Concerning his work in mathematics and physics I can speak with more confidence. He is the author of the Cartesian system of algebraic or analytic geometry, which has been so powerful an engine of research, far easier to wield than the old synthetic geometry. Without it Newton could never have written the _Principia_, or made his greatest discoveries. He might indeed have invented it for himself, but it would have consumed some of his life to have brought it to the necessary perfection. The principle of it is the specification of the position of a point in a plane by two numbers, indicating say its distance from two lines of reference in the plane; like the latitude and longitude of a place on the globe. For instance, the two lines of reference might be the bottom edge and the left-hand vertical edge of a wall; then a point on the wall, stated as being for instance 6 feet along and 2 feet up, is precisely determined. These two distances are called co-ordinates; horizontal ones are usually denoted by _x_, and vertical ones by _y_. If, instead of specifying two things, only one statement is made, such as _y_ = 2, it is satisfied by a whole row of points, all the points in a horizontal line 2 feet above the ground. Hence _y_ = 2 may be said to represent that straight line, and is called the equation to that straight line. Similarly _x_ = 6 represents a vertical straight line 6 feet (or inches or some other unit) from the left-hand edge. If it is asserted that _x_ = 6 and _y_ = 2, only one point can be found to satisfy both conditions, viz. the crossing point of the above two straight lines. Suppose an equation such as _x_ = _y_ to be given. This also is satisfied by a row of points, viz. by all those that are equidistant from bottom and left-hand edges. In other words, _x_ = _y_ represents a straight line slanting upwards at 45°. The equation _x_ = 2_y_ represents another straight line with a different angle of slope, and so on. The equation x^2 + y^2 = 36 represents a circle of radius 6. The equation 3x^2 + 4y^2 = 25 represents an ellipse; and in general every algebraic equation that can be written down, provided it involve only two variables, _x_ and _y_, represents some curve in a plane; a curve moreover that can be drawn, or its properties completely investigated without drawing, from the equation. Thus algebra is wedded to geometry, and the investigation of geometric relations by means of algebraic equations is called analytical geometry, as opposed to the old Euclidian or synthetic mode of treating the subject by reasoning consciously directed to the subject by help of figures.