Three questions get asked of every roughing pass: how much metal comes off, how long it takes, and what it costs the insert. The first two are multiplications. The third is not, and it is the one that surprises people.

$Q = V f a_p$ — read aloud Q equals V f a-p. $Q$ is the material removal rate in cm³/min. $V$ is the cutting speed in m/min. $f$ is the feed per revolution in millimetres — a lathe feeds per rev, where a mill feeds per tooth. $a_p$ is the depth of cut in millimetres, subscript p for the German Schnitttiefe convention every catalogue uses. Keep $V$ in m/min against $f$ and $a_p$ in mm and the answer lands straight in cubic centimetres a minute, with nothing to convert.

The three terms are equal in that product and nothing like equal in what they cost. Depth is nearly free — it barely touches tool life. Feed costs you surface finish. Speed costs you the insert, and it costs a great deal.

$t_m = \dfrac{L}{f N}$ — t-m equals L over f N. $t_m$ is the machining time for one pass in minutes, $L$ is the length of cut in millimetres INCLUDING approach and overrun, and $f N$ is how far the tool advances in a minute. Honest warning: this is cutting time only, and on most real parts it is the smaller half of the cycle. Loading, indexing, rapids, gauging and tool changes are the other half, and quotations that forget them lose money quietly.

Now the one that bites. $V T^{\,n} = C$ — V T to the n equals C. $V$ is the cutting speed, $T$ is the tool life in MINUTES — the equation counts nothing else — $n$ is the Taylor exponent, a bare number set by the tool material (about 0.1 for high-speed steel, 0.25 for carbide), and $C$ is the Taylor constant, the speed that would burn the edge out in exactly one minute. $C$ wears the same units as $V$, so a constant fitted in m/min will not serve a shop working in surface feet.

Solve it for life and the sting appears: $T = \left(\dfrac{C}{V}\right)^{1/n}$. With carbide at $n = 0.25$, that exponent is FOUR. Twenty per cent more speed costs you three fifths of the tool life. Double the speed and a sixteenth of it survives. Frederick Taylor spent twenty-six years of cutting trials establishing this, and it is still the arithmetic behind every decision to run hard and change inserts often, or run gently and leave them in.