Euclid. # Euclid's Elements of geometry, the first six books, chiefly from the text of Dr. Simson, with explanatory notes; a series of questions on each book ... Designed for the use of the junior classes in public and private schools online

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half the difference of the unequal parts.

The proof of Prop. x. may be deduced from Euc. ii. 4, 7, as Prop. ix.

Prop. X. Algebraically.

Let the line AB contain 2a linear units, of which its half AC or CB

will contain a units ;

and let BD contain m units.

Then the whole line and the part produced will contain ^a + m units,

and half the line and the part produced will contain a + m units,

.'. (2a + my = ia^ + 4am + m^,

add m^ to each of these equals,

/. (2a + my + m^ = 4a2 + 4am + 2wt^

Again, (a + my = a* + 2atn + m^,

add a^ to each of these equals,

.'. {a + my + a'^ = 2a^ + 2am + m\

and doubling these equals,

.-. 2 (a + my + 2d^ = ^a^ + 4am + 2m^

But (2a + my + mÂ« = 4a* + 4am + 2mÂ«.

Hence .'. (2a + m)' + m* = 2a2 + 2 (a + my.

That is, If a number be divided into two equal parts, and the whole

number and one of the parts be increased by the addition of another num-

ber, the squares of the whole number thus increased, and of the number

by which it is increased, are equal to double the squares of half the num-

ber, and of half the number increased.

The algebraical results of Prop, ix, and Prop, x, are identical, (the

enunciations of the two Props, arising, as in Prop, v, and Prop, vi, from

the two ways of exhibiting the difference between two lines) ; and both

may be included under the following proposition : The square on the

sum of two lines and the square on their difference, are together equal to

double the sum of the squares on the two lines.

Prop. xr. Two series of lines, one series decreasing and the other

series increasing in magnitude, and each line divided in the same man-

ner may be found by means of this proposition.

(1) To find the decreasing series.

In the fig. Euc. ii, 11, AB ^ AH + BR,

and since AB . BH = AH\ .'. {AH + BH) . BH = AH\

.-. BH^ = AH""- AH.BH=AH.(AH- BH).

If now in HJ, HL be taken equal to BH,

then HL^ = AH {AH - HL), or AH . AL = HU :

that is, AH is divided in L, so that the rectangle contained by the whole

line AH and one part, is equal to the square on the other part HL. IBy a

similar process, HL may be so divided ; and so on, by always taking from

the greater part of the divided line, a part equal to the less.

(2) To find the increasing series.

From the fig. it is obvious that CF . FA = CA^,

Hence CF is divided in A, in the same manner as AB is divided in H,

by adding AF a. line equal to the greater segment, to the given line CA

NOTES TO BOOK II. 107

IB. And by successively adding to the last line thus divided, its

greater segment, a series of lines increasing in magnitude may be found

similarly divided to JB.

It may also be shewn that the squares on the whole line and on the less

segment are equal to three times the square on the greater segment.

(Euc. XIII. 4.)

To solve Prop, xi, algebraically, or to find the point H in AB such

that the rectangle contained by the whole line AB and the part HB shall

be equal to the square on the other part AH.

Let JB contain a linear units, and AH one of the unknown parts con-

tain X units,

then the other part HB contains a â€” x units.

And .*. a(a â€” x) = x^, by the problem,

or x'* + ax = a*, a quadratic equation.

Whence x = â€¢= o â€¢

The former of these values of x determines the point H.

So that X = ^r â€” . AB = AH, one part,

and a â€” X = a â€” AH = . AB =HB, the other part.

It may be observed, that the parts AH and HB cannot be numerically

expressed by any rational number. Approximation to their true values

in terms of AB, may be made to any required degree of accuracy, by ex-

tending the extraction of the square root of 5 to any number of decimals.

To ascertain the meaning of the other result x = . a.

z

In the equation a (a â€” x) = x'^,

for X write â€” x, then a{a + x) = x*,

which when translated into words gives the following problem.

To find the length to which a given line must be produced so that the

rectangle contained by the given line and the line made up of the given line

and the part produced, may be equal to the square on the part produced.

Or, the problem may also be expressed as follows :

To find two lines having a given difference, such that the rectangle con-

tained by the difference and one of them may be equal to the square on

the other.

It may here be remarked, that Prop. xi. Book ii, affords a simple

Geometrical construction for a quadratic equation.

Prop. XII. Algebraically.

Assuming the truth of Euc. i. 47.

Let BC, CA, AB contain a, b, c linear units respectively,

and let CD, DA, contain m, n Tinits,

then BD contains a + m units.

And therefore, c* = (a + my + n', from the right-angled triangle ABD,

also b^ = m^ + n^ from A CD ;

.'. c'^ â€” 6* = (a + my â€” m?

= a^ + 1am + m- â€” m^ ,

108

EUCLID S ELEMEiMS.

= a' + 2a ?n,

/. c' = 6^ + a* + 2am,

that is, c- is greater than V^ + or by 2am.

Prop. XIII. Case ii. may be proved more simply as foUo-vvs.

Since BD is divided into two parts in the point Z>,

therefore the squares on CB, BD are equal to twice the rectangle con-

tained by CB, BD and the square on CD ; (ii. 7.)

add the square on AD to each of these equals ;

therefore the squares on CB, BD, DA are equal to twice the rectangle

CB, BD, and the squares on CD and DA,

but the squares on BDy DA are equal to the square on AB, (i. 47.)

and the squares on CD, DA are equal to the square on AC^

therefore the squares on CB, BA are equal to the square on AC, and

twice the rectangle CB, BD. That is, &c.

Prop. XIII. Algebraically.

Let BC, CA, AB contain respectively o, 6, c linear units, and let BD

and AD also contain m and n units.

Case I. Then DC contains a â€” m units.

