Problem 7 - A History of Mathematics: An Introduction

Question

Suppose Thales found that at the time a stick of length 6 feet cast a shadow of 9 feet, there was a length of 342 feet from the edge of the pyramid's side to the tip of its shadow. Suppose further that the length of a side of the pyramid was 756 feet. Find the height of the pyramid. (Assuming that the pyramid is laid out so the sides are due north-south and due east-west, this method requires that the sun be exactly in the south when the measurement is taken. When does this occur? \(^{20}\) )

Step-by-Step Solution

Answer

Answer: The height of the pyramid is 228 feet. The sun is exactly in the south at solar noon, which is when the sun is at its highest point in the sky, directly over the observer's local meridian or the observer's due south. The exact solar noon time may vary depending on location and date, so one should consult solar calculators or local weather resources to find the specific solar noon time for a particular place and day.

Step 1: Identify the similar triangles

Since the sun shines at the same angle for both the stick and the pyramid, we can assume that the triangles formed by their shadows are similar. Thus, we will compare the height and shadow length ratios between the stick and the side of the pyramid.

Step 2: Establish the proportion

Considering the similar triangles, we can create a proportion between the height of the stick (6 feet) and the length of its shadow (9 feet) to the height of the pyramid (unknown) and the length from the edge of the pyramid's side to the tip of its shadow (342 feet). So, let h represent the height of the pyramid. Then, we have: \(\frac{h}{342} = \frac{6}{9}\)

Step 3: Solve for the height of the pyramid

To find h, we can cross-multiply and then divide both sides by 3 to solve for h: \(h = \frac{6 \times 342}{9}\) Calculating the value: \(h = \frac{2052}{9} = 228\) feet So, the height of the pyramid is 228 feet.

Step 4: Determine when the sun is exactly in the south

As mentioned in the problem, the method used for this exercise assumes that the sun is directly in the south when the measurements are taken. Under normal circumstances, this occurs at solar noon, which is when the sun is at its highest point in the sky, directly over the observer's local meridian or the observer's due south. Note that solar noon may not match exactly with 12:00 PM local time, as local time is averaged across a time zone. To find the exact solar noon, one may consult solar calculators or local weather resources which give the solar noon time for a specific location and date.

Key Concepts

Similar Triangles

Similar triangles are a fundamental concept in geometry which occurs when two or more triangles have the same shape, but possibly different sizes. This similarity is defined by corresponding angles being equal and the sides around these angles being in proportion. When we look at Thales' theorem in the context of similar triangles, it pertains to right-angled triangles formed by a perpendicular object, like a stick or a pyramid, and the length of their respective shadows.

In the case of our exercise, the stick and the pyramid form two similar triangles because the angles of the sun's rays are constant, creating corresponding angles of the same measure in both triangles. Thales' keen observation of these shared angles allows us to compare ratios of corresponding sides, the very essence of similar triangles.

Proportion in Mathematics

Proportion is the heart of many geometrical concepts and practical applications. It describes the relationship between parts of a whole or between two quantities. When two ratios are equal, they are said to be in proportion. In mathematical terms, this can be represented as \( \frac{a}{b} = \frac{c}{d} \) which reads as 'a is to b as c is to d'.

Using proportions, one can solve for unknown values as seen in our historical exercise where the unknown height of the pyramid (h) can be discovered by setting up a proportion with the known heights and shadows of the stick. This not only provides an accurate measurement but also demonstrates the power of mathematical concepts in practical situations.

Triangle Shadow Problems

Triangle shadow problems revolve around using the properties of similar triangles to determine the lengths of objects based on the lengths of their shadows. This type of problem takes advantage of the predictability of similar shapes. Because the shadow of an object is essentially the hypotenuse of a right triangle, and the object's height is one of the legs, the second leg would be the length of the shadow on the ground.

Triangle shadow problems are a classic illustration of how simple geometrical principles can be applied to figure out real-world measurements without direct measurement of the object in question. This technique is beautifully illustrated in the shadow of the stick and the pyramid, where the proportionality between shadows and real heights provides us with the necessary data to find the height of the pyramid.

Historical Mathematics Problems

Historical mathematics problems offer a glimpse into the ingenuity of early mathematicians who developed fundamental concepts that form the bedrock of modern-day math. Through such problems, one can explore the application of these concepts to measure and understand the world in ancient times. The problem related to measuring the height of the pyramid using Thales' theorem is an excellent example of how ancient mathematicians engaged with their environment and utilized their theoretical insights.

The pursuit to measure grand structures like pyramids pushed the development of geometry and practical applications of mathematical ratios and proportions. These challenges not only solved immediate practical needs but also inspired subsequent generations of mathematical thought.

Measuring Techniques in Mathematics History

Measuring techniques in the history of mathematics showcase the innovative methods used by our ancestors to quantify dimensions without access to modern equipment. These techniques often utilized geometrical shapes, shadows, and fundamental principles of proportionality. They could estimate dimensions of inaccessible objects, navigate the seas, outline land for development, and construct architectural marvels.

The method used by Thales to measure the height of the pyramid is a prime example of how these ancient techniques could yield remarkably accurate results. This form of indirect measurement was crucial in a time when direct methods were impractical, such as the heights of mountains, depths of valleys, and the distances of celestial bodies. Mathematics thus served not just as a tool for theoretical speculation but as a practical necessity, shaping the world we live in today.