Don't limit yourself to using four different numbers. Think of a 3-4-5 right triangle and the Pythagorean Theorem. 9 + 16 = 25. If we can get two of the numbers summing to 9 and two summing to 16, then all four numbers will add to 25. The trick is to use the number 8 three times and the number 1 once. There are actually an infinite number of solutions to this problem (multiples of 3-4-5).

Incidentally, Dudeney's solution was 386 and 2,114 and 3,970 and 10,430. He offers no explanation for how he arrived at this answer. It would be interesting to learn how he did it long before computers. How did he even know it was possible?

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