*1293 This appeal requires us to decide whether the district court misinterpreted our mandate in an earlier appeal and misapplied the defense of fair use, 17 U.S.C. § 107 , in a dispute between three academic publishers and Georgia State University about the University's practice of distributing to students digital excerpts of copyrighted works without paying the publishers. After a bench trial, the district court ruled that the publishers established a prima facie case that 48 digital excerpts infringed their copyrights but that the University prevailed on an affirmative defense of fair use for 43 of those excerpts. The publishers appealed. We upheld the district court's analysis of the first and fourth fair-use factors, including its finding that widespread unlicensed use of 31 excerpts for which licenses existed could cause substantial harm to the potential markets. But we reversed and remanded for the district court to correct specific errors in its analysis of fair use. One such error was the use of a mathematical formula to balance the four statutory fair-use factors. Another was the insufficient weight the district court gave to the severe threat of market substitution. On remand, the district court ruled that the University prevailed on its fair-use defense for 44 of the 48 excerpts. Contrary to our instructions, the district court again applied a mathematical formula to balance the factors. It also revisited its market-harm analysis for the 31 licensed excerpts and found that the threat of market harm supported fair use in all but six of the 48 instances. Although the district court did not abuse its discretion when it declined to reopen the record to allow the publishers to present new evidence about the availability of digital licenses, the district court misinterpreted our earlier decision and misapplied the statutory test of fair use. We affirm in part, vacate in part, and remand with instructions.

I. BACKGROUND

We divide the background in three parts. First, we describe the facts and the original proceedings in the district court. Second, we describe our decision in the earlier appeal. Third, we describe the proceedings on remand.

A. The Facts and the Original Proceedings in the District Court

Cambridge University Press, Oxford University Press, Inc., and Sage Publications, Inc. publish academic works. Cambridge Univ. Press v. Patton ( Cambridge II ), 769 F.3d 1232 , 1238 (11th Cir. 2014). The publishers "market their books to professors who teach at universities and colleges" so that the professors will "assign them as required reading" for their courses and "students will purchase them." Id . at 1238-39. The publishers also sell "licenses to use excerpts of [their copyrighted] works." Id . at 1276. In the past, professors commonly assigned-and students purchased-paper "coursepacks" of licensed excerpts. Id . at 1239. But it has *1294 become more common for universities to distribute digital excerpts electronically. Id . The publishers license users "to photocopy [and to] digitally reproduce portions of their works." Id. at 1240 . They offer such licenses, called "permissions," both directly and through the Copyright Clearance Center. Id .

Georgia State University, a public university in Atlanta, Georgia, provides several ways to distribute excerpts of copyrighted works to students. The University pays to use licensed excerpts in paper coursepacks that the University bookstore assembles and sells to students. Id . at 1241 & n.4. The University also maintains two systems for electronic distribution of course materials, "ERes" and "uLearn." Id . at 1239. Both programs enable University personnel to upload digital copies of excerpts to University servers and allow students enrolled in a course to download the excerpts. Id . Neither the University, see id. at 1238 , nor the students, see id. at 1240 , pay for the use of the digital excerpts. This approach is "popular." Id ."For example, during the Spring 2009 term, paper coursepacks were offered for only about fifteen courses, while instructors in hundreds of courses made readings available on ERes." Id.

In 2008, the publishers filed a complaint against officials of the University for direct, contributory, and vicarious copyright infringement, 17 U.S.C. §§ 106 , 501(a) ; see also Metro-Goldwyn-Mayer Studios Inc. v. Grokster, Ltd. , 545 U.S. 913 , 125 S.Ct. 2764 , 162 L.Ed.2d 781 (2005) (acknowledging limited third-party liability for copyright violations), and sought declaratory and injunctive relief. Cambridge II , 769 F.3d at 1241-42 & n.6. The publishers alleged a pattern and practice of distributing substantial unlicensed excerpts of their copyrighted works, and they sought to prove their claims based on a representative sample of infringements. Id. at 1241-42 . The University asserted an affirmative defense of fair use, 17 U.S.C. § 107 . Cambridge II , 769 F.3d at 1242 .

