to one third a right angle.

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Problem A13.From the vertex of a given triangle, to draw to the base a straight line which shallbe less than the greater side by a given line.

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Problem A14.Given the perpendicular and the equal side of an isosceles triangle, to construct it.

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Problem A15.Through a given point P , to draw a straight line which shall cut off equal partsfrom two straight lines AB and AC , cutting one another in A.

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Problem A16.Given the vertical angle and the perpendicular height of an isosceles triangle, toconstruct it.

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Problem A17.Given the base, the lesser angle at the base, and the difference of the sides of atriangle, to construct it.

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Problem A18.From a given point, to draw the shortest line possible to a given straight line.

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Problem A19.In a straight line given in position, but indefinite in length, to find a point, whichshall be equidistant from each of two given points, on the same side of the givenline, and in the same plane with it.

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Problem A20.To draw a straight line from a given point to meet another straight line, which shallmake with it an angle equal to a given rectilineal angle.

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Problem A21.To find a point, in either of the equal sides of a given isosceles triangle, from which,if a straight line be drawn, perpendicular to that side, so as to meet the other sideproduced, it shall be equal to the base of the triangle.

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Problem A22.From a given point to draw a straight line cutting two parallel straight lines, so thatthe part of it intercepted between them shall be equal to a given finite straight line,not less than the perpendicular distance of the two parallels.

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Problem B1.To describe a circle which shall touch a given straight line in a given point, andalso touch a given circle.

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Problem B2.Given the base, the vertical angle, and the perpendicular height of a triangle, toconstruct it.

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Problem B3.To determine the position of a point of observation, D, at which lines drawn fromthree objects (A, B, C), whose distances from each other are given, shall make witheach other angles equal to given angles; that is, let the angles ADB and ADC begiven angles.

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Problem B4.Through a given point within a circle, to draw a chord which shall be bisected inthat point.

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Problem B5.To draw a tangent to a circle, which shall be parallel to a given straight line.

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Problem B6.Given the hypotenuse and the perpendicular let fall upon it from the opposite angle,to construct the right-angled triangle.

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Problem B7.Divide the circumference of a circle into six equal parts.

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Problem B8.From two given points on the same side of a line given in position, to draw twostraight lines which shall contain a given angle, and be terminated in that line.

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Problem B9.To describe a circle which shall pass through a given point, and touch a given linein another given point.

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Problem B10.To describe a circle which shall have a given radius, and which shall touch twogiven lines, not parallel to each other.

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Problem B11.To describe a circle, which shall touch a straight line in a given point, and alsotouch a given straight line.

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Problem B12.To describe two circles, each having a given radius, which shall touch each other,and the same given straight line on the same side of it.

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Problem B13.To describe three circles of equal diameters, which shall touch each other.

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Problem B14.To describe a circle which shall pass through a given point, have a given radius,and touch a given straight line.

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Problem B15.To describe a circle which shall pass through two given points, and touch a givenstraight line.

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Problem B16.To describe a circle the center of which may be in the perpendicular of a givenright-angled triangle, and the circumference pass through the right angle and touchthe hypotenuse.

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Problem B17.Through a given point to draw a straight line, the part of which intercepted by thecircle, shall be equal to a given line, which is not greater than the diameter.

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Problem B18.To draw a straight line which shall touch two given circles.

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Problem B19.In the diameter of a circle produced, to determine a point from which a tangentdrawn to the circumference shall be equal to the diameter.

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Problem B20.Upon a given straight line to describe a segment of a circle, which shall be similarto a given segment of another circle.

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Problem B21.The perimeter, the vertical angle, and the altitude of a. triangle being given, toconstruct it.

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Problem B22.Two circles being given in position and magnitude, to draw a straight line cuttingthem so that the chords in each circle may be equal to a given line, not greater tha.nthe diameter of the smaller circle.

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Problem B23.To describe a rectangle which shall be equal to a given square, and have its adjacentsides together equal to a given line.

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Problem B24.Having given the distance of the centers of two equal circles which cut each other,to inscribe a square in the space included between the two circumferences.

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Problem B25.In a given circle to inscribe a rectangle equal to a given rectilineal figure.

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Problem B26.To inscribe a circle in a given quadrant.

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Problem B27.The vertical angle, the base, and the sum of the three sides of a triangle being given,to construct it.

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Problem B28.The vertical angle, the base, and the excess of the greater of the two remainingsides, of a triangle, above the less, being given, to construct the triangle.

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Problem B29.To produce a given line, so that the rectangle contained by the whole line thusproduced, and the part of it produced, shall be equal to a given square.