Therefore c* = n^ + m- from the right-angled triangle ABD,

and 6Â« = n^ + (a - tnf from ADC ;

.*. c^ â€” P = m^ â€” {a â€” my

= m^ â€” a^ -{â– 2am â€” m*

= â€” a^ + 2am,

/. Â«Â« + c8 = ^8 + 2am,

or i* + 2am = a^ + c*,

that is, h^ is less than a' + c^ by 2am.

Case II. DC = m â€” a units,

/, c* = m^ + w* from the right-angled triangle ABB,

and b"^ = (m â€” ay + n- from ACDf

.'. c^ â€” b^ = m^ â€” {m â€” ay,

= m* â€” m^ + 2am â€” a^

= 2am â€” a',

/. aÂ« + c' = 62 + 2am,

or b^ + 2am = a* + c',

that is, S'' is less than a* + c^ by 2am.

Case III. Here m is equal to a.

And 6"^ + a* = c^, from the right-angled triangle ABC,

Add to each of these equals a^,

.-. b^ + 2aÂ« = cÂ« + aS

that is, 6' is less than c* + a' by 2a2, or 2aa.

These two propositions, Euc. ii. 12, 13, with Euc. i. 47, exhibit the

relations which subsist between the sides of an obtuse-angled, an acute-

angled, and right-angled triangle respectively.

( 109 )

NOTE OX THE ABBREVIATIONS AND ALGEBRAICAL

SYMBOLS EMPLOYED D^ GEOMETRY.

The ancient Geometry of the Greeks admitted no symbols besides the

diagrams and ordinary language. In later times, after symbols of opera-

tion had been devised by writers on Algebra, they were very soon adopted

and employed on account of their brevity and convenience, in writings

purely geometrical. Dr. Barrow was one of the first who introduced

algebraical symbols into the language of Elementary Geometry, and dis-

tinctly states in the preface to his Euclid, that his object is "to content

the desires of those who are delighted more with symbolical than verbal

demonstrations." As algebraical symbols are employed in almost all

works on the mathematics, whether geometrical or not, it seems proper

in this place to give some brief account of the marks which may be re-

garded as the alphabet of symbolical language.

The mark = was first used by Robert Recorde, in his treatise on Algebra

entitled, " The Whetstone of Witte," 1557. He remarks ; '* And to avoide

the tediouse repetition of these woordes : is equalle to : I will sette as I

doe often in woorke use, a paire of paralleles, or Gemowe lines of one

lengtbe, thus : =, bicause noe 2 thynges can be more equalle." It was

employed by him as simply affirming the equality of two numerical or

algebraical expressions. Geometrical equality is not exactly the same

as numerical equality, and when this symbol is used in geometrical reason-

ings, it must be understood as having reference to pure geometrical

equality.

The signs of relative magnitude, > meaning, is greater than, and < , is

less than, were first introduced into algebra by Thomas Harriot, in his

*' Artis Analyticas Praxis," which was published after his death in 1631.

The signs + and â€” were first employed by Michael Stifel, in his "Arith-

metica Integra," which was published in 1544. The sign + was employed

by him for the word plus, and the sign â€” , for the word minus. These

signs were used by Stifel strictly as the arithmetical or algebraical signs

of addition and subtraction.

The sign of multiplication x was first introduced by Oughtred in his

**Clavis Mathematica," which was published in 1631. In algebraical

multiplication he either connects the letters which form the factors of a

product by the sign x , or writes them as words without any sign or mark

between them, as had been done before by Harriot, who first introduced

the small letters to designate known and unknown quantities. However

concise and convenient the notation AB x BC or AB . BC may be in.

practice for ** the rectangle contained by the lines AB and BC" ; the student

is cautioned against the use of it, in the early part of his geometrical

studies, as its use is likely to occasion a misapprehension of Euclid's

meaning, by confounding the idea of Geometrical equality with that of

Arithmetical equality. Later writers on Geometry who employed the

Latin language, explained the notation AB x BC, by ** AB ductum in

BC" ; that is, if the line AB be carried along the line BC in a normal

position to it, until it come to the end C, it will then form with BC, the

rectangle contained by AB and BC. Dr. Barrow sometimes expresses

*Hhe rectangle contained by AB and BC" by **the rectangle ABC."

Michael Stifel was the first who introduced integral exponents to

denote the powers of algebraical symbols of quantity, for which he em-

ployed capital letters. Vieta afterwards used the vowels to denote known,

and the consonants, unknown quantities, but used words to designate the

i

110 Euclid's elements.

powers. Simon Stevin, in his treatise on Algebra, which was published

m 1605, improved the notation of Stifel, by placing the figures that in-

dicated the powers within small circles. Peter Ramus adopted the

initial letters /, q, c, bq oi latus, quadratus, cubus, biquadratus, as the nota-

tion of the first four powers. Harriot exhibited the difi'erent powers of

algebraical symbols by repeating the symbol, two, three, four, &c. times,

according to the order of the power. Descartes restored the numerical

exponents of powers, placing them at the right of the numbers, or symbols

of quantity, as at the present time. Dr. Barrow employed the notation

ABg, for " tJie square on the line AB," in his edition of Euclid. The

notations AB"'^ AB^, for " the square and cube on the line whose extremities

are A and J5," as well as AB x BC, for *'the rectangle contained by AB

a7id BC," are used as abbreviations in almost all works on the Mathe-

matics, though not wholly consistent with the algebraical notations a*

and a^.

The symbol V, being originally the initial letter of the word radix, was

first used by Stifel to denote the square root of the number, or of the

symbol, before which it is placed.

The Hindus, in their treatises on Algebra, indicated the ratio of two

numbers, or of two algebraical symbols, by placing one above the other,

without any line of separation. The line was first introduced by the Ara-

bians, from whom it passed to the Italians, and from them to the rest of

Europe. This notation has been employed for the expression of geome-

trical ratios by almost all writers on the Mathematics, on account of its

great convenience. Oughtred first used points to indicate proportion ;

thus, a:b::c:d, means that a bears the same proportion to 6, as c does to d.