After a bench trial, the district court ruled that the University infringed the copyrights of the publishers in five instances. See Cambridge Univ. Press v. Becker ( Cambridge I ), 863 F.Supp.2d 1190 , 1363-64 (N.D. Ga. 2012). The district court found that the publishers established a prima facie case for 48 instances of infringement. See Cambridge II , 769 F.3d at 1237 . But it ruled that the University established its fair-use defense for 43 of the excerpts. Id . We summarize its conclusions as relevant to this appeal.

The Copyright Act enumerates four "factors to be considered" in finding that "the use made of a work in any particular case is a fair use" instead of an infringement. 17 U.S.C. § 107 . The first factor is "the purpose and character of the use, including whether [it] is of a commercial nature or is for nonprofit educational purposes." Id. § 107(1). The second factor is "the nature of the copyrighted work." Id. § 107(2). The third factor is "the amount and substantiality of the portion used in relation to the copyrighted work as a whole." Id. § 107(3). And the fourth factor is "the effect of the use upon the potential market for or value of the copyrighted work." Id. § 107(4).

The district court found that "factor one strongly favor[ed]" a finding of fair use for every excerpt because the University's copying was for nonprofit educational uses. Cambridge I , 863 F.Supp.2d at 1225 . It found that factor two "favor[ed]" fair use for every excerpt because the academic works were "informational in nature." Id . at 1226-27. It found that factor three favored fair use for 35 excerpts each of which was less than 10 percent or one chapter of the original work but favored infringement for 13 excerpts that exceeded *1295 those bounds. Cambridge II , 769 F.3d at 1251 . And it found that factor four "strongly favored" a finding of copyright infringement for 31 excerpts for which digital permissions were available and "favored" fair use for 17 excerpts for which the publishers failed to prove the availability of digital permissions. Id.

In its overall analysis of fair use for each excerpt, the district court "g[ave] each of the four factors equal weight" and "treat[ed] the four factors as a simple mathematical formula." Id. at 1260 . That is, the district court simply "add[ed] up" its findings whether each factor favored fair use or infringement for each excerpt. Id . It found that "fair use applied whenever at least three of the four factors favored [the University]." Id. at 1251 . But when it initially weighed the factors together in its overall analysis for some excerpts, the district court found a tie, with factor one strongly favoring fair use, factor two favoring fair use, factor three favoring infringement, and factor four strongly favoring infringement. See Cambridge I , 863 F.Supp.2d at 1260, 1263, 1268, 1274, 1278, 1358, 1362 . For seven such excerpts, the district court broke the ties by reconsidering the weight of the third factor, the fourth factor, or both. See id. at 1260, 1263-64, 1268-69, 1274, 1278, 1358-59, 1362-63 . It found that evidence of the amount of past permissions revenues "strengthen[ed]" the showing in favor of infringement on the fourth factor in four instances, id. at 1260, 1268, 1278, 1363 , but that it "undercut[ ]" the weight of the showing in two instances, id. at 1264, 1274 . It did not change any of its findings that factor four favored infringement.

The district court granted partial declaratory and injunctive relief to the publishers, but it awarded attorney's fees and costs to the University, 17 U.S.C. § 505 .

B. The Earlier Appeal

In the earlier appeal, we reversed, vacated, and remanded. Cambridge II , 769 F.3d at 1284 . As relevant here, we upheld the district court's analysis of factors one and four. See id. at 1267, 1279, 1281, 1283 . But we instructed the district court to correct its "erroneous application of factors two and three" and its errors in "[weighing and] balancing the four fair use factors" in its overall analysis. Id. at 1283 .

We held that the district court got two things right in Cambridge I : its analysis of the first factor and its analysis of the fourth factor. On the first factor, we agreed that "the nonprofit educational nature" of the University's use favored the fair-use defense. Id. at 1267 ; see also id. at 1283 ("The District Court did not err in holding that the first factor ... favors fair use."). But we cautioned that the University's "nontransformative" use, verbatim copying that served "the same intrinsic purpose" for which the works were originally published, id. at 1263 , made "the threat of market substitution ... significant," id. at 1267 .

We also held "that the District Court's analysis under the fourth factor was correct." Id. at 1279 ; see also id. at 1281 ("[T]he District Court did not err in its application of the fourth factor."); id. at 1283 ("With regard to the fourth factor ... the District Court did not err."). But we held that "the District Court erred by not affording the fourth factor additional weight in its overall fair use calculus." Id. at 1281 . Indeed, we found important errors in "the District Court's overarching fair use methodology." Id. at 1259 .

