Back in the 1600s, a French scientist, Rene Descartes ("De-cart"), devised a systematic way of labeling each point on a flat plane by a pair of numbers. You may well be already familiar with it.

The system is based on two straight lines ("axes"), perpendicular to each other, each of them marked with the distances from the point where they meet ("origin")--distances to the right of the origin and above it, the origin being taken as positive and on the other sides as negative (see **drawing** below).

The distance on one axis is named "x" and on the other axis "y". Given then a point P, one draws from it lines **parallel** to the axes, and the values of x and y at their intersections completely define the point. In honor of Descartes, this way of labeling points is known as a **cartesian system** and the two numbers (x,y) that define the position of any point are its **cartesian coordinates**.

Graphs use this system, as do maps.

The two representations are closely related. From the definitions of the sine and cosine

x = r cos f

y = r sin f

That allows (x,y) to be derived from polar coordinates. To go in the opposite direction and derive (r,f) from (x,y), note that from the above equations (or from the **theorem of Pythagoras**) one can **derive r**:

r^{2} = x^{2} + y^{2}

Once r is known, the rest is easy

cos f = x*/ * sr

sin f = y*/ * r

These relations fail only at the origin, where x = y = r = 0. At that point, f is undefined and one can choose for it whatever one pleases.

## Coordinates in Three Dimensions

Up to here numbers were used to specify points on a flat plane, for instance a sheet of paper. The real world, however, is 3-dimensional, and sometimes it is necessary to label points in 3-dimensional space. The cartesian (x,y) labeling can be extended to 3 dimensions by adding a third coordinate z. If (x,y) is a point on the sheet, then the point (x,y,z) in space is reached by moving to (x,y) and then rising a distance z above the paper (points below it have negative z).

Very simple and clear, once a decision is made **on which side** of the sheet z is positive. By common agreement the positive branches of the (x,y, z) axes, in that order, follow the thumb and the first two fingers of the **right** hand when extended in a way that they make the largest angles with each other.

Again, the cartesian labeling (x,y,z) is nicely symmetric. Sometimes, though, it is convenient to follow the style of polar coordinates and label distance and and direction separately. Distance is easy: you take the line OP from the origin to the point and measure its length. You can even show from the theorem of Pythagoras that in this case

r^{2} = x^{2} + y^{2} + z^{2}

But what about direction?

Look again at the surveyor's telescope: the direction to which it points is completely specified by **two** angles--its **azimuth** f, the angle by which its base is rotated on a horizontal table-- and its **elevation** l (small Greek L), the angle by which it is lifted above the horizontal (if it looks down, l is negative). The two angles together can in principle specify any direction: f ranges from 0 to 360^{0}, and l from -90^{0} (straight down or "nadir") to +90^{0} (straight up or "zenith").

Again, one needs to decide from what direction is the azimuth measured--that is, where is azimuth zero? The rotation of the heavens (and the fact most humanity lives north of the equator) suggests (for surveyor-type measurements) the northward direction, and this is indeed the usual zero point. The azimuth angle (viewed from the north) is measured *counterclockwise*.

## The First Point in Aries

Given the system of celestial coordinates it is now possible to measure the declination and right ascension of stars, and to draw maps showing their positions and the constellations which they form. One problem here is that one cannot map the surface of a sphere onto flat paper without distortion: the same problem arises in mapping the surface of the Earth, and map-makers have long ago devised a variety of mapping methods, each with its advantages and drawbacks. This matter is sidestepped here by assuming the heavens to be mapped onto a **globe**, and in fact such globes do exist.

The star chart now resembles the celestial sphere, with the difference that instead of being at the center looking out, we are on the outside looking in. Polaris will be very near the north pole, and the equator will be halfway between the poles--for instance, the constellation of Orion straddles it and one of the 3 stars in Orion's "belt" is practically on top of it. And the ecliptic, equally long, cuts the equator at an angle of 23.5 degrees (see **drawing**)

Yet one essential detail is still missing. Just as azimuth is measured from the northward direction, right ascension must have a zero mark **somewhere**. And it does: by common agreement, it is one of the points at which the ecliptic cuts the equator, the one occupied by the Sun in spring.

Since that point is on the ecliptic, it is expected to belong to one of the constellations of the zodiac. Two thousand years ago or so, when the positions of stars were first measured, it was in the constellation of Aries, the ram. Over the centuries, it has slowly migrated (as is discussed in the next section) to the neighboring constellation of Pisces, the fish. However the old name is still used, the "**first point in Aries**," though some also refer to it as the "spring equinox" or "vernal equinox."