EDWIN F. HUNTER, Jr., District Judge:

Owen Cook was convicted by a jury on five counts of submitting false and fictitious claims on government contracts in violation of 18 U.S.C. § 287. Cook was sentenced to consecutive terms of five years imprisonment under two counts of the indictment and fined a total of $25,000 on the remaining three counts. He appeals his conviction asserting numerous errors in the trial below.

On May 1,1975 appellant Cook purchased Mikelco, Inc. from Carl Dunlap. The company, which was located in Bandera, Texas, was engaged in overhauling and repairing aerospace ground equipment for the United States Air Force under government contract. Cook renamed the company Mikelco of Bandera, Inc.

On May 12, May 29, June 12, June 24, and September 15, 1975, allegedly false invoices were submitted to the government pursuant to the contract. The scheme generally involved charging the government for ostensibly new parts that actually had been stolen from equipment previously sent for repair. The company also overbilled for legitimate parts and labor or created invoices from fictitious companies indicating purchases never actually made.

Appellant Cook raises five main issues on appeal. After careful consideration, finding no merit in any of the assignments of error, we AFFIRM.

I. SUFFICIENCY OF THE INDICTMENT

As his first ground for reversal, appellant argues that the indictment is insufficient because it fails to allege that he acted willfully. In United States v. Beasley, 550 F.2d 261 (5th Cir.), cert, denied, 434 U.S. 938, 98 S.Ct. 427, 54 L.Ed.2d 297 (1977), the defendant therein advanced the same argument that willfulness is an essential to a false claim violation under 18 U.S.C. § 287. While not reaching the issue of whether willfulness is a necessary element because the concurrent sentence doctrine rendered the issue moot, this court did discount the same authority the appellant now offers to support his contention:

Appellants rely heavily upon United States v. Mekjian, 505 F.2d 1320 (5th Cir. 1975) and Johnson v. United States, 410 F.2d 38 (8th Cir. 1968), cert, denied 396 U.S. 822, 90 S.Ct. 63, 24 L.Ed.2d 72 (1969). These cases do not provide the comfort sought. Mekjian arose under 18 U.S.C. § 1001, where both willfully and knowingly are required by the words of the statute. In Johnson the court seems to be holding that when one is charged with knowingly making a false claim and evidence is adduced showing knowledge of the requirements of the contract with the governmental agency and the complete disregard for such, and thereafter the jury is properly charged regarding the necessity of the government proving that such claims were false, fictitious and fraudulent and made knowingly and intentionally, “willfulness” has been established.

Id. at 270, n. 12.

Appellant does correctly point out that 18 U.S.C. § 1001, a companion statute to 18 U.S.C. § 287, expressly requires willfulness. However, United States v. Bramblett, 348 U.S. 503, 75 S.Ct. 504, 99 L.Ed. 594 (1955), wherein an exhaustive analysis of the origin and history of changes in §§ 287 and 1001 is developed, indicates that section 1001 *575 should not be employed to construe section 287’s requirements. Additionally, Congress’ inclusion of the term “willfully” in section 1001 while omitting the term in section 287 suggests an intent to exclude this state of mind from the essential elements of the latter offense. This judicial assessment of legislative intent is substantiated by the fact that the congressionally approved Form 10 of the Federal Rules of Criminal Procedure designed for section 287 indictments omits willfulness as an element of the offense. The indictment in the instant case in all pertinent respects substantially follows that form.

Alternatively, appellant Cook argues that even if willfulness is not required to validly charge the false claim offense, his being charged with “causing” the submission of the false claims activates 18 U.S.C. § 2 in which willfulness is an element. 18 U.S.C. § 2 is merely a codification of the law of principals. An individual can be convicted as an aider and abettor of an offense without being indicted in those terms, United States v. Young, 527 F.2d 1334 (5th Cir. 1976), and can be convicted as a principal even though the evidence establishes that he has only aided and abetted. United States v. Trollinger, 415 F.2d 527 (5th Cir. 1969). Persons accused only of causing another to do an illegal act need not be formally charged under § 2 and an indictment is sufficient if it alleges the criminal intent required for the substantive offense. Londos v. United States, 240 F.2d 1 (5th Cir.), cert, denied, 353 U.S. 949, 77 S.Ct. 860, 1 L.Ed.2d 858 (1957). Since the court finds that willfulness is not an essential element of 18 U.S.C. § 287, appellant’s first attack must fail.