We held that the district court erred when it "g[ave] each of the four factors equal weight, essentially taking a mechanical 'add up the factors' approach," and that it should not have "treat[ed] the four factors as a simple mathematical formula."

*1296 Id. at 1260 . We explained that "a given factor may be more or less important ... under the specific circumstances of [a particular] case." Id. In this case, we emphasized that the district court "should have afforded [factor four] more significant weight in its overall fair use analysis" because "the threat of market substitution is severe." Id . at 1283 ; accord id. at 1275 , 1276 n.31, 1281. We also reminded the district court that "the four statutory factors may not be treated in isolation, one from another. All are to be explored, and the results weighted together, in light of the purposes of copyright." Id. at 1260 (alteration adopted) (quoting Campbell v. Acuff-Rose Music, Inc. , 510 U.S. 569 , 578, 114 S.Ct. 1164 , 127 L.Ed.2d 500 (1994) ).

We also held that the district court erred in its analyses of the second and third factors. Although the district court initially found that factor two favored fair use and weighed equally with the other three factors, we explained that "the District Court should have held that the second factor was neutral, or even weighted against fair use," for excerpts in which "evaluative, analytical, or subjectively descriptive material" predominated, and that "the second fair use factor is of relatively little importance in this case." Id. at 1270. As for the third factor, we held that the district court erred in creating "a 10 percent-or-one-chapter safe harbor" for fair use. Id. at 1275. We explained that the district court was required to assess each excerpt "individually, considering the quantity and the quality of the material taken." Id. And we explained that it was required to determine "whether that taking was excessive in light of" not only "the educational purpose of the use" but also "the threat of market substitution." Id. at 1275. We also mentioned the "ample precedents that explain that excessive verbatim copying weighs against fair use under factor three." Id. at 1274 n.30.

We vacated the orders that granted partial declaratory and injunctive relief to the publishers and awarded attorney's fees and costs to the University. Id . at 1283-84. Because the relief that the district court ordered was based on its erroneous fair-use analysis, we remanded for the district court to correct its analysis to be "consistent with [our] opinion." Id. at 1284.

C. The Proceedings on Remand

On remand, the publishers moved to reopen the record to introduce new evidence that digital licenses had been available for the 17 excerpts for which the publishers did not establish license availability at trial. The district court dismissed the motion without prejudice. The publishers later filed the declaration of an officer of the Copyright Clearance Center, Debra J. Mariniello, stating that digital licenses were available in 2009 for the 17 excerpts. The district court refused to admit the declaration and struck all references to it from the publishers' brief. Cambridge Univ. Press v. Becker ( Cambridge III ), No. 1:08-cv-1425, slip op. at 18, 2016 WL 3098397 (N.D. Ga. Mar. 31, 2016).

The district court changed its fair-use analysis in response to our decision in Cambridge II . In the new analysis, factor one "favor[ed] fair use in all cases," but "not strongly." Id . at 11 (internal quotation marks omitted); see also id. at 18. The district court did not revisit its earlier first-factor analysis for any of the excerpts. Instead, in its new fair-use analysis for each excerpt, the district court simply repeated the finding that "[f]actor one favors fair use." Id. at 24, 29, 34, 39, 43, 47, 53, 56, 60, 64, 70, 73, 79, 83, 88, 91, 95, 98, 101, 103, 106, 110, 114, 120, 124, 130, 133, 137, 140, 144, 146, 150, 155, 158, 161, 165, 170, 174, 179, 183, 187, 191, 196, 201, 205 ; see also id . at 20, 52, 69.

*1297 The district court changed its second-factor analysis for each excerpt. It found that factor two favored a finding of fair use in two instances, see id. at 131, 145, that it was neutral in 34 instances, see id. at 22, 26, 30, 44, 57, 61, 70, 74, 80, 89, 93, 100, 102, 105, 108, 111, 115, 121, 135, 139, 142, 151, 156, 163, 166, 176, 180, 184, 190, 193, 203, 206 ; see also id. at 52, 69, and that it favored infringement in 12 instances, see id. at 37, 40, 50, 54, 67, 85, 97, 125, 148, 159, 172, 198.

