## The ratio between diameter and circumference in a circle demonstrated by angles, and Euclid's theorem, proposition 32, book 1, proved to be fallacious |

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A B C absurd admit answer appears argument arithmetical value assertion assume attempt base Book calculation centre chord circle of diameter circle of radius circular circum circumference circumscribing communication construction correct CORRESPONDENT DEAR SIR decimals demonstration denote describe diagram diameter unity difference divided dodecagon draw equal equation Euclid exactly expressed fact figure finite follows four geometrical Gibbons give given greater half Hence hexagon inscribed isosceles triangle JAMES SMITH join length less Letter Liverpool Logarithms March Mathematicians Mathematics mean measure meet never observe perimeter polygon proof Prop proportion prove quantity question radius ratio readers reasoning reference regular regular hexagon regular inscribed reply represented right angle right-angled triangle self-evident sides sine square straight line subtended symbol Tables theorem trigonometrical true truly unit Whitworth

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Page 25 - If a straight line be divided into any two, parts, the square on the whole line is equal to the sum of the rectangles contained by the whole and each of the parts*.

Page xiv - ... mathematicians on this question, is plainly expressed in the Report of MM. Demogeot and Montucci. ..." I should have to quote very largely indeed if I wished to draw attention to every hazardous statement which has been advanced; I must therefore severely restrain myself. Consider the following : " Unquestionably the best teachers depart largely from his words, and even from his methods. That is, they use the work of Euclid, but they would teach better without it. And this is especially true...

Page i - Now you are here,' said the patient, • I shall be obliged to you, Sir Richard, if you will tell me how I must live, what I may eat, and what not.' My directions as to that point,' replied Sir Richard,

Page 295 - ... 23".5. 40. Since angles at the centre of a circle are to each other as the arcs of the circumference intercepted between their sides (Geom., Prop. XVII. Bk. III.), these arcs may be regarded as the measures of the angles, and the number of units of arc intercepted on the circumference may be used to express both the arc and the corresponding angle. 41. A degree of arc...

Page 22 - If a straight line be divided into any two parts, the rectangles contained by the whole and each of the parts, are together equal to the square of the whole line. Let the straight line AB be divided...

Page 243 - In any proportion, the product of the means is equal to the product of the extremes.

Page xiv - ... leads me to the conclusion that the syllogistic form instead of being an impediment is really a great assistance, especially to early students. " Unsuggestiveness " has been urged as a fault in Euclid ; which is interpreted to mean that it does not produce ability to solve problems. We are told : " Everybody recollects, even if he have not the daily experience, how unavailable for problems a boy's knowledge of Euclid generally is. Yet this is the true test of geometrical knowledge ; and problems...