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= a + 2am,
c= + aa + 2am,
that is, c is greater than 6% + a' by 2am. Prop. XIII. Case 11. may be proved more simply as follows.
Since BD is divided into two parts in the point D, therefore the squares on CB, BD are equal to twice the rectangle contained by CB, BD and the square on CD; (11. 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 BD, DA are equal to the square on AB, (1. 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 a, b, c linear units, and let BD and AD also contain m and n units. Case 1. Then DC contains a
- m units.
and 32 = + (a – m) from ADC;
= ma a® + 2am – ma
a® + 2am, .. a+ c
62 + 2am, 72 + 2am a +ca,
that is, b? is less than a® + c? by 2am. Case II.
= m - a units,
and b = (m – a)* + n2 from ACD,
= ma – ma + 2am a
= 2am – a, .. a® + c = 62 + 2am,
or 82 + 2am = a® + Co,
that is, 62 is less than a + ca by 2am. Case III. Here m is equal to a. And + ao = cm, from the right-angled triangle ABC.
Add to each of these equals a”,
::.72 + 2a = c + a*, that is, 5* is less than c + a2 by 2a”, or 2aa. These two propositions, Euc. 11, 12, 13, with Euc. 1. 47, exhibit the relations which subsist between the sides of an obtuse-angled, an acuteangled, and right-angled triangle respectively.
NOTE ON THE ABBREVIATIONS AND ALGEBRAICAL
SYMBOLS EMPLOYED IN GEOMETRY.
The ancient Geometry of the Greeks admitted no symbols besides the diagrams and ordinary language In later times, after symbols of operation 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 distinctly 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 regarded as the alphabet of symbolical language.
Themark = 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 lengthe, 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 Analyticæ Praxis,” which was published after his death in 1631.
The signs + and were first employed by Michael Stifel, in his “Arithmetica Integra,” which was published in 1544. The sign + was employed by him for the word plus, and the sign 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 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 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 “the 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 employed capital letters. Vieta afterwards used the vowels to denote known, and the consonants, unknown quantities, but used words to designate the
powers. Simon Stevin, in his treatise on Algebra, which was published in 1605, improved the notation of Stifel, by placing the figures that indicated the powers within small circles. Peter Ramus adopted the initial letters l, q, c, bą of latus, quadratus, cubus, biquadratus, as the notation of the first four powers. Harriot exhibited the different 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 ABq, for the square on the line AB,” in his edition of Euclid. The notations ABS, AB3, for “ the square and cube on the line whose extremities are A and B,” as well as AB x BC, for “the rectangle contained by AB and BC,” are used as abbreviations in almost all works on the Mathematics, though not wholly consistent with the algebraical notations a? and as
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 Arabians, from whom it passed to the Italians, and from them to the rest of Europe. This notation has been employed for the expression of geometrical ratios by almost all writers on the Mathematics, on account of its great convenience. Oughtred first used points to indicate proportion ; thus, a:6::c:d, means that a bears the same proportion to b, as c does to d.
QUESTIONS ON BOOK II. 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 rectangle is said to be contained, and those by which it is bounded?
4. Define a gnomon. How many gnomons appear from the same construction 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 coincide ?
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 ofthe factors and the units of the product. Shew generally that a rectangle whose adjacent sides are represented by the integers a ando, is represented by ab. Also shew, that in the same sense, ah
a 6 the rectangle is represented by if the sides be represented by
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 mensuration 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:-"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, 11. 3, where must be the point of division of the line, so that the rectangle contained by the two parts may be a maximum? Exemplify in the case where the line is 12 inches long.
16. How may the demonstration of Euclid 11. 4, be legitimately shortened? 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. 11. 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 possible? 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. 11. 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 rectangles. Given the areas of the two which have not common sides : find the areas of the other two.
29. In how many ways may the difference of two lines be exhibited ? Enunciate the propositions in Book 11. which depend on that circumstance.
30. How may a series of lines be found similarly divided to the line AB in Euc. II. 11 :
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. 11. 11, find the greater.
33. Enunciate the Arithmetical theorems expressed by the following Algebraical formulæ, (a + b)2 = a* + 2ab + 62 : a– b* = (a+b) (a – 6): (a-6)* = a* – 2ab + 6%, and state the corresponding Geometrical propositions. 34. Shew that the first of the Algebraical propositions,
(a + x) (a – x) + x = a* : (a + x)2 +. (a – x)2 = 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, one proof.
37. In the second case of Euc. 11. 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. II. 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 68 inches, the triangle is acute-angled, but if it be 676 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.
4. If the original figure, Euc. 11. 14, were a right-angled triangle, whose 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 the 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 va can be represented by a line, if the unit be a line.
47. Could any of the propositions of the Second Book be made corollaries to other propositions, with advantage Point out any such propositions, and give your reasons for the alterations you would make.