By Thomas Heath
Quantity 1 of an authoritative two-volume set that covers the necessities of arithmetic and comprises each landmark innovation and each vital determine. This quantity gains Euclid, Apollonius, others.
Read or Download A History of Greek Mathematics, Volume 1: From Thales to Euclid PDF
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Extra info for A History of Greek Mathematics, Volume 1: From Thales to Euclid
So there. The sticks are just exactly as long as they are. But the first one is twice as long as the second. The other way to go is to say that since the units don’t matter, we’ll choose whatever unit is convenient. For example, I could S I Z E A N D S H A P E 33 choose the second stick to be my unit, or ruler, so that the lengths come out nice. The first stick has length 2, the second stick has length 1. I could just as easily say the lengths are 4 1 and 2, 6 and 3, or 1 and -2 . It just doesn’t matter.
For instance, the regular hexagon (six sides) and octagon (eight sides) are relatively easy to measure, whereas the heptagon (seven sides) is quite spectacularly difficult. Can you measure the diagonals and areas of the regular hexagon and octagon? Another one you might enjoy measuring is the regular dodecagon (twelve sides). Can you measure the diagonals and area of the regular dodecagon? S I Z E A N D S H A P E 53 One of the most beautiful (and challenging) problems in geometry is the measurement of the regular pentagon.
We can just count the m rows of n squares each. But what if the sides don’t come out even? How can we measure the area of a rectangle if we can’t chop it up nicely into unit squares? Here are two rectangles of the same height. I like to think of the second one as a “stretched” version of the first. Is it clear that their areas are in the same proportion as their lengths? Stretching in one direction is called dilation. What we’re saying is that dilation of a rectangle by a certain factor multiplies its area by that factor.