[Bill S (OP)]

While trying to make sense of mathematical dimensions I found:

https://aeon.co/essays/how-many-dimensions-are-there-and-what-do-they-do-to-reality

 This is the best (simple) explanation I’ve found, but I ran into a problem with:
“Descartes discovered that with this framework he could link geometric shapes and equations. Thus, a circle with a radius of 1 can be described by the equation x² + y² =1.”

This seemed to say: x=1, so x² =1;    y=1, so  y² =1.  Therefore, 1+1=2,  but the circle has radius 1.

Obviously, I’ve gone adrift here, somewhere. 

[jeffreyH]

When the radius is along the x-axis then y must be zero. As the radial line is rotated away from the horizontal in an anti-clockwise direction x decreases (becomes less than one) and y increases (becomes greater than zero) so that it is a sum of the squares of fractional components.

[Colin2B]

As @jeffreyH says.
Also think about it as - at any angle where radius is not along horizontal or vertical axes then radius will be hypotenuse of rt angle triangle with x&y as other 2 sides.

[Janus]

While trying to make sense of mathematical dimensions I found:

https://aeon.co/essays/how-many-dimensions-are-there-and-what-do-they-do-to-reality

 This is the best (simple) explanation I’ve found, but I ran into a problem with:
“Descartes discovered that with this framework he could link geometric shapes and equations. Thus, a circle with a radius of 1 can be described by the equation x² + y² =1.”

This seemed to say: x=1, so x² =1;    y=1, so  y² =1.  Therefore, 1+1=2,  but the circle has radius 1.

Obviously, I’ve gone adrift here, somewhere. 

In this equation, x and y are interdependent variables.   You cannot just assign any values you want to x and y.   So what the equation means that only values of x and y, which, when their squares are added, sums to 1 work for this equation.  In other words,the only values for x and y that work are those that make the equation true.
If you take the equation and rearrange it to y= sqrt(1-x^2) and plot y for different values of x you get a circle.

[Bill S (OP)]

Thanks, folks.  Putting those responses together, I think I can grasp Wertheim's article, without having that nasty feeling that there was something in the middle that didn't make sense.

[jeffreyH]

So y = sqrt(1 - x^2). If we let x range from -1 to 1 we can select a fixed number of values. Say -1, -3/4, -1/2, -1/4, 0, 1/4, 1/2, 3/4 and 1. Remembering that a square is always positive y will never exceed the value of 1. This function relies on the fact that the result of the square root function can be positive or negative so it does not automatically describe a circle and is open to interpretation.

[Bored chemist]

This seemed to say: x=1, so x2 =1;    y=1, so  y2 =1.  Therefore, 1+1=2,  but the circle has radius 1.

A cirvle with radius 1 (centred on the origin) doesn't go through the point (1,1).
It does go through ( root(2)/2, root (2)/2 ). If you do the arithmetic, it works.