QUESTIONS ON BOOK 11.

1. Is rectangle the same as rectus angulus? Explain the distinction,

and give the corresponding Greek terms.

2. What is meant by the sum of two, or more than two straight lines

in Geometry ?

3. Is there any difference between the straight lines by which a rect-

angle is said to be contained, and those by which it is bounded ?

4. Define a gnomon. How many gnomons appear from the same con-

struction in the same rectangle ? Find the difference between them.

5. What axiom is assumed in proving the first eight propositions of

the Second Book of Euclid ?

6. Of equal squares and equal rectangles, which must necessarily coin-

cide ? ^ .

7. How may a rectangle be dissected so as to form an equivalent

rectangle of any proposed length ?

8. When the adjacent sides of a rectangle are commensurable, the area

of the rectangle is properly represented by the product of the number^ of

units in two adjacent sides of the rectangle. Illustrate this by considering

the case when the two adjacent sides contain 3 and 4 units respectively,

and distinguish between the units of the factors and the units of the product.

Shew generally that a rectangle whose adjacent sides arerepresentedby the

integers a and'6, is represented by ab. Also shew, that in the same sense,

ah J 1. ^ ^

the rectangle is represented by â€” , if the sides be represented by -, - â€¢

QUESTIONS ON BOOK IT. Ill

9. Why may not Algebraical or Arithmetical proofs be substituted (as

being shorter) for the demonstrations of the Propositions in the Second

Book of Euclid ?

10. In what sense is the area of a triangle said to be equal to half the

product of its base and its altitude ? What two propositions of Euclid

may be adduced to prove it ?

11. How do you shew that the area of a rhombus is equal to half the

rectangle contained by the diagonals ?

12. How may a rule be deduced for finding a numerical expression

for the area of any parallelogram, when two adjacent sides are given?

13. The area of a trapezium which has two of its sides parallel is equal

to that of a rectangle contained by its altitude and half the sum of its

parallel sides. What propositions of the First and Second Books of Euclid

are employed to prove this ? Of what service is the above in the men-

suration of fields with irregular borders ?

14. From what propositions of Euclid may be deduced the following

rule for finding the area of any quadrilateral figure : â€” *' Multiply the sum

of the perpendiculars drawn from opposite angles of the figure upon the

diagonal joining the other two angles, and take half the product."

15. In Euclid, II. 3, where must be the point of division of the line, so

that the rectangle contained by the two parts may be a maximum ? Ex-

emplify in the case where the line is 12 inches long.

16. How may the demonstration of Euclid ii. 4, be legitimately short-

ened ? Give the Algebraical proof, and state on what suppositions it can

be regarded as a proof.

17. Shew that the proof of Euc. ii. 4, can be deduced from the two

previous propositions without any geometrical construction.

18. Shew that if the two complements be together equal to the two

squares, the given line is bisected.

19. If the line AB^ as in Euc. ii. 4, be divided into any three parts,

enunciate and prove the analogous proposition.

20. Prove geometrically that if a straight line be trisected, the square

on the whole line equals nine times the square on a third part of it.

21. Deduce from Euc. ii. 4, a proof of Euc. i. 47.

22. If a straight line be divided into two parts, when is the rectangle

contained by the parts, the greatest possiblef and when is the sum of the

squares of the parts, the least possible ?

23. Shew that if a line be divided into two equal parts and into two

unequal parts ; the part of the line between the points of section is equal

to half the difference of the unequal parts.

24. If half the sum of two unequal lines be increased by half their

difference, the sum will be equal to the greater line : and if the sum of

two lines be diminished by half their difference, the remainder will be

equal to the less line.

25. Explain what is meant by the internal and external segments of a

line ; and show that the sum of the external segments of a line or the

difference of the internal segments is double the distance between the

points of section and bisection of the line.

26. Shew how Euc. ii. 6, may be deduced immediately from the

preceding Proposition.

27. Prove Geometrically that the squares on the sum and difference

of two lines are equal to twice the squares on the lines themselves.

28. A given rectangle is divided by two straight lines into four rect-

angles. Given the areas of the two which have not common sides : find

the areas of the other two.

112 Euclid's elements.

29. In how many ways may the difference of two lines be exhibited?

Enunciate the propositions in Book ii. which depend on that circumstance.

30. How may a series of lines be found similarly divided to the line

^^ in Euc. II. n ?

31. Divide Algebraically a given line {a) into two parts, such that

the rectangle contained by the whole and one part may be equal to the

square of the other part. Deduce Euclid's construction from one so-

lution, and explain the other.

32. Given the lesser segment of a line, divided as in Euc. ii. 11,

find the greater.

33. Enunciate the Arithmetical theorems expressed by the following

Algebraical formulae,

(a + by = aÂ« + 2ab + b^ -.a" - b^ = (a + b) (a-b) : {a-bf = a" - 2ab + b\

and state the corresponding Geometrical propositions.

34. Shew that the first of the Algebraical propositions,

{a + x) {a-x) +x^=:a^:ia + xy 4- (a - xy = 2a^ + 2x\

is equivalent to the two propositions v. and vi., and the second of them,

to the two propositions ix. and x. of the Second Book of Euclid.

35. Prove Euc. ii. 12, when the perpendicular BE is drawn from

B on AC produced to E, and shew that the rectangle BC, CD is equal

to the rectangle AC, CE.

36. Include the first two cases of Euc. ii. 13, in one proof.

37. In the second case of Euc. ir. 13, draw a perpendicular CE from

the obtuse angle C upon the side AB, and prove that the square on AB

is equal to the rectangle AB, AE together with the rectangle BC, BD.

38. Enunciate Euc. ir. 13, and give an Algebraical or Arithmetical

proof of it.

39. The sides of a triangle are as 3, 4, 5. Determine whether the

angles between 3, 4 ; 4, 5 ; and 3, 5 ; respectively are greater than, equal

to, or less than, a right angle.