II. VARIANCE BETWEEN THE INDICTMENT AND THE PROOF

Appellant alleges that there was a variance between the indictment and proof because the government at trial proved that the fraudulent claims were sent to the General Services Administration in Dallas where they were checked and certified before being forwarded to the United States Air Force for payment. On the other hand, the indictment charges that the claims were submitted for payment to the United States Air Force, an agency of the United States. The appellant therefore maintains that the government should have included “through GSA” in the indictment.

It is clear that not every variance is fatal to an indictment. Berger v. United States, 295 U.S. 78, 55 S.Ct. 629, 79 L.Ed. 1314 (1935); United States v. Lambert, 501 F.2d 943 (5th Cir. 1974). A variance between the allegation and proof to require reversal of a conviction must affect the substantial rights of the accused. Such rights are not affected where the defendant is adequately apprised of the charges against him so that he may prepare his defense without surprise at trial, and may protect himself against another prosecution for the same offense. Berger v. United States, supra, 295 U.S. at 82, 55 S.Ct. at 630.

Prejudice to the defendant and possibilities of double jeopardy are negligible here because the indictment listed five individual offenses, specifically identifying each claim by date, number and item. The technical variance encountered here is best relegated to the “immaterial” category.

III. SUFFICIENCY OF THE EVIDENCE

Cook next maintains that the evidence is insufficient to support a verdict of guilty. The standard for review is that “the verdict of a jury must be sustained if there is substantial evidence, taking the view most favorable to the Government, to support it.” Glasser v. United States, 315 U.S. 60, 80, 62 S.Ct. 457, 469, 86 L.Ed. 680 (1942); United States v. Vomero, 567 F.2d 1315, 1316 (5th Cir. 1978). In the instant case there is substantial evidence to support the verdict.

At trial defense counsel did not challenge the existence of the fraudulent scheme but attempted to prove that Cook had no personal knowledge of the false billing practices and never instructed anyone to bill in that manner. The evidence of Cook’s par *576 ticipation is largely circumstantial. However, the standard for review of the sufficiency of evidence to support conviction is the same whether the evidence is direct or circumstantial. See Holland v. United States, 348 U.S. 121, 75 S.Ct. 127, 99 L.Ed. 150 (1954); United States v. Palacios, 556 F.2d 1359 (5th Cir. 1977); United States v. Warner, 441 F.2d 821 (5th Cir.), cert, denied, 404 U.S. 829, 92 S.Ct. 65, 30 L.Ed.2d 58 (1971).

Owen Cook visited the Bandera plant four or five times during May and once or twice a month thereafter until the plant closed down in early December. Cook also spoke regularly via telephone with Robert Bean, a previous owner, and Richard Weber, the material control manager, who actually performed most of the falsification at Bean’s command.

The testimony of Richard Weber, the government’s key witness, came closest to directly implicating Cook. Weber testified that during the routine phone calls he would get on the line after Bean and Cook had conversed. Cook would inquire, “what are we going to do this week, Richard?” Weber’s response would always be a dollar figure, representing his estimate of “whatever we could spend that week on falsified invoices” — the difference between amount actually spent on labor and parts and the higher figure submitted to the government as an inflated estimate of what the repairs would cost.

Weber also testified that he retained a stock of blank invoices from 19 or 20 supply companies, many of which were fictitious. One of these companies was Abtex Sales & Service of Abilene, Texas, either owned or officered by Cook. The five claims at issue in the government’s case were each supported by invoices from Abtex which Weber testified he had totally invented. Weber further stated that on one occasion Cook handed him a brown envelope, commenting that he heard Weber “needed more of these.” Weber indicated that the envelope contained blank Abtex invoices. There was also some evidence that Cook personally used the proceeds of Mikelco checks that paid for “purchases” from Abtex.

