The Quadrature of the Circle: The Square Root of Two, and the Right-angled Triangle |
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Page 44
... variable be made to approach as its constant the true radius , so near that the dif- ference between it and the true radius shall be less than any assignable quantity , so that it may be taken for the 44 INTRODUCTION .
... variable be made to approach as its constant the true radius , so near that the dif- ference between it and the true radius shall be less than any assignable quantity , so that it may be taken for the 44 INTRODUCTION .
Page 45
... variable be made to approach as its constant the true circumference , so near that the difference be- tween it and the true circumference shall be made less than any assign- able quantity , so that it may be taken for the true ...
... variable be made to approach as its constant the true circumference , so near that the difference be- tween it and the true circumference shall be made less than any assign- able quantity , so that it may be taken for the true ...
Page 55
... variable quantity , or simply , a variable , is a quantity which has different successive values . II . When the successive values of a variable , under the conditions im- posed upon it , approach more and more nearly to the value of ...
... variable quantity , or simply , a variable , is a quantity which has different successive values . II . When the successive values of a variable , under the conditions im- posed upon it , approach more and more nearly to the value of ...
Page 56
... variable quantities are always equal to each other and each approaches to a limit , the two limits are necessarily equal . For , two variables always equal to each other present in fact but one value , and it is evidently impossible ...
... variable quantities are always equal to each other and each approaches to a limit , the two limits are necessarily equal . For , two variables always equal to each other present in fact but one value , and it is evidently impossible ...
Page 125
... variable difference A - a or a — A , between a variable and its limit , is an infin- itesimal . It may be made less than any designated quantity , but can never become Or more generally , an infinitesimal is a variable whose limit is ...
... variable difference A - a or a — A , between a variable and its limit , is an infin- itesimal . It may be made less than any designated quantity , but can never become Or more generally , an infinitesimal is a variable whose limit is ...
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The Quadrature of the Circle, the Square Root of Two, and the Right-Angled ... William Alexander. Myers No preview available - 2013 |
Common terms and phrases
apothem arc cutting Archimedes ARTICLE assumed circumference assumed diameter Bisect chord circumscribed double triangle circumscribed polygon consequently cosine cumference curve decimal places deducted demonstration diagonal difference discovery division and cancellation double the number draw expressed extracting the square figures geometricians geometry give given arc given circle given polygon given radius given triangle half the number hyperbola hypothenuse hypothesis infinite inscribed and circumscribed inscribed double triangle inscribed polygon inscribed square James Gregory less limit mathematical mean proportional method multiplied number of sides Oronce Finée parabola perimeter perpendicular Plate polygon of double Polygons are Carried problem PROPOSITION quadrature quantity radius rectangle contained regular polygon result already established right angle right line right-angled triangle Scholium secant sine solution square described square root straight line subtracted tangent theorem triangle DAC trigonometry true circumference true ratio truth unity variable
Popular passages
Page 65 - A plane rectilineal angle is the inclination of two straight lines to one another, which meet together, but are not in the same straight line.
Page 39 - It furnishes art with all her materials, and without it judgment itself can at best but " steal wisely : " for art is only like a prudent steward that lives on managing the riches of nature.' Whatever praises may be given to works of judgment, there is not even a single beauty in them to which the invention...
Page 71 - AXIOMS. 1. Things which are equal to the same thing are equal to one another. 2. If equals be added to equals, the wholes are equal. 3. If equals be taken from equals, the remainders are equal.
Page 39 - Nor is it a wonder if he has ever been acknowledged the greatest of poets, who most excelled in that which is the very foundation of poetry. It is the invention that in different degrees...
Page 78 - COR. From this it is manifest, that the perpendicular drawn from the right angle of a right-angled triangle to the base, is a mean proportional between the segments of the base; and also that each of the sides is a mean proportional between the base, and...
Page 71 - If a straight line meets two straight lines, so as to " make the two interior angles on the same side of it taken " together less than two right angles...
Page 39 - And perhaps the reason why common critics are inclined to prefer a judicious and methodical genius to a great and fruitful one, is, because they find it easier for themselves to pursue their observations through an uniform and bounded walk of art, than to comprehend the vast and various extent of nature.
Page 31 - IN a right-angled triangle, if a perpendicular be drawn from the right angle to the base, the triangles on each side of it are similar to the whole triangle, and to one another.
Page 74 - In any right-angled triangle, the square which is described on the side subtending the right angle is equal to the sum of the squares described on the sides which contain the right angle.
Page 69 - The circumference of every circle is supposed to be divided into 360 equal parts, • called degrees, each degree into 60 minutes, and each minute into 60 seconds, etc.