The district court also changed its earlier third-factor analysis for each excerpt. It found that factor three favored fair use for 37 excerpts, 34 of which had benefited from its earlier 10-percent-or-one-chapter safe harbor. See id. at 22, 26, 31, 44, 52, 54, 57, 61, 69, 71, 75, 80, 94, 100, 102, 105, 108, 111, 116, 121, 126, 132, 135, 142, 145, 148, 152, 156, 160, 164, 167, 172, 177, 180, 185, 193, 203. The district court also found that the third factor favored fair use for three excerpts that exceeded its earlier safe harbor. Compare id. at 57, 131-32, 176-77, with Cambridge I , 863 F.Supp.2d at 1273-74, 1310, 1346 . For two excerpts, the district court found that the price of the unpaid permissions "would have been excessive" and that the high price "allow[ed]" the district court "to look more favorably" on the unpaid use of a greater "quantity" of material. Cambridge III , slip op. at 116, 176 (emphases omitted).

The district court changed its fourth-factor analysis for the 31 excerpts for which it had originally found that "factor four strongly disfavored fair use." Id. at 11. Under its new analysis, the district court stated that factor four would only "initially favor [the publishers]" in those instances. Id. at 12. Then, based on its reading of Cambridge II , the district court offered the University three ways to prove that the fourth factor actually favored fair use in a particular instance:

[1] [T]he Court of Appeals held that Defendants may seek to prove that in fact, the demand for excerpts of a particular copyrighted work was so limited that repetitive unpaid copying of excerpts from that work would have been unlikely even if unpaid copying of excerpts was a widespread practice in colleges and universities. In such a case the actions of Defendants in using unpaid excerpts would not have caused substantial damage to the potential market for the copyrighted work to such a degree that Plaintiffs would lose the incentive to publish the work. [2] Defendants may also seek to prove that their actions (even assuming widespread availability of unpaid excerpts) did not substantially affect the value of the copyrighted work in 2009.... [3] Defendants may also seek to prove that the portion of the market captured by unpaid use is so slight that it would have had no effect on the author's or the Plaintiffs' decision to propagate the work in the first place.

Id. at 12-13. The district court suggested that the relevant evidence for these showings would generally come from "the records of permissions sales for excerpts from the book." Id. at 13.

The district court also adjusted its method of balancing the four factors. It explained that the "initial, approximate respective weights of the four factors" were "25% for factor one, 5% for factor two, 30% for factor three, and 40% for factor four." Id. at 14. "If a particular factor ha[d] noteworthy strength or weakness, the weight of that factor w[ould] be adjusted" in the district court's new analysis. Id. at 13.

Despite these changes to its analysis, the district court reached similar bottom-line results. It again found fair use for each of the 43 excerpts for which it had originally done so. And it found fair use in *1298 one instance where it had previously found infringement. Compare Cambridge I , 863 F.Supp.2d at 1359 (finding that the University's use of two chapters from The Power Elite , by C. Wright Mills, was unfair), with Cambridge III , slip op. at 200-01 (finding that the use of The Power Elite was fair).

The district court again found fair use for the vast majority of excerpts in large part because it reversed most of its original fourth-factor findings. On remand, the district court found that the fourth factor supported fair use in 42 of 48 instances. See Cambridge III , slip op. at 24, 28-29, 34, 42, 46, 52, 55, 60, 63, 69, 73, 83, 87, 91, 95, 98, 101, 103, 106, 109, 114, 119, 124, 129, 132, 136, 140, 143, 146, 149, 154, 158, 160, 164, 169, 174, 177, 182, 186, 190, 195, 200. The district court left unchanged only six of its 31 original findings that the threat of market harm weighed against fair use for excerpts for which licenses were available. See id. at 38, 52, 68, 78, 204, 210.

The district court granted partial declaratory and injunctive relief against the University. But it ruled that the University was the "prevailing side and [was] entitled to an award of costs and attorneys' fees." Id. at 212. After the Supreme Court decided Kirtsaeng v. John Wiley & Sons, Inc. , --- U.S. ----, 136 S.Ct. 1979 , 195 L.Ed.2d 368 (2016), the publishers moved for reconsideration of the fee award, but the district court denied the motion.

II. STANDARDS OF REVIEW

"Fair use involves both questions of law and questions of fact." Cambridge II , 769 F.3d at 1255 . "After a bench trial, we review a district court's conclusions of law de novo and a district court's factual findings for clear error." Id. (alterations adopted) (quoting Proudfoot Consulting Co. v. Gordon , 576 F.3d 1223 , 1230 (11th Cir. 2009) ). "We review de novo the district court's interpretation and application of this court's mandate" in an earlier appeal. Cox Enters., Inc. v. News-Journal Corp ., 794 F.3d 1259 , 1272 (11th Cir. 2015) (emphasis added). And we review the denial of a motion to reopen the record for abuse of discretion. See Elston v. Talladega Cty. Bd. of Educ. , 997 F.2d 1394 , 1405 (11th Cir. 1993).

III. DISCUSSION

We divide our discussion in two parts. First, we explain that the district court misinterpreted our mandate and misapplied the test of fair use. Second, we explain that the district court did not abuse its discretion when it declined to reopen the record. We affirm in part, vacated in part, and remand with instructions.

A. The District Court Erred when It Made Its New Findings of Fair Use.

The district court misinterpreted our mandate and misapplied the test of fair use on remand. In Cambridge II , we instructed the district court to correct its "erroneous application of factors two and three" and its errors in weighing the four factors in its overall analysis of fair use for each excerpt. Cambridge II , 769 F.3d at 1283 . But the district court exceeded this limited mandate, revisiting its 31 earlier findings that factor four favored infringement and reversing all but six of them. The district court again applied a mathematical formula in its overall analysis of fair use. And it erroneously considered the high price of permissions when it found that factor three favored fair use for two excerpts.

1. The District Court Erred when It Revisited Factor Four.

The publishers argue that the district court "failed ... to follow this Court's *1299 remand instructions" when it revisited its earlier findings that factor four favored infringement. The University responds that the district court correctly changed its approach to reflect our decision in Cambridge II . We agree with the publishers.

In deciding this appeal, we are bound by the decision in Cambridge II , just as the district court was bound to apply its mandate. The doctrine of the law of the case "preclude[s] courts from revisiting issues that were [already] decided." Schiavo ex rel. Schindler v. Schiavo , 403 F.3d 1289 , 1291 (11th Cir. 2005). In particular, "findings of fact and conclusions of law by an appellate court are generally binding in all subsequent proceedings in the same case in the trial court or on a later appeal." Transamerica Leasing, Inc. v. Inst. of London Underwriters , 430 F.3d 1326 , 1331 (11th Cir. 2005) (quoting Heathcoat v. Potts , 905 F.2d 367 , 370 (11th Cir. 1990) ). And under the mandate rule, which "is nothing more than a specific application of the 'law of the case' doctrine," id. (quoting Piambino v. Bailey , 757 F.2d 1112 , 1120 (11th Cir. 1985) ), "[a] district court[,] when acting under an appellate court's mandate, 'cannot vary it, or examine it for any other purpose than execution; or give any other or further relief; ... or intermeddle with it, further than to settle so much as has been remanded,' " Litman v. Mass. Mut. Life Ins. Co. , 825 F.2d 1506 , 1510-11 (11th Cir. 1987) (quoting In re Sanford Fork & Tool Co. , 160 U.S. 247 , 255, 16 S.Ct. 291 , 40 L.Ed. 414 (1895) ). "The trial court must implement both the letter and the spirit of the mandate, taking into account the appellate court's opinion and the circumstances it embraces." Piambino , 757 F.2d at 1119 (citations omitted).

The district court interpreted our earlier decision to instruct it to change its fourth-factor analysis for the 31 excerpts for which digital permissions were available, Cambridge III , slip op. at 11-13, but the district court misinterpreted our mandate. In its original decision, the district court found that the fourth factor "weigh[ed] heavily" against fair use in all 31 instances where the publishers made digital licenses available. See Cambridge I , 863 F.Supp.2d at 1239 . Our decision in Cambridge II upheld those 31 findings. We held that "the District Court did not err in its application of the fourth factor." Cambridge II , 769 F.3d at 1281 ; see also id. at 1279 ("[T]he District Court's analysis under the fourth factor was correct...."); id. at 1283 ("With regard to the fourth factor ... the District Court did not err."). The only error we identified in the district court's treatment of the fourth factor was that, in weighing and balancing the relative importance of the factors, it undervalued the "severe" threat of market harm posed by the University's "nontransformative" copying. Id. at 1283 ; see also id. at 1267, 1275, 1281 .

