By Morris Kline
During this e-book Kline examines the improvement of arithmetic as our strongest device for exploring the actual international. He probes our present global of arithmetic and illuminates its workings as a technology permitting us to penetrate the secrets and techniques of the world's common phenomena.
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Since this radius is roughly two million times that of the garden we considered, one might expect that the additional length of the roadway should be about two million times the additional length of fence required to enclose the garden. The latter quantity was just 2p feet. Hence an intuitive argument for the additional length of roadway would seemingly lead to the figure of 2,000,000 X 2p feet. Whether or not you would agree to this argument, you would almost certainly estimate that the length of the roadway would be very much greater than the circumference of the Earth.
The principle involved in these examples THE FAILINGS OF THE SENSES AND INTUITION 27 and in the theory of linear perspective is that lines in the actual scene that recede directly from the observer must appear to come together at a distant point called the vanishing point. A simple example is furnished by the appearance of two parallel railroad tracks (Figure 15) that appear to meet at a distant point. FIGURE 15 28 MATHEMATICS AND THE SEARCH FOR KNOWLEDGE FIGURE 16 FIGURE 17 THE FAILINGS OF THE SENSES AND INTUITION 29 The influence of perspective is particularly apparent in Figure 16, where the usual perspective lines are drawn to suggest a scene.
Thus far the two jobs seem to be equally good. However, on the second job the increases start after the first six months, whereas on the first job they do not start until after one year has elapsed. Hence the second job will pay more during the latter six months of each year. Let us consider another simple problem. Suppose that a merchant sells his apples at two for five cents and his oranges at three for five cents. Being somewhat annoyed with having to do considerable arithmetic on each sale, the merchant decides to commingle apples and oranges and to sell any five pieces of fruit for ten cents.