Theoretic Arithmetic, in Three Books: Containing the Substance of All that Has Been Written on this Subject by Theo of Smyrna, Nicomachus, Iamblichus, and Boetius, Together with Some Remarkable Particulars Respecting Perfect, Amicable, and Other Numbers, which are Not to be Found in the Writings of Any Ancient Or Modern Mathematicians. Likewise, a Specimen of the Manner in which the Pythagoreans Philosophized about Numbers, and a Development of Their Mystical and Theological Arithmetic
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according added addition aggregate amicable numbers analogy angles anonymous appears appellation arising arithmetical arranged become bers body breadth called cause CHAPTER compared composite considered consists contains contrary cube decad denominated diatessaron difference divided divine division double duad duple energy equal essence evenly-even evident exceeds extremes fifth figures former four fourth geometrical gives gnomon greater greatest half harmonic Hence hexagon increase instance intellect interval kind knowledge latter length less likewise manner mathematical mean measure middle monad multiplied multitude nature numbers longer observes odd numbers omitted pentagon perfect number Plato possesses preceding principle prior proceed produced proportion proportionality Pythagoreans quadruple quantity ratio reason receive remain respect rest says sensible sesquialter sesquitertian seven side similar solid soul species square subsistence superparticular surpasses symphony take place tetrad things third tion triad triangle triple twice unity universe whole
Page 186 - was not only principally honoured by the Pythagoreans, because all symphonies are found to exist within it, but also because it appears to contain the nature of all things.
Page 152 - ... and the only alteration which they experience is a mutation of figure, and variation of light at different periods ; but in the sublunary region, while the spheres of the elements remain, on account of their subsistence as wholes, always according to nature, the parts of...
Page iii - The theorems of philosophy are to be enjoyed as much as possible, as if they were ambrosia and nectar. For the pleasure arising from them is genuine, incorruptible, and divine. They are also capable of producing magnanimity; and though they cannot make us eternal beings, yet they enable us to obtain a scientific knowledge of eternal natures.
Page 13 - Similarly it may be shown that Now AE = EC, and GE = EF ; therefore DF • FC = DG • GA. And if the rectangle under the extremes is equal to the rectangle under the means, the four straight lines are proportional ; therefore FD : DG = AG : CF.
Page 151 - ... some in a primary and others in a secondary degree; for the universe being a perfect whole, must have a first, a middle, and a last part. But its first parts, as having the most excellent subsistence, must always exist according to nature; and its last...
Page 6 - ... to itself. But in the sensible universe, the first monad is the world itself, which comprehends in itself all the multitude of which it is the cause (in conjunction with the cause of all). The second monad is the inerratic sphere. In the third place, the spheres of the planets succeed, each of which is also a monad, comprehending an appropriate multitude. And in the fourth and last place are the spheres of the elements, which are in a similar manner monads. All these monads likewise are denominated...
Page 152 - For in these periods, a fertility or sterility of men, animals, and plants, takes place ; so that in fertile periods, mankind will be both more numerous, and upon the whole, superior in mental and bodily endowments, to the men of a barren period. And a similar reasoning must be extended to animals and plants. The so much celebrated heroic age, was the result of one of these fertile periods, in which men transcending the herd of mankind, both in practical and intellectual virtue, abounded on the earth....
Page i - Theoretic Arithmetic, in three Books ; containing the Substance of all that has been written on the subject by Theo of Smyrna, Nicomachus, Jamblichus, and Boetius. — Together with some remarkable particulars respecting Perfect, Amicable, and other Numbers, which are not to be found in the writings of any ancient or modern Mathematicians.
Page 185 - They considered a point as analogous to the monad, a line to the duad, a superficies to the triad, and a body to the tetrad.
Page xxxiv - ... divinity, and proceeds together with him through the universe ; that the shining traces of her feet are conspicuous only in form; and that in the dark windings of matter she left nothing but a most obscure and fleeting resemblance of herself. This delusive phantom, however, the man of modern science ardently explores, unconscious that he is running in profound darkness and infinite perplexity, and that he is hastening after an object, which eludes all detection and mocks all pursuit.