It is clear that appellant through this assignment of error, as evidenced by his brief, is merely attempting to argue the credibility of the witnesses and the weight to be afforded their testimony. That responsibility rests exclusively with the trier of facts, in this case the jury, and those who reside in the ivory towers of appellate review must resist the temptation to retry cases.

In a related vein to the sufficiency of the evidence issue, appellant contends the trial court’s deferral of the ruling on the motion for acquittal as to one count is reversible error. At the close of the government’s case, Cook moved for acquittal. The trial judge denied the motion with respect to four counts but deferred ruling on the motion regarding Count II. The failure to grant the motion by deferring a ruling does not necessarily require a reversal. Montoya v. United States, 402 F.2d 847 (5th Cir. 1968). This court in United States v. Guinn, 454 F.2d 29, 33 (5th Cir.), cert, denied, 407 U.S. 911, 92 S.Ct. 2437, 32 L.Ed.2d 685 (1972), explained the scope of the deferred-ruling doctrine:

The doctrine means only that a defendant is entitled to a judgment of acquittal if the evidence at the close of the Government’s case is insufficient, and that the trial judge may not, be (sic) deferring decision on the defendant’s motion, allow sufficient evidence to be supplied by the defense or by the prosecution on rebuttal.

Appellant’s reliance here on the deferred-ruling doctrine is misplaced. As previously illustrated, the evidence was sufficient to support a conviction when the prosecution rested. Thus, any error in the district court’s failure to rule promptly on the motion was harmless.

As a corollary to his sufficiency argument, Cook criticizes the trial judge for delivering three Allen -type charges to the jury. See generally Allen v. United States, 164 U.S. 492, 17 S.Ct. 154, 41 L.Ed. 528 (1896). The first time the judge mentioned any of the Allen language was during the initial jury charge. (R. Vol. VI, 1158-59). However, it would be unfair to characterize this as a traditional Allen charge. The trial *577 judge merely asked each of the jurors to avoid announcing an emphatic opinion until listening to the others. He also requested them to re-examine their views, while admonishing them not to surrender an honest conviction simply for the purpose of returning a verdict. This language is unobjectionable, given its mild form and context within the entire charge.

The second objected to instruction, reproduced in its entirety below, 1 is more properly delineated an Allen charge. This charge was delivered after a weekend break in the deliberations and a note from the jurors that they remained divided and did not feel further discussion would resolve their differences. This charge was given without objection from defense counsel. In fact, defense counsel agreed that the jury should continue deliberations (R. Vol. VI, 1175) before the charge was given. (R. Vol. VI, 1177-79). We confidently conclude that this mild and balanced charge is within the *578 boundaries sanctioned by United States v. Bailey, 468 F.2d 652, rehearing en banc, 480 F.2d 518 (5th Cir. 1973), and the numerous cases cited and discussed therein.

The last criticized court communication with the jury, reproduced below, 2 is not an Allen charge. Rather, the trial court, by written note, simply observed that the jury had reached a verdict on one count of the indictment which was evidence of progress and asked the jurors to continue their deliberation. This final communication was approved prior to its delivery by both appellant and his counsel (R. Vol. VI, 1183-84). Having obtained the verdict he sought, but obviously not the one he desired, appellant Cook cannot now validly complain about the meticulously correct procedures employed by the trial court in meeting his wishes.

IV. PROSECUTORIAL MISCONDUCT

Cook asserts that the prosecutor’s closing argument was so prejudicial as to deny him a fair trial. The test to be applied in resolving this issue is well settled: Does the prosecutor’s argument, taken as a whole in the context of the entire case, prejudicially affect substantial rights of the defendant? Berger v. United States, 295 U.S. 78, 55 S.Ct. 629, 79 L.Ed. 1314 (1935); United States v. Rodriquez, 503 F.2d 1370 (5th Cir. 1974).

The appellant isolates five specific portions of the Government’s closing argument as detrimental. During the course of his argument, the prosecutor commented that “If [the equipment] ran, then Owen Cook was defrauding the Government of money. If they didn’t, then he was not only defrauding the Government of money, he was endangering the lives of servicemen in our national defense.” (R. Vol. VI, 1074). Of the several arguments which appellant attacks on appeal, only this one was considered serious enough by defense counsel to object at trial. Upon objection, the trial judge instructed the jury not to consider the argument:

Well, I will instruct the jury not to consider the argument, although I don’t think counsel said that there was any evidence to that effect. He was simply drawing a conclusion if there had been evidence to that effect he would have been endangering the safety of those people who worked on it, but members of the jury, I will instruct you not to consider the argument.

(R. Vol. VI, 1075).

Any potential prejudice was thus dispelled by this curing instruction as in United States v. Arteaga-Limones, 529 F.2d 1183, 1191 (5th Cir.), cert, denied, 429 U.S. 920, 97 S.Ct. 315, 50 L.Ed.2d 286 (1976).

Defense counsel did not interject an objection at trial to the other four complained of portions of the Government’s closing statement, reproduced below. 3 *579 Since no objection was voiced at the time these statements were made, we must affirm in the absence of “plain error.” United States v. Juarez, 566 F.2d 511, 516-17 (5th Cir. 1978); Dupoint v. United States, 388 F.2d 39, 45 (5th Cir. 1967). Our review of this record as a whole persuades us that the prosecutor’s concededly improper comments were harmless beyond a reasonable doubt.

V. INSTRUCTIONS TO THE JURY

Appellant asserts three errors regarding the jury instructions: 1) the trial court refused to give requested instructions on mistake of fact; 2) the court erroneously instructed the jury that the requirement of knowledge was met if defendant acted with “reckless disregard” for the truth; and 3) the court incorrectly charged the jury in reference to his “supervisory capacity” as President of Mikelco of Bandera, Inc.

In order to prevail under his first ground, appellant must demonstrate that his requested instruction was a correct one and that it had not been substantially delivered by the trial court. Pine v. United States, 135 F.2d 353 (5th Cir. 1943). This the appellant cannot demonstrate. Appellant’s requested instruction on mistake of fact was substantially given. The trial court did instruct the jury that “[a]n act is done knowingly if done voluntarily and intentionally and not because of mistake or accident or other innocent reason.” Shortly thereafter, the court reiterated in its definition of “willfully” that the act had to be committed voluntarily, intentionally, with knowledge that it was prohibited by law and with the purpose “to disobey or to disregard the law and not by mistake, accident or in good faith.” In reviewing a jury charge, the charge must be read and examined in its entirety. United States v. McCoy, 539 F.2d 1050 (5th Cir. 1976), cert, denied, 431 U.S. 919, 97 S.Ct. 2185, 53 L.Ed.2d 230 (1977). In light of the court’s inclusion of mistake and accident in its charge together with its standard instructions on the presumption of innocence and burden of proof with a definition of reasonable doubt, appellant’s vigorous contention that the trial judge did not adequately apprise the jury of Cook’s good faith belief in the supplying of parts must fail.

Appellant objects to the standard of knowledge employed in the jury charge. The charge read in pertinent part:

It is necessary that you find beyond a reasonable doubt that any act you may have found beyond a reasonable doubt to have been committed by the defendant was done or committed knowingly. In this connection, however, you are instructed that a person who makes a claim or causes a claim to be made with reckless disregard for the truthfulness of the claim and with the conscious purpose to avoid learning the truthfulness of the claim is deemed to have knowledge of the claim and its truthfulness or lack thereof.

