Now that all magnitudes are cut or divided by the same wherewith they are bounded, is conceived out of the definition of Continuum, e. 4. For if the common band to containe and couple together the parts of a Line, surface, & Body, be a Point, Line, and Surface, it must needes bee that a section or division shall be made by those common bandes: And that to bee dissolved which they did containe and knitt together.
7. A point is an undivisible signe in a magnitude.
A Point, as here it is defined, is not naturall and to bee perceived by sense; Because sense onely perceiveth that which is a body; And if there be any thing lesse then other to be perceived by sense, that is called a Point. Wherefore a Point is no Magnitude: But it is onely that which in a Magnitude is conceived and imagined to bee undivisible. And although it be voide of all bignesse or Magnitude, yet is it the beginning of all magnitudes, the beginning I meane potentiâ, in powre.
8. Magnitudes commensurable, are those which one and the same measure doth measure: Contrariwise, Magnitudes incommensurable are those, which the same measure cannot measure. 1, 2. d. X.
Magnitudes compared betweene themselves in respect of numbers have Symmetry or commensurability, and
Reason or rationality: Of themselves, Congruity and Adscription. But the measure of a magnitude is onely by supposition, and at the discretion of the Geometer, to take as pleaseth him, whether an ynch, an hand breadth, foote, or any other thing whatsoever, for a measure. Therefore two magnitudes, the one a foote long, the other two foote long, are commensurable; because the magnitude of one foote doth measure them both, the first once, the second twice. But some magnitudes there are which have no common measure, as the Diagony of a quadrate and his side, 116. p. X. actu, in deede, are Asymmetra, incommensurable: And yet they are potentiâ, by power, symmetra, commensurable, to witt by their quadrates: For the quadrate of the diagony is double to the quadrate of the side.
9. Rationall Magnitudes are those whose reason may bee expressed by a number of the measure given. Contrariwise they are irrationalls. 5. d. X.
Ratio, Reason, Rate, or Rationality, what it is our Authour (and likewise Salignacus) have taught us in the first Chapter of the second booke of their Arithmetickes: Thither therefore I referre thee.
Data mensura, a Measure given or assigned, is of Euclide called Rhetè, that is spoken, (or which may be uttered) definite, certaine, to witt which may bee expressed by some number, which is no other then that, which as we said, was called mensura famosa, a knowne or famous measure.
Therefore Irrationall magnitudes, on the contrary, are understood to be such whose reason or rate may not bee expressed by a number or a measure assigned: As the side of the side of a quadrate of 20. foote unto a magnitude of two foote; of which kinde of magnitudes, thirteene sorts are mentioned in the tenth booke of Euclides Elements: such are the segments of a right line proportionally cutte, unto the whole line. The Diameter in a circle is rationall: