opinion: Darrell TAYLOR, d/b/a Darrell Taylor Topographic Charts, Plaintiff-Appellee, v. Joseph B. MEIRICK, d/b/a Lakes Illustrated, Defendant-Appellant.
  No. 83-1098.
  United States Court of Appeals, Seventh Circuit.
  Argued May 9, 1983.
  Decided July 7, 1983.
  Rehearing Denied Aug. 9,1983.
  James Van Santen, Hill, Van Santen, Steadman, Chiara & Simpson, Chicago, 111., for defendant-appellant.
  Curtis Eric Edlund, Park Ridge, 111., for plaintiff-appellee.
  Before CUMMINGS, Chief Judge, POS-NER, Circuit Judge, and ROSENN, Senior Circuit Judge.
  Hon. Max Rosenn of the Third Circuit, sitting by designation.
  POSNER, Circuit Judge.
  This appeal from a judgment for the plaintiff in a suit for copyright infringement raises questions regarding the statute of limitations in the Copyright Act of 1976 and the methods for determining damages in copyright and, by implication, other business-tort cases.
  In 1974 the plaintiff, Taylor, doing business as Darrell Taylor Topographic Charts, copyrighted maps for use by fishermen of three Illinois lakes. The validity of his copyrights is conceded, as is the fact that the defendant, Meirick, doing business as Lakes Illustrated, copied the maps in 1976 and 1977 without Taylor’s authorization. But this suit was not filed till May 8,1980, and Meirick argues that it is barred by the three-year statute of limitations in the Copyright Act of 1976 (which is conceded to govern this case). The magistrate who tried the case with the consent of the parties disagreed, and awarded Taylor damages of $22,700 and attorney’s fees of $10,-000.
  The Copyright Act provides that “no civil action shall be maintained under the provisions of this title unless it is commenced within three years after the claim accrued.” 17 U.S.C. § 507(b). If Meirick had quit selling the infringing maps before May 8,1977, and had taken reasonable steps by then to get them back from his dealers before the maps were resold to consumers, this suit would be time-barred even if — as happened — the maps were still being sold years later by dealers who had bought them from Meirick before May 8, 1977, but had not gotten, or had ignored, the message and continued selling them to the public. See Mount v. Book-of-the-Month Club, Inc., 555 F.2d 1108, 1110 (2d Cir.1977); Maloney v. Stone, 171 F.Supp. 29, 32 (D.Mass.1959) (Wyzanski, J.). But knowing that he had placed infringing copies in the hands of his dealers Meirick could not sit on his hands while they sold them. A tortfeasor has a duty to assist his victim. The initial injury creates a duty of aid and the breach of the duty is an independent tort. See Restatement (Second) of Torts § 322, Comment c (1965). This principle applies to a statutory tort such as copyright infringement. So if Meirick failed to take reasonable steps to prevent infringement by his dealers he became a participant in their infringements. Since there was abundant evidence that retailers were selling the infringing maps even after the trial began, it was Meirick’s burden to prove that the selling had continued despite his reasonable efforts to recall the maps. The magistrate found that Meirick had not carried this burden and we cannot say that her finding was clearly erroneous.
  And this assumes that Meirick really did stop selling the maps before May 8, 1977, as he testified. The magistrate believed Taylor’s testimony that he ordered and received infringing maps from Lakes Illustrated in 1979. This testimony was vague, self-serving, and uncorroborated; the maps themselves were not put into evidence; and Taylor’s general credibility as a witness was undermined by the testimony he gave about the costs he would have incurred in selling more maps had there been no infringement — of which more anon. But it makes no difference whether Meirick himself was selling infringing maps in 1979, since his retailers were and he made insufficient efforts to prevent them from doing so.
  The irrelevance of when Meirick himself stopped infringing is further shown by the uncontested fact that Taylor did not learn of the infringements until 1979. Often, whether or not the defendant has done anything to conceal from the plaintiff the existence of the cause of action, the statute of limitations is tolled until the plaintiff learned or by reasonable diligence could have learned that he had a cause of action. Consider liability for defective products: although the tort is complete when the victim is injured, if the etiology of the injury is mysterious — as is often the case with injuries from drugs and chemicals — the tendency in modern law is to toll the statute of limitations until the victim could reasonably have discovered the cause of his woe. See, e.g., Needham v. White Laboratories, Inc., 639 F.2d 394, 399 (7th Cir.1981); Ricciuti v. Voltarc Tubes, Inc., 277 F.2d 809, 813 (2d Cir.1960). The approach is not limited to personal-injury cases. For example, Pollock v. Hafner, 108 Ill.App.3d 410, 64 Ill.Dec. 156, 439 N.E.2d 85 (1982), applies it to architectural malpractice. Although we cannot find a copyright case on point, a similar principle may apply in such cases. The fact that a publisher loses sales to a competitor is not in itself a clue to copyright infringement, since there is vigorous competition among copyrighted works. So we doubt that every time the sales of a publication dip, the publisher must, to preserve his right to sue for copyright infringement, examine all of his competitors’ publications to make sure none is infringing any of his copyrights. Probably it should be enough to toll the statute of limitations that a reasonable man would not have discovered the infringement; and there is no evidence that Taylor was unreasonable in failing to discover the infringing maps before 1979.
  In any event, there is no doubt that the copyright statute of limitations is tolled by “fraudulent concealment” of the infringement. See, e.g., Prather v. Neva Paperbacks, Inc., 446 F.2d 338, 340-41 (5th Cir.1971); Charlotte Telecasters, Inc. v. Jefferson-Pilot Corp., 546 F.2d 570, 573-74 (4th Cir.1976). Although these are not decisions under the 1976 Copyright Act, the statute of limitations in that Act was taken without material change from the one that had been added to the previous Act in 1957. See Act of Sept. 7, 1957, Pub.L. 85-313, 71 Stat. 633; H.Rep. No. 1476, 94th Cong., 2d Sess. 164 (1976), U.S.Code Cong. & Admin. News 1976, p. 5659. The term “fraudulent concealment” implies active misconduct, but there was that here. Meirick had put his own copyright notice on his copies of Taylor’s maps. This was calculated to throw purchasers, and Taylor himself, off the scent; only a close inspection of Meirick’s maps would have revealed that they were copies. Modem maps of the same area resemble each other closely — it would be most unsettling if, like medieval maps, they did not! The features that made Taylor’s maps copyrightable — and we repeat that the validity of his copyrights is not contested — were subtle and would easily escape notice with another’s name affixed as copyright holder.
  Although many cases state that mere ignorance of a cause of action does not toll the statute of limitations, in context these statements invariably mean only that the plaintiff has a duty of diligence: it is not enough that he did not discover he had a cause of action, if a reasonable man in his shoes would have. See, e.g., Campbell v. Upjohn Co., 676 F.2d 1122, 1127 (6th Cir. 1982). The significance of fraudulent concealment (as where price-fixers take steps to conceal their conspiracy from buyers) is that it frustrates even diligent inquiry. In a case such as this, where even if there had been no active concealment by the tortfeasor the injured party would have had no reason to suspect that he was the victim of a tort, there may be no duty of inquiry at all. But even if there is, Meirick’s action in putting his own copyright on maps he had copied from Taylor, an action calculated to obstruct any inquiry that Taylor might have made into the reasons for his losing sales, would toll the statute of limitations.
  The next question is whether Taylor can complain about infringing sales that occurred more than three years before he sued. He invokes the supposed rule that only the last infringing act need be within the statutory period. Oddly, considering how often the issue must arise, we have found little mention of such a rule in the cases. Two district court cases uphold the rule (none reject it), see Baxter v. Curtis Industries, Inc., 201 F.Supp. 100 (N.D.Ohio 1962); Cain v. Universal Pictures Co., 47 F.Supp. 1013, 1018 (S.D.Cal.1942), but Cain is not very explicit. However, there is no doubt of the rule’s validity if it is regarded not as something peculiar to copyright law but as the application to that law of the general principle that the statute of limitations does not begin to run on a continuing wrong till the wrong is over and done with (this seems to be the approach in Cain). See, e.g., Berkey Photo, Inc. v. Eastman Kodak Co., 603 F.2d 263, 295 (2d Cir.1979); Railing v. United Mine Workers of America, 429 F.2d 780, 783 (4th Cir.1970).
  The principle strikes a balance between the plaintiffs interest in being spared having to bring successive suits, and the two distinct interests, Gates Rubber Co. v. USM Corp., 508 F.2d 603, 611 (7th Cir.1975), that statutes of limitations serve. One is evidentiary — to reduce the error rate in legal proceedings by barring litigation over claims relating to the distant past. The other is repose — to give people the assurance that after a fixed time they can go about their business without fear of having their liberty or property taken through the legal process. Apart from the harmful effect of uncertainty on planning, it is more painful to lose what you have come to think of as your own than it is gratifying to get back something you wrote off many years ago and have grown accustomed to doing without. See Holmes, The Path of the Law, 10 Harv.L.Rev. 457, 477 (1897); cf. Hirshleifer, Price Theory and Applications 61 (1976). When the final act of an unlaw- ' ful course of conduct occurs within the statutory period, these purposes are adequately served, in balance with the plaintiff’s interest in not having to bring successive suits, by requiring the plaintiff to sue within the statutory period but letting him reach back and get damages for the entire duration of the alleged violation. Some of the evidence, at least, will be fresh. And the defendant’s uncertainty as to whether he will be sued at all will be confined to the statutory period. His uncertainty about the extent of his liability may be greater, but that is often true in litigation.
  Meirick’s infringement was a continuing wrong. He copied Taylor’s maps in 1976 and 1977 and either sold copies till 1979, which was well within three years of the bringing of this suit, or at least became party to infringements by his dealers which continued till after the suit was filed. The initial copying was not a separate and completed wrong but simply the first step in a course of wrongful conduct that continued till the last copy of the infringing map was sold by Meirick or with his connivance.
  Alternatively, since Taylor was unaware of the infringements until 1979, and (in part because of Meirick’s efforts at concealment) could not have been expected to discover them earlier by the exercise of reasonable vigilance, either of the tolling principles discussed earlier would allow him to collect damages for acts of infringement more than three years in the past, at least if he acted promptly once he discovered them, and he did. He notified Meirick of the infringement as soon as he discovered it in 1979 and Meirick sent back a soothing and, the magistrate thought, false assurance that the infringing plates had been destroyed long ago. Taylor sued promptly when he discovered that infringing maps were still being sold with the complicity of, and possibly directly by, Meirick.
  The next question is whether damages were calculated correctly. The $22,700 in damages that the magistrate awarded to Taylor consisted of two items — Taylor’s losses ($19,300), and Meirick’s profits ($3,300) (we have rounded off all figures to the nearest $100). The losses were calculated as follows. In 1974 Taylor’s gross revenue from sales of the maps infringed by Meirick beginning in 1976 was $4,700 and in 1975 it was $4,900. The average revenue for the two years — the pre-infringement period — was thus $4,800. Taylor assumed that but for the infringement he would have continued to gross this amount every year throughout the damage period, which is to say till the end of 1980 (he dropped his original contention that Meirick had infringed beyond then). This made his projected revenue for the five-year period 1976 through 1980, $23,900. His actual revenue was $4,600, and $19,300 is the difference.
  Taylor calculated Meirick’s profits from selling the infringing maps as follows. During the damage period Meirick sold 150 different maps; the three infringing maps represent 2 percent of this figure; 2 percent of Meirick’s gross profits for the years 1976 through 1980 as shown on his federal income tax returns for those years is $3,300.
  The damage calculations contain a number of serious errors which bring to mind Judge Friendly’s cautionary remarks in a related context about “an array of figures conveying a delusive impression of exactness.” Herman Schwabe, Inc. v. United Shoe Machinery Corp., 297 F.2d 906, 912 (2d Cir.1962). To begin with, there is double counting. Taylor’s basic damage theory was that Meirick had sold maps on which Taylor would have made $19,300 in profits, yet the $3,300 figure is Taylor’s estimate of Meirick’s profits on the same maps. The Copyright Act of 1976 entitles the copyright owner “to recover the actual damages suffered by him or her as a result of the infringement, and any profits of the infringer that are attributable to the infringement and are not taken into account in computing the actual damages.” 17 U.S.C. § 504(b). But the clause we have italicized may not be ignored. If the profits the owner would have made but for the infringement are equal to the profits the infringer made by selling the copyrighted item, and the owner proves up his lost profits, the “not taken into account” clause, eliminating an ambiguity with regard to the possibility of double recovery under the 1909 Act, see Sid & Marty Krofft Television Productions, Inc. v. McDonald’s Corp., 562 F.2d 1157, 1175-77 (9th Cir.1977), bars the owner from receiving an additional award of damages based on the infringer’s profits. See H.Rep. No. 1476, supra, at 161; 3 Nimmer on Copyright, ¶ l4.01[A], at p. 14-4 (1982); but see Miller v. Universal City Studios, Inc., 650 F.2d 1365, 1376 (5th Cir. 1981) (dictum).
  It is true that if the infringer makes greater profits than the copyright owner lost, because the infringer is a more efficient producer than the owner or sells in a different market, the owner is allowed to capture the additional profit even though it does not represent a loss to him. It may seem wrong to penalize the infringer for his superior efficiency and give the owner a windfall. But it discourages infringement. By preventing infringers from obtaining any net profit it makes any would-be infringer negotiate directly with the owner of a copyright that he wants to use, rather than bypass the market by stealing the copyright and forcing the owner to seek compensation from the courts for his loss. Since the infringer’s gain might exceed the owner’s loss, especially as loss is measured by a court, limiting damages to that loss would not effectively deter this kind of forced exchange. This analysis also implies that some of the “windfall” may actually be profit that the owner would have obtained from licensing his copyright to the infringer had the infringer sought a license.
  But Taylor presented no evidence that selling the infringing maps was more profitable to Meirick than selling more of the original maps would have been to himself. True, he would not have had to present such evidence if he were seeking to recover Meirick’s profits as the sole item of damages, as the statute permitted him to do. But since he was trying to recover both his lost profits and Meirick’s profits, he had to show what part of Meirick’s profits he, Taylor, would not have earned had the infringement not occurred; in other words, he had to subtract his profits from Meirick’s. He did not do this; and the fact that his estimate of Meirick’s profits is far lower than his estimate of his own lost profits shows the radical impossibility of what he was trying to do.
  All we have shown so far is that Taylor must choose between $19,300 and $3,300 (an easy choice). But in fact neither figure was computed correctly. $19,300 is, at best, an estimate of lost sales, not of lost profits. Even as an estimate of lost sales it is questionable. The use of before-and-after comparisons is a hallowed but coarse-grained and often abused method of estimating lost sales. The fact that Taylor’s sales of the three maps that were infringed fell by 45 percent in 1976, the first infringing year, and a further 88 percent the next year, is only slight evidence of the extent of the loss in sales caused by the infringement; post hoc ergo propter hoc is a naive theory of causation. But the declines are some evidence, made a little stronger by the fact that the bigger decline was in the second year of the infringement, as one would expect: for while we do not know when in 1976 Meirick began selling the infringing maps, presumably it was not on the first of the year; and even if it was, it would take a while to divert substantial sales from Taylor. The evidence of declining sales was enough to shift to Meirick the burden of producing some contrary evidence, which he attempted but failed to do. He showed there was a substantial price difference between Taylor’s maps and his, which could be powerful evidence that the maps were in different markets and therefore that Meirick’s infringing was unlikely to hurt Taylor — except that the price difference works in the wrong direction for Meirick. His maps sell at retail for $2.50 to $3, Taylor’s for $10. People who were buying Taylor’s maps for $10 would be delighted to be able to buy the same map for a fourth as much — though it would not be quite the same map; part of the difference in price reflects a difference in the quality of the materials used in making the maps. Meirick also showed that Taylor drastically reduced his advertising in 1977. If this had happened in 1976 it might be significant. But after Taylor’s sales nose-dived in 1976 it was only prudent for him to curtail his advertising.
  So we think Taylor adequately though not amply demonstrated his lost sales. But a loss of revenue is not the same thing as a loss of profits. If you sell less of your product you will have lower costs, and the cost savings is a gain that must be offset against the loss of revenues in computing lost profit. Taylor did not subtract a penny from the $19,300 that he estimated to be his lost sales revenues. This implies, improbably, that he could have made all the lost sales at zero cost. He did testify that he had printed a large batch of the maps before 1976 and would not have had to print any more in order to make the lost sales, but we find this testimony unbelievable. If it were true, either Taylor destroyed the maps when he found out that he could not sell them (and there is no evidence of that), or he still has them (but there is no evidence of that either) and they are an asset on which he can still realize a profit — now that Meirick has finally ceased infringing — which would have to be subtracted from his sales revenue in estimating his net loss from infringement. When a plaintiff cpntends that lost sales revenue would have been all profit, the contention is sufficiently improbable to require him to come forward with substantiating evidence, which Taylor failed to do. As we do not know what fraction of the $19,300 in lost sales represents profit, we cannot uphold the award of damages in this amount or even impose a remittitur.
  Taylor’s alternative method of computation, which yielded a figure of $3,300 from Meirick’s lost profits, was also deficient. Though labeled “gross profits,” the figure is actually sales revenue minus cost of goods sold. Other costs, such as commissions, rent, telephone, and repairs, were not deducted although clearly shown on Meirick’s income tax returns. When they are deducted it appears that Meirick had no net profits for the years in question. But it is possible, indeed likely, that he would have incurred at least some of these costs (rent, for example, and basic phone service) even if he had not sold the infringing maps; and to the extent this is so, his “gross profits” were real profits in the only sense relevant to damages calculation — they were his residual income after all costs necessary to generate the income were subtracted. Costs that would be incurred anyway should not be subtracted, because by definition they cannot be avoided by curtailing the profit-making activity. This principle is well established in the treatment of overhead costs in calculating damages for breach of contract. See Farnsworth, Contracts 854-55 (1982).
  Although subtracting just one cost item from the revenues shown on Meirick’s income tax returns was an extremely crude method of estimating Meirick’s total net profits, it satisfied Taylor’s burden of production. After all, he did subtract from the “gross profits” figure the cost most likely to vary with Meirick’s output, and it was up to Meirick to show what if any overhead items were really variable costs too. It is too much to ask a plaintiff who has proved infringement also to do the defendant’s cost accounting.
  But we cannot accept Taylor’s estimate that 2 percent of Meirick’s total net profits is allocable to the infringing maps. First, there is no evidence that the infringing maps represented 2 percent of Meirick’s total map sales as distinct from 2 percent of the titles in his inventory of maps. Second, half of Meirick’s business consists of sales of something called a “pH meter,” and Taylor lumped the profits on those sales in with the profits of Meirick’s map business to get the figure that he multiplied by 2 percent in order to estimate Meirick’s profits on the infringing maps. True, “in establishing the infringer’s profits, the copyright owner is required to present proof only of the infringer’s gross revenue, and the infringer is required to prove his or her deductible expenses and the elements of profit attributable to factors other than the copyrighted work.” 17 U.S.C. § 504(b). But all that means is that Taylor could have made out a prima facie case for an award of infringer’s profits by showing Meirick’s gross revenues from the sale of the infringing maps. It was not enough to show Meirick’s gross revenues from the sale of everything he sold, which is all, really, that Taylor did. If General Motors were to steal your copyright and put it in a sales brochure, you could not just put a copy of General Motors’ corporate income tax return in the record and rest your case for an award of infringer’s profits.
  The last issue is whether the magistrate erred in awarding Taylor his attorney’s fees. The Copyright Act of 1976 authorizes the trial court “to award a reasonable attorney’s fee to the prevailing party ....” 17 U.S.C. § 505. There is no suggestion that the $10,000 the magistrate awarded is unreasonably high in relation to the effort expended by Taylor’s counsel and there was abundant evidence that the infringement was willful, so the magistrate acted well within her discretion in awarding this sum to Taylor. The only question is whether it should make a difference that we are reversing the award of damages to Taylor and remanding for a new trial on damages. This may not make a difference in the end. The magistrate granted Taylor an injunction which Meirick has not even appealed, and attorney’s fees have long been awarded in cases of willful copyright infringement even though no actual damages were awarded or even sought. See, e.g., Shapiro, Bernstein & Co. v. Jerry Vogel Music Co., 161 F.2d 406, 410 (2d Cir. 1946); Twentieth Century Music Corp. v. Frith, 645 F.2d 6 (5th Cir.1981) (per curiam). However, since the amount of damages awarded could affect the magistrate’s exercise of her discretion in awarding fees, see 3 Nimmer on Copyright, supra, ¶ 14.10[C], at pp. 14-68 to 14-69, we shall vacate the fee award with directions to her to make a new award at the conclusion of the proceedings on remand.
  In summary, we affirm the magistrate’s decision on liability but reverse on damages, except that we uphold her finding that Taylor lost sales (not profits) of $19,300. We vacate the award of attorney’s fees, remand the case for a new trial limited to damages, and make no award of costs in this court.