Cook asserts that the charge is erroneous on the ground that reckless disregard of the falsity of the statement and a conscious effort to avoid learning the truth does not constitute “knowledge.” As an alternative, appellant suggests the charge should have contained the balancing instruction contemplated by the Model Penal Code. 4

A virtually identical charge to the one here questioned was lately approved by this *580 court in United States v. Evans, 5 559 F.2d 244 (5th Cir.), cert, denied, 434 U.S. 1015, 98 S.Ct. 731, 54 L.Ed.2d 759 (1978), a case involving a prosecution pursuant to 18 U.S.C. § 1001. A similar “deliberate ignorance” charge was also recently upheld in United States v. Restrepo-Granda, 575 F.2d 524 (5th Cir. 1978). The decisions from other circuits harmonize with the expressed conclusion that deliberate ignorance is the equivalent of knowledge under statutes similar to the one involved. 6

Finally, appellant contends that the court improvidently encouraged the jury to examine his conduct in his capacity as President of Mikelco of Bandera, Inc. This is so, it is argued, because the recklessness charge was given in connection with discussion of this supervisory capacity. The charge reads:

You are instructed that you cannot find a defendant guilty with respect to any count in the indictment merely from the fact, if you find it was a fact, that he was president of this Mikelco of Bandera, Inc. It is necessary that you find beyond a reasonable doubt that any act you may have found beyond a reasonable doubt to have been committed by the defendant was done or committed knowingly. In this connection, however, you are instructed that a person who makes a claim or causes it to be made with reckless disregard for the truthfulness of the claim and with the conscious purpose to avoid learning the truthfulness of the claim is deemed to have knowledge of the claim and its truthfulness or lack thereof.

The defendant faults the instruction because “[t]he court did not remind the jury that ‘in this connection’ (i. e. as president), only defendant’s willful actions could be considered, as section 2(b) requires.” The phrase “in this connection” logically does not relate back to appellant’s position as President of Mikelco. It refers to the sentence immediately preceding, where the jury was charged that it could not find Cook guilty unless it found beyond a reasonable doubt that he acted “knowingly.” 7

Having considered the arguments raised by appellant, we conclude that there was no reversible error. The judgment of the district court is AFFIRMED.

Notes

. The jury retired at 9:40 a. m. on Friday, January 20, 1978 to consider a verdict. Having failed to reach unanimity by 5:00 that afternoon, the jurors were instructed to return on Monday, January 23, 1978 for further deliberations. After less than two hours of deliberation on Monday, the jury communicated to the court that they were deadlocked. The following instruction was then delivered:

THE COURT: Good morning. Well, members of the jury, I have your note which says that you still remain divided after reviewing the material and voting again, and also this statement of your foreman to the effect that you do not feel that any further discussion will make any difference in the outcome.

Let me say a few additional words to you. This is an important case to the defendant, to the United States, and to the administration of justice. You know how long the case has been on trial. There is nothing to indicate that it can better be tried or presented at some other time any more exhaustively than it has been on both sides of the case during this trial. It is highly desirable from the standpoint of all concerned, the parties, the lawyers, the Court, and the public, that you agree upon a verdict if that is at all possible.

The Court does not want any juror to surrender his or her conscientious conviction. Each juror should perform his or her duty conscientiously and honestly according to the law and the evidence. Although the verdict to which a juror agrees, of course, must be his or her own verdict, the result of his or her own convictions, and not a mere acquiescence in the conclusions of other jurors, yet, in order to bring 12 minds to a unanimous result, you must consider the matters submitted to you with candor and with a proper regard for and deference to the opinions of each other.

You should consider that the case at some time must be decided and that you were selected in the same manner and from the same source from which any future jury must be selected. There is no reason to suppose that the case will ever be submitted to a jury more intelligent, more impartial or more competent to decide it or that more clear evidence will be produced on one side or the other.

In conferring together you ought to pay proper respect to each other’s opinions with an open minded disposition to be convinced by each other’s arguments. You should not, any of you, adopt a stubborn, close-minded attitude toward your responsibilities here. It is important that you should arrive at a verdict one way or the other, but only, of course, according to your convictions. You must not by reason of having taken a position at one time or another in the discussions, stubbornly adhere to that position without listening carefully to others and re-examining your views as well as the suggestions and ideas expressed by the other jurors with a view of ultimately reaching a verdict in which you can conscientiously join. This is very important, and I urge that you give full consideration to the matters I have suggested.