We did not instruct the district court to revisit its earlier findings that factor four favored infringement when we instructed it to reweigh the factors on remand. Indeed, we explained that, "[a]lthough ... the District Court's method for weighing the four factors against one another was erroneous, this does not mean that the District Court's reasoning under each of the four factors [was] also necessarily flawed." Id. at 1260 . And we repeatedly held that, although it should have weighed both factor one and factor four differently, the district court "did not err" in its application of those two factors. See id. at 1283 . On remand, the district court correctly declined to revisit its first-factor analysis for any of the excerpts. It followed our instruction to change only how "strongly" factor one weighed in favor of fair use in its overall analysis for each excerpt. See Cambridge III , slip op. at 11, 18. In contrast, the district court changed all but six of its original findings that factor four *1300 favored infringement instead of giving those findings "more significant weight in its overall fair use analysis," Cambridge II , 769 F.3d at 1283 .

The University contends that the district court correctly revisited its earlier findings that factor four favored infringement because Cambridge II announced a new standard of market harm, but the University misreads our earlier decision. It stresses that we described the "central question under the fourth factor" as whether a use "would cause substantial economic harm such that allowing it would frustrate the purposes of copyright by materially impairing [the publishers'] incentive to publish the work." Id. at 1276 (emphasis omitted). But we used this description of market harm-which merely echoed longstanding precedent, see Campbell , 510 U.S. at 590 , 114 S.Ct. 1164 ; Peter Letterese & Assoc., Inc. v. World Inst. of Scientology Enter. , Int'l , 533 F.3d 1287 , 1315 (11th Cir. 2008) -to explain that the earlier "analysis under the fourth factor was correct." Cambridge II , 769 F.3d at 1279 .

In short, we held in Cambridge II that the district court's original fourth-factor analysis was correct. That holding precluded the district court from revisiting the fourth factor in Cambridge III . On remand, the district court must reinstate its original findings that the fourth factor strongly disfavors fair use for the 31 excerpts for which the publishers proved the availability of digital licenses.

2. The District Court Erred when It Weighed and Balanced the Factors in Its Overall Calculus of Fair Use for Each Excerpt.

The publishers argue that the district court erred by again "appl[ying] a mathematical formula" in its overall analysis of fair use for each excerpt. The University contends that, "[a]lthough the district court provided approximate initial weights of the four factors at the outset of its remand decision," it adjusted that formula in its overall analysis for each excerpt. We again agree with the publishers.

In Cambridge II , we identified two distinct ways in which, in its original analysis, the district court failed to recognize that "a given factor may be more or less important ... under the specific circumstances of [a particular] case." Cambridge II , 769 F.3d at 1260 . First, we explained that "the District Court erred in giving each of the four factors equal weight." Id. Second, we explained that the district court erred "in treating the four factors as a simple mathematical formula," which we also described as an "arithmetic approach." Id.

On remand, the district court corrected the first of these errors but again committed the second. The district court assigned "initial, approximate respective weights of the four factors as follows: 25% for factor one, 5% for factor two, 30% for factor three, and 40% for factor four." Cambridge III , slip op. at 14. Although the district court heeded our instructions in Cambridge II when it recognized that some factors are more important than others, it failed to break free of its erroneous "arithmetic approach" and to give each excerpt the holistic review the Act demands.

As the Supreme Court has explained and as we reiterated in Cambridge II , "the four statutory factors may not be treated in isolation, one from another. All are to be explored, and the results weighed together, in light of the purposes of copyright." Cambridge II , 769 F.3d at 1260 (alteration adopted) (quoting Campbell , 510 U.S. at 578 , 114 S.Ct. 1164 ). We emphasized that "fair use is not a mechanical determination," id. (alteration adopted) (quoting Wright v. Warner Books, Inc. , 953 F.2d 731 , 740 (2d Cir. 1991) ), and that a court *1301 must "weigh[ ] ... the four factors in light of the facts of a given case," id. at 1259.

To be sure, the district court described its arithmetic weights as "initial" and "approximate," and it stated that it would "adjust[ ]" them when it found a "noteworthy strength or weakness" among the factors. Id. at 13. But the district court made such adjustments only four times, each time to bolster the importance of the third factor's weighing against fair use. See id. at 38, 68, 140, & 201. And, on those four occasions, the district court did nothing to adjust the other factors in the overall fair-use calculus. We conclude that the district court's quantitative rubric was an improper substitute for a qualitative consideration of each instance of copying in the light of its particular facts.

