"drawyng, measuring and proporcion"
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Note
The 47th propositionPythagoras's Theorem.
Gnomon: "the part of a parallelogram which remains after a similar parallelogram is taken away from one of its corners." (OED).
Pythagoras's Theorem Edmund Scarburgh, 1705
For generations of schoolchildren, geometry meant Euclid, and it meant in particular a handful of key theorems, of which we present one of the most famous here: Pythagoras's Theorem. Euclid had first been translated into English in 1570 (there is a section of the preface to that translation in Chapter 10); many other translations followed, often differing litde in their choice of words, diagrams, or notation but showing more variation in the annotations and supplements they provided. Scarburgh, as we see here, attempts to give some hint of how the theorem might have been first discovered: compare Joseph Fenn's imagined "first analysts" in Chapter 3.
Edmund Scarburgh, The English Euclide, being The First Six Elements of Geometry, Translated out of the Greek, with Annotations and useful Supplements (Oxford, 1705), pp. 108-109.
In a Right-angled Triangle, the Square of the side subtending the Right angle is equal to the Squares of the sides containing the Right angle
Let the Right-angled Triangle be ABC, having the Right angle BAC (see Figure 4.6). I say that the square of BC is equal to the squares of BA, AC.
For on BC, let be described the square BDEC, and on AB, AC, the squares GB, HC, and by A let AL be drawn parallel to either of the lines BD, CE, and let be joined AD, FC.
Now forasmuch as each of the angles BAC, BAG, is a Right angle, and to the straight line BA, and to a point in the same A, the two straight lines AC, AG, not lying the same way, make the consequent angles equal to two Right, therefore CA is j^aralleL to AG. By the same reason also AB is parallel to AH. And because the angle DBC is equal to the angle FBA, for each is a Right angle, let the angle ABC be added in common, therefore
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H
F
K
D        L E
Figure 4.6. The diagram for Pythagoras's Theorem.
the whole angle DBA is equal to the whole angle FBC. And because the two lines DB, BA, are equal to the two lines CB, BF, each to each, and the angle DBA is equal to the angle FBC, therefore the base AD is equal to the base FC, and the Triangle ABD is equal to the Triangle FBC.
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H D G
Figure 4.7. How Pythagoras's Theorem might have been first conceived. (Scarburgh, p. 109.)
Now the Parallelogram BL is double of the Triangle ABD, for they have the same base BD, and are in the same parallels DB, AL (Proposition 41). Also the square GB is double of the Triangle FBC, for they have the same base FB, and are in the same parallels FB, GC. Now the doubles of equals are equal to one another; therefore the Parallelogram BL is equal to the square GB.
In like manner, AE, BK being joined, may be proved that the Parallelogram CL is equal to the square HC; therefore the whole square BDEC is equal to the two squares GB, HC. And the square BDEC is described on BC, and GB, HC on BA, AC; wherefore the square of the side BC is equal to the squares of the sides BA, AC.
Therefore in Right-angled Triangles, the square of the side subtending the Right angle is equal to the squares of the sides containing the Right angle. Which was to be demonstrated.
This Proposition, among Geometricians most famous, is said to have been found out by Pythagoras, and the Invention publicly celebrated with a Sacrifice to the Muses. Yet the hint from whence the discovery of this Truth might first arise, seems to be very obvious.
For in this Figure (4.7) the square EFGH is apparently double of the square ABDC. But EFGH is described on EF, which is equal to
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BC, the side subtending the Right angle BAC of the isosceles. Triangle ABC; and the square ABDC is described on either of the sides AB, AC, containing the Right angle BAC, of the same isosceles. Triangle ABC. It is therefore hereupon very reasonable to conceive that the same property might likewise belong to Scalene Right-angled Triangles, and give the occasion of a farther enquiry into this matter.
Thus Geometricians often happen to discover a Truth before they have framed a legitimate demonstration of it, and find out their Propositions one way (which they usually conceal) but prove them in another. We have an Example of this kind in the Remains of Archimedes, who shows how first he found the Quadrature of a Parabola" Mechanically, as he calls it, and afterwards gives a Geometrical demonstration.
Note
Quadrature of a parabola: the area under a parabola.
Trigonometrical Definitions Edward Wells, 1714
It is very difficult to find approachable introductions to trigonometry from this period because—as this extract shows—the terms "sine," "cosine," and so on, were used for lengths in a particular construction involving a circle: not, as now, for ratios of lengths in a right-angled triangle. This way of using the terms remained in use until the second half of the nineteenth century, and it is not immediately obvious that—as is in fact the case—it is numerically equivalent to the modern way.
Edward Wells's training was in divinity and his profession that of an Anglican clergyman, but he had an abiding interest in education and put together textbooks ("more voluminous than distinguished," according to a recent biographer) on subjects from mathematics to geography, history, and religion. His trigonometry is, for its period, unusually lucid.
Edward Wells (1667-1727), The Young Gentleman's Trigonometry, Mechanicks, and Opticks. Containing such Elements of the said Arts or Sciences as are most Useful and Easy to be known. (London, 1714), pp. 1-5.