In making this statement I again emphasize that no jurors should surrender his or her conscientious conviction, and a verdict arrived at and to which a juror agrees must be his or her own verdict, the result of his or her own finally reached conclusions and convictions and not a mere acquiescence in the conclusion of others.

I am going to suggest to you that you retire again and carefully consider all of the evidence in the light of the Court’s previous instructions, and after giving due and full consideration of the instructions that I have now given you.

I am obliged to ask that you again retire, and the Court will wait for a further message from you after your continued deliberations.

Now, members of the jury, it is about 11:00 o’clock. When you want to go to lunch, you just tell the marshal and he will take you, but remember the instructions of the Court: do not discuss the case among yourselves or with anyone else and do not permit anyone to discuss it with you when you are out of the jury room. When you have completed your lunch, the marshal will bring you back to the jury room, and, marshal, it will be all right if you will leave the exhibits there and lock the jury room door when the jurors leave.

Ladies and gentlemen, you may now be excused to continue your deliberations.

(R. Vol. VI, 1177-79).

. At 2:30 p. m. on Monday, January 23, 1978, the jury informed the court that it had reached a unanimous decision on one count of the indictment. With appellant’s and his counsel’s consent the court sent the following note to the jury:

Ladies and gentlemen of the jury, I have received your last note in which you indicate the jury has reached a unanimous verdict with respect to one count of the indictment. Even though you had previously indicated that you had been deadlocked, it is obvious that you have made some progress. Under the circumstances, I am going to ask you to please continue your deliberations.

(R. Vol. VI, 1186).

On that same afternoon at 4:20 p. m. the jury reached a unanimous decision on all five counts of the indictment.

. (1) “The most important thing, though: what about the W-2’s? He testified he was a commission agent. He reported some of his money on his income tax returns. I lay you odds, ladies and gentlemen, he didn’t report the $13,-000 cash he took out of that. No W-2’s to the government that Owen Cook, a commissioned agent, earned blank amount in commission and sales or income this year. Not a bit, no withholding, nothing. Legitimate company? Incredible, does not merit belief.”

R. Vol. VI, 1066.

(2) Prosecutor mentioned testimony elicited from Mr. Cook “when he finally took the stand.”

R. Vol. VI, 1124.

(3) “[Tjhat is what we have proven to you, people — a man that was, according to his secretary, making $70,000 a year out of an insur-

*579 anee business — and good for him if that was legitimate . . .”

R. Vol. VI, 1135-36.

(4) “I don’t care whether [the machines] worked — well, I do care whether it worked. I’d like to think very much that it worked. I would like to think very much that the Air Force has got a piece of equipment that may have worked; it may have had a Sherwin-Williams overhaul; as long as it worked the Air Force doesn’t care, really.”

R. Vol. VI, 1128.

. Model Penal Code § 2.02(7) on which appellant bases his argument provides in part: “When knowledge of the existence of a particular fact is an element of an offense, such knowledge is established if, a person is aware of a high probability of its existence, unless he actually believes that it does not exist." [Emphasis supplied].

. In United States v. Evans, 559 F.2d 244, 246 (5th Cir.), cert, denied, 434 U.S. 1015, 98 S.Ct. 731, 54 L.Ed.2d 759 (1978) the sanctioned charge was:

However, a person who makes a statement with reckless disregard of the truthfulness of the statement and with the conscious purpose to avoid learning the truthfulness of the statement, is deemed to have knowledge of the statement and its truthfulness or lack thereof.

Except for the substitution in the case at bar of the work “claim” for the word “statement” in the charge approved in Evans, the two charges are identical.

. In United States v. Jacobs, 475 F.2d 270, 287-288 (2nd Cir. 1973), the court reasoned that the Supreme Court’s approval of the Model Penal Code definition of knowledge implies approval of an instruction that the knowledge requirement is satisfied by proof of a “conscious purpose to avoid learning the truth.”

. This identical “supervisory capacity” charge was requested by appellant.