40. Two sides of a triangle are 4 and 5 inches in length, if the

third side be 6i^6 inches, the triangle is acute-angled, but if it be 60

inches, the triangle is obtuse-angled.

41. A triangle has its sides 7, 8, 9 units respectively; a strip of

breadth 2 units being taken off all round from the triangle, find the

area of the remainder.

42. If the original figure, Euc. 11. 14, were a right-angled triangle,

v.'hose sides were represented by 8 and 9, what number would represent

the side of a square of the same area ? Shew that the perimeter of the

square is less than the perimeter of the triangle.

43. If tl:e sides of a rectangle are 8 feet and 2 feet, what is the side

of the equivalent square ?

44. "All plane rectilineal figures admit of quadrature." Point out

the succession of steps by which Euclid establishes the truth of this

proposition.

45. Explain the construction (without proof) for making a square

equal to a plane polygon.

46. Shew from Euc. 11. 14, that any algebraical surd as VÂ« can be

represented by a line, if the unit be a line.

47. Could any of the propositions of the Second Book be made co-

rollaries to other propositions, with advantage ? Point out any such pro-

positions, and give your reasons for the alterations you would make.

GEOMETRICAL EXERCISES ON BOOK 11.

PROPOSITION I. PROBLEM.

Divide a given straight line into two parts such, that their rectangle mag

he equal to a given square ; and determine the greatest square which the

rectangle can equal.

Let AB he the given straight line, and let M be the side of the

given square.

It is required to divide the line AB into two parts, so that the

rectangle contained by them may be equal to the square on 31.

D E

M.

A i' C B

Bisect AB in C, with center C, and radius CA or CB, describe the

semicircle ADB.

At the point B draw BJE at right angles to AB and equal to 3f.

Through JEJ, draw J3D parallel to AB and cutting the semicircle

inZ);

and draw DJP parallel to JEB meeting AB in F.

Then AB is divided in F, so that the rectangle AF, FB is equal

to the square on M. (il. 14.)

The square will be the greatest, when ED touches the semicircle,

or when 3Â£ is equal to half of the given line AB.

PROPOSITION II. THEOREM.

The square oti the excess of one straight line above another is less than the

squares on the tico lines by tioice their rectangle.

Let AB, BChe the two straight lines, wliose difference is AC.

Then the square on ^Cis less than the squares on AB and BChy

twice the rectangle contained by AB and BC.

A c B

K

g/

F E

Constructing as in Prop. 4. Book II.

Because the complement ^ 6^ is equal to GF,

add to each CK,

therefore the whole ^^is equal to the whole CF .

114 GEOMETRICAL EXERCISES

and AK, CE together are double of AK-,

but AK, CE are the gnomon AKF ond CK,

and AK is the rectangle contained by AB, JBC;

therefore the gnomon AKF and CKy

are equal to twice the rectangle AB, BC,

but AE, CK are equal to the squares on AB, BC;

taking the former equals from these equals,

therefore the difference of ^^ and the gnomon AKF is equal to

the difference between the squares on AB.BC.Rud twice the rectangle

JB,BC',

but the difference AE and the gnomon AKF is the figure SF

which is equal to the square on A C.

Wherefore the square on AC is equal to the difference between the

squares on AB, BC, and twice the rectangle AB, BC.

PROPOSITION III. THEOREM.

In any triangle the squares on the two sides are together double of the

squares on half the base and on the straight line joining its bisectio7i with the

opposite angle.

Let ABChe 2l triangle, and^Z) the line drawn from the vertex A

to the bisection D of the base BC.

From A draw AE perpendicular to BC.

Then, in the obtuse-angled triangle ABE, (ll. 12.) ;

the square on AB exceeds the squares on AE, EB, by twice the

rectangle BE, EE:

and in the acute-angled triangle AEC, (ll. 13.) ;

the square on ^C is less than the squares on AE, EC, by twice

the rectangle CE, EE:

wherefore, since the rectangle BE, EE is equal to the rectangle CE,

EE ; it follows that the squares on AB, A C are double of the

squares on AE, EB.

PROPOSITION IV. THEOREM.

If straight lines be drawn from each angle of a tria,ngle bisecting the

opposite sides, four times the sum of the squares 07i these lines is equal to

three titnes the sum of the squares on the sides of the triangle.

Let ABC he any triangle, and let AE, BE, CF be drawn from

A, B, C, to E, E, F, the bisections of the opposite sides of the tri-

angle : di-aw AG perpendicular to BC.

115

Then the square on AB is equal to the squares on JBD, DA together

with twice the rectangle J3D, DG, (ll. 12.)

and the square on A C is equal to the squares on CD, DA dimi-

nished by twice the rectangle CD, DG: (it. 13.)

therefore the squares on AH, A C are equal to twice the square on

BD, and twice the square on AD ; for 2) C is equal to BD :

and twice the squares on AB, AC are equai to the square on BC,

and four times the square on ^D : for ^C is twice BD.

Similarly, twdce the squares on AB, BC aie equal to the square on

A C, and four times the square on BD :

also twice the squares on B C, CA are equal to the square on AB,

and four times the square on FC:

hence, by adding these equals,

four times the squares on AB, AC, BC are equal to four times the

squares on AD, BD, Ci^together with the squares on AB, A C,BC:

and taking the squares on AB, AC, BC from these equals,

therefore three times the squares on AB, AC, BC scie equal to foui'

times the squares on AD, BD, CF.

PROPOSITION V. THEOREM.

The sum of the perpendiculars let fall from any point within an equila-

teral triangle, will he equal to the perpendicular let fall from one of its

ayigles tcpon the opposite side. Is this propositio7i true when the point is in

one of the sides of the triangle ? In what mamier must the p?'opositio?i be

enunciated when the point is without the triangle f

Let ABC he an equilateral triangle, and P any point within it:

and from P let fall PD,PE, PPperpendiculars on the sides AB, BC,

The proof of Prop. x. may be deduced from Euc. ii. 4, 7, as Prop. ix.

