If we plot a graph of the radiogenic ²⁰⁶Pb/²³⁸U ratio against the radiogenic ²⁰⁷Pb/²³⁵U ratio for concordant (closed) systems _of all ages_, we obtain the curved line shown in the figure below. The curve is the locus of _all_ concordant U-Pb ages and is called _Concordia_. Then if we test two or more particular zircons of the same age that have lost different amounts of lead, at about the same time, the plot of their ²⁰⁶Pb/²³⁸U ratios against their ²⁰⁷Pb/²³⁵U ratios will fall on a straight line that is a chord of the Concordia curve. The upper intersection of this chord with the curve then will mark the true age of the zircons. This is an elaborate technique utilizing difficult chemical procedures, but it has proved invaluable in solving some important geologic problems. [Illustration: _The Concordia curve offers a useful way of analyzing results of age determinations on the mineral zircon._] Hornblende The mineral hornblende provides another useful system. Hornblende is a complex silicate of sodium, calcium, iron, magnesium, and aluminum, and usually contains a few tenths of 1% of potassium. It is unusual in that it tenaciously retains its radiogenic argon, even at relatively high temperatures. Sanidine Still another good system is the rare feldspar, sanidine, which is excellent for both potassium-argon and rubidium-strontium age determination. Sanidine usually is found in volcanic ash falls and has been important in the establishment of the geologic time scale, as we shall see. Whole Rocks Finally there is still another way of obtaining a closed system by using the whole rock, not just a crystal of a single mineral within it. A large body of granite or similar rock may contain a number of minerals, some or none of which may be closed systems. Yet as long as this body of rock remains impermeable to solutions (which in nature means mostly to water), no substance will be able to move very far in it because diffusion in solids is so slow. Consequently it will remain a closed system, as a whole, regardless of what happens to the individual mineral grains. If we take a piece from near the middle of this body of rock and if this piece is much larger than the largest constituent grain in it, then we have a fair sample of a closed system—the whole rock. The only difficulty arises from the fact that few rocks are sufficiently impermeable to solutions to retain argon, and many rocks contain so much common strontium that rubidium-strontium analysis is impractical. Still, we can use the rapid survey methods as for feldspar, selecting the few rocks that would be useful. This work has been done frequently, and the results have been fruitful for rubidium-strontium analysis. The whole-rock rubidium-strontium age dates the time when the rock became impermeable. SOME INTERESTING RESULTS The Old Man from Olduvai