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Problem B30.From the obtuse angle of an obtuse-angled triangle, to draw a straight line to thebase, the square of which shall be equal to the rectangle contained by the segments,into which it divides the base.

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Problem B31.A flag-staff having a given height stands on the top of a tower whose height is alsogiven; at what point in the horizontal line, drawn from the foot of the tower, willthe flag-staff appear under the greatest angle?

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Problem B32.Given the vertical angle, the difference of the two aides containing it, and the dif-ference of the segments or the base made by a perpendicular from the vertex; toconstruct the triangle.

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Problem C1. [JZSS3.2].Now given a cow, a horse and a sheep have eaten up the seedlings of someone’sfield. The landlord demands 5 dou of millet as compensation. The shepherd says:“My sheep eats half as much as the horse”. The horse owner says: “My horse eatshalf as much as a cow”. The compensation is to be distributed according to therate. How much should each repay?

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Problem C2. [JZSS3.4].Now given a skillful weaver, who doubles her product every day. In 5 days sheproduces [a cloth of] 5 chi. How much does she weave in each successive day?

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Problem C3. [JZSS3.20].Now given one lends 1000 coins at a monthly interest of 30 coins. Given one lends750 coins for 9 days. How much interest?Answer: 63

4 coins.Method: Take one month as 30 days to multiply 1000 coins as divisor. L1 Multiplythe interest 30 by the number of coins lent, [and] again by 9 days as dividend.Divide, giving the number of coins. L2

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Problem C4. [JZSS6.16].Now given a guest on horseback rides 300 li a day. The guest leaves his clothesbehind. The host discovers them after 1

3 day, and he starts out with the clothes. Assoon as he catches up with the guest, the host gives back the clothes and returnshome in 3

4 days. Assume the host rides without a stop.Question: How far can he go in a day?

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Problem C5. [JZSS6.21].Now A starts from Zhang An to Qi taking 5 days. B starts from Qi to Zhang Antaking 7 days. Assume B starts 2 days earlier than A. When will they meet?Answer: 2 1

12 days.

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Problem C6. [JZSS6.22].Now one person makes 38 prostrate tiles or 76 supine tiles a day. Assume he makesan equal number of both kinds of tiles in a day. How many tiles [of each kind] canhe make?Answer: 251

3 tiles [of each kind].

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Problem C7. [JZSS6.23].Now a person can straighten 50 arrow shafts in one day, or pack feathers for 30arrows, or install 15 arrow heads. Assume he straightens shafts, packs feathers andinstalls arrow heads single-handedly. How many arrows can he prepare in a day?

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Problem D.1 [JZSS8.7].Now there are 5 cattle [and] 2 sheep costing 10 liang of silver. 2 cattle [and] 5sheep costs 8 liang of silver. What is the cost of a cow and a sheep, respectively?

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Problem D.2 [JZSS8.8].Now sell 2 cattle [and] 5 sheep to buy 13 pigs. Surplus 1000 cash. Sell 3 cattle[and] 3 pigs to buy 9 sheep. [There is] exactly cash. Sell 6 sheep [and] 8 pigs.Then buy 5 cattle. [There is] 600 coins deficit. What is the price of a cow, a sheep,and a pig, respectively?

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Problem D.3 [JZSS8.9].Now there are 5 sparrows [and] 6 swallows. Weigh. Sparrows are heavy andswallows are light. Exchange 1 sparrow and 1 swallow. Their weights are exactlybalanced. The swallows and the sparrows together weigh 1 jun. How much does aswallow and a sparrow each weigh?

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Problem D.4 [JZSS8.10].Now there are 2 persons A and B. Each has an unknown amount of coins. A gets12 of B’s, then [has] 50 coins. B gets 1

3 of A’s then [has] 50 coins also. What is theamount of coins A and B has each?

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Problem D.5 [JZSS8.11].Now there are 2 horses [and] 1 cow. Their price exceeds by 10000 the price of halfa horse. The price of 1 horse [and] 2 cattle is less than 10000 by the price of 1

2 acow. Cow and horse price, how much is each?

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Problem D.6 [JZSS8.12].Now there are 1 military horse, 2 ordinary horses [and] 3 inferior horses. Each[group] carries 40 dan to a slope and cannot climb. The military horse with 1ordinary horse, the ordinary horse with 1 inferior horse [and] the inferior horseswith 1 military horse, can then climb. How much pulling force does 1 military,ordinary [and] inferior horse each have?