Prop. X. Algebraically.

Let the line AB contain 2a linear units, of which its half AC or CB

will contain a units ;

and let BD contain m units.

Then the whole line and the part produced will contain ^a + m units,

and half the line and the part produced will contain a + m units,

.'. (2a + my = ia^ + 4am + m^,

add m^ to each of these equals,

/. (2a + my + m^ = 4a2 + 4am + 2wt^

Again, (a + my = a* + 2atn + m^,

add a^ to each of these equals,

.'. {a + my + a'^ = 2a^ + 2am + m\

and doubling these equals,

.-. 2 (a + my + 2d^ = ^a^ + 4am + 2m^

But (2a + my + mÂ« = 4a* + 4am + 2mÂ«.

Hence .'. (2a + m)' + m* = 2a2 + 2 (a + my.

That is, If a number be divided into two equal parts, and the whole

number and one of the parts be increased by the addition of another num-

ber, the squares of the whole number thus increased, and of the number

by which it is increased, are equal to double the squares of half the num-

ber, and of half the number increased.

The algebraical results of Prop, ix, and Prop, x, are identical, (the

enunciations of the two Props, arising, as in Prop, v, and Prop, vi, from

the two ways of exhibiting the difference between two lines) ; and both

may be included under the following proposition : The square on the

sum of two lines and the square on their difference, are together equal to

double the sum of the squares on the two lines.

Prop. xr. Two series of lines, one series decreasing and the other

series increasing in magnitude, and each line divided in the same man-

ner may be found by means of this proposition.

(1) To find the decreasing series.

In the fig. Euc. ii, 11, AB ^ AH + BR,

and since AB . BH = AH\ .'. {AH + BH) . BH = AH\

.-. BH^ = AH""- AH.BH=AH.(AH- BH).

If now in HJ, HL be taken equal to BH,

then HL^ = AH {AH - HL), or AH . AL = HU :

that is, AH is divided in L, so that the rectangle contained by the whole

line AH and one part, is equal to the square on the other part HL. IBy a

similar process, HL may be so divided ; and so on, by always taking from

the greater part of the divided line, a part equal to the less.

(2) To find the increasing series.

From the fig. it is obvious that CF . FA = CA^,

Hence CF is divided in A, in the same manner as AB is divided in H,

by adding AF a. line equal to the greater segment, to the given line CA

NOTES TO BOOK II. 107

IB. And by successively adding to the last line thus divided, its

greater segment, a series of lines increasing in magnitude may be found

similarly divided to JB.

It may also be shewn that the squares on the whole line and on the less

segment are equal to three times the square on the greater segment.

(Euc. XIII. 4.)

To solve Prop, xi, algebraically, or to find the point H in AB such

that the rectangle contained by the whole line AB and the part HB shall

be equal to the square on the other part AH.

Let JB contain a linear units, and AH one of the unknown parts con-

tain X units,

then the other part HB contains a â€” x units.

And .*. a(a â€” x) = x^, by the problem,

or x'* + ax = a*, a quadratic equation.

Whence x = â€¢= o â€¢

The former of these values of x determines the point H.

So that X = ^r â€” . AB = AH, one part,

and a â€” X = a â€” AH = . AB =HB, the other part.

It may be observed, that the parts AH and HB cannot be numerically

expressed by any rational number. Approximation to their true values

in terms of AB, may be made to any required degree of accuracy, by ex-

tending the extraction of the square root of 5 to any number of decimals.

To ascertain the meaning of the other result x = . a.

z

In the equation a (a â€” x) = x'^,

for X write â€” x, then a{a + x) = x*,

which when translated into words gives the following problem.

To find the length to which a given line must be produced so that the

rectangle contained by the given line and the line made up of the given line

and the part produced, may be equal to the square on the part produced.

Or, the problem may also be expressed as follows :

To find two lines having a given difference, such that the rectangle con-

tained by the difference and one of them may be equal to the square on

the other.

It may here be remarked, that Prop. xi. Book ii, affords a simple

Geometrical construction for a quadratic equation.

Prop. XII. Algebraically.

Assuming the truth of Euc. i. 47.

Let BC, CA, AB contain a, b, c linear units respectively,

and let CD, DA, contain m, n Tinits,

then BD contains a + m units.

And therefore, c* = (a + my + n', from the right-angled triangle ABD,

also b^ = m^ + n^ from A CD ;

.'. c'^ â€” 6* = (a + my â€” m?

= a^ + 1am + m- â€” m^ ,

108

EUCLID S ELEMEiMS.

= a' + 2a ?n,

/. c' = 6^ + a* + 2am,

that is, c- is greater than V^ + or by 2am.

Prop. XIII. Case ii. may be proved more simply as foUo-vvs.

Since BD is divided into two parts in the point Z>,

therefore the squares on CB, BD are equal to twice the rectangle con-

tained by CB, BD and the square on CD ; (ii. 7.)

add the square on AD to each of these equals ;

therefore the squares on CB, BD, DA are equal to twice the rectangle

CB, BD, and the squares on CD and DA,

but the squares on BDy DA are equal to the square on AB, (i. 47.)

and the squares on CD, DA are equal to the square on AC^

therefore the squares on CB, BA are equal to the square on AC, and

twice the rectangle CB, BD. That is, &c.

Prop. XIII. Algebraically.

Let BC, CA, AB contain respectively o, 6, c linear units, and let BD

and AD also contain m and n units.

Case I. Then DC contains a â€” m units.

Therefore c* = n^ + m- from the right-angled triangle ABD,

and 6Â« = n^ + (a - tnf from ADC ;

.*. c^ â€” P = m^ â€” {a â€” my

= m^ â€” a^ -{â– 2am â€” m*

= â€” a^ + 2am,

/. Â«Â« + c8 = ^8 + 2am,

or i* + 2am = a^ + c*,

that is, h^ is less than a' + c^ by 2am.

