Two New Sciences
With regard to this subject I must tell you of a remarkable property which just now occurs to me and which will explain the vast alteration and change of character which a finite quantity would undergo in passing to infinity. [84] Very briefly: this book is the authoritative English translation of Galileo's masterpiece, after a long history of notoriously poor and incomplete translations. (60) In short the infinite number of indivisible sides of the greater circle with their infinite number of indivisible retrogressions, made during the infinite number of instantaneous delays of the infinite number of vertices of the smaller circle, together with the infinite number of progressions, equal to the infinite number of sides in the smaller circle -- all these, I say, add up to a line equal to that described by the smaller circle, a line which contains an infinite number of infinitely small superpositions, thus bringing about a thickening or contraction without any overlapping or interpenetration of finite parts. It appears therefore that we may equate the circumference of a large circle to a single point. whose sides about the right angle has the same length as AC and the other is equal to the perimeter of the polygon itself; it is then manifest that the circumscribed polygon bears to the circle the same ratio which its perimeter bears to the circumference of the circle, or to the perimeter of the polygon B which is, by hypothesis, equal to the circumference of the circle. More than anything else here, we see the process of the curious mind discovering physical truth incrementally through experimentation. But the diameters are proportional to the circumferences and the circumferences are proportional to the areas of cylinders of equal height; hence the line AB is to E as the surface of the cylinder CD is to the surface of the cylinder AF. Now this exposition which I have given in the case of these hexagons must be understood to be applicable to all other polygons, whatever the number of sides, provided only they are In the case of the polygon, the vertices remain at rest during an interval of time which bears to the period of one complete rotation the same ratio which one side bears to the perimeter; likewise, in the case of the circles, the delay of each of the infinite number of vertices is merely instantaneous, because an instant is such a fraction of a finite interval as a point is of a line which contains an infinite number of points. 0921332505
He covers a lot of content in relatively few pages.
A large marble column was laid out so that its two ends rested each upon a piece of beam; a little later it occurred to a mechanic that, in order to be doubly sure of its not breaking in the middle by its own weight, it would be wise to lay a third support midway; this seemed to all an excellent idea; but the sequel showed that it was quite the opposite, for not many months passed before the column was found cracked and broken exactly above the new middle support. On account of these superpositions, which are the excesses of the sides of the larger over the smaller polygon, each net advance is equal to one side of the smaller polygon and, during one complete rotation, these amount to a straight line equal in length to the perimeter of the smaller polygon. And why should we feel greater repugnance, seeing that, in our search after the infinite among numbers we found it in unity? (15) First Day: pages 61 … It is the work written by Galileo in the last years of his life, while under house arrest by the Inquisition for suspected heresy.
Is it not clear that this is precisely the kind of resistance which one meets in the case of a thick hemp rope where the fibres form thousands and thousands of similar spirals?
This done, imagine the larger polygon to roll upon As the title suggests, the dialogue serves to highlight a shift in thought and the format does prove suitable to allow ideas and opinions to clash freely. of AE.
Sagredo and Salviati seem like mouthpieces for conflicting ideas with which Galileo himself had to reckon to arrive at his conclusions which are given in the text written by the "AcTwo New Sciences is definitely a unique physical treatise in that it is written as a Platonic style dialogue. For the separation of the plates precedes the formation of the vacuum which is produced as a consequence of this separation; and since it appears to me that, in the order of nature, the cause must precede the effect, even though it appears to follow in point of time, and since every positive effect must have a positive cause, I do not see how the adhesion of two plates and their resistance to separation -- actual facts -- can be referred to a vacuum as cause when this vacuum is yet to follow. Or the theorem presented by Galileo on the area of a circle as 'a mean proportional between any two regular and similar polygons of which one circumscribes it and the other is isoperimetric with it'. One of my many favourite parts of the dialogues was Galileo’s account of the speed of light using flickering lanterns. And as the cutting plane comes near the top, the two solids (always equal) as well as their bases (areas which are also equal) finally vanish, one pair of them degenerating into the circumference of a circle, the other into a single point, namely, the upper edge of the bowl and the apex of the cone.
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