We examine William Herschel's hypothesis that solar-cycle variation of the Sun's irradiance has a modulating effect on the Earth's climate and that this is, specifically, manifested as an anticorrelation between sunspot number and the market price of wheat. Since Herschel first proposed his hypothesis in 1801, it has been regarded with both interest and skepticism. Recently, reports have been published that either support Herschel's hypothesis or rely on its validity. As a test of Herschel's hypothesis, we seek to reject a null hypothesis of a statistically random correlation between historical sunspot numbers, wheat prices in London and the United States, and wheat farm yields in the United States. We employ binary-correlation, Pearson-correlation, and frequency-domain methods. We test our methods using a historical geomagnetic activity index, well known to be causally correlated with sunspot number. As expected, the measured correlation between sunspot number and geomagnetic activity would be an unlikely realization of random data; the correlation is "statistically significant." On the other hand, measured correlations between sunspot number and wheat price and wheat yield data would be very likely realizations of random data; these correlations are "insignificant." Therefore, Herschel's hypothesis must be regarded with skepticism. We compare and contrast our results with those of other researchers. We discuss procedures for evaluating hypotheses that are formulated from historical data.