Case II. DC = m â€” a units,

/, c* = m^ + w* from the right-angled triangle ABB,

and b"^ = (m â€” ay + n- from ACDf

.'. c^ â€” b^ = m^ â€” {m â€” ay,

= m* â€” m^ + 2am â€” a^

= 2am â€” a',

/. aÂ« + c' = 62 + 2am,

or b^ + 2am = a* + c',

that is, S'' is less than a* + c^ by 2am.

Case III. Here m is equal to a.

And 6"^ + a* = c^, from the right-angled triangle ABC,

Add to each of these equals a^,

.-. b^ + 2aÂ« = cÂ« + aS

that is, 6' is less than c* + a' by 2a2, or 2aa.

These two propositions, Euc. ii. 12, 13, with Euc. i. 47, exhibit the

relations which subsist between the sides of an obtuse-angled, an acute-

angled, and right-angled triangle respectively.

( 109 )

NOTE OX THE ABBREVIATIONS AND ALGEBRAICAL

SYMBOLS EMPLOYED D^ GEOMETRY.

The ancient Geometry of the Greeks admitted no symbols besides the

diagrams and ordinary language. In later times, after symbols of opera-

tion had been devised by writers on Algebra, they were very soon adopted

and employed on account of their brevity and convenience, in writings

purely geometrical. Dr. Barrow was one of the first who introduced

algebraical symbols into the language of Elementary Geometry, and dis-

tinctly states in the preface to his Euclid, that his object is "to content

the desires of those who are delighted more with symbolical than verbal

demonstrations." As algebraical symbols are employed in almost all

works on the mathematics, whether geometrical or not, it seems proper

in this place to give some brief account of the marks which may be re-

garded as the alphabet of symbolical language.

The mark = was first used by Robert Recorde, in his treatise on Algebra

entitled, " The Whetstone of Witte," 1557. He remarks ; '* And to avoide

the tediouse repetition of these woordes : is equalle to : I will sette as I

doe often in woorke use, a paire of paralleles, or Gemowe lines of one

lengtbe, thus : =, bicause noe 2 thynges can be more equalle." It was

employed by him as simply affirming the equality of two numerical or

algebraical expressions. Geometrical equality is not exactly the same

as numerical equality, and when this symbol is used in geometrical reason-

ings, it must be understood as having reference to pure geometrical

equality.

The signs of relative magnitude, > meaning, is greater than, and < , is

less than, were first introduced into algebra by Thomas Harriot, in his

*' Artis Analyticas Praxis," which was published after his death in 1631.

The signs + and â€” were first employed by Michael Stifel, in his "Arith-

metica Integra," which was published in 1544. The sign + was employed

by him for the word plus, and the sign â€” , for the word minus. These

signs were used by Stifel strictly as the arithmetical or algebraical signs

of addition and subtraction.

The sign of multiplication x was first introduced by Oughtred in his

**Clavis Mathematica," which was published in 1631. In algebraical

multiplication he either connects the letters which form the factors of a

product by the sign x , or writes them as words without any sign or mark

between them, as had been done before by Harriot, who first introduced

the small letters to designate known and unknown quantities. However

concise and convenient the notation AB x BC or AB . BC may be in.

practice for ** the rectangle contained by the lines AB and BC" ; the student

is cautioned against the use of it, in the early part of his geometrical

studies, as its use is likely to occasion a misapprehension of Euclid's

meaning, by confounding the idea of Geometrical equality with that of

Arithmetical equality. Later writers on Geometry who employed the

Latin language, explained the notation AB x BC, by ** AB ductum in

BC" ; that is, if the line AB be carried along the line BC in a normal

position to it, until it come to the end C, it will then form with BC, the

rectangle contained by AB and BC. Dr. Barrow sometimes expresses

*Hhe rectangle contained by AB and BC" by **the rectangle ABC."

Michael Stifel was the first who introduced integral exponents to

denote the powers of algebraical symbols of quantity, for which he em-

ployed capital letters. Vieta afterwards used the vowels to denote known,

and the consonants, unknown quantities, but used words to designate the

i

110 Euclid's elements.

powers. Simon Stevin, in his treatise on Algebra, which was published

m 1605, improved the notation of Stifel, by placing the figures that in-

dicated the powers within small circles. Peter Ramus adopted the

initial letters /, q, c, bq oi latus, quadratus, cubus, biquadratus, as the nota-

tion of the first four powers. Harriot exhibited the difi'erent powers of

algebraical symbols by repeating the symbol, two, three, four, &c. times,

according to the order of the power. Descartes restored the numerical

exponents of powers, placing them at the right of the numbers, or symbols

of quantity, as at the present time. Dr. Barrow employed the notation

ABg, for " tJie square on the line AB," in his edition of Euclid. The

notations AB"'^ AB^, for " the square and cube on the line whose extremities

are A and J5," as well as AB x BC, for *'the rectangle contained by AB

a7id BC," are used as abbreviations in almost all works on the Mathe-

matics, though not wholly consistent with the algebraical notations a*

and a^.

The symbol V, being originally the initial letter of the word radix, was

first used by Stifel to denote the square root of the number, or of the

symbol, before which it is placed.

The Hindus, in their treatises on Algebra, indicated the ratio of two

numbers, or of two algebraical symbols, by placing one above the other,

without any line of separation. The line was first introduced by the Ara-

bians, from whom it passed to the Italians, and from them to the rest of

Europe. This notation has been employed for the expression of geome-

trical ratios by almost all writers on the Mathematics, on account of its

great convenience. Oughtred first used points to indicate proportion ;

thus, a:b::c:d, means that a bears the same proportion to 6, as c does to d.

QUESTIONS ON BOOK 11.

