=1832.= Beyond question, Egyptian geometry, such as it was, was eagerly studied by the early Greek philosophers, and was the germ from which in their hands grew that magnificent science to which every Englishman is indebted for his first lessons in right seeing and thinking.--GOW, JAMES. _A Short History of Greek Mathematics (Cambridge, 1884), p. 131._ =1833.= A figure and a step onward: Not a figure and a florin. --MOTTO OF THE PYTHAGOREAN BROTHERHOOD. _W. B. Frankland: Story of Euclid (London, 1902), p. 33._ =1834.= The doctrine of proportion, as laid down in the fifth book of Euclid, is, probably, still unsurpassed as a masterpiece of exact reasoning; although the cumbrousness of the forms of expression which were adopted in the old geometry has led to the total exclusion of this part of the elements from the ordinary course of geometrical education. A zealous defender of Euclid might add with truth that the gap thus created in the elementary teaching of mathematics has never been adequately supplied. --SMITH, H. J. S. _Presidential Address British Association for the Advancement of Science (1873); Nature, Vol. 8, p. 451._ =1835.= The Definition in the Elements, according to Clavius, is this: Magnitudes are said to be in the same Reason [ratio], a first to a second, and a third to a fourth, when the Equimultiples of the first and third according to any Multiplication whatsoever are both together either short of, equal to, or exceed the Equimultiples of the second and fourth, if those be taken, which answer one another.... Such is Euclid’s Definition of Proportions; that _scare_-Crow at which the over modest or slothful Dispositions of Men are generally affrighted: they are modest, who distrust their own Ability, as soon as a Difficulty appears, but they are slothful that will not give some Attention for the learning of Sciences; as if while we are involved in Obscurity we could clear ourselves without Labour. Both of which Sorts of Persons are to be admonished, that the former be not discouraged, nor the latter refuse a little Care and Diligence when a Thing requires some Study.--BARROW, ISAAC. _Mathematical Lectures (London, 1734), p. 388._