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Problem D.7 [JZSS8.18].Now given 9 dou of sesame, 7 dou of wheat, 3 dou of soya beans, 2 dou of smallbeans, 5 dou of corn cost 140 coins in total;7 dou of sesame, 6 dou of wheat, 4 dou of soya beans, 5 dou of small beans and 3dou of corn cost 128 coins;3 dou of sesame, 5 dou of wheat, 7 dou of soya beans, 6 dou of small beans and 4dou of corn cost 116 coins;2 dou of sesame, 5 dou of wheat, 3 dou of soya beans, 9 dou of small beans and 4dou of corn cost 112 coins;1 dou of sesame, 3 dou of wheat, 2 dou of soya beans, 8 dou of small beans and 5dou of corn cost 95 coins.Question: what is the price of 1 dou [of each kind of grain]?

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Problem E1. [JZSS 9.11].Now given a door, whose height exceeds the width by 6 chi 8 cun. Two [opposite]corners are 1 zhang apart.Question: What are the height and the width of the door?

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Problem E2. [JZSS 9.13].A 1 zhang bamboo breaks and its top reaches ground, 3 chi from the bamboo.Question: How tall is the broken bamboo ?

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Problem E3. [JZSS 9.17].Now a given city 200 bu square, with gates opening in the middle of each side. 15bu from the east gate there is a tree.Question: At how many bu from the south gate with one see the tree?

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Problem E4. [JZSS 9.21].A city is a 10 li 1 square. At the center of each side is a gate. A and B part from thecenter of the city. B walks eastwards and A begins southwards for some distanceand then turns (in some northeast direction) through a corner of the city to meet B.Their speeds are in the ratio 5:3. How long has each walked ?Answer: A walks 800 bu from the south gate and then northeastward 48871

2 buuntil he meets B. B walks 43121

2 bu eastwards.

11 li = 300 bu.

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Problem E5.Given the sum of gou and hypotenuse, and the sum of gu and hypotenuse, solvethe right triangle using the method of the Nine Chapters. 1

1ZHU Shijie’s Siyuan yujian (1303). Hint: Imitate the solution of Problem IX.12 of the NineChapters.

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Problem E6.Use the method of the Nine Chapters to solve the following two problems in Dio-phantus’ Arithmetica:

VI.1 To find a (rational) right-angled triangle such that the hypotenuse minuseach of the sides gives a cube.

VI.2 To find a right-angled triangle such that the hypotenuse added to each ofthe sides gives a cube.

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Problem E7. The Chinese tangram.

It is known that exactly 13 convex polygons can be formed from the tangram.1

Show them. 2

1F. T. Wang and C. C. Hsiung, A theorem on the tangram, Amer. Math. Monthly, 49 (1942)596–599.

2There are 1 triangle, 6 quadrilaterals, 2 pentagons, and 4 hexagons.

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Problem F.1 [HDSJ 4].Now, for [the purpose of] looking into a deep valley, set up a carpenter’s square ofheight 6 chi on the lip of the valley. Sight on the bottom of the valley from the tipof the carpenter’s square; the bottom is seen at a point 9 chi 1 cun along the base ofthe carpenter’s square. Set up another similar carpenter’s square above [the first];the distance between the bases of the [two] squares is 3 zhang. Sight on the bottomof the valley from the tip of the upper square; the bottom is now seen at a point 8chi 5 cun along the base of the upper square. How deep is the valley?1

1Answer: 41 zhang 9 chi.

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Problem F.2 [HDSJ 5].Now for [the purpose of] observing a building on the level ground from a mountain,erect a carpenter’s square of height 6 chi on the mountain. Sight on the foot ofthe building from the tip of the square at a downward angle; the foot is seen ata point 1 zhang 2 chi along the base of this square. Set up again another similarcarpenter’s square above [the first]; the distance between the bases of [upper andlower] squares is 3 zhang. Sight on the foot of the building from the tip of theupper square in a downward manner; the foot is now seen at a point 1 zhang 1 chi4 cun along the upper base. Erect another small pole at the observation entry [onthe upper base]. Sight once again from the tip of the upper square in a downwardmanner; the pointed gable of the building is observed at a point 8 cun along theerected pole. What is the height of the building? 1

1Answer: 8 zhang.

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Problem F.3 [HDSJ 6].Now, for [the purpose of] looking toward the southeast at the mouth of a river, erecttwo poles in the north-south direction; the distance between them is 9 zhang, andthey are joined along the ground by a string. Face the west and move away 6 zhangfrom the north pole to observe the southern bank of the mouth of the river at groundlevel; the bank is seen at 4 zhang 2 cun from the north end of the string. Sight onthe northern bank [from the same position]; the bank is seen [along the string] 1zhang 2 chi from the previous observation measurement. Again, move away 13zhang 5 chi from the pole and observe the southern bank of the river’s mouth; it isseen that [the bank] coincides with the pole in the south. What is the width of theriver’s mouth? 1

1Answer: 1 li 200 bu.