1. Is rectangle the same as rectus angulus? Explain the distinction,

and give the corresponding Greek terms.

2. What is meant by the sum of two, or more than two straight lines

in Geometry ?

3. Is there any difference between the straight lines by which a rect-

angle is said to be contained, and those by which it is bounded ?

4. Define a gnomon. How many gnomons appear from the same con-

struction in the same rectangle ? Find the difference between them.

5. What axiom is assumed in proving the first eight propositions of

the Second Book of Euclid ?

6. Of equal squares and equal rectangles, which must necessarily coin-

cide ? ^ .

7. How may a rectangle be dissected so as to form an equivalent

rectangle of any proposed length ?

8. When the adjacent sides of a rectangle are commensurable, the area

of the rectangle is properly represented by the product of the number^ of

units in two adjacent sides of the rectangle. Illustrate this by considering

the case when the two adjacent sides contain 3 and 4 units respectively,

and distinguish between the units of the factors and the units of the product.

Shew generally that a rectangle whose adjacent sides arerepresentedby the

integers a and'6, is represented by ab. Also shew, that in the same sense,

ah J 1. ^ ^

the rectangle is represented by â€” , if the sides be represented by -, - â€¢

QUESTIONS ON BOOK IT. Ill

9. Why may not Algebraical or Arithmetical proofs be substituted (as

being shorter) for the demonstrations of the Propositions in the Second

Book of Euclid ?

10. In what sense is the area of a triangle said to be equal to half the

product of its base and its altitude ? What two propositions of Euclid

may be adduced to prove it ?

11. How do you shew that the area of a rhombus is equal to half the

rectangle contained by the diagonals ?

12. How may a rule be deduced for finding a numerical expression

for the area of any parallelogram, when two adjacent sides are given?

13. The area of a trapezium which has two of its sides parallel is equal

to that of a rectangle contained by its altitude and half the sum of its

parallel sides. What propositions of the First and Second Books of Euclid

are employed to prove this ? Of what service is the above in the men-

suration of fields with irregular borders ?

14. From what propositions of Euclid may be deduced the following

rule for finding the area of any quadrilateral figure : â€” *' Multiply the sum

of the perpendiculars drawn from opposite angles of the figure upon the

diagonal joining the other two angles, and take half the product."

15. In Euclid, II. 3, where must be the point of division of the line, so

that the rectangle contained by the two parts may be a maximum ? Ex-

emplify in the case where the line is 12 inches long.

16. How may the demonstration of Euclid ii. 4, be legitimately short-

ened ? Give the Algebraical proof, and state on what suppositions it can

be regarded as a proof.

17. Shew that the proof of Euc. ii. 4, can be deduced from the two

previous propositions without any geometrical construction.

18. Shew that if the two complements be together equal to the two

squares, the given line is bisected.

19. If the line AB^ as in Euc. ii. 4, be divided into any three parts,

enunciate and prove the analogous proposition.

20. Prove geometrically that if a straight line be trisected, the square

on the whole line equals nine times the square on a third part of it.

21. Deduce from Euc. ii. 4, a proof of Euc. i. 47.

22. If a straight line be divided into two parts, when is the rectangle

contained by the parts, the greatest possiblef and when is the sum of the

squares of the parts, the least possible ?

23. Shew that if a line be divided into two equal parts and into two

unequal parts ; the part of the line between the points of section is equal

to half the difference of the unequal parts.

24. If half the sum of two unequal lines be increased by half their

difference, the sum will be equal to the greater line : and if the sum of

two lines be diminished by half their difference, the remainder will be

equal to the less line.

25. Explain what is meant by the internal and external segments of a

line ; and show that the sum of the external segments of a line or the

difference of the internal segments is double the distance between the

points of section and bisection of the line.

26. Shew how Euc. ii. 6, may be deduced immediately from the

preceding Proposition.

27. Prove Geometrically that the squares on the sum and difference

of two lines are equal to twice the squares on the lines themselves.

28. A given rectangle is divided by two straight lines into four rect-

angles. Given the areas of the two which have not common sides : find

the areas of the other two.

112 Euclid's elements.

29. In how many ways may the difference of two lines be exhibited?

Enunciate the propositions in Book ii. which depend on that circumstance.

30. How may a series of lines be found similarly divided to the line

^^ in Euc. II. n ?

31. Divide Algebraically a given line {a) into two parts, such that

the rectangle contained by the whole and one part may be equal to the

square of the other part. Deduce Euclid's construction from one so-

lution, and explain the other.

32. Given the lesser segment of a line, divided as in Euc. ii. 11,

find the greater.

33. Enunciate the Arithmetical theorems expressed by the following

Algebraical formulae,

(a + by = aÂ« + 2ab + b^ -.a" - b^ = (a + b) (a-b) : {a-bf = a" - 2ab + b\

and state the corresponding Geometrical propositions.

34. Shew that the first of the Algebraical propositions,

{a + x) {a-x) +x^=:a^:ia + xy 4- (a - xy = 2a^ + 2x\

is equivalent to the two propositions v. and vi., and the second of them,

to the two propositions ix. and x. of the Second Book of Euclid.

35. Prove Euc. ii. 12, when the perpendicular BE is drawn from

B on AC produced to E, and shew that the rectangle BC, CD is equal

to the rectangle AC, CE.

36. Include the first two cases of Euc. ii. 13, in one proof.

37. In the second case of Euc. ir. 13, draw a perpendicular CE from

the obtuse angle C upon the side AB, and prove that the square on AB

is equal to the rectangle AB, AE together with the rectangle BC, BD.

38. Enunciate Euc. ir. 13, and give an Algebraical or Arithmetical

proof of it.

39. The sides of a triangle are as 3, 4, 5. Determine whether the

angles between 3, 4 ; 4, 5 ; and 3, 5 ; respectively are greater than, equal

to, or less than, a right angle.

