The second sort of principles are those of construction, usually called _postulata_, or petitions. As for those _notiones communes_, called _axioms_, they are from the definitions of their terms demonstrable, though they be so evident as they need not demonstration. These petitions are by Euclid called Ἀιτήματα, such as are granted by favour, that is, simply petitions, whereas by axiom is understood that which is claimed as due. So that between Ἀξίωμα and Ἀίτημα there is this other difference, that this latter is simply a petition, the former a petition of right. Of petitions simply, the first is, _that from any point to any point may be drawn a straight line_. The second, _that a finite straight line may be produced_. The third, _that upon any centre at any distance may be described a circle_. All which are both evident and necessary to be granted. And by all these a man may easily perceive that Euclid in the definitions of a point, a line, and a superficies, did not intend that a point should be nothing, or a line be without latitude, or a superficies without thickness; for if he did, his petitions are not only unreasonable to be granted, but also impossible to be performed. For lines are not drawn but by motion, and motion is of body only. And therefore his meaning was, that the quantity of a point, the breadth of a line, and the thickness of a superficies were not to be _considered_, that is to say, not to be reckoned in the demonstration of any theorems concerning the quantity of bodies, either in length, superficies, or solid. ========== OF THE FAULTS THAT OCCUR IN DEMONSTRATION. TO THE SAME EGREGIOUS PROFESSORS OF THE MATHEMATICS IN THE UNIVERSITY OF OXFORD. LESSON II. There be but two causes from which can spring an error in the demonstration of any conclusion in any science whatsoever; and those are ignorance or want of understanding, and negligence. For as in the adding together of many and great numbers, he cannot fail that knoweth the rules of addition, and is also all the way so careful, as not to mistake one number or one place for another; so in any other science, he that is perfect in the rules of logic, and is so watchful over his pen, as not to put one word for another, can never fail of making a true, though not perhaps the shortest and easiest demonstration.