The district court failed to give each excerpt the holistic review that the Act demands. On this remand, the district court must not apply a mathematical formula at any step of its analysis. We reiterate our holding in Cambridge II that "the fourth factor looms large in the overall fair use analysis" for each excerpt in this appeal, 769 F.3d at 1275 , but we instruct the district court to evaluate the four factors qualitatively, not quantitatively, and to take care to consider them holistically "in light of the purposes of copyright." Campbell , 510 U.S. at 578 , 114 S.Ct. 1164 .

3. The District Court Erred when It Considered the Cost of Purchasing Licenses in Finding that the Third Factor Favored Fair Use.

The district court twice erred in applying the third factor of the statutory test of fair use when it considered whether the cost of licensing was "excessive" in the light of the publishers' "marginal cost for authorizing digital copies ..., [which] would not vary no matter how many digital copies were authorized." Cambridge III , slip op. at 116, 176. The district court reasoned that high prices "allow[ed] [it] to look more favorably on the quantity of [the University's] use than it otherwise would." Id. at 116, 176 (emphases omitted). In these two instances, the district court deviated from the language of the Act.

The third factor of the statutory fair-use test is "the amount and substantiality of the portion used in relation to the copyrighted work as a whole." 17 U.S.C. § 107 (3). This provision of the Act does not direct courts to consider the price of the unpaid use. If it did, then the district court's reasoning could tilt the third factor in favor of fair use even in cases of extensive verbatim copying. After all, it is always the case that a publisher's "marginal cost for authorizing digital copies would be virtually nil, and would not vary no matter how many digital copies were authorized." Cambridge III , slip op. at 116, 176. When we instructed the district court to correct its analysis under the third factor on remand, we did not include this consideration. See Cambridge II , 769 F.3d at 1275 . The district court erred when it twice considered the price of the unpaid use as relevant to the third factor.

B. The District Court Did Not Abuse Its Discretion when It Declined To Reopen the Record.

The publishers argue that they should have been permitted to supplement the record on remand with evidence that digital permissions were available for the 17 works for which it did not previously provide this evidence. They argue that this evidence "was offered to fill a purported evidentiary gap that existed through no fault of [the publishers]" because "[t]he district court failed to put [them] on notice" of the need to prove license availability at trial. They stress that the availability of digital licenses took on greater importance *1302 after our decision in Cambridge II clarified the importance of factor four.

These arguments are unpersuasive. The question whether to reopen the record on remand is "left to the sound discretion of the trial court." Jones & Laughlin Steel Corp. v. Pfeifer , 462 U.S. 523 , 551, 103 S.Ct. 2541 , 76 L.Ed.2d 768 (1983). And the district court was entitled to conclude that it would be inappropriate to admit new evidence of license availability that was "offered years after the close of the trial and entry of the judgment and after review by the Court of Appeals." Cambridge III , slip op. at 18. Nor can the publishers excuse their delay by asserting that they were ignorant of the importance of this evidence. Indeed, they similarly "complain[ed]" in the earlier appeal "that they were not notified at trial that they were required to show license availability for 2009 in particular." Cambridge II , 769 F.3d at 1278 n.33. And we responded that the publishers "were-or should have been-aware that the relevant evidentiary question was whether a market existed at the time of the purported infringements." Id . The district court was not required to reopen the record to allow the publishers to introduce new evidence on this topic.

IV. CONCLUSION

We AFFIRM in part, VACATE in part, and REMAND for proceedings consistent with this opinion. We affirm the order denying the publishers' request to reopen the record, but we vacate the judgment entered on remand. The district court must reinstate its earlier findings that factor four strongly disfavors fair use for 31 of the 48 excerpts. The district court must eschew a quantitative approach to the weighing and balancing of the fair-use factors and give each excerpt the holistic, qualitative, and individual analysis that the Act demands. And the district court must omit any consideration of price from its analysis of the third factor. Because the district court's award of attorney's fees and costs was based on its erroneous fair-use analysis, we also VACATE that award and the underlying determination that the University is the prevailing party.