40. Two sides of a triangle are 4 and 5 inches in length, if the

third side be 6i^6 inches, the triangle is acute-angled, but if it be 60

inches, the triangle is obtuse-angled.

41. A triangle has its sides 7, 8, 9 units respectively; a strip of

breadth 2 units being taken off all round from the triangle, find the

area of the remainder.

42. If the original figure, Euc. 11. 14, were a right-angled triangle,

v.'hose sides were represented by 8 and 9, what number would represent

the side of a square of the same area ? Shew that the perimeter of the

square is less than the perimeter of the triangle.

43. If tl:e sides of a rectangle are 8 feet and 2 feet, what is the side

of the equivalent square ?

44. "All plane rectilineal figures admit of quadrature." Point out

the succession of steps by which Euclid establishes the truth of this

proposition.

45. Explain the construction (without proof) for making a square

equal to a plane polygon.

46. Shew from Euc. 11. 14, that any algebraical surd as VÂ« can be

represented by a line, if the unit be a line.

47. Could any of the propositions of the Second Book be made co-

rollaries to other propositions, with advantage ? Point out any such pro-

positions, and give your reasons for the alterations you would make.

GEOMETRICAL EXERCISES ON BOOK 11.

PROPOSITION I. PROBLEM.

Divide a given straight line into two parts such, that their rectangle mag

he equal to a given square ; and determine the greatest square which the

rectangle can equal.

Let AB he the given straight line, and let M be the side of the

given square.

It is required to divide the line AB into two parts, so that the

rectangle contained by them may be equal to the square on 31.

D E

M.

A i' C B

Bisect AB in C, with center C, and radius CA or CB, describe the

semicircle ADB.

At the point B draw BJE at right angles to AB and equal to 3f.

Through JEJ, draw J3D parallel to AB and cutting the semicircle

inZ);

and draw DJP parallel to JEB meeting AB in F.

Then AB is divided in F, so that the rectangle AF, FB is equal

to the square on M. (il. 14.)

The square will be the greatest, when ED touches the semicircle,

or when 3Â£ is equal to half of the given line AB.

PROPOSITION II. THEOREM.

The square oti the excess of one straight line above another is less than the

squares on the tico lines by tioice their rectangle.

Let AB, BChe the two straight lines, wliose difference is AC.

Then the square on ^Cis less than the squares on AB and BChy

twice the rectangle contained by AB and BC.

A c B

K

g/

F E

Constructing as in Prop. 4. Book II.

Because the complement ^ 6^ is equal to GF,

add to each CK,

therefore the whole ^^is equal to the whole CF .

114 GEOMETRICAL EXERCISES

and AK, CE together are double of AK-,

but AK, CE are the gnomon AKF ond CK,

and AK is the rectangle contained by AB, JBC;

therefore the gnomon AKF and CKy

are equal to twice the rectangle AB, BC,

but AE, CK are equal to the squares on AB, BC;

taking the former equals from these equals,

therefore the difference of ^^ and the gnomon AKF is equal to

the difference between the squares on AB.BC.Rud twice the rectangle

JB,BC',

but the difference AE and the gnomon AKF is the figure SF

which is equal to the square on A C.

Wherefore the square on AC is equal to the difference between the

squares on AB, BC, and twice the rectangle AB, BC.

PROPOSITION III. THEOREM.

In any triangle the squares on the two sides are together double of the

squares on half the base and on the straight line joining its bisectio7i with the

opposite angle.

Let ABChe 2l triangle, and^Z) the line drawn from the vertex A

to the bisection D of the base BC.

From A draw AE perpendicular to BC.

Then, in the obtuse-angled triangle ABE, (ll. 12.) ;

the square on AB exceeds the squares on AE, EB, by twice the

rectangle BE, EE:

and in the acute-angled triangle AEC, (ll. 13.) ;

the square on ^C is less than the squares on AE, EC, by twice

the rectangle CE, EE:

wherefore, since the rectangle BE, EE is equal to the rectangle CE,

EE ; it follows that the squares on AB, A C are double of the

squares on AE, EB.

PROPOSITION IV. THEOREM.

If straight lines be drawn from each angle of a tria,ngle bisecting the

opposite sides, four times the sum of the squares 07i these lines is equal to

three titnes the sum of the squares on the sides of the triangle.

Let ABC he any triangle, and let AE, BE, CF be drawn from

A, B, C, to E, E, F, the bisections of the opposite sides of the tri-

angle : di-aw AG perpendicular to BC.

115

Then the square on AB is equal to the squares on JBD, DA together

with twice the rectangle J3D, DG, (ll. 12.)

and the square on A C is equal to the squares on CD, DA dimi-

nished by twice the rectangle CD, DG: (it. 13.)

therefore the squares on AH, A C are equal to twice the square on

BD, and twice the square on AD ; for 2) C is equal to BD :

and twice the squares on AB, AC are equai to the square on BC,

and four times the square on ^D : for ^C is twice BD.

Similarly, twdce the squares on AB, BC aie equal to the square on

A C, and four times the square on BD :

also twice the squares on B C, CA are equal to the square on AB,

and four times the square on FC:

hence, by adding these equals,

four times the squares on AB, AC, BC are equal to four times the

squares on AD, BD, Ci^together with the squares on AB, A C,BC:

and taking the squares on AB, AC, BC from these equals,

therefore three times the squares on AB, AC, BC scie equal to foui'

times the squares on AD, BD, CF.

PROPOSITION V. THEOREM.

The sum of the perpendiculars let fall from any point within an equila-

teral triangle, will he equal to the perpendicular let fall from one of its

ayigles tcpon the opposite side. Is this propositio7i true when the point is in

one of the sides of the triangle ? In what mamier must the p?'opositio?i be

enunciated when the point is without the triangle f

Let ABC he an equilateral triangle, and P any point within it:

and from P let fall PD,PE, PPperpendiculars on the sides AB, BC,

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