Arithmetic, 3

Both Number and Magnitude have a certain original seed (as it were) of incredible property, which man is never able fully to declare. Of Number, a Unit, and of Magnitude, a Point, seem to be much alike as original causes: but the difference, nevertheless, is great. We defined a Unit to be a mathematically indivisible thing; a Point, likewise, we said to be a mathematically indivisible thing. And further, that a Point may have a certain determined situation: that is, that we may assign and prescribe a Point to be here, there, or over there, etc. In this, behold, our Unit is free, and can abide no bondage, nor be tied to any place or seat, divisible or indivisible. Again, because a Point may have a situation limited to it, a certain motion (to a place and from a place) is incident and belonging to a Point. But a Unit cannot be imagined to have any motion. A Point, by its motion, produces mathematically a line (as we said before), which is the first kind of magnitudes and most simple: a Unit cannot produce any number. A Line, though it is produced by a Point moved, yet does not consist of points: Number, though it is not produced by a Unit, yet does consist of units as a material cause. But formally,

Number.

Number is the union and unity of units. Which uniting and knitting is the workmanship of our mind, which, from distinct and discrete units, makes a Number, by a uniformity resulting from a certain multitude of units. And so every number may have its least part given, namely a unit: but of a Magnitude (no, not even of a Line) can the least part be given, because an infinite division of it may be conceived. All Magnitude is either a Line, a Plane, or a Solid. Which Line, Plane, or Solid can be perceived by no sense, nor exactly represented by hand in any way, nor produced by Nature: but as Number (by degrees) came to our perception, so by visible forms we are helped to imagine what our mathematical Line is, what our Point is. So precise are our magnitudes that one Line is no broader than another, for they have no breadth; nor our planes any thickness; nor yet our bodies any weight, be they ever so large of dimension. Our bodies we can have smaller than either Art or Nature can produce any, and greater also than all the world can contain. Our least magnitudes can be divided into as many parts as the greatest. As a Line one inch long (with us) may be divided into as many parts as may the diameter of the whole world from East to West, or extended any way: what privileges, above all manual Art and the might of Nature, have our two mathematical sciences, to exhibit and to deal with things of such power, liberty, simplicity, purity, and perfection? And in them to proceed so certainly, so orderly, so precisely: as excellent is that mechanical workman judged who can most nearly approach the representing of works mathematically demonstrated?

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And our two sciences, remaining pure and absolute in their proper terms and in their own matter, to have and allow only such demonstrations as are plain, certain, universal, and of an eternal truth?

Both number and magnitude have a certain original seed (as it were) of incredible property, which man is never able to declare fully. For number, a unit, and for magnitude, a point, seem to be very much alike as original causes; but the difference is nevertheless great. We defined a unit to be a mathematical thing indivisible; a point, likewise, we said to be a mathematical thing indivisible. Further, a point may have a certain fixed situation: that is, we may assign and set a point to be here, there, over there, etc. Here, notice, our unit is free, and can endure no bondage, nor be tied to any place or seat, divisible or indivisible. Again, because a point may have a situation limited to it, a certain motion (to a place and from a place) belongs to a point. But a unit cannot be imagined to have any motion. A point, by its motion, produces mathematically a line (as we said before), which is the first kind of magnitudes and the simplest; a unit cannot produce any number. A line, though it is produced by a point moved, does not consist of points; number, though it is not produced by a unit, does consist of units as a material cause. But formally,

Number.

Number is the union and unity of units. This uniting and joining is the workmanship of our mind, which, out of distinct and separate units, makes a number by the uniformity resulting from a certain multitude of units. And so every number can be given its least part, namely a unit; but of a magnitude (not even of a line) can the least part be given, because its division can be conceived without end. All magnitude is either a line, a plane or a solid. A line, plane or solid can be perceived by no sense, nor represented exactly by hand in any way, nor produced by nature; but as number came to our perception by degrees, so we are helped by visible forms to imagine what our mathematical line is, and what our point is. Our magnitudes are so precise that one line is no broader than another, for they have no breadth; nor do our planes have any thickness; nor do our bodies have any weight, however large in dimension. We can have bodies smaller than anything art or nature can produce, and greater also than all the world can contain. Our least magnitudes can be divided into as many parts as the greatest. For a line an inch long (with us) can be divided into as many parts as the diameter of the whole world from east to west, or extended any way. What privileges, above all manual art and nature's power, have our two mathematical sciences: to exhibit and deal with things of such power, liberty, simplicity, purity and perfection; and in them to proceed so certainly, so orderly, so precisely? Excellent is that mechanical workman judged who can come nearest to representing works mathematically demonstrated.

And our two sciences, remaining pure and absolute in their proper terms and in their own matter, to have and allow only such demonstrations as are plain, certain, universal and of eternal truth?