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Problem G.1.Within a given circle place six equal circles touching one another and the givencircle.

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Problem G.2.Inscribe a square in a given segment of a circle.

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Problem G.3.Construct the common tangents of 2 given circles.

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Problem G.4.Construct a square equal to a regular hexagon.

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Problem G.5.Construct a regular hexagon equal to a given square.

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Problem G.6.Bisect a triangle by a line drawn parallel to one of its sides.

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Problem G.7.Describe a square, having given the difference between the diagonal and a side.

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Problem G.8.Inscribe a square in a given regular pentagon.

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Problem G.9.Construct a triangle, given one side, its opposite angle, and the ratio of the othertwo sides.

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Problem G.10.Through a given point describe a circle touching a given line and a given circle.

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Problem G.11.Use the following diagram to dissect a square into 6 pieces which can be assembledinto three squares of equal areas.

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Problem G12.We dissect a given equilateral triangle (say each side of length 2) and reassemblethe pieces into a square. Since the area of the triangle is

√3, each side of the

square has length � = 4√

3. The following diagram shows a dissection in which Yand Z are the midpoints of AC and AB. We want to choose X and T on BC andsuitably so that when ZBXP is rotated about Z through 180◦ to ZAX ′B′ andwhen Y QTC is rotated about Y through 180◦ to Y AQ′T ′,(i) PP ′ and PQ′ are two sides of the squares, and(ii) X′, A, T ′ are collinear and(iii) the piece XQT can be translated to complete the square PP′SQ′ so thatXT = X ′T ′.Show that (a) XT = 1, (b) QT = �

2 , and (c) XY = �.

S

Q'

T'X'

P'

P

Q

TX

Z Y

A

BC

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Problem G13.

A circle is approximated by a regular octagon obtained by cutting out cornersfrom its circumscribed square. What is the approximate value of π?

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Problem G14.Archimedes’ Axiom: Given two positive numbers a > b, there exists a positiveinteger n such that a < nb. Make use of Archimedes’ axiom to prove the followingtwo proposition.

(a) If a is a nonnegative real number such that a < 1n for each positive integer

n, then a = 0;(b) If a, b, c are positive numbers satisfying a ≥ b ≥ c and if for every positive

integer n, there is a triangle �(n) with sides an, bn, cn, then a = b (andevery triangle �(n) is isosceles).

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Problem H.1. [JZSS 5.1].Now given an excavation of 10,000 [cubic] chiof mud.Question: How much rammed earth or loam comes from it?

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Problem H.2. [JZSS 5.2].given a city wall with a lower breadth of 4 zhang, and upper breadth of 2 zhang, analtitude of 5 zhang and a length of 126 zhang 5 chi.Question: What is the volume ?

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Problem H.3. [JZSS 5.3].Now given a wall with a lower breadth of 4 chi, and upper breadth of 2 chi, analtitude of 1 zhang 2 chi and a length of 22 zhang 5 chi 8 cun.Question: What is the volume ?

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Problem H.4. [JZSS 5.4].Now given a dyke with a lower breadth of 2 zhang, and upper breadth of 8 chi, analtitude of 4 chi 2 chi and a length of 12 zhang 7 chi.Question: What is the volume ?

Each laborer’s standard winter quota for earthworks is 444 [cubic] chi.Question: How many laborers are required?

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Problem H.5. [JZSS 5.5].Now given a trench with an upper breadth of 1 zhang 5 chi, a lower breadth of 1zhang, a depth of 5 chi and a length of 7 zhang.Question: What is the volume?

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Problem H.6. [JZSS 5.6].Now given a moat with an upper breadth of 1 zhang 6 chi 3 cun, a lower breadth of1 zhang, a depth of 6 chi 3 cun and a length of 13 zhang 2 chi 1 cun.Question: What is the volume?

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Problem H.7. [JZSS 5.7].Now given a canal with an upper breadth of 1 zhang 8 chi, a lower breadth of 3 chi6 cun, a depth of 1 zhang 8 chi and a length of 51824 chi.Question: What is the volume?

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Problem H.8. [JZSS 5.8].Now given a square fort, 1 zhang 6 chi square and 1 zhang 5 chi tall.Question: What is the volume ?

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Problem H.9. [JZSS 5.9].Now given a circular fort with a circumference of 4 zhang 8 chi and an altitude of1 zhang 1 chi.Question: What is the volume ?