The machine that did algebra, not just arithmetic. The MIR (МИР — Mashina dlya Inzhenernykh Raschyotov, “machine for engineering calculations”) — MIR-1 (1965), MIR-2 (1969) — from V. M. Glushkov’s Institute of Cybernetics in Kyiv carried a high-level language, Analitik, close to the hardware, and manipulated formulae symbolically. This panel is runnable: parse an expression over x, differentiate it exactly with the power, sum and product rules, then check the answer two ways.
source The MIR series, Institute of Cybernetics, Academy of Sciences of the Ukrainian SSR, Kyiv, under V. M. Glushkov; MIR-1 (1965), MIR-2 (1969), input language Analitik. Room: THE MACHINE. Glushkov also designed OGAS — see the-ogas. Rendered, not quoted.
Viktor Mikhailovich Glushkov’s Institute of Cybernetics in Kyiv designed the MIR-1 (1965) for engineering offices, teaching and research, then the MIR-2 (1969). Each carried a high-level language close to the hardware — the MIR-2’s was called Analitik.
MIR = Mashina dlya Inzhenernykh Raschyotov. Analitik developed the built-in language of the MIR-1 and added the formulation of tasks as analytic transformations of formulae — algebra done in firmware.
A desk calculator returns a number. The MIR returned a formula. It manipulated fractions, polynomials, derivatives and integrals as symbols — exact, not rounded — in hardware, in the 1960s.
That is the difference this sphere makes concrete: d/dx as an exact symbolic transform with integer coefficients, next to the finite-difference a numeric machine is stuck with. The engine in the centre does the symbolic thing.
Symbolic computation on a machine → the computer algebra systems (Reduce, Macsyma, later Mathematica / Maple) that do exactly what Analitik reached for: transform expressions, not just evaluate them.
And Glushkov himself → OGAS, the proposed nationwide economic-computing network. The MIR is one machine in that larger cybernetic project. Cross-link: the-ogas.
Type a small expression in x — integer coefficients, sums, products, and non-negative integer powers. The engine parses it into a tree, then differentiates that tree symbolically.
Allowed: x, integers, + − *, ^n (n ≥ 0), parentheses. The result is always a polynomial with integer coefficients — no rounding anywhere in the symbolic step.
THE RULES — applied to the tree, exactly:
d/dx(c)=0 · d/dx(x)=1 · power d/dx(xn)=n·xn−1 · sum is linear · product d(uv)=u'v+uv'.
symbolic derivative f'(x) = —
CHECK 1 — coefficient-for-coefficient vs the known analytic answer
| power | symbolic d/dx | analytic | = |
|---|
CHECK 2 — against a central finite difference (f(x+h)−f(x−h))/2h
| x | f'(x) symbolic | central diff | |err| |
|---|
halving h shrinks the error by ~4× — order h2: —
Anchor: d/dx(x^3) = — · product rule on (x^2)(x^3) = — (= d/dx x^5) · d/dx(42) = —. Every witness re-checks live.
An engineer’s machine that did the algebra a slide rule and a calculator could not — a step toward the personal workstation: high-level language, close to the hardware, on a desk.
“Computers only crunch numbers.” Cut. The MIR did exact symbolic mathematics — derivatives of formulae, not decimals — in firmware in the 1960s. The centre reproduces it and checks it.
“Symbolic algebra started with 1980s desktop software.” Cut — earlier. Analitik on the MIR-2 (1969) transformed formulae in hardware a decade before the home CAS.
“A finite difference is as good as a derivative.” Cut. It only approximates — error O(h2), never exact. The symbolic answer is exact. Check 2 shows the gap shrink but never close.
The red move: replace the product rule with the classic freshman error — d(uv)=u'·v' (multiply the derivatives) instead of u'v+uv'.
Engage it and the derivative of (x^2)(x^3) becomes 2x·3x^2 = 6x^3 instead of 5x^4. Both checks break: the coefficients stop matching the analytic answer and the finite difference disagrees. The witness (5) turns red. The product rule is u'v+uv', not u'v'. Caught.